diff --git "a/big_dataset_feyman.txt" "b/big_dataset_feyman.txt" new file mode 100644--- /dev/null +++ "b/big_dataset_feyman.txt" @@ -0,0 +1,78368 @@ + + + + +--- Trang 1 --- +Fey1‹a1 +LECTURES ON PHYSICS +Feynman - Leighton - Sands +--- Trang 2 --- +le Fey7 g1 +LECTURESON +NEW MILLENNIUM EDITION +FEYNMANsLEIGHTONsSANDS +BASIC BOOKS VOLUME I +--- Trang 3 --- +Copyright © 1963, 2006, 2010 by California Institute of 'Technology, +Michael A. Gottlieb, and Rudolf Pfeifer +Published by Basie Books, +A Member of the Perseus Books Group +AII rights reserved. Printed ín the United States of America. +No part oŸ this boolk may be reproduced in any manner whatsoever without written permission +except in the case of brief. quotations embodied in critical articles and reviews. +For Information, address Basic Books, 250 West 57th Street, 15th Floor, New York, NY 10107. +Books published by Basic Books are available at special discounts for bu]k purchases +1n the United States by corporations, Institutions, and other organizations. +For more information, please contact the Special Markets Department at the +Perseus Books Group, 2300 Chestnut Street, Suite 200, Philadelphia, PA 19103, +or call (S00) 810-4145, ext. 5000, or e-mail special.markets()perseusbooks.com. +A CTIP catalog record for the hardcover edition of +this book is available from the Lñbrary of Congress. +LCCN: 2010938208 +Hardcover ISBN: 978-0-465-02414-8 +E-book ISBN: 978-0-465-02569-6 +--- Trang 4 --- +Abouwt Hicher‹[ Foggrernterrt +Born in 1918 in New York City, Richard P. Feynman received his Ph.D. +from Princeton in 1942. Despite his youth, he played an important part in the +Manhattan Project at Los Alamos during World War II. Subsequently, he taught +at Cornell and at the California Institute of Technology. In 1965 he received the +Nobel Prize in Physics, along with Sin-ltiro Tomonaga and Julian Schwinger, Íor +his work in quantum electrodynamics. +Dr. Feynman won his Nobel Prize for successfully resolving problems with the +theory oŸ quantum electrodynamics. He also created a mathematical theory that +accounts for the phenomenon of superfuidity ín liquid helium. Thereafter, with +Murray Gell-Mamn, he did fundamental work in the area of weak interactions such +as beta decay. In later years Feynman played a key role in the development of +quark theory by putting forward his parton model of high energy proton collision +DTOC©SSGS. +Beyond these achievements, Dr. Feynman introduced basic new computa- +tional techniques and notations into physics—above all, the ubiquitous Eeynman +diagrams that, perhaps more than any other formalism in recent scientifc history, +have changed the way in which basic physical processes are conceptualized and +calculated. +teynman was a remarkably efective educator. Of all his numerous awards, +he was especially proud of the Oersted Medal for Teaching, which he won In +1972. The Fewwman Lectures on Phụsics, originally published in 1963, were +described by a reviewer in Scientific American as “tough, but nourishing and full +of favor. After 25 years it is fhe guide for teachers and for the best of beginning +students.” In order to increase the understanding of physics among the lay public, +Dr. Feynman wrote The Character oƒ Phụsical[ Lau and QED: The Strange +Theor oƒ Light and Matter. He also authored a number of advanced publications +that have become classic references and textbooks for researchers and students. +Richard Feynman was a constructive publie man. His work on the Challenger +commission is well known, especially his famous demonstration of the susceptibility +of the O-rings to cold, an elegant experiment which required nothing more than +a glass of ice water and a C-clamp. Less well known were Dr. Feynman's eforts +on the California State Curriculum Committee in the 1960s, where he protested +the mediocrity of textbooks. +--- Trang 5 --- +A recital of Richard Feynman's myriad scientific and educational accomplish- +ments cannot adequately capture the essence of the man. Âs any reader of +even his most technical publications knows, Feynman's lively and multi-sided +personality shines through all his work. Besides being a physicist, he was at +various times a repairer of radios, a picker of locks, an artist, a dancer, a bongo +player, and even a decipherer of Mayan Hieroglyphics. Perpetually curious about +his world, he was an exemplary empiricist. +Richard Feynman died on February 15, 1988, in Los Angeles. +--- Trang 6 --- +}Proftic© ếo tho /Voec Willorartrtrrit cÏfffornt +Nearly fifty years have passed since Richard Eeynman taught the introductory +physics course at Caltech that gave rise to these three volumes, 7 he Fewman +Lectures on Phụsics. In those fñẨty years our understanding of the physical +world has changed greatly, but The Feynman Lectures on Phạs¿cs has endured. +teynman”s lectures are as powerful today as when frst published, thanks to +teynman”s unique physics insights and pedagogy. “They have been studied +worldwide by novices and mature physicists alike; they have been translated +into at least a dozen languages with more than 1.5 millions copies prinwed in the +linglish language alone. Perhaps no other set of physics books has had such wide +Impact, for so long. +This Neu Miienmium Edition ushers in a new era for The Feunman Lectures +ơn Phụsics (FLP): the twenty-flrst century era of electronic publishing. LP +has been converted to eFP, with the text and equations expressed in the IATEX +electronic typesetting language, and all ñgures redone using modern drawing +SOftware. +The consequences for the przn# version of this edition are øœø‡ startling; it +looks almost the same as the original red books that physics students have known +and loved for decades. The main differences are an expanded and improved index, +the correction of 885 errata found by readers over the fve years since the frst +printing of the previous edition, and the ease of correcting errata that future +readers may fñnd. To this I shall return below. +The eBook Version of this edition, and the Enhanced blectronic Version are +electronic innovations. By contrast with most eBook versions of 20th century tech- +nical books, whose equations, fñgures and sometimes even text become pixellated +when one tries to enlarge them, the IHÃIEX manuscript of the Me Mllenniun +bdition makes 1Ý possible to create eBooks of the highest quality, in which all +--- Trang 7 --- +features on the page (except photographs) can be enlarged without bound and +retain their precise shapes and sharpness. And the nhønced Electronic Version, +with its audio and blackboard photos from Feynmans original lectures, and its +links to other resources, is an innovation that would have given Feynman great +pleasure. +IMorweertos oŸ Foggrtrtterrt "s Eoe£mr'os +These three volumes are a selEcontained pedagogical treatise. They are also a +historical record of Feynman's 1961-64 undergraduate physics lectures, a course +required of all Caltech freshmen and sophomores regardless of their majors. +Readers may wonder, as [ have, how Feynmans lectures impacted the students +who attended them. Eeynman, in his Preface to these volumes, offered a somewhat +negative view. “I don”t think I did very well by the students,” he wrote. Matthew +Sands, in his memoir in Feywman's Tips on Phụsics expressed a far more posifive +view. Out of curiosity, in spring 2005 I emailed or talked to a quasi-random set +oŸ 17 students (out of about 150) from Feynman's 1961-63 class—some who had +great dificulty with the class, and some who mastered it with ease; majors In +biology, chemistry, engineering, geology, mathematics and astronomy, as well as +in physics. +The intervening years might have glazed their memories with a euphoric tỉnt, +but about 80 percent recall Feynman'”s lectures as highlights of their college years. +“H was like going to church.” The lectures were “a transformational experience, ” +“the experience of a lifetime, probably the most important thing I got from +Caltech.” “[ was a biology major but Feynman's lectures stand out as a high +point in my undergraduate experience ... though I must admit I couldn't do +the homework at the time and I hardly turned any of it in” “[ was among the +least promising of students in this course, and Ï never missed a lecture.... Ï +remember and can still feel Eeynman's joy of discovery.... His lectures had an +... emotional impact that was probably lost in the printed Lectures.” +By contrast, several of the students have negative memories due largely to two +issues: (1) “You couldn't learn to work the homework problems by attending the +lectures. Feynman was too slick——=he knew tricks and what approximations could +be made, and had intuition based on experience and genius that a beginning +student does not possess.” Feynman and colleagues, aware of this aw ¡in the +course, addressed it in part with materials that have been incorporated into +teụwmans Tips on Phụsïcs: three problem-solving lectures by Feynman, and +--- Trang 8 --- +a set of exercises and answers assembled by Robert B. Leighton and Rochus +Vogt. (1) “The insecurity of not knowing what was likely to be discussed in +the next lecture, the lack of a text book or reference with any connection to +the lecture material, and consequent inability for us to read ahead, were very +frustrating.... I found the lectures exciting and understandable in the hall, but +they were Sanskrit outside [when T tried to reconstruct the details|” 'This problem, +of course, was solved by these three volumes, the printed version of The Feunman +Lectures on Phụsics. They became the textbook from which Caltech students +studied for many years thereafter, and they live on today as one of Feynman's +greatest legacies. +A HHistorg, o£ FErraÉ( +The Feunwman Lectures on Phụsics was produced very quickly by Feynman +and his co-authors, Robert B. Leighton and Matthew Sands, working from +and expanding on tape recordings and blackboard photos of Feynman”s course +lecturesề (both of which are incorporated into the Enhưnccd Electronic Version +of this Weu Mllennium Edition). Given the hiph speed at which Feynman, +Leighton and Sands worked, it was inevitable that many errors crept into the first +edition. Feynman accumulated long lists of claimed errata over the subsequent +years—errata found by students and faculty at Caltech and by readers around +the world. In the 1960°s and early 70's, Feynman made time in his intense liíe +to check most but not all of the claimed errata for Volumes I and II, and insert +corrections into subsequent printings. But Feynman”s sense of duty never rose +high enough above the excitement of discovering new things to make him deal +with the errata in Volume III.† After his untimely death in 1988, lists of errata +for all three volumes were deposited in the Caltech Archives, and there they lay +forgotten. +In 2002 Ralph Leighton (son of the late Robert Leighton and compatriot of +Feynman) informed me of the old errata and a new long list compiled by Ralph”s +* Eor descriptions of the genesis of Feynman”s lectures and of these volumes, see Feynmans +Preface and the Forewords to each of the three volumes, and also Matt Sands" Memoir in +Feuwmanws Tips on Phụs¿cs, and the Special Preface to the Commemoradtiue Editöion of FLP, +written in 1989 by David Goodstein and Gerry Neugebauer, which also appears in the 2005 +Deftnittue Edition. +† Im 1975, he started checking errata for Volume III but got distracted by other things and +never finished the task, so no corrections were made. +--- Trang 9 --- +triend Michael Gottlieb. Leighton proposed that Caltech produce a new edition +of The Feunman Lectures with all errata corrected, and publish it alongside a new +volume of auxiliary material, Feynwmans Tiips on Phụs¿cs, which he and Gottlieb +WCT€ DI€paring. +teynman was my hero and a close personal friend. When I saw the lists of +errata and the content of the proposed new volume, I quickly agreed to oversee +this project on behalf of Caltech (EFeynman”s long-time academic home, to which +he, Leighton and Sands had entrusted all rights and responsibilities for 75e +Feunman Lectures). After a year and a halŸ oŸ meticulous work by Gottlieb, and +careful serutiny by Dr. Michael Hartl (an outstanding Caltech postdoc who vetted +all errata plus the new volume), the 2005 Oefimitiue Edilion oƒƑ The Feuuman +Lectures on Phụsics was born, with about 200 errata corrected and accompanied +by teunwmans Tips on Phụsics by Feynman, Gottlieb and Leighton. +T thought that edition was goïng to be “Defnitive”. What I did no antic- +Ipate was the enthusiastic response of readers around the world to an appeal +trom Gottlieb to identify further errata, and submit them via a website that +Gottlieb created and continues to maintain, 7 he Feunwman Lectures Website, +www. eynmanlectures.info. In the fve years since then, 965 new errata have +been submitted and survived the meticulous scrutiny of Gottlieb, Hartl, and Nate +Bode (an outstanding Caltech physics graduate sbudent, who succeeded Hartl +as Caltech?s vetter of errata). Of these, 965 vetted errata, 80 were corrected in +the fourth printing of the Defimitiue Edition (August 2006) and the remaining +885 are correcbed in the first printing of this Neu Mllenmum Edition (332 in +volume I, 263 in volume II, and 200 in volume TIT). For details of the errata, see +www. feynmanlectures. in£o. +Clearly, making The Fewman Lectures on Phụs¿cs error-free has become a +world-wide community enterprise. Ôn behalf of Caltech I thank the 50 readers +who have contributed since 2005 and the many more who may contribute over the +coming years. The names of all contributors are posted at www..feynmanlectures. +info/flp_errata.htm1l. +Almost all the errata have been of three types: (1) typographical errors +in prose; (ii) typographical and mathematical errors in equations, tables and +fgures—sign errors, incorrect numbers (e.g., a 5 that should be a 4), and missing +subscripts, summation signs, parentheses and terrmms in equations; (i11) incorrecE +cross references to chapters, tables and fgures. 'Phese kinds of errors, though +not terribly serious to a mature physicist, can be frustrating and confusing to +Feynman”s primary audience: students. +--- Trang 10 --- +Tt is remarkable that among the 1165 errata corrected under my auspices, +only several do Ï regard as true errors in physics. An example is Volume TT, +page 5-9, which now says “... no static distribution of charges inside a closed +grounded conductor can produce any |electric] fields outside” (the word grounded +was omitted in previous editions). This error was pointed out to Feynman by a +number of readers, including Beulah Elizabeth Cox, a student at The College of +William and Mary, who had relied on Feynmanˆs erroneous passage in an exam. +To Ms. Cox, Feynman wrote in 1975,* “Your instructor was right not bo give +you any points, Íor your answer was wrong, as he demonstrated using Gauss's +law. You should, in science, believe logic and arguments, carefully drawn, and +not authorities. You also read the book correctly and understood it. I made a +mistake, so the book is wrong. I probably was thinking oŸ a grounded conducting +sphere, or else of the fact that moving the charges around in diferent places +inside does not affect things on the outside. I am not sure how T dịd it, but +goofed. And you goofed, too, for believing me.” +Note thís 'Votr Wĩiliortrtrtrrtre EcÏfffOre Ấ (qiaớc Éo lồo +Between November 2005 and July 2006, 340 errata were submitted to 7 he +teunman Lectures Website www.feynman1ectures.info. Remarkably, the bulk +of these came from one person: Dr. Rudolf Pfeifer, then a physics postdoctoral +fellow at the Ủniversity of Vienna, Austria. The publisher, Addison Wesley, fixed +80 errata, but balked at fixing more because of cost: the books were being printed +by a photo-offset process, working ữom photographic images of the pages om +the 1960s. Correcting an error involved re-typesetting the entire page, and to +ensure no new errors crept in, the page was re-typeset twice by two diferent +people, then compared and proofread by several other people—a very costÌy +process indeed, when hundreds of errata are involved. +Gottlieb, Pfeifer and Ralph Leighton were very unhappy about this, so they +formulated a plan aimed at facilitating the repair of all errata, and also aimed +at produeing eBook and enhanced electronic versions of The henman Lectures +on Phụsics. They proposed their plan to me, as Caltech”s representative, in +2007. I was enthusiastic but cautious. After seeing further details, including a +one-chapter demonstration of the Enhanced PElectronic Version, Ï recommended +* Pages 288-289 of Perfectu Reasonable Dcuialions jrom the Beaten Track, The Letters oŸ +lRichard P. Fewman, ed. Michelle EFeynman (Basic Books, New York, 2005). +--- Trang 11 --- +that Caltech cooperate with Gottlieb, Pfeifer and Leighton in the execution of +their plan. 'Phe plan was approved by three successive chairs of Caltech's Division +of Physics, Mathematics and Astronomy——Tom 'Tombrello, Andrew Lange, and +Tom Soifer—and the complex legal and contractual details were worked out by +Caltech's Intellectual Property Counsel, Adam Cochran. With the publication of +this Neu Millenn¿um Edition, the plan has been executed successfully, despite +1ts complexity. Specifically: +Pfeifer and Gottlieb have converted into IAIEX all three volumes of LP +(and also more than 1000 exercises from the Feynman course for incorporation +into FeWwmans Tips on Phụsics). The FPLP figures were redrawn in modern +electronic form in India, under guidance of the ƑLP German translator, Henning +Heinze, for use in the German edition. Gottlieb and Pfeifer traded non-exclusive +use of their IXTIEX equations in the German edition (published by Oldenbourg) +for non-exclusive use of Heinze”s fñgures in this Weu MiiHennium English edition. +Pfeifer and Gottlieb have meticulously checked all the IÃTERX text and equations +and all the redrawn fñgures, and made corrections as needed. Nate Bode and +1, on behalf of Caltech, have done spot checks of text, equations, and figures; +and remarkably, we have found no errors. Pfeifer and Gottlieb are unbelievably +meticulous and accurate. Gottlieb and Pfeifer arranged for John Sullivan at the +Huntington Library to digitize the photos of Eeynmans 1962-64 blackboards, +and for George Blood Audio to digitize the lecture tapes—with financial support +and encouragement from Caltech Professor Carver Mead, logistical support from +Caltech Archivist Shelley Erwin, and legal support om Cochran. +The legal issues were serious: In the 1960s, Caltech licensed to Addison Wesley +rights to publish the print edition, and in the 1990s, rights to distribute the audio +of Feynman's lectures and a variant of an electronic edition. In the 2000s, through +a sequence of acquisitions of those licenses, the print rights were transferred to +the Pearson publishing group, while rights to the audio and the electronic version +were transferred to the Perseus publishing group. Cochran, with the aid of Ike +'Williams, an attorney who specializes in publishing, succeeded in uniting all of +these rights with Perseus (Basic Books), making possible this Neu Miilenniumn +kdition. +Ackreo:r-loclqgrraorsÉs +Ơn behalf of Caltech, I thank the many people who have made this Neu +MMiilenniuưun PEdition possible. Specifically, I thank the key people mentioned +--- Trang 12 --- +above: Ralph Leighton, Michael Gottlieb, Tom Tombrello, Michael Hartl, Rudolf +Pfeifer, Henning Heinze, Adam Cochran, Carver Mead, Nate Bode, Shelley Erwin, +Andrew Lange, Tom Soifer, Ike Williams, and the 50 people who submitted errata +(listed at www.feynmanlectures.info). And I also thank Michelle Feynman +(daughter of Richard Feynman) for her continuing support and advice, Alan Rice +for behind-the-scenes assistance and advice at Caltech, Stephan Puchegger and +Calvin Jackson for assistance and advice to Pfeifer about conversion of #'LP to +HATEX, Michael Figl, Manfred Smolik, and Andreas Stangl for discussions about +corrections of errata; and the Staff of Perseus/Basic Books, and (for previous +editions) the staff of Addison Wesley. +lip S. Thorne +The Feynman Professor of Theoretical Physics, Emeritus +California Institute of 'Technology October 2010 +--- Trang 13 --- +MAINLY MECHANICS, RADIATION, AND HEAT +RICHARD P. FEYNMAN +Richard Chace Tolhman Professor oƒ Theoretical Physics +California Institute oƒ Technology +ROBERT B. LEIGHTON +Professor 0ƒ Physics +California Institute oƒ Technology +MATTHEW SANDS +Professor 0ƒ Physics +California Institute oƒ Technology +--- Trang 14 --- +Copyright © 1963 +CALIFORNIA INSTITUTE OF TECHNOLOGY +Primted in the United States oƒ America +ALL RIGHTS RESERVED. THIS BOOK, OR PARTS THEREOF +MAY NOT BE REPRODUCED IN ANY FORM WITHOUT +WRITTEN PERMISSION OF THE COPYRIGHT HOLDER. +Library oƒ Congress Catalog Card No. 63-20717 +Sixth priming, February 1977 +TSBN 0-201-02010-6-H +0-201-02116-1-P +CCDDEEFFGG-MU-89 +--- Trang 15 --- +WA IS +\\. (4O +TUẾ ¡ +Mroggrtraerrt`s Profqe© -" +These are the lectures in physics that I gave last year and the year before +to the freshman and sophomore classes at Caltech. 'Phe lectures are, of cOUrse, +not verbatim——they have been edited, sometimes extensively and sometimes less +so. The lectures form only part of the complete course. The whole group of 180 +students gathered in a big lecture room twice a week to hear these lectures and +then they broke up into small groups of 15 to 20 students in recitation sections +under the guidance of a teaching assistant. In addition, there was a laboratory +Session once a week. +The special problem we tried to get at with these lecbures was to maintain +the interest of the very enthusiastic and rather smart students coming out of +the high schools and into Caltech. They have heard a lot about how interesting +and exciting physics is—the theory of relativity, quantum mechanics, and other +modern ideas. By the end of two years of our previous course, many would be +very discouraged because there were really very few grand, new, modern ideas +presented to them. They were made to study inclined planes, electrostatics, and +so forth, and after bwo years it was quite stultifying. The problem was whether +or not we could make a course which would save the more advanced and excited +student by maintaining his enthusiasm. +'The lectures here are not in any way meant to be a survey course, but are very +serious. I thought to address them to the most intelligent in the class and to make +sure, if possible, that even the most intelligent student was unable to completely +encompass everything that was in the lectures—by putting in suggestions of +--- Trang 16 --- +applications of the ideas and concepts in various directions outside the main line +of attack. For this reason, though, I tried very hard to make all the statements +as accurate as possible, to point out in every case where the equations and ideas +ñtted into the body of physics, and how—when they learned more—things would +be modifed. T also felt that for such students ï§ is Important to indicate what +1E is that they should—if they are sufficiently clever—be able to understand by +deduction from what has been said before, and what is beïng put in as something +new. When new ideas came in, Ï would try either to deduce them ïf they were +deducible, or to explain that it øas a new idea which hadn't any basis in terms of +things they had already learned and which was not supposed to be provable—but +was Just added in. +At the start of these lectures, Ï assumed that the students knew something +when they came out of high school——such things as geometrical opties, simple +chemistry ideas, and so on. I also didn't see that there was any reason to make the +lectures in a defnite order, in the sense that I would not be allowed to mention +something until Ï was ready to discuss i% in detail. 'Phere was a great deal of +mention of things to come, without complete discussions. 'Phese more complete +discussions would come later when the preparation became more advanced. +Examples are the discussions of inductance, and of energy levels, which are at +ñrst brought in in a very qualitative way and are later developed more completely. +At the same time that Ï was aiming at the more active student, I also wanted +to take care of the fellow for whom the extra fireworks and side applications are +merely disquieting and who cannot be expected to learn most of the material +in the lecture at all. For such students I wanted there to be at least a central +core or backbone of material which he could get. ven if he didn't understand +everything in a lecture, Ï hoped he wouldn't get nervous. I didn”t expect him to +understand everything, but only the central and most direct features. It takes, +of course, a certain intelligence on his part to see which are the central theorems +and central ideas, and which are the more advanced side issues and applications +which he may understand only in later years. +In giving these lectures there was one serious difficulty: in the way the course +was given, there wasn't any feedback from the students to the lecturer to indicate +how well the lectures were going over. 'This is indeed a very serious difliculty, and +T don't know how good the lectures really are. 'Phe whole thỉng was essentially +an experiment. Ảnd ïf I did it again I wouldn”t do it the same way——I hope Ï +đorft have to do it againl I think, though, that things worked out——so far as the +physics is concerned——qulte satisfactorily in the first year. +--- Trang 17 --- +In the second year Ï was not so satisfed. In the frst part of the course, dealing +with electricity and magnetism, I couldnˆt think of any really unique or diferent +way of doing it —oŸ any way that would be particularly more exciting than the +usual way of presenting it. So I don't think I did very much in the lectures on +electricity and magnetism. At the end of the second year I had originally intended +to go on, after the electricity and magnetism, by giving some more lecbures on +the properties of materials, but mainly to take up things like fundamental modes, +solutions of the difusion equation, vibrating systems, orthogonal functions, ... +developing the frst stages of what are usually called “the mathematical methods +of physics.” In retrospect, I think that if Ï were doing i% again I would go back +to that original idea. But since it was not planned that I would be giving these +lectures again, it was suggested that it might be a good idea to try to give an +introduection to the quantum mechanics—what you will ñnd in Volume THỊ. +Tt is perfectly clear that students who will major in physics can wait until +theïr third year for quantum mechanics. Ôn the other hand, the argument was +made that many of the students in our course study physics as a background for +theïr primary interest in other fields. And the usual way of dealing with quantum +mnechanics makes that subJect almost unavailable for the great majJority of students +because they have to take so long to learn it. Yet, in i6s real applications—— +especially in its more complex applications, such as in electrical engineering +and chemistry—the full machinery of the diferential equation approach is not +actually used. So ÏI tried to describe the prineiples of quantum mechanics In +a way which wouldnˆt require that one first know the mathematics of partial +diferential equations. Even for a physicist I think that is an interesting thing +to try to do—to present quantum mechanics in this reverse fashion——for several +reasons which may be apparent in the lectures themselves. However, I think that +the experiment in the quantum mechanies part was not completely successful——in +large part because I really did not have enough time at the end (I should, for +Instance, have had three or four more lectures in order to deal more completely +with such matters as energy bands and the spatial dependence of amplitudes). +Also, I had never presented the subject this way before, so the lack of feedback was +particularly serious. Ï now believe the quantum mechaniecs should be given at a +later time. Maybe lI have a chance to do it again someday. Then Ƒl] do it right. +The reason there are no lectures on how to solve problems 1s because there +were recitation sections. Although I did put in three lectures in the first year on +how to solve problems, they are not included here. Also there was a lecture on +inertial guidance which certainly belongs after the lecture on rotating systems, +--- Trang 18 --- +but which was, unfortunately, omitted. 'Phe fñifth and sixth lectures are actually +due to Matthew Sands, as Ï was out of town. +'The question, of course, is how well this experiment has succeeded. My own +point of view——which, however, does not seem to be shared by most of the people +who worked with the students——is pessimistic. I don't think I did very well by +the students. When I look at the way the majority of the students handled the +problems on the examinations, I think that the system is a failure. Of course, +my fiends point out to me that there were one or ÿ6wo dozen students who—very +surprisingly——understood almost everything in all of the lectures, and who were +quite active in working with the material and worrying about the many points +in an excited and interested way. Thhese people have now, l believe, a first-rate +background in physics—and they are, after all, the ones Ï was trying to get at. +But then, “The power of instruction is seldom of mụch efflcacy except in those +happy dispositions where it is almost superfuous.” (Gibbon) +Stil, I didn't want to leave any student completely behind, as perhaps I did. +T think one way we could help the students more would be by putting more hard +work into developing a set of problems which would elucidate some of the ideas +in the lectures. Problems give a good opportunity to fll out the material of the +lectures and make more realistic, more complete, and more settled in the mind +the ideas that have been exposed. +1 think, however, that there isnˆt any solution to this problem of education +other than to realize that the best teaching can be done only when there is a +direct individual relationship between a student and a good teacher—a situation +in which the student discusses the ideas, thinks about the things, and talks about +the things. It's impossible to learn very much by simply sitting in a lecture, or +even by simply doing problems that are assigned. But in our modern tỉmes we +have so many students to teach that we have to try to fnd some substitute for +the ideal. Perhaps my lectures can make some contribution. Perhaps in some +small place where there are individual teachers and students, they may get some +inspiration or some ideas from the lectures. Perhaps they will have fun thinking +them through——or goïng on to develop some of the ideas further. +RICHARD P. FEEYNMAN +Jưne, 1963 +--- Trang 19 --- +orosror-‹l +This book is based upon a course of lectures in introductory physics given +by Prof. R. P. Feynman at the California Institute of Technology during the +academic year 1961-62; it covers the frst year of the Ewo-year introductory course +taken by all Caltech freshmen and sophomores, and was followed in 1962-63 by +a similar series covering the second year. The lectures constitute a major part of +a fundamental revision of the introductory course, carried out over a Íour-year +period. +'The need for a basic revision arose both from the rapid development of physics +in recent decades and from the fact that entering freshmen have shown a steady +increase in mathematical ability as a result of improvements in high school mathe- +matics course content. We hoped to take advantage of this improved mathematical +background, and also to introduce enough modern subject matter to make the +course challenging, interesting, and more representative of present-day physics. +In order to generate a variety of ideas on what material to include and how to +present it, a substantial number of the physics faculty were encouraged to offer +theïr ideas in the form of topical outlines for a revised course. Several of these were +presented and were thoroughly and critically discussed. It was agreed almost at +once that a basic revision of the course could not be accomplished either by merely +adopting a diferent textbook, or even by writing one øb ?m2fio, but that the new +course should be centered about a set of lectures, to be presented at the rate of +two or three per week; the appropriate text material would then be produeced as a +secondary operation as the course developed, and suitable laboratory experiments +would also be arranged to fit the lecture material. Accordinply, a rough outline of +--- Trang 20 --- +the course was established, but this was recognized as being incomplete, tentative, +and subject to considerable modification by whoever was to bear the responsibility +for actually preparing the lectures. +Concerning the mechanism by which the course would fnally be brought +to life, several plans were considered. “These plans were mostly rather similar, +involving a cooperative efort by Ñ staff members who would share the total +burden symmetrically and equally: each man would take responsibility for 1/N of +the material, deliver the lectures, and write text material for his part. However, +the unavailability of suficient staf, and the dificulty of maintaining a uniform +point of view because of diferences in personality and philosophy of individual +participants, made such plans seem unworkable. +The realization that we actually possessed the means to create not jusÈ a +new and diferent physics course, but possibly a unique one, came as a happy +inspiration to Professor Sands. He suggested that Professor R. P. Feynman pre- +pare and deliver the lectures, and that these be tape-recorded. When transcribed +and edited, they would then become the textbook for the new course. This is +essentially the plan that was adopted. +lt was expected that the necessary editing would be minor, mainly consisting +of supplying fgures, and checking punctuation and grammair; it was to be done by +one or two graduate students on a part-time basis. nfortunately, this expectation +was short-lived. It was, In fact, a major editorial operation to transform the +verbatim transcript into readable form, even without the reorganization or revision +of the subject matter that was sometimes required. Purthermore, it was not a Job +for a technical editor or for a graduate student, but one that required the close +attention of a professional physicist for from ten to twenty hours per lecturel +The dificulty of the editorial task, together with the need to place the material +in the hands of the students as soon as possible, set a strict limit upon the amount +of “polishing” of the material that could be accomplished, and thus we were +forced to aim toward a preliminary but technically correct product that could +be used immediately, rather than one that might be considered fñnal or ñnished. +Because of an urgent need for more copies for our students, and a heartening +Interest on the part of instructors and students at several other institutions, we +decided to publish the material in its preliminary form rather than wait for a +further major revision which might never occur. We have no illusions as to the +completeness, smoothness, or logical organization of the material; in fact, we +plan several minor modifications in the course in the immediate future, and we +hope that it will not become static in form or content. +--- Trang 21 --- +In addition to the lectures, which constitute a centrally important part of the +COurse, it was necessary also to provide suitable exercises to develop the students' +experience and ability, and suitable experiments to provide first-hand contact +with the lecture material in the laboratory. Neither of these aspecfs 1s in as +advanced a state as the lecture material, but considerable progress has been made. +Some exercises were made up as the lectures progressed, and these were expanded +and amplifed for use in the following year. However, because we are not yet +satisfed that the exercises provide sufficient variety and depth of application of +the lecture material to make the student fully aware of the tremendous power +being placed at his disposal, the exercises are published separately in a less +permanent form in order to encourage frequent revision. +A number of new experiments for the new course have been devised by +Professor H. V. Neher. Among these are several which utilize the extremely +low friction exhibited by a gas bearing: a novel linear air trough, with which +quantitative measurements of one-dimensional motion, impacts, and harmonic +motion can be made, and an air-supported, air-driven Maxwell top, with which +accelerated rotational motion and gyroscopic precession and nutation can be +studied. "The development of new laboratory experiments is expected to continue +for a considerable period of time. +The revision program was under the direction of Professors R. B. Leighton, +H. V. Neher, and M. Sands. Officially participating in the program were Professors +R. P. Feynman, Œ. Neugebauer, R. M. Sutton, H. P. Stabler,* F. Strong, and R. +Vogt, from the division of Physics, Mathematics and Astronomy, and Professors +T. Caughey, M. Plesset, and C. H. Wilts from the division of Engineering Science. +The valuable assistance of all those contributing to the revision program is +gratefully acknowledged. We are particularly indebted to the Eord Eoundation, +without whose financial assistance this program could not have been carried out. +HROBERT B. LEIGHTON +Juhụ, 1968 +* 1961-62, while on leave from Williams College, Williamstown, Mass. +--- Trang 22 --- +or~ÉœrtÉs +CHAPTER 1. ATOMS IN MOTION +1-3 Atomic processes. . . . . . . . . . Q Q Q Q Q *+*k*Š + >2 >>> T-8 +CHAPTER 2. BASIC PHYSICS +CHAPTER 3. “HE RELATION OF PHYSICS TO ÔTHER SCIENCES +bì 5 o9 HH... la a a( (: dd 3-1 +3-3 Biology....... Q Q Q Q Q Q H Q k v kk k x x . th +3-4 ASETOHOMYV...... . . Q Q Q Q HQ HQ n1 1355113 x + e0 +CHAPTER 4. CONSERVATION OF ENERGY +--- Trang 23 --- +CHAPTER 5ð. TIME AND DISTANCE +B MW) aáaaaaa a Ha Ta HA +5 0b Số .aaa. ——=s=<... +5-7 Shortdistances...... . . . HQ Q1 12123232 + +. O=14 +CHAPTER 6. PROBABILITY +CHAPTER 7. 'THE THEORY OF GRAVITATION +7-2 lKeplerlaws ... . . . . HQ HQ K23 33+ + + “2 +HN, sa › `. ... “8ä ...g77a“<“aaad13...Á«Á +7-8 Gravity and relativitly...... . . . Q.20 +CHAPTER 8. MOTION +8-2 Speed ...... . HQ ng vn V1 1113151114232 +. Ñ=4 +--- Trang 24 --- +CHAPTER 9._ NEWTON”S LAWS OF DYNAMICS +CHAPTER 10. CONSERVATION OF MOMENTUM +CHAPTER 11. VECTORS +II 9. a .... .. . . TT L - aa &š&ä-: +CHAPTER 12. CHARACTERISTICS OF EORƠE +12-1 WhatisaÍOrce?....... . . . LH Q2 22222222222 22+ 12-1 +--- Trang 25 --- +CHAPTER 13. WORK AND POTENTIAL ENERGY (A) +CHAPTER 14. WORK AND POTENTIAL ENERGY (CONGLUSION) +14-3 Conservative ÍOrC@S....... . . . Ặ Q HQ HQ HQ 12353 + >> 14-5 +CHAPTER 15. “HE SPECIAL THEORY OF RÑELATIVITY +CHAPTER 16. RELATIVISTIC EBNERGY AND MOMENTUM +CHAPTER 17. SPACE-TIME +--- Trang 26 --- +CHAPTER 18. ROTATION IN TWO DIMENSIONS +CHAPTER 19. CENTER OF MASS; MOMENT OF Í[NERTIA +CHAPTER 20. ROTATION IN SPACE +CHAPTER 21. “HE HARMONIC Ô§CILLATOR +CHAPTER 22. ALGEBRA +--- Trang 27 --- +CHAPTER 23. RESONANCE +CHAPTER 24. 'ERANSIENTS +CHAPTER 25. LINEAR SYSTEMS AND REVIEW +CHAPTER 26. (OPTICS: THE PRINCIPLE OF LEAST TIME +26-1 Light....... . . Q1 1111111514554 55423 + + 26-1 +CHAPTER 27. GEOMETRICAL OPTICS +--- Trang 28 --- +CHAPTER 28. ELECTROMAGNETIC RADIATION +CHAPTER 29. ÍÏNTEREFERENCE +CHAPTER 30. DIFFRACTION +CHAPTER 31. “HE ORIGIN OF THE REFRACTIVE [NDEX +CHAPTER 32. RADIATION DAMPING. LIGHT SCATTERING +--- Trang 29 --- +CHAPTER 33. POLARIZATION +33-5 Optical activity...... . . . Q Q Q Q Q Q k + k> „ «.„ „38-10 +CHAPTER 34. RELATIVISTIC EFFECTS IN RADIATION +34-7 The œ,k ÍOUT-VeCEOT........ . Ặ Q23 23+ „+. « 34-16 +CHAPTER 3ð. COLOR VISION +CHAPTER 36. MECHANISMS OF SEEING +--- Trang 30 --- +CHAPTER 37. (QQUANTUM BEHAVIOR, +CHAPTER 38. “HE RELATION OF WAVE AND PARTICLE VIEWPOINTS +CHAPTER 39. “HE KINETIC 'HEORY OF GASES +CHAPTER 40. “HE PRINCIPLES OF STATISTICAL MECHANICS +--- Trang 31 --- +CHAPTER 41. “HE BROWNIAN MOVEMENT +CHAPTER 42. APPLICATIONS OEF KINETIC THEORY +CHAPTER 43. DIFEUSION +CHAPTER 44. “HE LAWS OF THERMODYNAMICS +CHAPTER 45. lLLUSTRATIONS OF THERMODYNAMICS +--- Trang 32 --- +CHAPTER 46. RATCHET AND PAWL +CHAPTER 47. SOUND. HE WAVE EQUATION +CHAPTER 48. BEATS +48-1 Adding ÉWO WaVeS...... . Q.33 313333 + + +. 48-1 +CHAPTER 49. MODES +CHAPTER 50. HARMONICS +--- Trang 33 --- +CHAPTER 5l. WAVES +51-1 PowwWaves........ . HQ Q Q Q Q Q Q22 2212222222222 SIm] +CHAPTER 52. SYMMETRY IN PHYSICAL LAWS +ÏNDEX +NAME ÏNDEX +LIST OF SYMBOLS +--- Trang 34 --- +Aforms tra WẪoffOGre +1-1 Introduction +'This two-year course In physics is presented from the point of view that you, +the reader, are going to be a physicist. This is not necessarily the case Of course, +but that is what every professor in every subject assumesl IÝ you are going to be +a physicist, you will have a lot to study: two hundred years of the most rapidly +developing field of knowledge that there is. 5o much knowledge, in fact, that you +might think that you cannot learn all of it in four years, and trulÌy you cannot; +you will have to go to graduate school tool +Surprisingly enouph, ¡in spite of the tremendous amount of work that has been +done for all this time it is possible to condense the enormous mass of results to +a large extent—that is, to fñnd /œws which summarize all our knowledge. Even +so, the laws are so hard to grasp that it is unfair to you to start exploring this +tremendous subject without some kind oŸ map or outline of the relationship of one +part of the subject of science to another. Following these preliminary remarks, +the first three chapters wiïll therefore outline the relation of physics to the rest +of the sciences, the relations of the sciences to each other, and the meaning of +seience, to help us develop a “feel” for the subJect. +You might ask why we cannot teach physics by just giving the basic laws on +page one and then showing how they work in all possible circumstances, as we +do in Euclidean geometry, where we state the axioms and then make all sorts of +deductions. (So, not satisfied to learn physics in Íour years, you want to learn it +in four minutes?) We cannot do it in this way for two reasons. First, we do not +yet knou all the basic laws: there is an expanding frontier of ignorance. Second, +the correct statement of the laws of physics involves some very unfamiliar ideas +which require advanced mathematies for their description. 'Pherefore, one needs a +considerable amount of preparatory training even to learn what the +0ords mean. +No, it is not possible to do it that way. We can only do it piece by piece. +--- Trang 35 --- +lach piece, or part, of the whole of nature is always merely an œpprozữmation +to the complete truth, or the complete truth so far as we know it. In fact, +everything we know is only some kind of approximation, because +0e kno+ that +tue do not knou aÌl the laus as yet. Therefore, things must be learned only to be +unlearned again or, more likely, to be corrected. +'The principle of science, the defnition, almost, is the following: The test oƒ +gÌÌ knouledqe is ezperimnent. xperiment 1s the sole 7udge of scientific “truth.” +But what ¡is the source of knowledge? Where do the laws that are to be tested +come from? Experiment, itself, helps to produce these laws, in the sense that it +gives us hints. But also needed is #maginalion to create from these hints the great +generalizations—to guess at the wonderful, simple, but very strange patterns +beneath them all, and then to experiment to check again whether we have made +the right guess. This Imagining process is so dificult that there is a division of +labor in physics: there are #2eoreficœl physicists who imagine, deduece, and guess +at new laws, but do not experiment; and then there are ezperữmnental physicists +who experiment, imagine, deduce, and øuess. +W© said that the laws of nature are approximate: that we fñrst ñnd the “wrong” +ones, and then we ñnd the “right” ones. Now, how can an experiment be “wrong”? +first, in a trivial way: 1ƒ something is wrong with the apparatus that you did +not notice. But these things are easily fxed, and checked back and forth. So +without snatching at such minor things, how can the results oŸ an experiment +be wrong? Only by being inaccurate. For example, the mass oŸ an object never +seems to change: a spinning top has the same weight as a still one. So a “law” +was invented: mass is constant, independent of speed. That “law” is now found +to be incorrect. Mass is found to increase with velocity, but appreciable increases +require velocities near that of light. A frue law is: if an objecb moves with a +speed of less than one hundred miles a second the mass is constant to within one +part in a million. In some such approximate form this is a correct law. So 1n +practice one might think that the new law makes no significant diference. Well, +yes and no. Eor ordinary speeds we can certainly forget it and use the simple +constant-mass law as a good approximation. But for high speeds we are wrong, +and the higher the speed, the more wrong we are. +Finally, and most interesting, ph?losophácallu tue are cormnpletclg trong with +the approximate law. Our entire picture of the world has to be altered even +though the mass changes only by a little bít. This is a very peculiar thing about +the philosophy, or the ideas, behind the laws. Even a very small efect sometimes +requires profound changes In our ideas. +--- Trang 36 --- +Now, what should we teach first? Should we teach the correc£ but unfamiliar +law with its strange and difficult conceptual ideas, for example the theory of +relativity, four-dimensional space-time, and so on? Ôr should we first teach the +simple “constant-mass” law, which is only approximate, but does not involve such +diffcult ideas? “The first is more exciting, more wonderful, and more fun, but +the second is easier to get at first, and is a first step to a real understanding of +the fñrst idea. This point arises again and again in teaching physics. At diferent +times we shall have to resolve 1% in diferent ways, but at each stage it is worth +learning what is now known, how accurate It is, how it fits into everything else, +and how it may be changed when we learn more. +Let us now proceed with our outline, or general map, of our understanding of +science today (in particular, physics, but also of other sciences on the periphery), +so that when we later concentrate on some particular point we will have some +idea. of the background, why that particular point is interesting, and how it fts +Into the big structure. So, what 7s our over-all picture of the world? +1-2 Matter is made of atoms +T, in some cataclysm, all of scientifie knowledge were to be destroyed, and +onÌy one sentence passed on to the next generations of creatures, what statement +would contain the most information in the fewest words? I believe i% is the +atomäc hụpothesis (or the atomic ƒfact, or whatever you wish to call it) that ail +thứngs are mmade oƒ atormns—lifle particles that moue around ?ín perpetual motion, +ttracting cach other t”hen theU are a litie distance apart, Du repelling tupon +being squeczcd ¡no one another. In that one sentence, you wilÏ see, there is an +€norrmmous amount of information about the world, 1 Just a little imagination and +thinking are applied. +To illustrate the power of the atomic idea, suppose that we have a drop +of water a quarter of an ¡inch on the side. If we look at it very closely we see +nothing but water—smooth, continuous water. Even iŸ we magnify it with the +best optical microscope available—roughly ©wo thousand times—then the water +drop will be roughly forty feet across, about as big as a large room, and if we +looked rather closely, we would s#ji see relatively smooth water——but here and +there small football-shaped things swimming back and forth. Very interesting. +These are paramecia. You may stop at this point and get so curious about the +paramecia with their wiggling cilia and twisting bodies that you go no further, +except perhaps to magnify the paramecia still more and see inside. 'This, of +--- Trang 37 --- +C) C | i O +so hQS +D -&® lÓO +O4. - @® +` C3 —&WV ) +WATER MAGNIFIED ONE BILLION TIMES +Figure 1-1 +course, is a subject for biology, but for the present we pass on and look still +more closely at the water material itself, magnifying it two thousand times again. +Now the drop of water extends about fñfteen miles across, and if we look very +closely at i we see a kind of teeming, something which no longer has a smooth +appearance——it looks something like a crowd at a football game as seen from a +very great distance. In order to see what this teeming is about, we will magnify +it another two hundred and ffty times and we will see something similar to what +is shown in Fig. I-I. This is a picbure of water magnified a billion times, but +1dealized in several ways. In the first place, the particles are drawn in a simple +manner with sharp edges, which is inaccurate. Secondly, for simplicity, they are +sketched almost schematically in a ©wo-dimensional arrangement, but oŸ course +they are moving around in three dimensions. Notice that there are two kinds of +“blobs” or circles to represent the aboms of oxygen (black) and hydrogen (white), +and that each oxygen has two hydrogens tied to it. (Each little group oŸ an +oxygen with its two hydrogens is called a molecule.) The picture is idealized +further in that the real particles in nature are continually jiggling and bouncing, +turning and twisting around one another. You will have to imagine this as a +dynamic rather than a static picture. Another thing that cannot be illustrated in +a drawing is the fact that the particles are “stuck together”—that they attract +cach other, this one pulled by that one, etc. The whole group is “glued together,” +so to speak. Ôn the other hand, the particles do not squeeze through each other. +T you try to squeeze two of them too close together, they repel. +The atoms are 1 or 2 x 10” em in radius. NÑow 10~Š em is called an angstrom +(just as another name), so we say they are 1 or 2 angstroms (Ä) in radius. Another +way to remember theïr size is this: if an apple is magnified to the size of the earth, +then the atoms in the apple are approximately the size of the original apple. +--- Trang 38 --- +Now imagine this great drop of water with all of these jiggling particles stuck +together and tagging along with each other. 'Phe water keeps its volume; it does +not fall apart, because of the attraction of the molecules for each other. Tf the +drop is on a sÌope, where it can move from one place to another, the water will +fow, but it does not just disappear—things do not just ñy apart——because of the +molecular attraction. Now the jiggling motion is what we represent as heaf#: when +we increase the temperature, we increase the motion. lf we heat the water, the +Jiggling increases and the volume between the atoms increases, and if the heating +continues there comes a time when the pull bebween the molecules is not enough +to hold them together and they đo ñy apart and become separated from one +another. OŸ course, this is how we manufacture steam out of water——by increasing +the temperature; the particles ñy apart because of the increased motion. +STEAM +Figure 1-2 +In Eig. I-2 we have a picture of steam. 'Phis picture of steam fails in one +respect: at ordinary atmospheric pressure there certainly would not be as many +as three water molecules in this fgure. Most squares this size would contain +none—but we accidentally have two and a half or three in the picture (just +so it would not be completely blank). Now in the case of sieam we see the +characteristic molecules more clearly than in the case of water. For simplicity, the +molecules are drawn so that there is a 120° angle between the hydrogen atoms. In +actual fact the angle is 1053”, and the distance between the center of a hydrogen +and the center of the oxygen is 0.957 Ä, so we know this molecule very well. +Let us see what some of the properties of steam vapor or any other gas are. +'The molecules, being separated from one another, will bounce against the walls. +Imagine a room with a number of tennis balls (a hundred or so) bouncing around +in perpetual motion. When they bombard the wall, this pushes the wall away. +--- Trang 39 --- +` \ế \ +‡ “ Ả— +Figure 1-3 +(Of course we would have to push the wall back.) This means that the gas exerts +a Jittery force which our coarse senses (not being ourselves magnified a billion +times) feel only as an ø0erage push. In order to confine a gas we must apply +a pressure. Figure l-3 shows a siandard vessel for holding gases (used in all +textbooks), a cylinder with a piston in it. Now, it makes no diference what the +shapes of water molecules are, so for simplicity we shall draw them as tennis balls +or little dots. These things are in perpetual motion in all directions. So many of +them are hitting the top piston all the time that to keep it from being patiently +knocked out of the tank by this continuous banging, we shall have to hold the +piston down by a certain force, which we call the pressure (really, the pressure +times the area is the force). Clearly, the force is proportional to the area, for If +we increase the area but keep the number of molecules per cubic centimeter the +same, we increase the number of collisions with the piston in the same proportion +as the area was increased. +Now let us put 0wice as many molecules in this tank, so as to double the +density, and let them have the same speed, ¡.e., the same temperature. Then, to +a close approximation, the number of collisions will be doubled, and since each +will be just as “energetic” as before, the pressure is proportional to the density. +Tf we consider the true nature of the forces between the atoms, we would expect +a slight decrease in pressure because of the attraction between the atoms, and +a slipht Increase because of the fnite volume they occupy. Nevertheless, to an +excellent approximation, if the density is low enough that there are not many +atoms, £he pressure ¡s proportional to the densit. +We can also see something else: lÝ we increase the temperature without +changing the density of the gas, I.e., iŸ we increase the speed of the atoms, what +1s goïng to happen to the pressure? Well, the atoms hit harder because they are +--- Trang 40 --- +moving faster, and in addition they hit more often, so the pressure increases. +You see how simple the ideas of atomie theory are. +Let us consider another situation. Suppose that the piston moves inward, so +that the atoms are slowly compressed into a smaller space. What happens when +an atom hits the moving piston? Evidently it picks up speed from the collision. +You can try it by bouncing a ping-pong ball from a forward-moving paddle, for +example, and you will fnd that ít comes of with more speed than that with +which ¡9 struck. (Special example: iŸ an atom happens to be standing still and +the piston hits it, it will certainly move.) So the atoms are “hotter” when they +come away from the piston than they were before they struck it. Therefore all +the atoms which are in the vessel wiïll have picked up speed. “This means that +tuhen tue compress œ gas sÏloulụ, the temperature oƒ the gas ?ncreases. So, under +SÌlOWw compression, a gas wiÌ] ?merease in temperature, and under sÌOw ezpdnsion +1t will đecrease in temperature. +'We now return to our drop of water and look in another direction. Suppose +that we decrease the temperature of our drop of water. Suppose that the jiggling +of the molecules of the atoms in the water is steadily decreasing. We know that +there are forces of attraction between the atoms, so that after a while they will +not be able to jiggle so well. What will happen at very low temperatures 1s +indicated in Fig. 1-4: the molecules lock into a new pattern which is ?cc. This +particular schematic diagram of ice is wrong because it is in two dimensions, but +1t 1s right qualitatively. The interesting point is that the material has a defnite +pÌace for cuer œtom, and you can easily appreciate that If somehow or other +we were to hold all the atoms at one end of the drop in a certain arrangement, +cach atom in a certain place, then because of the structure of interconnections, +which is rigid, the other end miles away (at our magnified scale) will have a +ý Qua gô -—c% ` +cv @@- +Figure 1-4 +--- Trang 41 --- +defnite location. So if we hold a needle of ice at one end, the other end resists +our pushing it aside, unlike the case of water, in which the structure is broken +down because of the increased jiggling so that the atoms all move around In +diÑerent ways. The diference between solids and liquids is, then, that in a solid +the atoms are arranged in some kind of an array, called a crstalline arrau, and +they do not have a random position at long distances; the position of the atoms +on one side of the crystal is determined by that of other atoms millions of atoms +away on the other side of the crystal. Pigure 1-4 is an invented arrangement Íor +ice, and although it contains many of the correct features oŸ ice, i is not the true +arrangement. One of the correct features is that there is a part of the symmetry +that is hexagonal. You can see that iŸ we turn the picture around an axis by 60, +the picture returns to itself. 5o there is a sựmưmnefrw in the ice which accounts for +the six-sided appearance of snowflakes. Another thing we can see from Eig. l-4 is +why ice shrinks when it melts. The particular crystal pattern of ice shown here +has many “holes” in it, as does the true ice structure. When the organization +breaks down, these holes can be occupied by molecules. Most simple substances, +with the exception of water and type metal, ezpand upon melting, because the +atoms are closely packed in the solid crystal and upon melting need more room +to jiggle around, but an open structure collapses, as In the case of water. +Now although ice has a “rigid” crystalline form, its temperature can change—— +ice has heat. IÝ we wish, we can change the amount of heat. What is the heat in +the case of ice? "The atoms are not standing still. They are jiggling and vibrating. +So even thouph there is a defnite order to the crystal—a defnite structure——all of +the atoms are vibrating “in place” As we increase the temperature, they vibrate +with greater and greater amplitude, until they shake themselves out of place. We +call this melting. As we decrease the temperature, the vibration decreases and +decreases until, at absolute zero, there is a minimum amount of vibration that +the atoms can have, but noøý zero. This minimum amount of motion that atoms +can have is not enough to melt a substance, with one exception: helium. Helium +merely decreases the atomic motions as much as it can, but even at absolute zero +there is still enough motion to keep it from freezing. Helium, even at absolute +zero, does not freeze, unless the pressure is made so great as to make the atoms +squash together. IÝ we increase the pressure, we cøn make it solidIfy. +1-3 Atomic processes +So mụuch for the description of solids, liquids, and gases from the atomic point +of view. However, the atomic hypothesis also describes ørocesses, and so we shall +--- Trang 42 --- +°Ồ s +Q cm 6®. .( +WATER EVAPORATING IN AIR +® ° 2 +©XYGEN HYDROGEN NITROGEN +Figure 1-5 +now look at a number of processes from an atomie standpoint. The first process +that we shall look at is associated with the surface of the water. What happens at +the surface of the water? We shall now make the picture more complieated—=and +more realistic—by imagining that the surface is in air. Eigure I-5 shows the +surface of water in air. We see the water molecules as before, forming a body of +liquid water, but now we also see the surface of the water. Above the surface +we fnd a number of things: First of all there are water molecules, as in steam. +Thìs is 0øfer 0apor, which is always found above liquid water. (There is an +cquilibrium between the steam vapor and the water which will be described later.) +Tn addition we ñnd some other molecules—here two oxygen atoms stuck together +by themselves, forming an ozgen rnolecule, there two nitrogen atoms also stuck +together to make a nitrogen molecule. Air consists almost entirely of nitrogen, +oxygen, some water vapor, and lesser amounts of carbon dioxide, argon, and +other things. So above the water surface is the air, a gas, containing some water +vapor. Now what is happening in this picture? 'Phe molecules in the water are +always jiggling around. Erom time to time, one on the surface happens to be hit a +little harder than usual, and gets knocked away. It is hard to see that happening +in the picture because ït is a sfZll picture. But we can imagine that one molecule +near the surface has Just been hit and is Ñying out, or perhaps another one has +been hit and is fying out. Thus, molecule by molecule, the water disappears——1t +evaporates. But if we ciose the vessel above, after a while we shall fnd a large +number of molecules of water amongst the air molecules. From tỉme to time, one +of these vapor molecules comes fÑying down to the water and gets sbuck again. +So we see that what looks like a dead, uninteresting thing—a glass of water with +--- Trang 43 --- +a cover, that has been sitting there for perhaps twenty years—really contains +a dynamic and interesting phenomenon which is goïng on all the time. 'To our +eyes, our crude eyes, nothing 1s changing, but if we could see it a billion times +magnifed, we would see that from its own point oŸ view it is always changing: +mmolecules are leaving the surface, molecules are coming back. +Why do +0e see no change? Because just as many molecules are leaving as are +coming backl In the long run “nothing happens.” If we then take the top of the +vessel of and blow the moist air away, replacing it with dry air, then the number +of molecules leaving is just the same as it was before, because this depends on +the jiggling of the water, but the number coming back is greatly reduced because +there are so many fewer water molecules above the water. Thherefore there are +more going out than coming in, and the water evaporates. Hence, If you wish to +evaporate water turn on the fanl +Here is something else: Which molecules leave? When a molecule leaves 1t +is due to an accidental, extra accumulation of a little bit more than ordinary +energy, which it needs iÝ it is to break away from the attractions of its neighbors. +'Therefore, since those that leave have more energy than the average, the ones that +are left have iess average motion than they had before. 5o the liquid gradually +cools 1ƒ it evaporates. Of course, when a molecule of vapor comes from the air to +the water below there is a sudden great attraction as the molecule approaches the +surface. 'Phis speeds up the incoming molecule and results in generation of heat. +So when they leave they take away heat; when they come back they generate +heat. Of course when there is no net evaporation the result is nothing—the +water is not changing temperature. lf we blow on the water so as to maintain a +continuous preponderance in the number evaporating, then the water is cooled. +Hence, blow on soup to cool it† +Of course you should realize that the processes just described are more +complicated than we have indicated. Not only does the water go into the air, but +also, from time to time, one of the oxygen or nitrogen molecules will come in and +“get lost” in the mass of water molecules, and work its way into the water. Thus +the air dissolves in the water; oxygen and nitrogen molecules will work their way +into the water and the water will contain air. If we suddenly take the air away +from the vessel, then the air molecules will leave more rapidly than they come ïn, +and in doïng so will make bubbles. 'Phis is very bad for divers, as you may know. +Now we go on to another process. In Fig. I-6 we see, from an atomie point of +view, a solid dissolving in water. lf we put a crystal of salt in the water, what +will happen? 5alt is a solid, a crystal, an organized arrangement oŸ “salt atoms.” +--- Trang 44 --- +S ) SS C seo \ °, +có )›ề Ả J@” x© +®, Co © +®œ® s= @ +SALT DISSOLVING IN WATER +® CHLORINE C SODIUM +Figure 1-6 +Jigure 1-7 is an ilHustration of the three-dimensiona]l structure oŸ common salt, +sodium chloride. Strictly speaking, the crystal is not made of atoms, but oŸ what +we call jons. An ion is an atom which either has a few extra electrons or has lost +a few electrons. In a salt crystal we find chlorine ions (chlorine atoms with an +extra electron) and sodium ions (sodium atoms with one electron missing). The +1ons all stick together by electrical attraction in the solid salt, but when we put +them in the water we fñnd, because of the attractions of the negative oxygen and +positive hydrogen for the ions, that some of the ions jiggle loose. In Eig. 1-6 we +see a chlorine ion getting loose, and other atoms foating in the water in the form +of lons. This picture was made with some care. Notice, for example, that the +hydrogen ends of the water molecules are more likely to be near the chlorine ion, +. _ˆ 8 +Rode S52 LEEL. +Sylvine K ClI | 6.28 È +Ag | Cl | 5.54 +H55 +Pb | Se | 6.14 đd mi +Pb | Te | 6.34 ù Ò © +Nearest neighbor +distance d = a/2 +Figure 1-7 +--- Trang 45 --- +while near the sodium ion we are more likely to ñnd the oxygen end, because the +sodium is positive and the oxygen end of the water is negative, and they attract +electrically. Can we tell from this picture whether the salt is đ/ssolưing ín water +or crstallizing out of water? Of course we cønnot tell, because while some of +the atoms are leaving the crystal other atoms are rejoining it. The process is a +đụngmïc one, just as in the case of evaporation, and it depends on whether there +is more or less salt in the water than the amount needed for equilibrium. By +cquilibrium we mean that situation in which the rate at which atoms are leaving +Just matches the rate at which they are coming back. If there is almost no salt in +the water, more atoms leave than return, and the salt dissolves. If, on the other +hand, there are too many “salt atoms,” more return than leave, and the salt is +crystallizing. +In passing, we mention that the concept of a rmmolecule oŸ a substanece is onÌy +approximate and exists only for a certain class of substances. It is clear in the +case of water that the three atoms are actually stuck together. lt is not so clear in +the case of sodium chloride in the solid. 'Phere is just an arrangement of sodiun +and chlorine Ions in a cubic pattern. There is no natural way to group them as +“molecules of salt.” +Returning to our discussion of solution and precipitation, if we increase the +temperature of the salt solution, then the rate at which atoms are taken away +1s increased, and so is the rate at which atoms are brought back. It turns out +to be very diflcult, in general, to predict which way it is going to go, whether +more or less of the solid will dissolve. Most substances dissolve more, but some +substances dissolve less, as the temperature increases. +1-4 Chemical reactions +In all of the processes which have been described so far, the atoms and the +lons have not changed partners, but of course there are cireumstances in which +the atoms do change combinations, forming new molecules. 'This is ilustrated in +Eig. I-8. Á process in which the rearrangement of the atomic partners OcCUTS is +what we call a chemjcal reaction. The other processes so far described are called +physical processes, but there is no sharp distinction bebween the bwo. (Nature +does not care what we call it, she just keeps on doïng it.) Thịs figure is supposed +to represent carbon burning in oxygen. In the case oŸ oxygen, #o oxygen atoms +sbick together very stronply. (Why do not #hree or even ƒour stick together? That +is one of the very peculiar characteristics of such atomic processes. Atoms are +--- Trang 46 --- +`/{Š Ý +¬ K.ey 0559555 956 +CARBON BURNING IN OXYGEN +Figure 1-8 +very special: they like certain particular partners, certain particular directions, +and so on. lt is the job of physics to analyze why each one wants what it wants. +At any rate, two oxygen atoms form, saturated and happy, a molecule.) +The carbon atoms are supposed to be in a solid crystal (which could be +graphite or diamond*). Now, for example, one of the oxygen molecules can come +over to the carbon, and each atom can pick up a carbon atom and go fying of +in a new combination—“carbon-oxygen”—which is a molecule of the gas called +carbon monoxide. It is given the chemical name CO. It is very simple: the letters +“CO” are practically a picbure of that molecule. But carbon attracts oxygen +much more than oxygen attracts oxygen or carbon attracts carbon. 'Pherefore +in this process the oxygen may arrive with only a little energy, but the oxygen +and carbon will snap together with a tremendous vengeance and commotion, and +everything near them will pick up the energy. A large amount of motion energy, +kinetic energy, is thus generated. This of course 1s burnzng; we are getting hea +from the combination oŸ oxygen and carbon. The heat is ordinarily in the form +of the molecular motion of the hot gas, but in certain circumstances it can be so +enormous that it generates /2gh. That is how one gets fiames. +In addition, the carbon monoxide is not quite satisfed. It is possible for it to +attach another oxygen, so that we might have a much more complicated reaction +in which the oxygen is combining with the carbon, while at the same time there +happens to be a collision with a carbon monoxide molecule. Ône oxygen atom +could attach itself to the CO and ultimately form a molecule, composed of one +carbon and two oxygens, which is designated COsa and called carbon dioxide. lf +we burn the carbon with very little oxygen in a very rapid reaction (for example, +in an automobile engine, where the explosion 1s so fast that there is not time +* One can burn a diamond in air. +--- Trang 47 --- +for it to make carbon dioxide) a considerable amount of carbon monoxide is +formed. In many such rearrangements, a very large amount oŸ energy is released, +forming explosions, Ñames, etc., depending on the reactions. Chemists have +studied these arrangements of the atoms, and found that every substance is some +type OŸ arrangement oƒ atoms. +To illustrate thịs idea, let us consider another example. lIf we go into a fñeld +of small violets, we know what “that smell” is. It is some kind of molecule, or +arrangement of atoms, that has worked Its way into our noses. First of all, hou +dịd it work its way in? That is rather easy. If the smell is some kind of molecule +in the aïr, jiggling around and being knocked every which way, it might have +accidentallu worked its way into the nose. Certainly it has no particular desire to +get into our nose. lt is merely one helpless part of a jostling crowd of molecules, +and in its aimless wanderings this particular chunk of matter happens to fñnd +1tself in the nose. +Now chemists can take special molecules like the odor of violets, and analyze +them and tell us the ezøc# arrangement of the atoms in space. We know that +the carbon dioxide molecule is straight and symmetrical: O—C——O. (That can +be determined easily, too, by physical methods.) However, even for the vastly +more complicated arrangements of atoms that there are in chemistry, one can, +by a long, remarkable process of detective work, fnd the arrangements of the +atoms. Figure l-9 is a picture of the air in the neighborhood of a violet; again +we find nitrogen and oxygen in the air, and water vapor. (Why is there water +vapor? Because the violet is œef. AII plants transpire.) However, we also see +a “monster” composed of carbon atoms, hydrogen atoms, and oxygen atoms, +which have picked a certain particular pattern in which to be arranged. It is +a mụch more complicated arrangement than that of carbon dioxide; in fact, 1% +2 ©° 42 +4JDD4Đ +ODOR OF VIOLETS +Figure 1-9 +--- Trang 48 --- +CHa: CHs +N >c< HN ọ +CH:-CZ“ C—C=C—C—CH: +HẸ SỐ 4 —CHza +Fig. 1-10. The substance pictured Is œ-irone. +1s an enormously complicated arrangement. nfortunately, we cannot picture +all that is really known about it chemically, because the precise arrangement +of all the atoms is actually known in three dimensions, while our picture is In +only t§wo dimensions. The six carbons which form a rỉng do not form a fat ring, +but a kind of “puckered” ring. All of the angles and distances are known. So a +chemical ƒormula is merely a picture oŸ such a molecule. When the chemist writes +such a thing on the blackboard, he is trying to “draw,” roughly speaking, in two +dimensions. Eor example, we see a “ring” of six carbons, and a “chain” of carbons +hanging on the end, with an oxygen second from the end, three hydrogens tied +to that carbon, two carbons and three hydrogens sticking up here, etc. +How does the chemist fnd what the arrangement is? He mixes bottles full of +stuf together, and if it turns red, it tells him that it consists of one hydrogen +and two carbons tied on here; 1Ý it turns blue, on the other hand, that is not +the way it is at all. Thịis is one of the most fantastic pieces of detective work +that has ever been done—organic chemistry. To discover the arrangement of the +atoms in these enorrmously complicated arrays the chemist looks at what happens +when he mixes two diferent substances together. The physicist could never quite +believe that the chemist knew what he was talking about when he described +the arrangement of the atoms. For about twenty years it has been possible, In +some cases, to look at such molecules (not quite as complicated as this one, but +some which contain parts of it) by a physical method, and it has been possible +to locate every atom, not by looking at colors, but by rmeasuring tuhere theU qre. +And lo and behold!, the chemists are almost aÌways correct. +Tt turns out, in fact, that in the odor oŸ violets there are three slightly diferent +mmolecules, which difÑfer only in the arrangement of the hydrogen atoms. +One problem of chemistry is to name a substance, so that we will know what +itis. Pind a name for this shapel Not only must the name tell the shape, but +--- Trang 49 --- +1 must also tell that here is an oxygen atom, there a hydrogen——exactly what +and where each atom is. So we can appreciate that the chemical names must +be complex In order to be complete. You see that the name of this thing In +the more complete form that will tell you the structure of it is 4-(2, 2, 3, 6 +tetramethy]-5-cyclohexeny])-3-buten-2-one, and that tells you that thìs is the +arrangement. We can appreciate the difficulties that the chemists have, and also +appreciate the reason for such long names. Ït is not that they wish to be obscure, +but they have an extremely dificult problem in trying to describe the molecules +in wordsl +How do we knou that there are atoms? By one of the tricks mentioned earlier: +we make the hựpothesis that there are atoms, and one after the other results come +out the way we prediect, as they ought to 1ƒ things are made of atoms. There is +also somewhat more direct evidence, a good example oŸ which is the following: +The atoms are so small that you cannot see them with a light microscope——in +fact, not even with an electron microscope. (With a light microscope you can +only see things which are much bigger.) Now if the atoms are always in motion, +say in water, and we put a big ball of something in the water, a ball much bigger +than the atoms, the ball will jiggle around——much as in a push ball game, where +a great big ball is pushed around by a lot of people. “The people are pushing in +various directions, and the ball moves around the fñeld in an irregular fashion. +So, in the same way, the “large ball” will move because of the inequalities of the +collisions on one side to the other, from one moment to the next. Thherefore, if we +look at very tiny particles (colloids) in water through an excellent microscope, we +see a perpetual jiggling of the particles, which is the result of the bombardment +of the atoms. This ¡is called the PBrounian rnotion. +We can see further evidence for atoms in the structure of crystals. In many +cases the structures deduced by x-ray analysis agree in their spatial “shapes” +with the forms actually exhibited by crystals as they occur in nature. The angles +between the various “faces” of a crystal agree, within seconds of arc, with angles +deduced on the assumption that a crystal is made of many “layers” of atoms. +ueruthing ¡s made öƒ atoms. That 1s the key hypothesis. The most important +hypothesis ín all of biology, for example, is that cuerthing that animals do, atoms +đo. In other words, (here ¡s nothing that liuứng thíngs do that cannot be wnderstood +from the poin‡ oƒƑ uieuU that the are made oƒ atoms acting according to the lats +öƒ phụsics. This was not known from the beginning: it took some experimenting +and theorizing to suggest this hypothesis, but now it is accepted, and it is the +mmost useful theory for producing new ideas in the fñeld of biology. +--- Trang 50 --- +TÝ a piece of steel or a piece of salt, consisting of atoms one next to the other, +can have such interesting properties; iŸ water—which is nothing but these little +blobs, mile upon mile of the same thing over the earth—can form waves and +foam, and make rushing noises and strange patterns as it runs over cement; ïf all +of this, all the life of a stream of water, can be nothing but a pile of atoms, hou +tmuch more is possible? T instead oŸ arranging the atoms in some defñnite pattern, +again and again repeated, on and on, or even forming little lumps of complexity +like the odor of violets, we make an arrangement which is akh0øws đierent from +place to place, with difÑferent kinds of atoms arranged in many ways, continually +changing, not repeating, how much more marvelously is it possible that this thing +might behave? Is it possible that that “thing” walking back and forth in front of +you, talking to you, is a great glob of these atoms in a very complex arrangement, +such that the sheer complexity of it staggers the imagination as to what it can +do? When we say we are a pile of atoms, we do not mean we are merel a pile +of atoms, because a pile of atoms which is not repeated from one to the other +might well have the possibilities which you see before you in the mirror. +--- Trang 51 --- +M?qasic FPhạysữcs +2-1 Introduction +In this chapter, we shall examine the most fundamental ideas that we have +about physics—the nature of things as we see them at the present time. We shall +not discuss the history of how we know that all these ideas are true; you will +learn these details in due time. +'The things with which we concern ourselves in sclence appear in myriad forms, +and with a multitude of attributes. Eor example, if we stand on the shore and +look at the sea, we see the water, the waves breaking, the foam, the sloshing +motion of the water, the sound, the air, the winds and the clouds, the sun and +the blue sky, and light; there is sand and there are rocks of various hardness +and permanence, color and texture. There are animals and seaweed, hunger and +disease, and the observer on the beach; there may be even happiness and thought. +Any other spot in nature has a similar variety of things and infuences. Ït is always +as complicated as that, no matter where it is. Curiosity demands that we ask +questions, that we try to put things together and try to understand this multitude +Of aspects as perhaps resulting from the action of a relatively small number of +elemental things and forces acting in an infnite variety of combinations. +For example: Is the sand other than the rocks? That is, is the sand perhaps +nothing but a great number of very tiny stones? Is the moon a great rock? lf +we understood rocks, would we also understand the sand and the moon? Is the +wind a sloshing of the air analogous to the sloshing motion of the water in the +sea? What common features do diferent movements have? What is common to +diferent kinds of sound? How many diferent colors are there? And so on. In +this way we try gradually to analyze all things, to put together things which at +first sipht look diferent, with the hope that we may be able to reduce the number +Of đjƒerent things and thereby understand them better. +--- Trang 52 --- +A few hundred years ago, a method was devised to fnd partial answers to +such questions. seruation, reason, and ezperiment make up what we call the +sctentifltc mmethod. We shall have to limit ourselves to a bare description of our +basic view of what is sometimes called ƒundaœmnental phụsícs, or fundamental ideas +which have arisen from the application of the scientific method. +'What do we mean by “understanding” something? We can imagine that this +complicated array of moving things which constitutes “the world” is something +like a great chess game being played by the gods, and we are observers of the +game. We do not know what the rules of the game are; all we are allowed to do +1s tO œ0øứch the playing. Of course, iŸ we watch long enough, we may eventually +catch on to a few of the rules. The rules oƒ the game are what we mean by +fundamental phụsics. ven 1ƒ we knew every rule, however, we might not be able +to understand why a particular move is made in the game, merely because it is +too complicated and our minds are limited. If you play chess you must know +that it is easy to learn all the rules, and yet it is often very hard to select the +best move or to understand why a player moves as he does. So it is in nature, +only much more so; but we may be able at least to fñnd all the rules. Actually, +we do not have all the rules now. (Every once in a while something like castling +is going on that we still do not understand.) Aside from not knowing all oŸ the +rules, what we really can explain in terms of those rules is very limited, because +almost all situations are so enormously complicated that we cannot follow the +plays of the game using the rules, much less tell what is going to happen next. +Woe must, therefore, limit ourselves to the more basic question of the rules of the +game. lf we know the rules, we consider that we “understand” the world. +How can we tell whether the rules which we “guess” at are really right iŸ we +cannot analyze the game very well? There are, roughly speaking, three ways. +Pirst, there may be situations where nature has arranged, or we arrange nature, +to be simple and to have so few parts that we can predict exactly what will +happen, and thus we can check how our rules work. (In one corner of the board +there may be only a few chess pieces at work, and that we can fgure out exactly.) +A second good way to check rules is in terms of less specific rules derived from +them. For example, the rule on the move of a bishop on a chessboard is that 1t +moves only on the diagonal. One can deduce, no matter how many moves may +be made, that a certain bishop will always be on a red square. So, without being +able to follow the details, we can always check our idea about the bishop's motion +by fñnding out whether it is always on a red square. Of course it will be, for a long +tỉme, until all of a sudden we fñnd that it is on a biack square (what happened of +--- Trang 53 --- +course, is that in the meantime it was captured, another pawn crossed for queening, +and iÈ turned into a bishop on a black square). That is the way ïÈ is in physics. +For a long time we will have a rule that works excellently in an over-all way, even +when we cannot follow the details, and then some tỉme we may discOVer a nu +ru"e. From the point of view of basic physics, the most interesting phenomena, +are of course in the me places, the places where the rules do not work——not the +places where they đo workl "That is the way in which we discover new rules. +The third way to tell whether our ideas are right is relatively crude but +probably the most powerful of them all. 'That is, by rough approzzmafion. While +we may not be able to tell why Alekhine moves £Ö⁄4s particular piece, perhaps we +can rzøoughi understand that he is gathering his pieces around the king to protect +1t, more or less, since that is the sensible thing to do in the circumstances. In +the same way, we can often understand nature, more or less, without being able +to see what euerw liitle piece is doïng, ïn terms of our understanding of the game. +At first the phenomena of nature were roughly divided into classes, like heat, +electricity, mechanics, magnetism, properties of substances, chemical phenomena, +light or optics, x-rays, nuclear physics, gravitation, meson phenomena, etc. +However, the aim is to see cømplete nature as diferent aspects of one seÈ oŸ +phenomena. 'Phat is the problem in basic theoretical physics, today——to ƒnd +the laus behind ezperiment; to œmalgamate these classes. Historically, we have +always been able to amalgamate them, but as time goes on new things are found. +Woe were amalgamating very well, when all of a sudden x-rays were found. hen +we amalgamated some more, and mesons were found. 'Therefore, at any siage +of the game, it always looks rather messy. A great deal is amalgamated, but +there are always many wires or threads hanging out in all directions. That is the +situation today, which we shall try to describe. +Some historic exarmples of amalgamation are the following. First, take heat +and mechanics. When atoms are in motion, the more motion, the more heat the +system contains, and so hea£ and aÏÌ temperature e[ffects cœn be representcd bụ +the lats oƒ rmmechanics. Another tremendous amalgamation was the discovery of +the relation between electricity, magnetism, and light, which were found to be +diferent aspects of the same thing, which we call today the electrormnagnetic teld. +Another amalgamation is the unification of chemical phenomena, the various +properties of various substances, and the behavior of atomic particles, which 1s +in the quantwm rmmechanics oƑ chemistru. +The question is, of course, is it going to be possible to amalgamate euerwthing, +and merely discover that this world represents diferent aspects of ønme thing? +--- Trang 54 --- +NÑobody knows. All we know is that as we go along, we find that we can amalga- +mate pieces, and then we find some pieces that do not ft, and we keep trying to +put the jigsaw puzzle together. Whether there are a ñnite number of pieces, and +whether there is even a border to the puzzle, is of course unknown. It will never +be known until we fñnish the picture, If ever. What we wish to do here is to see +to what extent this amalgamation process has gone on, and what the situation +1s at present, in understanding basic phenomena in terms of the smallest set of +principles. 'To express ï§ in a simple manner, 0hœ‡ œre thứngs made öoƒ and hou) +ƒeu clements are there? +2-2 Physics before 1920 +Tt is a little dificult to begin at once with the present view, so we shall first +see how things looked in about 1920 and then take a few things out of that +picture. Before 1920, our world picture was something like this: The “stage” on +which the universe øgoes is the three-dimensional spøce of geometry, as described +by Euclid, and things change in a medium called #ne. The elements on the +stage are øarf/cles, for example the atoms, which have some properiies. Eirst, the +property of inertia: If a particle is moving it keeps on going in the same direction +unless ƒorces act upon it. The second element, then, is ƒorces, which were then +thought to be of two varieties: First, an enormously complicated, detailed kind +of interaction force which held the various atoms in diferent combinations in a +complicated way, which determined whether salt would dissolve faster or sÌlower +when we raise the temperature. The other force that was known was a long-range +interaction—a smooth and quiet attraction—which varied inversely as the square +of the distance, and was called graiation. 'Phis law was known and was very +simple. Whyụ things remain in motion when they are moving, or h# there is a +law of gravitation was, of course, not known. +A description of nature is what we are concerned with here. EFrom this point +of view, then, a gas, and indeed all matter, is a myriad of moving particles. Thus +many of the things we saw while standing at the seashore can immediately be +connected. Eirst the pressure: this comes from the collisions of the atoms with the +walls or whatever; the drift of the atoms, if they are all moving in one direction +on the average, is wind; the random internal motions are the heøf. Thhere are +wawves of excess density, where too many particles have collected, and so as they +rush of they push up piles of particles farther out, and so on. This wave of excess +--- Trang 55 --- +density is sound. It is a tremendous achievement to be able to understand so +much. Some of these things were described in the previous chapter. +What kznds of particles are there? "There were considered to be 92 at that +time: 92 different kinds of atoms were ultimately discovered. They had difÑferent +names associated with their chemical properties. +The next part of the problem was, 0haf are the short-range ƒorces? Why does +carbon attract one oxygen or perhaps wo oxygens, but not three oxygens? What +1s the machinery of interaction bebtween atoms? Is it gravitation? 'Phe answer +is no. Gravity is entirely too weak. But imagine a force analogous to gravity, +varying inversely with the square of the distance, but enormousÌly more powerful +and having one difference. In gravity everything attracts everything else, but +now imagine that there are £o kinds of “things,” and that this new force (which +is the electrical force, oŸ course) has the property that likes repel but unlikes +a#trac¿. The “thing” that carries this strong interaction is called charge. +'Then what do we have? Suppose that we have two unlikes that attract each +other, a plus and a minus, and that they stick very close together. Suppose we +have another charge some distance away. Would it feel any attraction? It would +feel pracficall none, because 1f the first two are equal in size, the attraction for +the one and the repulsion for the other balance out. Therefore there is very little +force at any appreciable distance. Ôn the other hand, if we get 0erw close with +the extra charge, œftraction arises, because the repulsion of likes and attraction +of unlikes will tend to bring unlikes closer together and push likes farther apart. +Then the repulsion will be /ess than the attraction. Thịs is the reason why the +atoms, which are constituted out of plus and minus electric charges, feel very +little force when they are separated by appreciable distance (aside from gravity). +'When they come close together, they can “see inside” each other and rearrange +their charges, with the result that they have a very strong interaction. 'Phe +ultimate basis of an interaction between the atoms is elecfrical. Since this force +1s so enormous, all the plusses and all minuses will normally come together in +as intimate a combination as they can. All things, even ourselves, are made of +ñne-grained, enormously strongly interacting plus and minus parts, all neatly +balanced out. Ônce in a while, by accident, we may rub of a few minuses or a +few plusses (usually it is easier to rub of minuses), and in those circumstances +we fñnd the force of electricity nbalanced, and we can then see the efects of these +electrical attractions. +To give an idea of how much stronger electricity is than gravitation, consider +two grains of sand, a millimeter across, thirty meters apart. If the force between +--- Trang 56 --- +them were not balanced, if everything attracted everything else instead of likes +repelling, so that there were no cancellation, how much force would there be? +'There would be a force of three rmillion tons between the twol You see, there is +very, 0er little excess or deficit of the number of negative or positive charges +necessary to produce appreciable electrical efects. 'This is, of course, the reason +why you cannot see the diference between an electrically charged or uncharged +thing—so few particles are involved that they hardly make a diference in the +weight or size of an object. +With this picture the atoms were easier to understand. 'They were thought +to have a “nucleus” at the center, which is positively electrically charged and +very massive, and the nucleus is surrounded by a certain number of “electrons” +which are very light and negatively charged. Now we go a little ahead in our +story to remark that in the nucleus itself there were found two kinds of particles, +protons and neutrons, almost of the same weight and very heavy. 'Phe protons +are electrically charged and the neutrons are neutral. If we have an atom with six +protons inside its nucleus, and this is surrounded by six electrons (the negative +particles in the ordinary world of matter are all electrons, and these are very +light compared with the protons and neutrons which make nuclei), this would be +atom number six in the chemical table, and i% is called carbon. Atom number +eight ¡is called oxygen, etc., because the chemical properties depend upon the +electrons on the ow#s¿đe, and in fact only upon hoa rmamy electrons there are. 5o +the chemzcal properties of a substance depend only on a number, the number +of electrons. (The whole list of elements of the chemists really could have been +called 1, 2, 3, 4, 5, etc. Instead of saying “carbon,” we could say “element six,” +meaning six electrons, but of course, when the elements were first discovered, it +was not known that they could be numbered that way, and secondly, it would +make everything look rather complicated. It is better to have names and symbols +for these things, rather than to call everything by number.) +More was discovered about the electrical force. The natural interpretation of +electrical interaction is that two objects simply attract each other: plus against +minus. However, this was discovered to be an inadequate idea to represent ït. +A more adequate representation of the situation is to say that the existence of +the positive charge, in some sense, distorts, or creates a “condition” in space, +so that when we put the negative charge in, it feels a force. 'Phis potentiality +for producing a force is called an electric ficld. When we put an electron in an +electric field, we say it is “pulled” We then have two rules: (a) charges make a +fñeld, and (b) charges in fields have forces on them and move. "The reason for +--- Trang 57 --- +this will become clear when we discuss the following phenomena: If we were +to charge a body, say a comb, electrically, and then place a charged piece of +paper at a distance and move the comb back and forth, the paper will respond by +always pointing to the comb. lf we shake it faster, it will be discovered that the +paper is a little behind, £Öere ?s a đelœw in the action. (At the frst stage, when +we move the comb rather slowly, we fnd a complication which is r =agnetism. +Magnetic inÑuences have to do with charges ?m relaliue mmotion, so magnetic +forces and electric forces can really be attributed to one field, as two diferent +aspects of exactly the same thing. A changing electric field cannot exist without +magnetism.) IÝ we move the charged paper farther out, the delay is greater. Then +an interesting thing is observed. Although the forces between two charged objects +should go inversely as the sguare of the distance, it is found, when we shake a +charge, that the inÑuenece extends uer rmuch farther ou‡ than we would guess at +first sipht. That is, the efect falls of more slowly than the inverse square. +Here is an analogy: If we are in a pool of water and there is a Ñoating cork +very close by, we can move it “directly” by pushing the water with another +cork. If you looked only at the bwo cor&s, all you would see would be that one +moved immediately in response to the motion of the other—there is some kind of +“;nteraction” between them. OÝ course, what we really do is to disturb the t0afer; +the ater then disturbs the other cork. We could make up a “law” that if you +pushed the water a little bit, an object close by in the water would move. Tf it +were farther away, of course, the second cork would scarcely move, for we move +the water /ocaliu. On the other hand, if we jiggle the cork a new phenomenon +1s involved, in which the motion of the water moves the water there, etc., and +tuaues travel away, so that by jiggling, there is an inÑuence 0erg rmuch ƒarther out, +an oscillatory infuence, that cannot be understood from the direct interaction. +Therefore the idea of direct interaction must be replaced with the existence of +the water, or in the electrical case, with what we call the electromagnetic field. +The electromagnetic field can carry waves; some of these waves are ljghứ, +others are used in radio broadcasis, but the general name is elecfromagnetic +tuaues. 'hese oscillatory waves can have various ƒreguencies. The only thing that +is really diferent from one wave to another is the ƒrequenec oj oscillalion. lỶ we +shake a charge back and forth more and more rapidly, and look at the efects, we +get a whole series of diferent kinds of efects, which are all unifed by specifying +but one number, the number of oscillations per second. 'Phe usual “pickup” that +we get from electric currents in the circuits in the walls of a building have a +frequency of about one hundred cycles per second. If we increase the frequency to +--- Trang 58 --- +Table 2-1 +The Electromagnetic Spectrum +trequency in Rough +oscillations/sec Name behavior +102 Electrical disturbance Eield +5 x 107 - 108 Radio broadcast +10Ẻ EFM—TV +1019 Radar Waves +5x 1012-10!” Light ' +1018 X-rays +10?! ^-rays, nuclear +10? ^-rays, “artificial” Particle +10? ^-rays, in cosmiCc rays +500 or 1000 kilocycles (1 kilocycle = 1000 cycles) per second, we are “on the air,” +for this is the requenecy range which is used for radio broadcasts. (Of course it +has nothing to do with the ør! W©e can have radio broadcasts without any air.) lÝ +we again increase the requency, we come into the range that is used for EM and +TV. Going still further, we use certain short waves, for example for rœdar. Still +higher, and we do not need an instrument to “see” the stuf, we can see it with +the human eye. In the range of frequeney from 5ð x 101! to 101 eyeles per second +our eyes would see the oscillation of the charged comb, 1Ÿ we could shake it that +fast, as red, blue, or violet light, depending on the frequency. Frequenecies below +this range are called infrared, and above it, ultraviolet. The fact that we can see +in a particular frequency range makes that part of the electromagnetic spectrum +no more impressive than the other parts from a physicist's standpoint, but from +a human standpoint, of course, i% ¡s more interesting. IÝ we go up even higher in +Írequency, we get x-rays. X-rays are nothing but very high-fequency light. If we +go still higher, we get gamma rays. These two terms, x-rays and gamma rays, +are used almost synonymously. Ũsually electromagnetic rays coming from nuclei +are called gamma rays, while those of high energy from atoms are called x-rays, +but at the same frequency they are indistinguishable physically, no matter what +their source. If we go to still higher frequencies, say to 102 eycles per second, we +fnd that we can make those waves artificially, for example with the synchrotron +--- Trang 59 --- +here at Caltech. We can fñnd electromagnetic waves with stupendously high +frequencies—with even a thousand times more rapid oscillation——in the waves +found in cosznic ras. These waves cannot be controlled by us. +2-3 Quantum physics +Having described the idea of the electromagnetic field, and that this fñeld +can carry waves, we soon learn that these waves actually behave in a strange +way which seems very unwavelike. At higher frequencies they behave much +more like particles! Tt 1s guantum rmechanics, discovered just after 1920, which +explains this strange behavior. In the years before 1920, the picture of space +as a three-dimensional space, and oŸ time as a separate thing, was changed by +Binstein, fñrst into a combination which we call space-time, and then still further +IntO a cur0ed space-time to represent gravitation. So the “stage” is changed +into space-time, and gravitation 1s presumably a modification of space-time. +'Then it was also found that the rules for the motions of particles were incorrect. +The mechanical rules of “inertia” and “forces” are romg——Newton's laws are +turong——in the world of atoms. Instead, it was discovered that things on a small +scale behave 0othing like things on a large scale. 'Phat is what makes physics +difcult—and very interesting. It is hard because the way things behave on a +small scale is so “unnatural”; we have no direct experience with it. Here things +behave like nothing we know of, so that it is impossible to describe this behavior +in any other than analytic ways. It is difcult, and takes a lot of imagination. +Quantum mechanics has many aspects. In the first place, the idea that a +particle has a defnite location and a defñnite speed is no longer allowed; that is +wrong. To give an example of how wrong classical physics is, there is a rule in +quantum mechanics that says that one cannot know both where something is and +how fast it is moving. The uncertainty of the momentum and the uncertainty +of the position are complementary, and the product of the two is bounded by +a small constant. We can write the law like this: Az Ap > ñ/2, but we shall +explain it in more detail later. This rule is the explanation of a very mysterious +paradox: If the atoms are made out of plus and minus charges, why don't the +minus charges simply sit on top oŸ the plus charges (they attract each other) and +get so close as to completely cancel them out? Whg are atoms so bñg? Wlhy ïs the +nueleus at the center with the electrons around it? It was first thought that this +was because the nucleus was so big; but no, the nucleus is 0er small An atom +has a diameter of about 10~ em. The nueleus has a diameter of about 10” !3 em. +--- Trang 60 --- +Tf we had an atom and wished to see the nucleus, we would have to magnify 1t +until the whole atom was the size of a large room, and then the nucleus would +be a bare speck which you could just about make out with the eye, but very +nearly aÏÏ the uueighf of the atom is in that infnitesimal nucleus. What keeps the +electrons from simply falling in? This principle: If they were in the nucleus, we +would know their position precisely, and the uncertainty principle would then +require that they have a very /arøe (but uncertain) momentum, i.e., a very large +kimelic energu. With this energy they would break away from the nucleus. They +make a compromise: they leave themselves a little room for this uncertainty and +then jiggle with a certain amount of minimum motion in accordance with this +rule. (Remember that when a crystal is cooled to absolute zero, we said that the +atoms do not stop moving, they still Jiggle. Why? TIf they stopped moving, we +would know where they were and that they had zero motion, and that is against +the uncertainty principle. We cannot know where they are and how fast they are +moving, so they must be continually wiggling ¡in therel) +Another most interesting change in the ideas and philosophy of science brought +about by quantum mechanies is this: it is not possible to predict ezacflu what will +happen in any circumstance. For example, it is possible to arrange an atom which +is ready to emit light, and we can measure when it has emitted light by picking up +a photon particle, which we shall describe shortly. We cannot, however, predict +tuhen 1% is goïng to emit the light or, with several atoms, œhúch ơne is goïng to. +You may say that this is because there are some internal “wheels” which we have +not looked at closely enough. No, there are no internal wheels; nature, as we +understand it today, behaves in such a way that it is ƒundamentall impossible +to make a precise prediction oŸ ezacfl that uiil happen in a given experiment. +This is a horrible thing; in fact, philosophers have said before that one of the +fundamental requisites of science is that whenever you set up the same conditions, +the same thing must happen. This is simply nøÝ frue, it is no£‡ a fundamental +condition of scienece. “Phe fact is that the same thing does not happen, that we +can ñnd only an average, statistically, as to what happens. Nevertheless, science +has not completely collapsed. Philosophers, incidentally, say a great deal about +what 1s œbsolutel necessar for seience, and 1t is always, so far as one can see, +rather naive, and probably wrong. Eor example, some philosopher or other said +1t is fundamental to the scientifc efort that If an experiment is performed in, say, +Stockholm, and then the same experiment is done in, say, Quito, the sœrne resulis +must occur. Phat is quite false. It is not necessary that sc¿ence do that; it may be +a fact oƒ czperience, but it is not necessary. For example, if one of the experiments +--- Trang 61 --- +1s to look out at the sky and see the aurora borealis in Stockholm, you do not see it +in Quito; that is a diferent phenomenon. “But,” you say, “that is something that +has to do with the outside; can you close yourself up in a box in Stockholm and +pull down the shade and get any diference?” Surely. If we take a pendulum on +a universal Joint, and pull it out and let go, then the pendulum will swing almost +in a plane, but not quite. Slowly the plane keeps changing in Stockholm, but not +in Quito. The blinds are down, too. The fact that this happened does not bring +on the destruction of science. What ¡s the fundamental hypothesis of science, +the fundamental philosophy? We stated it in the first chapter: he soÏe test oƒ +the 0ualiditU oƒ am tdea is czpertment. TÝ it turns out that most experiments work +out the same in Quito as they do in Stockholm, then those “most experiments” +will be used to formulate some general law, and those experiments which do not +come out the same we will say were a result of the environment near Stockholm. +W©e will invent some way to summarize the results of the experiment, and we do +not have to be told ahead of time what this way will look like. If we are told that +the same experiment will always produce the same result, that is all very well, +but ifƒ when we try it, i§ does no, then it does nmoøý. We just have to take what we +see, and then formulate all the rest of our ideas in terms of our actual experience. +Returning again to quantum mechanics and fundamenta] physics, we cannot øO +into details of the quantum-mechanical principles at this time, of course, because +these are rather dificult to understand. We shall assume that they are there, and +go on to describe what some of the consequences are. Ône of the consequences 1s +that things which we used to consider as waves also behawve like particles, and +particles behave like waves; in fact everything behaves the same way. Thhere is +no distinction between a wave and a particle. So quantum mechanics unifies the +idea of the field and its waves, and the particles, all into one. Now it is true that +when the frequeney is low, the fñeld aspect of the phenomenon is more evident, or +more useful as an approximate description in terms of everyday experiences. But +as the frequency increases, the particle aspects of the phenomenon become more +evident with the equipment with which we usually make the measurements. In +fact, although we mentioned many Írequencies, no phenomenon directly involving +a frequeney has yet been detected above approximately 1012 eyeles per second. +We© only deduce the higher Írequencies from the energy of the particles, by a rule +which assumes that the particle-wave idea of quantum mechanics is valid. +Thus we have a new view of electromagnetic interaction. We have a new kind +of parficle to add to the electron, the proton, and the neutron. hat new particle +1s called a pho£on. "The new view of the interaction of electrons and photons that +--- Trang 62 --- +1s electromagnetic theory, but with everything quantum-mechanically correct, is +called qguantum clectrodunamics. Thĩs fundamental theory of the interaction of +light and matter, or electric field and charges, is our greatest success so far In +physics. In this one theory we have the basic rules for all ordinary phenomena +except for gravitation and nuclear processes. For example, out of quantum +electrodynamiecs come all known electrical, mechanical, and chemical laws: the +laws for the collision of billiard balls, the motions of wires in magnetic fields, +the specifc heat of carbon monoxide, the color of neon signs, the density of salt, +and the reactions of hydrogen and oxygen to make water are all consequences +of this one law. All these details can be worked out if the situation is simple +enouph for us to make an approximation, which is almost never, but often we can +understand more or less what is happening. At the present từme no exceptions +are found to the quantum-electrodynamic laws outside the nucleus, and there we +do not know whether there is an exception because we simply do not know what +is goiïng on in the nucleus. +In principle, then, quantum electrodynamies is the theory of all chemistry, +and of lie, ïf life is ultimately reduced to chemistry and therefore Just to physics +because chemistry is already reduced (the part of physics which is involved in +chemistry being already known). Purthermore, the same quantum electrodynam- +1cs, this pgreat thing, predicts a lot of new things. In the first place, it tells the +properties of very high-energy photons, gamma rays, etc. It predicted another +very remarkable thing: besides the electron, there should be another particle of +the same mass, but of opposite charge, called a poszron, and these two, coming +together, could annihilate each other with the emission of light or gamma rays. +(After all, light and gamma rays are all the same, they are just different points +on a frequency scale.) The generalization of this, that for each particle there is an +antiparticle, turns out to be true. In the case of electrons, the antiparticle has an- +other name——it is called a positron, but for most other particles, ¡t 1s called anti-so- +and-so, like antiproton or antineutron. In quantum electrodynamies, ÉuUo mumnbers +are put in and most of the other numbers in the world are supposed to come out. +'The two numbers that are put in are called the mass of the electron and the charge +of the electron. Actually, that is not quite true, for we have a whole set of numbers +for chemistry which tells how heavy the nuclei are. Thhat leads us to the next part. +2-4 Nuclei and particles +What are the nuclei made of, and how are they held together? It ¡is found +that the nuclei are held together by enormous forces. When these are released, +--- Trang 63 --- +the energy released is tremendous compared with chemical energy, in the same +ratio as the atomic bomb explosion is to a NT explosion, because, of course, +the atomie bomb has to do with changes inside the nucleus, while the explosion +of TNT has to do with the changes of the electrons on the outside of the atoms. +'The question is, what are the forces which hold the protons and neutrons together +in the nucleus? Just as the electrical interaction can be connected to a particle, +a photon, Yukawa suggested that the forces between neutrons and protons also +have a field of some kind, and that when this fñeld jiggles it behaves like a +particle. Thus there could be some other particles in the world besides protons +and neutrons, and he was able to deduce the properties of these particles from +the already known characteristics of nuclear forces. For example, he predicted +they should have a mass of two or three hundred times that of an electron; and lo +and behold, in cosmic rays there was discovered a particle of the right massl But +1t later turned out to be the wrong particle. It was called a -meson, or muon. +However, a little while later, in 1947 or 1948, another particle was found, the +7-meson, or pion, which satisied Yukawa/s criterion. Besides the proton and +the neutron, then, in order to get nuclear forces we must add the pion. Now, +you say, “Oh greatl, with this theory we make quantum nucleodynamics using +the pions just like Yukawa wanted to do, and see if it works, and everything will +be explained” Bad luck. It turns out that the calculations that are involved in +this theory are so dificult that no one has ever been able to figure out what the +consequences of the theory are, or to check it against experiment, and this has +been going on now for aÌlmost twenty yearsl +So we are stuck with a theory, and we do not know whether it is right or +wrong, but we do know that it is a i2 wrong, or at least incomplete. While we +have been dawdling around theoretically, trying to calculate the consequences of +this theory, the experimentalists have been discovering some things. For example, +they had already discovered this -meson or muon, and we do not yet know where +it fts. Also, in cosmic rays, a large number of other “extra” particles were found. +Tt turns out that today we have approximately thirty particles, and it is very +difficult to understand the relationships of all these particles, and what nature +wants them for, or what the connections are from one to another. We do not +today understand these various particles as different aspects of the same thing, +and the fact that we have so many unconnected particles is a representation of +the fact that we have so much unconnected information without a good theory. +After the great successes of quantum electrodynamics, there is a certain amount +of knowledge of nuclear physics which is rough knowledge, sort of half experience +--- Trang 64 --- +and half theory, assuming a type of force between protons and neutrons andÌ seeing +what will happen, but not really understanding where the force comes from. Aside +from that, we have made very little progress. We have collected an enormous +number of chemical elements. In the chemical case, there suddenly appeared a +relationship among these elements which was unexpected, and which is embodied +in the periodic table of Mendeleev. For example, sodium and potassium are +about the same in their chemical properties and are found in the same colunn +in the Mendeleev chart. We have been seeking a Mendeleev-type chart for the +new particles. One such chart of the new particles was made independently by +Gell-Mamn in the U.S.A. and Nishijima in Japan. The basis of their classification +1s a new number, like the electric charge, which can be assigned to each particle, +called its “strangeness,” Š. 'This number is conserved, like the electric charge, in +reactions which take place by nuclear Íorces. +In Table 2-2 are listed all the particles. We cannot discuss them mụch at this +siage, but the table will at least show you how much we do not know. Ứnderneath +cach particle is mass is given in a certain unit, called the MeV. One MeV is +equal to 1.783 x 10~?” gram. The reason this unit was chosen is historical, and +we shall not go into it now. More massive particles are put higher up on the +chart; we see that a neutron and a proton have almost the same mass. In vertical +columns we have put the particles with the same electrical charge, all neutral +objects in one column, all positively charged ones to the right of this one, and all +negatively charged objects to the left. +Particles are shown with a solid line and “resonances” with a dashed one. +Several particles have been omitted from the table. 'hese include the important +zero-mass, zero-charge particles, the photon and the graviton, which do not fall +into the baryon-meson-lepton classiication scheme, and also some oŸ the newer +resonances (KŠ, ó, ). The antiparticles of the mesons are listed in the table, but +the antiparticles of the leptons and baryons would have to be listed in another +table which would look exactly like this one reflected on the zero-charge columm. +Although all of the particles except the electron, neutrino, photon, graviton, and +proton are unstable, decay products have been shown only for the resonances. +Strangeness assignments are not applicable for leptons, since they do not interact +strongly with nuclel. +AII particles which are together with the neutrons and protons are called +baruons, and the following ones exist: There is a “lambda,” with a mass of +1115 MeV, and three others, called sigmas, minus, neutral, and plus, with several +masses almost the same. 'Phere are groups or multiplets with almost the same +--- Trang 65 --- +Table 2-2 +Elementary Particles +MASS CHARGE GROUPING & +in MeV —e 0 +e STRANGENESS +1400p= Y[>AJtT- YỊ3A}LT? VỈ›AẰI s=-2 +T395 +=_- =0 S=-2 +1300 1319 TBIT +1200 _— >° >+ s=-llố +1196 1191 "T189 È +Ạ9 S=_-1|m +1100 1115 +—n - mi S=0 +839 938 +s00 023m S=0 +Đ—OT‡T 271m7 p°©X‡T S=0 +500 _KC KT gr s=ml| 2 +494 498 494 õ +TT— r0 ii S=0 +T39.6 Tâ5Ø 139.6 +_H—— œ +100 T0B.6 z +0 B5 ~8— +--- Trang 66 --- +mass, within one or two percent. Each particle in a multiplet has the same +strangeness. The first multiplet is the proton-neutron doublet, and then there is +a singlet (the lambda) then the sigma triplet, and ñnally the xi doublet. Very +recenfly, in 1961, even a few more particles were found. Ôr are they particles? +They live so short a time, they disintegrate almost instantaneouslÌy, as soon as +they are formed, that we do not know whether they should be considered as new +particles, or some kind of “resonance” interaction of a certain definite energy +between the Á and z produects into which they disintegrate. +In addition to the baryons the other particles which are involved in the nuclear +interaction are called rmesons. “Thore are first the pions, which come in three +varleties, positive, negative, and neutral; they form another multiplet. We have +also found some new things called K-mesons, and they occur as a doublet, KT +and K0. Also, every particle has its antiparticle, unless a particle is is ơun +antiparticle. Eor example, the x— and the z? are antiparticles, but the #2 is +its own antiparticle. The K~ and KT are antiparticles, and the KU and KD. +In addition, in 1961 we also found some more mesons or ?nø;/be mmesons which +disintegrate almost immediately. A thing called œ¡ which goes into three pions +has a mass 780 on this scale, and somewhat less certain is an object which +disintegrates into two pions. These particles, called mesons and baryons, and +the antiparticles of the mesons are on the same chart, but the antiparticles of +the baryons must be put on another chart, “reflected” through the charge-zero +column. +Just as Mendeleev's chart was very good, except for the fact that there were +a number oŸ rare earth elements which were hanging out loose from it, so we have +a number of things hanging out loose from this chart—particles which do not +interact strongly in nuclei, have nothing to do with a nuclear interaction, and do +not have a strong interaction (I mean the powerful kind of interaction of nuclear +energy). These are called leptons, and they are the following: there is the electron, +which has a very small mass on this scale, only 0.510 MeV. Then there is that +other, the /-meson, the muon, which has a mass mụch higher, 206 times as heavy +as an electron. So far as we can tell, by all experiments so far, the diference +bebween the electron and the muon is nothing but the mass. Everything works +exactly the same for the muon as for the electron, except that one is heavier than +the other. Why is there another one heavier; what is the use for it? We do not +know. In addition, there is a lepton which is neutral, called a neutrino, and this +particle has zero mass. In fact, it is now known that there are £#o diferent kinds +of neutrinos, one related to electrons and the other related to muons. +--- Trang 67 --- +Pinally, we have two other particles which do not interact strongly with the +nuclear ones: one is a photon, and perhaps, If the field of gravity also has a +quantum-mechanical analog (a quantum theory of gravitation has not yet been +worked out), then there will be a particle, a graviton, which will have zero mass. +What is this “zero mass”? "he masses given here are the masses of the +particles ø‡ resf. The fact that a particle has zero mass means, in a way, that it +cannot be at resf. Á photon is never at rest, i is always moving at 186,000 miles +a second. We will understand more what mass means when we understand the +theory of relativity, which will come in due time. +Thus we are confronted with a large number of particles, which together seem +to be the fundamental constituents of matter. Fortunately, these particles are +not all diferent in their zn#eraclions with one another. In fact, there seem to be +Just ƒour kinds of interaction between particles which, in the order of decreasing +strength, are the nuclear force, electrical interactions, the beta-decay interaction, +and gravity. The photon is coupled to all charged particles and the strength of the +interaction is measured by some number, which is 1/137. The detailed law of this +coupling is known, that is quantum electrodynamics. Gravity is coupled to all +cnergu, but its coupling is extremely weak, much weaker than that of electricity. +This law is also known. Then there are the so-called weak decays——beta. decay, +which causes the neutron to disintegrate into proton, electron, and neutrino, +relatively slowly. This law is only partly known. The so-called strong interaction, +the meson-baryon interaction, has a strength of 1 in this scale, and the law 1s +completely unknown, although there are a number of known rules, such as that +the number of baryons does not change in any reaction. +Table 2-3. Elementary Interactions +Coupling Strength” Law +Photon to charged particles ~ 1072 Law known +Gravity to all energy ~ 10? Law known +'Weak decays ~10" Law partly known +Mesons to baryons ~1 Law unknown (some rules known) +” The “strength” is a dimensionless measure of the coupling constant involved +in each interaction (~ means “of the order”). +--- Trang 68 --- +This then, is the horrible condition of our physics today. To summarize it, +I would say this: outside the nucleus, we seem to know all; inside it, quantum +mmechanics is valid——the principles of quantum mechanies have not been found to +fail. The stage on which we put all of our knowledge, we would say, is relativistic +space-time; perhaps gravity is involved in space-time. We do not know how the +universe got started, and we have never made experiments which check our ideas +OŸ space and time accurately, below some tỉny distance, so we only knou that our +ideas work above that distance. We should also add that the rules of the game +are the quantum-mechanical principles, and those principles apply, so far as we +can tell, to the new particles as well as to the old. "The origin of the forces In +nuelei leads us to new particles, but unfortunately they appear in great profusion +and we lack a complete understanding of their interrelationship, although we +already know that there are some very surprising relationships among them. We +seem gradually to be groping toward an understanding of the world of subatomic +particles, but we really do not know how far we have yet to go in this task. +--- Trang 69 --- +Tho Holeafforte of IPhịgsícs ío hon Scforeeos +3-1 Introduction +Physics is the most fundamental and all-inclusive of the sciences, and has had +a profound efect on all seientifc development. In fact, physics is the present-day +equivalent of what used to be called œ=ø‡ural philosophụ, from which most of our +mmodern sciences arose. Students of many fields ñnd themselves studying physics +because of the basic role it plays in all phenomena. In this chapter we shall +try to explain what the fundamental problems in the other sciences are, but of +course it is Impossible in so small a space really to deal with the complex, subtle, +beautiful matters in these other felds. Lack of space also prevents our discussing +the relation of physics to engineering, industry, society, and war, or even the +most remarkable relationship between mathematics and physics. (Mathematics is +not a science from our point of view, in the sense that it is not a nøÈurøÏ science. +The test of its validity is not experiment.) We must, incidentally, make it clear +from the beginning that iIf a thing is not a science, it is not necessarily bad. For +example, love is not a science. So, if something is said not to be a sclence, it does +not mean that there is something wrong with it; ¡9 just means that it is not a +Sclence. +3-2 Chemistry +The science which is perhaps the most deeply affected by physics is chemistry. +Historically, the early days of chemistry dealt almost entirely with what we now +call inorganic chemistry, the chemistry of substances which are not associated +with living things. Considerable analysis was required to discover the existence oŸ +the many elements and theïr relationships—how they make the various relatively +simple compounds found in rocks, earth, etc. This early chemistry was very +important for physics. 'Phe interaction between the two sciences was very great +--- Trang 70 --- +because the theory of atoms was substantiated to a large extent by experiments +in chemistry. “The theory of chemistry, i.e., of the reactions themselves, was +summarized to a large extent in the periodic chart of Mendeleev, which brings out +many strange relationships among the various elements, and it was the collection +of rules as to which substanece is combined with which, and how, that constituted +inorganic chemistry. All these rules were ultimately explained in principle by +quantum mechanics, so that theoretical chemistry 1s in fact physics. On the +other hand, it must be emphasized that this explanation is 7n pr/nciple. We have +already discussed the diference between knowing the rules of the game of chess, +and being able to play. So it is that we may know the rules, but we cannot play +very well. It turns out to be very dificult to predict precisely what will happen in +a given chemical reaction; nevertheless, the deepest part of theoretical chemistry +must end up in quantum mechanics. +There is also a branch of physics and chemistry which was developed by +both seiences together, and which is extremely important. This is the method +of statistics applied in a situation in which there are mechanical laws, which 1s +aptly called s¿aiistical mechanics. In any chemical situation a large number of +atoms are involved, and we have seen that the atoms are all jiggling around in +a very random and complicated way. If we could analyze each collision, and be +able to follow in detail the motion of each molecule, we might hope to figure out +what would happen, but the many numbers needed to keep track of all these +mmolecules exceeds so enormously the capacity of any computer, and certainly +the capacity of the mind, that it was important to develop a method for dealing +with such complicated situations. Statistical mechanics, then, is the science of +the phenomena. of heat, or thermodynamics. Inorganic chemistry is, as a science, +now reduced essentially to what are called physical chemistry and quantum +chemistry; physical chemistry to study the rates at which reactions occur and +what is happening in detail (How do the molecules hit? Which pieces fly of ñrst?, +etc.), and quantum chemistry to help us understand what happens in terms of +the physical laws. +The other branch of chemistry is organic cherm¿str, the chemistry of the +substances which are associated with living things. Eor a tỉme it was believed that +the substances which are associated with living things were so marvelous that +they could not be made by hand, from inorganic materials. "This is not at all true—— +they are just the same as the substances made in inorganie chemistry, but more +complicated arrangements of atoms are involved. Organic chemistry obviously +has a very close relationship to the biology which supplies its substances, and +--- Trang 71 --- +to industry, and furthermore, much physical chemistry and quantum mechanics +can be applied to organic as well as to inorganie compounds. However, the main +problems of organic chemistry are not in these aspects, but rather in the analysis +and synthesis of the substances which are formed in biological systems, in living +things. This leads imperceptibly, in steps, toward biochemistry, and then into +biology itself, or molecular biology. +3-3 Biology +Thus we come to the seience of b2ology, which is the study of living things. In +the early days of biology, the biologists had to deal with the purely descriptive +problem of ñnding out uha¿‡ living things there were, and so they just had to +count such things as the hairs of the limbs of Heas. After these matters were +worked out with a great deal of interest, the biologists went into the rmachiner +inside the living bodies, fñrst from a gross standpoint, naturally, because it takes +some efort to get into the fñner details. +There was an interesting early relationship between physics and biology in +which biology helped physics in the discovery oŸ the conserualion oƒ energu, which +was frst demonstrated by Mayer in connection with the amount of heat taken in +and given out by a living creature. +Tf we look at the processes of biology of living animals more cÌosely, we see +man physical phenomena: the circulation of blood, pumps, pressure, etc. There +are nerves: we know what is happening when we step on a sharp stone, and +that somehow or other the information goes om the leg up. Ït is interesting +how that happens. In their study of nerves, the biologists have come to the +conclusion that nerves are very fñne tubes with a complex wall which is very +thin; through this wall the cell pumps lons, so that there are positive ions on the +outside and negative ions on the inside, like a capacitor. Now this membrane has +an interesting property; if it “discharges” in one place, i.e., if some oŸ the ions +were able to move through one place, so that the electric voltage is reduced there, +that electrical inÑluence makes itself felt on the ions in the neighborhood, and it +affects the membrane in such a way that it lets the ions through at neighboring +points also. 'Phis in turn affects 1t farther along, etc., and so there is a wave of +“benetrability” of the membrane which runs down the fñber when it is “excited” +at one end by stepping on the sharp stone. PThis wave is somewhat analogous +to a long sequence of vertical dominoes; 1ƒ the end one 1s pushed over, that one +pushes the next, etc. Of course this will transmit only one message unless the +--- Trang 72 --- +dominoes are set up again; and similarly in the nerve cell, there are processes +which pump the ions slowly out again, to get the nerve ready for the next impulse. +So it is that we know what we are doïng (or at least where we are). Of course +the electrical efects associated with this nerve impulse can be picked up with +electrical instruments, and because there are electrical efects, obviously the +physics of electrical effects has had a great deal of inÑuence on understanding +the phenomenon. +The opposite efect is that, from somewhere in the brain, a message is sent +out along a nerve. What happens at the end of the nerve? “There the nerve +branches out into fñne little things, connected to a structure near a musele, called +an endplate. Eor reasons which are not exactly understood, when the impulse +reaches the end of the nerve, little packets of a chemical called acetylcholine are +shot of (five or ten molecules at a time) and they affect the muscle fiber and make +1b contract—how simplel What makes a muscle contract? A muscle is a very +large number of Ñbers close together, containing two diÑferent substances, myosin +and actomyosin, but the machinery by which the chemical reaction induced by +acetylcholine can modify the dimensions of the muscle is not yet known. Thus +the fundamental processes 1n the muscle that make mechanical motions are not +known. +Biology is such an enormously wide field that there are hosts of other problems +that we cannot mention at all—problems on how vision works (what the light +does in the eye), how hearing works, etc. (The way in which £h#nking works we +shall discuss later under psychology.) NÑow, these things concerning biology which +we have just discussed are, from a biological standpoint, really not fundamental, +at the bottom of life, in the sense that even 1ƒ we understood them we still would +not understand life itself. 'To illustrate: the men who study nerves feel their work +1s very important, because after all you cannot have animals without nerves. But +you cơn have ljƒe without nerves. Plants have neither nerves nor muscles, but +they are working, they are alive, just the same. 5o for the fundamental problerms +of biology we must look deeper; when we do, we discover that all living things +have a great many characteristics in common. “The most common feature is that +they are made of celis, within each of which is complex machinery for doïng +things chemically. In plant cells, for example, there is machinery for picking up +light and generating glucose, which is consumed in the dark to keep the plant +alive. When the plant ¡is eaten the glucose itself generates in the animal a series +of chemical reactions very closely related to photosynthesis (and its opposite +effect in the dark) in plants. +--- Trang 73 --- +In the cells of living systems there are many elaborate chemical reactions, +in which one compound is changed into another and another. To give some +impression of the enormous eforts that have gone into the study of biochemistry, +the chart in Fig. 3-1 summarizes our knowledge to date on just one small part of +the many series of reactlons which occur in cells, perhaps a percent or so OÝ ïf. +Here we see a whole series of molecules which change from one to another in a +sequence or cycle of rather small steps. It is called the Krebs cycle, the respiratory +cycle. Each of the chemicals and each of the steps is fairly simple, in terms of +what change ¡is made in the molecule, but——and this is a centrally important +discovery in biochemistry—these changes are relaiiuel djficult to accomplish +ứn a laboratoru. TIf we have one substance and another very similar substance, +the one does not just turn into the other, because the bwo forms are usually +separated by an energy barrier or “hill” Consider this analogy: If we wanted +to take an object from one place to another, at the same level but on the other +side of a hill, we could push it over the top, but to do so requires the addition of +some energy. Thus most chemical reactions do not occur, because there is what +acetyl coenzyme A +S61~mmseTT—s Hạ-COO- +!Ổ HỌC COO. +:COO-;? CoA-SH Ha-COO~ +Ox2loaeetate ciate no +ooˆ^ Oa,, DPNH+H +Ha DPN TC Ha-COO—~ +H-C-ÔH Đo, -COO- +SQ- 4% H-COO- +L-malate cis-aconitate +H:O<‡ FUMARASE ACONITASE Yrmo +hệ ll›-COO- +hếu : CITRIC ACID CYCLE KH cSo: +! COO-? HO-CH-COO~ +`" fimarate đ-isocitrate +Fe† Tflavin A. ped0BoetuosE TPNỶ +OO= TPNH+H+ +FeT Tflavin GP “« Hz-COO~ +H5 sọ chócöo” +SO- nã Oz¿-COO“ +Succinate _ &“ oxalosuccinate +cm V CoA-SH _— Ố + +mẻ. '..ˆn +HOPO, Ở hy h2 ¡ |ThPP,LAŠ Hệ ¡ +(DP);O_°_€ CoA! ci +succinyl coenzyme A ĐPNH+H+ ¬= tả Retöglutarate +Fig. 3-1. The Krebs cycle. +--- Trang 74 --- +1s called an øcfoalion energu in the way. In order to add an extra atom tO Our +chemical requires that we get it close enough that some rearrangement can OCCUT; +then it will stick. But if we cannot give it enough energy to get it close enough, +it will no go to completion, ¡i% will jusE go part way up the “hill” and back down +again. However, ¡Ÿ we could literally take the molecules in our hands and push +and pull the atoms around ïn such a way as to open a hole to let the new atom +in, and then let it snap back, we would have found another way, around the hill, +which would not require extra energy, and the reaction would go easily. Now +there actually are, in the cells, uerw large molecules, mụuch larger than the ones +whose changes we have been describing, which in some complicated way hold +the smaller molecules just right, so that the reaction can occur easily. 'Phese +very large and complicated things are called enzymes. (They were first called +ferments, because they were originally discovered in the fermentation oŸ sugar. +In fact, some of the first reactions in the cycle were discovered there.) In the +presence of an enzyme the reaction will go. +An enzyme is made of another substance called protein. Enzymes are very +big and complicated, and each one is diferent, each being built to control a +certain special reaction. 'Phe names of the enzymes are written in Fig. 3-1 at each +reaction. (Sometimes the same enzyme may control ©wo reactions.) We emphasize +that the enzymes themselves are not involved in the reaction directly. Thhey do +not change; they merely let an atom go from one place to another. Having done +so, the enzyme is ready to do it to the next molecule, like a machine in a factory. +Of course, there must be a supply oŸ certain atoms and a way of disposing of +other atoms. Take hydrogen, for example: there are enzymes which have special +units on them which carry the hydrogen for all chemical reactions. For example, +there are three or four hydrogen-reducing enzymes which are used all over our +cycle in difÑferent places. It is interesting that the machinery which liberates some +hydrogen at one place will take that hydrogen and use it somewhere else. +The most important feature of the cycle of Fig. 3-1 is the transformation +from GDP to GTTP (guanosine-di-phosphate to guanosine-tri-phosphate) because +the one substance has much more energy in ¡i% than the other. Just as there +is a “box” in certain enzymes Íor carrying hydrogen atoms around, there are +special energu-carrying “boxes” which involve the triphosphate group. So, TP +has more energy than GDP and ïf the cycle is goỉng one way, we are producing +molecules which have extra energy and which can go drive some other cycle +which reqguzres energy, for example the contraction of muscle. 'Phe muscle wïll +not contract unless there is GP. We can take musecle fiber, put it in water, and +--- Trang 75 --- +add GTEVP, and the fñbers contract, changing TP to GDP ïf the right enzymes +are present. So the real system is in the GDP-GTTP transformation; in the dark +the GTP which has been stored up during the day is used to run the whole cycle +around the other way. Ấn enzyme, you see, does not care in which direction the +reaction goes, for ïf it did it would violate one of the laws of physics. +Physics is of great importance in biology and other sciences for still another +reason, that has to do with ezperimental techniques. In fact, 1f it were not for +the great development of experimental physies, these biochemistry charts would +not be known today. The reason is that the most useful tool of all for analyzing +this fantastically complex system 1s to lœbel the atoms which are used in the +reactions. 'Thus, iŸ we could introduee into the cycle some carbon dioxide which +has a “green mark” on it, and then measure after three seconds where the green +mark is, and again measure after ten seconds, etc., we could trace out the course +of the reactions. What are the “green marks”? They are different ¡sotopes. We +recall that the chemical properties of atoms are determined by the number of +clectrons, not by the mass of the nucleus. But there can be, for example in +carbon, six neutrons or seven neutrons, together with the six protons which all +carbon nuelei have. Chemically, the two atoms C†!2 and C1 are the same, but +they diÑer in weight and they have different nuclear properties, and so they are +distinguishable. By using these isotopes of diferent weights, or even radioactive +isotopes like C!, which provide a more sensitive means for tracing very small +quantities, it is possible to trace the reactions. +NÑow, we return to the description of enzymes and proteins. All proteins are +not enzymes, but all enzymes are proteins. There are many proteins, such as the +proteins in muscle, the structural proteins which are, for example, in cartilage +and haïr, skin, etc., that are not themselves enzymes. However, proteins are a +very characteristic substance of life: first of all they make up all the enzymes, and +second, they make up much of the rest of living material. Proteins have a very +Interesting and simple structure. 'Phey are a series, or chain, of diferent ønino +acids. Thhere are twenty different amino acids, and they all can combine with each +other to form chains in which the backbone is CO-NH, etc. Proteins are nothing +but chains of various ones of these twenty amino acids. Each of the amino acids +probably serves some special purpose. Some, for example, have a sulfur atom +at a certain place; when two sulfur atoms are in the same protein, they form a +bond, that is, they tie the chain together at two points and form a loop. Another +has extra oxygen atoms which make it an acidic substance, another has a basic +characteristic. 5ome of them have big groups hanging out to one side, so that +--- Trang 76 --- +they take up a lot of space. One of the amino acids, called proline, is not really an +amino acid, but imino acid. 'There is a slight diference, with the result that when +proline is in the chain, there is a kink in the chain. If we wished to manufacture +a particular protein, we would give these instructions: put one of those sulfur +hooks here; next, add something to take up space; then attach something to +put a kink in the chain. In this way, we will get a complicated-looking chaïin, +hooked together and having some complex structure; this is presumably just the +mamner in which all the various enzymes are made. One of the great triumphs +in recent tỉmes (since 1960), was at last to discover the exacb spatial atomic +arrangement of certain proteins, which involve some fifty-six or sixty amino acids +in a row. Over a thousand atoms (more nearly two thousand, iŸ we count the +hydrogen atoms) have been located in a complex pattern in wo proteins. The +first was hemoglobin. Ône of the sad aspects of this discovery is that we cannot +see anything from the pattern; we do not understand why it works the way 1 +does. Of course, that is the next problem to be attacked. +Another problem is how do the enzymes know what to be? A red-eyed ly +makes a red-eyed fly baby, and so the information for the whole pattern of +enzymes to make red pigment must be passed from one fy to the next. 'This is +done by a substance in the nucleus oŸ the cell, not a protein, called DNA (short +for des-oxyribose nucleic acid). 'This is the key substance which is passed from one +cell to another (for instance sperm cells consist mostly of DNA) and carries the +information as to how to make the enzymes. DNA ¡is the “blueprint” What does +the blueprint look like and how does it work? First, the blueprint must be able +to reproduce itself. Secondly, it must be able to instruct the protein. Concerning +the reproduction, we might think that this proceeds like cell reproduction. Cells +simply grow bigger and then divide in half. Must it be thus with DNÑA molecules, +then, that they too grow bigger and divide in half? Every a‡omn certainly does +not grow bigger and divide in halfl No, it is impossible to reproduce a molecule +except by some more clever way. +The structure of the substance DNÑA was studied for a long time, first chemi- +cally to fnd the composition, and then with x-rays to fñnd the pattern in space. +The result was the following remarkable discovery: The DNA molecule is a pair +of chaïns, twisted upon each other. 'Phe backbone of each of these chains, which +are analogous to the chains of proteins but chemically quite different, is a series +of sugar and phosphate groups, as shown in FEig. 3-2. NÑow we see how the chain +can contain instructions, for if we could split this chain down the middle, we +would have a series ĐAADŒ... and every living thing could have a different +--- Trang 77 --- +SỐ | )—BIA— } SUên +Ho ° Q on +tot [ABC] H89 +Ho ° ào +3895E [ ap—C |] S85 +AC › z2 +Ho ° con +s2E [pc CC] N8 +2 0 S z9 +Ho ọ IG +SIỐN | }—CD—C }] SUêA +Fig. 3-2. Schematic diagram of DNA. +series. Thus perhaps, in some way, the specIfc ?nstrucfions for the manufacture +Of proteins are contained in the specifc ser2es of the DNA. +Attached to each sugar along the line, and linking the two chains together, +are certain pairs of cross-links. However, they are not all of the same kind; there +are four kinds, called adenine, thymine, cytosine, and guanine, but let us call +them 4, Ø, C, and D. 'The interesting thing is that only certain pairs can sit +opposite each other, for example A with and Œ with 2. These pairs are put on +the two chains in such a way that they “ñt together,” and have a sirong energy +Of interaction. However, will not ft with A, and will not ñt with Œ; they +will only fit in pairs, A against and Œ against J. 'Therefore if one is Œ, the +--- Trang 78 --- +other must be D, etc. Whatever the letters may be in one chaïn, each one must +have its specifc complementary letter on the other chain. +'What then about reproduction? Suppose we split this chain in two. How can +we make another one just like it? lf, in the substances of the cells, there is a +manufacturing department which brings up phosphate, sugar, and A, 8, Œ, D +units not connected in a chain, the only ones which will attach to our split chain +will be the correct ones, the complements of ĐAADŒ..., namely, ABC... +'Thus what happens is that the chain splits down the middle during cell division, +one half ultimately to go with one cell, the other half to end up in the other cell; +when separated, a new complementary chain is made by each hal£chain. +NÑext comes the question, precisely how does the order of the A, , Œ, D units +determine the arrangement of the amino acids in the protein? 'This is the central +unsolved problem in biology today. "The first clues, or pieces of information, +however, are these: There are in the cell tiny particles called ribosomes, and it is +now known that that ¡is the place where proteins are made. But the ribosomes +are not in the nucleus, where the DNA and its instructions are. Something seerms +to be the matter. However, it is also known that little molecule pieces come of +the DNA——not as long as the big DNA molecule that carries all the information +itself, but like a small section of it. This is called RNA, but that is not essential. +lt is a kind of copy of the DNA, a short copy. The RNA, which somehow carries +a message as to what kind of protein to make goes over to the ribosome; that is +known. When it gets there, protein is synthesized at the ribosome. 'Phat is also +known. However, the details of how the amino acids come in and are arranged in +accordance with a code that is on the RNA are, as yet, still unknown. We do +not know how to read it. IÝ we knew, for example, the “lineup” A, Ö,CŒ,CŒ, A, +we could not tell you what protein is to be made. +Certainly no subject or fñeld is making more progress on so many fronts at +the present moment, than biology, and iŸ we were to name the most powerful +assumption of all, which leads one on and on in an attempt to understand life, it +1s that all things are mmade oƒ atorms, and that everything that living things do +can be understood in terms of the jigglings and wigglings oŸ atoms. +3-4 Astronomy +In this rapid-fire explanation of the whole world, we must now turn to astron- +omy. Astronomy is older than physics. In fact, it got physics started by showing +the beautiful simplicity of the motion of the stars and planets, the understanding +--- Trang 79 --- +of which was the beginnzng of physics. But the most remarkable discovery in all +OŸ astronomy is that the sfars are made oƒ atoms oƒ the same kind as those on the +carth.* How was this done? Atoms liberate light which has defnite frequencies, +something like the timbre of a musical instrument, which has defnite pitches or +Írequencies of sound. When we are listening to several different tones we can tell +them apart, but when we look with our eyes at a mixture of colors we cannot tell +the parts from which iÿ was made, because the eye is nowhere near as discerning +as the ear in this connection. However, with a spectroscope we cøn analyze the Íre- +quencies of the light waves and in this way we can see the very tunes oŸ the atoms +that are in the diferent stars. As a matter of fact, 6wo of the chemical elements +were discovered on a star before they were discovered on the earth. Helium was +discovered on the sun, whence its name, and technetium was discovered in certain +cool stars. This, of course, permits us to make headway in understanding the +stars, because they are made of the same kinds of atoms which are on the earth. +Now we know a great deal about the atoms, especially concerning their behavior +under conditions of high temperature but not very great density, so that we +can analyze by statistical mechanics the behavior of the stellar substance. Even +though we cannot reproduce the conditions on the earth, using the basic physical +laws we often can tell precisely, or very closely, what will happen. So it is that +physics aids astronomy. 5trange as iÿ may seem, we understand the distribution of +matter in the interior of the sun far better than we understand the interior of the +carth. What goes on ns2đde a star is better understood than one might guess from +the dificulty of having to look at a little dot of light through a telescope, because +we can cdlculate what the atoms in the stars should do in most circumstances. +One of the most impressive discoveries was the origin of the energy of the +stars, that makes them continue to burn. Ône of the men who discovered this +* How Im rushing through this! How much each sentence in this brief story contains. “The +stars are made of the same atoms as the earth.” I usually pick one small topic like this to give a +lecture on. Poets say science takes away from the beauty of the stars—mere globs of gas atoms. +Nothing is “mere.” I too can see the stars on a desert night, and feel them. But do ÏI see less +or more? “The vastness of the heavens stretches my imagination—stuck on this carousel my +little eye can catch one-million-year-old light. A vast pattern—of which I am a part —perhaps +my stuff was belched from some forgotten star, as one is belching there. Or see them with the +greater eye of Palomar, rushing all apart from some common starting point when they were +perhaps all together. What is the pattern, or the meaning, or the œh¿/? It does not do harm to +the mystery to know a little about it. Eor far more marvelous is the truth than any artists of +the past imaginedl Why do the poets of the present not speak of it? What men are poets who +can speak of Jupiter if he were like a man, but if he is an immense spinning sphere of methane +and ammonia must be silent? +--- Trang 80 --- +was out with his girl friend the night after he realized that nœuclear reaclions +must be going on in the stars in order to make them shine. She said “Look at +how pretty the stars shinel” He said “Yes, and right now I am the only man in +the world who knows h# they shine.” She merely laughed at him. She was not +impressed with being out with the only man who, at that moment, knew why +stars shine. Well, it is sad to be alone, but that is the way it is in this world. +Tt is the nuclear “burning” of hydrogen which supplies the energy of the sun; +the hydrogen is converted into helium. Eurthermore, ultimately, the manufacture +Of various chemical elements proceeds in the centers of the stars, from hydrogen. +'The stuff of which +e are made, was “cooked” once, in a star, and spit out. How +do we know? Because there is a clue. The proportion of the diferent isotopes—— +how much C†?, how much CÌỞ, etc., is something which is never changed by +chemical reactions, because the chemical reactions are so mụch the same for the +two. The proportions are purely the result oŸ uclear reactions. By looking +at the proportions of the isotopes in the cold, dead ember which we are, we +can discover what the ƒurznace was like in which the stuf of which we are made +was formed. That furnace was like the stars, and so it is very likely that our +elements were “made” in the stars and spit out in the explosions which we call +novae and supernovae. Astronomy is so close to physics that we shall study many +astronomical things as we go along. +3-5 Goeology +We© turn now to what are called earth sciences, or geologụ. First, meteorology +and the weather. Of course the wstruwments of meteorology are physical instru- +ments, and the development of experimental physics made these instruments +possible, as was explained before. However, the theory of meteorology has never +been satisfactorily worked out by the physicist. “Well,” you say, “there is nothing +but aïr, and we know the equations of the motions of air.” Yes we do. “So 1Í we +know the condition of air today, why can't we figure out the condition oŸ the air +tomorrow?” Eirst, we do not rzeallu know what the condition is today, because +the air is swirling and twisting everywhere. It turns out to be very sensitive, and +even unstable. If you have ever seen water run smoothly over a dam, and then +turn into a large number of blobs and drops as it falls, you wiïll understand what +Ï mean by unstable. You know the condition of the water before it goes over the +spillway; 1È is perfectly smooth; but the moment it begins to fall, where do the +drops begin? What determines how big the lumps are goïng to be and where +--- Trang 81 --- +they will be? That is not known, because the water is unstable. ven a smooth +mmoving mass 0Ý air, in going over a mountain turns into complex whirlpools and +eddies. In many fields we fñnd this situation of #wrbulent fiou that we cannot +analyze today. Quickly we leave the subject of weather, and discuss geologyl +The question basic to geology is, what makes the earth the way it is? 'Phe +most obvious processes are in front of your very eyes, the erosion processes of +the rivers, the winds, etc. Ït is easy enough to understand these, but for every +bit of erosion there is an equal amount of something else going on. Mountains +are no lower today, on the average, than they were in the past. There must +be mountain- forrming processes. You will ñnd, if you study geology, that there +re mountain-forming processes and volcanism, which nobody understands but +which is half of geology. The phenomenon of volcanoes is really not understood. +What makes an earthquake is, ultimately, not understood. It is understood +that 1ƒ something is pushing something else, it snaps and will slide—that is all +ripht. But what pushes, and why? The theory is that there are currents inside +the earth—circulating currents, due to the diference in temperature inside and +outside—which, in their motion, push the surface slightly. Thus if there are two +opposite circulations next to each other, the matter will collect in the region +where they meet and make belts of mountains which are in unhappy stressed +conditions, and so produce volcanoes and earthquakes. +'What about the inside of the earth? A great deal is known about the speed of +cearthquake waves through the earth and the density of distribution of the earth. +However, physicists have been unable to get a good theory as to how dense a +substance should be at the pressures that would be expected at the center of the +earth. In other words, we cannot fñgure out the properties of matter very well in +these circumstances. We do much less well with the earth than we do with the +conditions of matter in the stars. The mathematics involved seems a little too +dificult, so far, but perhaps it will not be too long before someone realizes that +1t is an important problem, and really works it out. The other aspect, of course, +is that even iƒ we did know the density, we cannot fgure out the circulating +currents. Nor can we really work out the properties of rocks at high pressure. +W© cannot tell how fast the rocks should “give”; that must all be worked out by +experIment. +3-6 Psychology +Next, we consider the science of ps/chology. Incidentally, psychoanalysis +1s not a science: i§ is at best a medical process, and perhaps even more like +--- Trang 82 --- +witch-doctoring. It has a theory as to what causes disease—lots of different +“spirits,” etc. The witch doctor has a theory that a disease like malaria is caused +by a spirit which comes into the air; i§ is not cured by shaking a snake over it, +but quinine does help malaria. So, If you are sick, Ï would advise that you go to +the witch doctor because he is the man in the tribe who knows the most about +the disease; on the other hand, his knowledge is not science. Psychoanalysis has +not been checked carefully by experiment, and there is no way to fnd a list of +the number of cases in which it works, the number of cases in which it does not +WOrk, ©WC. +The other branches of psychology, which involve things like the physiology +of sensation——what happens in the eye, and what happens in the brain—are, If +you wish, less interesting. But some small but real progress has been made in +studying them. One of the most interesting technical problems may or may not +be called psychology. The central problem of the mind, if you will, or the nervous +system, is this: when an animal learns something, it can do something diferent +than i% could before, and its brain cell must have changed too, if it is made out +of atoms. nw +0ha£t tuay is ?t djƒerent? We do not know where to look, or what +%o look for, when something is memorized. We do not know what iÿ means, or +what change there is in the nervous system, when a fact is learned. 'This is a +very important problem which has not been solved at all. Assuming, however, +that there is some kind of memory thing, the brain is such an enormous mass +of interconnecting wires and nerves that it probably cannot be analyzed in a +straightforward manner. There is an analog of this to computing machines and +computfing elements, in that they also have a lot of lines, and they have some +kind of element, analogous, perhaps, to the synapse, or connection oŸ one nerve +to another. 'Phis is a very interesting subject which we have not the time to +discuss further——the relationship between thinking and computing machines. lt +must be appreciated, of course, that this subject will tell us very little about the +real complexities of ordinary human behavior. All human beings are so diferent. +It will be a long time before we get there. We must start much further back. If +we could even fñgure out how a đoøg works, we would have gone pretty far. Dogs +are easier to understand, but nobody yet knows how dogs work. +3-7 How did it get that way? +In order for physics to be useful to other selences in a #heoretical way, other +than in the invention of instruments, the science in question must supply to the +--- Trang 83 --- +physicist a description of the object in a physicist's language. They can say “why +does a frog jump?,” and the physicist cannot answer. T they tell hm what a frog +1s, that there are so many molecules, there is a nerve here, etc., that is diferent. +Tf they will tell us, more or less, what the earth or the stars are like, then we +can figure it out. In order for physical theory to be of any use, we must know +where the atoms are located. In order to understand the chemistry, we must +know exactly what atoms are present, for otherwise we cannot analyze it. That +1s but one limitation, of course. +'There is another kinđ of problem in the sister seiences which does not exist in +physics; we might call it, for lack of a better term, the historical question. How +dịd it get that way? IÝ we understand all about biology, we will want to know how +all the things which are on the earth got there. There is the theory of evolution, +an important part of biology. In geology, we not only want to know how the +mmountains are forming, but how the entire earth was formed in the beginning, +the origin of the solar system, etc. 'That, of course, leads us to want to know +what kind of matter there was in the world. How did the stars evolve? What +were 0he initial conditions? “That is the problem of astronomical history. A great +deal has been found out about the formation of stars, the formation of elements +from which we were made, and even a little about the origin of the universe. +There is no historical question being studied in physics at the present time. +W© do not have a question, “Here are the laws of physics, how did they get that +way?” We do not imagine, at the moment, that the laws of physics are somehow +changing with time, that they were diferent in the past than they are a% present. +Of course they may be, and the moment we ñnd they øre, the historical question +of physics will be wrapped up with the rest of the history of the universe, and then +the physicist will be talking about the same problems as astronomers, geologists, +and biologists. +Finally, there is a physical problem that is commmon to many fields, that is very +old, and that has not been solved. It is not the problem of nding new fundamental +particles, but something left over from a long time ago—over a hundred years. +Nobody in physics has really been able to analyze it mathematically satisfactorily +in spite of its importance to the sister sciences. Ït is the analysis of c#rculafing or +turbulent ffưids. TÝ we watch the evolution of a star, there comes a point where we +can deduce that it is goïing to start convection, and thereafter we can no longer +deduce what should happen. AÁ few million years later the star explodes, but we +cannot fñgure out the reason. We cannot analyze the weather. We do not know +the patterns of motions that there should be inside the earth. The simplest form +--- Trang 84 --- +of the problem is to take a pipe that is very long and push water through i% at +high speed. We ask: to push a given amount of water through that pipe, how +much pressure is needed? No one can analyze it from first principles and the +properties of water. If the water ows very slowly, or if we use a thick goo like +honey, then we can do it nicely. You will ñnd that in your textbook. What we +really cannot do is deal with actual, wet water running through a pipe. That is +the central problem which we ought to solve some day, and we have not. +A poet once said, “The whole universe is in a glass of wine” We will probably +never know in what sense he meant that, for poets do not write to be understood. +But it is true that if we look at a glass of wine closely enough we see the +entire universe. 'There are the things of physics: the twisting liquid which +evaporates depending on the wind and weather, the refections in the glass, and +our imagination adds the atoms. 'Phe glass is a distillation of the earth”s rocks, +and in its composition we see the secrets of the universe's age, and the evolution +Of stars. What strange array of chemicals are in the wine? How did they come +to be? 'Phere are the ferments, the enzymes, the substrates, and the products. +There in wine is found the great generalization: all life is fermentation. Nobody +can discover the chemistry of wine without discovering, as did Louis Pasteur, the +cause of much disease. How vivid is the claret, pressing its existence into the +consciousness that watches it! TỶ our small minds, for some convenience, divide +this glass of wine, this universe, into parts—physics, biology, geology, astronomy, +psychology, and so on—remember that nature does not know itl So let us put +it all back together, not forgetting ultimately what it is for. Let it give us one +more fñinal pleasure: drink it and forget it all +--- Trang 85 --- +(©ortsor-'terffore œŸ F rt©r'JgJ/ +4-1 What is energy? +In this chapter, we begin our more detailed study of the diferent aspects of +physics, having fñnished our description of things in general. To ilustrate the +ideas and the kind of reasoning that might be used in theoretical physics, we shall +now examine one of the most basic laws of physics, the conservation of energy. +There is a fact, or if you wish, a ia, governing all natural phenomena that +are known to date. There is no known exception to this law—it is exact so far as +we know. 'Phe law is called the conseruation oƒ energ. It states that there is +a certain quantity, which we call energy, that does not change In the manifold +changes which nature undergoes. hat is a most abstract idea, because 1W is a +mathematical principle; 1% says that there is a numerical quantity which does +not change when something happens. Ït is not a description of a mechanism, or +anything concrete; it is just a strange fact that we can calculate some number and +when we fnish watching nature go through her tricks and calculate the number +again, it is the same. (Something like the bishop on a red square, and after a +number of moves—details unknown——it is still on some red square. ÏIt is a law of +this nature.) Since it is an abstract idea, we shall illustrate the meaning of it by +an analogy. +TImagine a child, perhaps “Dennis the Menace,” who has blocks which are +absolutely indestructible, and cannot be divided into pieces. Each is the same +as the other. Let us suppose that he has 28 blocks. His mother puts him with +his 28 blocks into a room at the beginning of the day. At the end of the day, +beiïng curious, she counts the blocks very carefully, and discovers a phenomenal +law—no matter what he does with the blocks, there are always 28 remainingl +This continues for a number of days, until one day there are only 27 blocks, +but a little investigating shows that there is one under the rug—she must look +everywhere to be sure that the number of blocks has not changed. One day, +--- Trang 86 --- +however, the number appears to change—there are only 26 blocks. Careful +investigation indicates that the window was open, and upon looking outside, the +other two blocks are found. Another day, careful count indicates that there are +30 blocksl "This causes considerable consternation, until it is realized that Bruce +came to visit, bringing his blocks with him, and he left a few at Dennis' house. +After she has disposed of the extra blocks, she closes the window, does not let +Bruee in, and then everything is going along all right, until one time she counts +and finds only 25 blocks. However, there is a box in the room, a toy box, and +the mother goes to open the toy box, but the boy says “No, do not open my toy +box,” and sereams. Mother is not allowed to open the toy box. Being extremely +curious, and somewhat ingenious, she invents a schemel She knows that a block +weighs three ounces, so she weighs the box at a time when she sees 28 blocks, +and it weighs 16 ounces. The next time she wishes to check, she weighs the box +again, subtracts sixteen ounces and divides by three. She discovers the following: +( nunber of ) R (weight of box) — 16 ounces _ constant. (41) +ocks seen 3 ounces +There then appear to be some new deviations, but careful study indicates that +the dirty water in the bathtub is changing its level. The child is throwing blocks +Into the water, and she cannot see them because It is so dirty, but she can fnd +out how many blocks are in the water by adding another term to her formula. +Since the original height of the water was 6 inches and each bloeck raises the water +a quarter of an inch, this new formula would be: +number of (weight of box) — 16 ounces +mm n) 3 ounces ++ (height TT 6 inches _ constant. (4:2) +In the gradual increase in the complexity of her world, she ñnds a whole series +of terms representing ways of calculating how many blocks are In places where +she is not allowed to look. As a result, she finds a complex formula, a quantity +which has to be computed, which always stays the same in her situation. +'What is the analogy of this to the conservation of energy? 'The most remark- +able aspect that must be abstracted from this picture is that ứhere are mo blocks. +Take away the first terms in (4.1) and (4.2) and we find ourselves calculating +--- Trang 87 --- +more or less abstract things. he analogy has the following points. First, when +we are calculating the energy, sometimes some of it leaves the system and goes +away, or sometimes some comes in. Ín order to verify the conservation oŸ energy, +we must be careful that we have not put any ¡in or taken any out. Second, the +energy has a large number of đjƒƒeren‡ ƒorms, and there is a formula for each +one. 'These are: gravitational energy, kinetic energy, heat energy, elastic energy, +electrical energy, chemical energy, radiant energy, nuclear energy, InasS ©I©Tgy. +TỶ we total up the formulas for each of these contributions, it will not change +except for energy going in and out. +Tt is important to realize that in physics today, we have no knowledge of what +energy 2s. We do not have a picture that energy comes in little blobs of a delnite +amount. lt is not that way. However, there are formulas for calculating some +numerical quantity, and when we add ït all together it gives “28”——always the +same number. Ït is an abstract thing in that it does not tell us the mechanism +or the reøsons for the various formulas. +4-2 Gravitational potential energy +Conservation of energy can be understood only If we have the formula for +all of its forms. I wish to discuss the formula for gravitational energy near the +surface of the Earth, and I wish to derive this formula in a way which has nothing +to do with history but is simply a line of reasoning invented for this particular +lecture to give you an ïllustration of the remarkable fact that a great deal about +nature can be extracted from a few facts and close reasoning. It is an illustration +of the kind of work theoretical physicists become involved in. It is patterned after +a mmost excellent argument by Mr. Carnot on the efficiency of steam engines.Š +Consider weight-lifting machines—machines which have the property that +they lift one weight by lowering another. Let us also make a hypothesis: that +there is no such thứng as perpetudl motion with these weight-lifting machines. (In +fact, that there is no perpetual motion at all is a general statement of the law of +conservation oŸ energy.) WWe must be careful to delne perpetual motion. Eirst, +let us do it for weight-lifting machines. If, when we have lifted and lowered a lot +of weights and restored the machine to the original condition, we fnd that the +net result is to have Ùjfed œ ueight, then we have a perpetual motion machine +because we can use that lifted weight to run something else. 'That is, prouided the +* Qur point here is not so mụuch the result, (4.3), which in fact you may already know, as +the possibility of arriving at it by theoretical reasoning. +--- Trang 88 --- +Fig. 4-1. Simple weight-lifting machine. +machine which lifted the weight is brought back to its exact original condition, +and furthermore that it is completely self-con#ained—that it has not received +the energy to lift that weight from some external source—like Bruce's blocks. +A very simple weight-lifting machine is shown in Fig. 4-1. This machine lifts +weights three units “strong.” We place three units on one balance pan, and one +unit on the other. However, in order to get it actually to work, we must liÍt a +little weight of the left pan. Ôn the other hand, we could lift a one-unit weight +by lowering the three-unit weight, If we cheat a little by hfting a little weight +of the other pan. Of course, we realize that with any ac£ual lifting machine, we +must add a little extra to get it to run. This we disregard, #emporardi. Ideal +machines, although they do not exist, do not require anything extra. A machine +that we actually use can be, in a sense, œử”mos‡ reversible: that is, if it will Hft +the weight of three by lowering a weight of one, then i% will also lift nearly the +weight of one the same amount by lowering the weight of three. +W© imagine that there are two classes of machines, those that are oø‡ reversible, +which ineludes all real machines, and those that are reversible, which of course +are actually not attainable no matter how careful we may be in our design +of bearings, levers, etc. We suppose, however, that there is such a thing—a +reversible machine—which lowers one unit of weight (a pound or any other unit) +by one unit of distance, and at the same time lifts a three-unit weight. Call this +reversible machine, Machine A. Suppose this particular reversible machine lifts +the three-unit weight a distance X. Then suppose we have another machine, +Machine , which is not necessarily reversible, which also lowers a unit weight a +unit distance, but which lifts three units a distance Y.. We can now prove that +Y is not higher than X; that is, it is impossible to build a machine that will lift +a weight an higher than it will be lifted by a reversible machine. Let us see +why. Let us suppose that Y' were higher than X. We take a one-unit weight and +lower it one unit height with Machine Ö, and that lifts the three-unit weight up +a distance Y.. Thhen we could lower the weight rom Y to X, obfaining [ree pouer, +and use the reversible Machine 4, running backwards, to lower the three-unit +--- Trang 89 --- +weight a distance à and lift the one-unit weight by one unit height. This will +put the one-unit weight back where it was before, and leave both machines ready +to be used again! We would therefore have perpetual motion if Y' were higher +than X, which we assumed was impossible. With those assumptions, we thus +deduce that Y ¡s not higher than ÄÃ, so that oŸ alÏ machines that can be designed, +the reversible machine is the best. +We can also see that all reversible machines must liẾt to ezactlU the same +height. Suppose that were really reversible also. The argument that Y is not +higher than Ä is, of course, Just as good as it was before, but we can also make +our argument the other way around, using the machines in the opposite order, and +prove that ÄX ¡s no‡ húgher than Y.. This, then, is a very remarkable observation +because it permits us to analyze the height to which diferent machines are +going to lift something u#thout looking at the tnterior mmechanism. We know at +once that if somebody makes an enormously elaborate series of levers that lift +three units a certain distance by lowering one unit by one unit distance, and we +compare it with a simple lever which does the same thing and is fundamentally +reversible, his machine will lift it no higher, but perhaps less high. If his machine +1s reversible, we also know exactly ho high it will lit. To summarize: every +reversible machine, no matter how it operates, which drops one pound one foot +and lifts a three-pound weight always lifts it the same distance, X. 'This is clearly +a universal law of great utility. he next question is, of course, what is X? +5uppose we have a reversible machine which is going to lift this distance X, +three for one. We set up three balls in a rack which does not move, as shown +in Eig. 4-2. One ball is held on a stage at a distance one foot above the ground. +The machine can lift three balls, lowering one by a distance 1. Now, we have +arranged that the platform which holds three balls has a Ñoor and ©wo shelves, +exactly spaced at distance X, and further, that the rack which holds the balls +is spaced at distance X, (a). Eirst we roll the balls horizontally from the rack +to the shelves, (b), and we suppose that this takes no energy because we do no +change the height. “The reversible machine then operates: i% lowers the single +ball to the foor, and it lifts the rack a distance X, (c). Ñow we have ingeniously +arranged the rack so that these balls are again even with the platforms. Thus we +unload the balls onto the rack, (d); having unloaded the balls, we can restore the +machine to is original condition. NÑow we have three balls on the upper three +shelves and one at the bottom. But the strange thing is that, in a certain way +of speaking, we have not lifted #uo of them at all because, after all, there were +balls on shelves 2 and 3 before. The resulting efect has been to lHiÍt ome bajl a +--- Trang 90 --- +1ft. _ + +(a) START (b) LOAD BALLS +(c) 1 lb. LIFTS 3lb. A (d) UNLOAD BALLS +DISTANCE X +1ft. _ +_=—=—— T x +(e) REARRANGE (f) END +Fig. 4-2. A reversible machine. +distance 3X. Now, if 3X exceeds one foot, then we can louer the ball to return +the machine to the initial condition, (), and we can run the apparatus again. +Therefore 3X cannot exceed one foot, for If 3X exceeds one foot we can make +perpetual motion. Likewise, we can prove that ơne ƒoot cannot czcccd 3X, by +making the whole machine run the opposite way, since it is a reversible machine. +Therefore 3X is neither greater nor less than a foot, and we discover then, by +argument alone, the law that X = $ foot. The generalization is clear: one pound +falls a certain distance in operating a reversible machine; then the machine can +li p pounds this distance divided by p. Another way of putting the result is +that three pounds times the height lifted, which in our problem was X, is equal +to one pound times the distance lowered, which is one foot in this case. lÝ we +take all the weights and multiply them by the heights at which they are now, +above the floor, let the machine operate, and then multiply all the weights by all +the heights again, £here tuiiÏ be no change. (WG have to generalize the example +where we moved only one weight to the case where when we lower one we lift +several đifferent ones——but that is easy.) +--- Trang 91 --- +'W© call the sum of the weights times the heights grauitational potential energU—— +the energy which an objecE has because of its relationship in space, relative to +the earth. The formula for gravitational energy, then, so long as we are not Eoo +far rom the earth (the force weakens as we go higher) is +gravitational +potential energy | = (weight) x (height). (4.3) +for one object +Tt is a very beautiful line of reasoning. The only problem is that perhaps it is +not true. (After all, nature does not høơue to go along with our reasoning.) For +example, perhaps perpetual motion is, in fact, possible. Some of the assumptions +may be wrong, or we may have made a mistake in reasoning, so it is always +necessary to check. χ turns ouÈ ezperimentaliu, in fact, to be true. +The general name of energy which has to do with location relative to something +else is called po#enlal energy. In this particular case, of course, we call it +grauftatlional potential energu. TỶ ït is a question oŸ electrical forces against which +we are working, instead of gravitational forces, if we are “lifting” charges away +from other charges with a lot of levers, then the energy content is called elecfrical +potential energu. The general principle is that the change In the energy is the +force times the distance that the force is pushed, and that this is a change In +energy in general: +Am ") = (force) x bàn van ) (4.4) +cenergy acts through +W©e will return to many of these other kinds of energy as we continue the course. +'The principle of the conservation of energy is very useful for deducing what +will happen in a number of circumstances. In high school we learned a lot of +laws about pulleys and levers used in diferent ways. W©e can now see that these +“laws” are dÌl the same thứng, and that we dịd not have to memorize 7ð rules to +figure i% out. A simple example is a smooth inclined plane which is, happily, a +three-four-five triangle (Fig. 4-3). We hang a one-pound weight on the inclined +plane with a pulley, and on the other side of the pulley, a weight W. We want +to know how heavy W must be to balance the one pound on the plane. How +can we fgure that out? If we say it is just balanced, it is reversible and so can +move up and down, and we can consider the following situation. In the initial +circumstanece, (a), the one pound weight is at the bottom and weight W is at +--- Trang 92 --- +tạ 1b, +VN ` N +(a) () +Fig. 4-3. Inclined plane. +the top. When Wƒ has slipped down in a reversible way, we have a one-pound +weight at the top and the weight W/ the slant distance, (b), or five feet, from the +plane in which it was before. We ij#ed the one-pound weight only #hree feet and +we lowered W pounds by ƒØioe feet. herefore W = Ỷ of a pound. Note that we +deduced this trom the conseruation oƒ energu, and not from force components. +Cleverness, however, is relative. It can be deduced in a way which is even more +brilliant, discovered by 5tevinus and inscribed on his tombstone. Figure 4-4 +explains that it has to be Ỷ of a pound, because the chain does not go around. +lt is evident that the lower part of the chain is balanced by itself, so that the +pull of the ñve weights on one side must balance the pull of three weights on the +other, or whatever the ratio of the legs. You see, by looking at this diagram, that +W must be š of a pound. (Tf you get an epitaph like that on your gravestone, +you are doing fine.) +Let us now illustrate the energy principle with a more complicated problem, +the screw jack shown in Eig. 4-5. A handle 20 inches long is used to turn the +Fig. 4-4. The epitaph of Stevinus. +--- Trang 93 --- +10 =đRĐ— +INCH E= +20” —Ị +Fig. 4-5. A screw Jack. +serew, which has 10 threads to the inch. We would like to know how mụuch force +would be needed at the handle to lift one ton (2000 pounds). IÝ we want to lift +the ton one ¡nch, say, then we must turn the handle around ten times. When it +goes around once i% goes approximately 126 inches. The handle must thus travel +1260 inches, and If we used various pulleys, etc., we would be lifting our one ton +with an unknown smaller weight W/ applied to the end of the handle. So we find +out that W is about 1.6 pounds. 'Phis is a result of the conservation of energy. +° /s0\_ joo +_______... —_ +Fig. 4-6. Weighted rod supported on one end. +Thake now the somewhat more complicated example shown in Eig. 4-6. A rod +or bar, 8 feet long, is supported at one end. In the middle of the bar is a weight +of 60 pounds, and at a distance of two feet from the support there is a weight of +100 pounds. How hard do we have to lift the end of the bar in order to keep 1§ +balanced, disregarding the weight of the bar? Suppose we put a pulley at one end +and hang a weight on the pulley. How big would the weight W/ have to be in order +for it to balance? We imagine that the weight falls any arbitrary distance—tO +make 1% easy for ourselves suppose it goes down 4 inches—how high would the +two load weights rise? 'Phe center rises 2 inches, and the point a quarter of the +way from the fxed end lifts 1 inch. "Therefore, the principle that the sum of +the heights times the weights does not change tells us that the weight W times +4 inches down, plus 60 pounds times 2 inches up, plus 100 pounds times 1 inch +has to add up to nothing: +— 4W + (2)(60) + (1)(100) =0, W =5ä lb. (4.5) +--- Trang 94 --- +Thus we must have a 55-pound weight to balance the bar. In this way we can +work out the laws of “balance”——the statics oŸ complicated bridge arrangements, +and so on. 'Phis approach is called the principle oƒ uirtual tuork, because in order +to apply this aregument we had to #nag¿ne that the structure moves a little—even +though ït is not reølu moving or even rmooabile. We use the very small imagined +motion to apply the principle of conservation of energy. +4-3 Kinetic energy +To illustrate another type of energy we consider a pendulum (Fig. 4-7). TỶ we +pull the mass aside and release it, it swings back and forth. In its motion, it Ìoses +height in goïng from either end to the center. Where does the potential energy +go? Gravitational energy disappears when it is down at the bottom; nevertheless, +it will climb up again. The gravitational energy must have gone into another +form. Evidently i§ is by virtue of Its mofZon that 16 is able to climb up again, +so we have the conversion of gravitational energy into some other form when it +reaches the bottom. +¬¬+—X +Fig. 4-7. Pendulum. +We must get a formula for the energy of motion. Now, recalling our arguments +about reversible machines, we can easily see that in the motion at the bottom +must be a quantity of energy which permits it to rise a certain height, and which +has nothing to do with the machzner by which it comes up or the pa#h by which +it comes up. So we have an equivalence formula something like the one we wrote +for the child's blocks. We have another form to represent the energy. Ít is easy +to say what it is. The kinetic energy at the bottom equals the weight times the +height that it could go, corresponding to its velocity: K.E. = WH. What we +need is the formula which tells us the height by some rule that has to do with the +motion of objects. If we start something out with a certain velocity, say straight +up, it wïll reach a certain height; we do not know what it is yet, but it depends +--- Trang 95 --- +on the velocity—there is a formula for that. 'Phen to ñnd the formula for kinetic +energy for an object moving with velocity V, we must calculate the height that +it could reach, and multiply by the weight. We shall soon ñnd that we can write +1t this way: +K.E. =WV3/2g. (4.6) +OŸÝ course, the fact that motion has energy has nothing to do with the fact that +we are in a gravitational fñeld. It makes no diference +øhere the motion came +from. 'This is a general formula for various velocities. Both (4.3) and (4.6) are +approximate formulas, the fñrst because it is incorrect when the heights are great, +1.e., when the heights are so high that gravity is weakening: the second, because +of the relativistic correction at high speeds. However, when we do fñnally get the +exact formula for the energy, then the law of conservation of energy 1s correct. +4-4 Other forms of energy +W© can continue in this way to ïllustrate the existence of energy in other forms. +First, consider elastic energy. If we pull down on a spring, we must do some work, +for when we have it down, we can lift weights with it. Therefore in its stretched +condition it has a possibility of doing some work. If we were to evaluate the suns +of weights times heights, it would not check out——we must add something else +to account for the fact that the spring is under tension. Elastic energy is the +formula for a spring when it is stretched. How much energy is it? If we let go, the +elastic energy, as the spring passes through the equilibrium poïint, is converted +to kinetic energy and it goes back and forth bebtween compressing or stretching +the spring and kinetic energy of motion. (There is also some gravitational energy +going in and out, but we can do this experiment “sideways” if we like.) It keeps +going until the losses—Ahal We have cheated all the way through by putting +on little weights to move things or saying that the machines are reversible, or +that they go on forever, but we can see that things do stop, eventually. Where is +the energy when the spring has fnished moving up and down? This brings in +another form oŸ energy: heat energ. +Inside a spring or a lever there are crystals which are made up oŸ lots of atoms, +and with great care and delicacy in the arrangement of the parts one can try to +adjust things so that as something rolls on something else, none of the atoms do +any jiggling at all. But one must be very careful. Ordinarily when things roll, +there is bumping and jiggling because of the irregularities of the material, and +--- Trang 96 --- +the atoms start to wiggle inside. So we lose track of that energy; we find the +atoms are wigegling inside in a random and confused manner after the motion +slows down. There is still kinetic energy, all right, but iE is not associated with +visible motion. What a dreaml How do we knou there is still kinetic energy? lt +turns out that with thermometers you can find out that, in fact, the spring or +the lever is t0armer, and that there is really an increase of kinetic energy by a +defñnite amount. We call this form oŸ energy heø‡ enerø, but we know that it +is not really a new form, it is just kinetic energy——internal motion. (Ône of the +diñiculties with all these experiments with matter that we do on a large scale +1s that we cannot really demonstrate the conservation of energy and we cannot +really make our reversible machines, because every time we move a large clump of +stuf, the atoms do not remain absolutely undisturbed, and so a certain amount +of random motion goes into the atomic system. We cannot see it, but we can +measure it with thermometers, etc.) +There are many other forms of energy, and oŸ course we cannot describe +them in any more detail just now. There is electrical energy, which has to +do with pushing and pulling by electric charges. 'There is radiant energy, the +energy of light, which we know is a form of electrical energy because light can be +represented as wigglings in the electromagnetic field. 'There is chemical energy, +the energy which is released in chemical reactions. Actually, elastic energy is, +to a certain extent, like chemical energy, because chemical energy is the energy +of the attraction of the atoms, one for the other, and so is elastic energy. Qur +modern understanding is the following: chemical energy has bwo parts, kinetic +energy of the electrons inside the atoms, so part of it is kinetic, and electrical +energy of interaction of the electrons and the protons——the rest of it, therefore, +1s electrical. Next we come to nuclear energy, the energy which is involved with +the arrangement of particles inside the nucleus, and we have formulas for that, +but we do not have the fundamental laws. We know that it is not electrical, not +gravitational, and not purely chemical, but we do not know what it is. It seems +to be an additional form of energy. Pinally, associated with the relativity theory, +there is a modifcation of the laws of kinetic energy, or whatever you wish to call +it, so that kinetic energy is combined with another thing called mass energy. An +object has energy from its sheer ezisfence. Tf Ï have a positron and an electron, +standing still doing nothing—never mind gravity, never mind anything—and +they come together and disappear, radiant energy will be liberated, in a defnite +amount, and the amount can be calculated. All we need know is the mass of the +object. It does not depend on what 1% is—we make two things disappear, and +--- Trang 97 --- +we get a certain amount of energy. 'Phe formula was first found by Binstein; it +is E = mcŸ. +Tt is obvious from our discussion that the law of conservation of energy is +enormously useful in making analyses, as we have illustrated in a few examples +without knowing all the formulas. If we had all the formulas for all kinds of +energy, we could analyze how many processes should work without having to go +into the details. 'Pherefore conservation laws are very interesting. The question +naturally arises as to what other conservation laws there are in physics. There +are two other conservation laws which are analogous to the conservation oŸ energy. +One is called the conservation of linear momentum. The other is called the +conservation of angular momentum. We will ñnd out more about these later. +In the last analysis, we do not understand the conservation laws deeply. We do +not understand the conservation of energy. We do not understand energy as a +certain number oŸ little blobs. You may have heard that photons come out in +blobs and that the energy of a photon is Planck's constant times the frequency. +That is true, but since the frequency of light can be anything, there is no law +that says that energy has to be a certain defnite amount. nlike Dennis' blocks, +there can be any amount of energy, at least as presently understood. So we do +not understand this energy as counting something at the moment, but just as a +mmathematical quantity, which is an abstract and rather peculiar circumstance. +In quantum mechanics it turns out that the conservation of energy is very closely +related to another mmportant property of the world, ¿hings do not depend on +the absolute từmec. YWWe can set up an experiment at a given moment and try i§ +out, and then do the same experiment at a later moment, and it will behave in +exactly the same way. Whether this is strictly true or not, we do not know. lf +we assume that it 7s true, and add the principles of quantum mechanics, then we +can deduce the principle of the conservation of energy. It is a rather subtle and +interesting thing, and it is not easy to explain. 'Phe other conservation laws are +also linked together. 'The conservation of momentum is associated in quantum +mechanics with the proposition that 1 makes no diference where you do the +experiment, the results will always be the same. As independence in space has +to do with the conservation of momentum, independence of time has to do with +the conservation of energy, and finally, If we #uzn our apparatus, this too makes +no diference, and so the invariance of the world to angular orientation is related +to the conservation of anguÏar rnomentum. Besides these, there are three other +conservation laws, that are exact so far as we can tell today, which are much +simpler to understand because they are in the nature of counting bloecks. +--- Trang 98 --- +The first of the three is the conseruation oƒ charge, and that merely means +that you count how many positive, minus how many negative electrical charges +you have, and the number is never changed. You may get rid oŸ a positive with a +negative, but you do not create any net excess of positives over negatives. 'ÏWo +other laws are analogous to this one——one is called the conseruation oj bar0ons. +There are a number of strange particles, a neutron and a proton are examples, +which are called baryons. In any reaction whatever in nature, if we count how +many baryons are coming into a process, the number of baryons# which come +out will be exactly the same. 'Phere is another law, the conseruation oƒ leptons. +W© can say that the group of particles called leptons are: electron, mu meson, +and neutrino. 'Phere is an antielectron which is a positron, that is, a —1 lepton. +Counting the total number of leptons in a reaction reveals that the number in +and out never changes, at least so far as we know at present. +'These are the six conservation laws, three of them subtle, involving space and +time, and three of them simple, in the sense of counting something. +With regard to the conservation of energy, we should note that auailable +energy is another matter—there is a lot of jiggling around in the atoms of the +water of the sea, because the sea has a certain temperature, but it is impossible +to get them herded into a deñnite motion without taking energy from somewhere +else. That is, although we know for a fact that energy is conserved, the energy +avajlable for human utility is not conserved so easily. The laws which govern how +much energy is available are called the laus oƒ thermodWnœmics and involve a +concept called entropy for irreversible thermodynamic processes. +Finally, we remark on the question oŸ where we can get our supplies oŸ energy +today. Our supplies of energy are from the sun, rain, coal, uranium, and hydrogen. +The sun makes the rain, and the coal also, so that all these are from the sun. +Although energy is conserved, nature does not seem to be interested ïn it; she +liberates a lot of energy from the sun, but only one part in two billion falls on the +earth. Nature has conservation of energy, but does not really care; she spends a +lot of it in all directions. We have already obtained energy from uranium; we can +also get energy from hydrogen, but at present only in an explosive and dangerous +condition. Tf it can be controlled in thermonuclear reactions, it turns out that +the energy that can be obtained from 10 quarts of water per second is equal to +all of the electrical power generated in the United States. With 150 gallons of +running water a minute, you have enough fuel to supply all the energy which is +* Counting antibaryons as —1 baryon. +--- Trang 99 --- +used in the United States today! "Therefore it is up to the physicist to figure out +how to liberate us from the need for having energy. It can be done. +--- Trang 100 --- +Tĩn+© (ra3eổl ÍÌsÉcrrtc© +5-1 Motion +In this chapter we shall consider some aspects of the concepts of #ne and +đistance. It has been emphasized earlier that physics, as do all the sciences, +depends on øbseruøiion. One might also say that the development of the physical +sciences to their present form has depended to a large extent on the emphasis +which has been placed on the making of quøaniitati»e observations. Only with +quantitative observations can one arrive at quantitative relationships, which are +the heart of physics. +Many people would like to place the beginnings of physics with the work done +350 years ago by Galileo, and to call him the first physicist. Ủntil that time, the +study of motion had been a philosophical one based on arguments that could be +thought up in one”s head. Most of the arguments had been presented by Aristotle +and other Greek philosophers, and were taken as “proven.” Galileo was skeptical, +and did an experiment on motion which was essentially this: He allowed a ball +to roll down an inclined trough and observed the motion. He did not, however, +Jjust look; he measured hou ƒar the ball went in hou long a từme. +'The way to measure a distance was well known long before Galileo, but there +wWere no accurate ways of measuring time, particularly short times. Although +he later devised more satisfactory clocks (though not like the ones we know), +Galileo”s first experiments on motion were done by using his pulse to count off +cequal intervals of time. Let us do the same. +'We may count of beats of a pulse as the ball rolls down the track: “one... +make a small mark at the location of the ball at each count; we can then measure +the đZstance the ball travelled from the point of release in one, or two, or three, +etc., equal intervals of time. Galileo expressed the result of 52s observations in +this way: If the location of the ball is marked at 1, 2, 3, 4,... units of time +--- Trang 101 --- +“STARTE -'ONE" Dœt +Lm ` c~'THREE” +Fig. 5-1. A ball rolls down an inclined track. +from the instant of its release, those marks are distant from the starting point in +proportion to the numbers 1, 4, 9, 16,... Today we would say the distance 1s +proportional to the square of the time: +'The study of motion, which is basic to all of physics, treats with the questions: +where? and when? +5-2 Time +Let us consider first what we mean by me. What ¡s time? It would be nice +1ƒ we could fnd a good defnition of time. Webster defines “a time” as “a period,” +and the latter as “a time,” which doesnt seem to be very useful. Perhaps we +should say: ““Dime is what happens when nothing else happens.” Which also +doesn't get us very far. Maybe it is just as well if we face the fact that tỉme is +one oŸ the things we probably cannot define (in the dictionary sense), and just +say that it is what we already know it to be: it is how long we waitl +'What really matters anyway is not how we đeƒfne time, but how we measure +it. One way of measuring time is to utilize something which happens over and +over again in a regular fashion—something which is periodic. For example, a day. +A day seems to happen over and over again. But when you begin to think about +1%, you might well ask: “Are days periodic; are they regular? Are all days the +same length?” One certainly has the impression that days in summer are longer +than days in winter. Of course, some of the days in winter seem to get awfully +long 1ƒ one is very bored. You have certainly heard someone say, “My, but this +has been a long day!” +Tt does seem, however, that days are about the same length ơn the œuerage. +ls there any way we can test whether the days are the same length—either from +--- Trang 102 --- +one day to the next, or at least on the average? One way is to make a comparison +with some other periodic phenomenon. Let us see how such a comparison might +be made with an hour glass. With an hour glass, we can “create” a periodic +Occurrence ¡iŸ we have someone standing by it day and night to turn it over +whenever the last grain of sand runs out. +We could then count the turnings oŸ the glass from each morning to the next. +We would fnd, this time, that the number of “hours” (¡.e., turnings of the glass) +was not the same each “day.” We should distrust the sun, or the glass, or both. +After some thoupht, ¡it might occur to us to count the “hours” from noon to noon. +(Nooøn is here defined øø# as 12:00 o'clock, but that instant when the sun is at its +highest point.) We would fnd, this tỉme, that the number of “hours” each day is +the same. +WS now have some confidence that both the “hour” and the “day” have a +regular periodicity, 1.e., mark of successive equal intervals of time, although we +have not proued that either one is “really” periodic. Someone might question +whether there might not be some omnipotent being who would slow down the +fow of sand every night and speed it up during the day. Our experiment does +not, OŸ course, give us an answer to this sort of question. All we can say is that +we fñnd that a regularity of one kind fits together with a regularity of another +kind. We can just say that we base our đefinztzon of tỉme on the repetition of +some apparently periodic event. +5-3 Short tỉmes +We should now notice that in the process of checking on the reproducibility +of the day, we have received an important by-produect. We have found a way of +measuring, more accurately, ƒracfions of a day. We have found a way of counting +time in smaller pieces. Can we carry the process further, and learn to measure +even smaller intervals of tỉme? +Galileo decided that a given pendulum always swings back and forth in equal +intervals of time so long as the size of the swing is kept small. Á test comparing +the number of swings of a pendulum in one “hour7” shows that such is indeed +the case. We can in this way mark fractions of an hour. lÝ we use a mechanical +device to count the swings—and to keep them going——we have the pendulum +clock of our grandfathers. +Let us agree that iŸ our pendulum oscillates 3600 times in one hour (and if +there are 24 such hours in a day), we shall call each period of the pendulum +--- Trang 103 --- +one “second.” We have then divided our original unit of time into approximately +10 parts. We can apply the same prineiples to đivide the second into smaller +and smaller intervals. lt is, you will realize, not practical to make mechanical +pendulums which go arbitrarily fast, but we can now make electricøal pendulums, +called oscillators, which can provide a periodie occurrence with a very short +period of swing. In these electronie oscillators it is an electrical current which +swings to and fro, in a manner analogous to the swinging of the bob of the +pendulum. +W© can make a series of such electronie oscillators, each with a period 10 times +shorter than the previous one. We may “calibrate” each oscillator against the next +slower one by counting the number oŸ swings it makes for one swing oŸ the slower +oscillator. When the period of oscillation of our clock is shorter than a fraction +of a second, we cannot count the oscillations without the help of some device +which extends our powers of observation. One such device is the electron-beam +oscilloscope, which acts as a sort of microscope for short times. 'This device plots +on a fuorescent screen a graph of electrical current (or voltage) versus time. By +connecting the oscilloscope to two of our oscillators in sequence, so that it plots a +graph first of the current in one of our oscillators and then of the current in the +other, we get two graphs like those shown in Eig. 5-2. We can readily determine +the number of periods of the faster oscillator in one period of the slower oscillator. +With modern electronic techniques, oscillators have been built with periods +as short as about 1012 second, and they have been calibrated (by comparison +methods such as we have described) in terms of our standard unit oŸ tỉìme, the +second. With the invention and perfection of the “laser,” or light amplifier, in the +past few years, it has become possible to make oscillators with even shorter periods +than 10~!2 second, but it has not yet been possible to calibrate them by the +mmethods which have been described, although ï© will no doubt soon be possible. +Times shorter than 107!2 second have been measured, but by a diferent +technique. In efect, a diferent definmition of “time” has been used. One way has +been to observe the đistønce between two happenings on a moving object. lỸ, for +example, the headlights of a moving automobile are turned on and then of, we +can fgure out ho+ long the lights were on if we know œere they were turned on +and off and how fast the car was moving. The time is the distance over which +the lights were on divided by the speed. +Within the past few years, just such a technique was used to measure the +lifetime of the x?-meson. By observing in a microscope the minute tracks left in +a photographic emulsion in which 0-mesons had been created one saw that a +--- Trang 104 --- +Ị hị I NNẽ. +II IIITIIIIIIIIIIIH +IH[IIIIIIIIIITIIHIIHIIIIHIIHIII +IIIIIIIIIIIIIIIITITITIIIII +HÍIƒHHÍTH.HIHTHHIHHHHHHI +II IHHI lÌ IIfIIIIIIIIIIIII +IlliilfllllHÚI +WlÌ Ũ Ili +Fig. 5-2. Two views of an oscilloscope screen. In (a) the oscilloscope +is connected to one oscillator, in (b) it is connected to an oscillator with +a period one-tenth as long. +U-meson (known to be travelling at a certain speed nearly that of light) went +a distance of about 10” meter, on the average, before disintegrating. It lived +for only about 10718 sec. It should be emphasized that we have here used a +somewhat diferent defnition of “time” than before. 5o long as there are no +Inconsistencies in our understanding, however, we feel fairly confdent that our +defñnitions are sufficiently equivalent. +--- Trang 105 --- +By extending our techniques—and if necessary our defnitions—still further +we can infer the time duration of still faster physical events. We can speak of +the period of a nuclear vibration. We can speak of the lifetime of the newly +discovered strange resonances (particles) mentioned in Chapter 2. 'Their complete +life occupies a time span of only 10?“ second, approximately the time it would +take light (which moves at the fastest known speed) to cross the nucleus of +hydrogen (the smallest known object). +'What about still smaller times? Does “time” exist on a still smaller scale? +Does it make any sense to speak of smaller times iŸ we cannot measure——Or +perhaps even think sensibly about—something which happens in a shorter time? +Perhaps not. These are some of the open questions which you will be asking and +perhaps answering in the next twenty or thirty years. +5-4 Long tỉmes +Let us now consider times longer than one day. Measurement of longer times +1s easy; we just count the days—so long as there is someone around to do the +counting. Eirst we fnd that there is another natural periodicity: the year, about +365 days. We have also discovered that nature has sometimes provided a counter +for the years, in the form of tree rings or river-bottom sediments. In some cases +we can use these natural time markers to determine the time which has passed +Sỉnce some earÌy event. +When we cannot count the years for the measurement of long times, we +must look for other ways to measure. One of the most successful is the use of +radioactive material as a “clock.” In this case we do not have a periodic occurrence, +as for the day or the pendulum, but a new kind of “regularity.” We fñnd that the +radioactivity of a particular sample of material decreases by the same ƒraction +for successive equal increases in its age. If we plot a graph of the radioactivity +observed as a function of time (say in days), we obtain a curve like that shown +in Eig. 5-3. We observe that if the radioactivity decreases to one-half in 7' days +(called the “half-life”), then it decreases to one-quarter in another 7” days, and so +on. In an arbitrary time interval £ there are #/7' “halfFlives,” and the fraction +left after this time # is (3)!⁄. +T we knew that a piece of material, say a piece of wood, had contained an +amount A of radioactive material when it was formed, and we found out by a +đirect measurement that it now contains the amount Ö, we could compute the +--- Trang 106 --- +TIMES +YEARS SECONDS LIFE OF +77777??? +1018 Age of universe +109 Aqe of earth U238 +106 Earliest men +1012 Aqe of pyramids +Ra226 +Age of U.S. +109 Life of a man HŠ +One day +103 Light goes from sun to earth Neutron +1 One heart beat +103 Period of a sound wave +1086 Period of radiowave Muon +7*-meson +109 Light travels one foot +1012 Period of molecular rotation +10-15 Period of atomic vibration +70-meson +1018 Light crosses an atom +Period of nuclear vibration +10-2 Light crosses a nucleus Strange +particle +77777??? +--- Trang 107 --- +RADIOACTIVITY +1/2+—-—— ` +1/4 ——— 1 _——_—> +0 T 2T 3T TIME +Fig. 5-3. The decrease with time of radioactivity. The activity de- +creases by one-half in each “half-life,” 7. +age of the object, ý, by solving the equation +(1) = BA. +There are, fortunately, cases in which we can know the amount of radioactivity +that was in an object when it was formed. We know, for example, that the carbon +dioxide in the aiïr contains a certain small fraction of the radioactive carbon +isotope C1 (replenished continuously by the action of eosmie rays). I we measure +the #oføÏ carbon content of an object, we know that a certain fraction of that +amount was originally the radioactive C!“; we know, therefore, the starting +amount 4 to use in the formula above. Carbon-14 has a half-life of 5000 years. +By careful measurements we can measure the amount left after 20 half-lives or +so and can therefore “date” organic objects which grew as long as 100,000 years +W©e would like to know, and we think we do know, the life of stïll older things. +Much of our knowledge is based on the measurements oŸ other radioactive isobopes +which have diferent half-lives. lf we make measurements with an isotope with a +longer half-life, then we are able to measure longer times. Uranium, for example, +has an isotope whose half-life is about 109 years, so that if some material was +formed with uranium in it 10 years ago, only half the uranium would remain +today. When the uranium disintegrates, it changes into lead. Consider a piece of +rock which was formed a long time ago in some chemical process. Lead, being of +a chemical nature diferent from uranium, would appear in one part of the rock +and uranium would appear in another part of the rock. The uranium and lead +would be separate. If we look at that piece of rock today, where there should only +--- Trang 108 --- +be uranium we will now find a certain fraction of uranium and a certain fraction +of lead. By comparing these ractions, we can tell what percent of the uranium +disappeared and changed into lead. By this method, the age of certain rocks has +been determined to be several billion years. An extension of this method, not +using particular rocks but looking at the uranium and lead in the oceans and +using averages over the earth, has been used to determine (within the past few +years) that the age of the earth itself is approximately 4.5 billion years. +Tt is encouraging that the age of the earth is found to be the same as the age +of the meteorites which land on the earth, as determined by the uranium method. +lt appears that the earth was formed out of rocks Ñoating in space, and that the +meteorites are, quite likely, some of that material left over. At some time more +than fñve billion years ago, the universe started. It is now believed that at least +our part of the universe had its beginning about ten or twelve billion years ago. +W©e do not know what happened before then. In fact, we may well ask again: +Does the question make any sense? Does an earlier tỉme have any meaning? +5-5 Units and standards of tỉme +W©e have implied that it is convenient if we start with some standard unit of +time, say a day or a second, and refer all other times to some multiple or fraction +of this unit. What shall we take as our basic standard of time? Shall we take the +human pulse? If we compare pulses, we fnd that they seem to vary a lot. Ôn +comparing ©wo clocks, one fnds they do not vary so much. You might then say, +well, let us take a clock. But whose clock? 'Phere 1s a story of a 5wiss boy who +wanted all of the clocks in his town to ring noon at the same time. So he went +around trying to convince everyone oŸ the value of this. Everyone thought it was +a marvelous idea so long as all of the other clocks rang noon when his didl lt is +rather difficult to decide whose clock we should take as a standard. Fortunately, +we all share one clock—the earth. Eor a long time the rotational period of the +carth has been taken as the basic standard of time. As measurements have been +made more and more precise, however, it has been found that the rotation of the +earth is not exactly periodic, when measured ïn terms of the best clocks. 'These +“best” clocks are those which we have reason to believe are accurate because they +agree with each other. We now believe that, for various reasons, some days are +longer than others, some days are shorter, and on the average the period of the +earth becomes a little longer as the centuries pass. +--- Trang 109 --- +Until very recently we had found nothing much better than the earth's period, +so all clocks have been related to the length of the day, and the second has been +defned as 1/86,400 of an average day. Recently we have been gaining experience +with some natural oscillators which we now believe would provide a more constant +time reference than the earth, and which are also based on a natural phenomenon +available to everyone. 'These are the so-called “atomic clocks.” 'Their basic internal +period is that of an atomiec vibration which is very insensitive to the 6emperature +or any other external efects. Thhese clocks keep time to an accuracy of one part +in 102 or better. Within the past two years an improved atomic clock which +operates on the vibration of the hydrogen atom has been designed and built by +Professor Norman Ramsey at Harvard University. He believes that this clock +might be 100 times more accurate still. Measurements now in progress will show +whether this is true or not. +We may expect that since it has been possible to build clocks mụuch more +accurate than astronomical time, there will soon be an agreement among scientists +to defñne the unit of tỉme in terms of one oŸ the atomiec clock standards. +5-6 Large distances +Let us now turn to the question oŸ đjs‡ønece. How far, or how bịg, are things? +lverybody knows that the way you measure distance is to start with a stick and +count. Or start with a thumb and count. You begin with a unit and count. How +does one measure smaller things? How does one subdivide distance? In the same +way that we subdivided time: we take a smaller unit and count the number of +such units it takes to make up the longer unit. So we can measure smaller and +smaller lengths. +But we do not always mean by distance what one gets by counting of with +a meter stick. It would be difcult to measure the horizontal distance between +two mountain tops using only a meter stick. We have found by experience that +distance can be measured in another fashion: by triangulation. Althouph this +mmeans that we are really using a diferent definition of distance, when they can +both be used they agree with each other. Space 1s more or less what Euclid +thought it was, so the two types of defnitions of distance agree. Since they do +agree on the earth it gïves us some confdence in using triangulation for still larger +distances. Eor example, we were able to use triangulation to measure the height +of the first Sputnik. We found that it was roughly 5 x 10” meters high. By more +careful measurements the distance to the moon can be measured in the same +--- Trang 110 --- +V CÀ ccccccceeccceeefng +ễP6Ệ555=Eœ +Fig. 5-4. The height of a Sputnik ¡is determined by triangulation. +way. wo telescopes at diferent places on the earth can give us the two angles +we need. It has been found in this way that the moon is 4 x 10 meters away. +W© cannot do the same with the sun, or at least no one has been able to yet. +"The accuracy with which one can fÍocus on a given point on the sun and with which +one can measure angles is not good enough to permit us to measure the distance +to the sun. 'PThen how can we measure the distance to the sun? We must invent an +extension of the idea of triangulation. We measure the relative distances of all the +planets by astronomical observations of where the planets appear to be, and we +get a picture of the solar system with the proper relate distances of everything, +but with no absolu£e distance. Ône absolute measurement is then required, which +has been obtained in a number of ways. One of the ways, which was believed +until recently to be the most accurate, was to measure the distance from the +earth to Eros, one of the small planetoids which passes near the earth every now +and then. By triangulation on this little object, one could get the one required +scale measurement. Knowing the relative distances of the rest, we can then tell +the distance, for example, from the earth to the sun, or tom the earth to Pluto. +'Withim the past year there has been a big improvement in our knowledge of +the scale of the solar system. At the Jet Propulsion Laboratory the distance from +the earth to Venus was measured quite accurately by a direct radar observation. +'This, of course, is a still diferent type of inferred distance. We say we know the +specd at which light travels (and therefore, at which radar waves travel), and we +assume that ï is the same speed everywhere between the earth and Venus. We +send the radio wave out, and count the time until the relected wave comes back. +trom the #ữne we infer a đis‡ønce, assuming we know the speed. We have really +another defñnition of a measurement of distance. +How do we measure the distance to a star, which is much farther away? +tFortunately, we can go back to our triangulation method, because the earth +--- Trang 111 --- +ASTAR +TT T~` +⁄ SUN N +ÁSE1fssmsx_)annanEAslồ +^ ¬ - — ~Z ⁄ +Fig. 5-5. The distance of nearby stars can be measured by triangula- +tion, using the diameter of the earth's orbit as a baseline. +moving around the sun gives us a large baseline for measurements of objecEs +outside the solar system. lÝ we focus a telescope on a star in summer and in +winter, we might hope to determine these two angles accurately enough to be +able to measure the distance to a star. +'What ïf the stars are too far away for us to use triangulation? Astronomers +are always inventing new ways of measuring distance. They fnd, for example, +that they can estimate the size and brightness of a star by its color. The color and +brightness of many nearby stars—whose distances are known by triangulation—— +have been measured, and ït is found that there is a smooth relationship between the +color and the intrinsic brightness of stars (¡in most cases). IÝone now measures the +color ofa distant star, one may use the color-brightness relationship to determine +the intrinsic brightness of the star. By measuring how bright the star øppears to +us at the earth (or perhaps we should say how đớn it appears), we can compute +how far away it is. (Eor a given intrinsic brightness, the apparent brightness +decreases with the square of the distance.) A nice confirmation of the correctness +of this method of measuring stellar distances is given by the results obtained for +groups of stars known as globular clusters. A photograph of such a group is shown +in Eig. 5-6. Just from looking at the photograph one is convinced that these +stars are all together. The same result is obtained from distance measurements +by the color-brightness method. +A study of many globular clusters gives another important bit of information. +Tt is found that there is a high concentration oŸ such clusters in a certain part of +--- Trang 112 --- +° ệ Ề k be +c AI l sẽ. s- +"`"... “... +Fig. 5-6. A cluster of stars near the center of our galaxy. 'Their +distance from the earth is 30,000 light-years, or about 3 x 1022 meters. +the sky and that most of them are about the same distance from us. Coupling +this Information with other evidence, we conclude that this concentration of +clusters marks the center of our galaxy. We then know the distance to the center +of the galaxy——about 1029 meters. +lnowing the size of our own galaxy, we have a key to the measurement of +stiilH larger distances—the distances to other galaxies. Eigure 5-7 is a photograph +of a galaxy, which has much the same shape as our own. Probably it is the +same size, too. (Other evidence supports the idea that galaxies are all about the +same size.) IÝ it is the same size as ours, we can tell its distance. We measure +the angle it subtends in the sky; we know its diameter, and we compute its +distance—triangulation againl +Photographs of exceedingly distant galaxies have recently been obtained with +the giant Palomar telescope. One is shown in Pig. 5-8. It is now believed that +some of these galaxies are about halfway to the limit of the universe—10”8 meters +away——the largest distance we can contemplatel +--- Trang 113 --- +`° ® * tàc . +r 7 _< k 5 +.*® s về +* ® ý * +T : >- £œ ° +Fig. 5-7. A spiral galaxy like our own. Presuming that its diameter +Is similar to that of our own galaxy, we may compute Its distance from +its apparent size. lt is 30 million light-years (3 x 1023 meters) from the +earth. +5-7 Short distances +Now lets think about smaller distances. Subdividing the meter is easy. +'Without mụuch dificulty we can mark of one thousand equal spaces which add up +to one meter. With somewhat more difficulty, but in a similar way (using a good +microscope), we can mark off a thousand equal subdivisions of the millimeter +to make a scale of microns (millionths of a meter). It ¡is dificult to continue to +smaller scales, because we cannot “see” obJects smaller than the wavelength of +visible light (about 5 x 10~7 meter). +W© need not stop, however, at what we can see. With an electron microscope, +we can continue the process by making photographs on a still smaller scale, +say down to 10” meter (Eig. 5-9). By indirect measurements—by a kind of +triangulation on a microscopic scale—we can continue to measure to smaller and +smaller scales. First, from an observation oŸ the way light of short wavelength (x- +radiation) is reflected from a pattern oŸ marks of known separation, we determine +--- Trang 114 --- +k 20t § +Ẳ° S° +ó ° +Fig. 5-8. The most distant object, 3C295 in BOOTES (indicated by +the arrow), measured by the 200-inch telescope to date (1960). +the wavelength of the light vibrations. Then, from the pattern of the scattering +of the same light from a crystal, we can determine the relative location of the +atoms in the crystal, obtaining results which agree with the atomic spacings aÌso +determined by chemical means. We fñnd in this way that atoms have a diameter +of about 10~1 meter. +There is a large “gap” in physical sizes between the typical atomie dimension +of about 10~10 meter and the nuclear dimensions 10~!5 meter, 10—5 times smaller. +For nuclear sizes, a diferent way of measuring size becomes convenient. We +measure the øpparen‡ area, ơ, called the efective cross secfion. lf we wish the +radius, we can obtain it from ø = ør2, since nuclei are nearly spherical. +Measurement of a nuclear cross section can be made by passing a beam of +high-energy particles through a thin slab of material and observing the number +of particles which do not get through. 'These high-energy particles will plow right +through the thin cloud of electrons and will be stopped or deflected only If they +hit the concentrated weight of a nucleus. Suppose we have a piece of material +1 centimeter thick. There will be about 10Ẻ atomic layers. But the nuclei are +so small that there is little chance that any nucleus will lie behind another. We +--- Trang 115 --- +DISTANCES +LIGHT-YEARS METERS +???7?217??? +Edge of universe +106 To nearest neighbor galaxy +To center of our galaxy +To nearest star +Radius of orbit of Pluto +To the sun +To the moon +Height of a Sputnik +Height of a TV antenna tower +1 Height of a child +A grain of salt +A virus +Radius of an atom +10-15 Radius of a nucleus +???7?217??? +--- Trang 116 --- +l0 01 lu 0n (đRk v. 127271 T2 cải +4210122121777: 14) ÔNG .L2: U70 lệ. +: SN rat LAI Tự. /A A72. 21 sả si s +Tổ nó ¿ sử xế 452) ` ý # “xi + xi giếy Jzt .í +Š1%)) Vý\ báu cự 2sý: “Hÿ Đột 1P 45-; +200/2” LỄ H4 1/2876 .7171,Ê +p2 nh Suy lo K2) 17 27.) VU +.ÝSV lật TU, lẻ + 4120 À ÁN +_Ắ' M25 13 hn +IỄ” T2 n. à DỀ 22 +vỆ' # NV. t7Ằ Ị +2770109 0 1n e W7? 26 +Z3 (13v Ý XÃ, VU: đo vT +PA A$- (š KP} Sv IA 200 014 tà 2y +1+ VAT có v74? (464, Ÿ2 76v sảờ: . +7 cự ĐI ÁA CHẾ (2 hy Cá” 4, K p +: X22) 524 j¡t X vày - VG) 'c4...‹ .“ +.Š s.ế t^v vẤa ` .... W2 z vây C: , +H Ba & v4 D14 3: Á độ đ #›, 2 ` * " +3£ ta gả cổ M7... xế. Ma +M2. 0. NR.. Ẻ V' l +Fig. 5-9. Electron micrograph of some virus molecules. The “large” +sphere is for calibration and is known to have a diameter of 2x 10” meter +(2000 Ä). +might #nagine that a highly magnified view of the situation——looking along the +particle beam——would look like Eig. 5-10. +Fig. 5-10. lmagined view through a block of carbon 1 cm thick If only +the nuclei were observed. +'The chance that a very small particle will hit a nuecleus on the trip through is +Just the total area covered by the profiles of the nuclei divided by the total area +in the picture. Suppose that we know that in an area A of our slab of material +there are W atoms (each with one nucleus, of course). Then the fraction of the +--- Trang 117 --- +area “covered” by the nuclei is Nơ/A. Now let the number of particles of our +beam which arrive at the slab be ø+ and the number which come out the other +side be mạ. The fraction which do nø£ get through is (m¡ — m2)/m+, which should +just equal the fraction of the area covered. We can obtain the radius of the +nucleus from the equationF +_.-ˆ.... +N T1 +trom such an experiment we fnd that the radii of the nuclei are from about +1 to 6 times 1015 meter. The length unit 1015 meter is called the ƒermi, in +honor of Enrico Fermi (1901-1954). +What do we fnd If we go to smaller distances? Can we measure smaller +distances? Such questions are not yet answerable. It has been suggested that +the still unsolved mystery of nuclear forces may be unravelled only by some +modifcation of our idea. oŸ space, or measurement, at such small distances. +Tt might be thought that ít would be a good idea to use some natural length as +our unit o£ length—say the radius of the earth or some fraction of it. 'Phe meter +was originally intended to be such a unit and was defned to be (/2) x 10—7 times +the earth”s radius. I% is neither convenient nor very accurate to determine the +unit of length in this way. For a long tỉme it has been agreed internationally that +the meter would be defined as the distance between two scratches on a bar kept +in a special laboratory in France. More recently, ¡it has been realized that this +defnition is neither as precise as would be useful, nor as permanent or universal as +one would like. It is currently beïng considered that a new defnition be adopted, +an agreed-upon (arbitrary) number of wavelengths of a chosen spectral line. +Measurements of distance and of time give results which depend on the +observer. 'Wwo observers moving with respect to each other will not measure +the same distances and times when measuring what appear to be the same +things. Distances and time intervals have diferent magnitudes, depending on the +coordinate system (or “frame of reference”) used for making the measurements. +W© shall study this subJect in more detail in a later chapter. +* 'Phis equation is right only if the area covered by the nuclei is a small fraction of the total, +1.e., 1Ÿ (mị — 2)/mị is much less than 1. Otherwise we must make a correction for the fact that +some nuclei will be partly obscured by the nuclei in front of them. +--- Trang 118 --- +Perfectly precise measurements of distances or times are not permitted by +the laws of nature. We have mentioned earlier that the errors in a measurement +of the position of an obJect must be at least as large as +Az> h/2Ab, +where ñ is a small fundamental physical constant called the reduced Planck +constant and Ấp 1s the error in our knowledge of the momentum (mass times +velocity) of the object whose position we are measuring. It was also mentioned +that the uncertainty in position measurementfs is related to the wave nature of +particles. +The relativity of space and time implies that time measurements have aÌso a +minimum error, given in fact by +At>h/2AE, +where A is the error in our knowledge of the energy of the process whose tỉme +period we are measuring. lf we wish to know rmore precisely hen something +happened we must know less about +0ha# happened, because our knowledge of +the energy involved will be less. The time uncertainty is also related to the wave +nature of matter. +--- Trang 119 --- +PProberbrlrty +“The true logic of this world is in the calculus of probabilities.” +— James Clerk Maxwell +6-1 Chance and likelihood +“Chance” is a word which is in common use in everyday living. The radio +reports speaking of tomorrow's weather may say: “There is a sixty percent chance +of rain” You might say: ““There is a small chance that I shall live to be one +hundred years old.” Scientists also use the word chance. A seismologist may be +interested ¡in the question: “What ¡is the chance that there will be an earthquake +of a certain size in Southern California next year?” A physicist might ask the +question: “What is the chance that a particular geiger counter will register bwenty +counts in the next ten seconds?” A politician or statesman might be interested +in the question: “What is the chance that there will be a nuclear war within +the next ten years?” You may be interested in the chance that you will learn +something from this chapter. +By chance, we mean something like a guess. Why do we make guesses? We +make guesses when we wish to make a judgment but have incomplete information +or uncertain knowledge. We want to make a guess as to what things are, or what +things are likely to happen. Often we wish to make a guess because we have to +make a decision. For example: Shall I take my raincoat with me tomorrow? For +what earth movement should I design a new building? Shall I build myself a +fallout shelter? Shall I change my stand in international negotiations? Shall I go +to class today? +Sometimes we make guesses because we wish, with our limited knowledge, +to say as much as we cøn about some situation. Really, any generalization is +in the nature of a guess. Any physical theory is a kind of guesswork. There +are good guesses and there are bad guesses. “The theory of probability is a +--- Trang 120 --- +system for making better guesses. The language of probability allows us to speak +quantitatively about some situation which may be highly variable, but which +does have some consistent average behavior. +Let us consider the flipping of a coïn. If the toss—and the coin——are “honest,” +we have no way of knowing what to expect for the outcome of any particular toss. +Yet we would feel that in a large number of tosses there should be about equal +numbers oŸ heads and tails. We say: “The probability that a toss will land heads +is 0.5.” +W© speak of probability only for observations that we contemplate being made +in the future. Pựụ the “probabtlitU” oƒƑ a parlicular outcome oƒ an obser0atlion tue +mean our' estimate for the most likelU [raclion oƒ a tuumber öƒ repeated obseruations +that tuiil uicld that particular ou‡come. TÝ we imagine repeating an observatlon—— +such as looking at a freshly tossed coin——/ times, and ïf we call WA our estimafe +of the most likely number of our observations that will give some specified result A, +say the result “heads,” then by P(4), the probability of observing 4, we mean +P(A) = NẠ/N. (6.1) +Our defnition requires several comments. FEirst of all, we may speak of a +probability of something happening only if the occurrenece is a possible outcome +of some repeafabie observation. It is not clear that ít would make any sense to +ask: “What is the probability that there is a ghost in that house?” +You may object that no situation is ezacfly repeatable. That is right. Every +diferent observation must at least be at a diferent tỉme or place. All we can say +1s that the “repeated” observations should, for our intended purposes, øøøear +to be cquiuadlent. We should assume, at least, that each observation was made +from an equivalentÌy prepared situation, and especially with the same degree of +ignorance at the start. (If we sneak a look at an opponent°s hand in a card game, +our estimate of our chances of winning are different than if we do notÏ) +W©e should emphasize that NÑ and N4 in Eq. (6.1) are of intended to rep- +resent numbers based on actual observations. a4 is our best esfma#e of what +tuould occur in Ý ?magined observations. Probability depends, therefore, on our +knowledge and on our ability to make estimates. In efect, on our common sensel +Fortunately, there is a certain amount of agreement in the common sense oÝ many +things, so that diferent people will make the same estimate. Probabilities need +not, however, be “absolute” numbers. Since they depend on our ignorance, they +may become different if our knowledge changes. +--- Trang 121 --- +You may have noticed another rather “subJective” aspect of our defnition of +probability. We have referred to a as “our estimate of the most likely number +...” We do not mean that we expect to observe ezøctu Na, but that we expect +a number øcar Na, and that the number WA is more lkelu than any other +number in the vicinity. lf we toss a coin, say, 30 times, we should expect that the +number of heads would not be very likely to be exactly 15, but rather only some +number near to 1ð, say 12, 13, 14, 15, 16, or 17. However, if we must choose, +we would decide that 15 heads is more l2kely than any other number. We would +write P(heads) = 0.5. +Why did we choose 15 as more likely than any other number? We must have +argued with ourselves in the following manner: lf the most likely number of heads +1s Nụ in a total number of tosses , then the most likely number of tails M„~ +1s (N — NH). (WG are assuming that every toss gives e/ther heads ør tails, and +no “other” resultl) But if the coin is “honest,” there is no preference for heads or +tails. Until we have some reason to think the coin (or 6oss) is dishonest, we musb +give equal likelihoods for heads and tails. 5o we must set W_- = Nh. It follows +that Nr = Nụ = N/2, or P(H) = P(T) = 05. +W© can generalize our reasoning to ømw situation in which there are w different +but “equivalent” (that is, equally likely) possible results of an observation. TÍ +an observation can yield mm. diferent results, and we have reason to believe that +any one of them is as likely as any other, then the probability of a particular +outcome 4 is P(4) = 1/m. +Tf there are seven diferent-colored balls in an opaque box and we pick one +out “at random” (that is, without looking), the probability of getting a ball +of a particular color 1s #- The probability that a “blind draw” from a shuffled +deck of 52 cards will show the ten of hearts is sB- The probability of throwing a +double-one with dice is z.. +In Chapter 5 we described the size of a nucleus in terms of is apparent area, or +“cross section.” When we did so we were really talking about probabilities. When we +shoot a high-energy particle at a thin slab of material, there is some chance that it will +pass right through and some chance that it will hit a nucleus. (Since the nucleus is so +small that we cannot see it, we cannot aim right at a nucleus. We must “shoot blind”) +Tf there are m atoms in our slab and the nucleus of each atom has a cross-sectional +area ø, then the total area “shadowed” by the nuclei is nơ. In a large number of +random shots, we expect that the number of hits )œ of some nucleus will be in the +--- Trang 122 --- +ratio to /N as the shadowed area is to the total area of the slab: +NGƒN = nơ/A. (6.2) +We may say, therefore, that the probab7lztụ that any one projectile particle will sufer a +collision in passing through the slab is +Pe= T5: (6.3) +where ?+/A4 is the number of atoms per unit area in our slab. +6-2 Fluctuations +We would like now to use our ideas about probability to consider in some +greater detail the question: “How many heads do I really ezpec£ to get If Ï toss a +coin Ñ times?” Before answering the question, however, let us look at what does +happen in such an “experiment.” Figure 6-1 shows the results obtained in the first +three “runs” of such an experiment in which = 30. 'The sequences of “heads” +and “tails” are shown just as they were obtained. 'Phe first game gave 11 heads; +the second also 11; the third 16. In three trials we did not once get 15 heads. +Should we begin to suspect the coin? Or were we wrong in thinking that the +most likely number of “heads” in such a game is 15? NÑinety-seven more runs +were made to obtain a total of 100 experiments of 30 tosses each. The results of +the experiments are given in Table 6-1. +XXXx XXX X XXXXx +XXXXXXXXXXXXX XXXX XX- +x Xx X X XXX X XX x +_XXXX XXXX XX XXXX XX XXX-. +X XXX XX X XXX XXXXXX +x Xx X XX XX XX XXxX Xxx +Fig. 6-1. Observed sequences of heads and tails in three games of +30 tosses each. +* After the first three games, the experiment was actually done by shaking 30 pennies +violently in a box and then counting the number of heads that showed. +--- Trang 123 --- +Table 6-1 +Number of heads in successive trials of 30 tosses of a coïỉn. +11 16 17 15 17 16 19 18 15 13 +11 17 17 12 20 23 11 16 17 14 +16 12 15 10 18 17 13 15 14 lỗ +16 12 11 22 12 20 12 lỗ 16 12 +16 10 15 13 14 16 15 16 13 18 100 trial +14 14 13 16 15 19 21 14 12 lỗ may +16 11 16 14 17 14 11 16 17 16 +19 15 14 12 18 l5 14 21 11 16 +17 17 12 13 14 17 9 13 19 13 +14 12 15 17 14 10 17 17 12 11 +H \ |>——— OBSERVED IN THIS +; \ EXPERIMENT +NUMBER OF / ` +GAMES IN ; : +WHICH THE_ 10 / \ +SCORE WAS H \ +OBTAINED r \ +j \„—PROBABLE NUMBER +5 , Ñ +0 -. TẾ +0 5 10 † 20 25 30 +k = NUMBER OF HEADS +Fig. 6-2. Summary of the results of 100 games of 30 tosses each. +The vertical bars show the number of games In which a score of k heads +was obtained. The dashed curve shows the expected numbers of games +with the score k obtained by a probability computation. +--- Trang 124 --- +Looking at the numbers in Table 6-1, we see that most of the results are +“near” 15, in that they are between 12 and 18. We can get a better feeling for +the details of these results if we plot a graph of the đ¿str?bufion of the results. +W© count the number of games in which a score of k was obtained, and plot this +number for each &. Such a graph is shown in Fig. 6-2. A score of 15 heads was +obtained in 13 games. A score of 14 heads was also obtained 13 times. Scores of +16 and 17 were each obtained more than 13 times. Are we to conclude that there +is some bias toward heads? Was our “best estimate” not good enough? Should +we conclude now that the “most likely” score for a run of 30 tosses is really +16 heads? But waitl In all the games taken together, there were 3000 tosses. +And the total number of heads obtained was 1493. The fraction of tosses that +gave heads is 0.498, very nearly, but slightly iess than half. We should certainly +no‡ assume that the probability of throwing heads is greater than 0.5! "The fact +that one øarticular set of observations gave 16 heads most often, is a fÏuctuation. +We still expect that the rmost likely number of heads is 1ã. +W©e may ask the question: “What ¿s the probability that a game of 30 tosses +will yield 15 heads——or 16, or any other number?” We have said that in a game +of one toss, the probability of obtaining øne head is 0.5, and the probability of +obtaining no head is 0.5. In a game of two tosses there are ƒour possible outcomes: +HH, HT, TH, TT. Since each of these sequences is equally likely, we conelude +that (a) the probability of a score of two heads is +, (b) the probability of a score +of one head is 2, (c) the probability of a zero score is +. There are /o ways oŸ +obtaining one head, but only one of obtaining either zero or bwo heads. +Consider now a game of 3 tosses. The third toss is equally likely to be heads +or tails. There is only one way to obtain 3 heads: we znusứ have obtained 2 heads +on the first two tosses, and then heads on the last. "There are, however, £hree +ways of obtaining 2 heads. We could throw tails after having thrown two heads +(one way) or we could throw heads after throwing only one head in the frst two +%osses (two ways). So Íor scores of 3-J, 2-H, 1-H, 0-H we have that the number +of equally likely ways 1s 1, 3, 3, 1, with a total of 8 diferent possible sequences. +'The probabilities are 8) ẩ› Š› §- +The argument we have been making can be summarized by a diagram like +that in Fig. 6-3. It is clear how the diagram should be continued for games with +a larger number of tosses. Figure 6-4 shows such a diagram for a game of 6 tosses. +The number of “ways” to any point on the diagram is just the number of diferent +“paths” (sequences of heads and tails) which can be taken from the starting point. +The vertical position gives us the total number of heads thrown. “The set of +--- Trang 125 --- +WAYS WAYS WAYS SCORE_ PROB. +H 1 3H 1/8 +n1 h 3 2H 3/8 +< SỰ 2S +.. „>7 1H 3/8 +FIRST ị ' 1 0H 1/8 +TOSS SECOND Ị +'TOSS 'THIRD +Fig. 6-3. A diagram for showing the number of ways a score of 0, 1, +2, or 3 heads can be obtained In a game of 3 tosses. +SCORE +1 4 15 4 +1 3 10 +< 2 6 20 3 +1 3 10 +1 4 15 2 +Fig. 6-4. A diagram like that of Fig. 6-3, for a game of 6 tosses. +numbers which appears in such a diagram is known as Pascals triangle. The +numbers are also known as the Ùznormal coefficien‡s, because they also appear In +the expansion of (ø + Ö)”. If we call ø the number of tosses and k the number of +heads thrown, then the numbers in the diagram are usually designated by the +symbol (): W©e may remark in passing that the binomial coeflicients can also be +computed from +LÀN nÌ (6.4) +kj — kl{n— k)!' l +where ml, called “n-factorial,” represents the produet (n)(w®— 1)(m—2) - - - (3)(2)(1). +W© are now ready to compute the probability P{k,m) of throwing k heads in +?+ tosses, using our defnition Eq. (6.1). The total number of possible sequences +is 2" (since there are 2 outcomes for cach toss), and the number of ways of +--- Trang 126 --- +obtaining & heads is (0). all equally likely, so we have +P(k,n) = Sạ (6.5) +Since P(k,m) is the raction of games which we expect to yield k heads, then +in 100 games we should expect to ñnd k heads 100 - P(k,n) times. The dashed +curve in Fig. 6-2 passes through the points computed rom 100 - P(k,30). We +see that we ezpect to obtain a score of 15 heads in 14 or 15 games, whereas this +score was observed in 13 games. We ezpec£ a score of 16 in 13 or 14 games, but +we obtained that score in 16 games. Such ñuctuations are “part of the game.” +'The method we have just used can be applied to the most general situation +in which there are only two possible outcomes of a single observation. Let us +designate the two outcomes by W (for “win”) and E (for “lose”). In the general +case, the probability of W or in a single event need not be equal. Let p +be the probability of obtaining the result W/. 'Phen g, the probability of Ù, 1s +necessarily (1 — ø). In a set of ø trials, the probability P(k,nø) that W will be +obtained & times is +PŒ,n) = (0)p*a"~Ẻ. (6.6) +'This probability function is called the Bernoulli or, also, the binomizal probability. +6-3 The random walk +There is another interesting problem in which the idea of probability is +required. It ¡is the problem of the “random walk.” In its simplesÈ version, we +imagine a “game” in which a “player” starts at the point z = 0 and at each “move” +is required to take a step e#her forward (toward +z) or backward (toward —z). +'The choïce is to be made randomiu, determined, for example, by the toss of a coïn. +How shall we describe the resulting motion? In is general form the problem is +related to the motion of atoms (or other particles) in a gas—called Brownian +motion——and also to the combination oŸ errors in measurements. You will see +that the random-walk problem is closely related to the coin-tossing problem we +have already discussed. +Flirst, let us look at a few examples of a random walk. We may characterize +the walker°s progress by the net distance /)„ traveled in steps. We show In +the graph of Fig. 6-5 three examples of the path of a random walker. (We have +--- Trang 127 --- +(DISTANCE FROM d = SN Cu Hư +—5 NA n ⁄ NI +0 10 20 30 +N (STEPS TAKEN) +Fig. 6-5. The progress made ¡in a random walk. The horizontal coor- +dinate ÑN ¡s the total number of steps taken; the vertical coordinate D„ +Is the net distance moved from the starting position. +used for the random sequence of choices the results of the coin tosses shown in +Eig. 6-1.) +'What can we say about such a motion? We might fñrst ask: “How far does he +get on the average?” We must ezpect that his average progress will be zero, since +he is equally likely to go either forward or backward. But we have the feeling +that as / increases, he is more likely to have strayed farther from the starting +point. We might, therefore, ask what is his average distance travelled in absolufe +0alue, that is, what is the average of |J|. It is, however, more convenient to deal +with another measure of “progress,” the square of the distanece: 22 is positive +for either positive or negative motion, and is therefore a reasonable rneasure of +such random wandering. +We can show that the expected value of DẶ, is just /Ụ, the number of steps +taken. By “expected value” we mean the probable value (our best guess), which +we can think of as the ezpected average behavior in man repeø‡ed sequences. We +represent such an expected value by (D'$,), and may refer to it also as the “mean +square distance.” After one step, DĐ is always +1, so we have certainly (DŸ) = 1. +--- Trang 128 --- +(AI distances will be measured in terms of a unit of one step. We shall not +continue to write the units of distance.) +The expected value of DẶ, for > 1 can be obtained from y_¡. lf, after +(N — 1) steps, we have 2„y_, then after Ñ steps we have 2y = DA_-¡ +1 +or Dạ =DN_¡ — 1. Eor the squares, +DẶ_-¡+2DN_1 +1, +DẬ = or (6.7) +DẶ_¡—2Dy_¡ +1. +In a number of independent sequences, we expect to obtain each value one-half +of the time, so our average expectation is just the average of the wo possible +values. The expected value of DẶ, is then DẶ,_¡ +1. In general, we should ezpect +for DẬ,_ ¡ its “expected value” (D$,_¡) (by defnition!). So +(Dậ) = (Dậ_¡) +1. (6.8) +We have already shown that (D?) = 1; it follows then that +(Dậ) = N, (6.9) +a particularly simple resultl +lf we wish a number like a distance, rather than a distance squared, to +represent the “progress made away from the origin” in a random walk, we can +use the “root-mean-square distance” Jyụs: +Dym¿ = (D32) =vN. (6.10) +We have pointed out that the random walk is closely similar in its mathematics +to the coin-tossing game we considered at the beginning of the chapter. lÝ we +imagine the direction of each step to be in correspondence with the appearance of +heads or tails in a coin toss, then J is Just )ự„ — „+, the diference in the number +of heads and tails. Since W„ + W„p = ÑN, the total number of steps (and tosses), +we have = 2N} — N. Woe have derived earlier an expression for the expected +distribution of Ä„ (also called &) and obtained the result of Eq. (6.5). Since W +is just a constant, we have the corresponding distribution for . (Since for every +head more than /2 there is a tail “missing,” we have the factor of 2 between +--- Trang 129 --- +Nhu and D.) The graph of Fig. 6-2 represents the distribution of distances we +might get in 30 random steps (where k = lỗ is to be read 2 = 0; k= 16, D= 2; +etc.). +The variation of Wy from its expected value N/2 is +Ñn— — =—. 6.11 +H—S=5 (6.11) +'The rms deviation 1s +(xz _ 3) =5VN. (6.12) +2 T1nS +According to our result for Dym;, we expect that the “typical” distanece in +30 steps ought to be w⁄30 = 5.5, or a typical k should be about 5.5/2 = 2.8 units +from 15. We see that the “width” of the curve in Eig. 6-2, measured from the +center, is just about 3 units, in agreement with this result. +W© are now in a position to consider a question we have avoided until now. +How shall we tell whether a coin is “honest” or “loaded”? We can give now at +least a partial answer. EFor an honest coin, we expect the fraction of the times +heads appears to be 0.5, that is, +———— = 035. 6.13 +_ (6.13) +W© aÏso expect an actual Ấy to deviate from Ñ/2 by about v N/2, or the ƒfraction +to deviate by +1LvN 1 +N 2 2vN. +The larger is, the closer we ezpect the fractlon Wg/N to be to one-half. +In Eig. 6-6 we have plotted the fraction Vy/N for the coïin tosses reported ear- +lier in this chapter. We see the tendency for the fraction of heads to approach 0.5 +for large W. Unfortunately, for any given run or combination of runs there is no +guarantee that the observed deviation will be even øcør the ezpected deviation. +'There is always the finite chance that a large fuctuation—a long string of heads +or tails—will give an arbitrarily large deviation. All we can say is that jƒ the +deviation is near the expected 1/2VWN (say within a factor of 2 or 3), we have +no reason to suspect the honesty of the coin. lÝ it is much larger, we may be +suspicious, but cannot prove, that the coïn is loaded (or that the 6osser is cleverl). +--- Trang 130 --- +1.0 +FRACTION +HEADS Ö~? +0.5 < +0 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 +N (COIN TOSSES) +Fig. 6-6. The fraction of the tosses that gave heads in a particular +sequence of Ñ tosses of a penny. +W©e have also not considered how we should treat the case of a “coin” or +sơme similar “chancy” object (say a stone that always lands in either of two +positions) that we have good reason to believe should have a diferent probability +for heads and tails. We have deined P(H) = (Nn)/N. How shall we know what +to ezpec£ for N„? In some cases, the best we can do is to observe the number +of heads obtained in large numbers of tosses. For want of anything better, we +must set (N;;) = Nư(observed). (How could we expect anything else?) We must +understand, however, that in such a case a diferent experiment, or a diferent +observer, might conclude that P(H) was diferent. We would ezpect, however, +that the various answers should agree within the deviation 1/2VN [if P(H) is +near one-half]. An experimental physicist usually says that an “experimentally +determined” probability has an “error,” and writes +P(H) N + 2VN (6.14) +'There is an implication in such an expression that there 7s a “true” or “correc$” +probability which couwld be computed If we knew enouph, and that the observation +may be in “error” due to a Ñuctuation. 'There is, however, no way to make such +thinking logically consistent. It is probably better to realize that the probability +concept is in a sense subjective, that it is always based on uncertain knowledge, +and that its quantitative evaluation is subject to change as we obtain more +Information. +--- Trang 131 --- +6-4 A probability distribution +Let us return now to the random walk and consider a modification of it. +Suppose that in addition to a random choïce of the đecføn (+ or —) of each +step, the /ength of each step also varied in some unpredictable way, the only +condition being that on the auerage the step length was one unit. 'Phis case is +more representative of something like the thermal motion of a molecule in a gas. +Tí we call the length of a step Š, then Š may have any value at all, but most often +will be “near” 1. To be specifc, we shall let (S2) = 1 or, equivalently, Sz„;„ = 1. +Our derivation for (D?) would proceed as before except that Eq. (6.8) would be +changed now to read +(DẬ) = (DẶ +) + (52) = (DẶ_ ¡) +1. (6.15) +W© have, as before, that +(DV})=N. (6.16) +'What would we expect now for the distribution of distances D? What is, for +example, the probability that J2 = 0 after 30 steps? The answer is zerol 'Phe +probability is zero that D will be amw particular value, since there is no chance at +all that the sum of the backward steps (of varying lengths) would exactly equal +the sum oŸ forward steps. We cannot plot a graph like that of Eig. 6-2. +W© can, however, obtain a representation similar to that of Fig. 6-2, if we ask, +not what is the probability of obtaining D exactly equal to 0, 1, or 2, but instead +what is the probability of obtaining D near 0, 1, or 2. Let us define P(z, Az) +as the probability that D will lie in the interval Az located at z (say from ø +to z-+ Az). We expect that for small Az+ the chance of DĐ landing in the interval +is proportional to Az, the width of the interval. So we can write +Pí(œ, Az) = p(œ) Az. (6.17) +The function ø(x) is called the przobabilitụ densitg. +The form oŸ p(+) will depend on , the number of steps taken, and also on the +distribution of individual step lengths. We cannot demonstrate the proofs here, +but for large W, p(#) is the sarme for all reasonable distributions in individual +step lengths, and depends only on ÑW. W© plot (+) for three values oŸ Ý in +Fig. 6-7. You will notice that the “halfwidths” (typical spread from # = 0) of +these curves is v(, as we have shown it should be. +--- Trang 132 --- +PROBABILITY DENSITY +N = 10,000 STEPS +40,000 STEPS +160,000 STEPS +—700 —600 —500—400—300—200-100 0 100 200 300 400 500 600. 700 +D = DISTANCE FROM START +Fig. 6-7. The probability density for ending up at the distance 2 from +the starting place in a random walk of N steps. (D is measured in units +of the rms step length.) +You may notice also that the value oŸ ø0() near zero is inversely proportional +to VN. This comes about because the curves are all of a similar shape and theïr +areas under the curves must all be equal. Since ø(#) Az is the probability of +fñnding Din Az when Az is small, we can determine the chance of finding D +sơmcuhere inside an arbitrary interval from # to #a, by cutting the interval in +a number of small increments Az and evaluating the sum of the terms ø() Az +for each increment. “The probability that D lands somewhere between #ø and za, +which we may write P{z„ < D < za), is equal to the shaded area in Eig. 6-8. +The smaller we take the increments Az, the more correct is our result. We can +write, therefore, +P(œạ< D< z:) = À`p(z) Ax= J p(ø) da. (6.18) +The area under the whole curve is the probability that Ð lands somewhere +(that is, has sormme value between ø = —œ and # = +œc). That probability is +--- Trang 133 --- +X1 X2 x +Fig. 6-8. The probability that the distance D traveled in a random +walk is between xị and xa is the area under the curve of p(x) from xị +to Xa. +surely 1. We must have that +J p(z) dz = 1. (6.19) +Since the curves in Fig. 6-7 get wider in proportion to W, theïr heights must +be proportional to 1/WN to maintain the total area equal to 1. +The probability density function we have been describing is one that is +encountered most commonly. It is known as the n=ormal or gausstøn probability +density. It has the mathematical form +p(&) = —— 9/2, (6.20) +where ø is called the s¿øndard deuiafion and is given, in our case, by ơ = VN or, +1f the rms step size is diferent from 1, by ơ = VN mạ. +We remarked earlier that the motion of a molecule, or of any particle, in a +gas is like a random walk. Suppose we open a bottle of an organic compound +and let some of its vapor escape Into the air. If there are air currents, so that +the air is circulating, the currents will also carry the vapor with them. But even +in perfeclu si azr, the vapor will gradually spread out—will difuse—until it +has penetrated throughout the room. We might detect it by its color or odor. +The individual molecules of the organic vapor spread out in still air because of +the molecular motions caused by collisions with other molecules. If we know the +average “step” size, and the number of steps taken per second, we can fnd the +--- Trang 134 --- +probability that one, or several, molecules wiïll be found at some distance from +their starting point after any particular passage of time. Äs time passes, more +steps are taken and the gas spreads out as in the successive curves of Fig. 6-7. +In a later chapter, we shall fnd out how the step sizes and step Ífrequencies are +related to the temperature and pressure of a gas. +Barlier, we said that the pressure of a gas is due to the molecules bouneing +against the walls of the container. When we come later to make a more quanti- +tative description, we will wish to know how fast the molecules are going when +they bounce, since the impact they make will depend on that speed. We camnot, +however, speak oŸ #he speed of the molecules. Ït is necessary %o use a probability +description. A molecule may have any speed, but some speeds are more likely +than others. We describe what is going on by saying that the probability that +any particular molecule will have a speed between 0 and ø + Ao is p(o) Ao, +where Øø(0), a probability density, is a given funection of the speed ø. We shall see +later how Maxwell, using common sense and the ideas of probability, was able to +ñnd a mathematical expression for ø(0). The form of the function ø(0) is shown +in Fig. 6-9. Velocities may have any value, but are most likely to be near the +most probable value 0;. +N-p(v) +Vp VỊ V2 V +Fig. 6-9. The distribution of velocities of the molecules In a gas. +We often think of the curve of Fig. 6-9 in a somewhat different way. lf +we consider the molecules in a typical container (with a volume of, say, one +liter), then there are a very large number of molecules present ( + 1022). +Since ø(0) Ao is the probability that øøe molecule will have its velocity in Áo, +* Maxwell's expression is (0) = Cu2c—*°Ỷ, where œ is a constant related to the temperature +and Œ is chosen so that the total probability is one. +--- Trang 135 --- +by our defnition oŸ probability we mean that the ezpecfed number (AN) to be +found with a velocity in the interval Au is given by +(AM) = Np(o) Ao. (6.21) +We call N p(o) the “distribution in velocity.” The area under the curve bebween +two velocitles 0 and 0a, for example the shaded area in Fig. 6-9, represents [for +the curve ý ø(ø)| the expected number of molecules with velocities bebween 0 +and 0a. 5ince with a gas we are usually dealing with large numbers of molecules, +we expect the deviations from the expected numbers to be small (like 1/v `), so +we often neglect to say the “expected” number, and say instead: “The number oŸ +mmolecules with velocitles between 0 and 0a 2s the area under the curve.” We should +remember, however, that such statements are always about probable numbers. +6-5 The uncertainty principle +The ideas of probability are certainly useful in describing the behavior of +the 1022 or so molecules in a sample of a gas, for it is clearly impractical even to +attempt to write down the position or velocity of each molecule. When probability +was first applied to such problems, 1 was considered to be a conwuenience—a +way of dealing with very complex situations. We now believe that the ideas of +probability are essenfial to a description of atomie happenings. According to +quantum mechaniecs, the mathematical theory of particles, there is always some +uncertainty in the specifcafion of positions and velocities. We can, at best, say +that there is a certain probability that any particle will have a position near some +coordinate z. +We can give a probability density ø+(#), such that ø+(#) Az is the probability +that the particle will be found between + and z-+ Az. T the particle is reasonably +well localized, say near zoọ, the function ø1(z) might be given by the graph of +Eig. 6-10(a). Similarly, we must specify the velocity of the particle by means of +a probability density pa(), with pa(0) Ao the probability that the velocity will +be found between 0 and ø + Áo. +lt is one of the fundamental results of quantum mechanics that the two +functions ø¡(#) and øa(0) cannot be chosen independently and, in particular, +cannot both be made arbitrarily narrow. lf we call the typical “width” of the +ø1(z) curve [Az], and that of the øa(ø) curve [Aö| (as shown in the figure), +nature demands that the product of the two widths be at least as big as the +--- Trang 136 --- +pa(v) ' +Fig. 6-10. Probability densities for observatlon of the position and +velocity of a particle. +number #/2m, where mm is the mass of the particle. We may write this basic +relationship as +[Az] - [Aul > h/2m. (6.22) +This equation is a statement of the He¿senberg uncertaintụ principle that we +mentioned earlier. +Since the right-hand side of Eq. (6.22) is a constant, this equation says that +1Í we try to “pin down” a particle by forcing it to be at a particular place, 1% +ends up by having a high speed. Or if we try to fÍorce it to go very slowly, or at a +precise velocity, it “spreads out” so that we do not know very well just where ït +1s. Particles behave in a funny wayl +'The uncertainty principle describes an inherent fuzziness that must exist in +any attempt to describe nature. Ôur most precise description of nature rwusf +ben terms of probabilöties. There are some people who do not like this way of +describing nature. They feel somehow that if they could only tell what is reallu +going on with a particle, they could know its speed and position simultaneously. +In the early days of the development of quantum mechanics, Einstein was quite +worried about this problem. He used to shake his head and say, “But, surely God +--- Trang 137 --- +Fig. 6-11. A way of visualizing a hydrogen atom. The density (white- +ness) of the cloud represents the probability density for observing the +electron. +does not throw dice in determining how electrons should go!” He worried about +that problem for a long time and he probably never really reconciled himself to +the fact that this is the best description of nature that one can give. There are +still one or two physicists who are working on the problem who have an intuitive +conviction that it is possible somehow to describe the world in a different way +and that all of this uncertainty about the way things are can be removed. No +one has yet been successful. +The necessary uncertainty in our specifcation of the position of a particle +becomes most important when we wish to describe the structure of atoms. In the +hydrogen atom, which has a nucleus of one proton with one electron outside of +the nucleus, the uncertainty in the position of the electron is as large as the atom +itselft We cannot, therefore, properly speak of the electron moving in some “orbit” +around the proton. The most we can say is that there is a certain chanee p(r) AV, +of observing the electron in an element of volume AV at the distance r from +the proton. The probability density p(z) is given by quantum mechanics. For +an undisturbed hydrogen atom p(z) = Ae~?*/*, The number ø is the “typical” +radius, where the function is decreasing rapidly. 5ince there is a small probability +of fñnding the electron at distances from the nucleus mụuch greater than ø, we +may think of ø as “the radius of the atom,” about 10—†19 meter. +W© can form an image of the hydrogen atom by imagining a “cloud” whose +density is proportional to the probability density for observing the electron. A +--- Trang 138 --- +sample of such a cloud is shown in EFig. 6-11. “Thus our best “picture” of a +hydrogen atom is a nucleus surrounded by an “electron cloud” (although we +reall mean a “probability cloud”). The electron is there somewhere, but nature +permits us to know only the chance of fñnding it at any particular place. +In its eforts to learn as much as possible about nature, modern physics has +found that certain things can never be “known” with certainty. Much of our +knowledge must always remain uncertain. "The mos we can know is in terms of +probabilities. +--- Trang 139 --- +Tho Thoortg ©Ÿ Ấnrcrtff(rffOre +7-1 Planetary motions +In this chapter we shall diseuss one of the most far-reaching generalizations oŸ +the human mỉnd. While we are admiring the human mind, we should take some +time of to stand in awe of a na#ure that could follow with such completeness +and generality such an elegantly simple principle as the law of gravitation. What +1s this law of gravitation? It is that every object in the universe attracts every +other obJect with a force which for any two bodies is proportional to the mass +of each and varies inversely as the square of the distance between them. “This +statement can be expressed mathematically by the equation +F=G——.. +T to this we add the fact that an object responds to a force by accelerating in +the direction of the force by an amount that is inversely proportional to the mass +of the object, we shall have said everything required, for a sufficiently talented +mathematician could then deduce all the consequences of these two principles. +However, since you are not assumed to be sufficiently talented yet, we shall +discuss the consequences in more detail, and not just leave you with only these +two bare principles. We shall brieRy relate the story of the discovery of the +law of gravitation and discuss some of its consequences, its efects on history, +the mysteries that such a law entails, and some reñnements of the law made +by Einstein; we shall also discuss the relationships of the law to the other laws +of physics. All this cannot be done in one chapter, but these subjects will be +treated in due time in subsequent chapters. +The story begins with the ancients observing the motions of planets among +the stars, and finally deducing that they went around the sun, a fact that was +rediscovered later by Copernicus. Exactly ho the planets went around the sun, +--- Trang 140 --- +with exactly t0høt motion, took a littÌe more work to discover. In the beginning +of the fñfteenth century there were great debates as to whether they really went +around the sun or not. 'Eycho Brahe had an idea that was diferent from anything +proposed by the ancients: his idea was that these debates about the nature of the +motions of the planets would best be resolved if the actual positions of the planets +in the sky were measured sufficiently accurately. IÝ measurement showed exactly +how the planets moved, then perhaps it would be possible to establish one or +another viewpoint. 'This was a tremendous idea—that to fñnd something out, it is +better to perform some careful experiments than to carry on deep philosophical +areuments. Pursuing this idea, Tycho Brahe studied the positions of the planets +for many years in his observatory on the island of Hven, near Copenhagen. He +made voluminous tables, which were then studied by the mathematician Kepler, +after Iycho's death. Kepler discovered from the data some very beautiful and +remarkable, but simple, laws regarding planetary motion. +7-2 Kepler?s laws +First of all, Kepler found that each planet goes around the sun in a curve +called an ellpse, with the sun at a focus of the ellipse. An ellipse is not just an +oval, but is a very specifc and precise curve that can be obtained by using two +tacks, one at each focus, a loop oŸ string, and a pencil; more mathematically, it +1s the locus oŸ all points the sum oŸ whose distances from two fixed points (the +foci) is a constant. Ôr, iŸ you will, it is a foreshortened circle (Fig. 7-1). +“EN | +rị + ra = 2a +Fig. 7-1. An ellipse. +Jepler°s second observation was that the planets do not go around the sun at +a uniform speed, but move faster when they are nearer the sun and more sÌowly +when they are farther from the sun, in precisely this way: Suppose a planet +--- Trang 141 --- +Z7 +2 Z7 +Fig. 7-2. Kepler's law of areas. +1s observed at any two successive times, let us say a week apart, and that the +radius vector* is drawn to the planet for each observed position. The orbital are +traversed by the planet during the week, and the two radius vectors, bound a +certain plane area, the shaded area shown in Fig. 7-2. If two similar observations +are made a week apart, at a part of the orbit farther from the sun (where the +planet moves more slowly), the similarly bounded area is exactly the same as in +the first case. So, in accordance with the second law, the orbital speed of each +planet is such that the radius “sweeps out” equal areas in equal times. +tPinally, a third law was discovered by Kepler much later; this law is of a +diferent category from the other two, because it deals not with only a single +planet, but relates one planet to another. 'This law says that when the orbital +period and orbit size of any two planets are compared, the periods are proportional +to the 3/2 power of the orbit sỉze. In this statement the period is the tỉme interval +1t takes a planet to go completely around ïits orbit, and the size is measured by +the length of the greatest diameter of the elliptical orbit, technically known as +the major axis. More simply, If the planets went in circles, as they nearly do, the +time required to go around the circle would be proportional to the 3/2 power of +the diameter (or radius). Thus Kepler”s three laws are: +I. Each planet moves around the sun in an ellipse, with the sun at one focus. +TL. The radius vector from the sun to the planet sweeps out equal areas in +equal intervals of time. +TH. The squares of the periods of any two planets are proportional to the ceubes +of the semimajor axes of their respective orbits: 7' œ a3⁄2. +—* A radius vector is a line drawn from the sun to any point in a planet”s orbit. +--- Trang 142 --- +7-3 Development of dynamics +While Kepler was discovering these laws, Galileo was studying the laws of +motion. The problem was, what makes the planets go around? (In those days, +one of the theories proposed was that the planets went around because behind +them were invisible angels, beating their wings and driving the planets forward. +You will see that this theory is now modifiedL It turns out that in order to keep the +planets going around, the invisible angels must fly in a diÑerent direction and they +have no wings. Otherwise, it is a somewhat similar theoryl) Galileo discovered a +very remarkable fact about motion, which was essential for understanding these +laws. That is the principle oŸ 7nmerf2a—lŸ something is moving, with nothing +touching it and completely undisturbed, it will go on forever, coasting at a +uniform speed in a straight line. (W2, does i% keep on coasting? We do not +know, but that is the way it is.) +Newton modifed this idea, saying that the only way to change the motion +of a body is to use ƒorce. lf the body speeds up, a force has been applied #n +the direction oƒ motion. On the other hang, 1f its motion is changed to a new +đircction, a force has been applied s7deuazs. Newton thus added the idea that +a Íforce is needed to change the speed ør ¿he direcfion of motion of a body. For +example, IŸ a stone is attached to a string and is whirling around in a circle, i% +takes a force to keep 1È in the circle. We have to pull on the string. In fact, the +law is that the acceleration produced by the force is inversely proportional to the +mass, or the force is proportional to the mass times the acceleration. The more +massive a thing is, the stronger the force required to produce a given acceleration. +(The mass can be measured by putting other stones on the end of the same string +and making them go around the same circle at the same speed. In this way 1 +1s found that more or less force is required, the more massive object requiring +more force.) The brilliant idea resulting from these considerations is that no +tangeniial force is needed to keep a planet in its orbit (the angels do not have to +ñy tangentially) because the planet would coast in that direction anyway. lÝ there +were nothing at all to disturb it, the planet would go of in a sứraight line. But +the actual motion deviates from the line on which the body would have gone if +there were no force, the deviation beiïng essentially at right angles to the motion, +not in the direction of the motion. In other words, because of the principle of +inertia, the force needed to control the motion of a planet arownd the sun is not +a force around the sun but #øouard the sun. (TỶ there is a force toward the sun, +the sun might be the angel, oŸ coursel) +--- Trang 143 --- +7-4 Newton?s law of gravitation +tFrom his better understanding of the theory of motion, Newton appreciated +that the sun could be the seat or organization of forces that govern the motion of +the planets. NÑewton proved to himself (and perhaps we shall be able to prove it +soon) that the very fact that equal areas are swept out in equal times is a precise +sign post of the proposition that all deviations are precisely rœd¿al—that the law +Of areas 1s a direct consequence of the idea that all of the forces are directed +exactly £ouard the sun. +Next, by analyzing Kepler's third law it is possible to show that the farther +away the planet, the weaker the forces. If two planets at diferent distances +from the sun are compared, the analysis shows that the forces are inversely +proportional to the squares of the respective distances. With the combination +of the two laws, Newton concluded that there must be a force, inversely as the +square of the distance, directed in a line between the two obJects. +Being a man of considerable feeling for generalities, NÑewton supposed, of +course, that this relationship applied more generally than just to the sun holding +the planets. It was already known, for example, that the planet Jupiter had +moons going around it as the moon of the earth goes around the earth, and +Newton felt certain that each planet held its moons with a force. He already knew +of the force holding us on the earth, so he proposed that this was a unớuersal +ƒorce—that cuerWthing pulls cueruthing cls. +The next problem was whether the pull of the earth on its people was the +“same” as its pull on the moon, 1.e., inversely as the square of the distance. If +an objJect on the surface of the earth falls 16 feet in the first second after it 1s +released from rest, how far does the moon fall in the same time? We might say +that the moon does not fall at all. But if there were no force on the moon, 1§ +would go of in a straight line, whereas it goes in a circle instead, so it really +ƒalls ín Írom where i% would have been ïf there were no force at all. We can +calculate from the radius of the moon's orbit (which is about 240,000 miles) and +how long it takes to go around the earth (approximately 29 days), how far the +mmoon moves in its orbit in 1 second, and can then calculate how far it falls in +one second.* 'This distance turns out to be roughly 1/20 of an inch in a second. +That fts very well with the inverse square law, because the earth”s radius 1s +4000 miles, and if something which is 4000 miles from the center of the earth +* 'That is, how far the circle of the moon”s orbit falls below the straight line tangent to it at +the point where the moon was one second before. +--- Trang 144 --- +falls 16 feet in a second, something 240,000 miles, or 60 times as far away, should +fall only 1/3600 of 16 feet, which also is roughly 1/20 of an inch. Wishing to +put this theory of gravitation to a test by similar calculations, Newton made his +calculations very carefully and found a discrepancy so large that he regarded the +theory as contradicted by facts, and did not publish his results. Six years later +a new measurement of the size of the earth showed that the astronomers had +been using an incorrect distance to the moon. When Newton heard of this, he +made the calculation again, with the corrected figures, and obtained beautiful +agrccment. +This idea that the moon “falls” is somewhat confusing, because, as you see, +1 does not come any cÍoser. “The idea is suficiently interesting to merit further +explanation: the moon falls in the sense that # ƒalls auaw jrom the straight line +that ?t tuould pursue ?ƒ there tuere mo [orces. Let us take an example on the surface +of the earth. An object released near the earth”s surface will fall 16 feet in the +first second. An object shot out horizon‡ali will also fall 16 feet; even though it +is moving horizontally, it stilHl falls the same 16 feet in the same time. Pigure 7-3 +shows an apparatus which demonstrates this. Ôn the horizontal track ¡is a ball +which is going to be driven forward a little distance away. At the same height is +a ball which is going to fall vertically, and there is an electrical switch arranged +so that at the moment the first ball leaves the track, the second ball is released. +That they come to the same depth at the same time is witnessed by the fact +that they collide in midair. An object like a bullet, shot horizontally, might go +a long way in one second——perhaps 2000 feet——but it will still fall 16 feet 1Í it +1s aimed horizontally. What happens if we shoot a bullet faster and faster? Do +not forget that the earth's surface is curved. If we shoot i% fast enough, then +đZenemee _ Z +lk⁄ h hị = hạ +Fig. 7-3. Apparatus for showing the independence of vertical and +horizontal motions. +--- Trang 145 --- +Fig. 7-4. Acceleration toward the center of a circular path. From +plane geometry, x/S = (2R — S)/x 2R/x, where is the radius of +the earth, 4000 miles; x Is the distance “travelled horizontally” in one +second; and S is the distance “fallen” in one second (16 feet). +when it falls 16 feet it may be at just the same height above the ground as it +was before. How can that be? It still falls, but the earth curves away, so it falls +“around” the earth. The question is, how far does it have to go in one second so +that the earth is 16 feet below the horizon? In Fig. 7-4 we see the earth with +1ts 4000-mile radius, and the tangential, straight line path that the bullet would +take If there were no force. Now, IŸ we use one of those wonderful theorems In +geometry, which says that our tangent is the mean proportional between the two +parts of the diameter cut by an equal chord, we see that the horizontal distance +travelled is the mean proportional bebween the 16 feet fallen and the 8000-mile +điameter of the earth. The square root of (16/5280) x 8000 comes out very close +to 5 miles. Thus we see that if the bullet moves at 5 miles a second, it then will +continue to fall toward the earth at the same rate of 16 feet each second, but will +never get any closer because the earth keeps curving away from it. 'hus it was +that Mr. Gagarin maintained himself in space while going 25,000 miles around +the earth at approximately 5 miles per second. (He took a little longer because +he was a little higher.) +Any great discovery of a new law is useful only if we can take more out than +we put in. Now, Newton œseđd the second and third of Kepler?s laws to deduce +his law of gravitation. What did he predict? Eirst, his analysis of the moon”s +motion was a prediction because it connected the falling of objects on the earth”s +surface with that of the moon. Second, the question is, ¡s Éhe orb#t an cllipse? +W© shall see in a later chapter how it is possible to calculate the motion exactly, +--- Trang 146 --- +and indeed one can prove that it should be an ellipse,Š so no extra facE is needed +to explain Kepler”s #rs law. Thus Newton made his first powerful prediction. +'The law of gravitation explains many phenomena not previously understood. +For example, the pull of the moon on the earth causes the tides, hitherto mys- +terious. 'Phe moon pulls the water up under ¡i§ and makes the tides—people +had thought of that before, but they were not as clever as Newton, and so they +thought there ought to be only one tide during the day. 'Phe reasoning was that +the moon pulls the water up under it, making a high tide and a low tide, and since +the earth spins underneath, that makes the tide at one station go up and down +every 24 hours. Actually the tide goes up and down in 12 hours. Another school +of thought claimed that the high tide should be on the other side of the earth +because, so they argued, the moon pulls the earth away from the waterl Both of +these theories are wrong. It actually works like this: the pull of the moon for the +earth and for the water is “balanced” at the center. But the water which is closer +to the moon is pulled znmore than the average and the water which is farther away +from it is pulled /ess than the average. Eurthermore, the water can ow while the +more rigid earth cannot. The true picture is a combination of these two things. +What do we mean by “balanced”? What balances? If the moon pulls the +whole earth toward it, why doesn”t the earth fall right “up” to the moon? Because +the earth does the same trick as the moon, it goes in a circle around a point +which is inside the earth but not at its center. 'Phe moon does not just go around +the earth, the earth and the moon both go around a central position, each falling +toward this common position, as shown in Eig. 7-5. This motion around the +_~Z“MOON +HạO _«< +X POINT AROUND WHICH +EARTH & MOON ROTATE +C3 +Fig. 7-5. The earth-moon system, with tides. +* The proof is not given in this course. +--- Trang 147 --- +common center is what balances the fall of each. So the earth is not goïng in a +straight line either; it travels in a circle. The water on the far side is “unbalanced” +because the moon”s attraction there is weaker than it is at the center of the +carth, where it just balances the “centrifugal force.” 'Phe result of this imbalance +1s that the water rises up, away from the center of the earth. Ôn the near side, +the attraction from the moon is stronger, and the Iimbalance is in the opposite +direction in space, but again øøw from the center of the earth. The net result is +that we get £ưo tidal bulges. +7-5 Universal gravitation +'What else can we understand when we understand gravity? Everyone knows +the earth is round. Why is the earth round? That is easy; it is due to gravitation. +The earth can be understood to be round merely because everything attracts +everything else and so it has attracted itself together as far as it can! If we go +even further, the earth is not ezøcflu a sphere because it is rotating, and this +brings in centrifugal efects which tend to oppose gravity near the equator. ÏI§ +turns out that the earth should be elliptical, and we even get the right shape for +the ellipse. We can thus deduce that the sun, the moon, and the earth should be +(nearly) spheres, Just from the law of gravitation. +'What else can you do with the law of gravitation? If we look at the moons +of Jupiter we can understand everything about the way they move around that +planet. Incidentally, there was once a certain dificulty with the moons oŸ Jupiter +that is worth remarking on. 'Phese satellites were studied very carefully by +Roemer, who noticed that the moons sometimes seemed to be ahead of schedule, +and sometimes behind. (One can ñnd their schedules by waiting a very long tỉme +and fnding out how long ïÈ takes on the average for the moons to go around.) +Now they were øhead when Jupiter was particularly close to the earth and they +were 0ch¿nd when Jupiter was ƒfarther from the earth. This would have been +a very diffcult thing to explain according to the law of gravitation—it would +have been, in fact, the death of this wonderful theory If there were no other +explanation. If a law does not work even in ønwe pÌace where it ought to, it 1s +Just wrong. But the reason for this discrepancy was very simple and beautiful: ït +takes a little while to see the moons of Jupiter because of the time it takes light +to travel from Jupiter to the earth. When Jupiter is closer to the earth the time +1s a little less, and when ït is farther from the earth, the time is more. This is why +mmoons appear to be, on the average, a little ahead or a little behind, depending +--- Trang 148 --- +on whether they are closer to or farther om the earth. 'This phenomenon showed +that light does not travel instantaneously, and furnished the first estimate of the +speed of light. This was done in 1656. +T all of the planets push and pull on each other, the force which controls, +let us say, Jupiter in going around the sun is not just the force from the sun; +there is also a pull from, say, Saturn. This force is not really strong, since the +sun is much more massive than Saturn, but there is søzne pull, so the orbit +of Jupiter should not be a perfect ellipse, and it is not; it is slightly of, and +“wobbles” around the correct elliptical orbit. Such a motion 1s a little more +complicated. Attempts were made to analyze the motions of Jupiter, Saturn, +and Uranus on the basis of the law of gravitation. 'Phe efects of each of these +planets on each other were calculated to see whether or not the tiny deviations +and irregularities in these motions could be completely understood from this +one law. Lo and behold, for Jupiter and Saturn, all was well, but Ủranus was +“weïrd.” behaved in a very peculiar manner. It was not travelling in an exact +ellipse, but that was understandable, because of the attractions of Jupiter and +Saturn. But even ïf allowance were made for these attractions, Dranus si was +not going right, so the laws of gravitation were in danger of beïng overturned, +a possibility that could not be ruled out. Two men, Adams and Le Verrier, in +England and FErance, independently, arrived at another possibility: perhaps there +1s another planet, dark and invisible, which men had not seen. This planet, N, +could pull on Dranus. They calculated where such a planet would have to be in +order to cause the observed perturbations. They sent messages to the respective +observatories, saying, “Gentlemen, point your telescope to such and such a place, +and you will see a new planet.” It often depends on with whom you are working +as to whether they pay any attention to you or not. They did pay attention to +Le Verrier; they looked, and there planet W wasl "The other observatory then +also looked very quickly in the next few days and saw it too. +This discovery shows that Newton”s laws are absolutely right in the solar +system; but do they extend beyond the relatively small distances of the nearest +planets? 'The first test lies in the question, do s#ars attract cach other as well as +planets? We have defnite evidence that they do in the double stars. Figure 7-6 +shows a double star—Ewo stars very close together (there is also a third star in +the picture so that we will know that the photograph was not turned). "The stars +are also shown as they appeared several years later. We see that, relative to the +“ñxed” star, the axis of the pair has rotated, i.e., the bwo stars are going around +each other. Do they rotate according to NÑewton's laws? Careful measurements +--- Trang 149 --- +Fig. 7-6. A double-star system. +180° +° Xu +# » sẽ en ` +» » @ + +» sỲ sờ +b KỒ X» +270° S>—— 90° +äyw 1862 +° +% 3 KG +© %, @ +0 21 4 6 g1 10 12 +Ô,,, Ô +SCALE +Fig. 7-7. Orbit of Sirnus B with respect to Sirius A. +--- Trang 150 --- +of the relative positions of one such double star system are shown in Fig. 7-7. +There we see a beautiful ellipse, the measures starting in 1862 and going all the +way around to 1904 (by now it must have gone around once more). Ðverything +coincides with Newton?s laws, except that the sbar Sirius Á is no at the ƒocus. +'Why should that be? Because the plane of the ellipse is not in the “plane of the +sky.” We are not looking at right angles to the orbit plane, and when an ellipse is +viewed at a tilt, it remains an ellipse but the focus is no longer at the same place. +Thus we can analyze double stars, moving about each other, according to the +requirements of the gravitational law. +bi E v. vì XS %. 4444 +: LÊN Tủ « : "h.. “4 TÔ +Fig. 7-8. A globular star cluster. +That the law of gravitation is true at even bigger distances is indicated in +Hig. 7-8. lÝ one cannot see gravitation acting here, he has no soul. 'This fgure +shows one of the most beautiful things in the sky—a globular star cluster. AII +of the dots are stars. Although they look as ïf they are packed solid toward the +center, that is due to the fallibility of our instruments. Actually, the distances +between even the centermost stars are very great and they very rarely collide. +There are more stars in the Interior than farther out, and as we move outward +there are fewer and fewer. It is obvious that there is an attraction among these +stars. It is clear that gravitation exists at these enormous dimensions, perhaps +100,000 times the size of the solar system. Let us now go further, and look at an +--- Trang 151 --- +w : : : +: nhà, : ° +`... k---: +sử : : b kc¿ xả ¬ +ĂĂằ-< 'ẽẺ .. +Fig. 7-9. A galaxy. +cntire galaz, shown in Fìg. 7-9. 'The shape of this galaxy indicates an obvious +tendency for its matter to agglomerate. OÝ course we cannot prove that the +law here is precisely Inverse square, only that there ¡s still an attraction, at this +enormous dimension, that holds the whole thing together. One may say, “Well, +that is all very clever but why is it not Just a ball?” Because it is sp#mn#ng and +has angular rmnormmnentưm which it cannot give up as it contracts; it must contract +mostly in a plane. (Incidentally, if you are looking for a good problem, the +exact details of how the arms are formed and what determines the shapes of +these galaxies has not been worked out.) It is, however, clear that the shape of +the galaxy is due to gravitation even though the complexities of its structure +have not yet allowed us to analyze it completely. In a galaxy we have a scale +of perhaps 50,000 to 100,000 light years. The earth's distance from the sun 1s +8 light mưnutes, so you can see how large these dimensions are. +Gravity appears to exist at even bigger dimensions, as indicated by Fig. 7-10, +which shows many “little” things clustered together. This is a clusfer oƒ galazies, +Just like a star cluster. Thus galaxies attract each other at such distances that +they too are agglomerated into clusters. Perhaps gravitation exists even OVer +distances of tens oƒ mmillions of light years; so far as we now know, gravity seems +to go out forever inversely as the square of the distanee. +--- Trang 152 --- +Fig. 7-10. A cluster of galaxies. +lự% Z © NS ° > x R +Ỹ c® Ề % ỏ . «° 4 4 +l..C s pÔ cã ; đ. ° Lẻ 1 +"N... _—— ^ +h + Ẵ k : +% Huy Thị t * : : +Fig. 7-11. An interstellar dust cloud. +--- Trang 153 --- +Fig. 7-12. The formation of new stars? +Not only can we understand the nebulae, but from the law of gravitation we +can even get some ideas about the origin of the stars. If we have a big cloud of +dust and gas, as indicated in Eig. 7-11, the gravitational attractions of the pieces +of dust for one another might make them form little lumps. Barely visible in the +figure are “little” black spots which may be the beginning of the accumulations +of dust and gases which, due to their gravitation, begin to form stars. Whether +we have ever seen a star form or not is still debatable. Figure 7-12 shows the one +piece of evidence which suggests that we have. At the left is a picture oŸ a region +of gas with some stars in it taken in 1947, and at the right is another picture, +taken only 7 years later, which shows two new bright spots. Has gas accumulated, +has gravity acted hard enough and collected it into a ball big enough that the +stellar nuclear reaction starts in the interior and turns ¡it into a star? Perhaps, +and perhaps not. Ït is unreasonable that in only seven years we should be so +lucky as to see a star change itself into visible form; it is much less probable that +we should see #of +7-6 Cavendish°s experiment +Gravitation, therefore, extends over enormous distances. But ï1f there is a +force bebween ønw pair of objects, we ought to be able to measure the force +--- Trang 154 --- +bebween our own objects. Instead of having to watch the stars go around each +other, why can we not take a ball of lead and a marble and watch the marble go +toward the ball of lead? 'Phe difficulty of this experiment when done in such a +simple manner is the very weakness or delicacy of the force. It must be done with +extreme care, which means covering the apparatus to keep the air out, making +sure it is not electrically charged, and so on; then the force can be measured. +lt was first measured by Cavendish with an apparatus which is schematically +indicated in Eig. 7-13. 'This ñrst demonstrated the direct force bebween ©wo large, +fñxed balls of lead and two smaller balls of lead on the ends of an arm supported +by a very fñne fñber, called a torsion fber. By measuring how much the fñber +gets twisted, one can measure the strength of the force, verify that it is inversely +proportional to the square of the distance, and determine how strong it is. Thus, +one may accurately determine the coefficient G in the formula +AII the masses and distances are known. You say, “We knew it already for +the earth” Yes, but we did not know the rmass of the earth. By knowing G +from this experiment and by knowing how strongly the earth attracts, we can +indirectly learn how great is the mass of the earthl "This experiment has been +called “weighing the earth” by some people, and it can be used to determine the +coefficient G of the gravity law. 'This is the only way in which the mass of the +GÌ I Ww +Fig. 7-13. A simplified diagram of the apparatus used by Cavendish to +verify the law of universal gravitation for small objects and to measure +the gravitational constant G. +--- Trang 155 --- +earth can be determined. Œ turns out to be +6.670 x 10~!! newton - m”/kgŸ. +Tt is hard to exaggerate the importance of the efect on the history of sclence +produced by this great success of the theory of gravitation. Compare the confusion, +the lack of confidence, the ineomplete knowledge that prevailed in the earlier ages, +when there were endless debates and paradoxes, with the clarity and simplicity +of this law—this fact that all the moons and planets and stars have such a sữmnpÏle +ruïe to govern them, and further that man could understønd it and deduce how +the planets should movel "This is the reason for the success of the sciences in +following years, for it gave hope that the other phenomena of the world might +also have such beautifully simple laws. +7-7 What is gravity? +But is this such a simple law? What about the machinery ofit? All we have +done is to describe 5ou the earth moves around the sun, but we have not said +tuhat makes ?t go. Newton made no hypotheses about this; he was satisfed to ñnd +tuhøt it dịd without getting into the machinery ofit. No one has sincc giuen ang +tmachiner. Tt 1s characteristic of the physical laws that they have this abstract +character. 'Phe law of conservation of energy is a theorem concerning quantities +that have to be calculated and added together, with no mention of the machinery, +and likewise the great laws of mechanics are quantitative mathematical laws Íor +which no machinery is available. Why can we use mathematics to describe nature +without a mechanism behind it? No one knows. We have to keep going because +we fnd out more that way. +Many mechanisms for gravitation have been suggested. Ït is interesting to +consider one of these, which many people have thought of rom time to time. At +first, one is quite excited and happy when he “discovers” it, but he soon finds +that i% is not correct. lt was first discovered about 1750. Suppose there were +many particles moving in space at a very high speed in all directions and being +only slightly absorbed in going through matter. When they are absorbed, they +give an impulse to the earth. However, since there are as many going one wawy +as another, the impulses all balance. But when the sun 1s nearby, the particles +coming toward the earth through the sun are partially absorbed, so fewer of them +are coming from the sun than are coming from the other side. Therefore, the +--- Trang 156 --- +earth feels a net impulse toward the sun and it does not take one long to see +that it is inversely as the square of the distance—because of the variation of the +solid angle that the sun subtends as we vary the distance. What is wrong with +that machinery? It involves some new consequences which are noø‡ fruec. This +particular idea has the following trouble: the earth, in moving around the sun, +would impinge on more particles which are coming from i§s forward side than +from its hind side (when you run in the rain, the rain in your face is stronger +than that on the back of your headl). Therefore there would be more impulse +given the earth from the front, and the earth would feel a resistance ‡o motion +and would be slowing up in its orbit. One can calculate how long i9 would take +for the earth to stop as a result of this resistance, and it would not take long +enough for the earth to still be in its orbit, so this mechanism does not work. No +machinery has ever been invented that “explains” gravity without also predicting +some other phenomenon that does øœø exist. +Next we shall discuss the possible relation of gravitation to other forces. Thhere +is no explanation of gravitation in terms of other forces at the present time. lt +1s not an aspect of electricity or anything like that, so we have no explanation. +However, gravitation and other forces are very similar, and it is interesting to +note analogies. Eor example, the force of electricity between two charged obJects +looks just like the law of gravitation: the force of electricity is a constant, with a +minus sign, times the produet of the charges, and varies inversely as the square +of the distance. It is in the opposite direction——likes repel. But is it still not very +remarkable that the two laws Involve the same function of distance? Perhaps +gravitation and electricity are much more closely related than we think. Many +attempts have been made to unify them; the so-called unifñed fñeld theory is only +a very elegant attempt to combine electricity and gravitation; but, in comparing +gravitation and electricity, the most interesting thing is the relatioe strengths of +the forces. Any theory that contains them both must also deduce how strong the +gTAVIEYy 1s. +TỶ we take, in some natural units, the repulsion of two electrons (nature's +universal charge) due to electricity, and the attraction of 6wo electrons due to +their masses, we can measure the ratio of electrical repulsion to the gravitational +attraction. “The ratio is independent of the distance and is a fundamental constant +of nature. The ratio is shown in Fig. 7-14. 'Phe gravitational attraction relative +to the electrical repulsion bebween two electrons is 1 divided by 4.17 x 102! The +question is, where does such a large number come from? lt is not accidental, like +the ratio of the volume of the earth to the volume of a fea. We have considered +--- Trang 157 --- += 1⁄4 70, 2O, 000, ooo Sa, +-ạoe '098 +"Sao Đ00 oøo, +Fig. 7-14. The relative strengths of electrical and gravitational inter- +actions between two electrons. +two natural aspects of the same thing, an electron. This fantastic number is a +natural constant, so it Involves something deep in nature. Where could such +a tremendous number come from? Some say that we shall one day fnd the +“universal equation,” and ïn it, one of the roots will be this number. ϧ is very +dificult to ñnd an equation for which such a fantastic number is a natural root. +Other possibilities have been thought of; one is to relate it to the age of the +universe. Clearly, we have to fnd øanother large number somewhere. But do +we mean the age of the universe in eørs? No, because years are not “natural”; +they were devised by men. As an example of something natural, let us consider +the time it takes light to go across a proton, 102? second. If we compare this +time with the aøe oƒ the niuerse, 2 x 1010 years, the answer is 1072. ]t has +about the same number of zeros going of it, so it has been proposed that the +gravitational constant is related to the age of the universe. If that were the case, +the gravitational constant would change with time, because as the universe got +older the ratio of the age of the universe to the time which it takes for light to go +across a proton would be gradually increasing. Is it possible that the gravitational +constant ¡s changing with time? Of course the changes would be so small that it +1s quite difficult to be sure. +One test which we can think of is to determine what would have been the +effect of the change during the past 10 years, which is approximately the age +from the earliest life on the earth to now, and one-tenth of the age of the universe. +In this time, the gravity constant would have increased by about 10 percent. +Tt turns out that if we consider the structure of the sun—the balance bebween +--- Trang 158 --- +the weight of its material and the rate at which radiant energy ¡is generated +Inside it —we can deduce that if the gravity were 10 percent stronger, the sun +would be much more than 10 percent brighter—by the sizth pouer of the gravity +constantl If we calculate what happens to the orbit of the earth when the gravity +is changing, we find that the earth was then cỉoser 7n. Altogether, the earth +would be about 100 degrees centigrade hotter, and all of the water would not +have been in the sea, but vapor in the aïr, so life would not have started in the +sea. So we do ro now believe that the gravity constant is changing with the age +of the universe. But such arguments as the one we have just given are not very +convincing, and the subject is not completely closed. +lt is a fact that the force of gravitation is proportional to the mass, the +quantity which is fundamentally a measure of 7nerf2aœ—of how hard ït is to hold +something which is going around ïn a cirele. Therefore two obJects, one heavy +and one light, goïng around a larger object in the same cirele at the same speed +because of gravity, will stay together because to go in a circle reguzres a Íforce +which is stronger for a bigger mass. That is, the gravity is stronger Íor a given +mass in 7us( the right proportion so that the ©wo objects will go around together. +TỶ one object were inside the other it would sa inside; it is a perfect balance. +'Therefore, Gagarin or Titov would fñnd things “weightless” inside a space ship; 1Ý +they happened to let go of a piece of chalk, for example, it would go around the +earth in exactly the same way as the whole space ship, and so it would appear +to remain suspended before them in space. Ït is very interesting that this Íorce +1s eœøctu proportional to the mass with great precision, because 1Ý it were not +exactly proportional there would be some effect by which inertia and weight +would difer. The absence of such an efect has been checked with great accuracy +by an experiment done fñrst by Eötvös in 1909 and more recently by Dicke. Eor +all substances tried, the masses and weights are exactly proportional within 1 +part in 1,000,000,000, or less. This is a remarkable experiment. +7-8 Gravity and relativity +Another topic deserving discussion is Einstein's modification of Newton°s law +OŸ gravitation. In spite of all the excitement it created, Newton's law of gravitation +is not correctl It§ was modifed by Einstein to take into account the theory of +relativity. According to NÑewton, the gravitational efect is instantaneous, that +1s, IŸ we were to move a mass, we would at onece feel a new force because of the +new position of that mass; by such means we could send signals at infinite speed. +--- Trang 159 --- +Binstein advanced arguments which suggest that we cannot send signals ƒaster +than the specd oƒ light, so the law oŸ gravitation must be wrong. By correcting +1t to ©ake the delays into account, we have a new law, called Einstein's law of +gravitation. One feature of this new law which is quite easy to understand is this: +In the Einstein relativity theory, anything which has energy has mass—mass in +the sense that it is attracted gravitationally. Even light, which has an energy, +has a “mass.” When a light beam, which has energy ¡n it, comes past the sun +there is an attraction on i% by the sun. 'Phus the light does not go straight, but is +defected. During the eclipse of the sun, for example, the stars which are around +the sun should appear displaced from where they would be ïif the sun were not +there, and this has been observed. +Finally, let us compare gravitation with other theories. In recent years we have +discovered that all mass is made of tiny particles and that there are several kinds +of interactions, such as nuclear forces, etc. None of these nuclear or electrical +forces has yet been found to explain gravitation. 'The quantum-mechanical aspects +Of nature have not yet been carried over to gravitation. When the scale is sO +small that we need the quantum efects, the gravitational efects are so weak +that the need for a quantum theory of gravitation has not yet developed. Ôn +the other hand, for consistency in our physical theories it would be important to +see whether Newton's law modified to Einstein”s law can be further modifed to +be consistent with the uncertainty principle. 'Phis last modification has not yet +been completed. +--- Trang 160 --- +JMoffort +8-1 Description of motion +In order to fnd the laws governing the various changes that take place in +bodies as time goes on, we must be able to đescribe the changes and have some +way to record them. 'Phe simplest change to observe in a body is the apparent +change in its position with time, which we call motion. Let us consider some solid +object with a permanent mark, which we shall call a point, that we can observe. +We shall discuss the motion of the little marker, which might be the radiator cap +of an automobile or the center of a falling baill, and shall try to describe the fact +that it moves and how it moves. +These examples may sound trivial, bu many subtleties enter into the descrip- +tion of change. Some changes are more difficult to describe than the motion of a +point on a solid object, for example the speed of drift of a cloud that is drifting +very slowly, but rapidly forming or evaporating, or the change of a womans mind. +W© do not know a simple way to analyze a change of mind, but since the cloud +can be represented or described by many molecules, perhaps we can describe the +motion of the cloud in principle by describing the motion of all its individual +molecules. Likewise, perhaps even the changes in the mind may have a parallel +in changes of the atoms inside the brain, but we have no such knowledge yet. +At any rate, that is why we begin with the motion of points; perhaps we +should think of them as atom, but it is probably better to be more rough in +the beginning and simply to think of some kind of small obJects—smaill, that +1s, compared with the distance moved. For instance, in describing the motion +of a car that is going a hundred miles, we do not have to distinguish bebween +the front and the back of the car. To be sure, there are slight diferences, but +for rough purposes we say “the car,” and likewise it does not matter that our +points are not absolute points; for our present purposes it is not necessary to be +extremely precise. Also, while we take a first look at this subjecb we are goïng +--- Trang 161 --- +Table 8-1 E- 25000 +# (min) | s (ft) # 2oooo +0 0 D +1 1200 ụ 15000 +2 4000 3 +3 9000 ụị 10000 +4 9500 š +b) 9600 b 5000 +6 13000 5 +7 18000 2A4 6 8 q0 +§ 23500 TIME IN MINUTES +9 24000 Fig. 8-1. Graph of distance versus time for the car. +to forget about the three dimensions of the world. We shall just concentrate +on moving in one direction, as in a car on one road. We shall return to three +dimensions after we see how to describe motion in one dimension. Ñow, you may +say, ““This ¡is all some kind of trivia,” and indeed it is. How can we describe such a +one-dimensional motion——let us say, of a car? Nothing could be simpler. Among +many possible ways, one would be the following. To determine the position of +the car at diferent times, we measure its distance from the starting point and +record all the observations. In 'Table S-1, s represents the distance of the car, In +feet, from the starting point, and # represents the time in minutes. 'Phe first line +in the table represents zero distance and zero time—the car has not started yet. +After one minute it has started and has gone 1200 feet. Then in two minutes, it +goes farther——notice that it picked up more distance in the second minute——1§ +has accelerated; but something happened between 3 and 4 and even more so +at 5—it stopped at a light perhaps? Then ït speeds up again and goes 13,000 feet +by the end of 6 minutes, 18,000 feet at the end of 7 minutes, and 23,500 feet in +8 minutes; at 9 minutes it has advanced to only 24,000 feet, because in the last +minute it was stopped by a cop. +That is one way to describe the motion. Another way is by means of a +graph. H we plot the time horizontally and the distance vertically, we obtain a +curve something like that shown in Eig. 8-1. As the tỉme increases, the đistance +Increases, at first very slowly and then more rapidly, and very slowly again for a +little while at 4 minutes; then it increases again for a few minutes and ñnally, +at 9 minutes, appears to have stopped increasing. 'Phese observations can be +--- Trang 162 --- +Table 8-2 th +Z 300 +f (sec) | s (ft) Ế +0 0 £ 200 +1 16 Ờ +2 64 Ê 100 +3 144 ễ +5 400 : TIME IN SECONDS ˆ ; +Fig. 8-2. Graph of distance versus time for a falling +made from the graph, without a table. Obviously, for a complete description +one would have to know where the car is at the half-minute marks, too, but we +suppose that the graph means something, that the car has some position at all +the intermediate times. +'The motion of a car is complicated. Eor another example we take something +that moves in a simpler manner, following more simple laws: a falling ball. +Table 8-2 gives the time in seconds and the distance in feet for a falling body. +At zero seconds the ball starts out at zero feet, and at the end of 1 second it +has fallen 16 feet. At the end of 2 seconds, it has fallen 64 feet, at the end of +ở seconds, 14⁄4 feet, and so on; ïf the tabulated numbers are plotted, we get the +nice parabolic curve shown in Fig. 8-2. The formula for this curve can be written +s= 16. (8.1) +This formula enables us to calculate the distances at any time. You might say +there ought to be a formula for the first graph too. Actually, one may write such +a formula abstractly, as +s=ƒ/(0, (8.2) +meaning that s is some quantity depending on ý or, in mathematical phraseology, +ø is a function of . Since we do not know what the function is, there is no way +we can write it in defnite algebraic form. +We have now seen ÿwo examples of motion, adequately described with very +simple ideas, no subtleties. However, there øre subtleties—several of them. In +--- Trang 163 --- +the first place, what do we mean by f£#ne and space? It turns out that these deep +philosophical questions have to be analyzed very carefully in physics, and this +1s not so easy to do. 'Phe theory of relativity shows that our ideas of space and +time are not as simple as one might think at fñrst sight. However, for our present +purposes, for the accuracy that we need at first, we need not be very careful +about defning things precisely. Perhaps you say, “Phat's a terrible thing—I +learned that in seience we have to defñne cuerwthing precisely.” We cannot defne +gmything preciselyl TẾ we attempt to, we get into that paralysis of thought that +comes to philosophers, who sit opposite each other, one saying to the other, “You +don”? know what you are talking about!” “The second one says, “What do you +mean by knou? What do you mean by falking? What do you mean by ow#,” +and so on. In order to be able to talk constructively, we Just have to agree that +we are talking about roughly the same thing. You know as much about time as +we need for the present, but remember that there are some subtleties that have +to be discussed; we shall discuss them later. +Another subtlety involved, and already mentioned, is that ¡t should be possible +to imagine that the moving point we are observing is always located somewhere. +(Of course when we are looking at it, there it is, but maybe when we look away it +isn't there.) It turns out that in the motion of atoms, that idea also is false—we +cannot fnd a marker on an atom and watch it move. 'Phat subtlety we shall +have to get around in quantum mechanies. But we are first go¡ing to learn what +the problems are before introducing the complications, and ¿hen we shall be in a +better position to make corrections, in the light of the more recent knowledge +of the subject. We shall, therefore, take a simple point of view about time and +space. We know what these concepts are in a rough way, and those who have +driven a car know what speed means. +8-2 Speed +ven though we know roughly what “speed” means, there are still some +rather deep subtleties; consider that the learned Greeks were never able to +adequately describe problems involving velocity. The subtlety comes when we try +to comprehend exactly what is meant by “speed.” The Greeks got very confused +about this, and a new branch of mathematies had to be discovered beyond the +geometry and algebra of the Greeks, Arabs, and Babylonians. As an illustration +of the dificulty, try to solve this problem by sheer algebra: A balloon is being +infated so that the volume of the balloon is increasing at the rate of 100 em” +--- Trang 164 --- +per second; at what speed is the radius inereasing when the volume is 1000 em”? +'The Greeks were somewhat confused by such problems, being helped, of course, +by some very confusing Greeks. To show that there were difficulties in reasoning +about speed at the time, Zeno produced a large number of paradoxes, of which +we shall mention one to illustrate his point that there are obvious dificulties in +thinking about motion. “Listen,” he says, “to the following argument: Achilles +runs 10 times as fast as a tortoise, nevertheless he can never catch the tortoise. +For, suppose that they start in a race where the tortoise is 100 meters ahead +of Achilles; then when Achilles has run the 100 meters to the place where the +tortoise was, the tortoise has proceeded 10 meters, having run one-tenth as fast. +NÑow, Achiles has to run another 10 meters to catch up with the tortoise, but on +arriving at the end of that run, he ñnds that the tortoise is still 1 meter ahead +of him; running another meter, he fnds the tortoise 10 centimeters ahead, and +SO On, ød ?nƒfinøtum. Pherefore, at any moment the tortoise is always ahead of +Achilles and Achilles can never catch, up with the tortoise.” What is wrong with +that? It is that a finite amount of time can be divided into an infnite number of +pieces, just as a length of line can be divided into an infnite number of pieces +by dividing repeatedly by bwo. And so, although there are an infnite number +Of sbeps (in the argument) to the point at which Achilles reaches the tortoise, +it doesnt mean that there is an infnite amount of #me. We can see from this +example that there are indeed some subtleties in reasoning about speed. +In order to get to the subtleties in a clearer fashion, we remind you of a +Jjoke which you surely must have heard. At the point where the lady in the car +is caught by a cop, the cop comes up to her and says, “Lady, you were goiỉng +60 miles an hour!” She says, “hat ”s impossible, sir, I was travelling for only +seven minutes. It is ridiculous—how can I go 60 miles an hour when Ï wasn't +goïng an hour?” How would you answer her 1Ý you were the cop? Of course, If +you were really the cop, then no subtleties are involved; it is very sỉimple: you say, +“Tell that to the judge!” But let us suppose that we do not have that escape and +we make a more honest, intellectual attack on the problem, and try to explain to +this lady what we mean by the idea that she was going 60 miles an hour. .Jjust +what do we mean? We say, “What we mean, lady, is this: if you kept on going +the same way as you are going now, in the next hour you would go 60 miles.” She +could say, “Well, my foot was off the accelerator and the car was sÌlowing down, +so 1f Ï kept on going that way it would not go 60 miles” Ôr consider the falling +ball and suppose we want to know its speed at the time three seconds ïf the ball +kept on going the way it is going. What does that mean——kept on øccelerating, +--- Trang 165 --- +going faster? No—kept on goiỉng with the same øeloc#u. But that is what we are +trying to definel For if the ball keeps on going the way it is goïng, it will just +keep on going the way i% is going. Thus we need to defñne the velocity better. +'What has to be kept the same? 'Phe lady can also argue this way: “If I kept on +going the way m goiïng for one more hour, Ï would run into that wall at the end +of the streetl” It is not so easy to say what we mean. +Many physicists think that measurement is the only defnition of anything. +Obviously, then, we should use the instrument that measures the speed——the +speedometer——=and say, “Look, lady, your speedometer reads 60.” So she says, +“My speedometer is broken and didn”t read at all” Does that mean the car is +standing still? We believe that there is something to measure before we build +the speedometer. Only then can we say, for example, “The speedometer isn'$ +working right,” or “the speedometer is broken.” That would be a meaningless +sentence ïf the velocity had no meaning independent of the speedometer. So we +have in our minds, obviously, an idea that is independent of the speedometer, +and the speedometer is meant only to measure this idea. So let us see IŸ we can +get a better defnition of the idea. We say, “Yes, of course, before you went an +hour, you would hít that wall, but if you went one second, you would go 88 feet; +lady, you were going 88 feet per second, and ïf you kept on going, the next second +it would be 88 feet, and the wall down there is farther away than that.” She says, +“Ves, but there's no law against going 88 feet per secondl 'There ¡is only a law +against going 60 miles an hour.” “But,” we reply, “it's the same thing.” If it 2s +the same thing, it should not be necessary to go into this cireumlocution about +88 feet per second. In fact, the falling ball could not keep going the same way +even one second because it would be changing speed, and we shall have to delne +speed somehow. +Now we seem to be getting on the right track; it goes something like this: If +the lady kept on goïing for another 1/1000 of an hour, she would go 1/1000 of +60 miles. In other words, she does not have to keep on going for the whole hour; +the point is that for ø mmornent she is goïng at that speed. Now what that means +is that if she went just a little bit more in time, the extra distance she goes would +be the same as that of a car that goes at a s‡eady speed of 60 miles an hour. +Perhaps the idea of the 88 feet per second is right; we see how far she went In +the last second, divide by 88 feet, and ïf it comes out 1 the speed was 60 miles +an hour. In other words, we can fnd the speed in this way: We ask, how far +do we go in a very short time? We divide that distance by the time, and that +gives the speed. But the time should be made as short as possible, the shorter +--- Trang 166 --- +the better, because some change could take place during that time. If we take +the time of a falling body as an hour, the idea is ridiculous. lf we take it as a +second, the result is pretty good for a car, because there is not much change in +speed, but not for a falling body; so in order to get the speed more and more +accurately, we should take a smaller and smaller time interval. What we should +do is take a millionth oŸ a second, and divide that distance by a millionth of a +second. “The result gives the distance per second, which is what we mean by the +velocity, so we can defñne it that way. That is a successful answer for the lady, or +rather, that is the defnition that we are going to se. +The foregoing defñnition involves a new idea, an idea that was not available to +the Greeks in a general form. That idea was to take an ?nfinitesimal distance and +the corresponding ?nfin2tesimal time, form the ratio, and watch what happens +to that ratio as the time that we use gets smaller and smaller and smaller. In +other words, take a limit of the distance travelled divided by the time required, +as the time taken gets smaller and smaller, ød ?nfimuitưm. 'Phis idea was invented +by Newton and by Leibniz, independently, and is the beginning of a new branch +of mathematics, called the djferential calculus. Calculus was invented in order +to describe motion, and its first application was to the problem of defning what +is meant by going “60 miles an hour.” +Let us try to defne velocity a little better. Suppose that in a short time, c, +the car or other body goes a short distance z; then the velocity, 0, is defned as +U = đc, +an approximation that becomes better and better as the is taken smaller and +smaller. If a mathematical expression ¡is desired, we can say that the velocity +cquals the limit as the is made to go smaller and smaller in the expression #/c, +ø = lim `, (8.3) +ec>0 € +We cannot do the same thing with the lady in the car, because the table is +Iincomplete. We know only where she was at intervals of one minute; we can +get a rough idea that she was going 5000 ft/min during the 7th minute, but we +do not know, at exactly the moment 7 minutes, whether she had been speeding +up and the speed was 4900 ft/min at the beginning of the 6th minute, and is +now 5100 ft/min, or something else, because we do not have the exact details in +between. So only If the table were completed with an infnite number oŸ entries +could we really calculate the velocity from such a table. On the other hand, +--- Trang 167 --- +when we have a complete mathematical formula, as in the case of a falling body +(Eaq. 8.1), then it is possible to calculate the velocity, because we can calculate +the position at any time whatsoever. +Let us take as an example the problem of determining the velocity of the +falling ball at the particular time 5 seconds. Ône way to do this is to see from +Table 8-2 what it dịd in the 5th second; it went 400 — 256 = 144 Ít, so ï is goïng +144 ft/sec; however, that is wrong, because the speed is changing; øn the œuerage +1€ is 144 ft/sec during this interval, but the baill is speeding up and is really goïng +faster than 144 ft/sec. We want to lnd out ezactflU hou ƒast. The technique +involved in this process is the following: We know where the ball was at ð sec. +At 5.1 sec, the distance that it has gone all together is 16(5.1)2 = 416.16 ft (see +Eq. 8.1). At 5 sec it had already fallen 400 ft; in the last tenth of a second it +fell 416.16 — 400 = 16.16 ft. Since 16.16 ft in 0.1 sec is the same as 161.6 ft/sec, +that is the speed more or less, but it is not exactly correct. Is that the speed +at 5, or at 5.1, or halfway bebween at 5.05 sec, or when 7s that the speed? Never +mind—the problem was to fñnd the speed ø‡ 5 seconds, and we do not have exactly +that; we have to do a better job. So, we take one-thousandth of a second more +than ð sec, or 5.001 sec, and calculate the total fall as +s = 16(5.001)Ÿ = 16(25.010001) = 400.160016 ft. +In the last 0.001 sec the ball fell 0.160016 ft, and if we divide this number by +0.001 sec we obtain the speed as 160.016 ft/sec. That is closer, very close, but it is +siill not exact. Tt should now be evident what we must do to ñnd the speed exactly. +To perform the mathematics we state the problem a little more abstractly: to +find the velocity at a special time, ứọ, which in the original problem was ð sec. +Now the distance at fọ, which we call sọ, is 16fã, or 400 ft in this case. In order +to ñnd the velocity, we ask, “At the time £o + (a little bit), or fo +, where is +the body?” The new position is 16(fo + e)2 = 16fã + 32foe + 16c?. So it is farther +along than it was before, because before it was only 16/á. Thịis distance we shall +call so + (a little bit more), or sg + # (ïŸ z is the extra bit). Now if we subtract +the distance at ứo from the distance at fọ + c, we get z, the extra distance gone, +as # = 32fo -c-L 16e2. Qur first approximation to the veloeity is +b= h = 39fo + 16c. (8.4) +The true velocity is the value of this ratio, z/c, when e becomes vanishingly small. +In other words, after forming the ratio, we take the limit as e gets smaller and +--- Trang 168 --- +smaller, that is, approaches 0. “The equation reduces to, +9U (at time to) = 32to. +In our problem, #o = ð sec, so the solution is ø = 32 x 5 = 160 ft/sec. A few lines +above, where we took c as 0.1 and 0.001 sec successively, the value we got for 0 +was a little more than this, but now we see that the actual velocity is precisely +160 ft/sec. +8-3 Speed as a derivative +The procedure we have just carried out is performed so often in mathematics +that for convenience special notations have been assigned to our quantities +and z. In this notation, the used above becomes A£ and # becomes As. 'This +Af means “an extra bit of £,” and carries an implication that it can be made +smaller. The prefx A is not a multiplier, any more than sỉn Ø means s-i -n - Ø—it +simply defines a tỉme increment, and reminds us of its special character. As has +an analogous meaning for the distance s. Since A is not a factor, it cannot be +cancelled in the ratio As/Af to give s/f, any more than the ratio sin Ø/sin 20 +can be reduced to 1/2 by cancellation. In this notation, velocity is equal to the +limit of As/Af when Af gets smailler, or += lim —. 8.5 +_—- 5) +Thịis is really the same as our previous expression (8.3) with e and z, but it has +the advantage of showing that something is changing, and it keeps track of what +is changing. +Incidentally, to a good approximation we have another law, which says that +the change in distance of a moving point is the velocity times the time interval, +or As =0 Af. Thịs statement is true only if the velocity is not changing during +that time interval, and this condition is true only in the limit as Af goes to 0. +Physicists like to write it đs = 0 đf, because by đ£ they mean Af in circumstances +in which it is very small; with this understanding, the expression is valid to a close +approximation. If A£ is too long, the velocity might change during the interval, +and the approximation would become less accurate. Eor a time đÝ, approaching +zero, ds = 0 đf precisely. In this notation we can write (S.5) as +h As ds += lim -_— =-—. +T— Arso AE — đi +--- Trang 169 --- +The quantity đs/đ£ which we found above is called the “derivative of s with +respect to £” (this language helps to keep track of what was changed), and the +complicated process of ñnding ït is called ñnding a derivative, or diferentiating. +The đs's and đf£s which appear separately are called đjfereniials. To familiarize +you with the words, we say we found the derivative of the funetion 162, or the +derivative (with respect to £) of 16/2 is 32. When we get used to the words, the +ideas are more easily understood. Eor practice, let us fnd the derivative of a more +complicated funetion. We shall consider the formula s = 4£ + B + Œ, which +might describe the motion of a point. The letters 4, Ö, and Œ represent constant +numbers, as in the familiar general form oŸ a quadratic equation. Starting from +the formula for the motion, we wish to ñnd the velocity at any time. To ñnd the +velocity in the more elegant manner, we change # to ý + A# and note that s is +then changed to s-Ƒ some As; then we find the As in terms of A7. That is to say, +s+ As = A(+ At)Ỷ+ B(+ At)+Œ += Af + Bt+ CƠ +3Af? At+ BAt+3At(A9)? + A(AĐ, +but since +s= Af + Bt + C, +we fñnd that +As=3A/? At+ BAt+3At(A93 + A(A9. +But we do not want As—we want As divided by A¿. We divide the preceding +cquation by A£, getting +As 2 2 +Ar E34? + B+3AH(A0) + A(A0). +As Af goes toward 0 the limit of As/Af is đs/đf and is equal to +—=3A4/+D. +'This is the fundamental process of calculus, diferentiating functions. he process +1s even more simple than it appears. Observe that when these expansions contain +any term with a square or a cube or any higher power of A£, such terms may be +dropped at once, since they will go to 0 when the limit is taken. After a little +practice the process gets easier because one knows what to leave out. 'There are +many rules or formulas for differentiating various types of functions. 'Phese can +be memorized, or can be found in tables. A short list is found in Table 8-3. +--- Trang 170 --- +Table 8-3. A Short Table of Derivatives +8, tu, 0, t are arbitrary functions of ; ø, b, c, and m are arbitrary constants +Punction Derivative +=í" —=ni” +S—= Cu đs —=C€C đụ +có dt đt +s=ur+t0+1p+ ¬- `... +có dt dị. dị dt +s=c hiền 0 +s=tut te LG œdu bdu c du +có dc C\u dc dc. +” đt +8-4 Distance as an integral +Now we have to discuss the inverse problem. Suppose that instead of a table of +distances, we have a table of speeds at diferent times, starting from zero. Eor the +falling ball, such speeds and tỉmes are shown in Table 8-4. A similar table could +be constructed for the velocity of the car, by recording the speedometer reading +every minute or half-minute. If we know how fast the car is goiïng at any tỉme, +can we determine how far it goes? This problem is just the inverse of the one +solved above; we are given the velocity and asked to ñnd the distance. How can +we find the distance if we know the speed? If the speed of the car is not constant, +and the lady goes sixty miles an hour for a moment, then slows down, speeds up, +Table 8-4 +Velocity of a Falling Ball +£ (sec) | ® (ñ/sec) +--- Trang 171 --- +and so on, how can we determine how far she has gone? 'Phat is easy. We use +the same idea, and express the distance in terms of infnitesimals. Let us say, “In +the fñrst second her speed was such and such, and from the formula As = Af +we can calculate how far the car went the first second at that speed.” Now ín the +next second her speed is nearly the same, but slightly diferent; we can calculate +how far she went in the next second by taking the new speed times the time. +W©e proceed similarly for each second, to the end of the run. We now have a +number of little distances, and the total distance will be the sum of all these little +pieces. Thhat is, the distance will be the sum of the velocities times the times, +or s= 0 Af, where the Greek letter ồ” (sigma) is used to denote addition. To +be more precise, it is the sum of the velocity at a certain time, let us say the +¡-bh time, multipled by A¿. +s= » u(;) At. (8.6) +The rule for the times is that f;¿‡¡ = f¿ + A. However, the distance we obtain by +this method will not be correct, because the velocity changes during the time +interval A¿. TH we take the times short enough, the sum is precise, so we take +them smaller and smaller until we obtain the desired accuracy. 'Phe true s is +s= Am. » u(;) At. (8.7) +The mathematicians have invented a symbol for this limit, analogous to the +symbol for the diferemial. The A turns into a đ to remind us that the time is as +small as it can be; the velocity is then called 0 at the time f, and the addition is +written as a sum with a great “s,” ƒ (from the Latin sưmzna), which has become +distorted and is now unfortunately just called an integral sign. Thus we write +s= Tao đt. (8.8) +This process of adding all these terms together is called integration, and it 1s +the opposite process to diferentiation. "The derivative of this integral 1s 0, SO +one operabor (đ) undoes the other (ƒ). One can get formulas for integrals by +taking the formulas for derivatives and running them backwards, because they are +related to each other inversely. 'Thus one can work out his own table of integrals +--- Trang 172 --- +by diferentiating all sorts of functions. For every formula with a diferential, we +get an integral formula if we turn it around. +lvery function can be diferentiated analytically, i.e., the process can be +carried out algebraically, and leads to a defñnite function. But it is not possible +in a simple manner to write an analytical value for any integral at will. You can +calculate it, for instance, by doing the above sum, and then doïng it again with a +fñner interval A‡ and again with a fñiner interval until you have it nearly right. In +general, given some particular function, ït is not possible to find, analytically, what +the integral is. One may always try to ñnd a function which, when diferentiated, +gives some desired function; but one may not fnd ït, and it may not exist, In +the sense oŸ being expressible in terms oŸ functions that have already been given +names. +8-5 Acceleration +The next step in developing the equations of motion is to introduce another +idea which goes beyond the concept of velocity to that oŸ change of velocity, +and we now ask, “How does the velocity change?” In previous chapters we have +discussed cases in which forces produce changes in velocity. You may have heard +with great excitement about some car that can get from rest to 60 miles an hour +in ten seconds at. From such a performance we can see how fast the speed +changes, but only on the average. What we shall now discuss is the next level of +complexity, which is how fast the velocity is changing. In other words, by how +many feet per second does the velocity change in a second, that is, how many +feet per second, per second? We previously derived the formula for the velocity +of a falling body as = 32f, which is charted in Table 8-4, and now we want to +fnd out how much the velocity changes per second; this quantity ¡is called the +acceleration. +Acceleration is defined as the time rate of change of velocity. From the +preceding discussion we know enough already to write the acceleration as the +derivative đu /đứ, in the same way that the velocity is the derivative of the distance. +Tf we now diferentiate the formula = 32 we obtain, for a falling body, +a= TE 32. (8.9) +[To diferentiate the term 32 we can utilize the result obtained in a previous +problem, where we found that the derivative of Đứ is simply Ö (a constant). So +--- Trang 173 --- +by letting = 32, we have at once that the derivative of 32 is 32.] This means +that the velocity of a falling body is changing by 32 feet per second, per second +always. We also see from Table 8-4 that the velocity increases by 32 ft/sec in each +second. 'Phis is a very simple case, for accelerations are usually not constant. The +reason the acceleration is constant here is that the force on the falling body is +constant, and Newton”s law says that the acceleration is proportional to the force. +As a further example, let us find the acceleration in the problem we have +already solved for the velocity. Starting with +s= Af + Bt+ +we obtained, for ø = ds/dt, +0 =3Ai/2 + B. +Since acceleration is the derivative of the velocity with respect to the time, we +need to diferentiate the last expression above. Recall the rule that the derivative +of the two terms on the right equals the sum of the derivatives of the individual +terms. To diferentiate the first of these terms, instead of going through the +fundamental process again we note that we have already difÑferentiated a quadratic +term when we differentiated 16/2, and the efect was to double the numerical +coefficient and change the ¿2 to #; let us assume that the same thing will happen +this time, and you can check the result yourself. The derivative of 34/2 will +then be 64. Next we diferentiate , a constant term; but by a rule stated +previously, the derivative of Ö is zero; hence this term contributes nothing to +the acceleration. The final result, therefore, is ø = du/dt = 6At. +For reference, we state two very useful formulas, which can be obtained by +integration. If a body starts from rest and moves with a constant acceleration, ø, +its velocity 0 at any time £ is given by +U = gỉ. +The distance it covers in the same tỉme is +s= 3 gt2. +'Various mathematical notations are used in writing derivatives. 5ince velocity +1s ds/dt and acceleration is the time derivative of the velocity, we can also write +d (ds d2s +G=_— —— = _—xY (8.10) +đt \ dị d2 +which are common ways of writing a second derivative. +--- Trang 174 --- +W© have another law that the velocity is equal to the integral of the acceleration. +This is just the opposite of a = du/di; we have already seen that distance is +the integral of the velocity, so distance can be found by twice integrating the +acceleration. +In the foregoing discussion the motion was in only one dimension, and +space permits only a brief discussion of motion in three dimensions. Consider a +particle ? which moves in three dimensions in any manner whatsoever. At the +beginning of this chapter, we opened our discussion of the one-dimensional case +of a moving car by observing the distance of the car from its starting point at +various times. We then discussed velocity in terms of changes of these distances +with time, and acceleration in terms of changes In velocity. We can treat three- +dimensional motion analogously. It will be simpler to illustrate the motion on a +two-dimensional diagram, and then extend the ideas to three dimensions. We +establish a païir of axes at right angles to each other, and determine the position +of the particle at any moment by measuring how far it is from each of the two +axes. Thus each position is given in terms of an z-distance and a z-distance, and +the motion can be described by constructing a table in which both these distances +are given as functions of time. (Extension of this process to three dimensions +requires only another axis, at ripght angles to the first two, and measuring a third +distance, the z-distance. The distances are now measured from coordinate pÌanes +instead of lines.) Having constructed a table with z- and -distances, how can +we determine the velocity? We first fnd the components of velocity in each +direction. 'Phe horizontal part of the velocity, or z-component, is the derivative +of the z-distance with respect to the time, or +U„ = dø/dt. (8.11) +Similarly, the vertical part of the velocity, or -component, is +uụ = dụ /dt. (8.12) +In the third dimension, +Uy = đz/dl. (8.13) +Now, given the components of velocity, how can we fnd the velocity along +the actual path of motion? In the two-dimensional case, consider two successive +positions of the particle, separated by a short distance As and a short time +--- Trang 175 --- +M As A/(Ax)2 + (Ay)2 +Ayv/Atf— XỬ +Ax#ø#v„At +Fig. 8-3. Description of the motion of a body in two dimensions and +the computation of its velocity. +interval f¿ — fq = Ai. In the time A£ the particle moves horizontally a dis- +tance Az % 0„ Af, and vertically a distance A¿ uy At. (The symbol “+” is +read “is approximately.”) The actual distance moved is approximately +Asxz V(Az)2 + (Aø)2, (8.14) +as shown in EFig. 8-3. The approximate velocity during this interval can be +obtained by dividing by A£ and by letting A# go to 0, as at the beginning of the +chapter. We then get the velocity as +U= T= V(dz/đdt)? + (dụ/đE)? = vu + 0. (8.15) +For three dimensions the result is +Đ= \(02 + 02 + 0Ẻ. (8.16) +In the same way as we defned velocities, we can delne accelerations: we have +an #-component of acceleration ø„, which is the derivative of ø„, the z-component +of the velocity (that is, a„ = đ?z/d/2, the second derivative of z with respect +to £), and so on. +Let us consider one nice example of compound motion in a plane. We shall +take a motion in which a ball moves horizontally with a constant velocity w, and +at the same time goes vertically downward with a constant acceleration —g; what +is the motion? We can say d#/dt = 0u„ = u. Since the velocity 0x is constant, +# = tứ, (8.17) +--- Trang 176 --- +and since the downward acceleration —gø 1s constant, the distance # the objec +falls can be written as +ụ= —39Ÿ. (8.18) +'What is the curve of its path, i.e., what is the relation between and z? We can +eliminate £ from Eq. (8.18), since ý = z/u. When we make this substitution we +fñnd that : +This relation between ø and z may be considered as the equation of the path of +the moving ball. When this equation is plotted we obtain a curve that ¡is called a +parabola; any freely falling body that is shot out in any direction will travel in a +parabola, as shown In Fig. 8-4. +Fig. 8-4. The parabola described by a falling body with an initial +horizontal velocity. +--- Trang 177 --- +NWeosrfore s EL«ttfs ©œŸ` ÏÌggTt(i110fS +9-1 Momentum and force +The discovery of the laws of dynamies, or the laws of motion, was a dramatic +moment in the history ofscience. Before Newton's time, the motions of things like +the planets were a mystery, but after Newton there was complete understanding. +ven the slight deviations from Kepler”s laws, due to the perturbations of the +planets, were computable. "The motions of pendulums, oscillators with springs +and weights in them, and so on, could all be analyzed completely after Newton”s +laws were enunciated. So it is with this chapter: before this chapter we could +not calculate how a mass on a spring would move; much less could we calculate +the perturbations on the planet Uranus due to Jupiter and Saturn. After this +chapter we 0 be able to compute not only the motion of the oscillating mass, +but also the perturbations on the planet Ủranus produced by Jupiter and Saturnl +Galileo made a great advance in the understanding of motion when he dis- +coverecd the prznciple oƒ tmnertia: 1f an objJect is left alone, is not disturbed, 1$ +continues to move with a constant velocity in a straight line if it was originally +moving, or it continues to stand still iŸ it was just standing still. Of course this +never appears to be the case in nature, for if we slide a block across a table it +stops, but that is because it is no left to itself—it is rubbing against the table. +Tt required a certain imagination to ñnd the right rule, and that imagination was +supplied by Galileo. +Of course, the next thing which is needed is a rule for fnding how an object +changes 1s speed iŸ something ?s afecting it. Thhat is, the contribution oŸ Ñewton. +Newton wrote down three laws: The First Law was a mere restatement of the +Galilean principle of inertia just described. The 5econd baw gave a specifc way +of determining how the velocity changes under diferent infuences called ƒorces. +The Third Law describes the forces to some extent, and we shall discuss that +at another time. Here we shall discuss only the Second Law, which asserts that +the motion of an object is changed by forces in this way: the time-rate-oJ-change +--- Trang 178 --- +oƒ a quantitụ called momentum is proportional to the [orce. We shall state thìs +mmathematically shortly, but let us first explain the idea. +Momentum is not the same as 0elocit. A lot of words are used in physics, +and they all have precise meanings in physics, although they may not have such +precise meanings in everyday language. Momentum is an example, and we must +defne it precisely. IÝ we exert a certain push with our arms on an object that +1s light, it moves easily; if we push Just as hard on another object that is much +heavier in the usual sense, then it moves much less rapidly. Actually, we must +change the words from “light” and “heavy” to Ïess rnass?ue and more 1ndssiue, +because there is a diference to be understood between the 0øe¿ghf of an object +and its ?nerta. (How hard ïE is to get it goïng is one thing, and how much it +weighs is something else.) Weight and inertia are proportional, and on the earth”s +surface are often taken to be numerically equal, which causes a certain confusion +to the student. Ôn Mars, weights would be diferent but the amount of force +needed to overcome inertia would be the same. +We use the term rmøss as a quantitative measure of inertia, and we may +mmeasure mass, for example, by swinging an object in a circle at a certain speed +and measuring how much force we need to keep it in the circle. In this way we +ñnd a certain quantity of mass for every object. Now the mmơmentum of an object +1s a product of bwo parts: Its zmass and its 0elocit. Thus Newton”s Second Law +may be written mathematically this way: +t= qi0n9). (9.1) +Now there are several points to be considered. In writing down any law such as +this, we use many intuitive ideas, Implications, and assumptions which are at +first combined approximately into our “law.” Later we may have to come back +and study ïn greater detail exactly what each term means, but if we try to do this +too soon we shall get confused. Thhus at the beginning we take several things Íor +granted. First, that the mass of an object is consfand; it isn't really, but we shall +start out with the NÑewtonian approximation that mass is constant, the same all +the time, and that, further, when we put two objects together, their masses ødd. +These ideas were of course Implied by NÑewton when he wrote his equation, for +otherwise 1t is meaningless. For example, suppose the mass varied inversely as +the velocity; then the momentum would ne0er chơnge in any circumstanece, so +the law means nothing unless you know how the mass changes with velocity. At +first we say, ? does not chơœngc. +--- Trang 179 --- +Then there are some implications concerning force. Âs a rough approximation +we think of force as a kind of push or pull that we make with our museles, but +we can defñne it more accurately now that we have this law of motion. The most +Iimportant thing to realize is that this relationship involves not only changes in +the rmagnitude of the momentum or of the velocity but also in their đứccfion. TẾ +the mass is constant, then Eq. (9.1) can also be written as +t'=m - = ma. (9.2) +The acceleration ø is the rate of change of the velocity, and Newton°s Second +Law says more than that the efect of a given force varies inversely as the mass; +1t says also that the đirection of the change in the velocity and the đireclion of +the force are the same. Thus we must understand that a change in a velocity, or +an acceleration, has a wider meaning than in common language: The velocity +of a moving object can change by its speeding up, slowing down (when it sÌows +down, we say it accelerates with a negative acceleration), or changing its direction +of motion. An acceleration at right angles to the velocity was discussed in +Chapter 7. There we saw that an object moving in a circle of radius with a +certain speed œ along the cirele falls away from a straightline path by a distance +equal to 2(02/R)£2 if £ is very small. Thus the formula for acceleration at right +angles to the motion is +a = 02/R, (9.3) +and a force at right angles to the velocity will cause an objecE to move in a curved +path whose radius of curvature can be found by dividing the force by the mass +to get the acceleration, and then using (9.3). +9-2 Speed and velocity +In order to make our language more precise, we shall make one further +defnition in our use of the words speed and 0elocit. Ordinarily we think of speed +and velocity as being the same, and in ordinary language they are the same. But +in physics we have taken advantage of the fact that there øre two words and have +chosen to use them to distinguish two ideas. We carefully distinguish velocity, +which has both magnitude and direction, from speed, which we choose to mean +the magnitude oŸ the velocity, but which does not include the direction. We can +formulate this more precisely by describing how the z-, -, and z-coordinates +--- Trang 180 --- +F4 +DR — TT . +TƯ INNG +“~—-- | +I “ˆ. +SA AV l/ +/Ax___W +Fig. 9-1. A small displacement of an object. +of an object change with time. Suppose, for example, that at a certain instant +an object is moving as shown in Fig. 9-1. In a given small interval of time Af +it will move a certain distance Az in the zø-direction, A2 in the -direction, +and Az ïn the z-direction. The total efect of these three coordinate changes is a +displacement As along the diagonal of a parallelepiped whose sides are Az, A#, +and Az. In terms of the velocity, the displacement Az is the #-component of the +velocity times A#, and similarly for A¿ and Az: +Az = uy At, AU = uy At, Az =0; At. (9.4) +9-3 Components of velocity, acceleration, and force +In Eq. (9.4) te heœue resolued the uelocitụ imito components by telling how fast +the object is moving in the #ø-direction, the -direction, and the z-direction. The +velocity is completely specifed, both as to magnitude and direction, IÝ we give +the numerical values of its three rectangular components: +U„ = dw/dt, 0y = dụ/dt, Uy = dz/dl. (9.5) +On the other hand, the speed of the object 1s +ds/đdt = |u| = viuà + 02 + tỷ. (9.6) +Next, suppose that, because of the action of a force, the velocity changes +to some other direction and a diferent magnitude, as shown in Fig. 9-2. We +--- Trang 181 --- +(T—#! +„ 1 Ị mm I +z1 l Tý Hi I +(-1---1 ~TrJ +I ———_—kL - +L7 Lự +Vh_____# +Fig. 9-2. A change in velocity in which both the magnitude and +direction change. +can analyze this apparently complex situation rather simply 1Ÿ we evaluate the +changes in the z-, -, and z-components of velocity. The change in the component +of the velocity in the z-direction in a time Af is Au„ = a„ At, where a„ is what +we call the #-component of the acceleration. 5imilarly, we see that Auy = ay At +and Aø; = ø; Ai. In these terms, we see that NÑewton?s Second Law, in saying +that the force is in the same direction as the acceleration, is really three laws, In +the sense that the component of the force in the z-, -, or z-direction is equal to +the mass times the rate of change of the corresponding component of velocity: +F„, = m(du„/dt) = m(dŠ#/dt?) = ma, +F„ = m(duy/dt) = m(d®u/dt?) = may, (9.7) +F, = m(du; /dt) = m(dŠz (dt?) = ma,. +Just as the velocity and acceleration have been resolved into components by +projecting a line sepment representing the quantity, and its direction onto three +coordinate axes, so, in the same way, a force in a given direction is represented +by certain components in the z-, -, and z-directions: +Tạ —= F'cos(œ, F), +Tụ = Fcos(u, `), (9.8) +Ty = Fcos(z,F), +--- Trang 182 --- +where #' is the magnitude of the force and (z, #) represents the angle between +the z-axis and the direction of Ƒ', etc. +Newton?s Second Law is given in complete form in Bq. (9.7). IÝ we know +the forces on an object and resolve them into z-, -, and z-components, then +we can find the motion of the object from these equations. Let us consider a +simple example. Suppose there are no forces in the - and z-directions, the only +force being in the z-direction, say vertically. Equation (9.7) tells us that there +would be changes in the velocity in the vertical direction, but no changes in +the horizontal direction. “This was demonstrated with a special apparatus In +Chapter 7 (see Eig. 7-3). A falling body moves horizontally without any change +in horizontal motion, while it moves vertically the same way as it would move +1f the horizontal motion were zero. In other words, motions in the z-, -, and +z-directions are independent If the ƒorces are not connected. +9-4 What is the force? +In order to use Newton”s laws, we have to have some formula for the force; +these laws say pay aœftenlion to the ƒorces. TỶ an object 1s accelerating, some +agency is at work; ñnd it. Our program for the future of dynamiecs must be to +imd the laus for the Ƒorce. Newton himself went on to give some examples. In the +case of gravity he gave a specifc formula for the force. In the case of other forces +he gave some part of the information in his Third Law, which we will study in +the next chapter, having to do with the equality of action and reaction. +Extending our previous example, what are the forces on objects near the +earth”s surface? Near the earth's surface, the force in the vertical direction due to +gravity is proportional to the mass of the object and is nearly independent of height +for heights small compared with the carths radius l: ' = GmM/R2 = mg, +where g = GM/R>? is called the acceleration oƒ graoit. Thus the law of gravity +tells us that weight is proportional to mass; the force is in the vertical direction +and is the mass times g. Again we find that the motion in the horizontal direction +1s at constant velocity. The interesting motion is in the vertical direction, and +Newton's Second Law tells us +mg = m(d°z/dt?). (9.9) +Cancelling the rm”s, we ñnd that the acceleration in the z-direction is constant +and equal to g. 'Phis is of course the well known law of free fall under gravity, +--- Trang 183 --- +EQUILIBRIUM +: x POSITION +Fig. 9-3. A mass on a spring. +which leads to the equations +U„ = 0o + g, +# = #o + 0of + šg2. (9.10) +As another example, let us suppose that we have been able to build a gad- +get (Eig. 9-3) which applies a force proportional to the distance and directed +oppositely—a spring. If we forget about gravity, which is of course balanced out +by the initial stretch of the spring, and talk only about ezcess forces, we see that +1f we pull the mass down, the spring pulls up, while if we push it up the spring +pulls down. This machine has been designed carefully so that the force is greater, +the more we pull it up, in exact proportion to the displacement from the balanced +condition, and the force upward is similarly proportional to how far we pull down. +Tf we watch the dynamies of this machine, we see a rather beautiful motion——up, +down, up, down, ... 'Phe question is, will Newton”s equations correctly describe +this motion? Let us see whether we can exactly calculate how it moves with this +periodic oscillation, by applying Newton”s law (9.7). In the present instance, the +equation 1s +— kœ& = rm(du„/dt). (9.11) +Here we have a situation where the velocity in the z-direction changes at a rate +proportional to z. Nothing will be gained by retaining numerous constants, so +we shall imagine either that the scale of time has changed or that there is an +accident in the units, so that we happen to have &/mn = 1. Thus we sha]l try to +solve the equation +đuy (dt = —ø. (9.12) +To proceed, we must know what „ is, but oŸ course we know that the velocity is +the rate of change of the position. +--- Trang 184 --- +9-5 Meaning of the dynamical equations +Now let us try to analyze Just what Eq. (9.12) means. Suppose that at a given +time ý the object has a certain velocity „; and position z. What is the velocity +and what is the position at a slightly later time ý + c? If we can answer this +question our problem is solved, for then we can start with the given condition and +compute how 1% changes for the first instant, the next instant, the next instant, +and so on, and in this way we gradually evolve the motion. To be speciffic, let us +suppose that at the time ý = Ö we are given that z = 1 and ø„ =0. Why does +the object move at all? Because there is a ƒforce on it when it is at any position +except z = 0. lÝz >0, that force is upward. Therefore the velocity which 1s +zero starts to change, because of the law of motion. Ônce it starts to build up +some velocity the object starts to move up, and so on. Now at any time #, IÍ € is +very small, we may express the position at time £ + e in terms of the position at +time ý and the velocity at time £ to a very good approximation as +z(t + €) = z(t) + cuz(Ð). (9.13) +The smaller the c, the more accurate this expression is, but it is still usefully +accurate even 1Ý e is not vanishingly smaill. Now what about the velocity? In +order to get the velocity later, the velocity at the time # + c, we need to know +how the velocity changes, the øccelerai#ion. And how are we going to find the +acceleration? That is where the law of dynamics comes in. The law of dynamics +tells us what the acceleration is. It says the acceleration is —z. +0z(t + €) = 0x(É) + eax(£) (9.14) += 0„z(f) — ez(f). (9.15) +Equation (9.14) is merely kinematics; it says that a velocity changes because of +the presence of acceleration. But Eq. (9.15) is đựụnøœmics, because it relates the +acceleration to the force; it says that at this particular time for this particular +problem, you can replace the acceleration by —z(#). Therefore, if we know both +the z and 0 at a given time, we know the acceleration, which tells us the new +velocity, and we know the new position——this is how the machinery works. The +velocity changes a little bit because of the force, and the position changes a little +bit because of the velocity. +--- Trang 185 --- +9-6 Numerical solution of the equations +Now let us really solve the problem. Suppose that we take e = 0.100 sec. After +we do all the work iŸ we fnd that this is not small enough we may have to go back +and do it again with e = 0.010 sec. Starting with our initial value z(0) = 1.00, +what is (0.1)? It is the old position #(0) plus the velocity (which is zero) tỉmes +0.10 sec. Thus z(0.1) is still 1.00 because it has not yet started to move. But +the new velocity at 0.10 sec will be the old velocity ø(0) = 0 plus e times the +acceleration. The acceleration is —#(0) = —1.00. Thus +(0.1) = 0.00 — 0.10 x 1.00 = —0.10. +Now at 0.20 sec ++(0.2) = z(0.1) + eo(0.1) += 1.00 — 0.10 x 0.10 = 0.99 +0(0.2) = 0(0.1) + ea(0.1) += —0.10 — 0.10 x 1.00 = —0.20. +And so, on and on and on, we can calculate the rest of the motion, and that is just +what we shall do. However, for practical purposes there are some little trieks by +which we can increase the accuracy. IÝ we continued this calculation as we have +started it, we would fnd the motion only rather crudely because e —= 0.100 sec +is rather crude, and we would have to go to a very small interval, say e = 0.01. +Then to go through a reasonable total time interval would take a lot of cycles +of computation. So we shall organize the work in a way that will increase the +precision of our calculations, using the same coarse interval e = 0.10 sec. 'This +can be done iŸ we make a subtle improvement in the technique of the analysis. +Notice that the new position is the old position plus the time interval e times +the velocity. But the velocity œhenŸ The velocity at the beginning of the time +interval is one velocity and the velocity at the end of the time interval is another +velocity. Our improvement is to use the velocity halftuau betueen. ]Ý we know +the speed now, but the speed is changing, then we are not goỉng to get the right +answer by going at the same speed as now. We should use some speed between +the “now” speed and the “then” speed at the end of the interval. "The same +considerations also apply to the velocity: to compute the velocity changes, we +should use the acceleration midway between the two times at which the velocity +1s to be found. Thus the equations that we shall actually use will be something +--- Trang 186 --- +Table 9-1 +Solution of du„/dt = —z +Interval: e = 0.10 sec +£ % U„ đ„ +0.0 1.000 0.000 | —1.000 +—0.050 +0.1 0.995 —0.995 +—0.150 +0.2 0.980 —0.980 +—0.248 +0.3 0.955 —0.955 +—0.343 +0.4 0.921 —0.921 +—0.435 +0.5 0.877 —0.877 +—0.523 +0.6 0.825 —0.825 +—0.605 +0.7 0.764 —0.764 +—0.682 +0.8 0.696 —0.696 +—0.751 +0.9 0.621 —0.621 +—0.814 +1.0 0.540 —0.540 +—0.868 +1.1 0.453 —0.453 +—0.913 +1.2 0.362 —0.362 +—0.949 +1.3 0.267 —0.267 +—0.976 +1.4 0.169 —0.169 +—0.993 +1.5 0.070 —0.070 +—1.000 +1x .... +--- Trang 187 --- +like this: the position later is equal to the position before plus e times the velocity +d‡ the từme in the rmniddle oƒ the interudl. Simllarly, the velocity at this halfway +point is the velocity at a tỉme e before (which is in the middle of the previous +interval) plus e tỉimes the acceleration at the time ứ. That is, we use the equations +z(£ + e) = z() + cu( + c/2), +0(£ + /2) = u(t — c/2) + ca(t), (9.16) +a(£) = —z(Ð). +There remains only one slipght problem: what is 0(c/2)? At the start, we are +given 0(0), not ø(—e/2). To get our calculation started, we shall use a special +equation, namely, 0(e/2) = (0) + (e/2)a(0). +Now we are ready to carry through our calculation. EOor convenience, we +may arrange the work in the form of a table, with columns for the time, the +position, the velocity, and the acceleration, and the in-between lines for the +velocity, as shown in Table 9-1. Such a table is, of course, just a convenient way +Of representing the numerical values obtained from the set of equations (9.16), +and in fact the equations themselves need never be written. We just fill in the +various spaces in the table one by one. 'Phis table now gïves us a very good idea +of the motion: it starts from rest, fñrst picks up a little upward (negative) velocity +and it loses some of its distance. The acceleration is then a little bit less but +1t is still gaining speed. But as it goes on it gains speed more and more slowly, +until as it passes ø = 0 at about ý = 1.50 sec we can confidently predict that it +will keep goïing, but now it will be on the other side; the position # will become +negative, the acceleration therefore positive. Thưus the speed decreases. Ít 1s +interesting to compare these numbers with the function = cos¿, which is done +in Eig. 9-4. The agreement is within the three significant fgure accuracy of our +calculationl We shall see later that ø = cosứ is the exact mathematical solution +of our equation of motion, but i1 is an impressive illustration of the power of +numerical analysis that such an easy calculation should gïve such precise results. +9-7 Planetary motions +'The above analysis is very nice for the motion of an oscillating spring, but can +we analyze the motion of a planet around the sun? Let us see whether we can +arrive at an approximation to an ellipse for the orbit. We shall suppose that the +sun is infñnitely heavy, in the sense that we shall not inelude its motion. Suppose +--- Trang 188 --- +1.0 +0.5 +ọ 0.5 1.0 1.5é £ (sec) +Fig. 9-4. Graph of the motion of a mass on a spring. +a planet starts at a certain place and is moving with a certain velocity; it goes +around the sun in some curve, and we shall try to analyze, by Newton's laws of +motion and his law of gravitation, what the curve is. How? At a given moment it +1s at some position in space. lf the radial distance from the sun to this position +is called r, then we know that there is a force directed inward which, according +to the law of gravity, is equal to a constant times the product of the sun”s mass +and the planet's mass divided by the square of the distance. 'To analyze this +further we must fnd out what acceleration will be produced by this force. We +shall need the componenfs of the acceleration along two directions, which we call +z and . Thus iŸ we specify the position of the planet at a given moment by +giving z and (we shall suppose that z is always zero because there is no force +in the z-direction and, if there is no initial velocity 0;, there will be nothing to +make z other than zero), the force is direcbed along the line joining the planet to +the sun, as shown in Fig. 9-5. +y F„ PLANET (x,y) +Fig. 9-5. The force of gravity on a planet. +--- Trang 189 --- +trom this fgure we see that the horizontal component of the force is related +to the complete force in the same manner as the horizontal distance z is to the +complete hypotenuse z, because the bwo triangles are similar. Also, IÝ # is positive, +F, is negative. That is, F„/|F| = —z/r, or F„ = —|F|z/z == —GMmz/r3. Ñow +we use the dynamical law to fnd that this force component is equal to the mass +of the planet times the rate of change of its velocity in the z-direction. 'Phus we +ñnd the following laws: +m(du„/dt) = —GMma/rẺ, +m{(duy/đt) = —GMmy/rẺ, (9.17) +r= V+2 +92. +This, then, is the set of equations we must solve. Again, in order to simplify +the numerical work, we shall suppose that the unit of time, or the mass of the +sun, has been so adjusted (or luck is with us) that GẢM = 1. Eor our specifc +example we shall suppose that the initial position of the planet is at z = 0.500 +and = 0.000, and that the velocity is all in the, g-direction at the start, and +1s Of magnitude 1.630. Now how do we make the calculation? We again make +a table with columns for the time, the #-position, the z-velocity „, and the +-acceleration œ„; then, separated by a double line, three columns for position, +velocity, and acceleration in the -direction. In order to get the accelerations we +are going to need Ed. (9.17); it tells us that the acceleration in the z-direction +is —#/rỞ, and the acceleration in the z-direction is —#/rỞ, and that z is the +square root of z2 +”. Thus, given ø and ¿, we must do a little calculating on the +side, taking the square root of the sum of the squares to fnd r and then, to get +ready to calculate the two accelerations, it is useful also to evaluate 1/r3. This +work can be done rather easily by using a table of squares, cubes, and reciprocals: +then we need only multiply # by 1/r, which we do on a slide rule. +Our calculation thus proceeds by the following steps, using time intervals e = +0.100: Initial values at # = Ú: ++(0) = 0.500 (0)=_ 0.000 +0„(0) = 0.000 0„(0) = +1.630 +trom these we flnd: +r(0)= 0.500 1/r(0) = 8.000 +ø„ = —4.000 a„ = 0.000 +--- Trang 190 --- +Thus we may calculate the velocities 0„(0.05) and 0„(0.05): +0„(0.05) = 0.000 — 4.000 x 0.050 = —0.200; +0„(0.05) = 1.630 + 0.000 x 0.050=—=_ 1.630. +Now our main calculations begin: +z(0.1) = 0.500—0.20x0.1 =_ 0.480 +(0.1) = 0.0 + 1.63 x 0.1 =_ 0.163 +r= V0.4802+0.1632 =_ 0.507 +1/rỶ = 7.677 +ø„(0.1) = —0.480 x 7.677 = —3.685 +a„(0.1) = —0.163 x 7.677 = —1.250 +0„(0.15) = —0.200 — 3.685 x 0.1 = —0.568 +0u(0.15) = 1.680 — 1.250 x0.1 = 1.505 ++(0.2) = 0.480 — 0.568 x01 =_ 0.4238 +(0.2) = 0.163 + 1.505x0.1 = 0.313 +In this way we obtain the values given in Table 9-2, and in 20 steps or so we +have chased the planet halfway around the sunl In Eig. 9-6 are plotted the z- +and -coordinates given in Table 9-2. "The dots represent the positions at the +succession of times a tenth of a unit apart; we see that at the start the planet +moves rapidly and at the end it moves slowly, and so the shape of the curve 1s +determined. Thus we see that we real do know how to calculate the motion of +planetsl +Table 9-2 +Solution of du„/d‡ = —øœ/rẺ, duy/dt —= —ụ/rẺ, r = +2 + 92. +Interval: = 0.100 +Ởrbiữt uy = 1.63 „=0 z=05 =0 at =0 +‡ un Uz đ„ ụ Uy đụ r 1/rŠ +0.0 0.500 —4.000 0.000 0.000 |[ 0.500 | 8.000 +—0.200 1.630 +--- Trang 191 --- +Table 9-2 +t un Uz đ„ ụ Uy đụ r 1/rẺ +0.1 0.480 —3.685 0.163 —1.251 || 0.507 | 7.677 +—0.568 1.505 +0.2 0.423 —2.897 0.313 —2.146 || 0.527 | 6.847 +—0.858 1.290 +0.3 0.337 —1.958 0.443 —2.569 || 0.556 | 5.805 +—1.054 1.033 +0.4 0.232 —1.112 0.546 —2.617 || 0.593 | 4.794 +—1.165 0.772 +0.5 0.115 —0.454 0.623 —2.449 || 0.634 | 3.931 +—1.211 0.527 +0.6 | —0.006 -+0.018 0.676 —2.190 || 0.676 | 3.241 +—1.209 0.308 +0.7 | —0.127 +0.342 0.706 —1.911 || 0.718 | 2.705 +—1.175 0.117 +0.8 | —0.244 -+0.559 0.718 —1.646 || 0.758 | 2.292 +—1.119 —0.048 +0.9 | —0.356 +0.702 0.713 —1.408 || 0.797 | 1.974 +—1.048 —0.189 +1.0 | —0.461 -+0.796 0.694 —1.200 || 0.833 | 1.728 +—0.969 —0.309 +1.1 | —0.558 -+0.856 0.664 —1.019 || 0.867 | 1.536 +—0.883 —0.411 +1.2 | —0.646 -+0.895 0.623 —0.862 || 0.897 | 1.385 +—0.794 —0.497 +1.3 | —0.725 -+0.919 0.573 —0.726 || 0.924 | 1.267 +—0.702 —0.569 +1.4 | —0.795 -+0.933 0.516 —0.605 || 0.948 | 1.174 +—0.608 —0.630 +1.5 | —0.856 +0.942 0.453 —0.498 || 0.969 | 1.100 +—0.514 —0.680 +1.6 | —0.908 -+0.947 0.385 —0.402 || 0.986 | 1.043 +—0.420 —0.720 +1.7 | —0.950 -+0.950 0.313 —0.313 || 1.000 | 1.000 +—0.325 —0.7ð1 +1.8 | —0.982 +0.952 0.238 —0.230 || 1.010 | 0.969 +—0.229 —0.774 +1.9 | —1.005 -+0.953 0.160 —0.152 || 1.018 | 0.949 +--- Trang 192 --- +Table 9-2 +t un Uz đ„ ụ Uy đụ r 1/rẺ +—0.134 —0.790 +2.0 | —1.018 +0.955 0.081 —0.076 || 1.022 | 0.938 +—0.038 —0.797 +2.1 | —1.022 +0.957 0.002 —0.002 || 1.022 | 0.936 ++0.057 —0.797 +2.2 | —1.017 +0.959 || —0.078 +0.074 || 1.020 | 0.944 +—0.790 +2.3 +Crossed zø-axis at 2.101 sec, .'. period = 4.20 sec. +„ = 0 at 2.086 sec. +Cross ø at —1.022, .'. semimajor axis = ... = 0.761. +0y = 0.T9T. +Predicted time z(0.761)3⁄/2 = x(0.663) = 2.082. +=1.0 Ỹ +t= — _t=05 +t=15—N * 05 7 +t= 20^" =0 +—1.0 —0.5 SUN 0.5 x +Fig. 9-6. The calculated motion of a planet around the sun. +Now let us see how we can calculate the motion of Neptune, Jupiter, UỦranus, +or any other planet. lÝ we have a great many planets, and let the sun move +too, can we do the same thing? Of course we can. We calculate the force on +a particular planet, let us say planet number ¡, which has a position #¿, ¿, Z¿ +(2= 1 may represent the sun, ¿ = 2 Mercury, ¿ = 3 Venus, and so on). We must +know the positions of all the planets. The force acting on one is due to all the +other bodies which are located, let us say, at positions #;,;,z;. Therefore the +--- Trang 193 --- +equations are +mị TU — N¬_ GmimjVi S17) +đt = Tử +đu; Ằ ` Gm¿m (Ui /) +mị th = TT TH, (9.18) +J=I 19 +".. » _ CmunjG¡ —) +đt m Tỉ, +Further, we defne r¿¿ as the distance between the two planets ¿ and 7; this is +equal to +tụ = V8 — #7)? + (Mi — 9ý)” + (#¡ — 22). (9.19) +AIso, 3) means a sum over all values of j—all other bodies——except, of course, +for j ==. Thus all we have to do is to make more columns, /o2#s more columns. +W© need nine columns for the motions of Jupiter, nine for the motions of Saturn, +and so on. Then when we have all initial positions and velocities we can calculate +all the accelerations from Eq. (9.18) by first calculating all the distances, using +Eq. (9.19). How long will it take to do it? TỶ you do i9 at home, it will take a +very long timel But in modern times we have machines which do arithmetic +very rapidly; a very good computing machine may take 1 microsecond, that is, a +millionth of a second, to do an addition. To do a multiplication takes longer, say +10 microseconds. lt may be that in one cycle of calculation, depending on the +problem, we may have 30 multiplications, or something like that, so one cycle will +take 300 microseconds. 'Phat means that we can do 3000 cycles of computation +per second. In order to get an accuracy, of, say, one part in a billion, we would +need 4 x 105 eycles to correspond to one revolution of a planet around the sun. +That corresponds to a computation time of 130 seconds or about two minutes. +Thus it take only 6wo minutes to follow Jupiter around the sun, with all the +perturbations of all the planets correct to one part in a billion, by this methodl +(It turns out that the error varies about as the square of the interval e. lÝ we +make the interval a thousand times smaller, it is a million times more accurate. +So, let us make the interval 10,000 times smaller.) +So, as we said, we began this chapter not knowing how to calculate even +the motion of a mass on a spring. Now, armed with the tremendous power of +--- Trang 194 --- +Newton”s laws, we can not only calculate such simple motions but also, given +only a machine to handle the arithmetic, even the tremendously complex motions +of the planets, to as hipgh a degree of precision as we wishl +--- Trang 195 --- +I0 +(t©rtsorterff©ore @œŸ WQ@rt©refErrrrt +10-1 Newton?s Third Law +On the basis of NÑewton”s second law of motion, which gives the relation +between the acceleration of any body and the force acting on it, any problem in +mmechanies can be solved in principle. EFor example, to determine the motion of +a few particles, one can use the numerical method developed in the preceding +chapter. But there are good reasons to make a further study of Newton”s laws. +First, there are quite simple cases of motion which can be analyzed not only +by numerical methods, but also by direct mathematical analysis. For example, +although we know that the acceleration of a falling body is 32 ft/sec2, and +trom this fact could calculate the motion by numerical methods, i% is much +casier and more satisfactory to analyze the motion and fñnd the general solution, +8 = 8g + 0g + 162. In the same way, although we can work out the positions of a +harmonic oscillator by numerical methods, ït is also possible to show analytically +that the general solution is a simple cosine function of £, and so it is unnecessary +to go to all that arithmetical trouble when there is a simple and more accurate +way to get the result. In the same manner, although the motion of one body +around the sun, determined by gravitation, can be calculated point by point by +the numerical methods of Chapter 9, which show the general shape of the orbit, +1E is nice also to get the exact shape, which analysis reveals as a perfect ellipse. +Unfortunately, there are really very few problems which can be solved exactly +by analysis. In the case of the harmonic oscillator, for example, if the spring +force is not proportional to the displacement, but is something more complicated, +one must fall back on the numerical method. Ôr ïf there are two bodies goïng +around the sun, so that the total number of bodies is three, then analysis cannot +produce a simple formula for the motion, and in practice the problem must +be done numerically. “That ¡is the famous three-body problem, which so long +challenged human powers of analysis; it is very interesting how long it took +people to appreciate the fact that perhaps the powers of mathematical analysis +--- Trang 196 --- +were limited and ¡it might be necessary to use the numerical methods. Today an +enormous number oŸ problems that cannot be done analytically are solved by +numerical methods, and the old three-body problem, which was supposed to be +so difficult, is solved as a matter of routine in exactly the same manner that was +described in the preceding chapter, namely, by doing enough arithmetic. However, +there are also situations where both methods fail: the simple problems we can +do by analysis, and the moderately difcult problems by numerical, arithmetical +methods, but the very complicated problems we cannot do by either method. A +complicated problem 1s, for example, the collision of two automobiles, or even +the motion of the molecules of a gas. There are countless particles in a cubic +millimeter of gas, and ¡it would be ridiculous to try to make calculations with +so many variables (about 10!——a hundred million billion). Anything like the +motion of the molecules or atoms of a gas or a block or iron, or the motion of +the stars in a globular cluster, instead of just two or three planets goïing around +the sun——such problems we cannot do directly, so we have to seek other means. +In the situations in which we cannot follow details, we need to know some +general properties, that is, general theorems or principles which are consequences +of Newton's laws. One of these is the principle oŸ conservation of energy, which +was discussed in Chapter 4. Another is the principle of conservation oŸ momentum, +the subject of this chapter. Another reason for studying mechanics further is +that there are certain patterns of motion that are repeated in many diferent +circumstances, so iÈ is good to study these patterns in one particular cireumstance. +For example, we shall study collisions; diferent kinds of collisions have much +in common. In the fow of Ñuids, it does not make mụuch diference what the +fuid is, the laws of the fow are similar. Other problems that we shall study +are vibrations and oscillations and, in particular, the peculiar phenomena of +mmechanical waves—sound, vibrations of rods, and so on. +In our discussion of NÑewton”s laws it was explained that these laws are a kind +of program that says “Pay attention to the forces,” and that Newton told us only +two things about the nature of forces. In the case of gravitation, he gave us the +complete law of the force. In the case of the very complicated forces between +atoms, he was not aware of the right laws for the forces; however, he discovered +one rule, one general property of forces, which is expressed in his 'Third Law, and +that is the total knowledge that Newton had about the nature of forces—the law +of gravitation and this principle, but no other details. +'This principle is that acfon eguals reaction. +--- Trang 197 --- +'What is meant is something of this kind: Suppose we have ©wo small bodies, +say particles, and suppose that the first one exerts a force on the second one, +pushing it with a certain force. 'Then, simultaneously, according to Newton's +Thind Law, the second particle will push on the frst with an equal force, In +the opposite direction; furthermore, these forces efectively act in the same line. +This is the hypothesis, or law, that Newton proposed, and it seems to be quite +accurate, though not exact (we shall discuss the errors later). For the moment +we shall take it to be true that action equals reaction. Of course, If there is a +third particle, not on the same line as the other ©wo, the law does no mean that +the total force on the first one is equal to the total force on the second, since +the third particle, for instance, exerts its own push on each of the other two. +'The result is that the total efect on the first bwo is in some other direction, and +the forces on the first two particles are, in general, neither equal nor opposite. +However, the forces on each particle can be resolved into parts, there beïing one +contribution or part due to each other interacting particle. Then each pœ¿r of +particles has corresponding components of mutual interaction that are equal in +magnitude and opposite in direction. +10-2 Conservation of momentum +Now what are the interesting consequences of the above relationship? Suppose, +for simplicity, that we have just two interacting particles, possibly of diferent +mass, and numbered 1 and 2. “The forces between them are equal and opposite; +what are the consequences? According to Newton's Second Law, force is the +time rate of change of the momentum, so we conclude that the rate of change of +mmomentum ?Ø¡ of particle 1 is equal to minus the rate of change of momentum Øøs +of particle 2, or +Now lf the raf#e oƒ chønge is always equal and opposite, it follows that the £otal +chơngec In the momentum of particle 1 is equal and opposite to the #o‡øÏ change +in the momentum of particle 2; this means that if we add the momentum of +particle 1 to the momentum of particle 2, the rate of change of the sum of these, +due to the mutual forces (called internal forces) bebween particles, is zero; that is +đứm + p›)/dt = 0. (10.2) +There is assumed to be no other force in the problem. lỶ the rate of change of this +sum is always zero, that is Just another way of saying that the quantity (0 + Øa) +--- Trang 198 --- +does not change. (This quantity is also writben ?n10ị + mạøa, and is called the +total mormentum of the two particles.) We have now obtained the result that the +total momentum of the two particles does not change because of any mutual +interactions between them. This statement expresses the law of conservation of +mmomentum in that particular example. We conclude that if there is any kind +of force, no matter how complicated, between two particles, and we measure or +calculate m0 + ma0a, that is, the sum of the two momenta, both before and +after the forces act, the results should be equal, I.e., the total momentum is a +constant. +T we extend the argument to three or more interacting particles in more +complicated circumstances, it is evident that so far as internal forces are concerned, +the total momentum of all the particles stays constant, sỉince an increase in +mmomentum of one, due to another, is exactly compensated by the decrease of +the second, due to the first. That ¡s, all the internal forces will balance out, and +therefore cannot change the total momentum of the particles. Then If there are +no forces rom the outside (external forces), there are no forces that can change +the total momentum; hence the total momentum is a constant. +lt is worth describing what happens if there are forces that do nø£# come from +the mutual actions of the particles in question: suppose we isolate the interacting +particles. If there are only mutual forces, then, as before, the total momentum +of the particles does not change, no matter how complicated the forces. Ôn the +other hand, suppose there are also forces coming from the particles outside the +isolated group. Any force exerted by outside bodies on inside bodies, we call an +czternal force. We shall later demonstrate that the sum of all external forces +equals the rate of change of the total momentum of all the particles inside, a +very useful theorem. +'The conservation of the total momentum of a number of interacting particles +can be expressed as +THỊĐ1 + ThaUa + Tn303 + - - : = a constant, (10.3) +1f there are no net external forces. Here the masses and corresponding velocities +of the particles are numbered 1, 2, 3, 4,... The general statement of Ñewton”s +Second Law for each particle, +t—= qiữn9); (10.4) +is true specifically for the cømponen‡s of force and momentum in any given +--- Trang 199 --- +direction; thus the zø-component of the force on a particle is equal to the z- +component of the rate of change of momentum of that particle, or += qi_n9z), (10.5) +and similarly for the - and z-directions. Therefore Eq. (10.3) is really three +equations, one for each direction. +In addition to the law of conservation of momentum, there is another interesf- +ing consequence of NÑewton”s Second Law, to be proved later, but merely stated +now. 'Phis principle is that the laws of physics will look the same whether we are +standing still or moving with a uniform speed ïn a straight line. For example, +a child bouncing a ball in an airplane ñnds that the ball bounces the same as +though he were bouncing it on the ground. Even though the airplane is moving +with a very high velocity, unless it changes its velocity, the laws look the same +to the child as they do when the airplane is standing still. This is the so-called +rclatiuilụ priứnciple. As we use it here we shall call it “Galilean relativity” to +distinguish it rom the more careful analysis made by Binstein, which we shall +study later. +W© have just derived the law of conservation of momentum from Newton”s +laws, and we could go on from here to find the special laws that describe impacts +and collisions. But for the sake of variety, and also as an illustration of a kind of +reasoning that can be used in physics in other cireumstances where, for example, +one might not know Newton”s laws and might take a different approach, we shall +discuss the laws of impacts and collisions from a completely diferent point of +view. WWe shall base our discussion on the principle of Galilean relativity, stated +above, and shall end up with the law of conservation of momentum. +We shall start by assuming that nature would look the same if we run along at +a certain speed and watch it as it would iƒ we were standing still. Before discussing +collisions in which bwo bodies collide and stick together, or come together and +bounce apart, we shall ñrst consider 6wo bodies that are held together by a spring +or something else, and are then suddenly released and pushed by the spring or +perhaps by a little explosion. Eurther, we shall consider motion in only one +direction. First, let us suppose that the two obJects are exactly the same, are nice +symmetrical objects, and then we have a little explosion between them. After the +explosion, one of the bodies will be moving, let us say toward the right, with a +velocity ø. Then it appears reasonable that the other body is moving toward the +left with a velocity 0, because if the objects are alike there is no reason for right +--- Trang 200 --- +or left to be preferred and so the bodies would do something that is symmetrical. +Thịs is an illustration of a kind of thinking that is very useful in many problems +but would not be brought out if we just started with the formulas. +The first result from our experiment is that equal objects will have equal +speed, but now suppose that we have two objects made of diferent materials, say +copper and aluminum, and we make the two rmasses equal. We shall now suppose +that ïf we do the experiment with two masses that are equal, even though the +objects are not identical, the velocities will be equal. Someone might object: +“But you know, you could do it backwards, you did not have to swppose that. You +could đefne equal masses to mean two masses that acquire equal velocities In +this experiment.” We follow that suggestion and make a little explosion between +the copper and a very large piece of aluminum, so heavy that the copper flies out +and the aluminum hardly budges. That is too much aluminum, so we reduce the +amount until there is just a very tỉny piece, then when we make the explosion the +aluminum goes fying away, and the copper hardly budges. hat is not enough +aluminum. Evidently there is some right amount in between; so we keep adjusting +the amount until the velocities come out equal. Very well then——let us turn I§ +around, and say that when the velocities are equal, the masses are equal. 'This +appears to be just a defnition, and it seems remarkable that we can transform +physical laws into mere defnitions. Nevertheless, there øre some physical laws +Involved, and if we accept this definition of equal masses, we Immediately fñnd +one of the laws, as follows. +Suppose we know from the foregoing experiment that two pieces of matter, +A and B (of copper and aluminum), have equal masses, and we compare a +third body, say a piece of gold, with the copper in the same manner as above, +making sure that its mass is equal to the mass of the copper. lf we now make +the experiment between the aluminum and the gold, there is nothing in logic +that says fhese masses must be equal; however, the ezperữnent shows that they +actually are. So now, by experiment, we have found a new law. A statement of +this law might be: IỶ two masses are each equal to a third mass (as determined +by cqual velocities in this experiment), then they are equal to each other. (This +statement does noø‡ follow at all from a similar statement used as a postulate +regarding rmathematical quantities.) From this exarmple we can see how quickly we +start to infer things If we are careless. It is nmoøf just a delnition to say the masses +are equal when the velocities are equal, because to say the masses are equal is to +imply the mathematical laws of equality, which in turn makes a prediction about +an experiment. +--- Trang 201 --- +As a second example, suppose that A and Ö are found to be equal by doiïng +the experiment with one strength of explosion, which gives a certain velocity; If +we then use a stronger explosion, will it be true or not true that the velocities +now obtained are equal? Again, in logic there is nothing that can decide this +question, but experiment shows that it 7s true. So, here is another law, which +might be stated: If two bodies have equal masses, as measured by equal velocities +at one velocity, they will have equal masses when measured at another velocity. +trom these examples we see that what appeared to be only a deñnition really +involved some laws of physics. +In the development that follows we shall assume it is true that equal masses +have equal and opposite velocities when an explosion occurs between them. We +shall make another assumption in the inverse case: lÝ two identical obJects, +moving in opposite directions with equal velocities, collide and stick together by +some kind of glue, then which way will they be moving after the collision? 'Phis +1s again a symmetrical situation, with no preference between right and left, so +we assume that they stand still. We shall also suppose that any bwo objects of +cequal mass, even if the objects are made of diferent materials, which collide and +stick together, when moving with the same velocity in opposite directions will +come to rest after the collision. +10-3 Momentum ¿s conserved! +W©e can verify the above assumptions experimentally: first, that 1Ý bwo sta- +tionary objects of equal mass are separated by an explosion they will move apart +with the same speed, and second, if two obJects of equal mass, coming together +with the same speed, collide and stick together they will stop. 'This we can +do by means of a marvelous invention called an air trough,X which gets rid of +friction, the thing which continually bothered Galileo (Fig. 10-1). He could not +QEtS) HOLES +Ầ fïˆ S989 +TRÀ) Y. “Z7 +Fig. 10-1. End view of linear alr trough. +* HH. V. Neher and R. B. Leighton, Amer. Jour. oƒ Phụas. 31, 255 (1963). +--- Trang 202 --- +BUMPER SPRING TOY PISTOL CAP +SPARK ELECTRODE +CYLINDER PISTON BUMPER SPRING +Fig. 10-2. Sectional view of gliders with explosive Interaction cylinder +attachment. +do experiments by sliding things because they do not slide freely, but, by adding +a magic touch, we can today get rid oŸ friction. Our objects will slide without +diffculty, on and on at a constant velocity, as advertised by Galileo. 'This is +done by supporting the objects on air. Because air has very low Íriction, an +object glides along with practically constant velocity when there is no applied +force. First, we use 6wo glide blocks which have been made carefully to have +the same weight, or mass (their weight was measured really, bu we know that +this weight is proportional to the mass), and we place a small explosive cap in +a closed cylinder bebween the two blocks (Fig. 10-2). We shall start the blocks +from rest at the center point of the track and force them apart by exploding the +cap with an electric spark. What should happen? If the speeds are equal when +they fy apart, they should arrive at the ends of the trough at the same time. Ôn +reaching the ends they will both bounce back with practically opposite velocity, +and will come together and stop at the center where they started. lt is a good +test; when it is acbually done the result is Jjust as we have described (Eig. 10-3). +PB 4đ EỚớỚ} đe +~-——v về -~ +P &—1 =m: {]@œ) +tr — = hịc +>> V —V ~==— +EP 4+ _—1 E=iNH) 4144) +pm +ằĂHẶ} ẽ {e +Fig. 10-3. Schematic view of action-reaction experiment with equal +masses. +--- Trang 203 --- +VIEW FROM VIEW FROM +CENTER OF MASS MOVING CAR +(CAR VELOCITY = —v) +v => -——v 2v-> 0 +BEFORE COLLISION +v=0 V_> +AFTER COLLISION +Fig. 10-4. TWo views of an inelastic collision between equal masses. +Now the next thing we would like to fñgure out is what happens in a less +simple situation. Suppose we have ÿwo equal masses, one moving with velocity 0 +and the other standing still, and they collide and stick; what is goïing to happen? +There is a mass 2mm altogether when we are fñnished, drifting with an unknown +velocity. What velocity? 'That is the problem. 'To find the answer, we make the +assumption that if we ride along in a car, physics will look the same as if we +are standing still. We start with the knowledge that two equal masses, moving +in opposite directions with equal speeds 0, will stop dead when they collide. +Now suppose that while this happens, we are riding by in an automobile, at a +velocity —ø. Then what does ít look like? Since we are riding along with one +of the two masses which are coming together, that one appears to us to have +zero velocity. The other mass, however, going the other way with velocity , will +appear to be coming toward us at a velocity 20 (Eig. 10-4). Finally, the combined +masses after collision will seem to be passing by with velocity 0. We therefore +conclude that an object with velocity 2u, hitting an equal one at rest, will end +up with velocity 0, or what is mathematically exactly the same, an object with +velocity œ hitting and sticking to one at rest will produce an object moving with +velocity 0/2. Note that if we multiply the mass and the velocity beforehand and +add them together, mo + 0, we get the same answer as when we multiply the +mass and the velocity of everything afterwards, 2w times 0/2. So that tells us +what happens when a mass of velocity 0 hits one standing still. +In exactly the same manner we can deduce what happens when equal objects +having am two velocities hit each other. +Suppose we have two equal bodies with velocities ø¡ and 0s, respectively, +which collide and stick together. What is their velocity 0 after the collision? +Again we ride by in an automobile, say at velocity 0a, so that one body appears +to be at rest. The other then appears to have a velocity 0 — 0a, and we have +the same case that we had before. When it is all ñnished they will be moving +--- Trang 204 --- +VIEW FROM. “LAB” VIEW FROM CAR +VỊ va VỊ — Vạ= 0 +BEFORE COLLISION +v->~ 1/2(vị — v›)-> +AFTER COLLISION +Fig. 10-5. Two views of another inelastic collision between equal +masses. +ab 2(0ị — 0s) with respect to the car. What then is the acbual speed on the +ground? +TEis ø = 3(0Ị — 02) + 0a or š(0 + 92) (Fig. 10-5). Again we note that +T0 -Ƒ tmuua = 2m(01 + 0a) /2. (10.6) +'Thus, using this principle, we can analyze any kind of collision in which Ewo +bodies oŸ equal mass hit each other and stick. In fact, although we have worked +only in one dimension, we can fñnd out a great deal about mụuch more complicated +collisions by imagining that we are riding by ín a car in some oblique direction. +'The prineciple is the same, but the details get somewhat complicated. +In order to test experimentally whether an object moving with velocity 0, +colliding with an equal one at rest, forrms an object moving with velocity 0/2, +we may perform the following experiment with our air-trough apparatus. We +place in the trough three equally massive objects, two of which are initially joined +together with our explosive cylinder device, the third being very near to but +slightly separated from these and provided with a sticky bumper so that it will +stick to another object which hits it. Now, a moment after the explosion, we have +two objects of mass w moving with equal and opposite velocities ø. ÀA moment +after that, one of these collides with the third object and makes an objecE of +mass 2n moving, so we believe, with velocity 0/2. How do we test whether it +1s really 0/2? By arranging the initial positions oŸ the masses on the trough so +that the distances to the ends are not equal, but are in the ratio 2: 1. Thus our +first mass, which continues to move with velocity 0, should cover twice as much +distance in a given tỉme as the 6wo which are sbuck together (allowing for the +small distance travelled by the second object before ¡it collided with the third). +'The mass ?n and the mass 2n should reach the ends at the same time, and when +we try it, we find that they do (Fig. 10-6). +--- Trang 205 --- +NZA — = +[+^—2D+A——>Lm | m ][ m ]<—D_—>{Œ] +~= —v v> 0 +HD^~—22 Lm Lm ]}~—D->#¬ +"=. vi +LHm_] L_2m _W] +Fig. 10-6. An experiment to verify that a mass m. with velocIty œ +striking a mass mm, with zero velocity gives 2w with velocity 0/2. +'The next problem that we want to work out is what happens If we have two +diferent masses. Let us take a mass rm and a mass 2m and apply our explosive +Interaction. What will happen then? Tf, as a result of the explosion, ?nw mmoves +with velocity 0, with what velocity does 2n move? “The experiment we have +just done may be repeated with zero separation between the second and third +masses, and when we try it we get the same result, namely, the reacting masses +m and 2m attain velocities —u and 0/2. Thus the direct reaction between ?m +and 2m gives the same result as the symmetrical reaction between rn and m, +followed by a collision between rn and a third mass ?m in which they stick together. +Purthermore, we find that the masses rnm and 2n returning from the ends of the +trough, with their velocities (nearly) exactly reversed, sbop dead ïf they stick +together. +Now the next question we may ask is this. What will happen iIÝ a mass rn +with velocity 0, say, hits and sticks to another mass 2 at rest? 'This is very +easy to answer using our prineiple of Galilean relativity, for we simply watch +the collision which we have just described from a car moving with velocity —0/2 +(Fig. 10-7). Erom the car, the velocities are +U =0— 0(car) =0u+0/2=30/2 +0 = —0/2~ 0(car) = —0/2+/2=0. +After the collision, the mass 3n appears to us to be moving with velocity 0/2. +Thus we have the answer, I.e., the ratio of velocitles before and after collision +1s 3 to 1: if an object of mass mm collides with a stationary object of mass 2m, +then the whole thing moves of, stuck together, with a velocity 1/3 as mụuch. The +general rule again is that the sum of the produects of the masses and the velocities +--- Trang 206 --- +VIEW FROM VIEW FROM +CM SYSTEM CAR +vV —Vv/2 3v/2 0 +— —— — +BEFORE COLLISION +0 v/2-> +AFTER COLLISION +Fig. 10-7. TWo views of an inelastic collision between m and 2m. +stays the same: ?zø + 0 equals 3mm tỉmes 0/3, so we are gradually building up +the theorem of the conservation of momentum, piece by piece. +Now we have one against two. Ủsing the same arguments, we can predict the +result oŸ one against three, two against three, etc. The case of two against three, +starting from rest, is shown in Fig. 10-8. +0 Ø0 vw¿ 0 0 0 +0 -~——v v+> 0 0 +-——v/2 v/2> 0 +~——v/2 v/3-> +Fig. 10-8. Action and reaction between 2m and 3m. +In every case we find that the mass of the first obJect times its velocity, plus +the mass of the second object times its velocity, is equal to the total mass of the +fnal obJect times its velocity. Thhese are all examples, then, of the conservation +of momentum. Starting from simple, symmetrical cases, we have demonstrated +the law for more complex cases. We could, in fact, do 1 for any rational mass +ratio, and since every ratio is exceedingly close to a rational ratio, we can handle +every ratio as precisely as we wish. +10-4 Momentum and energy +All the foregoing examples are simple cases where the bodies collide and stick +together, or were initially stuck together and later separated by an explosion. +--- Trang 207 --- +However, there are situations in which the bodies do no cohere, as, for example, +two bodies of equal mass which collide with equal speeds and then rebound. Eor +a brief moment they are in contact and both are compressed. At the instant of +mmaximum compression they both have zero velocity and energy is stored in the +elastic bodies, as in a compressed spring. This energy is derived from the kinetic +energy the bodies had before the collision, which becomes zero at the instant +their velocity is zero. The loss of kinetic energy is only momentary, however. The +compressed condition is analogous to the cap that releases energy in an explosion. +The bodies are immediately decompressed in a kind of explosion, and fy apart +again; but we already know that case—the bodies ñy apart with equal speeds. +However, this speed of rebound is less, in general, than the initial speed, because +not all the energy is available for the explosion, depending on the material. If +the material is putty no kinetic energy is recovered, but IŸ it is something more +rigid, some kinetic energy is usually regained. In the collision the rest of the +kinetic energy is transformed into heat and vibrational energy——the bodies are +hot and vibrating. 'Phe vibrational energy also is soon transformed into heat. lt +is possible to make the colliding bodies rom highly elastic materials, such as +sieel, with carefully designed spring bumpers, so that the collision generates very +little heat and vibration. In these circumstances the velocities oŸ rebound are +practically equal to the initial velocities; such a collision is called elastic. +That the speeds 0efore and a/fter an elastic collision are equal is not a matter oŸ +conservation oŸ momentum, but a matter of conservation of kinefic energu. That +the veloeities of the bodies rebounding after a symmetrical collision are equal to +and opposite each other, however, is a matter of conservation of momentum. +We might similarly analyze collisions between bodies of diferent masses, +diferent initial velocities, and various degrees of elasticity, and determine the +ñnal velocities and the loss of kinetic energy, but we shall not go into the details +of these processes. +Bilastic collisions are especially interesting for systems that have no internal +“gears, wheels, or parts.” Then when there is a collision there is nowhere for the +energy to be impounded, because the objects that move apart are in the same +condition as when they collided. 'Therefore, bebween very elementary obJects, +the collisions are always elastic or very nearly elastic. For instance, the collisions +between atoms or molecules in a gas are said to be perfectly elastic. Although +this is an excellent approximation, even such collisions are not perƒectlu elastic; +otherwise one could not understand how energy in the form of light or heat +radiation could come out of a gas. Once in a while, in a gas collision, a low-energy +--- Trang 208 --- +infrared ray is emitted, but this occurrence is very rare and the energy emitted is +very small. So, for most purposes, collisions of molecules in gases are considered +to be perfectly elastic. +As an interesting example, let us consider an eÏasfic collision between two +objects of eguøl rmass. If they come together with the same speed, they would +come apart at that same speed, by symmetry. But now look at this in another +circumstanece, in which one oŸ them is moving with velocity ø and the other one +1s at rest. What happens? We have been through this before. We watch the +symmetrical collision from a car moving along with one of the objects, and we +ñnd that if a stationary body is struck elastically by another body of exactly the +same mass, the moving body stops, and the one that was standing still now moves +away with the same speed that the other one had; the bodies simply exchange +velocities. 'This behavior can easily be demonstrated with a suitable impaect +apparatus. More generally, If both bodies are moving, with diferent velocities, +they simply exchange velocity at impact. +Another example of an almost elastic interaction is magnetism. ÏIÝ we arrange +a païr of U-shaped magnets in our glide blocks, so that they repel each other, +when one drifts quietly up to the other, it pushes it away and stands perfectly +still, and now the other goes along, frictionlessly. +The principle of conservation of momentum is very useful, because it enables +us to solve many problems without knowing the details. We did not know the +details of the gas motions in the cap explosion, yet we could predict the velocities +with which the bodies came apart, for example. Another interesting example is +rocket propulsion. A rocket of large mass, ă, ejects a small piece, oŸ mass mm, +with a terrific velocity V relative to the rocket. After this the rocket, If it were +originally standing still, will be moving with a smaill velocity, ø. sing the +principle of conservation of momentum, we can calculate this velocity to be +b=T: V. +So long as material is being ejected, the rocket continues to pick up speed. Roecket +propulsion is essentially the same as the recoil of a gun: there is no need for any +air to push against. +10-5 Relativistic momentum +In modern times the law oŸ conservation of momentum has undergone certain +modifcations. However, the law is still true today, the modifications being mainly +--- Trang 209 --- +in the defñnitions of things. In the theory of relativity it turns out that we do have +conservation of momentum; the particles have mass and the momentum ïs still +given by ?m0, the mass times the velocity, bu the rmass changes tuïth the uelocit, +hence the momentum also changes. The mass varies with velocity according to +the law +m= TT —0——. (10.7) +V1=u2/° +where ?nọ is the mass of the body at rest and e is the speed of light. It is easy to +see from the formula that there is negligible diference between mm and mo unless +ò is very large, and that for ordinary velocities the expression for momentum +reduces to the old formula. +The components of momentum for a single particle are written as +Tĩì0U„ ThoUụ TH0uUz +mm... ——-. ¬=a.. . +where øŸ = 02+ D + 02. IÝ the z-components are summed over all the interacting +particles, both before and after a collision, the sums are equal; that is, momentum +1s conserved in the z-direction. The same holds true in any direction. +In Chapter 4 we saw that the law of conservation of energy is not valid unless +we recognize that energy appears in diferent forms, electrical energy, mechanical +energy, radiant energy, heat energy, and so on. In some of these cases, heat +energy for example, the energy might be said to be “hidden.” 'This example +might suggest the question, “Are there also hidden forms of momentum——perhaps +heat momentum?” 'Phe answer is that it is very hard to hide momentum for the +following reasons. +The random motions of the atoms of a body furnish a measure of heat energy, +1f the sguares of the velocities are summed. 'Phis sum will be a positive result, +having no directional character. The heat is there, whether or not the body moves +as a whole, and conservation oŸ energy in the form of heat is not very obvious. +On the other hand, 1ƒ one sums the 0eloczfzes, which have direction, and fñnds a +result that is not zero, that means that there is a drift of the entire body in some +particular direction, and such a gross momentum is readily observed. Thus there +is no random internal lost momentum, because the body has net momentum only +when i% moves as a whole. 'Therefore momentum, as a mechanical quantity, 1s +difcult to hide. Nevertheless, momentum cøø be hidden-—in the electromagnetic +ñeld, for example. 'This case is another efect of relativity. +--- Trang 210 --- +One of the propositions of Newton was that interactions at a distance are +instantaneous. Ït turns out that such is not the case; in situations involving elec- +trical forces, for instance, 1ƒ an electrical charge at one location is suddenly moved, +the efects on another charge, at another place, do not appear instantaneousÌy—— +there is a little delay. In those circumstances, even If the forces are equal the +momentum will not check out; there will be a short time during which there will +be trouble, because for a while the first charge will feel a certain reaction force, +say, and will picek up some momentum, but the second charge has felt nothing +and has not yet changed its momentum. lt takes time for the inÑuence tO cross +the intervening distance, which it does at 186,000 miles a second. In that tiny +time the momentum of the particles is not conserved. OÝ course after the second +charge has felt the efect of the first one and all is quieted down, the momentum +cequation will check out all right, but during that small interval momentum is not +conserved. We represent this by saying that during this interval there is another +kind of momentum besides that of the particle, mu, and that is momentum in +the electromagnetic field. If we add the feld momentum to the momentum of +the particles, then momentum is conserved at any moment all the time. “The +fact that the electromagnetic field can possess momentum and energy makes that +fñeld very real, and so, for better understanding, the original idea that there are +Jjust the forces bebween particles has to be modified to the idea that a particle +makes a field, and a field acts on another particle, and the field itself has such +familiar properties as energy content and momentum, just as particles can have. +To take another example: an electromagnetic fñeld has waves, which we call light; +it turns out that light also carries momentum with it, so when light impinges +on an object 1% carries in a certain amount of momentum per second; this is +equivalent to a force, because if the illuminated object is picking up a certain +amount of momentum per second, its momentum is changing and the situation +1s exactly the same as If there were a force on it. Light can exert pressure by +bombarding an object; this pressure is very small, but with sufficiently delicate +apparatus it is measurable. +Now in quantum mechanics it turns out that momentum is a diferent thing—— +1E is no longer rm0. It is hard to defñne exactly what is meant by the velocity oŸ a +particle, but momentum still exists. In quantum mechanies the diference is that +when the particles are represented as particles, the momentum 1s still ru, but +when the particles are represented as waves, the momentum is measured by the +number of waves per centimeter: the greater this number of waves, the greater +the momentum. In spite of the diferences, the law of conservation of momentum +--- Trang 211 --- +holds also in quantum mechanics. Even though the law #! = rma is false, and +all the derivations of NÑewton were wrong for the conservation oŸ momentum, in +quantum mechanics, nevertheless, in the end, that particular law maintains itselfl +--- Trang 212 --- +Weoe£or-s +11-1 Symmetry in physỉcs +In this chapter we introduce a subject that is technically known in physics +as sumưmnetrụ tín phụsical lau. The word “symmetry” is used here with a special +meaning, and therefore needs to be defñned. When is a thing symmetrical—how +can we defne it? When we have a picture that is symmetrical, one side 1s +somehow the same as the other side. Professor Hermann Weyl has given this +defnition of symmetry: a thing is symmetrical iŸ one can subject it to a certain +operation and it appears exactly the same after the operation. Eor instance, If +we look at a silhouette of a vase that is left-and-right symmetrical, then turn it +1802 around the vertical axis, it looks the same. We shall adopt the definition +of symmetry in Weyl's more general form, and in that form we shall discuss +symmetry of physical laws. +Suppose we bưild a complex machine in a certain place, with a lot of compli- +cated interactions, and balls bouneing around with forces between them, and so +on. Now suppose we build exactly the same kind of equipment at some other +place, matching part by part, with the same dimensions and the same orientation, +everything the same only displaced laterally by some distance. Khen, if we start +the two machines in the same initial circumstances, in exact correspondence, we +ask: will one machine behave exactly the same as the other? WIHI ít follow all the +motions in exact parallelism? Of course the answer may well be øø, because 1Í we +choose the wrong place for our machine it might be inside a wall and interferences +from the wall would make the machine not work. +AII of our ideas in physics require a certain amount of common sense in their +application; they are not purely mathematical or abstract ideas. We have to +understand what we mean when we say that the phenomena are the same when +we move the apparatus to a new position. We mean that we move everything +that we believe is relevant; 1f the phenomenon is not the same, we suggest that +--- Trang 213 --- +something relevant has not been moved, and we proceed to look for it. TỶ we +never fñnd it, then we claim that the laws of physics do not have this symmetry. +Ôn the other hand, we may find it—we expect to fnd it—ïf the laws of physics +do have this symmetry; looking around, we may discover, for instance, that the +wall is pushing on the apparatus. The basic question is, if we defñne things well +enough, If all the essential forces are included inside the apparatus, ïf all the +relevant parts are moved from one place to another, wiïll the laws be the same? +WIII the machinery work the same way? +Tt is clear that what we want to do is to move all the equipment and essenfial +Iinuences, but not cuerwthzng in the world—planets, stars, and all—for If we +do that, we have the same phenomenon again for the trivial reason that we are +ripht back where we started. No, we cannot move cuerwth”ng. But it turns out +in practice that with a certain amount of intelligence about what to move, the +machinery will work. In other words, if we do not go inside a wall, If we know +the origin of the outside forces, and arrange that those are moved too, then the +machinery 6 work the same in one location as in another. +11-2 Translations +We shall limit our analysis to just mechanics, for which we now have sufficient +knowledge. In previous chapters we have seen that the laws of mechanics can be +summarized by a set of three equations for each particle: +m(d°+/dt?) = F„, m(d®u/dt2) = Fụ, m(dÊz/d12) = F;. (11.1) +Now this means that there exists a way tO measure ø, ụ, and z on three perpen- +dicular axes, and the forces along those directions, such that these laws are true. +These must be measured from some origin, but œhere do t0e pu‡ the origin? All +that Newton would tell us at fñrst is that there ¡s some place that we can measure +from, perhaps the center of the universe, such that these laws are correct. But we +can show immediately that we can never ñnd the center, because if we use some +other origin it would make no diference. In other words, suppose that there are +two people—Joe, who has an origin in one place, and Moe, who has a parallel +system whose origin is somewhere else (Eig. II-I). Ñow when Joe measures the +location of the point in space, he fnds it at #z, , and z (we shall usually leave z +out because it is too confusing to draw in a picture). Moe, on the other hand, +when measuring the same point, will obtain a diferent + (in order to distinguish +--- Trang 214 --- +JOE_ |MOE +x x xí +Fig. 11-1. Two parallel coordinate systems. +it, we will call it +), and in principle a diferent , although in our example they +are numerically equal. So we have ++ =#— d, =ụ, z'=z. (11.2) +Now in order to complete our analysis we must know what Moe would obtain for +the forces. The force is supposed to act along some line, and by the force in the +z-direction we mean the part of the total which is in the z-direction, which is +the magnitude of the force times this cosine of its angle with the zø-axis. Now +we see that Moe would use exactly the same proJection as Joe would use, so we +have a set of equations +Hạ — Fựạ, Tự = Đụ, đà. —= F). (11.3) +'These would be the relationships between quantities as seen by jJoe and Moe. +The question 1s, if Joe knows Newton”s laws, and If Moe tries to write +down Newton's laws, will they also be correct for hữm? Does it make any +diference rom which origin we measure the points? In other words, assuming +that equations (11.1) are true, and the Bqs. (11.2) and (11.3) gïve the relationship +of the measurements, is iW or is it not true that +(a) m(d®z/di?) = F„„, +(b) m(d2y//4”) = Fạ, (114) +(c)_ m(d2z'/di?) = F„.? +In order to test these equations we shall diferentiate the formula for øˆ bwice. +First of all +dd ( ) d> — da +———= (#—d)=————.. +dt dt dt — dt +--- Trang 215 --- +NÑow we shall assume that Moe's origin is fxed (not moving) relative to Joe”s; +therefore ø is a constant and đa/dt = 0, so we fnd that +da /dt = da/dt +and therefore +d2z' /dt? = d°+/d); +therefore we know that Eq. (11.4a) becomes +m(d°+/dt?) = Fị›. +(W© also suppose that the masses measured by Joe and Moe are equal.) Thus +the acceleration times the mass is the same as the other fellow's. We have also +found the formula for F7, for, substituting from Ead. (11.1), we find that +Từ —= Fụ. +Therefore the laws as seen by Moe appear the same; he can write Newton's +laws too, with diferent coordinates, and they will still be right. That means that +there is no unique way to defne the origin of the world, because the laws will +appear the same, from whatever position they are observed. +'This 1s also true: ïf there is a piece of equipment in one place with a certain +kind of machinery in it, the same equipment in another place will behave in the +same way. Why? Because one machine, when analyzed by Moe, has exactly the +same equations as the other one, analyzed by Joe. Since the eguations are the +same, the phenornena appear the same. So the proof that an apparafus in a new +position behaves the same as it did in the old position is the same as the proof +that the equations when displaced in space reproduce themselves. 'Pherefore +we say that the laus oƒ phụsics are sụmmetrical [or translatlional đisplacemenis, +symmetrical in the sense that the laws do not change when we make a translation +of our coordinates. OÝ course it is quite obvious intuitively that this is true, but +1E is interesting and entertaining to discuss the mathematics of it. +11-3 Rotations +The above is the first of a series of ever more complicated propositions +concerning the symmetry of a physical law. The next proposition is that it should +make no diference in which đirecfion we choose the axes. In other words, If we +--- Trang 216 --- +build a piece of equipment in some place and watch it operate, and nearby we +buïld the same kind of apparatus but put it up on an angle, will it operate in the +same way? Obviously ¡it will not if it is a Grandfather clock, for examplel If a +pendulum clock stands upright, ¡it works fine, but ïf ¡it is tilted the pendulum falls +against the side of the case and nothing happens. The theorem is then false in +the case of the pendulum clock, unless we include the earth, which ¡is pulling on +the pendulum. Therefore we can make a prediction about pendulum clocks 1Ÿ we +believe in the symmetry of physical law for rotation: something else is involved in +the operation oŸ a pendulum clock besides the machinery of the clock, something +outside it that we should look for. We may also predict that pendulum clocks will +not work the same way when located in diÑferent places relative to this mysterious +Source of asymmetry, perhaps the earth. Indeed, we know that a pendulum clock +up ïn an artificial satellite, for example, would not tick either, because there is no +effective force, and on Mars it would go at a diferent rate. Pendulum clocks đo +involve something more than just the machinery inside, they involve something +on the outside. Onece we recognize this factor, we see that we must turn the earth +along with the apparatus. Of course we do not have to worry about that, it is easy +to do; one simply waits a moment or ©wo and the earth turns; then the pendulum +clock ticks again in the new position the same as it did before. While we are +rotating in space our angles are always changing, absolutely; this change does not +seem to bother us very much, for in the new position we seem to be in the same +condition as in the old. 'This has a certain tendency to confuse one, because 1 +1s true that in the new turned position the laws are the same as in the unturned +position, but it is nof true that as 0e turn a thíng ï€ follows the same laws as it +does when we are not turning it. IÝ we perform sufficiently delicate experiments, +we can tell that the earth ¡s rofa#ng, but not that it had rotated. In other words, +we cannot locate its angular position, but we can tell that it is changing. +Now we may discuss the efects of angular orientation upon physical laws. +Let us ñnd out whether the same game with Joe and Moe works again. 'This +time, to avoid needless complication, we shall suppose that Joe and Moe use the +same origin (we have already shown that the axes can be moved by translation +to another place). Assume that Moe”s axes have rotated relative to Joe's by an +angle Ø. The two coordinate systems are shown in Fig. l1I-2, which is restricted +to two dimensions. Consider any point P having coordinates (z,) in jJoe's +system and (z”,') in Moe's system. We shall begin, as in the previous case, by +expressing the coordinates zø“ and 3“ in terms of z, #, and Ø. To do so, we first +drop perpendiculars from ? to all four axes and draw 4Ö perpendicular to PQ. +--- Trang 217 --- +— (& ý) +ca N ysin8 (MOE) +S0 NI x +xcos0 ~Zr +PB (JOE) +Fig. 11-2. IWwo coordinate systems having different angular orienta- +tions. +Inspection of the fgure shows that #“ can be written as the sum of two lengths +along the ø-axis, and ø as the difference of two lengths along 4Ø. All these +lengths are expressed in terms of z, , and Ø in equations (11.5), to which we +have added an equation for the third dimension. ++“ = #øcos 8 + 1 sin 6, +ˆ = cos0 — zsin0, (11.5) +The next step is to analyze the relationship of forces as seen by the two observers, +following the same general method as before. Let us assume that a force #', which +has already been analyzed as having components „ and #2 (as seen by Joe), is +acting on a particle of mass rn, located at point Pín Fig. 11-2. For simplicity, +let us move both sets of axes so that the origin is at , as shown in Eig. l1-3. +Moe sees the components of #" along his axes as F and È;¿. F„ has components +along both the z/- and '-axes, and #„ likewise has components along both these +axes. To express #¿ in terms of F; and #„, we sum these components along the +z/-axis, and in a like manner we can express #2 in terms of #+ and F„. The +results are +tạ = Fạ cos Ø + Fý, sin Ø, +Tàu = Fy cosØ — F„ sìn Ú, (11.6) +đà, =F,. +Tt is interesting to note an accident of sorts, which is of extreme importance: the +formulas (11.5) and (11.6), for coordinates oŸ P and components of #", respectively, +are 0ƒ identical form. +--- Trang 218 --- +FyE-------=z F +_~“4 ụ +FT BÀ x: +Fig. 11-3. Components of a force in the two systems. +As before, Newton”s laws are assumed to be true in Joe°s system, and are +expressed by equations (11.1). The question, again, is whether Moe can apply +Newton”s laws—will the results be correct for his system of rotated axes? In +other words, if we assurne that Eqs. (11.5) and (11.6) give the relationship of the +mmeasurements, is it true or not true that +m(d°z! (dt?) = F:, +m(d2W' dt?) = Fụ, (11.7) +m(d°z! (dt?) = F..? +To test these equations, we calculate the left and right sides independently, and +compare the results. To calculate the left sides, we multiply equations (11.5) +by n, and diferentiate twice with respect to time, assuming the angle Ø to be +constant. This gives +m(d°+! (dt?) = m(dÊ+/df?) cos 9 + m(d®u /dt2) sin 0, +m(dSV /dt?) = m(d /đt2) cos 9 — rn(dŠ+/dt?) sìn 6, (11.8) +m(d°z! /dt?) = m(d°z/d12). +W© calculate the right sides of equations (11.7) by substituting equations (11.1) +into equations (11.6). This gives +F}„ = m(d°®+/đt?) eos 0 + m(d2u/d?) sìn 0, +Đụ = m(d°0/đt?) cos 9 — m(d2+/đf?) sìn 0, (11.9) +F.. = m(d°z/di?). +--- Trang 219 --- +Behold! The right sides of Eqs. (11.5) and (11.9) are identical, so we conclude +that if Newton's laws are correct on one set of axes, they are also valid on +any other set of axes. 'This result, which has now been established for both +translation and rotation of axes, has certain consequences: first, no one can +claim his particular axes are unique, but of course they can be more conwuen¿ent +for certain particular problems. Eor example, it is handy to have gravity along +one axis, but this is not physically necessary. Second, it means that any piece +of equipment which ¡is completely selfcontained, with all the force-generating +equipment completely inside the apparatus, would work the same when turned +at an angle. +11-4 Vectors +Not only Newton's laws, but also the other laws of physics, so far as we know +today, have the two properties which we call invariance (or symmetry) under +translation of axes and rotation of axes. These properties are so important that a +mathematical technique has been developed to take advantage of them in writing +and using physical laws. +The foregoing analysis involved considerable tedious mathematical work. To +reduce the details to a minimum in the analysis of such questions, a very powerful +mmathematical machinery has been devised. 'Phis system, called uector ønalJsis, +supplies the title of this chapter; strictly speaking, however, this is a chapter on +the symmetry of physical laws. By the methods of the preceding analysis we +were able to do everything required for obtaining the results that we sought, but +in practice we should like to do things more easily and rapidly, so we employ the +vector technique. +We began by noting some characteristics of two kinds of quantities that are +important in physics. (Acbtually there are more than two, but let us start out +with ©wo.) One of them, like the number of potatoes in a sack, we call an ordinary +quantity, or an undirected quantity, or a scøiar. Temperature is an example of +such a quantity. Other quantities that are important in physics do have direction, +for Instance velocity: we have to keep track of which way a body is going, not +Just its speed. Momentum and force also have direction, as does displacement: +when someone steps from one place to another in space, we can keep track of +how far he went, but if we wish also to know œhere he went, we have to specify a +direction. +All quantities that have a direction, like a step in space, are called 0ectors. +--- Trang 220 --- +A vector is three numbers. In order to represent a step in space, say from the +origin to some particular point whose location is (z,, z), we really need three +numbers, but we are going to invent a single mathematical symbol, r, which is +unlike any other mathematical symbols we have so far used.* It is no£ a single +number, it represents #hree numbers: z, ¿, and z. It§ means three numbers, but not +really only £hose three numbers, because If we were to use a different coordinate +system, the three numbers would be changed to 4, , and z”. However, we want +to keep our mathematics simple and so we are going to use the sœne rnark to +represent the three numbers (z,,2) and the three numbers (z',',z7). That +1s, we use the same mark to represent the first set of three numbers for one +coordinate system, but the second set oŸ three numbers if we are using the other +coordinate system. This has the advantage that when we change the coordinate +system, we do not have to change the letters of our equations. lfÝ we write an +equatfion in terms of #z, , z, and then use another system, we have to change to +',,Z, but we shall just write r, with the convention that it represents (#, , Z) +1Ÿ we use one set of axes, or (#,, z7) 1ƒ we use another seb oŸ axes, and so on. +The three numbers which describe the quantity in a given coordinate system are +called the componenfs oŸ the vector in the direction of the coordinate axes of that +system. 'That is, we use the same symbol for the three letters that correspond +to the sưme objecf, œs seen [rom difƒerent azes. The very fact that we can say +“the same object” implies a physical intuition about the reality of a step in space, +that is independent of the components in terms of which we measure it. So the +symbol ? will represent the same thing no matter how we turn the axes. +Now suppose there is another directed physical quantity, any other quantity, +which also has three numbers associated with it, like force, and these three +numbers change to three other numbers by a certain mathematical rule, iÝ we +change the axes. It must be the same rule that changes (z, , z) into (4,3,2). In +other words, any physical quantity associated with three numbers which transform +as do the components of a step in space is a vector. An equation like +would thus be true in am coordinate system ïf it were true in one. 'This equation, +Of course, stands for the three equations +hHụ — ø, Tụ —= U, h} —z, +* In type, vectors are represented by boldface; in handwritten form an arrow is used: ?* +--- Trang 221 --- +or, alternatively, for +Fyụ =a, Fụ =, Ty, =zZ. +The fact that a physical relationship can be expressed as a vector equation assures +us the relationship is unchanged by a mere rotation of the coordinate system. +'That is the reason why vectors are so useful in physics. +NÑow let us examine some of the properties of vectors. Âs examples of vecbors +we may mention velocity, momentum, force, and acceleration. For many purposes +1t is convenient to represent a vector quantity by an arrow that indicates the +direction in which it is acting. Why can we represent force, say, by an arrow? +Because it has the same mathematical transformation properties as a “step In +space.” We thus represent it in a diagram as If it were a step, using a scale such +that one unit of force, or one newton, corresponds to a certain convenient length. +Once we have done this, all forces can be represented as lengths, because an +cequation like +FP'=kr, +where & is some constant, is a perfectly legitimate equation. Thus we can always +represent forces by lines, which is very convenient, because once we have drawn +the line we no longer need the axes. Of course, we can quickly calculate the +three componentfs as they change upon turning the axes, because that is just a +geometric problem. +11-5 Vector algebra +Now we must describe the laws, or rules, for combining vecfors in various ways. +'The first such combination is the øđd/fzon oftwo vectors: suppose that œ is a vector +which in some particular coordinate system has the three components (đ„, đ„, đ;), +and that b is another vector which has the three components (b„, b„,b„). Ñow +let us invent three new numbers (a„ + Ö„,d„ + b„,a„ + b;). Do these form a +vector? “Well,” we might say, “they are three numbers, and every three numbers +form a vector.” Ño, no£ every three numbers form a vector! In order for it to be +a vector, not only must there be three numbers, but these must be associated +with a coordinate system in such a way that 1Ý we turn the coordinate system, +the three numbers “revolve” on each other, get “mixed up” in each other, by +the precise laws we have already described. So the question is, if we now rotate +the coordinate system so that (a„, #,,ø„) become (đx, đ„,øz:) and (b„, b„, b„) +--- Trang 222 --- +become (b„¿, b„/, b„;), what do (az„ + bạ, ay + bự, ø„ + b„) become? Do they become +(Ga + bạ, dự -E bu,a„; + by) or not? The answer is, oÝ course, yes, because +the prototype transformations of Eq. (11.5) constitute what we call a ii¿mear +transformation. If we apply those transformations to ø„ and Ö„ to get aœ„ + bạ, +we fnd that the transformed a„ -+ b„ is indeed the same as ø„; + b„;. When œ +and ö are “added together” in this sense, they will form a vector which we may +call e. We would write this as +c=œa+b. +Now e has the interesting property +c=b+ea, +as we can immediately see from i%s components. 'Thus also, +œ-+(b+c)=(a+b)+c. +W© can add vectors in any order. +What is the geometric significance of œ + b? Suppose that œ and b were +represented by lines on a piece of paper, what would e look like? 'This is shown +in Eig. II-4. We see that we can add the components of b to those oŸ œ most +conveniently if we place the rectangle representing the components of next to +that representing the components of ø in the manner indicated. Since b just +“fñts” into its rectangle, as does ø into its rectangle, this is the same as putting +the “tail” of b on the “head” of ø, the arrow from the “tail” of œ to the “head” +of b being the vector œ. OÝ course, if we added ø to b the other way around, we +___— ' +__— /. __—_— Ị +I 1 Xx +Fig. 11-4. The addition of vectors. +--- Trang 223 --- +would put the “tail” of œ on the “head” of b, and by the geometrical properties +of parallelograms we would get the same result for c. Note that vectors can be +added in this way without reference to any coordinate axes. +Suppose we multiply a vector by a number œ, what does this mean? We +deffne it to mean a new vector whose components are œø„, œa„, and œaz;. We +leave 1t as a problem for the student to prove that it 7s a vector. +Now let us consider vector subtraction. We may deñne subtraction in the +same way as addition, but instead of adding, we subtract the components. Ôr +we might defñne subtraction by defning a negative vector, —b = —1b, and then +we would add the components. ÏIt comes to the same thing. The result ¡is shown +in Eig. 11-5. This fñgure shows d = œ— b= ø-+ (—Ùb); we also note that the +diference ø — b can be found very easily from ø and b by using the equivalent +relatlon œ = b+ d. 'Thus the diference is even easier to find than the sum: we +Jjust draw the vector from b to ø, to get œ — bÌ +Fig. 11-5. The subtraction of vectors. +Next we discuss velocity. Why is velocity a vector? lÝ position is given +by the three coordinates (z,,z), what is the velocity? "The velocity is given +by dz/dt, dụ/dt, and dz/dt. Is that a vector, or not? We can fnd out by +differentiating the expressions in Eq. (11.5) to find out whether đ+ /đ transƒorms +in the ripht way. We see that the components đz/đt and dụ/dt do transform +according to the same law as # and , and therefore the time derivative 2s a +vector. 5o the velocity is a vector. We can write the velocity in an interesting +WayV aS += dr(dt. +What the velocity is, and why i% is a vector, can also be understood more +pictorially: How far does a particle move in a short time A£? Answer: Az, so if +a particle is “here” at one instant and “there” at another instant, then the vector +diference of the positions Am = rs — r, which is in the direction oŸ motion +--- Trang 224 --- +shown in Fig. 11-6, divided by the time interval Af = ‡a — f, is the “average +velocity” vector. +Ar = ra — +T2 1 +Fig. 11-6. The displacement of a particle in a short time interval Af = +ta — tì. +In other words, by vector velocity we mean the limit, as A# goes to 0, of the +diference between the radius vectors at the time £ + A£ and the time ý, divided +by Ai: +ò= lim (Ar/At) = dr/át. (11.10) +Thus velocity is a vector because it is the difference of two vectors. ÏIt is also the +right defnition of velocity because its components are d+/dt, dụ/dt, and dz/di. +In fact, we see from this argument that if we diferentiate amw vector with respect +to time we produce a new vector. So we have several ways of producing new +vectors: (1) multiply by a constant, (2) diferentiate with respect to time, (3) add +or subtract bwo vectOrs. +11-6 Newton°s laws in vector notation +In order to write Newton”s laws in vector form, we have to go Just one step +further, and defne the acceleration vector. 'This is the time derivative of the +velocity vector, and it is easy to demonstrate that its components are the second +derivatives of z, , and z with respect to ý: +đu đÀ (dr đ?r +dt dt dt đị2 +duy d2z đuy dầu dù; d2z +TC dị) C9 CA d) “5” đÐ — dự 112) +--- Trang 225 --- +With this defñnition, then, Newton's laws can be written in this way: +ma = F (11.13) +m(dŠr/dt?) = F. (11.14) +Now the problem of proving the invariance of Ñewton”s laws under rotation +of coordinates is this: prove that œ is a vector; this we have just done. Prove +that #' is a vector; we swppose it is. So 1Í Íforce is a vector, then, since we know +acceleration is a vector, q. (11.13) will look the same in any coordinate system. +Writing ¡it in a form which does not explicitly contain zø”s, 's, and zˆs has the +advantage that from now on we need not write #hree laws every tỉme we write +Newton”s equations or other laws of physics. We write what looks like one law, +but really, of course, it is the three laws for any particular set of axes, because +any vector equation involves the statement that cach oƒ the components is cqudl. +Fig. 11-7. A curved trajectory. +The fact that the acceleration is the rate of change of the vector velocity +helps us to calculate the acceleration in some rather complicated circumstances. +Suppose, for instance, that a particle is moving on some complicated curve +(Fig. 11-7) and that, at a given instant ứ, it had a certain velocity ơi, but that +when we go to another instant £a a little later, it has a diferent velocity 0a. What +is the acceleration? Answer: Acceleration is the diference in the velocity divided +by the small time interval, so we need the diference of the two velocities. How +do we get the diference of the velocities? '[o subtract two vectors, we put the +vector across the ends of 0a and 0; that is, we draw Ao as the diference of the +two vectors, right? /o/ That only works when the #øÏs of the vectors are in the +same placel It has no meaning if we move the vector somewhere else and then +--- Trang 226 --- += ~Í\ +V2 +Fig. 11-8. Diagram for calculating the acceleration. +draw a line across, so watch outl We have to draw a new diagram to subtract +the vectors. In Fig. 11-8, 0 and 0a are both drawn parallel and equal to their +counterparts in Fig. 11-7, and now we can discuss the acceleration. Of course the +acceleration is simply Aø/Af. Tt is interesting to nobe that we can compose the +velocity diference out of two parts; we can think of acceleration as having #uo +componenis, A0||, in the direction tangent to the path and Aø_ at right angles +to the path, as indicated in Eig. 11-8. 'Phe acceleration tangent to the path is, of +course, just the change in the lengfh of the vector, i.e., the change in the speed 0: +địị = du/dt. (11.15) +The other component of acceleration, at ripght angles to the curve, is easy %O +calculate, using Eigs. I1-7 and 11-8. In the short time Af let the change in angle +bebween Øø¡ and 0a; be the small angle A0. If the magnitude of the velocity is +called ø, then of course +AUL =uA0 +and the acceleration ø will be +ø¡ = 0(A0/At). +NÑow we need to know A6/A¿, which can be found thìs way: TẾ, at the given +mmoment, the curve is approximated as a circle of a certain radius #, then in a +time A£ the distance s is, of course, 0A, where 0 is the speed. +A0 =(uAt)/R, Or A0/At = u/R. +'Therefore, we find +ai =02/R, (11.16) +as we have seen before. +11-7 Scalar product of vectors +Now let us examine a little further the properties of vectors. Ï% is easy to see +that the lengfh of a step In space would be the same in any coordinate system. +--- Trang 227 --- +'That 1s, if a particular step 7 is represented by z#, , z, In one coordinate system, +and by 4,0,2” in another coordinate system, surely the distance z = |r| would +be the same in both. Ñow +r=VW#2+ 2+ z2 +and also ++ = \/„2 +2 -+- z2. +So what we wish to verify is that these two quantities are equal. It is mụch more +convenient not to bother to take the square root, so let us talk about the square +of the distance; that ïs, let us fnd out whether +z2? +?2+z?=z^2+^2+ z2. (11.17) +It had better be—and if we substitute Eq. (11.5) we do indeed ñnd that it is. +So we see that there are other kinds of equations which are true for any ÿWO +coordinate systems. +Something new is involved. We can produce a new quantity, a function of +z, , and z, called a scalar ƒunctlion, a quantity which has no direction but which +1s the same in both systems. Out of a vector we can make a scalar. We have to +ñnd a general rule for that. It is clear what the rule is for the case just considered: +add the squares of the components. Let us now define a new thing, which we +call œ- œ. 'This is not a vector, but a scalar; it is a number that is the same in all +coordinate systems, and it is defned to be the sum of the squares of the three +components of the vector: +qŒ-d = d2 + d2 + đệ. (11.18) +Now you say, “But with what axes?” It does not depend on the axes, the answer is +the same in euer set of axes. So we have a new kznởd of quantity, a new ?nuariant +or scalar produced by one vector “squared.” IÝÍ we now defñne the following quantity +for any two vectors œ and b: +œ-b= q„bÙ„ + aub„ + azÐz, (11.19) +we fñnd that this quantity, calculated in the primed and unprimed systems, also +stays the same. To prove it we note that it is true of ø - ø, b- b, and e- c, where +--- Trang 228 --- +c=øœ+b. Therefore the sum of the squares (a„ + b„)” + (œy + b„)Ÿ + (a; + b;)? +will be invarlant: +(a„ + b„)Ÿ + (ay + bụ)Ÿ + (ay + by)” = (a„ + bại)” ++ (dự; + bự)Ÿ + (az + b„.)Š. (11.20) +Tf both sides of this equation are expanded, there will be cross produects of Jjust the +type appearing in Eq. (11.19), as well as the sums of squares oŸ the components +of œø and b. The invariance of terms of the form of Eq. (11.18) then leaves the +cross product terms (11.19) invariant also. +The quantity œ - b is called the scalar product of two vectors, œ and b, and ït +has many interesting and useful properties. For instance, it is easily proved that +œ-(b+c)=a-b+eœ-c. (11.21) +AIlso, there is a simple geometrical way to calculate ø - b, without having to +calculate the components of œ and b: ø- b is the product of the length of œ and +the length of b times the cosine of the angle between them. Why? Suppose +that we choose a special coordinate system in which the z-axis lies along œ; in +those circumstances, the only component of œ that will be there 1s ø„, which is +of course the whole length of œ. Thus Eq. (11.19) reduces to ø- Ð = a„b„ for this +case, and this is the length of œ times the component of b in the direction of œ, +that is, bcos ổ: +œ-b = abcos 0. +Therefore, in that special coordinate system, we have proved that œ - b ¡is the +length of œ times the length of b times cosØ. But ?ƒ ?# ¡s truc ?ím one coordinate +sustem, tt 1s true ím œÏÏ, because œ - b is independent of the coordinate system; +that is our argument. +What good is the dot product? Are there any cases in physics where we +need it? Yes, we need it all the time. Eor instance, in Chapter 4 the kinetic +energy was called 3m03, but ïŸ the object is moving in space it should be the +velocity squared in the z-direction, the -direction, and the z-direction, and so +the formula for kinetic energy according to vector analysis is +K.E. = šm(0- 0) = šm(02 + 0y + 9)). (11.22) +Energy does not have direction. Momentum has direction; it is a vector, and 1$ +1s the mass times the velocity vector. +--- Trang 229 --- +Another example of a dot product is the work done by a force when something +1s pushed from one place to the other. We have not yet deined work, but 1 1s +equivalent to the energy change, the weights lifted, when a force #' acts through +a distance s: +Work = È'!- 3. (11.23) +Tt is sometimes very convenient to talk about the component of a vector in +a certain direction (say the vertical direction because that is the direction of +gravity). For such purposes, it is useful to invent what we call a uw2t 0ector in +the direction that we want to study. By a unit vector we mean one whose dot +product with itself is equal to unity. Let us call this unit vector ?; then 2 - s = 1. +'Then, if we want the component of some vector in the direction of 2, we see that +the dot product ø - ? will be acosØ, i.e., the component of ø in the direction +of 2. This is a nice way to get the component; in fact, it permits us to get øiÏ +the components and to write a rather amusing formula. Suppose that in a given +system of coordinates, z, , and z, we invent three vectors: 2, a unit vector In +the direction zø; 7, a unit vector in the direction ; and &, a unit vector in the +direction z. Note frst that 2-# = 1. What is z- 7? When ÿwo vectors are at right +angles, their dot product is zero. 'Phus +:‹2=0 77=1 +z-k=0 J3-k=0 k-k=l (11.24) +Now with these definitions, any vector whatsoever can be written this way: +œ = đ„9 + du) + a„k. (11.25) +By this means we can go from the components of a vector to the vector itself. +This discussion of vectors is by no means complete. However, rather than try +to go more deeply into the subject now, we shall first learn to use in physical +situations some of the ideas so far discussed. 'Phen, when we have properly +mastered this basic material, we shall fñnd it easier to penetrate more deeply into +the subject without getting too confused. We shall later fnd that it is useful +to defñne another kind of produet of two vectors, called the vector product, and +written as œ x Ö. However, we shall undertake a discussion of such matters in a +later chapter. +--- Trang 230 --- +( her'rcforrsÉfcs ©@Ê Foree©e +12-1 What is a force? +Although it is interesting and worth while to study the physical laws simply +because they help us to understand and to use nature, one ought to sÈop every +onece in a while and think, “What do they really mean?” 'Phe meaning of any +statement is a subjJect that has interested and troubled philosophers from time +Immemorial, and the meaning of physical laws is even more interesting, because +1t is generally believed that these laws represent some kind of real knowledge. +The meaning of knowledge is a deep problem in philosophy, and ït is always +Iimportant to ask, “What does it mean?” +Let us ask, “What is the meaning of the physical laws of Newton, which we +write as ` =ma? What is the meaning of force, mass, and acceleration?” Well, +we can intuitively sense the meaning of mass, and we can đefine acceleration ïŸ +we know the meaning of position and time. We shall not discuss those meanings, +but shall concentrate on the new concept of ƒorce. 'Phe answer is equally simpIle: +“Ha body is accelerating, then there is a force on it.” That is what Newton's laws +say, so the most precise and beautiful defnition of force imaginable might simply +be to say that force is the mass of an object times the acceleration. 5uppose we +have a law which says that the conservation of momentum is valid if the sum +of all the external forces 1s zero; then the question arises, “What does it mean, +that the sum of all the external forces is zero?” A pleasant way to define that +statement would be: “When the total momentum is a constant, then the sum of +the external forces is zero.” There must be something wrong with that, because it +is Just not saying anything new. If we have discovered a fundamental law, which +asserts that the force is equal to the mass times the acceleration, and then defne +the force to be the mass times the acceleration, we have found out nothing. We +could also defñne force to mean that a moving object with no force acting on i§ +continues to move with constant velocity in a straight line. If we then observe an +object not moving in a straight line with a constant velocity, we might say that +--- Trang 231 --- +there is a force on it. Now such things certainly cannot be the content of physics, +because they are defnitions going In a circle. The Newtonian statement above, +however, seems to be a most precise definition of force, and one that appeals to +the mathematician; nevertheless, it is completely useless, because no prediction +whatsoever can be made from a definition. One might sit in an armchair all +day long and deñne words at will, but to ñnd out what happens when two balls +push against each other, or when a weight is hung on a spring, is another matter +altogether, because the way the bodies behøaue 1s something completely outside +any choice of definitions. +For example, if we were to choose to say that an object left to itself keeps its +position and does not move, then when we see something drifting, we could say +that must be due to a “gorce”——a gorce is the rate of change of position. Now we +have a wonderful new law, everything stands still except when a gorce is acting. +You see, that would be analogous to the above definition of force, and it would +contain no information. 'The real content of Newton”s laws is this: that the force +1s supposed to have some ?mdependent properties, in addition to the law P — ma; +but the speczfc independent properties that the force has were not completely +described by Newton or by anybody else, and therefore the physical law ` = na is +an ineomplete law. It implies that if we study the mass times the acceleration and +call the product the force, i.e., 1 we study the characteristics of force as a program +of interest, then we shall fnd that forces have some simplicity; the law is a good +program for analyzing nature, it is a suggestion that the forces will be simple. +Now the first example of such forces was the complete law of gravitation, +which was given by Newton, and ín stating the law he answered the question, +“What is the force?” If there were nothing but gravitation, then the combination +of this law and the force law (second law of motion) would be a complete theory, +but there is mụch more than gravitation, and we want to use Newton's laws in +many different situations. 'Therefore in order to proceed we have to tell something +about the properties of force. +For example, in dealing with force the tacit assumption is always made that +the force is equal to zero unless some physical body is present, that If we fnd a +force that is not equal to zero we also find something ¡in the neighborhood that is +a source of the force. 'PThis assumption is entirely diferent from the case of the +“gorce” that we introduced above. Ône of the most important characteristics of +force is that it has a material origin, and this is nø£ just a defñnition. +Newton also gave one rule about the force: that the forces between interacting +bodies are equal and opposite—action equals reaction; that rule, it turns out, +--- Trang 232 --- +1s not exactly true. In fact, the law = ma is not exactly true; iÝ iE were a +defnition we should have to say that 1t is øløas exactly true; but ït is not. +The student may object, “[ do not like this imprecision, I should like to have +everything defned exactly; in fact, it says in some books that any science 1s +an exact subject, in which cuerwthing is deñned.” TỶ you insist upon a precise +defnition of force, you will never get itl Pirst, because Newton”s Second Law is +not exact, and second, because in order to understand physical laws you must +understand that they are all some kind of approximation. +Any simple idea is approximate; as an illustration, consider an object,... what +is an object? Philosophers are always saying, “Well, Just take a chair for example.” +"The moment they say that, you know that they do not know what they are talking +about any more. What ¿s a chair? Well, a chair is a certain thing over there.... +certain?, how certain? 'Phe atoms are evaporating from ït from time to tỉme——=not +many atoms, but a few—dirt falls on it and gets dissolved in the paint; so to +defne a chaïr precisely, to say exactly which atoms are chaïir, and which atoms are +air, or which atoms are dirt, or which atoms are paint that belongs to the chaïr is +impossible. So the mass of a chair can be defned only approximately. In the same +way, to delñne the mass of a single object is impossible, because there are not any +single, left-alone objects in the world—every object is a mixture of a lot of things, +so we can deal with it only as a series Of approximations and idealizations. +'The trick is the idealizations. 'To an excellent approximation of perhaps one +part in 1010, the number of atoms in the chair does not change in a minute, and +1Í we are not too precise we may idealize the chair as a defñnite thing; in the same +way we shall learn about the characteristics of force, in an ideal fashion, if we +are not too precise. Ône may be dissatisied with the approximate view of nature +that physics tries to obtain (the attempt is always to increase the accuracy of the +approximation), and may prefer a mathematical definition; but mathematical +defnitions can never work in the real world. A mathematical definition will be +good for mathematics, in which all the logic can be followed out completely, but +the physical world is complex, as we have indicated in a number of examples, such +as those of the ocean waves and a glass of wine. When we try to isolate pieces of it, +to talk about one mass, the wine and the glass, how can we know which is which, +when one dissolves in the other? 'Phe forces on a single thing already involve +approximation, and if we have a system of discourse about the real world, then that +system, at least for the present day, must involve approximations of some kind. +'This system ïs quite unlike the case of mathematics, in which everything can +be defñned, and then we do not knou what we are talking about. In fact, the glory +--- Trang 233 --- +of mathematies is that 0e do no‡ hœue to sa that tue are talking abou‡. The gÌory +1s that the laws, the arguments, and the logic are independent of what “it” is. lÝ +we have any other set of objects that obey the same system of axioms as Buclid”s +geometry, then if we make new defñnitions and follow them out with correct logic, +all the consequences will be correct, and it makes no diference what the subject +was. In nature, however, when we draw a line or establish a line by using a light +beam and a theodolite, as we do in surveying, are we measuring a line in the sense +of Euclid? No, we are making an approximation; the cross hair has some width, +but a geometrical line has no width, and so, whether Buclidean geometry can be +used for surveying or not is a physical question, not a mathematical question. +However, from an experimental standpoint, not a mathematical standpoint, we +need to know whether the laws of Euelid apply to the kind of geometry that we +use in measuring land; so we make a hypothesis that it does, and it works pretty +well; but it is not precise, because our surveying lines are not really geometrical +lines. Whether or not those lines of Euclid, which are really abstract, apply to +the lines of experience is a question for experienece; it is not a question that can +be answered by sheer reason. +In the same way, we cannot just call ?' = rmma a defnition, deduce everything +purely mathematically, and make mechanics a mathematical theory, when me- +chanics is a description of nature. By establishing suitable postulates it is always +possible to make a system of mathematics, just as Euclid did, but we cannot make +a mathematics of the world, because sooner or later we have to fnd out whether +the axioms are valid for the objects of nature. Thus we immediately get involved +with these complicated and “dirty” objects of nature, but with approximations +©V€T lnCreasing In accuracy. +12-2 Eriction +The foregoing considerations show that a true understanding of NÑewton'°s laws +requires a discussion of forces, and ït is the purpose of this chapter to introduce +such a discussion, as a kind of completion of Newton's laws. We have already +studied the defnitions of acceleration and related ideas, but now we have to +study the properties of force, and this chapter, unlike the previous chapters, will +not be very precise, because forces are quite complicated. +To begin with a particular force, let us consider the drag on an airplane fying +through the air. What is the law for that force? (Surely there is a law Íor every +force, we rmus‡ have a lawl) One can hardly think that the law for that force will +--- Trang 234 --- +be simple. 'Iry to imagine what makes a drag on an airplane flying through the +air—the air rushing over the wings, the swirling in the back, the changes going +on around the fuselage, and many other complications, and you see that there is +not going to be a simple law. Ôn the other hand, it is a remarkable fact that the +drag force on an airplane is approximately a constant times the square of the +velocity, or F` cu. +Now what is the status of such a law, is i9 analogous to F' = ma? Not at +all, because in the first place this law is an empirical thing that is obtained +roughly by tests in a wind tunnel. You say, “Well ' = rmø might be empirical +too.” That is not the reason that there is a diference. The difference is not that +1t is empirical, but that, as we understand nature, this law is the result of an +enormous complexity of events and is not, fundamentally, a simple thíng. ITf +we continue to study it more and more, measuring more and more accurately, +the law will continue to become more complicated, not /ess. In other words, as +we study this law of the drag on an airplane more and more closely, we find +out that it is “falser” and “falser,” and the more deeply we study it, and the +more accurately we measure, the more complicated the truth becomes; so in that +sense we consider it not to result from a simple, fundamental process, which +agrees with our original surmise. Eor example, if the velocity 1s extremely low, +so low that an ordinary airplane is not ñying, as when the airplane is dragged +slowly through the aïr, then the law changes, and the drag friction depends more +nearly linearly on the velocity. To take another example, the frictional drag on +a ball or a bubble or anything that is moving slowly through a viscous liquid +like honey, is proportional to the velocity, but for motion so fast that the fÑuid +swirls around (honey does not but water and air do) then the drag becomes more +nearly proportional to the square of the velocity (F' = cø”), and if the velociEy +continues to increase, then even this law begins to fail. People who say, “Well +the coefficient changes slightly,” are dodging the issue. Second, there are other +great complications: can this force on the airplane be divided or analyzed as a +force on the wings, a force on the front, and so on? Indeed, this can be done, +1ƒ we are concerned about the torques here and there, but then we have to get +special laws for the force on the wings, and so on. It is an amazing fact that +the force on a wing depends upon the other wing: in other words, if we take +the airplane apart and put just one wing in the air, then the force is not the +same as If the rest of the plane were there. The reason, of course, is that some +of the wind that hits the front goes around to the wings and changes the force +on the wings. Ít seems a miracle that there is such a simple, rough, empirical +--- Trang 235 --- +law that can be used in the design of airplanes, but this law is not in the same +class as the basic laws of physics, and further study of it will only make it more +and more complicated. AÁ study of how the coefficient e depends on the shape +of the front of the airplane is, to put ¡1% mildly, frustrating. 'Phere jusÈ is no +simple law for determining the coefficient in terms of the shape of the airplane. +In contrast, the law of gravitation is simple, and further study only indicates its +greater simplicity. +We have just discussed bwo cases of friction, resulting from fast movement in +air and slow movement in honey. There is another kind of friction, called dry +triction or sliding friction, which occurs when one solid body slides on another. +In this case a force is needed to maintain motion. 'This is called a frictional force, +and its origin, also, is a very complicated matter. Both surfaces of contact are +irregular, on an atomie level. 'Phere are many points of contact where the atoms +seem to cling together, and then, as the sliding body is pulled along, the atoms +snap apart and vibration ensues; something like that has to happen. Formerly +the mechanism of this friction was thought to be very simple, that the surfaces +were merely full of irregularities and the friction originated in liting the slider +over the bumps; but this cannot be, for there is no loss of energy in that process, +whereas power is in facE consumed. “The mechanism of power loss is that as +the slider snaps over the bumps, the bumps deform and then generate waves +and atomic motions and, after a while, heat, in the two bodies. NÑow 1È 1s very +remarkable that again, empirically, this friction can be described approximately +by a simple law. 'This law is that the force needed to overcome friction and to +drag one object over another depends upon the normal force (i.e., perpendicular +to the surface) between the two surfaces that are in contact. Actually, to a fairly +good approximation, the frictional force is proportional to this normal force, and +has a more or less constant coefficient; that is, +†=uN, (12.1) +where / is called the coeffficient oƒ fricion (Eig. 12-1). Although this coeflicient +1s not exactly constant, the formula is a good empirical rule for Judging approxi- +mately the amount of force that will be needed in certain practical or engineering +circumstances. If the normal force or the speed of motion gets too big, the law +fails because of the excessive heat generated. lt is important to realize that each +of these empirical laws has its limitations, beyond which ¡it does not really work. +'That the formula #' = uN is approximately correct can be demonstrated by +a simple experiment. We set up a plane, inclined at a small angle Ø, and place a +--- Trang 236 --- +—>= DIRECTION OF MOTION +Fig. 12-1. The relation between frictional force and the normal force +for sliding contact. +block of weight W/ on the plane. We then tilt the plane at a steeper angle, until +the block just begins to slide from its own weight. The component of the weight +downward along the plane is W sinØ, and this must equal the frictional force #! +when the block is sliding uniformly. 'Phe component of the weight normail to the +plane is W cosØ, and this is the normal force /Ú. With these values, the formula +becomes Wƒ sin Ø = W cosØ, from which we get = sỉn Ø/ cos Ø = tan Ø. TỶ this +law were exactly true, an object would start to slide at some defñnite inclination. +T the same block is loaded by putting extra weight on it, then, although W ¡is +increased, all the forces in the formula are increased in the same proportion, +and W canecels out. If ð stays constant, the loaded block will slide again at the +same slope. When the angle Ø is determined by trial with the original weight, it +is found that with the greater weight the block will slide at about the same angle. +This will be true even when one weight is many times as great as the other, and +so we conclude that the coefficient of friction is independent of the weight. +In performing this experiment it is noticeable that when the plane ïs tilted +at about the correct angle Ø, the block does not slide steadily but in a halting +fashion. At one place it may stop, at another it may move with acceleration. This +behavior indicates that the coefficient of friction is only roughly a constant, and +varies from place to place along the plane. The same erratic behavior is observed +whether the block is loaded or not. Such variations are caused by diferent degrees +of smoothness or hardness of the plane, and perhaps dirt, oxides, or other foreign +matter. The tables that list purported values of for “steel on sbeel,” “copper +on copper,” and the like, are all false, because they ignore the factors mentioned +above, which really determine . “The friction is never due to “copper on copper,” +etc., but to the impurities clinging to the copper. +In experiments of the type described above, the friction is nearly independent +of the velocity. Many people believe that the friction to be overcome to get +--- Trang 237 --- +something started (static friction) exceeds the force required to keep it sliding +(sliding friction), but with dry metals it is very hard to show any diference. The +opinion probably arises from experiences where small bits of oil or lubricant are +present, or where blocks, for example, are supported by springs or other fexible +supports so that they appear to bind. +Tt ¡is quite dificult to do accurate quantitative experiments In friction, and +the laws of friction are still not analyzed very well, in spite of the enormous +engineering value of an accurate analysis. Although the law #' = ðN is fairly +accurate once the surfaces are standardized, the reason for this form of the law +is not really understood. To show that the coeficient is nearly independent of +velocity requires some delicate experimentation, because the apparent friction 1s +much reduced ïf the lower surface vibrates very fast. When the experiment is done +at very hiph speed, care must be taken that the obJjects do not vibrate relative +to one another, since apparent decreases of the friction at hipgh speed are often +due to vibrations. At any rate, this friction law is another of those semiempirical +laws that are not thoroughly understood, and in view of all the work that has +been done it is surprising that more understanding of this phenomenon has not +come about. At the present time, in fact, it is impossible even to estimate the +coeflicient of friction between two substances. +lt was pointed out above that attempts to measure by sliding pure substances +such as copper on copper will lead to spurious results, because the surfaces in +contact are not pure copper, but are mixtures of oxides and other impurities. lf +we try to get absolutely pure copper, iŸ we clean and polish the surfaces, outgas +the materials in a vacuum, and take every conceivable precaution, we sfill do +not get . Eor If we tilt the apparatus even to a vertical position, the slider will +not fall of —the two pieces of copper stick togetherl The coeficient , which is +ordinarily less than unity for reasonably hard surfaces, becomes several times +unity! The reason for this unexpected behavior is that when the atoms in contact +are all of the same kind, there is no way for the atoms to “know” that they are in +diferent pieces of copper. When there are other atoms, in the oxides and greases +and more complicated thin surface layers of contaminants in between, the atoms +“know” when they are not on the same part. When we consider that it is Íorces +between atoms that hold the copper together as a solid, it should become clear +that it is impossible to get the right coefficient of friction for pure metals. +The same phenomenon can be observed In a simple home-made experiment +with a fat glass plate and a glass tumbler. If the tumbler is placed on the plate +and pulled along with a loop of string, it slides fairly well and one can feel the +--- Trang 238 --- +coefficient of friction; it is a little irregular, but it is a coefficient. If we now wet +the glass plate and the bottom of the tumbler and pull again, we fnd that it +binds, and if we look closely we shall ñnd scratches, because the water is able +to lift the grease and the other contaminants of the surface, and then we really +have a glass-to-glass contact; this contact is so good that it holds tight and resists +separation so much that the gÌass 1s torn apart; that is, it makes scratches. +12-3 Molecular forces +We shall next discuss the characteristics of molecular forces. These are Íorces +between the atoms, and are the ultimate origin of friction. Molecular forces have +never been satisfactorily explained on a basis of classical physies; it takes quantum +mechanics to understand them fully. Empirically, however, the force between +atoms is illustrated schematically in Eig. 12-2, where the force ' between two +atoms is plotted as a function of the distance r between them. “There are diferent +cases: in the water molecule, for example, the negative charges sit more on the +oxygen, and the mean positions of the negative charges and of the positive charges +are not at the same point; consequently, another molecule nearby feels a relatively +large force, which is called a dipole-dipole force. However, for many systems the +charges are very much better balanced, in particular for oxygen gas, which 1s +perfectly symmetrical. In this case, although the minus charges and the plus +charges are dispersed over the molecule, the distribution is such that the center +of the minus charges and the center of the plus charges coincide. A molecule +where the centers do not coincide is called a polar molecule, and charge times +the separation between centers is called the dipole moment. A nonpolar molecule +F REPULSION +ATTRACTION +Fig. 12-2. The force between two atoms as a function of their distance +of separation. +--- Trang 239 --- +1s one in which the centers of the charges coincide. Eor all nonpolar molecules, in +which all the electrical forces are neutralized, it nevertheless turns out that the +force at very large distances is an attraction and varies inversely as the seventh +power of the distance, or ` = k/r7, where k is a constant that depends on the +molecules. Why this is we shall learn only when we learn quantum mechanics. +'When there are dipoles the forces are greater. When atoms or molecules get too +close they repel with a very large repulsion; that is what keeps us from falling +through the foorl +These molecular forces can be demonstrated in a fairly direct way: one of +these is the friction experiment with a sliding glass tumbler; another is to take +two very carefully ground and lapped surfaces which are very accurately Ẩat, +so that the surfaces can be brought very close together. An example of such +surfaces is the Johansson blocks that are used in machine shops as standards for +making accurate length measurements. If one such block is slid over another very +carefully and the upper one ïs lifted, the other one will adhere and also be lifted +by the molecular forces, exemplifying the direct attraction bebween the atoms on +one block for the atoms on the other block. +Nevertheless these molecular forces of attraction are still not fundamental in +the sense that gravitation is fundamental; they are due to the vastly complex +interactions of a]l the electrons and nuclei in one molecule with all the electrons +and nuclei in another. Any simple-looking formula we get represents a summation +of complications, so we still have not got the fundamental phenomena. +Since the molecular forces attract at large distances and repel at short distances, +as shown In EFig. 12-2, we can make up solids in which all the atoms are held +together by their attractions and held apart by the repulsion that sets in when +they are too close together. At a certain distance ở (where the graph in Fig. 12-2 +crosses the axis) the forces are zero, which means that they are all balanced, so +that the molecules stay that distance apart from one another. If the molecules are +pushed closer together than the distance đ they all show a repulsion, represented +by the portion of the graph above the r-axis. To push the molecules only slightly +closer together requires a great force, because the molecular repulsion rapidly +becomes very great at distances less than ở. If the molecules are pulled slightly +apart there is a slight attraction, which increases as the separation increases. If +they are pulled sufficiently hard, they will separate permanently——the bond 1s +broken. +Tf the molecules are pushed only a øer small distance closer, or pulled only +a 0erU small distance farther than đ, the corresponding distance along the curve +--- Trang 240 --- +of Eig. 12-2 is also very small, and can then be approximated by a straight line. +'Therefore, in many circumstaneces, if the displacement is not too great the ƒorce +¡s proportional to the đisplacemenf. Thĩs principle 1s known as Hooke”s law, or +the law of elasticity, which says that the force in a body which tries to restore the +body to its original condition when ït is distorted is proportional to the distortion. +This law, of course, holds true only if the distortion is relatively small; when 1$ +gets too large the body will be torn apart or crushed, depending on the kind of +distortion. "The amount of force for which Hooke's law is valid depends upon +the material; for instance, for dough or putty the force is very small, but for +steel it is relatively large. Hookeˆs law can be nicely demonstrated with a long +coil spring, made of steel and suspended vertically. A suitable weight hung on +the lower end of the spring produces a tỉny twist throughout the length of the +wire, which results in a small vertical deflection in each turn and adds up to a +large displacement iŸ there are many turns. lf the total elongation produced, +say, by a 100-gram weight, is measured, it is found that additional weights of +100 grams will each produce an additional elongation that is very nearly equal +to the stretch that was measured for the frst 100 grams. This constant ratio +of force to displacement begins to change when the spring is overloaded, i.e., +Hooke's law no longer holds. +12-4 Eundamental forces. Fields +We shall now discuss the only remaining forces that are fundamental. We +call them fundamental in the sense that their laws are fundamentally simple. We +shall first discuss electrical force. ObJects carry electrical charges which consist +simply of electrons or protons. If any t£wo bodies are electrically charged, there +1s an electrical force between them, and if the magnitudes of the charges are +q¡ and qa, respectively, the force varies inversely as the square of the distance +between the charges, or ' = (const)giqa/r2. Eor unlike charges, this law is like +the law of gravitation, but for 2e charges the force is repulsive and the sign +(direction) is reversed. The charges g¡ and qs can be intrinsically either positive +or negative, and in any specifc application of the formula the direction of the +force will come out right ïif the g's are given the proper plus or minus sign; the +force is directed along the line between the two charges. The constant in the +formula depends, of course, upon what units are used for the force, the charge, +and the distance. In current practice the charge is measured in coulombs, the +distance in meters, and the force in newtons. 'Then, in order to get the force +--- Trang 241 --- +to come out properly in newtons, the constant (which for historical reasons is +written 1/4zeo) takes the numerical value +cọ = 8.854 x 10~12 coul”/newton - m? +1/4meo = 8.99 x 109 N - m2/coulŸ. +Thus the force law for static charges is +F. = qiqar/4aegrẺ. (12.2) +In nature, the most important charge of all is the charge on a single elec- +tron, which is 1.60 x 10~†! coulomb. In working with electrical forces between +fundamental particles rather than with large charges, many people prefer the +combination (qei)Ÿ/4zeo, in which qe is deñned as the charge on an electron. This +combination occurs frequently, and to simplify calculations it has been defned +by the symbol eŸ; its numerical value in the mks system of units turns out to +be (1.52 x 10~14)2, The advantage of using the constant in this form is that the +force between two electrons in newtons can then be written simply as e2/z?, with +r in meters, without all the individual constants. Electrical forces are much more +complicated than this simple formula indicates, since the formula. gives the Íorce +between two objects only when the objects are standing still. We shall consider +the more general case shortÌy. +In the analysis oŸ forces oŸ the more fundamental kinds (not such forces as +friction, but the electrical force or the gravitational force), an interesting and very +Important concept has been developed. Since at first sipht the Íorces are very +much more complicated than ¡is indicated by the inverse-square laws and these +laws hold true only when the interacting bodies are standing still, an improved +method is needed to deal with the very complex forces that ensue when the bodies +start to move in a complicated way. Experience has shown that an approach +known as the concept of a “field” is of great utility for the analysis of forces of +this type. To illustrate the idea for, say, electrical force, suppose we have two +electrical charges, g¡ and qs, located at points ? and respectively. Then the +force between the charges is given by +F. = qiqar/4aegrẺ. (12.3) +To analyze this force by means of the field concept, we say that the charge g +at produces a “condition” at , such that when the charge ga is placed at ? +--- Trang 242 --- +1t “feels” the force. This is one way, strange perhaps, of describing 1t; we say +that the force # on ga at Tỉ can be written in two parts. lt is g¿ multiplied by a +quantity that would be there whether ga were there or not (provided we keep +all the other charges in their right places). # is the “condition” produced by q, +we say, and #' ¡is the response of ga to #. E/ is called an clectric feld, and ït 1s a +vector. The formula for the electric fñeld # that is produced at by a charge g +at P is the charge g¡ tỉimes the constant 1/4zeo divided by zŸ (z is the distance +from to ?#?), and it is acting in the direction of the radius vector (the radius +vector ? divided by its own length). The expression for # ¡is thus +E = qir/4negrở. (12.4) +We then write +P=qsE, (12.5) +which expresses the force, the field, and the charge in the field. What ¡is the point +of all this? "The point ¡is to divide the analysis into two parts. One part says that +something produces a field. 'Phe other part says that something is øcfed ơn by +the fñeld. By allowing us to look at the two parts independently, this separation +of the analysis simplifies the calculation of a problem in many situations. If many +charges are present, we first work out the total electrie feld produced at ?# by all +the charges, and then, knowing the charge that is placed at , we fñnd the force +On I1. +In the case of gravitation, we can do exactly the same thing. In this case, +where the force #' = —Œmqmar/rỞ, we can make an analogous analysis, as +follows: the force on a body in a gravitational fñeld is the mass of that body +times the field Œ. The force on rn¿ is the mass ma times the field Œ produced +by mị; that is, E! = mạ(C. Then the fñeld Œ produced by a body of mass rn +is Œ = —Œm?/rỞ and it is direcbed radially, as in the electrical case. +In spite of how it might at fñrst seem, this separation of one part from another +1s not a triviality. It would be trivial, jus6 another way of writing the same +thing, if the laws of force were simple, but the laws of force are so complicated +that it turns out that the fields have a reality that is almost independent of +the objects which create them. One can do something like shake a charge and +produce an effect, a field, at a distance; if one then stops moving the charge, the +field keeps track of all the past, because the interaction between two particles 1s +not instantaneous. lt is desirable to have some way to remember what happened +previously. If the force upon some charge depends upon where another charge +--- Trang 243 --- +was yesterday, which it does, then we need machinery to keep track of what went +on yesterday, and that is the character of a fñeld. So when the forces get more +complicated, the feld becomes more and more real, and this technique becomes +less and less of an artificial separation. +In analyzing forces by the use of ñelds, we need two kinds of laws pertaining +to fields. “The first is the response to a field, and that gives the equations of +motion. For example, the law of response oŸ a mass to a gravitational feld is +that the force is equal to the mass times the gravitational fñeld; or, If there 1s +also a charge on the body, the response of the charge to the electric feld equals +the charge times the electric feld. 'Phe second part of the analysis of nature in +these situations is to formulate the laws which determine the strength of the +fñeld and how it is produced. 'These laws are sometimes called the feld cquafions. +W© shall learn more about them in due time, but shall write down a few things +about them now. +First, the most remarkable fact of all, which ¡is true exactly and which can be +easily understood, is that the total electric field produced by a number of sources +1s the vector sum of the electric felds produced by the first source, the second +Source, and so on. In other words, if we have numerous charges making a fñeld, +and ïf all by itself one of them would make the field #7, another would make the +feld 2z, and so on, then we merely add the vectors to get the total feld. 'This +prineciple can be expressed as +t=Ei+Ea+Es+--- (12.6) +or, in view of the defnition given above, +đG;T;¡ +J— ——. 12.7 +» 47corỷ ( ) +Can the same methods be applied to gravitation? "The force between two +masses ?¡ and my was expressed by Newton as F' = -Gmạmar/rỶ. But +according to the field concept, we may say that ?mị creates a field Œ in all the +surrounding space, such that the Íorce on ?m¿ is given by +F'—=mạC. (12.8) +By complete analogy with the electrica] case, +--- Trang 244 --- +and the gravitational fñeld produced by several masses 1s +C =C+ạ+C¿+Ca+--- (12.10) +In Chapter 9, in working out a case of planetary motion, we used this principle +in essence. We simply added all the force vectors to get the resultant force on a +planet. If we divide out the mass oŸ the planet in question, we get Eq. (12.10). +Equations (12.6) and (12.10) express what is known as fhe principle oƒ +superposition of fields. 'Phis prineiple states that the total fñeld due to all the +sources is the sum of the fields due to each source. So far as we know today, +for electricity this is an absolutely guaranteed law, which is true even when +the force law is complicated because of the motions of the charges. There are +apparent violations, but more careful analysis has always shown these to be due +to the overlooking of certain moving charges. However, although the principle of +superposition applies exactly for electrical forces, it is not exact for gravity if the +fñeld is too strong, and NÑewton”s equation (12.10) is only approximate, according +to Binstein's gravitational theory. +Closely related to electrical force is another kind, called magnetic force, and +this too is analyzed in terms oŸ a field. Some of the qualitative relations bebween +electrical and magnetie forces can be ïllustrated by an experiment with an electron- +ray tube (Fig. 12-3). At one end of such a tube is a source that emits a stream +of electrons. Within the tube are arrangements for accelerating the electrons to +a high speed and sending some of them in a narrow beam to a Ñuorescent screen +at the other end of the tube. A spot of light glows in the center of the screen +where the electrons strike, and this enables us to trace the electron path. Ôn the +DI :M ++ __-—— +NÓ, —4 +I VN c— T—— Nị | 7 +_ I”IL] J1 +ELECTRON GUN ị É_-< Ự +HLCC TRƠN SoURCE V— — À~ 7 Ấ UORESCENT +Fig. 12-3. An electron-beam tube. +--- Trang 245 --- +way to the screen the electron beam passes through a narrow space between a +pair of parallel metal plates, which are arranged, say, horizontally. A voltage can +be applied across the plates, so that either plate can be made negative at will. +'When such a voltage is present, there is an electric fñeld between the plates. +The first part of the experiment is to apply a negative voltage to the lower +plate, which means that extra electrons have been placed on the lower plate. +Since like charges repel, the light spot on the screen instantly shifts upward. +(We could also say this in another way—that the electrons “felt” the ñeld, and +responded by deflecting upward.) We next reverse the voltage, making the upper +plate negative. The light spot on the screen now jumps below the center, showing +that the electrons in the beam were repelled by those in the plate above them. +(Or we could say again that the electrons had “responded” to the field, which is +now in the reverse direction.) +'The second part of the experiment is to disconnect the voltage from the plates +and test the efect ofa magnetic fñeld on the electron beam. 'This is done by means +of a horseshoe magnet, whose poles are far enough apart to more or less straddle +the tube. Suppose we hold the magnet below the tube in the same orientation +as the letter U, with its poles up and part of the tube in between. We note that +the light spot is deflected, say, upward, as the magnet approaches the tube from +below. So it appears that the magnet repels the electron beam. However, it is not +that simple, for If we invert the magnet without reversing the poles side-for-side, +and now approach the tube from above, the spot still moves øœrd, so the +electron beam is øøý repelled; instead, it appears to be attracted this time. Now +we start again, restoring the magnet to its original U orientation and holding +1t below the tube, as before. Yes, the spot is still defected upward; but now turn +the magnet 180 degrees around a vertical axis, so that ït is still in the Ù position +but the poles are reversed side-for-side. Behold, the spot now Jjumps downward, +and stays down, even if we invert the magnet and approach from above, as before. +'To understand this peculiar behavior, we have to have a new combination of +forces. We explain it thus: Across the magnet from one pole to the other there is a +magnetic field. Thịs fñeld has a direction which is always away from one particular +pole (which we could mark) and toward the other. Inverting the magnet did +not change the direction of the field, but reversing the poles side-for-side did +reverse is direction. For example, if the electron velocity were horizontal in the +z-direction and the magnetic field were also horizontal but in the ø-direction, the +magnetic force øn ‡he rnouïng clectrons would be in the z-direction, i.e., up or +down, depending on whether the fñeld was in the positive or negative -direction. +--- Trang 246 --- +Although we shall not at the present time give the correct law of force between +charges moving in an arbitrary manner, one relative to the other, because iE is +too complicated, we shall give one aspect of it: the complete law of the forces +£J the ields are knoumn. "The force on a charged object depends upon its motion; +1ƒ, when the objJect is standing still at a given place, there is some force, this +1s taken to be proportional to the charge, the coefficient being what we call +the electric field. When the object moves the force may be different, and the +correction, the new “piece” of force, turns out to be dependent exactly lineariu +on the 0elocitu, but at right angles to 9ø and to another vector quantity which we +call the magnetic induction ÐB. Tf the components of the electric fñeld # and the +magnetic induction Ö are, respectively, (E„, Ey, #;) and (B„, By, B,), and if the +velocity ø has the components (0z, 0y, 0„), then the total electric and magnetic +force on a moving charge g has the components +Ty = q(E„ +uyB, — 0„BỤ), +Tụ = q(Ey +u„B„ — 0y„B,), (12.11) +1; =q(E„ + uy„Bụ — 0y). +Tí, for instance, the only component of the magnetic feld were Ö„ and the only +component of the velocity were „, then the only term left in the magnetic force +would be a force in the z-direction, at right angles to both Ö and 0. +12-ã Pseudo forces +The next kind of force we shall discuss might be called a pseudo force. In +Chapter I1 we discussed the relationship between two people, Joe and Moe, who +use diferent coordinate systems. Let us suppose that the positions of a particle +as measured by Joe are z and by Moe are #; then the laws are as follows: +z—=#+$, U—=, z=#Z, +where s is the displacement of Moeˆs system relative to Joe's. If we suppose that +the laws of motion are correct for Joe, how do they look for Moe? We fnd frst, +d+/dt = da" (dt + ds/di. +Previously, we considered the case where s was constant, and we found that s +made no diference in the laws of motion, since đs/dt = 0; ultimately, therefore, +--- Trang 247 --- +the laws of physics were the same in both systems. But another case we can +take is that s — œ‡, where w is a uniform velocity In a straight line. “Then +ø is nob constant, and đs/đf is not zero, but is u, a constant. However, the +acceleration đ2z/đi2 is still the same as đˆz/đf”, because du/d‡ = 0. Thịis proves +the law that we used in Chapter 10, namely, that if we move in a straight line +with uniform velocity the laws of physics will look the same to us as when we are +standing still. 'Phat is the Galilean transformation. But we wish to discuss the +interesting case where s is still more complicated, say s = af2/2. Then ds/df = at +and đ2s/đi2 = aø, a uniform acceleration; or in a still more complicated case, the +acceleration might be a function of time. Thịs means that although the laws of +motion from the point of view of Joe would look like +m Tên đụ, +the laws of motion as looked upon by Moe would appear as +m = Hạ — h„ — ma. +'That is, since Moe”s coordinate system is accelerating with respect to Joe”s, the +extra term mø comes in, and Moe will have to correct his forces by that amount +in order to get Newton's laws to work. In other words, here is an apparent, +mmysterious new force of unknown origin which arises, of course, because Moe +has the wrong coordinate system. 'This is an example of a pseudo force; other +examples occur in coordinate systems that are rofating. +Another example of pseudo force is what is ofben called “centrifugal force.” +An observer in a rotating coordinate system, e.g., in a rotating box, will ñnd +mmysterious forces, not accounted for by any known origin oŸ force, throwing +things outward toward the walls. Thhese forces are due merely to the fact that +the observer does not have NÑewton's coordinate system, which is the simplest +coordinate system. +Pseudo force can be ïllustrated by an interesting experiment in which we +push a jar of water along a table, with acceleration. Gravity, of course, acts +downward on the water, but because of the horizontal acceleration there is also a +pseudo force acting horizontally and in a direction opposite to the acceleration. +'The resultant of gravity and pseudo force makes an angle with the vertical, and +during the acceleration the surface of the water will be perpendicular to the +--- Trang 248 --- +resultant force, ¡.e., inclined at an angle with the table, with the water standing +higher in the rearward side of the jar. When the push on the jar stops and the +jar decelerates because of friction, the pseudo force is reversed, and the water +stands higher in the forward side of the jar (Eig. 12-4). +_> ———————> -^=— +Fig. 12-4. lllustration of a pseudo force. +One very important feature of pseudo forces 1s that they are always Dropor- +tional to the masses; the same is true of gravity. The possibility exists, therefore, +that grauift ?selƒ ¡s a pseudo ƒorce. Ïs it not possible that perhaps gravitation is +due simply to the fact that we do not have the right coordinate system? After +all, we can always get a force proportional to the mass if we imagine that a +body is accelerating. Eor instance, a man shut up in a box that is standing +still on the earth ñnds himself held to the ñoor of the box with a certain force +that is proportional to his mass. But ïf there were no earth at all and the box +were sianding still, the man inside would foat in space. Ôn the other hand, If +there were no earth at all and something were puilmg the box along with an +acceleration ø, then the man in the box, analyzing physics, would ñnd a pseudo +force which would pull him to the foor, just as gravity does. +Binstein put forward the famous hypothesis that accelerations give an imitation +OoŸ gravitation, that the forces of acceleration (the pseudo forces) cœwnot be +địstinguished from those oŸ gravity; 1 is not possible to tell how much of a given +force is gravity and how much is pseudo force. +Tt might seem all right to consider gravity to be a pseudo force, to say that we +are all held down because we are accelerating upward, but how about the people +in Madagascar, on the other side of the earth—are they accelerating too? Einstein +found that gravity could be considered a pseudo force only at one point at a time, +and was led by his considerations to suggest that the geometrU oj the tuuorld 1s +more complicated than ordinary Euclidean geometry. The present discussion is +only qualitative, and does not pretend to convey anything more than the general +idea. To give a rough idea of how gravitation could be the result of pseudo fÍorces, +we present an ïllustration which is purely geometrical and does not represent the +--- Trang 249 --- +real situation. Suppose that we all lived in two dimensions, and knew nothing of +a thid. We think we are on a plane, but suppose we are really on the surface of a +sphere. And suppose that we shoot an object along the ground, with no forces on +it. Where will it go? It will appear to go ïn a straight line, but it has to remain +on the surface of a sphere, where the shortest distance between two poinfs 1s +along a great circle; so it goes along a great circle. If we shoot another object +similarly, but in another direction, it goes along another great circle. Because +we think we are on a plane, we expect that these two bodies will continue to +diverge linearly with time, but careful observation will show that if they go far +enough they move closer together again, as though they were attracting each +other. But they are nø£ attracting each other—there is just something “weird” +about this geometry. This particular ïllustration does not describe correctly the +way in which Einstein's geometry is “weird,” but ït illustrates that if we distort +the geometry sufficiently it is possible that all gravitation is related in some way +to pseudo forces; that is the general idea of the Einsteinian theory of gravitation. +12-6 Nuclear forces +W©e conclude this chapter with a brief discussion of the only other known +forces, which are called mœ%wclear ƒorces. These forces are within the nuclei of +atoms, and although they are much discussed, no one has ever calculated the +force between two nuelei, and indeed at present there is no known law for nuclear +forces. These forces have a very tiny range which is just about the same as +the size of the nucleus, perhaps 10~†13 centimeter. With particles so small and +at such a tiny distance, only the quantum-mechanical laws are valid, not the +Newtonian laws. In nuclear analysis we no longer think in terms of forces, and in +fact we can replace the force concept with a concept of the energy of interaction +of two particles, a subject that will be discussed later. Any formula that can +be written for nuclear forces is a rather crude approximation which omits many +complications; one might be somewhat as follows: forces within a nucleus do +not vary inversely as the square of the distance, but die off exponentially over a +cortain distance r, as expressed by #' = (1/z?) exp(—z/ro), where the distance 7o +is of the order of 10—13 centimeter. In other words, the forces disappear as soon +as the particles are any great distance apart, although they are very strong +within the 10~13 centimeter range. So far as they are understood today, the laws +of nuclear force are very complex; we do not understand them in any simple +way, and the whole problem of analyzing the fundamental machinery behind +--- Trang 250 --- +nuclear forces is unsolved. Attempts at a solution have led to the discovery of +numerous strange particles, the x-mesons, for example, but the origin of these +forces remains obscure. +--- Trang 251 --- +I2 +MVor'Ek (ra ốổl IPoforeffteal FErrorggg/ (Ì) +13-1 Energy of a falling body +In Chapter 4 we discussed the conservation of energy. In that discussion, we +địd not use Newton's laws, but i§ is, oÝ course, of great interest to see how 1 +comes about that energy is in fact conserved in accordance with these laws. For +clarity we shall start with the simplest possible example, and then develop harder +and harder examples. +The simplest example of the conservation of energy is a vertically falling +object, one that moves only in a vertical direction. An object which changes its +height under the inÑuence of gravity alone has a kinetic energy 7 (or K.E.) due +to its motion during the fall, and a potential energy ?møh, abbreviated (or +P.E.), whose sum is constant: +simu7 + mụgh = const, +K.E. P.E. +1'+UU = const. (13.1) +Now we would like to show that this statement is true. What do we mean, show ït +is true? Hrom Newton's Second Law we can easily tell how the objecE moves, and +1E is easy to fnd out how the velocity varies with time, namely, that it increases +proportionally with the time, and that the height varies as the square of the time. +So 1Ý we measure the height from a zero point where the object 1s stationary, 1W +1s no miracle that the height turns out to be equal to the square of the velocity +times a number of constants. However, let us look at it a little more closely. +Let us ñnd out đứrecfiu from Newtons Second Law how the kinetic energy +should change, by taking the derivative of the kinetic energy with respect to time +and then using Newton's laws. When we diferentiate smu2 with respect to time, +we obtain đT d đo đo +Trm (Sm02) = 3m20 Px... (13.2) +--- Trang 252 --- +since 7n is assumed constant. But from Newton”s Second Law, m(do/đf) = F}, so +đT/dt = Fo. (13.3) +In general, it will come out to be #'-ø, but in our one-dimensional case let us +leave 1 as the force times the velocity. +Now in our simple example the force is constant, equal to —?mng, a vertical +force (the minus sign means that it acts downward), and the velocity, oÝ course, +1s the rate of change of the vertical position, or heipht h, with time. Thus the +rate of change of the kinetic energy is —rng(dh/đf), which quantity, miracle of +miracles, is minus the rate of change of something elsel It is minus the time rate +of change of mmghl 'Therefore, as time goes on, the changes in kinetic energy and +in the quantity rmgh are equal and opposite, so that the sum of the two quantities +remains constant. Q.E.D. +W©e have shown, om Newton's second law of motion, that energy is con- +served for constant forces when we add the potential energy ?mgh to the kinetic +©n©rgy sinu2. Now let us look into this further and see whether it can be gener- +alized, and thus advance our understanding. Does it work only for a freely falling +body, or is it more general? We expect from our discussion of the conservation +of energy that it would work for an object moving from one point to another +in some kind of frictionless curve, under the inÑuence of gravity (Fig. 13-1). If +the obJect reaches a certain height h from the original height HỨ, then the same +formula should again be right, even though the velocity is now in some direction +other than the vertical. We would like to understand :ø0h# the law is still correct. +Let us follow the same analysis, ñnding the time rate of change of the kinetic +energy. This will again be rmø(du/đf), but rm(du/đf) is the rate of change of +the magnitude of the momentum, 1.e., the ƒorce ?n the đirection oƒ motion—the +Fig. 13-1. An object moving on a frictionless curve under the influence +Of gravity. +--- Trang 253 --- +tangential force ;¿. Thus +—= — = F0. +dc “ng +Now the speed is the rate of change of distance along the curve, đs/đf, and +the tangential force #‡ 1s not —rng but is weaker by the ratio of the vertical +distance đh to the distance đs along the path. In other words, +tỳ = —mmgsin 8 = —rmg —, +so that +m đs đhÀ ( ds dh +—— —= — T\|, —— —— =—= —†TT\( — +° “Áás J (ái “án” +since the đs°s cancel. Thus we get —rng(dh/đ£#), which is equal to the rate of +change of —rngh, as before. +Tn order to understand exactly how the conservation of energy works in general +in mechanics, we shall now discuss a number of concepts which will help us to +analyze it. +First, we discuss the rate of change of kinetic energy in general in three +dimensions. 'Phe kinetic energy in three dimensions is +T= ÿm(w + b2 +?). +'When we differentiate this with respect to time, we get three terrifying terms: +dđT duy đuy dù; +—= „—— —= +0; —_—- ]. 18.4 +dt mẮn Ai tờ TP x) 034) +But m(doz/đÐ) is the force F„ acting on the object in the z-direction. Thus the +right side of Eq. (13.4) is Fxu„ + Fyuy + Fxo„. We recall our vector analysis and +recognize this as #'- 0; therefore +đT /dt = F- 0. (13.5) +This result can be derived more quickly as follows: if œ and b are two vectOrs, +both of which may depend upon the time, the derivative of a - b is, in general, +d(œ - b)/dt = a- (db/df) + (da/di) - b. (13.6) +--- Trang 254 --- +We then use this in the form œ = b = 0: +d($mœ2 d(3m®-® du S +"“ “_--=. _- .Ắ. (13.7) +Because the concepts of kinetic energy, and energy in general, are so important, +various names have been given to the important terms in equations such as these. +smu2 is, as we know, called kớứnetic energu. F`-0 is called pouer: the force acting +on an object times the velocity of the object (vector “dot” produet) is the power +beïng delivered to the obJect by that force. We thus have a marvelous theorem: +the rate 0ƒ change oƒ kinetic energ oƒ an object is cqual to the potuer ezpended +bụ the forces acting on tt. +However, to study the conservation oŸ energy, we want to analyze this still +more closely. Let us evaluate the change in kinetic energy in a very short tỉme đi. +If we multiply both sides of Bq. (18.7) by đý, we ñnd that the diferential change +in the kinetic energy is the force “dot” the diferential distance moved: +đi = F':- d3. (13.8) +TỶ we now integrate, we get +AT= II F'- ds. (13.9) +What does this mean? lt means that if an object is moving 7n am wøœw under +the infuence of a force, moving in some kind of curved path, then the change +in K.E. when it goes from one poïnt to another along the curve is equal to the +integral of the component of the force along the curve times the diferential +displacement đs, the integral being carried out from one point to the other. 'This +integral also has a name; it is called the tuork done bụ the ƒorce ơn the object. VWe +see Immediately that pouer equals tuork done per second. W©e also see that 1t 1s +only a component oŸ force #n the direclfion oƒ motion that contributes to the work +done. In our simple example the forces were only vertical, and had only a single +component, say #;, equal to —mng. No matter how the obJect moves in those +circumstances, falling in a parabola for example, È' : s, which can be written +as F„ dz + Eụ dụ + F> dz, has nothing left of it but F; dz = —?rng đz, because the +other components of force are zero. Therefore, in our simple case, +2 Z2 +J F`-ds—= J —ng đz = —rng(za — Z1), (13.10) +1 Z1 +--- Trang 255 --- +so again we fñnd that it is only the 0ertical height from which the object falls +that counts toward the potential energy. +A word about units. Since forces are measured in newtons, and we multiply +by a distanece in order to obtain work, work is measured in øeufon - meters (Ñ-m), +but people do not like to say newton-meters, they prefer to say jøuwes (J). A +newton-meter is called a joule; work is measured in joules. Power, then, is joules +per second, and that is also called a øø## (W). IÝ we multiply watts by time, the +result is the work done. 'Phe work done by the electrical company in our houses, +technically, is equal to the watts times the time. That is where we get things like +kilowatt hours, 1000 watts times 3600 seconds, or 3.6 x 108 joules. +Now we take another example of the law of conservation of energy. Consider +an object which initially has kinetic energy and is moving very fast, and which +slides against the Hoor with friction. It stops. At the start the kinetic energy +1s mo‡ zero, but at the end it 2s zero; there is work done by the forces, because +whenever there is friction there is always a component of force in a direction +opposite to that of the motion, and so energy is steadily lost. But now let us +take a mass on the end of a pivot swinging in a vertical plane in a gravitational +feld with no friction. What happens here is diferent, because when the mass is +goïing up the force is downward, and when it is coming down, the force is also +downward. Thus #'- đs has one sign going up and another sign coming down. At +each corresponding point of the downward and upward paths the values of F' - đs +are exactly equal in size but of opposite sign, so the net result of the integral +will be zero for this case. Thus the kinetic energy with which the mass comes +back to the bottom is the same as it had when it left; that is the principle of the +conservation of energy. (Note that when there are friction forces the conservation +of energy seems at first sipght to be invalid. We have to fnd another ƒorm of +energy. Ït turns out, in fact, that heaf 1s generated in an object when ï§ rubs +another with friction, but at the moment we supposedly do not know that.) +13-2 Work done by gravity +The next problem to be discussed 1s mụch more difficult than the above; it +has to do with the case when the forces are not constant, or simply vertical, as +they were in the cases we have worked out. We want to consider a planet, for +example, moving around the sun, or a satellite in the space around the earth. +W© shall first consider the motion of an object which starts at some point 1 +and falls, say, đirecfu toward the sun or toward the earth (Fig. 13-2). WilI there +--- Trang 256 --- +s c——=———s +Fig. 13-2. A small mass mm falls under the influence of gravity toward +a large mass Mĩ. +be a law oŸ conservation of energy in these circumstances? The only difference is +that in this case, the force is changing as we go along, it is not just a constant. +As we know, the force is —GŒM/rẺ tỉmes the mass ?m, where ?m is the mass that +moves. Now certainly when a body falls toward the earth, the kinetic energy +Increases as the distance fallen increases, just as it does when we do not wOorry +about the variation of force with height. The question is whether it is possible to +ñnd another formula for potential energy diferent from rmgh, a diferent function +of distance away from the earth, so that conservation of energy will still be true. +This one-dimensional case is easy to treat because we know that the change +in the kinetic energy is equal to the integral, from one end of the motion to the +other, of —ŒMmn/r2 times the displacement đr: +1;—7¡== | GMm —>- (13.11) +'There are no cosines needed for this case because the force and the displacement +are in the same direction. It is easy to integrate dr/z2; the result is —1/z, so +Eq. (13.11) becomes +Tạ — Tì =+GMm[ - ¬) (13.12) +T2 TỊ +Thus we have a diferent formula for potential energy. Equation (13.12) tells us +that the quantity ($zmø2 — GMm/r) calculated at poïnt 1, at poïnt 2, or at any +other place, has a constant value. +W©e now have the formula for the potential energy in a gravitational fñeld for +vertical motion. NÑow we have an interesting problem. Can we make perpetual +tmotion in a gravitational fñeld? "The gravitational field varies; in diferent places +1t is in diferent directions and has diferent strengths. Could we do something +like this, using a fxed, frictionless track: start at some point and lift an object +out to some other point, then move it around an arc to a third point, then lower +1t a certain distance, then move it in at a certain slope and pull it out some other +way, so that when we bring it back to the starting point, a certain amount of work +--- Trang 257 --- +has been done by the gravitational force, and the kinetic energy of the object +is increased? Can we design the curve so that it comes back moving a little bit +faster than it did before, so that it goes around and around and around, and gives +us perpetual motion? Since perpetual motion is impossible, we ought to fnd +out that this is also impossible. We ought to discover the following proposition: +since there is no friction the object should come back with neither higher nor +lower velocity——it should be able to keep going around and around any closed +path. 5tated in another way, he totaÏ tuork đone ín goïng arouwnd a complete +cụcÌe should be zero for gravity forces, because ïÍ it is not zero we can get energy +out by going around. (Tf the work turns out to be less than zero, so that we +get less speed when we go around one way, then we merely go around the other +way, because the forces, of course, depend only upon the position, not upon the +direction; if one way is plus, the other way would be minus, so unless it is zero +we will get perpetual motion by goỉng around either way.) +° : 6 +M3 4 +Fig. 13-3. A closed path ¡in a gravitational field. +ls the work really zero? Let us try to demonstrate that it is. First we shall +explain more or less why it is zero, and then we shall examine it a little better +mathematically. Suppose that we use a simple path such as that shown In +Fig. 13-3, in which a small mass is carried from point 1 to point 2, and then is +made to go around a circle to 3, back to 4, then to 5, 6, 7, and 8, and fñnally +back to 1. AlI of the lines are either purely radial or circular, with ă as the +center. How much work is done in carrying m around this path? Between points +1 and 2, it is GŒMm tìmes the difference of 1/r between these bwo points: +Wha =Í Esds= | -GMm =GMm( = ^) +1 1 r T2 TỊ +tHrom 2 to 3 the force is exactly at right angles to the curve, so that W2¿ = 0. +The work from 3 to 4 is +Mai = ƒ E-ds= GAm( TC — n): +3 T4 T3 +--- Trang 258 --- +In the same fashion, we find that Was = 0, Wss = GMm(1/re — 1/rs), Wsy =0, +W7s = GMm(1/rs — 1/r;), and Wsy =0. Thus +1 1 1 1 1 1 1 1 +W=GMm( + TT tam} +T2 — T1 PA Tạ T6 T5 T§ã Tĩ +But we note that ra = 73, 74 — 7s, re =r7;, and rs =r\. Therefore W =0. +° +lo x Llb +Fig. 13-4. A “smooth” closed path, showing a magnified segment of +It approximated by a series of radial and circumferential steps, and an +enlarged view of one step. +Of course we may wonder whether this is too trivial a curve. What iÝ we use +a real curve? Let us try i9 on a real curve. First of all, we might like to assert +that a real curve could always be imitated sufficiently well by a series of sawtooth +Jiggles like those of Fig. 13-4, and that therefore, etc., Q.E.D., but without a +little analysis, it is not obvious at first that the work done going around even a +small triangle is zero. Let us magnify one of the triangles, as shown in EFig. 13-4. +1s the work done in going from ø to b and ö to c on a triangle the same as the +work done in going directly from a to c? Suppose that the force is acting in a +certain direction; let us take the triangle such that the side be is in this direction, +Just as an example. We also suppose that the triangle is so small that the force +1s essentially constant over the entire triangle. What is the work done in goïng +from ø to c? lt is +W. = J E'-ds = Fscos0, +since the force is constant. Now let us calculate the work done in going around the +other ©wo sides of the triangle. Ôn the vertical side œb the force is perpendicular +--- Trang 259 --- +to đs, so that here the work is zero. Ôn the horizontal side be, +MS F'-ds = Ea. +Thus we see that the work done in going along the sides of a small triangle is +the same as that done going on a slant, because scosØ is equal to ø. We have +proved previously that the answer is zero for any path composed of a series of +notches like those of Fig. 13-3, and also that we do the same work iŸ we cut across +the corners instead oŸ going along the notches (so long as the notches are ñne +enough, and we can always make them very fine); therefore, Éhe Uork done ïn +goïng around œnụ path ímn a grauitatlional field ts zero. +'This 1s a very remarkable result. It tells us something we did not previousÌy +know about planetary motion. It tells us that when a planet moves around the +sun (without any other objects around, no other forces) it moves in such a manner +that the square of the speed at any point minus some constants divided by the +radius at that point is always the same at every point on the orbit. Eor example, +the closer the planet is to the sun, the faster it is going, but by how much? By +the following amount: if instead of letting the planet go around the sun, we were +to change the direction (but not the magnitude) of its velocity and make it move +radially, and then we let ¡it fall from some special radius to the radius of interest, +the new speed would be the same as the speed it had in the actual orbit, because +this is just another example of a complicated path. So long as we come baeck to +the same distance, the kinetic energy will be the same. 5o, whether the motion is +the real, undisturbed one, or is changed in direction by channels, by frictionless +constraints, the kinetic energy with which the planet arrives at a point will be +the same. +Thus, when we make a numerical analysis of the motion of the planet in is +orbit, as we did earlier, we can check whether or not we are making appreciable +errors by calculating this constant quantity, the energy, at every step, and I1§ +should not change. For the orbit of Table 9-2 the energy does change,* it changes +by some 1.5 percent from the beginning to the end. Why? Either because for +the numerical method we use fñnite intervals, or else because we made a slight +mistake somewhere in arithmetic. +Let us consider the energy in another case: the problem of a mass on a spring. +When we displace the mass from its balanced position, the restoring fÍorce is +* 'The energy per unit mass is Hơi: + 92) — 1/zr in the units of Table 9-2. +--- Trang 260 --- +proportional to the displacement. In those circumstances, can we work out a law +for conservation of energy? Yes, because the work done by such a force is +H5 % +w= Paz= | —k# dư = —šk#Ÿ. (13.13) +'Therefore, for a mass on a spring we have that the kinetic energy of the oscillating +mass plus skz? 1s a constant. Let us see how this works. We pull the mass down; +1 is standing still and so is speed is zero. But zø is not zero, + is at is maximum, +so there is some energy, the potential energy, of course. Now we release the mass +and things begin to happen (the details not to be discussed), but at any instant +the kinetie plus potential energy must be a constant. Eor example, after the mass +1s on its way past the original equilibrium point, the position + equals zero, but +that is when it has its biggest ø2, and as it gets more #2 it gets less 02, and so +on. So the balance of z7 and œ2 is maintained as the mass goes up and down. +'Thus we have another rule now, that the potential energy for a spring is skz, 1Í +the force is —k#. +13-3 Summation of energy +Now we go on to the more general consideration of what happens when there +are large numbers of objects. Suppose we have the complicated problem of many +objects, which we label ¿ = 1, 2, 3,..., all exerting gravitational pulls on each +other. What happens then? We shall prove that if we add the kinetic energies +of all the particles, and add to this the sum, over all pøirs of particles, of their +mutual gravitational potential energy, —GMm/r;¡;, the total is a constant: +1 2 Gm¿m; +» 51n;U; + » TT g = cCOnSf. (13.14) +? (pairs 27) Ñ +How do we prove it? We diferentiate each side with respect to time and get +zero. When we diferentiate 1m02, we fnd derivatives of the velocity that are +the forces, Just as in Eq. (13.5). We replace these forces by the law of force that +we know from Newton”s law of gravity and then we notice that what is left is +minus the time derivative of +» Gmm; +palrs Tj +--- Trang 261 --- +The time derivative of the kinetic energy is +d 1 2 dù; +n2 5n =2 min X“. +=) Fiui (13.15) +Gm1n;T'; +D j 1ÿ +The time derivative of the potential energy is +d Gm¿m; _— Gm¿m; đĩ;; +3 ` xa, ' +pairs palrs +2 +Tịj = Vị — #7)” + (Mì — 9)” + (3í — 2): +so that +đĩ;; 1 d+; d+; +—=“=_—_— |2(z;—z;)| —-_— “” +dE — 2ny | ứ “0Í dt — đi ) +đụi — đụ; +2(— ;)| — —--Sˆ ++36, =1) (SE — Sự) +đzi¿ — dz; +2(z;¿T—z;)| - ° ++3 Hi dt — dị )| +¿ — Ðÿ +—= ¿7 * ———————— +— +22 T¡j Xà) Ti +since T¡j — TTj¡, while T¡j — Ti. 'Thus +d Gm¿m; Gm1m;1T'; Gm;1n¿T;¡ +dt » có Tụ 2> | Tử _v® THỊ 3.16) +pairs pairs +2 4! +Now we must note carefully what 3 ){Š)} and 3` mean. In Eq. (13.15), 3 {5`} +? 3 pairs Ũ 3 +means that ? takes on all values ? = 1, 2, 3,... in turn, and for each value oŸ ¿, +--- Trang 262 --- +the index 7 takes on all values except ?. Thus if ¿ = 3, 7 takes on the values 1, 2, +In Eq. (13.16), on the other hand, Ề` means that given values of ? and 7 +occur only once. 'Phus the particle pair 1 and 3 contributes only one term to the +sum. 'To keep track of this, we might agree to let ¿ range over all values 1, 2, +3,..., and for each ¿ let 7 range only over values greø‡er than ¿. Thus iŸ ¿ = 3, 7 +could only have values 4, 5, 6,... But we notice that for each z, j7 value there are +©wo contributions to the sum, one involving ¿, and the other ø;, and that these +terms have the same appearance as those of Eq. (13.15), where ai values of ? +and 7 (except 2 = 7) are included in the sum. Therefore, by matching the terms +one by one, we see that Eqs. (13.16) and (13.15) are precisely the same, but of +opposite sign, so that the time derivative of the kinetic plus potential energy is +indeed zero. 'Thus we see that, for many objects, ¿he kứnetic energụ is the sum +0ƒ the contributions [rom cach ?ndiuidual ob7ect, and that the potential energy +1s also simple, it being also just a sum of contributions, the energies between +all the pairs. We can understand œh¿ it should be the energy of every pair this +way: Suppose that we want to fnd the total amount of work that must be done +to bring the objects to certain distances from each other. We may do this in +several steps, bringing them in from infinity where there is no force, one by one. +First we bring in number one, which requires no work, since no other objects +are yet present to exert$ force on i§. Next we bring in number two, which does +take some work, namely W/1a = —Œmmyma/r+s. NÑow, and thìs is an important +point, suppose we bring in the next object to position three. Ất any moment the +force on number 3 can be written as the sum of two forces—the force exerted by +number 1 and that exerted by number 2. 'Therefore fhe tuork done is the sum oƒ +the tuorks done bụ cach, because 1Ÿ F'z can be resolved into the sum of two forces, +tạ = Fla + F›a, +then the work is +[Fi-dẽ= Í Fúycdst | Ea cds= Ha ti +That is, the work done is the sum of the work done against the fñrst force and the +second force, as if each acted independently. Proceeding in this way, we see that +the total work required to assemble the given confguration of objects is precisely +the value given in Eq. (13.14) as the potential energy. It is because gravity obeys +--- Trang 263 --- +the prineiple of superposition of forces that we can write the potential energy as +a sum over each pair of particles. +13-4 Gravitational ñeld of large objects +Now we shall calculate the fñelds which are met in a few physical circumnstances +involving đistributions oƒ mass. We have not so far considered distributions of +mass, only particles, so it is interesting to calculate the forces when they are +produced by more than just one particle. Pirst we shall fnd the gravitational +force on a mass that is produced by a plane sheet of material, infñnite in extent. +'The force on a unit mass at a given point , produced by this sheet of material +(Fig. 13-5), will of course be directed toward the sheet. Leb the disbance of the +point from the sheet be ø, and let the amount of mass per unit area of this huge +sheet be u. We shall suppose / to be constant; it is a uniform sheet of material. +Now, what small fñeld đŒ is produced by the mass đm lying between ø and ø+ đo +from the point Ó of the sheet nearest point: P? Answer: đŒ = —GŒ(dmr/r3). But +this field ¡is directed along ?, and we know that only the z-component of it will +remain when we add all the little vector đŒ”s to produce Œ. “The z-component +Of dC is +dŒ, =—G = =-G CHỊ +Now all masses đi. which are at the same distance r from will yield the +same đŒ„, so we may at once write for đmn the total mass in the ring between /ø +and ø + đo, namely đừn = u2mp dp (27p dp 1s the area oŸ a rỉng oŸ radius ø and +width đø, if đo < ø). Thus +đŒy = —GMu27p ng, +~IdPF— ø —IO +dm ` a +Fig. 13-5. The gravitational field C at a mass point produced by an +Iinfinite plane sheet of matter. +--- Trang 264 --- +Then, since rŸ = øŸ + a”, odo =rdr. Therefore, +Œy = —>nGua | là = ->nGua(^ — —) = —27GU. (13.17) +“ T a ®° +Thus the force is independent of distance al! Why? Have we made a mistake? +One might think that the farther away we go, the weaker the force would be. But +nol TỶ we are close, most of the matter is pulling at an unfavorable angle; if we +are far away, more of the matter is situated more favorably to exert a pull toward +the plane. At any distance, the matter which is most efective lies in a certain +cone. When we are farther away the force is smaller by the inverse square, but +in the same cone, in the same angle, there 1s much rnore matter, larger by just +the square of the distancel “This analysis can be made rigorous by just noticing +that the diferential contribution in any given cone is in fact independent of the +distance, because of the reciprocal variation of the strength of the force from a +given mass, and the amount oŸ mass included in the cone, with changing distance. +The force is not really constant of course, because when we go on the other side +of the sheet it is reversed in sign. +We have also, in effect, solved an electrical problem: if we have an electrically +chargcd plate, with an amount ø of charge per unit area, then the electric feld +at a poinÈ outside the sheet is equal to ø/2eo, and is in the outward direction If +the sheet is positively charged, and inward ïf the sheet is negatively charged. To +prove this, we merely note that —G, for gravity, plays the same role as 1/47o +for electricity. +Now suppose that we have two plates, with a positive charge +ơ on one and +a negative charge —ơ on another at a distance Ù from the frst. What is the +fñeld? Outside the two plates it is zero. Why? Because one attracts and the other +repels, the force being ?ndependent oƒ đistance, so that the two balanece outl Also, +the fñeld befteen the two plates is clearly twice as great as that from one plate, +namely # = ø/co, and is directed from the positive plate to the negative one. +Now we come to a most interesting and important problem, whose solution +we have been assuming all the time, namely, that the force produced by the earth +at a point on the surface or outside it is the same as if all the mass of the earth +were located at its center. The validity of this assumption is not obvious, because +when we are close, some of the mass is very close to us, and some is farther away, +and so on. When we add the efects all together, it seems a miracle that the net +force is exactly the same as we would get iŸ we put all the mass in the middlel +--- Trang 265 --- +Fig. 13-6. A thịn spherical shell of mass or charge. +We now demonstrate the correctness of this miracle. In order to do so, +however, we shall consider a thin uniform hollow shell instead of the whole earth. +Let the total mass of the shell be rn, and let us calculate the potental energu of +a particle oŸ mass mm“ a distance ?‡ away from the center of the sphere (Eig. 13-6) +and show that the potential energy is the same as it would be if the mass ?n were +a point at the center. (The potential energy is easier to work with than is the +fñeld because we do not have to worry about angles, we merely add the potential +energies of all the pieces of mass.) IÝ we call z the distance of a certain plane +section from the center, then all the mass that is in a slice dz is at the same +distance ? from , and the potential energy due to this rỉng is —Œm đm/r. How +much mass is in the small slice dz? An amount +2 đ 2 d +đĩn —= 2ml ds — “uhet ¬..... 2na_u đa, +sin 8 Ụ +where / = rm/4a? is the surface density of mass on the spherical shell. (It is a +general rule that the area oŸ a zone of a sphere is proportional to its axial width.) +'Therefore the potential energy due to đn is +đW =— Gm đm " Gm'2maụu da +But we see that +r? =g2+(R—z)°=2++?+ R—2Ra +=a?+R”—2R+. +2rdr = —2Rd+z +dy — dĩ +--- Trang 266 --- +'Therefore, +Gm'2ma_u đr +dW = P › +and so , Rịca +W- Gm2malu J dự +t Tì+a +Œmm'2mxaju 2 Œm(4ma?) += R TT R += R. (13.18) +Thus, for a thin spherical shell, the potential energy of a mass ?m/, external to +the shell, is the same as though the mass of the shell were concentrated at its +center. The earth can be imagined as a series of spherical shells, each one of +which contributes an energy which depends only on its mass and the distance +from its center to the particle; adding them all together we get the £o‡œl mmass, +and therefore the earth acts as though all the material were at the centerl +But notice what happens if our point is on the ?ws¿de of the shell. Making +the same calculation, but with ? on the inside, we still get the diference of the +©wo r's, but now in the form a— jÈ— (œ-+ R) = —2, or minus twice the distance +from the center. In other words, W comes out to be W = —Œmmrn/a, which is +¿ndependen‡t of F and independent of position, ¡.e., the same energy no matter +tohere we are inside. 'Pherefore no force; no work is done when we move about +inside. Tf the potential energy is the same no matter where an object is placed +inside the sphere, there can be no force on it. So there is no force inside, there 1s +only a force outside, and the force outside is the same as though the mass were +all at the center. +--- Trang 267 --- +MVor'k (ra ốổl IPo£ortfterl Froorggg, (c©ortecltrsrom) +14-1 Work +In the preceding chapter we have presented a great many new ideas and +results that play a central role in physics. 'These ideas are so important that 1t +seems worth while to devote a whole chapter to a closer examination of them. +In the present chapter we shall not repeat the “proofs” or the specifc tricks by +which the results were obtained, but shall concentrate instead upon a discussion +of the ideas themselves. +In learning any subject of a technical nature where mathematics plays a role, +one is confronted with the task of understanding and storing away in the memory +a huge body of facts and ideas, held together by certain relationships which can +be “proved” or “shown” to exist between them. It is easy to confuse the proof +1tself with the relationship which it establishes. Clearly, the important thing +to learn and to remember is the relationship, not the proof. In any particular +circumstance we can either say “it can be shown that” such and such is true, or +we can show it. In almost all cases, the particular proof that is used is concocted, +ñirst of all, in such form that it can be written quickly and easily on the chalkboard +or on paper, and so that it will be as smooth-looking as possible. Consequently, +the proof may look deceptively simple, when in fact, the author might have +worked for hours trying diferent ways of calculating the same thing until he has +found the neatest way, so as to be able to show that it can be shown in the +shortest amount of timel 'The thing to be remembered, when seeing a proof, is +not the proof itself, but rather that it can be shoun that such and such is true. +Of course, if the proof involves some mmathematical procedures or “tricks” that +one has not seen before, attention should be given not to the trick exactly, but +to the mathematical idea. involved. +Tt is certain that in all the demonstrations that are made in a course such +as this, not one has been remembered from the time when the author studied +--- Trang 268 --- +freshman physics. Quite the contrary: he merely remembers that such and such +1s true, and to explain how it can be shown he invents a demonstration at the +mmoment ¡% is needed. Anyone who has really learned a subject should be able +to follow a similar procedure, but it is no use remermbering the proofs. 'That is +why, in this chapter, we shall avoid the proofs of the various statements made +previously, and merely sumnmarize the results. +The frst idea that has to be digested is t0ork dơne bụ a force. The physical +word “work” is not the word in the ordinary sense of “Workers of the world +unitel,” but is a different idea. Physical work is expressed as ƒ F': ds, called “the +line integral of P' dot đs,” which means that if the force, for instance, is In one +direction and the object on which the force is working is displaced in a certain +direction, then omlu the component oƒ force ïn the dicction oƒ the displacement +does any work. If, for instance, the force were constant and the displacement +were a finite distance As, then the work done in moving the object through that +distance is only the component of force along As times Az. The rule is “force +times distance,” but we really mean only the component of force in the direction +of the displacement tỉimes As or, equivalently, the component of displacement in +the direction of force times #'. It is evident that no work whatsoever is done by +a force which is at right angles to the displacement. +Now 1ƒ the vector displacement As is resolved into components, in other +words, if the actual displacement is As and we want %o consider i% efectively +as a component of displacement Az in the z-direction, A# ïn the -direction, +and Az in the z-direction, then the work done in carrying an object from one +place to another can be calculated in three parts, by calculating the work done +along z, along , and along z. The work done in goïing along # involves only that +component of force, namely #„, and so on, so the work is F„ Az + tụ Au+ ty Az. +'When the force is not constant, and we have a complicated curved motion, then +we must resolve the path into a lot of little As”s, add the work done in carrying +the object along each As, and take the limit as As goes to zero. Thịs is the +meaning of the “line integral.” +Everything we have just said is contained in the formula W = Ƒ#'- ds. It +is all very well to say that it is a marvelous formula, but it is another thing to +understand what it means, or what some of the consequences are. +The word “work” in physics has a meaning so diferent from that of the word +as it is used in ordinary circumstances that it must be observed carefully that +there are some peculiar circumstances in which it appears not to be the same. +For example, according to the physical defnition of work, if one holds a hundred- +--- Trang 269 --- +pound weight of the ground for a while, he is doing no work. Nevertheless, +everyone knows that he begins to sweat, shake, and breathe harder, as If he were +running up a fÑight of stairs. Yet running upsfairs 7s considered as doïng work +(in running đowønstøirs, one gets work out of the world, according to physics), +but in simply holding an object in a fñxed position, no work is done. Clearly, the +physical defnition of work difers from the physiological defñnition, for reasons +we shall briely explore. +Tt is a fact that when one holds a weight he has to do “physiological” work. +'Why should he sweat? Why should he need to consume food to hold the weight +up? Why is the machinery inside him operating at full throttle, just to hold +the weight up? Actually, the weight could be held up with no efort by just +placing it on a table; then the table, quietly and calmly, without any supply of +energy, is able to maintain the same weight at the same heightl The physiological +situation is something like the following. There are two kinds of muscles in the +human body and in other animals: one kind, called strøted or skeletal muscle, 1s +the type of muscle we have in our arms, for example, which is under voluntary +control; the other kind, called srmmoo£h musele, is like the muscle in the intestines +or, in the clam, the greater adductor musecle that closes the shell. 'Phe smooth +museles work very slowly, but they can hold a “set”; that 1s to say, if the clam +tries to close its shell in a certain position, it will hold that position, even if there +is a very great force trying 0o change it. It will hold a position under load for +hours and hours without getting tired because it is very much like a table holding +up a weight, it “sets” into a certain position, and the molecules just lock there +temporarily with no work being done, no efort being generated by the clam. +The fact that we have to generate efort to hold up a weight is simply due to the +design of striated muscle. What happens is that when a nerve impulse reaches a +mmuscle fiber, the fñber gives a little twitch and then relaxes, so that when we hold +something up, enormous volleys of nerve impulses are coming in to the muscle, +large numbers oŸ twitches are maintaining the weight, while the other fñbers relax. +W© can see this, of course: when we hold a heavy weight and get tired, we begin +to shake. “The reason is that the volleys are coming irregularly, and the muscle +1s tired and not reacting fast enough. Why such an ineficient scheme? We do +not know exactly why, but evolution has not been able to develop ƒøs¿ smooth +muscle. Smooth muscle would be mụuch more efective for holding up weights +because you could just stand there and it would lock in; there would be no work +involved and no energy would be required. However, it has the disadvantage that +1t 1s very slow-operating. +--- Trang 270 --- +Returning now to physics, we may ask +0 we want 6o calculate the work +done. The answer is that it is interesting and useful to do so, since the work done +on a particle by the resultant of all the forces acting on it is exactly equal to the +change in kinetic energy of that particle. That is, iŸ an object is being pushed, it +picks up speed, and +A(0?)= ¬ .As. +14-2 Constrained motion +Another interesting feature of forces and work is this: suppose that we have +a sloping or a curved track, and a particle that must move along the track, but +without friction. Ôr we may have a pendulum with a string and a weight; the +string constrains the weight to move in a circle about the pivot point. “The pivot +point may be changed by having the string hit a peg, so that the path of the +weight is along two circles of diferent radii. Thhese are examples of what we call +liacd, [riclionless constraints. +In motion with a ñxed frictionless constraint, no work is done by the constraint +because the forces of constraint are always at right angles to the motion. By +the “forces of constraint” we mean those forces which are applied to the object +directly by the constraint itself—the contact force with the track, or the tension +in the string. +'The forces involved in the motion of a particle on a slope moving under the +inÑuence of gravity are quite complicated, since there is a constraint Íorce, a +gravitational force, and so on. However, if we base our calculation of the motion +on conservation of energy and the grauftational ƒorce alone, we get the right result. +This seems rather strange, because it is not strictly the right way to do it—we +should use the resulfamt force. Nevertheless, the work done by the gravitational +force alone will turn out to be the change in the kinetic energy, because the work +done by the constraint part of the force is zero (Eig. 14-1). +FORCE OF ` +CONSTRAINT FORCE OE +GRAVITY +Fig. 14-1. Forces acting on a sliding body (no friction). +--- Trang 271 --- +The important feature here is that if a force can be analyzed as the sum of +two or more “pieces” then the work done by the resultant force in going along a +certain curve is the sum of the works done by the various “component” forces into +which the force is analyzed. 'Thus if we analyze the force as being the vector sum +of several efects, gravitational plus constraint forces, etc., or the ø-component of +all forces and the -component of all forces, or any other way that we wish to +split it up, then the work done by the net force is equal to the sum of the works +done by all the parts into which we have divided the force in making the analysis. +14-3 Conservative Íorces +In nature there are certain forces, that of gravity, for example, which have a +very remarkable property which we call “eonservative” (no political ideas involved, +1E is again one oŸ those “crazy words”). IÝ we calculate how much work is done +by a force in moving an object from one poiïnt to another along some curved +path, in general the work depends upon the curve, but in special cases it does +not. lf it does not depend upon the curve, we say that the force is a conservative +force. In other words, if the integral of the force times the distance in goỉng from +position 1 to position 2 in Eig. 14-2 is calculated along curve 4 and then along Ö, +we get the same number of Joules, and if this is true for this pair of points on +cucrU curue, and 1f the same propositlon works no matter thích pa#r öŸ poin‡s +we use, then we say the fÍorce is conservative. In such circumstances, the work +integral going from 1 to 2 can be evaluated in a simpÌe manner, and we can give +a formula for the result. Ordinarily it is not this easy, because we also have to +specify the curve, but when we have a case where the work does not depend on +the curve, then, of course, the work depends only upon the pos/fzons of 1 and 2. +P C Z—>ce 2 +Fig. 14-2. Possible paths between two points ¡in a field of force. +To demonstrate this idea, consider the following. We take a “standard” +point Ð, at an arbitrary location (Fig. 14-2). Then, the work line-integral rom 1 +to 2, which we want to calculate, can be evaluated as the work done in goiỉng +--- Trang 272 --- +from 1 to plus the work done in going from ? to 2, because the forces are +conservative and the work does not depend upon the curve. Now, the work done +in goïng from position ? to a particular position in space is a function of that +position in space. Of course it really depends on ? also, but we hold the arbitrary +point P fñxed permanently for the analysis. IÝ that is done, then the work done +in goiïng from point ? to poiïnt 2 is some function of the ñnal position of 2. It +depends upon where 2 is; if we go to some other point we get a different answer. +We shall call this function oŸ position —(z, g, z), and when we wish to refer +%o some particular point 2 whose coordinates are (Za, 2, Z2), we shall write (2), +as an abbreviation for (zas,a,za2). The work done in going from point 1 to +point ? can be written also by going the ø/her t0øy along the integral, reversing +all the ds”s. That is, the work done in going from 1 to ? is mnus the work done +in going from the point P to 1: +P 1 1 +J Esds= [ E+(—dg) =— F- ds. +1 P P +Thus the work done in going from to 1 is —U(1), and from P to 2 the work +is —U(2). Therefore the integral from 1 to 2 is equal to —U(2) plus [—U(1) +backwards]l, or +U(1) — U(2): +0q) == [ T- ds, U@) =~ [ T- ds, +II t-ds = U(1) — UD(). (14.1) +The quantity U(1) — (2) is called the change in the potential energy, and +we call Ư the potential energy. We shall say that when the object is located +at position 2, it has potential energy (2) and at position 1 it has potential +energy (1). If it is located at position ?, it has zero potential energy. IÝ we had +used any other point, say Q, instead o£ P, it would turn out (and we shall leave it +to you to demonstrate) that the pofenfial energụ ¡s changed onhụ bụ the addition +öoƒ a cons‡ơønt. Since the conservation of energy depends only upon chønges, 1Ề +does not matter if we add a constant to the potential energy. Thus the poïint ? +1s arbitrary. +Now, we have the following two propositions: (1) that the work done by a force +is equal to the change in kinetic energy of the particle, but (2) mathematically, +--- Trang 273 --- +for a conservative force, the work done is minus the change in a function which +we call the potential energy. Âs a consequence of these two, we arrive at the +proposition that #ƒ on conseruatiue jorces aœcl, the kinetlic energu T' pÌus the +potential energụ Ù remains constant: +7'+U = constant. (14.2) +Let us now discuss the formulas for the potential energy for a number oÝ cases. +TỶ we have a gravitational field that is uniform, iŸ we are not going to heights +comparable with the radius of the earth, then the force is a constant vertical +force and the work done is simply the force times the vertical distance. 'Phus +D{(z) = mụgz, (14.3) +and the point P? which corresponds to zero potential energy happens to be any +point in the plane z = 0. We could also have said that the potential energy +1s rmwg(z — 6) iŸ we had wanted to—all the results would, of course, be the same in +our analysis except that the value oŸ the potential energy at z = 0 would be —rng6. +lt makes no diference, because only đjfƒerences In potential energy count. +The energy needed to compress a linear spring a distance z from an equilibrium +point 1s +U(œ) = škzŸ, (14.4) +and the zero of potential energy is at the point z = 0, the equilibrium position of +the spring. Again we could add any constant we wish. +'The potential energy of gravitation for point masses ⁄ and rn, a distance z +apart, 1s +U(r) =—GMm/r. (14.5) +The constant has been chosen here so that the potential is zero at inñnity. Of +course the same formula applies to electrical charges, because it ¡is the same law: +U(r) = qiqa/4mcqr. (14.6) +Now let us actually use one of these formulas, to see whether we understand +what it means. Question: How fast do we have to shoot a rocket away from the +earth in order for it to leave? Solutzon: The kinetie plus potential energy must +be a constant; when it “leaves,” it will be millions of miles away, and ïŸ it is just +barely able to leave, we may suppose that it is moving with zero speed out there, +--- Trang 274 --- +Just barely going. Let œ be the radius of the earth, and Mƒ its mass. The kinetic +plus potential energy is then initially given by D1 — ŒGmM/a. At the end of +the motion the two energies must be equal. The kinetic energy is taken to be zero +at the end of the motion, because it is supposed to be just barely drifting away +at essentially zero speed, and the potential energy is GmMĩ divided by infnity, +which is zero. 5o everything is zero on one side and that tells us that the square +of the veloeity must be 2ŒGÄ/a. But ŒAf/a2 is what we call the acceleration of +gravity, g. Thus +UŠ = 2ga. +At what speed must a satellite travel in order to keep going around the earth? +We worked this out long ago and found that øŸ = GŒA//a. Therefore to go øa +from the earth, we need v⁄2 times the velocity we need to just go arownd the +carth near its surface. We need, in other words, tướce œs rnuch energu (because +energy goes as the square of the velocity) to leave the earth as we do to go around +it. Thherefore the first thíng that was done historically with satellites was to get +one to øo around the earth, which requires a speed of five miles per second. The +next thing was to send a satellite away from the earth permanently; this required +twice the energy, or about seven miles per second. +Now, continuing our discussion of the characteristics of potential energy, let +us consider the interaction of bwo molecules, or two atoms, ÿwo oxygen atoms +for instance. When they are very far apart, the Íorce is one of attraction, which +varies as the inverse seventh power of the distance, and when they are very close +the force is a very large repulsion. If we integrate the inverse seventh power to +fnd the work done, we fnd that the potential energy Ứ, which is a function of +the radial distance between the two oxygen atoms, varies as the inverse sixth +power of the distance for large distances. +TỶ we sketch the curve of the pobential energy (7) as in Fig. 14-3, we thus +start out at large r with an inverse sixth power, but IŸ we come in sufficiently +near we reach a point đ where there is a minimum of potential energy. “The +minimum of potential energy at r = đ means this: if we start at đ and move +a small distance, a very small distance, the work done, which is the change In +potential energy when we move this distance, is nearly zero, because there is +very little change in potential energy at the bottom of the curve. Thus there is +no force at this point, and so it is the equilibrium point. Another way to see +that it is the equilibrium poïnt is that it takes work to move away from đin +either direction. When the two oxygen atoms have settled down, so that no more +--- Trang 275 --- +U(r) œ 1/rŠ +(IF r> đ) +Fig. 14-3. The potential energy between two atoms as a function of +the distance between them. +energy can be liberated from the force between them, they are in the lowest +energy state, and they will be at this separation d. 'This is the way an oxygen +mmolecule looks when it is cold. When we heat it up, the atoms shake and move +farther apart, and we can in fact break them apart, but to do so takes a certain +amount of work or energy, which is the potential energy difference bebween r = đ +and r = œ. When we try to push the atoms very close together the energy goes +up very rapidly, because they repel each other. +The reason we bring this out is that the idea of force is not particularly +suitable for quantum mechanics; there the idea of energw is most natural. We fnd +that although forces and velocities “dissolve” and disappear when we consider +the more advanced forces between nuclear matter and between molecules and so +on, the energy concept remains. 'Pherefore we fnd curves of potential energy In +quantum mechanies books, but very rarely do we ever see a curve for the Íorce +between two molecules, because by that time people who are doing analyses are +thinking in terms of energy rather than of force. +Next we note that if several conservative Íorces are acting on an object at the +same time, then the potential energy of the object is the sum of the potential +energies from each of the separate forces. This is the same proposition that we +mentioned before, because i1f the force can be represented as a vector sum of +forces, then the work done by the total force is the sum of the works done by +the partial forces, and it can therefore be analyzed as changes in the potential +energies of each of them separately. Thus the total potential energy 1s the sum +of all the little pieces. +W© could generalize this to the case oŸ a system of many objects interacting +with one another, like Jupiter, Saturn, Ủranus, etc., or oxygen, nitrogen, carbon, +--- Trang 276 --- +etc., which are acting with respect to one another in pairs due to forces all of +which are conservative. In these circumstances the kinetic energy in the entire +system is simply the sum of the kinetic energies of all of the particular atoms or +planets or whatever, and the potential energy of the system is the sum, over the +pairs of particles, of the potential energy of mutual interaction of a single pair, +as though the others were not there. (This is really not true for molecular forces, +and the formula is somewhat more complicated; it certainly is true for NÑewtonian +gravitation, and ït is true as an approximation for molecular forces. For molecular +forces there is a potential energy, but it is sometimes a more complicated function +of the positions of the atoms than simply a sum oŸ terms from pairs.) In the +special case of gravity, therefore, the potential energy is the sum, over all the +pairs ? and 7, of —ŒGm¿m;/r¡;, as was indicated in Eq. (135.14). Equation (13.14) +expressed mathematically the following proposition: that the total kinetic energy +plus the total potential energy does not change with time. As the various planets +wheel about, and turn and twist and so on, IŸ we calculate the total kinetic energy +and the total potential energy we fñnd that the total remains constant. +14-4 Nonconservative Íorces +We have spent a considerable time discussing conservative forces; what about +nonconservative forces? We shall take a deeper view of this than is usual, and state +that there are no nonconservative forcesl As a matter of fact, all the fundamental +forces in nature appear to be conservative. This is not a consequence of Ñewton”s +laws. In fact, so far as Newton himself knew, the forces could be nonconservative, +as Íriction apparently is. When we say friction øpparenfly 1s, we are taking a +modern view, in which it has been discovered that all the deep forces, the forces +between the particles at the most fundamental level, are conservative. +Tí, for example, we analyze a system like that great globular star cluster that +we saw a picture of, with the thousands of stars all interacting, then the formula +for the total potential energy is simply one term plus another term, etc., summed +over all pairs oŸ stars, and the kinetic energy is the sum of the kinetic energies of +all the individual stars. But the globular cluster as a whole is drifting in space +too, and, if we were far enough away from it and did not see the details, could +be thought of as a single object. Then if forces were applied to i%, some of those +forces might end up driving it forward as a whole, and we would see the center +of the whole thing moving. Ôn the other hand, some of the forces can be, so to +speak, “wasted” in increasing the kinetic or potential energy of the “particles” +--- Trang 277 --- +inside. Let us suppose, for instance, that the action of these forces expands the +whole cluster and makes the particles move faster. 'Phe total energy of the whole +thing is really conserved, but seen from the outside with our crude eyes which +cannot see the confusion of motions inside, and just thinking of the kinetic energy +of the motion of the whole object as though it were a single particle, it would +appear that energy is not conserved, but this is due to a lack of appreciation of +what it is that we see. And that, it turns out, is the case: the total energy of the +world, kinetic plus potential, is a constant when we look closely enough. +'When we study matter in the ñnest detail at the atomic level, it is no always +cøs+ to separate the total energy of a thing into two parts, kinetic energy and +potential energy, and such separation is not always necessary. It is aửnos‡ always +possible to do it, so let us say that it 7s always possible, and that the potential- +plus-kinetic energy of the world is constant. 'Phus the total potential-plus-kinetic +energy inside the whole world is constant, and if the “world” is a piece of isolated +material, the energy is constant if there are no external forces. But as we have +seen, some of the kinetic and potential energy of a thing may be internal, for +instance the internal molecular motions, in the sense that we do not notice ït. +W©e know that in a glass of water everything is jiggling around, all the parts are +moving all the time, so there is a certain kinetic energy inside, which we ordinarily +may not pay any attention to. We do not notice the motion of the atoms, which +produces heat, and so we do not call it kinetic energy, but heat is primarily +kinetic energy. Internal potential energy may also be in the form, for instance, of +chemical energy: when we burn gasoline energy is liberated because the potential +energies of the atoms in the new atomic arrangement are lower than in the old +arrangement. Tt is not strictly possible to treat heat as being pure kinetic energy, +for a little of the potential gets in, and vice versa for chemical energy, so we +put the ©wo together and say that the total kinetic and potential energy inside +an object is partly heat, partly chemical energy, and so on. Anyway, all these +diferent forms of internal energy are sometimes considered as “lost” energy in the +sense described above; this will be made clearer when we study thermodynamics. +As another example, when friction is present it is not true that kinetic energy +is lost, even though a sliding object stops and the kinetic energy seems to be lost. +The kinetic energy is not lost because, of course, the atoms inside are jiggling +with a greater amount of kinetic energy than before, and although we cannot +see that, we can measure it by determining the temperature. Of course IÝ we +disregard the heat energy, then the conservation of energy theorem will appear +to be false. +--- Trang 278 --- +Another situation in which energy conservation appears to be false is when +we study only part of a system. Naturally, the conservation of energy theorem +will appear not to be true iŸ something is interacting with something else on the +outside and we neglect to take that interaction into account. +In classical physics potential energy involved only gravitation and electricity, +but now we have nuclear energy and other energies also. Light, for example, +would involve a new form oŸ energy in the classical theory, bu we can aÌso, iÍ we +want to, imagine that the energy of light is the kinetic energy of a photon, and +then our formula (14.2) would still be right. +14-5 Potentials and ñelds +W© shall now discuss a few of the ideas associated with potential energy and +with the idea of a fieid. Suppose we have two large objects A4 and Ö and a +third very small one which is attracted gravitationally by the bwo, with some +resultant force #'. We have already noted in Chapter 12 that the gravitational +force on a particle can be written as its mass, mm, times another vector, C, which +1s debendent only upon the øoszfzon of the particle: +F' —nC. +W© can analyze gravitation, then, by imagining that there is a certain vector Ơ +at every position in space which “acts” upon a mass which we may place there, +but which is there itself whether we actually supply a mass for it to “act” on +or not. has three components, and each of those components is a function +6Ÿ (z,, z), a funetion oŸ position in space. Such a thing we call a #eld, and we +say that the objects A and Ö generafe the field, ï.e., they “make” the vector Ơ. +When an obJect is put in a field, the force on it is equal to 10s mass times the +value of the fñeld vector at the point where the object is put. +W©e can also do the same with the potential energy. Since the potential +energy, the integral of (—force) - (ds) can be written as ?m times the integral of +(—ñeld) - (4s), a mere change of scale, we see that the potential energy (+, , 2) +of an object located at a poïnt (z, , z) in space can be written as ?m tỉmes another +function which we may call the pofenfial W. The integral ƒ C - ds = —Ù, just +as [ F'- dø = —U; there is only a scale factor between the two: +U== | E+ds— =m | C -ds — mộ, (14.7) +--- Trang 279 --- +By having this function (z,,z) at every point in space, we can immedi- +ately calculate the potential energy of an object at any point in space, namely, +U(z, , z2) = mmW(z, 0, z)—rather a trivial business, it seems. But it is not really +trivial, because it is sometimes muụch nicer to describe the field by giving the +value of W everywhere in space instead of having to give Ơ. Instead of having to +write three complicated components of a vector function, we can give instead the +scalar function . Furthermore, it is much easier to calculate than any given +component of Ý when the field is produced by a number of masses, Íor since +the potential is a scalar we merely add, without worrying about direction. Also, +the fñeld Œ can be recovered easily from , as we shall shortly see. Suppose we +have point masses ?n, ma, ... at the points 1, 2,... and we wish to know the +potential W at some arbitrary point p. 'This is simply the sum of the potentials +at ø due to the individual masses taken one by one: +ữ() » ANG. 1,2,... (14.8) +In the last chapter we used this formula, that the potential is the sum of +the potentials from all the diferent objects, to calculate the potential due to a +spherical shell of matter by adding the contributions to the potential at a poïnt +trom all parts of the shell. "The result of this calculation is shown graphically +in EFig. 14-4. It is negative, having the value zero at r = oo and varying as l/r +down to the radius ø, and then is constant inside the shell. Outside the shell +the potential is —Œmm/r, where rm is the mass of the shell, which is exactly the +same as it would have been ïif all the mass were located at the center. But it is +not eueruhere exactly the same, for inside the shell the potential turns out to +be —Œm/a, and is a constantl WZhen the potential is constant, there is no Jield, +or when the potential energy is constant there is no force, because iŸ we move an +$(r) = —Gm/r +ở(r) = CONSTANT = —Gm/a +Fig. 14-4. Potential due to a spherical shell of radius a. +--- Trang 280 --- +object from one place to another anywhere Iinside the sphere the work done by +the force is exactly zero. Why? Because the work done in moving the object from +one place to the other is equal to minus the change in the potential energy (or, +the corresponding field integral is the change of the potential). But the potential +energy is the sœme at any two points inside, so there is zero change in potential +energy, and therefore no work is done in goïng between any ÿwo points inside the +shell. The only way the work can be zero for all directions of displacement is +that there is no force at all. +This gives us a clue as to how we can obtain the force or the field, given the +potential energy. Let us suppose that the potential energy of an object is known +at the position (z,,z) and we want to know what the force on the object is. +lt will not do to know the potential at only this one point, as we shall see; 1% +requires knowledge of the potential at neighboring points as well. Why? How +can we calculate the #-cormponent of the force? (If we can do this, oŸ course, we +can also find the - and z-components, and we will then know the whole force.) +Now, if we were to move the object a small distance Az, the work done by the +force on the object would be the z-component of the force times Az, if Az is +sufficiently small, and this should equal the change in potential energy in going +from one point to the other: +AW =_—AU = lạ Az. (14.9) +We have merely used the formula ƒ F'- ds = —AU, but for a 0erw short path. +NÑow we divide by Az and so fnd that the force is +t„ =—AAU/Az. (14.10) +Of course this is not exact. What we really want is the limit of (14.10) +as Az gets smaller and smaller, because it is only ezacfửu right in the limit of +infinitesimal Az. “This we recognize as the derivative of U with respect to #, +and we would be inclined, therefore, to write —đŨ/dz+. But U depends on z, ở, +and z, and the mathematicians have invented a different symbol to remind us to +be very careful when we are diferentiating such a function, so as to remember +that we are considering that onh + 0aries, and and z do not vary. Instead +of a d they simply make a “backwards 6,” or Ø. (A Ø should have been used +in the beginning of calculus because we always want to cancel that đ, but we +never want to cancel a Øl) So they write ØỮ/Øz, and furthermore, in moments oŸ +duress, if they want to be øerw careful, they put a line beside it with a little z +--- Trang 281 --- +at the bottom (ØU/Øz|y;), which means “Take the derivative of U with respect +to #, keeping and z constant.” Most often we leave out the remark about what +1s kept constant because it is usually evident from the context, so we usually do +not use the line with the and z. However, øas use a Ø instead of a d as a +warning that it is a derivative with some other variables kept constant. 'This is +called a partial derduatiue; ïW 1s a derivative in which we vary only z. +Therefore, we find that the force in the z-direction is minus the partial +derivative of Ư with respect to #: +t„ = —ØU/Ôz. (14.11) +In a similar way, the force in the -direction can be found by diferentiating U +with respect to ø, keeping z and z constant, and the third component, of course, +is the derivative with respect to z, keeping and z constant: +tụ = —8U/0, ty = —8U/Ôz. (14.12) +This 1s the way to get from the potential energy to the force. We get the fcld +from the po#ential in exactly the same way: +Œ„ = —8/Ôz, Œy = —Ø9/Ô, Œ, =—8/Ôz. (14.13) +Incidentally, we shall mention here another notation, which we shall not +actually use for quite a while: Since Œ is a vector and has z-, -, and z-components, +the symbolized Ø/Øz, Ø/Øụ, and Ø/9z which produee the ø-, -, and z-components +are something like vectors. 'Phe mathematicians have invented a glorious new +symbol, V, called “grad” or “gradient”, which is not a quantity but an operator +that makes a vector from a scalar. It has the following “components”: 'Phe ø- +component of this “grad” is Ø/Øz the -component is Ø/Øy, and the z-component +is Ø/Øz, and then we have the fun of writing our formulas this way: +t'=_—-YNU, Œ =-VỪ. (14.14) +Using V gives us a quick way of testing whether we have a real vector equation +or not, but actually Eqs. (14.14) mean precisely the same as Eqs. (14.11), (14.12) +and (14.13); it is just another way of writing them, and since we do not want to +write three equations every time, we just write VŨ instead. +One more example of fields and potentials has to do with the electrical case. +In the case of electricity the force on a stationary object is the charge times the +--- Trang 282 --- +electric fñeld: ' = g#. (In general, of course, the #-component of foree in an +electrical problem has also a part which depends on the magnetic field. It is easy +to show from Eq. (12.11) that the force on a particle due to magnetic fields is +always at right angles to its velocity, and also at right angles to the ñeld. 5ince +the force due to magnetism on a moving charge is at right angles to the velocity, +no t0ork is done by the magnetism on the moving charge because the motion is at +right angles to the force. Therefore, in calculating theorems of kinetic energy in +electric and magnetic ñelds we can disregard the contribution tom the magnetic +fñeld, since it does not change the kinetic energy.) We suppose that there is only +an electric ñeld. Then we can calculate the energy, or work done, in the same way +as for gravity, and calculate a quantity ó which is minus the integral of # - ds, +from the arbitrary ñxed point to the point where we make the calculation, and +then the potential energy in an electric ñeld is just charge times this quantity ¿: +ðf)== [ Eds +U = qọ. +Let us take, as an example, the case of two parallel metal plates, each with a +surface charge of +øơ per unit area. 'This is called a parallel-plate capacitor. We +found previously that there is zero force outside the plates and that there is a +constant electric field between them, directed from + to — and of magnitude ø/eg +(Fig. 14-5). We would like to know how much work would be done in carrying a +charge from one plate 6o the other. The work would be the (force) - (4s) integral, +which can be written as charge times the potential value at plate 1 minus that at +plate 2: +wr= | đ - ds = q(01 — 92). +W© can actually work out the integral because the force is constant, and 1Ÿ we +Tin non: +1/111) ị +TNnnnnnn" +Fig. 14-5. Field between parallel plates. +--- Trang 283 --- +call the separation of the plates đ, then the integral is easy: +l E.ds= =mỊ dự = T”, +1 €0 J1 €0 +The diference in potential, Aø = ơd/(cọ, is called the 0olfage đifJerence, and ở is +measured in volts. When we say a pair of plates is charged to a certain voltage, +what we mean is that the diference in electrical potential of the bwo plates is +So-and-so many volts. For a capacitor made of two parallel plates carrying a +surface charge -+ơ, the voltage, or difference in potential, of the pair of plates +is ơd/eọ. +--- Trang 284 --- +Tho Spocrerl Thoorgg of lĩocl(fitftgy +15-1 The principle of relativity +For over 200 years the equations of motion enunciated by NÑNewton were believed +to describe nature correctly, and the fñrst time that an error in these laws was +discovered, the way to correct it was also discovered. Both the error and its +correction were discovered by Einstein in 1905. +Newton?s Second Law, which we have expressed by the equation +†' = d(mu)/dt, +was siated with the tacit assumption that ?m is a constant, but we now know +that this is not true, and that the mass of a body increases with velocity. In +Binstein”s corrected formula ?m has the value +m=—————., (15.1) +v1— 12/c2 +where the “rest mass” rnọ represents the mass of a body that is not moving and é +is the speed of light, which is about 3 x 105 km -see~1 or about 186,000 mi - sec—1. +For those who want to learn just enough about it so they can solve problems, +that is all there is to the theory of relativity——it just changes Newton's laws by +introducing a correction factor 6o the mass. From the formula itself it is easy +to see that this mass increase is very small in ordinary circumstances. Tf the +velocity 1s even as great as that of a satellite, whiích goes around the earth at +5 mi/sec, then ø/c = 5/186,000: putting this value into the formula shows that +the correction to the mass is only one part in two to three billion, which is nearly +impossible to observe. Actually, the correcbness of the formula has been amply +confirmed by the observation oŸ many kinds of particles, moving at speeds ranging +up to practically the speed of light. However, because the efect is ordinarily +--- Trang 285 --- +so small, it seems remarkable that it was discovered theoretically before it was +discovered experimentally. Empirically, at a sufficiently high velocity, the efect +is very large, but it was not discovered that way. Therefore it is interesting to see +how a law that involved so delicate a modification (at the time when it was firsb +discovered) was brought to light by a combination of experiments and physical +reasoning. Contributions to the discovery were made by a number of people, the +ñnal result of whose work was Einstein's discovery. +There are really two Hinstein theories of relativity. 'This chapter is concerned +with the Special Theory of Relativity, which dates from 1905. In 1915 Einstein +published an additional theory, called the General 'Pheory of Relativity. This +latter theory deals with the extension of the Special Theory to the case of the +law of gravitation; we shall not discuss the General 'Pheory here. +The principle of relativity was first stated by Newton, in one of his corollaries +to the laws of motion: ““The motions of bodies included in a given space are +the same among themselves, whether that space 1s at rest or moves uniformly +forward in a straight line.” 'Phis means, for example, that 1Ý a space ship is drifting +along at a uniform speed, all experiments performed in the space ship and all the +phenomena in the space ship will appear the same as if the ship were not moving, +provided, of course, that one does not look outside. 'That is the meaning of the +principle of relativity. This is a simple enough idea, and the only question 1s +whether it is £rue that in all experiments performed inside a moving system the +laws of physics will appear the same as they would if the system were standing +still. Let us frst investigate whether Newton's laws appear the same in the +1noving system. +3uppose that Moe is moving in the z-direction with a uniform velocity , and +he measures the position of a certain point, shown in Fig. 15-1. He designates +the “z-distance” of the point in his coordinate system as #”. Joe is at rest, and +JOE MOE (x,y',z") +ụ e«.P or +(x,y.Z) +Fig. 15-1. TWo coordinate systems In uniform relative motion along +thelr x-axes. +--- Trang 286 --- +measures the position of the same point, designating its #ø-coordinate in his +system as ø. The relationship of the coordinates in the two systems is clear from +the diagram. After time ý Moe's origin has moved a distance œ#, and if the two +systems originally coincided, +zh—=#— tt, +Ă (15.2) +zZ —=#, +TÝ we substitute this transformation of coordinates into NÑewton's laws we fnd +that these laws transform to the same laws in the primed system; that is, the laws +of Newton are of the same form in a moving system as in a stationary system, +and therefore it is impossible to tell, by making mechanical experiments, whether +the system is moving or not. +The principle of relativity has been used in mechanies for a long time. lt +was employed by various people, in particular Huygens, to obtain the rules for +the collision of billiard balls, in much the same way as we used it in Chapter 10 +to discuss the conservation of momentum. In the 190h century interest in iE +was heightened as the result of investigations into the phenomena. of electricity, +magnetism, and light. A long series of careful studies of these phenomena by +many people culminated in Maxwells equations of the electromagnetic field, +which describe electricity, magnetism, and light in one uniform system. However, +the Maxwell equations did øœø# seem to obey the principle of relativity. That +is, IÝ we transform Maxwells equations by the substitution of equations (15.2), +theñr ƒorm does no‡ remain the same; therefore, in a moving space ship the +electrical and optical phenomena should be diferent from those in a stationary +ship. Thus one could use these optical phenomena to determine the speed of +the ship; in particular, one could determine the absolute speed of the ship by +making suitable optical or electrical measurements. One of the consequences of +Maxwells equations is that if there is a disturbance in the fñeld such that light is +generated, these electromagnetic waves go out in all directions equally and at +the same speed c, or 186,000 mi/sec. Another consequence oŸ the equations is +that 1f the source of the disturbance 1s moving, the light emitted goes through +space at the same speed c. 'This is analogous to the case of sound, the speed of +sound waves being likewise independent of the motion of the source. +'This independenece of the motion of the source, in the case of light, brings up +an interesting problem: +--- Trang 287 --- +Suppose we are riding in a car that is going at a speed , and light trom the +rear is going past the car with speed e. Diferentiating the frst equation in (15.2) +da /dt = d+/dt — tu, +which means that according to the Galilean transformation the apparent speed +of the passing light, as we measure it in the car, should not be é but should +be ce—u. For instance, if the car is going 100,000 mi/sec, and the light is +going 186,000 mi/sec, then apparently the light going past the car should go +86,000 mi/sec. In any case, by measuring the speed of the light going past the car +(ïf the Galilean transformation is correct for light), one could determine the speed +of the car. A number of experiments based on this general idea were performed to +determine the velocity of the earth, but they all failed—they gave no uelocitU +dÏl. We shall discuss one of these experiments in detail, to show exactly what was +done and what was the matter; something +0øs the matter, of course, something +was wrong with the equations of physics. What could it be? +15-2 The Lorentz transformation +'When the failure of the equations of physics in the above case came to light, +the fñrst thought that occurred was that the trouble must lie in the new Maxwell +equations of electrodynamics, which were only 20 years old at the time. It seemed +almost obvious that these equations must be wrong, so the thing to do was to +change them in such a way that under the Galilean transformation the principle +of relativity would be satisfied. When this was tried, the new terms that had to +be put into the equations led to predictions of new electrical phenomena that did +not exist at all when tested experimentally, so this attempt had to be abandoned. +'Then it gradually became apparent that Maxwell's laws of electrodynamics were +correct, and the trouble must be sought elsewhere. +In the meantime, H. A. Lorentz noticed a remarkable and curious thing when +he made the following substitutions in the Maxwell equations: +„h= % — Uuử +V1=u2/' +Ụ =U,; +z2, (15.3) +rằ t— u#/c2 +1_— u2/c2` +--- Trang 288 --- +namely, Maxwells equations remain in the same form when this transformation +is applied to theml Equations (15.3) are known as a Eoreniz transformation. +Hinstein, following a suggestion originally made by Poincaré, then proposed that +dÌl the phụs¿cal laes should be oŸ such a kind that they remain unchơngcd under +Loreniz transformation. In other words, we should change, not the laws of +electrodynamics, but the laws of mechanics. How shall we change Newton”s laws +so that £hew will remain unchanged by the Lorentz transformation? lf this goal is +set, we then have to rewrite Newton”s equations in such a way that the conditions +we have imposed are satisied. As it turned out, the only requirement is that the +mass ?m in Newton”s equations must be replaced by the form shown in Eq. (15.1). +'When this change is made, Newton's laws and the laws of electrodynamiecs will +harmomize. 'Phen if we use the Lorentz transformation in comparing Moe's +measurements with Joe”s, we shall never be able to detect whether either is +moving, because the form of all the equations wiïll be the same in both coordinate +systemsl +Tt is interesting to discuss what it means that we replace the old transformation +between the coordinates and time with a new one, because the old one (Galilean) +seems to be self-evident, and the new one (Lorentz2) looks peculiar. We wish +to know whether it is logically and experimentally possible that the new, and +not the old, transformation can be correct. 'To find that out, i% is not enough +to study the laws of mechanics but, as Einstein did, we too must analyze our +ideas of space and f#me in order to understand this transformation. We shall +have to discuss these ideas and their implications for mechanics at some length, +So we say in advance that the efort will be justifed, since the results agree with +experIment. +15-3 The Michelson-Morley experiment +As mentioned above, attempts were made to determine the absolute velocity +of the earth through the hypothetical “ether” that was supposed to pervade all +space. The most famous of these experiments is one performed by Michelson and +Morley in 1887. It was 18 years later before the negative results oŸ the experiment +were fñnally explained, by Einstein. +The Michelson-Morley experiment was performed with an apparatus like that +shown schematically in Fig. 15-2. This apparatus is essentially comprised of a +light source A, a partially silvered glass plate Ö, and two mirrors Œ and #, all +mounted on a rigid base. The mirrors are placed at equal distances Ù from ÿÖ. +--- Trang 289 --- +L4 +Souce “|À 2 \⁄8» Là +xx”Š5————‹>— = +‹ .. lÍ +Waves 3 S Waves out +in phase S < LG of phase +: : Š € Đa +DF Dị! +Fig. 15-2. Schematic diagram of the Michelson-Morley experiment. +The plate splits an oncoming beam of light, and the two resulting beams +continue in mutually perpendicular directions to the mirrors, where they are +reflected back to . Ôn arriving back at , the two beams are recombined as +two superposed beams, D and ?'. Tf the time taken for the light to go from ? +to È and back is the same as the time from ?Ö to Œ and back, the emerging +beams D and # wïll be in phase and will reinforce each other, but If the two +times difer slightly, the beams will be slightly out of phase and interference will +result. If the apparatus is “at rest” in the ether, the times should be precisely +cqual, but iŸ it is moving toward the right with a velocity w, there should be a +diference in the times. Let us see why. +Pirst, let us calculate the time required for the light to go from to and +back. Let us say that the time for light to go from plate Ö to mirror 2 is É\, +and the time for the return is ¿¿. Now, while the light is on its way from +to the mirror, the apparatus moves a distance œ#, so the light must traverse a +distance Ù + œ#t, at the speed c. We can also express this distance as c1, so we +cị = Ù+uớa, Or tạ = L/(c— 0). +(This result is also obvious from the point of view that the velocity of light relative +to the apparatus is e— , so the time is the length Ƒ divided by c— 0.) In a like +manner, the time #s can be calculated. During this time the plate Ö advances a +--- Trang 290 --- +distance œ£¿, so the return distance of the light is Ù — uứ¿. Then we have +ca —= ÙL— tua, OT tạ = L/(c+ 0). +Then the total time is +t + ta = 2Lc/(c2 — u2). +For convenience in later comparison of times we write this as +fi+tạ= ——>—a- 15.4 +¬—.. /cœ2 05-4) +Our second calculation will be of the time ¿¿ for the light to go trom to +the mirror Œ. As before, during tỉme ¿ the mirror Œ moves to the right a +distance œ‡a to the positlon C”; in the same time, the light travels a distance ca +along the hypotenuse oŸ a triangle, which is ĐC”. For this right triangle we have +(cts)? = L2 + (u£z)Ÿ +L2 = c”1 — u?tạ = (c?— u2)tã, +from which we get +tạ = L/VWc2— u2. +For the return trip from C” the distance is the same, as can be seen from the +symmetry of the fñgure; therefore the return time is also the same, and the total +time is 2f¿. With a little rearrangement of the form we can write +2L 2L/c +2fz = ————p — —2Hkc _. (15.5) +ve2—u2 v1-—u2/c2 +WS are now able to compare the times taken by the two beams of light. In +expressions (15.4) and (15.5) the numerators are identical, and represent the +time that would be taken ïf the apparatus were at rest. In the denominators, the +term ”/c? will be small, unless is comparable in size to c. The denominators +represent the modifications in the times caused by the motion of the apparatus. +And behold, these modifcations are no the sœme—the time to go to C and +back is a little less than the time to # and back, even though the mirrors are +equidistant from 7, and all we have to do is to measure that diference with +Drecision. +--- Trang 291 --- +Here a minor technical point arisessuppose the two lengths Ù are not exactÌy +cqual? In fact, we surely cannot make them exactly equal. In that case we simply +turn the apparatus 90 degrees, so that BƠ is in the line of motion and +is perpendicular to the motion. Any small diference in length then becomes +unimportant, and what we look for is a shØf in the interference Íringes when we +rotate the apparatus. +In carrying out the experiment, Michelson and Morley oriented the apparatus +so that the line BE was nearly parallel to the earth's motion in its orbit (at +certain times of the day and night). This orbital speed is about 18 miles per +second, and any “ether drift” should be at least that much at some time of the day +or night and at some time during the year. 'Phe apparatus was amply sensitive +to observe such an efect, but no time difference was found—the velocity of the +carth through the ether could not be detected. The result of the experiment was +'The result of the Michelson-Morley experiment was very puzzling and most +disturbing. The frst truitful idea for fñnding a way out oŸ the impasse came from +Lorentz. He suggested that material bodies contract when they are moving, and +that this foreshortening is only in the direction of the motion, and also, that 1Í +the length is họ when a body is at rest, then when it moves with speed œ parallel +to its length, the new length, which we call Lịị (L-parallel), is given by +Lị = LoV1— u2/c2. (15.6) +'When this modification is applied to the Michelson-Morley interferometer appa- +ratus the distance from #Ö to Œ does not change, but the distance trom #Ö to # +is shortened to 4/1 — u2/c2. Therefore Eq. (15.5) is not changed, but the of +Edq. (15.4) must be changed in accordance with Eq. (15.6). When this is done we +obtain +take (2L/c)w1—u2/c2 —— 2Lƒc 15.7 +"ốm. d5.) +Comparing this result with Eq. (15.5), we see that £+-+ta = 24. So ifthe apparatus +shrinks in the manner just described, we have a way of understanding why the +Michelson-Morley experiment gives no efect at all. Although the contraction +hypothesis successfully accounted for the negative result of the experiment, it was +open to the objection that it was invented for the express purpose of explaining +away the difficulty, and was too artificial. However, in many other experiments +to discover an ether wind, similar difficulties arose, until it appeared that nature +--- Trang 292 --- +was in a “conspiracy” to thwart man by introducing some new phenomenon ©o +undo every phenomenon that he thought would permit a measurement of ứ. +It was ultimately recognized, as Poincaré pointed out, that ø comgplete con- +spfrac is iselƒ a lau oƒ naturel Poincaré then proposed that there ¡s such a law +of nature, that it is not possible to discover an ether wind by øng experiment; +that is, there is no way to determine an absolute velocity. +15-4 Transformation of tỉme +In checking out whether the contraction idea is in harmony with the facts +in other experimentfs, i% turns out that everything is correct provided that the +tmes are also modifiled, in the manner expressed in the fourth equation of the +set (15.3). That is because the time £z, calculated for the trip trom Ö to Œ and +back, is not the same when calculated by a man performing the experiment in a +moving space ship as when calculated by a stationary observer who is watching +the space ship. To the man ín the ship the time is simply 2/e, but to the other +observer it is (21/c)/1— u2/c2 (Eq. 15.5). In other words, when the outsider +sees the man in the space ship lighting a cigar, all the actions appear to be +slower than normal, while to the man inside, everything moves at a normal rate. +So not only must the lengths shorten, but also the time-measuring instruments +(“clocks”) must apparently slow down. That is, when the clock in the space ship +records 1 second elapsed, as seen by the man in the ship, it shows 1/4/1 — u2/c2 +second to the man outside. +This slowing of the clocks in a moving system is a very peculiar phenomenon, +and is worth an explanation. In order to understand this, we have to watch the +machinery of the clock and see what happens when it is moving. Since that 1s +rather dificult, we shall take a very simple kind of clock. The one we choose is +rather a silly kind of clock, but it will work in principle: it is a rod (meter stick) +with a mirror at each end, and when we start a light signal between the mirrors, +the light keeps going up and down, making a click every time it comes down, +like a standard ticking clock. We build ©wo such clocks, with exactly the same +lengths, and synchronize them by starting them together; then they agree always +thereafter, because they are the same in length, and light always travels with +speed c. We give one of these clocks to the man to take along in his space ship, +and he mounts the rod perpendicular to the direction of motion of the ship; then +the length of the rod will not change. How do we know that perpendicular lengths +do not change? The men can agree to make marks on each other's -meter stick +--- Trang 293 --- +as they pass each other. By symmetry, the 6wo marks must come at the same 1 +and #-coordinates, since otherwise, when they get together to compare results, +one mark will be above or below the other, and so we could tell who was really +1noving. +Now let us see what happens to the moving clock. Before the man took 1§ +aboard, he agreed that it was a nice, standard clock, and when he goes along in +the space ship he will not see anything peculiar. If he did, he would know he +was moving—If anything at all changed because of the motion, he could tell he +was moving. But the prineiple of relativity says this is impossible in a uniformly +moving system, so nothing has changed. Ôn the other hand, when the external +observer looks at the clock going by, he sees that the light, in going from mirror +to mirror, is “really” taking a zigzag path, since the rod is moving sidewise all +the while. We have already analyzed such a zigzag motion in connection with +the Michelson-Morley experiment. lfin a given time the rod moves forward +a distance proportional to in Eig. 15-3, the distance the light travels in the +same tỉme is proportional to e, and the vertical distance is therefore proportional +to W2 — u2. +That is, it takes a longer tứmne for light to go tom end to end in the moving +clock than in the stationary clock. Therefore the apparent time bebween clicks is +longer for the moving clock, in the same proportion as shown in the hypotenuse +of the triangle (that is the source of the square root expressions in our equations). +trom the figure it is also apparent that the greater œ is, the more slowly the +moving clock appears to run. Not only does this particular kind of clock run +more slowly, but if the theory of relativity is correct, any other clock, operating +on any principle whatsoever, would also appear to run slower, and in the same +proportion—we can say this without further analysis. Why is this so? +To answer the above question, suppose we had two other clocks made exactÌy +alike with wheels and gears, or perhaps based on radioactive decay, or something +else. Then we adjust these clocks so they both run in precise synchronism with +our frst clocks. When light goes up and back in the frst clocks and announces +1ts arrival with a click, the new models also complete some sort of cycle, which +they simultaneously announce by some doubly coincident flash, or bong, or other +signal. One of these clocks is taken into the space ship, along with the fñrst kind. +Perhaps #h2s clock will not run slower, but will continue to keep the same time +as its stationary counterpart, and thus disagree with the other moving clock. Ah +no, 1ƒ that should happen, the man ïn the ship could use this mismatch between +his two clocks to determine the speed of his ship, which we have been supposing +--- Trang 294 --- +Mirror +Sr system D +xí Ề | +Photocell +¬ reflected ========e +.. | 3UT _"" +S system + +x x4 N5 D +Pulse _—> bàce-g Pulse +emitted () received +wc2— ư] +Fig. 15-3. (a) A “tight clock” at rest in the S” system. (b) The same +clock, moving through the S system. (c) Illustration of the diagonal +path taken by the light beam In a moving “light clock.” +--- Trang 295 --- +is Impossible. We need not knou anthing about the machinerg of the new clock +that might cause the efect——we simply know that whatever the reason, it will +appear to run slow, just like the first one. +Now ïf alÏ moving clocks run sÌower, iÝ no way oŸ measuring time gives anything +but a slower rate, we shall Just have to say, In a certain sense, that #ữne ?‡sclƒ +appears to be slower in a space ship. All the phenomena there—the man”s +pulse rate, his thought processes, the time he takes to light a cigar, how long 1 +takes to grow up and get old—all these things must be slowed down in the same +proportion, because he cannot tell he is moving. The biologists and medical men +sometimes say it is not quite certain that the time it takes for a cancer to develop +will be longer in a space ship, but from the viewpoint of a modern physicist +1t is nearly certain; otherwise one could use the rate oŸ cancer development to +determine the speed of the ship! +A very interesting example of the slowing of time with motion is furnished by +mu-mesons (muons), which are particles that disintegrate spontaneously after an +avcrage lifetime of 2.2 x 10”8 sec. They come to the earth in cosmic rays, and +can also be produced artifcially in the laboratory. 5ome of them disintegrate +in midaïr, but the remainder disintegrate only after they encounter a piece of +material and stop. It is clear that in is short lifetime a muon cannot travel, +even at the speed of light, mụch more than 600 meters. But although the muons +are created at the top of the atmosphere, some 10 kilometers up, yet they are +actually found in a laboratory down here, in cosmic rays. How can that be? +The answer is that diferent muons move at various speeds, some of which are +very close to the speed of light. While from their own point of view they live +only about 2 /sec, from our point of view they live considerably longer—enough +longer that they may reach the earth. 'Phe factor by which the tỉme is increased +has already been given as 1/4/1 — ^2/c2. The average life has been measured +quite accurately for muons of diferent velocities, and the values agree closely +with the formula. +W©e do not know why the meson disintegrates or what its machinery 1s, but +we do know its behavior satisfes the principle of relativity. That is the utility of +the principle of relativity——it permits us to make predictions, even about things +that otherwise we do not know mụch about. Eor example, before we have any +idea at all about what makes the meson disintegrate, we can still predict that +when it is moving at nine-tenths of the speed of light, the apparent length of time +that it lasts is (2.2 x 10”8)/4/1 — 92/102 sec; and our prediction works—that is +the good thing about ït. +--- Trang 296 --- +15-5 The Lorentz contraction +Now let us return to the Lorentz transformation (15.3) and try 6o get a better +understanding of the relationship between the (z,,z,f) and the (z',,z,t) +coordinate systems, which we shall call the S and 5” systems, or Joe and Moe +systems, respectively. We have already noted that the first equation is based on +the Lorentz suggestion of contraction along the z-direction; how can we prove +that a contraction takes place? In the Michelson-Morley experiment, we now +appreciate that the fransuerse arm BC cannot change length, by the principle +of relativity; yet the null result of the experiment demands that the £#mes must +be equal. So, in order for the experiment to give a null result, the longitudinal +am BE must appear shorter, by the square root 4/1 — u2/c2. What does thìs +contraction mean, in terms of measurements made by Joe and Moe? Suppose +that Moe, moving with the Š” system in the z-direction, is measuring the #“- +coordinate of some point with a meter stick. He lays the stick down #“ tỉmes, so +he thinks the distance is ø“ meters. From the viewpoint of Joe in the Š system, +however, Moe is using a foreshortened ruler, so the “real” distance measured is +#“V1— u2/c2 meters. Then if the 5“ system has travelled a distance uý away +from the Š system, the Š observer would say that the same point, measured in +his coordinates, is at a distance œ = #/4/1— u2/c2 + uÈ, or +; % — UuÈ += — ma. n.') +V1= u2/e +which is the first equation of the Lorentz transformation. +15-6 Simultaneity +In an analogous way, because of the difference in time scales, the denominator +expression is introduced into the fourth equation of the Lorentz transformation. +The most interesting term in that equation is the #/c in the numerator, because +that is quite new and unexpected. Now what does that mean? If we look at the +situation carefully we see that events that occur at two separated places at the +same time, as seen by Moe in ®”, do nø‡ happen at the same tỉme as viewed by +Joe in 6. lf one event occurs at point zø+ at time #o and the other event at #s +and £o (the same time), we ñnd that the two corresponding times /¡ and £2 difer +by an amount +tứ — u(Œ1 — +2) /c? +¿2 V1—u2/c2 ` +--- Trang 297 --- +'This circumstance ïs called “failure of simultaneity at a distance,” and to make +the idea a little clearer let us consider the following experiment. +Suppose that a man moving ïn a space ship (system ,5”) has placed a clock at +each end of the ship and is interested in making sure that the two clocks are in +synchronism. How can the clocks be synchronized? There are many ways. One +way, involving very little calculation, would be frst to locate exactly the midpoint +between the clocks. hen from this station we send out a light signal which will +go both ways at the same speed and will arrive at both clocks, clearly, at the +same time. 'Phis simultaneous arrival of the signals can be used to synchronize +the clocks. Let us then suppose that the man in Š” synchronizes his clocks by +this particular method. Let us see whether an observer in system Š would agree +that the two clocks are synchronous. The man in Š” has a right to believe they +are, because he does not know that he is moving. But the man in Š reasons that +since the ship is moving forward, the clock in the front end was running away +trom the light signal, hence the light had to go more than halfway in order to +catch up; the rear clock, however, was advancing to meet the light signal, so this +distance was shorter. Therefore the signal reached the rear clock frst, although +the man in S5” thought that the signals arrived simultaneously. We thus see that +when a man in a space ship thinks the times at two locations are simultaneous, +cqual values of # in his coordinate system must correspond to đjferent values +of # in the other coordinate systeml +1ã-7 EFour-vectors +Let us see what else we can discover in the Lorentz transformation. Ït is +interesting to note that the transformation between the #'s and £”s is analogous +in form to the transformation of the #ø's and s that we studied in Chapter 11 +for a rotation oŸ coordinates. We then had +, : +; #cos ổ -Ƒ sìn ổ, (15.8) +ự —=cos8 — zsin0, +in which the new #“ mixes the old z and , and the new ˆ also mixes the old ++ and ; similarly, in the Lorentz transformation we fnd a new zø“ which is a +mixture of z and ý, and a new £ which is a mixture of ¿ and z. So the Lorentz +transformation is analogous to a rotation, only it is a “rotation” in spœce and +time, which appears to be a strange concept. A check of the analogy to rotation +--- Trang 298 --- +can be made by calculating the quantity +a2 + 2 + z2 — c?U2 = x? + 02+ z? — c?£. (15.9) +In this equation the first three terms on each side represent, in three-dimensional +geometry, the square of the distance between a point and the origin (surface +of a sphere) which remains unchanged (invariant) regardless of rotation of the +coordinate axes. Similarly, Øq. (15.9) shows that there is a certain combination +which includes time, that is invariant to a Lorentz transformation. Thus, the +analogy to a rotation is complete, and is of such a kind that vectors, 1.e., quantities +involving “components” which transform the same way as the coordinates and +time, are also useful in connection with relativity. +'Thus we contemplate an extension of the idea of vectors, which we have so far +considered to have only space components, to ineclude a time component. That +1s, we expect that there will be vectors with four components, three of which are +like the components of an ordinary vector, and with these will be associated a +fourth component, which is the analog of the time part. +This concept will be analyzed further in the next chapters, where we shall +fínd that ïf the ideas of the preceding paragraph are applied to momentum, +the transformation gives three space parts that are like ordinary momentum +components, and a fourth component, the time part, which is the energ. +15-8 Relativistic dynamics +W©S are now ready to investigate, more generally, what form the laws of +mechanics take under the Lorentz transformation. [We have thus far explained +how length and time change, but not how we get the modifed formula for ?n +(Eq. 15.I). We shall do this in the next chapter.]} To see the consequences +of Hinstein's modification of m for Newtonian mechanics, we start with the +Newtonian law that force is the rate of change of momentum, or +Ƒ' = d(mo)/dt. +Momentum is still given by rm, but when we use the new ?n this becomes +ÐĐ= 1ẽU= ——————. 15.10 +v1— 032/c2 ( ) +--- Trang 299 --- +This is Einstein's modification of Newton”s laws. Under this modification, 1Í +action and reaction are still equal (which they may not be in detail, but are in +the long run), there will be conservation oŸ momentum in the same way as before, +but the quantity that is being conserved is not the old z2 with its constant mass, +but instead is the quantity shown in (15.10), which has the modifed mass. When +this change is made in the formula for momentum, conservation of momentum +still works. +Now let us see how momentum varies with speed. In NÑewtonian mechanics it +is proportional to the speed and, according (15.10), over a considerable range of +speed, but small compared with e, it is nearly the same in relativistic mechanics, +because the square-root expression differs only slightly from 1. But when 0 is +almost equal to e, the square-root expression approaches zero, and the momentum +therefore goes toward infnity. +'What happens i a constant force acts on a body for a long time? In Newtonian +mechanies the body keeps picking up speed until it goes faster than light. But +this is impossible in relativistic mechanics. In relativity, the body keeps picking +up, not speed, but momentum, which can continually increase because the mass +1s increasing. After a while there is practically no acceleration in the sense of a +change of velocity, but the momentum continues to increase. Of course, whenever +a force produces very little change in the velocity of a body, we say that the body +has a great deal oŸ inertia, and that is exactly what our formula for relativistic +mass says (see lq. 15.10)—it says that the inertia is very great when ø is nearly +as great as c. Ás an example of this efect, to defect the high-speed electrons +in the synchrotron that is used here at Caltech, we need a magnetic ñeld that +1s 2000 times stronger than would be expected on the basis of Newton”s laws. +In other words, the mass of the electrons in the synchrotron is 2000 times as +great as their normal mass, and is as great as that of a protonl That m should +be 2000 times mọ means that 1 — ø2/c2 must be 1/4,000,000, and that means +that 0 difers from c by one part in 8,000,000, so the electrons are getting pretty +close to the speed of light. If the electrons and light were both to start from +the synchrotron (estimated as 700 feet away) and rush out to Bridge Lab, which +would arrive first? The light, of course, because light always travels faster.* How +mụuch earlier? 'Phat is too hard to tell—instead, we tell by what distance the +light is ahead: ¡it is about 1/1000 of an inch, or hì the thickness of a piece of +* 'Phe electrons would actually win the race versus 0¿s2ble light because of the index of +refraction of air. A gamma ray would make out better. +--- Trang 300 --- +paperl When the electrons are going that fast their masses are enormous, but +their speed cannot exceed the speed oŸ light. +Now let us look at some further consequences of relativistic change of mass. +Consider the motion of the molecules in a small tank of gas. When the gas +is heated, the speed of the molecules is increased, and therefore the mass 1s +also increased and the gas is heavier. An approximate formula to express the +Increase of mass, for the case when the velocity is small, can be found by +expanding mo/1 — 02/c2 = mọ(1 — 02/c2)~1⁄2 in a power series, using the +binomial theorem. We get +mạ(1L— t2/cÈ) 8 = mạ(1 + }08/cÊ + 304/et +), +W© see clearly from the formula that the series converges rapidly when 0 is small, +and the terms after the first two or three are negligible. So we can write +~ 1 s( 1 +m nọ + srnoU 2 (15.11) +in which the second term on the ripght expresses the increase of mass due to +molecular velocity. When the temperature inereases the ø2 increases proportion- +ately, so we can say that the increase in mass is proportional to the increase +in temperature. But since singu? 1s the kinetic energy in the old-fashioned +Newtonian sense, we can also say that the increase In mass of all this body of gas +is equal to the inerease in kinetic energy divided by e2, or Am = A(K.B.)/€2. +15-9 Equivalence of mass and energy +The above observation led Einstein to the suggestion that the mass of a body +can be expressed more simply than by the formula (15.1), if we say that the mass +is equal to the total energy content divided by c2. If Eq. (15.11) is multiplied +by c? the result is +mcŸ = mạc” + 3m0” + - -- (15.12) +Here, the term on the left expresses the total energy oŸ a body, and we recognize +the last term as the ordinary kinetic energy. Einstein interpreted the large +constant term, ?moe2, to be part of the total energy of the body, an intrinsic +energy known as the “rest energy.” +Let us follow out the consequences of assuming, with Einstein, that ứhe +energu oƒ a bod aluas eguals mc2. As an interesting result, we shall find the +--- Trang 301 --- +formula (15.1) for the variation of mass with speed, which we have merely assumed +up to now. W© start with the body at rest, when its energy is mọc”. Then we +apply a force to the body, which starts it moving and gives it kinetic energy; +therefore, since the energy has increased, the mass has increased——this is Implieit +in the original assumption. So long as the force continues, the energy and the +mass both continue to increase. We have already seen (Chapter 13) that the rate +of change of energy with time equals the force times the velocity, or +—— —= È'`-0. 15.13 +di ° 5.13) +W© also have (Chapter 9, Eq. 9.1) that ' = d(mo)/dt. When these relations are +put together with the delnition of , Eq. (15.13) becomes +d(mc2) d(mb) +————=U'—.. 15.14 +dc ””” đ 5.14) +We wish to solve this equation for ?mm. To do this we first use the mathematical +trick of multiplying both sides by 2mm, which changes the equation to +đm d(ựm?®) +2m) —— = 2m0 - ———. 15.15 +cm) d‡ TT ' ) +W© need to get rid of the derivatives, which can be accomplished by integrating +both sides. The quantity (2m) đưm/đf can be recognized as the từne derivative +of m2, and (2m0) - d(mo) /dt is the tỉme derivative of (mø)2. So, Eq. (15.15) is +the same as (m2) (m202) +d(m d(m“u +———=—.. 15.16 +“— dị di (15.16) +Tf the derivatives of two quantities are equal, the quantities themselves differ at +most by a constant, say Œ. 'Phis permits us to write +m°c? = m?u° + Œ. (15.17) +W© necd to delne the constant Œ more explicitly. Since Eq. (15.17) must be true +for all velocities, we can choose a special case where ø = 0, and say that in this +case the mass is mo. Substituting these values into Eq. (15.17) gives +mặc? =0+ Œ. +--- Trang 302 --- +W© can now use thịs value of Ở in Eq. (15.17), which becomes +mỸc2 = mm 2u” + mặc”. (15.18) +Dividing by e? and rearranging terms gives +mÊ(1 — 0Ê/c2) = mã, +from which we get +m = mo/W1— 02/c2. (15.19) +This is the formula (15.1), and is exactly what is necessary for the agreement +between mass and energy in Eq. (15.12). +Ordinarily these energy changes represent extremely slight changes in mass, +because most of the time we cannot generate much energy from a given amount +of material; but in an atomie bomb of explosive energy equivalent to 20 kilotons +of NT, for example, it can be shown that the dirt after the explosion is lighter +by 1 gram than the initial mass of the reacting material, because of the energy +that was released, I.e., the released energy had a mass of l1 gram, according to +the relationship AE = A(mc2). Thịis theory of equivalence of mass and energy +has been beautifully verified by experiments in which matter is annihilated—— +convcrted totally to energy: An electron and a positron come together at rest, +cach with a rest mass mmọ. When they come together they disintegrate and bwo +øamma rays emerge, each with the measured energy of mọc2. This experiment +furnishes a direct determination of the energy associated with the existence of +the rest mass of a particle. +--- Trang 303 --- +I6 +Miolqfitisfic Freorgjgg «areeÏ WẤC@rtt©reftrrtt +16-1 Relativity and the phỉilosophers +In this chapter we shall continue to discuss the principle of relativity of +Binstein and Poincaré, as it afects our ideas of physics and other branches of +human thought. +Poincaré made the following statement of the principle of relativity: “Ac- +cording to the principle of relativity, the laws of physical phenomena must be +the same for a fixed observer as for an observer who has a uniform motion of +translation relative to him, so that we have not, nor can we possibly have, any +mmeans of discerning whether or not we are carried along in such a motion.” +When this idea descended upon the world, it caused a great stir among +philosophers, particularly the “cocktail-party philosophers,” who say, “Oh, it is +very simple: Einstein's theory says all is relativel” In fact, a surprisingly large +number oŸ philosophers, not only those found at cocktail parties (but rather than +embarrass them, we shall Just call them “cocktail-party philosophers”), will say, +“That all is relative is a consequence of Einstein, and it has profound infuences +on our ideas.” In addition, they say “lt has been demonstrated in physics that +phenomena depend upon your frame of reference.” We hear that a great deal, +but ï is dificult to ñnd out what it means. Probably the frames of reference +that were originally referred to were the coordinate systems which we use in the +analysis of the theory of relativity. So the fact that “things depend upon your +frame of reference” is supposed to have had a profound efect on modern thought. +One might well wonder why, because, after all, that things depend upon one”s +point oŸ view is so simple an idea that it certainly cannot have been necessary to +go to all the trouble of the physical relativity theory in order to discover it. That +what one sees depends upon his tame of reference is certainly known to anybody +who walks around, because he sees an approaching pedestrian first from the +tront and then from the back; there is nothing deeper in most of the philosophy +--- Trang 304 --- +which is said to have come from the theory of relativity than the remark that “A +person looks diferent from the front than from the back.” 'Phe old story about +the elephant that several blind men describe in different ways is another example, +perhaps, of the theory of relativity from the philosopher”s point of view. +But certainly there must be deeper things in the theory of relativity than +Just this simple remark that “Á person looks điferent rom the front than from +the back.” Of course relativity is deeper than this, because +0e cøn rmmauke deftnife +prediclions tuïth dt. It certainly would be rather remarkable if we could predict +the behavior of nature from such a simple observation alone. +'There is another school of philosophers who feel very uncomfortable about the +theory of relativity, which asserts that we cannot determine our absolute velocity +without looking at something outside, and who would say, “lt is obvious that +one cannot measure his velocity without looking outside. It is self-evident that it +1s mmeaningless to talk about the velocity of a thing without looking outside; the +physieists are rather stupid for having thought otherwise, but ït has Just dawned on +them that this is the case. If only we philosophers had realized what the problems +were that the physicists had, we could have decided immediately by brainwork +that it is impossible to tell how fast one is moving without looking outside, and +we could have made an enormous contribution ©o physics.” 'These philosophers +are always with us, struggling in the periphery $o try to tell us something, but +they never really understand the subtleties and depths of the problem. +Our inability to detect absolute motion is a result of ezperiment‡ and not a +result of plain thought, as we can easily illustrate. In the fñrst place, Newton +believed that it was true that one could not $ell how fast he is going If he 1s +moving with uniform velocity in a straight line. In fact, Newton frst stated +the principle of relativity, and one quotation made in the last chapter was a +statement of Newtons. Why then did the philosophers not make all this fuss +about “all is relative,” or whatever, in Newton's time? Because it was not until +Maxwells theory of electrodynamics was developed that there were physical laws +that suggested that one couldd measure his velocity without looking outside; soon +1ÿ was found ezperimenfallu that one could noi. +Now, 7s it absolutely, defñnitely, philosophically øœ=ecessøar that one should +not be able $o tell how fast he is moving without looking outside? One of the +consequences of relativity was the development of a philosophy which said, “You +can only define what you can measurel Sinece it is self-evident that one cannot +measure a velocity without seeing what he is measuring it relative to, therefore +1t 1s clear that there is no rmeønzng to absolute velocity. 'he physicists should +--- Trang 305 --- +have realized that they can talk only about what they can measure.” But tha£ +¡s the tphole problem: whether or not one can define absolute velocity is the +same as the problem of whether or not one can defect ïn an ezpertữment, without +looking outside, whether he is moving. In other words, whether or not a thing +1s measurable is not something to be decided ø pz7or¿ by thought alone, but +something that can be decided only by experiment. Given the fact that the +velocity of light is 186,000 mi/sec, one will ñnd few philosophers who will calmly +state that it is selEevident that if light goes 186,000 mi/sec inside a car, and +the car is going 100,000 mi/sec, that the light also goes 186,000 mi/sec past an +observer on the ground. That is a shocking fact to them; the very ones who claïim +1E is obvious ñnd, when you give them a specifc fact, that it is not obvious. +Finally, there is even a philosophy which says that one cannot detect ønw +motion except by looking outside. It is simply not true in physics. True, one +cannot perceive a wn⁄form motion in a strøigh# line, but 1f the whole room were +rotating we would certainly know it, for everybody would be thrown to the wall— +there would be all kinds of “centrifugal” efects. That the earth is turning on its +axis can be determined without looking at the stars, by means of the so-called +Foucault pendulum, for example. 'Therefore it is not true that “all is relative”; it +is only wniform 0elocitu that cannot be detected without looking outside. Uniform +rotation about a fxed axis cơn be. When this is told to a philosopher, he is very +upset that he did not really understand it, because to him 1% seems Impossible +that one should be able to determine rotation about an axis without looking +outside. If the philosopher is good enough, after some time he may come back +and say, “I understand. We really do not have such a thing as absolute rotation; +we are really rotating relatiue to the stars, you see. And so some infuence exerted +by the stars on the object must cause the centrifugal force.” +Now, for all we know, that is true; we have no way, at the present time, of +telling whether there would have been centrifugal force if there were no stars and +nebulae around. We have not been able to do the experiment of removing all the +nebulae and then measuring our rotation, so we simply do not know. We must +admit that the philosopher may be right. He comes back, therefore, in delight and +says, “lt is absolutely necessary that the world ultimately turn out to be this way: +absolute rotation means nothing; it is only rela#zue to the nebulae.” Then we say to +him, “ Nou, my friend, is it or is it not obvious that uniform velocity in a straight +line, relate to the nebulae should produce no efects inside a car?” Now that the +motion is no longer absolute, but is a motion relatibe to the nebulae, it becomes +a mysterious question, and a question that can be answered only by experiment. +--- Trang 306 --- +'What, then, are the philosophiec inÑuences of the theory of relativity? If we +limit ourselves to infuences in the sense of t0ha‡ kứnd oƒ neu tdeas ơnd suggestions +are made to the physicist by the prineiple of relativity, we could describe some of +them as follows. The first discovery is, essentially, that even those ideas which +have been held for a very long time and which have been very accurately veriflled +might be wrong. Ït was a shocking discovery, of course, that NÑewton”s laws are +wrong, after all the years in which they seemed to be accurate. Of course iW is +clear, not that the experiments were wrong, but that they were done over only a +limited range of velocities, so smaill that the relativistic efects would not have +been evident. But nevertheless, we now have a mụch more humble point of view +of our physical laws——everything can be wrongl +Secondly, if we have a set of “strange” ideas, such as that time goes sÌower +when one moves, and so forth, whether we /Zke them or do nöø# like them is an +Irrelevant question. 'The only relevant question is whether the ideas are consistent +with what is found experimentally. In other words, the “strange ideas” need only +agree with ezperimemt, and the only reason that we have to discuss the behavior +of clocks and so forth is to demonstrate that although the notion of the time +dilation is strange, 1È is conszs‡ten£ with the way we measure time. +Finally, there is a third suggestion which is a little more technical but which +has turned out to be of enormous utility in our study of other physical laws, +and that is to look at the sụmmetru oƒ the la+s or, more specifcally, to look for +the ways in which the laws can be transformed and leave their form the same. +When we discussed the theory of vectors, we noted that the fundamental laws +of motion are not changed when we rotate the coordinate system, and now we +learn that they are not changed when we change the space and time variables In +a particular way, given by the Lorentz transformation. 5o this idea of studying +the patterns or operations under which the fundamental laws are not changed +has proved to be a very useful one. +16-2 The twin paradox +To continue our discussion of the Lorentz transformation and relativistic +efects, we consider a famous so-called “paradox” of Peter and Paul, who are +supposed to be twins, born at the same time. When they are old enough to +drive a space ship, Paul flies away at very hiph speed. Because Peter, who is left +on the ground, sees Paul goïing so fast, all of Paul”s clocks appear to go sÌower, +his heart beats go slower, his thoughts go slower, everything goes sÌlower, from +--- Trang 307 --- +Peter”s point of view. Of course, Paul notices nothing unusual, but if he travels +around and about for a while and then comes back, he will be younger than Peter, +the man on the groundl “That is actually right; it is one of the consequences +of the theory of relativity which has been clearly demonstrated. đJust as the +mmu-mesons last longer when they are moving, so also wiïll Paul last longer when +he is moving. 'This is called a “paradox” only by the people who believe that the +principle of relativity means that aÏl motion 1s relative; they say, “Heh, heh, heh, +from the point of view of Paul, can't we say that Pe‡er was moving and should +therefore appear to age more slowly? By symmetry, the only possible result is +that both should be the same age when they meet.” But in order for them to +come back together and make the comparison, Paul must either stop at the end +of the trip and make a comparison of clocks or, more simply, he has to come +back, and the one who comes back must be the man who was moving, and he +knows this, because he had to turn around. When he turned around, all kinds of +unusual things happened in his space ship—the rockets went of, things Jjammed +up against one wall, and so on—while Peter felt nothing. +So the way to state the rule is to say that the man tpho has [elt the accelerations, +who has seen things fall against the walls, and so on, is the one who would be +the younger; that ¡is the diference between them in an “absolute” sense, and 1t +1s certainly correct. When we discussed the fact that moving mu-mesons live +longer, we used as an example their straight-line motion in the atmosphere. But +we can also make mu-mesons in a laboratory and cause them to go in a curve +with a magnet, and even under this accelerated motion, they last exactly as much +longer as they do when they are moving in a straight line. Although no one has +arranged an experiment explicitly so that we can get rid of the paradox, one +could compare a mu-meson which ¡s left standing with one that had gone around +a complete circle, and it would surely be found that the one that went around +the cirele lasted longer. Although we have not actually carried out an experiment +using a complete circle, it is really not necessary, of course, because everything +ñts together all right. This may not satisfy those who insist that every single fact +be demonstrated directly, but we confidently predict the result of the experiment +in which Paul goes in a complete cirele. +16-3 Transformation of velocities +The main diference bebween the relativity of Einstein and the relativity of +Newton is that the laws of transformation connecting the coordinates and times +--- Trang 308 --- +between relatively moving systems are diferent. The correct transformation law, +that of Lorentz, is +„h= % — tu +v1 u2/c2' +Ụ =1, +16.1 +m (16.1) +rằ t— u#/c2 +v1—*2/c2 +These equations correspond to the relatively simple case in which the relative +motion of the two observers 1s along their common z-axes. Of course other +directions of motion are possible, but the most general Lorentz transformation is +rather complicated, with all four quantities mixed up together. We shall continue +to use this simpler form, since it contains all the essential features of relativity. +Let us now discuss more of the consequences of this transformation. First, it +1s Interesting to solve these equations in reverse. hat is, here is a set of linear +equations, four equations with four unknowns, and they can be solved in reverse, +for #, , z, in terms of #”,3/,z”,f. The result is very interesting, since it tells us +how a system of coordinates “at rest” looks from the point of view of one that is +“moving.” ÔÝ course, since the motions are relative and of uniform velocity, the +man who is “moving” can say, if he wishes, that it is really the other fellow who +is moving and he himself who is at rest. And since he is moving in the opposite +direction, he should get the same transformation, but with the opposite sign of +velocity. Thhat is precisely what we ñnd by manipulation, so that is consistent. lÝ +it địd not come out that way, we would have real cause to worryl ++ + uf! +#=——————p, +v1—2/c2 +, (16.2) +". Ứ + tua! /c2 +V1—u2/c2: +Next we discuss the interesting problem of the addition of velocities in relativity. +We© recall that one of the original puzzles was that light travels at 186,000 mi/sec +in all systems, even when they are in relative motion. 'Phis is a special case of +--- Trang 309 --- +the more general problem exermplified by the following. Suppose that an object +inside a space ship is going at 100,000 mi/sec and the space ship itself is goïing at +100,000 mi/sec; how fast is the object inside the space ship moving from the point +of view of an observer outside? We might want to say 200,000 mi/sec, which is +faster than the speed of light. This is very unnerving, because it is not supposed +to be going faster than the speed of lightl "The general problem is as follows. +Let us suppose that the object inside the ship, from the point of view of +the man inside, is moving with velocity 0, and that the space ship Itself has a +velocity œ with respect to the ground. We want to know with what velocity 0; +this object is moving from the point of view of the man on the ground. 'Thịs is, +of course, still but a special case in which the motion is in the z-direction. “There +will also be a transformation for velocities in the ¿-direction, or for any angle; +these can be worked out as needed. Inside the space ship the velocity 1s ơ„, +which means that the displacement z” is equal to the velocity times the tỉme: ++ = Uy. (16.3) +Now we have only to calculate what the position and time are from the point of +view of the outside observer for an object which has the relation (16.2) between ++“ and #. So we simply substitute (16.3) into (16.2), and obtain +... (16.4) +v1—u?/c2 +But here we fnd z expressed in terms of f“. In order to get the velocity as seen +by the man on the outside, we must divide hús distance by hás từne, not by the +other rmmans timef So we must also calculate the £#me as seen from the outside, +which 1s +# ;# 2 +"x. “dã (16.5) +v1— 1u2/c2 +Now we must fnd the ratio of z to , which is +H5 + -Ƒ Đại +=—==——a, 16.6 +"`" /c 066) +the square roots having cancelled. 'This is the law that we seek: the resultant +velocity, the “summing” of two velocities, is not just the algebraic sum oŸ wo +--- Trang 310 --- +velocities (we know that it cannot be or we get in trouble), but is “corrected” +by 1+ uo/c. +Now let us see what happens. Suppose that you are moving inside the space +ship at half the speed of light, and that the space ship itself is goiïng at half the +specd of light. 'Thus % is 2C and 0 is Ọ€, but ïn the denominator œo is one-fourth, +so that +›€C + ›C 4e +“` 1+1. 5 +So, in relativity, “half” and “half” does not make “one,” it makes only “4/5” Of +course low velocities can be added quite easily in the familiar way, because so +long as the velocities are small compared with the speed of light we can forget +about the (1 + œø/e2) factor; but things are quite diferent and quite interesting +at high velocity. +Let us take a limiting case. Jjust for fun, suppose that inside the space ship +the man was observing lgh# ?tsejf In other words, ø = c, and yet the space ship +is moving. How will it look to the man on the ground7? 'Phe answer will be +tu -+EC u+C +" 1+ ue/c2 _..aẽ. +Therefore, if something is moving at the speed of light inside the ship, it will +appear to be moving at the speed of light from the point of view of the man +on the ground tool "This is good, for it is, in fact, what the Einstein theory of +relativity was desiegned to do in the frst place—so it had beffer workl +Of course, there are cases in which the motion is not in the direction of the +uniform translation. For example, there may be an object inside the ship which +is just moving “upward” with the velocity 0y; with respect to the ship, and the +ship is moving “horizontally.” Now, we simply go through the same thing, only +using #s instead of z's, with the result +Ụ=1/ = 0t, +so that 1 0„: = 0, +Uụ = : =0wW1— u2/c2. (16.7) +Thus a sidewise velocity is no longer ⁄, but 0yv⁄/1— u2/c?. We found this +result by substituting and combining the transformation equations, but we can +--- Trang 311 --- +ưN 'Ộ IN +ZN ¡y4 NÊGHT +£ N lư h N +# X f 1 \ +: "u N # h Lư N +Fig. 16-1. Trajectorles described by a light ray and particle inside a +moving clock. +also see the result directly from the principle of relativity for the following reason +(it is always good to look again to see whether we can see the reason). We have +already (Pig. 15-3) seen how a possible clock might work when it is moving; the +light appears to travel a% an angle at the speed c in the fxed system, while 1t +simply goes vertically with the same speed in the moving system. We found that +the 0erfical componen¿ of the velocity in the ñxed system is less than that of +light by the facbor 4⁄1 — w2/c2 (see Bq. 15.3). But now suppose that we let a +material particle go back and forth in this same “clock,” but at some integral +fraction 1/ø of the speed of light (Eig. 16-1). Then when the particle has gone +back and forth once, the light will have gone exactly ? times. That is, each “click” +of the “particle” elock will coincide with each øœth “click” of the light clock. 7s +ƒact tmwust si be true tuhen the tuhole sụstem ¡s mmoving, because the physical +phenomenon of coinecidence will be a coincidenee in any frame. Therefore, since +the speed ey is less than the speed of light, the speed „ of the particle must be +slower than the corresponding speed by the same square-root ratiol That is why +the square root appears in any vertical velocity. +16-4 Relativistic mass +We learned in the last chapter that the mass of an object increases with +velocity, but no demonstration of this was given, in the sense that we made no +arguments analogous to those about the way clocks have to behave. However, +we cøn show that, as a consequence of relativity plus a few other reasonable +assumptions, the mass must vary in this way. (W© have to say “a few other +assumptions” because we cannot prove anything unless we have some laws which +--- Trang 312 --- +we assume to be true, if we expect to make meaningful deductions.) To avoid +the need to study the transformation laws of force, we shall analyze a collision, +where we need know nothing about the laws of force, except that we shall assume +the conservation of momentum and energy. Also, we shall assume that the +momentum of a particle which is moving is a vector and is always directed in the +direction oŸ the velocity. However, we shall not assume that the momentum is a +constønt tìmes the velocity, as Newton did, but only that it is some ƒwncfion of +velocity. We thus write the momentum vector as a certain coefficient times the +vector velocity: +Dp~Tn,„0®. (16.8) +We©e put a subscript ø on the coeficient to remind us that it is a function of +velocity, and we shall agree to call this coefficient m„ the “mass.” Of course, +when the velocity is small, it is the same mass that we would measure in the +slow-moving experiments that we are used to. Now we shall try to demonstrate +that the formula for rm„ must be rmo/4/1 — 02/2, by arguing from the principle +of relativity that the laws of physics must be the same in every coordinate system. +1 9/2 6/2 +¡ l 9/2 g/2 +1 1 (b) +Fig. 16-2. Two views of an elastic collision between equal obJects +moving at the same speed In opposite directions. +Suppose that we have two particles, like two protons, that are absolutely equal, +and they are moving toward each other with exactly equal velocities. Theïir total +mmomentum is zero. Now what can happen? After the collision, their directions +of motion must be exactly opposite to each other, because If they are not exactly +opposite, there will be a nonzero total vector momentum, and momentum would +not have been conserved. Also they must have the same speeds, since they are +exactly similar objects; in fact, they must have the same speed they started with, +since we suppose that the energy is conserved in these collisions. 5o the diagram +of an elastic collision, a reversible collision, will look like Fig. 16-2(a): all the +arrows are the same length, all the speeds are equal. We shall suppose that such +--- Trang 313 --- +collisions can always be arranged, that any angle Ø can occur, and that any speed +could be used in such a collision. Next, we notice that this same collision can be +viewecd diferently by turning the axes, and just for convenience we sÖø/l turn +the axes, so that the horizontal splits i evenly, as in Fig. 16-2(b). It is the same +collision redrawn, only with the axes turned. +„v2 : +œ œ x tị tị x +tu tu '94 œ +w 1⁄⁄v v51 +() 1ÍŸ1 (b) +Fig. 16-3. Two more views of the collision, from moving cars. +Now here is the real trick: let us look at this collision from the point of view of +someone riding along ín a car that is moving with a speed equal to the horizontal +component of the velocity of one particle. Then how does the collision look? +Tt looks as though particle 1 is just going straight up, because it has lost its +horizontal component, and it comes straight down again, also because i% does +not have that component. That is, the collision appears as shown in Fig. 16-3(a). +Particle 2, however, was going the other way, and as we ride pastf 1% appears tO +ñy by at some terrifc speed and at a smaller angle, but we can appreciate that +the angles before and after the collision are the sœme. Let us denote by u the +horizontal component of the velocity of particle 2, and by œ the vertical velocity +of particle 1. +Now the question is, what is the vertical velocity œtan œ? If we knew that, we +could get the correct expression for the momentum, using the law of conservation +of momentum in the vertical direction. Clearly, the horizontal component of the +1mmomentum is conserved: ¡% is the same before and after the collision for both +particles, and is zero for particle 1. So we need use the conservation law only +for the upward velocity wutanœ. But we cøn get the upward velocity, simply by +looking at the same collision going the other wayl If we look at the collision of +Eig. 16-3(a) from a car moving to the left with speed ứ, we see the same collision, +except “turned over,” as shown in Eig. 16-3(b). Ñow particle 2 is the one that goes +up and down with speed œ, and particle 1 has picked up the horizontal speed ứ. +--- Trang 314 --- +Of course, now we &noœ what the velocity wœtanœ is: iE is 04/1 — w2/c2 (see +Eq. 16.7). We know that the change in the vertical momentum of the vertically +moving particle is +Ap = 2m„+ +(2, because it moves up and back down). The obliquely moving particle has a +certain velocity ø whose components we have found to be w and +04/1 — u2/c2, +and whose mass is m„. The change in øerf2cøl momentum of this particle is +therefore AjÈ' = 2m„+0v/1— u2/c2 because, in accordance with our assumed +law (16.8), the momentum component is always the mass corresponding to the +magnitude of the velocity times the component of the velocity in the direction of +interest. Thus in order for the total momentum to be zero the vertical momenta, +must cancel and the ratio of the mass moving with speed ø and the mass moving +with speed + must therefore be +— = V1 u2/e2. (16.9) +°U +Let us take the limiting case that +0 is inÑnitesimal. lÝ u is very tiny indeed, it +1s clear that ø and œ are practically equal. In this case, m„„ —> nọ and rn„ —> my. +The grand result is +¬ .. (16.10) +“ v1—u?/c2 +]t is an interesting exercise now to check whether or not Eq. (16.9) is indeed true +for arbitrary values of +, assuming that Eq. (16.10) is the right formula for the +mass. Note that the velocity ø needed in Eq. (16.9) can be calculated rom the +right-angle triangle: +0U —= uẺ + 00Ẻ(1 — u2/c2). +Tlt will be found to check out automatically, although we used it only in the limit +Of smaill ơi. +Now, let us accept that momentum is conserved and that the mass depends +upon the velocity according to (16.10) and go on to find what else we can conclude. +Let us consider what is commonly called an ?melastic collision. For simplicity, we +shall suppose that two objects of the same kind, moving oppositely with equal +speeds +0, hit each other and stick together, to become some new, statlonary +object, as shown in Eig. 16-4(a). The mass ?n of each corresponds to +0, which, as +we know, is ?mo/4/1 — w2/c2. T we assume the conservation oŸ momentum and +--- Trang 315 --- +"5.1. ——: +(@) — M mạ *Ủ @ +Fig. 16-4. Two views of an Inelastic collision between equally massive +objects. +the principle of relativity, we can demonstrate an interesting fact about the mass +of the new object which has been formed. We imagine an infinitesimal velocity u +at right angles to œ0 (we can do the same with ñnite values of œ, but iE is easier to +understand with an infinitesimal velocity), then look at this same collision as we +ride by in an elevator at the velocity —w. What we see is shown in Fig. 16-4(b). +The composite object has an unknown mass M. Now object 1 moves with an +upward component of velocity œ and a horizontal component which is practically +cqual to +œø, and so also does object 2. After the collision we have the mass Ä⁄ƒ +moving upward with velocity œ, considered very small compared with the speed +of light, and also small compared with +. Momentum must be conserved, so let +us estimate the momentum ¡in the upward direction before and after the collision. +Before the collision we have p 2m„„u, and after the collision the momentum is +evidently ø' = M„u, but Ä⁄„ is essentially the same as Äíức because œ is so small. +These momenta must be equal because of the conservation of momentum, and +therefore +Mẹ = 2m. (16.11) +The rnass oƒ the object thích ¡s ƒormecd hen tuo equal objects collide rmmust be +tuñce the mmass 0ƒ the objects tuhích come together. You might say, “Yes, oŸ course, +that is the conservation of mass.” But not “Yes, of course,” so easily, because +these mmasses hœue been enhanced over the masses that they would be if they were +standing still, yet they still contribute, to the total Ä, not the mass they have +when standing still, but more. Astonishing as that may seem, in order for the +conservation of momentum to work when two objects come together, the mass +that they form must be greater than the rest masses of the objects, even though +the objects are at rest after the collision! +16-5 Relativistic energy +In the last chapter we demonstrated that as a result of the dependence of +the mass on velocity and Newton's laws, the changes in the kinetic energy oŸ an +--- Trang 316 --- +object resulting from the total work done by the forces on it always comes out to +AT' = (mụ — mọ)c? = ——9_- mạọc?. (16.12) +V1— u2/e° +We even went further, and guessed that the total energy is the total mass tỉmes cẺ. +Now we continue this discussion. +Suppose that our bwo equally massive objects that collide can still be “seen” +inside Mĩ. Eor instance, a proton and a neutron are “stuck together,” but are still +moving about inside of Mƒ. Then, although we might at fñrst expect the mass MỸ to +be 2m, we have found that it is not 2mọ, but 2?n¿„. Since 2?n„„ is what is put ïn, +but 2mọ are the rest masses of the things inside, the ezcess mass of the composite +obJject is equal to the kinetic energy broughtin. This means, of course, that energu +has zmertia. In the last chapter we discussed the heating of a gas, and showed that +because the gas molecules are moving and moving things are heavier, when we put +energy into the gas its molecules move faster and so the gas gets heavier. But in +fact the argument is completely general, and our discussion of the inelastie collision +shows that the mass is there whether or not i is knetc energy. In other words, if +two particles come together and produce potential or any other form of energy; if +the pieces are slowed down by climbing hills, doing work against internal Íorces, or +whatever; then it is still true that the mass is the total energy that has been put in. +So we see that the conservation of mass which we have deduced above is equivalent +to the conservation of energy, and therefore there is no place in the theory of +relativity for strictly inelastic collisions, as there was in Newtonian mechanics. +According to Newtonian mechaniœs it is all right for two things to collide and so +form an object of mass 2mọ which is in no way distinct from the one that would +result from putting them together slowly. Of course we know from the law of +conservation of energy that there is more kinetic energy ¡nside, but that does not +affect the mass, according to Newton”s laws. But now we see that this is impossible; +because of the kinetic energy involved in the collision, the resulting object will +be heauier; therefore, it will be a đjƒerent object. When we put the objects +together gently they make something whose mass is 2m; when we put them +together forcefully, they make something whose mass is greater. When the mass is +diferent, we can #ell that it is diferent. 5o, necessarily, the conservation of energy +must go along with the conservation of momentum in the theory of relativity. +This has interesting consequences. For example, suppose that we have an +object whose mass Ä⁄ƒ is measured, and suppose something happens so that it fies +--- Trang 317 --- +into two equal pieces moving with speed +, so that they each have a mass Tn¿„. +Now suppose that these pieces encounter enough material to slow them up until +they stop; then they will have mass mọ. How much energy will they have given +to the material when they have stopped? Each will give an amount (m„„ — mọ)€Ẻ, +by the theorem that we proved before. 'Phis much energy is left in the material +in some form, as heat, potential energy, or whatever. Now 2m¿„ —= M, so the +liberated energy is # = (ÁMf — 2mo)c?. This equation was used to estimate how +much energy would be liberated under fssion in the atomic bomb, for example. +(Although the ragments are not exactly equal, they are nearly equal.) The mass +of the uranium atom was known——it had been measured ahead of time—and the +atoms into which ït split, iodine, xenon, and so on, all were of known mass. By +masses, we do not mean the masses while the atoms are moving, we mean the +mmasses when the atoms are ø res. In other words, both MỸ and rmọ are known. +So by subtracting the two numbers one can calculate how much energy will be +released 1f Mƒ can be made to split in “half” Eor this reason poor old Einstein +was called the “father” of the atomie bomb in all the newspapers. Of course, all +that meant was that he could tell us ahead of time how much energy would be +released if we told him what process would occur. The energy that should be +liberated when an atom of uranium undergoes fission was estimated about six +months before the frst direct test, and as soon as the energy was in fact liberated, +Someone measured it directly (and if Einsteins formula had not worked, they +would have measured it anyway), and the moment they measured it they no +longer needed the formula. Of course, we should not belittle Einstein, but rather +should criticize the newspapers and many popular descriptions of what causes +what in the history of physics and technology. The problem of how to get the +thing to occur in an efective and rapid manner is a completely diÑferent matter. +The result is just as significant in chemistry. Eor instance, if we were to weigh +the carbon dioxide molecule and compare its mass with that of the carbon and +the oxygen, we could ñnd out how much energy would be liberated when carbon +and oxygen form carbon dioxide. The only trouble here is that the diferences In +mmasses are so small that it is technically very difcult to do. +Now let us turn to the question of whether we should add mạc? to the kinetic +energy and say from now on that the total energy of an objeect is me2. First, iŸ we +can still see the component pieces of rest mass ?mọ inside M, then we could say +that some of the mass Ä⁄ of the compound object is the mechanical rest mass of +the parts, part of it is kinetic energy of the parts, and part of it is potential energy +of the parts. But we have discovered, in nature, particles of various kinds which +--- Trang 318 --- +undergo reactions just like the one we have treated above, in which with all the +study ín the world, we cannot sec the parts ứnside. For instance, when a K-meson +disintegrates into two pions it does so according to the law (16.11), but the idea +that a K is made out of 2 s is a useless idea, because it also disintegrates into +'Therefore we have a neu idea: we do not have to know what things are made +of inside; we cannot and need not identify, inside a particle, which of the energy 1s +rest energy of the parts into which it is goïng to disintegrate. It is not convenient +and often not possible to separate the total me? energy of an object into rest +energy of the inside pieces, kinetic energy of the pieces, and potential energy of +the pieces; instead, we simply speak of the £oføœÏ energu of the particle. We “shift +the origin” of energy by adding a constant mọc? to everything, and say that the +total energy of a particle is the mass in motion times c2, and when the object is +standing still, the energy is the mass at rest times cẺ. +Finally, we fnd that the velocity 0, momentum ?, and total energy # are +related in a rather simple way. That the mass in motion at speed 0 is the mass mo +at rese divided by 4⁄1 — 02/c?, surprisingly enough, is rarely used. Instead, the +following relations are easily proved, and turn out to be very useful: +E2 — P?c? = mặc" (16.13) +Pc= EuÍc. (16.14) +--- Trang 319 --- +Spereco- Time© +17-1 The geometry of space-time +The theory of relativity shows us that the relationships of positions and +times as measured in one coordinate system and another are not what we would +have expected on the basis of our intuitive ideas. It is very important that we +thoroughly understand the relations oŸ space and time implied by the Lorentz +transformation, and therefore we shall consider this matter more deeply in this +chapter. +The Lorentz transformation between the positions and tỉimes (#,, z,È) as +measured by an observer “standing still,” and the corresponding coordinates and +tìme (4, /, z, ) measured inside a “moving” space ship, moving with velocity u +; % — tu +# =———p, +v1—*2/c2 +Ụ =U, +17.1 +Xa (11) +TH... a +v1—*2/c2 +Let us compare these equations with Eq. (11.5), which also relates measurements +in two systems, one of which in this instance is ro/a£#ed relative to the other: ++ = #øcos 0 + 1 sin 0, += cosØ — #sin 0, (17.2) +z' =z. +In this particular case, Moe and Jjoe are measuring with axes having an angle Ø +between the z- and z-axes. In each case, we note that the “primed” quantities +--- Trang 320 --- +are “mixtures” of the “unprimed” ones: the new # is a mixture oŸ # and , and +the new # is also a mixture oŸ z and . +An analogy is useful: When we look at an object, there is an obvious thing we +might call the “apparent width,” and another we might call the “depth” But the +two ideas, width and depth, are not ƒundœmenfal properties of the object, because +1ƒ we step aside and look at the same thing from a different angle, we get a different +width and a diferent depth, and we may develop some formulas for computing the +new ones rom the old ones and the angles involved. Equations (17.2) are these +formulas. One might say that a given depth is a kind of “mixture” of all depth +and all width. If it were impossible ever to move, and we always saw a given +object from the same position, then this whole business would be irrelevant——we +would always see the “true” width and the “true” depth, and they would appear +to have quite diferent qualities, because one appears as a subtended optical angle +and the other involves some focusing of the eyes or even intuition; they would +seem to be very different things and would never get mixed up. lt is because we +can walk around that we realize that depth and width are, somehow or other, +Jjust two different aspects of the same thiỉng. +Can tue no‡ look at the Loren‡z transƒformations ín the sơme tua? Here aÌso +we have a mixture—of positions and the time. A diference between a space +mmneasurement and a time measurement produces a new space measurement. Ïn +other words, in the space measurements of one man there is mixed in a little bit +of the time, as seen by the other. Our analogy permits us to generate this idea: +The “reality” of an object that we are looking at is somehow greater (speaking +crudely and intuitively) than its “width” and its “depth” because £#e depend +upon ho we look at it; when we move to a new position, our brain immediately +recalculates the width and the depth. But our brain does not immediately +recaleulate coordinates and time when we move at high speed, because we have +had no efective experience of going nearly as fast as light to appreciate the +fact that time and space are also of the same nature. It is as though we were +always stuck in the position oŸ having to look at just the width of something, +not beïng able to move our heads appreciably one way or the other; if we could, +we understand now, we would see some of the other man ”s tine—we would see +“behind,” so to speak, a little bít. +Thus we shall try to think of objects in a new kind of world, of space and time +mixed together, in the same sense that the objects in our ordinary space-world are +real, and can be looked at from different directions. We shall then consider that +obJects occupying space and lasting for a certain length of time occupy a kind of +--- Trang 321 --- +(a)| /Œ) +X0 x +Fig. 17-1. Three particle paths in space-time: (a) a particle at rest +at x = xo; (b) a particle which starts at x = xo and moves with constant +speed; (c) a particle which starts at high speed but slows down; (d) a +light path. +a “blob” in a new kind of world, and that we look at this “blob” from difÑferent +points of view when we are moving at diferent velocities. Thhis new world, this +geometrical entity in which the “blobs” exist by occupying position and taking +up a certain amount of time, is called space-tme. A given point (z,,z,É) in +space-time is called an cuenứ. Imagine, for example, that we plot the #-positions +horizontally, and z in two other directions, both mutually at “right angles” and +at “ripht angles” to the paper (P), and time, vertically. Now, how does a moving +particle, say, look on such a diagram? If the particle is standing stiH, then it has +a certain ø, and as time goes on, it has the same ø, the same #, the same 4; sO +its “path” is a line that runs parallel to the f-axis (Eig. 17-1 a). On the other +hang, if it drifts outward, then as the time goes on z# increases (Eig. 17-1 b). +So a particle, for example, which starts to drift out and then slows up should +have a motion something like that shown in Fig. 17-I(c). A particle, in other +words, which is permanent and does not disintegrate is represented by a line in +space-time. Á particle which disintegrates would be represented by a forked line, +because it would turn into 6wo other things which would start from that poïnt. +'What about light? Light travels at the speed e, and that would be represented +by a line having a certain ñxed slope (Eig. 17-1 d). +Now according to our new idea, ïÝ a given event occurs to a particle, say IÝ +it suddenly disintegrates at a certain space-time point into two new ones which +follow some new tracks, and this interesting event occurred at a certain value +of z and a certain value of £, then we would expect that, if this makes any sense, +we just have to take a new pair of axes and turn them, and that will give us the +new ý and the new # in our new system, as shown in Fig. 17-2(a). But this is +wrong, because Eq. (17.1) is not ezacflu the same mathematical transformation +--- Trang 322 --- +cíí ct ⁄“ +\Y. “ +Xx x Xx +(a)NOT CORRECT (b) CORRECT +Fig. 17-2. Two views of a disintegrating particle. +as Bq. (17.2). Note, for example, the difference in sign between the two, and the +fact that one is written in terms of cos Ø and sin Ø, while the other is written with +algebraic quantities. (Of course, iÈ is not impossible that the algebraic quantities +could be written as cosine and sine, but actually they cannot.) But still, the two +expressions øre very similar. As we shall see, i% is not really possible bo think +Of space-time as a real, ordinary geometry because of that diference in sign. In +fact, although we shall not emphasize this point, it turns out that a man who is +moving has to use a set of axes which are inclined equally to the light ray, using +a special kind of projection parallel to the z/- and f/-axes, for his ø“ and #, as +shown in EFig. I7-2(b). We shall not deal with the geometry, since it does not +help much; it is easier to work with the equations. +17-2 Space-time intervals +Although the geometry of space-time is not Buclidean in the ordinary sense, +there 7s a geometry which is very similar, but peculiar in certain respects. If +this idea of geometry is right, there ought to be some functions of coordinates +and time which are independent of the coordinate system. Eor example, under +ordinary rotations, if we take two points, one at the origin, for simplicity, and +the other one somewhere else, both systems would have the same origin, and +the distance from here to the other point is the same in both. That is one +property that is independent of the particular way of measuring it. The square +of the distanece is #2 + 2 + z”. Now what about space-time? It is not hard to +demonstrate that we have here, also, something which stays the same, namely, the +combination e?£2 — #2 — 92 — z is the same before and after the transformation: +c2t2 — „2 — 2 — z2 = c3? — g2 — g2 — z3, (17-3) +This quantity is therefore something which, like the distance, is “real” in some +sense; it is called the 7m‡eruøl between the two space-time points, one of which is, +--- Trang 323 --- +in this case, at the origin. (Actually, oŸ course, it is the interval squared, just +as #2 -Ƒ 2 + z2 is the distance squared.) We give it a diferent name because it +1s In a diferent geometry, but the interesting thing is only that some signs are +reversed and there is a c in it. +Let us get rid of the œ; that is an absurdity iŸ we are goïng to have a wonderful +space with zˆ's and #'s that can be interchanged. One of the confusions that could +be caused by someone with no experience would be to measure widths, say, by the +angle subtended at the eye, and measure depth in a difÑferent way, like the strain +on the muscles needed to focus them, so that the depths would be measured in +feet and the widths in meters. Then one would get an enormously complicated +mness oŸ equations in making transformations such as (17.2), and would not be +able to see the clarity and simplicity of the thing for a very simple technical +reason, that the same thing is being measured in two diferent units. Now in Eqs. +(17.1) and (17.3) nature is telling us that time and space are equivalent; tỉme +becomes space; (he should be mmeasured ?ín the same uniis. What distance 1s a +“second”? It is easy to fgure out from (17.3) what it is. It is 3 x 10Ÿ meters, fhe +địstance that light tUould go ?m one second. In other words, iŸ we were to measure +all distances and times in the same units, seconds, then our unit of distance +would be 3 x 10 meters, and the equations would be simpler. Or another way +that we could make the units equal is to measure time in meters. What is a +meter of time? AÁ meter of tỉme is the time it takes for light to go one meter, +and is therefore 1/3 x 107 see, or 3.3 billionths of a second! We would like, in +other words, to put all our equations in a system of units in which c= 1. H time +and space are measured in the same units, as suggested, then the equations are +obviously much simplified. They are +AM... +g0 (17⁄4) +Z =Z, +trằ t— Uuz +t2 —ạt2 — 2 — y2 =12— g2 — 2 — z2, (17.5) +TÍ we are ever unsure or “frightened” that after we have this system with c= l +we shall never be able to get our equations right again, the answer is quite the +--- Trang 324 --- +opposite. Ït ¡is much easier to remember them without the c's in them, and ï£ 1s +always easy to put the đs back, by looking after the dimensions. For instance, +in V1 — 2, we know that we cannot subtract a velocity squared, which has units, +from the pure number 1, so we know that we must đivide u2 by e? in order to +make that unitless, and that is the way it goes. +'The diference between space-time and ordinary space, and the character of an +interval as related to the distance, is very interesting. According to formula (17.5), +1ƒ we consider a point which in a given coordinate system had zero time, and +only space, then the interval squared would be negative and we would have an +imaginary interval, the square root of a negative number. Intervals can be either +real or imaginary in the theory. The square of an interval may be either positive +or negafive, unlike distance, which has a positive square. When an interval is +imaginary, we say that the two points have a space-like ¿nterual between them +(instead of imaginary), because the inberval is more like space than like time. Ôn +the other hang, if two objects are at the same place in a given coordinate system, +but difer only in time, then the square of the time is positive and the distances +are zero and the interval squared is positive; this is called a fữne-like ¿mterual. In +our diagram of space-time, therefore, we would have a representation something +like this: at 452 there are ©wo lines (actually, in four dimensions these will be +“cones,” called light cones and points on these lines are all at zero interval from +the origin. Where light goes from a given point is always separated from it by a +zero interval, as we see rom Eq. (17.5). Incidentally, we have just proved that +1f light travels with speed c in one system, i% travels with speed c in another, +for 1ƒ the Interval is the same in both systems, i.e., zero in one and zero in the +other, then to state that the propagation speed of light is Invariant is the same +as saying that the interval is zero. +17-3 Past, present, and future +'The space-time region surrounding a given space-time point can be separated +into three regions, as shown in EFig. 17-3. Ín one region we have space-like +Intervals, and in two regions, time-like intervals. Physically, these three regions +into which space-time around a given poïnt is divided have an interesting physical +relationship to that point: a physical object or a signal can get om a point in +region 2 to the event @ by moving along at a speed less than the speed of light. +Therefore events in this region can afect the point Ó, can have an inÑuence +on i% from the past. In fact, of course, an object at ? on the negative f-axis 1s +--- Trang 325 --- +Tàu: LIGHT-CONE +b % R©) LIGHT-CONE +Fig. 17-3. The space-time region surrounding a point at the origin. +precisely in the “past” with respect to ; it is the same space-point as Ó, only +carlier. What happened there then, affects Ó now. (Unfortunately, that is the +way life is.) Another object at Q can get to Ó by moving with a certain speed +less than e, so if this object were in a space ship and moving, it would be, again, +the past of the same space-point. 'That is, in another coordinate system, the axis +of time might go through both @Ø and Q. So all points of region 2 are in the +“past” of Ó, and anything that happens in this region cøn afect Ó. 'Therefore +region 2 is sometimes called the øƒfectiue past, or affecting past; i is the locus of +all events which can afect point Ó in any way. +Region 3, on the other hand, is a region which we can affect from O, we can +“hit” things by shooting “bullets” out at speeds less than c. So this is the world +whose future can be afected by us, and we may call that the a[ƒectiue ƒuture. +Now the interesting thing about all the rest of space-time, I.e., region 1, ¡is that we +can neither affect it now from O, nor can it affect us now ø‡ Ó, because nothing +can go faster than the speed of light. Of course, what happens at Ï can affect us +later; that 1s, 1f the sun is exploding “right now,” it takes eight minutes before +we know about it, and it cannot possibly affect us before then. +What we mean by “right now” is a mysterious thing which we cannot deñne +and we cannot afect, but it can affect us later, or we could have afected it If +we had done something far enough in the past. When we look at the star Alpha +Centauri, we see ib as it was Íour years ago; we might wonder what ït is like +“now.” “NÑow” means at the same time from our special coordinate system. We +can only see Alpha Centauri by the light that has come from our past, up to four +years ago, but we do not know what i§ is doïng “now”; it will take Íour years +before what it is doing “now” can affect us. Alpha Centauri “now” is an idea +or concept of our mỉnd; it is not something that is really deñnable physically +at the moment, because we have to wait to observe it; we cannot even defne 1t +ripght “now.” Purthermore, the “now” depends on the coordinate system. ÏTÝ, for +--- Trang 326 --- +example, Alpha Centauri were moving, an observer there would not agree with +us because he would put his axes at an angle, and his “now” would be a đjƒerent +time. We have already talked about the fact that simultaneity is not a unique +thing. +'There are fortune tellers, or people who tell us they can know the future, and +there are many wonderful stories about the man who suddenly discovers that +he has knowledge about the afective future. Well, there are lots of paradoxes +produced by that because if we know something is going to happen, then we can +mmake sure we will avoid it by doing the right thing at the right time, and so on. +But actually there is no fortune teller who can even tell us the øresen#l 'There +is no one who can tell us what is really happening right now, at any reasonable +distance, because that is unobservable. We might ask ourselves this question, +which we leave to the student to try to answer: Would any paradox be produced +1f it were suddenly to become possible to know things that are in the space-like +intervals of region 17 +17-4 More about four-vectors +Let us now return to our consideration of the analogy of the Lorentz transfor- +mation and rotations of the space axes. We have learned the utility of collecting +together other quantities which have the same transformation properties as the +coordinates, to form what we call øec#ors, directed lines. In the case of ordinary +rotations, there are many quantities that transform the same way as z, , and z +under rotation: for example, the velocity has three components, an ø, , and z- +component; when seen in a diferent coordinate system, none of the components +1s the same, instead they are all transformed to new values. But, somehow or +other, the velocity “itself” has a greater reality than do any of its particular +components, and we represent it by a directed line. +We therefore ask: Is it or is it not true that there are quantities which +transform, or which are related, in a moving system and in a nonmoving system, +in the same way as ø, , z, and #? From our experience with vectors, we know that +three of the quantities, like ø, , z, would constitute the three components of an +ordinary space-vector, but the fourth quantity would look like an ordinary scalar +under space rotation, because it does not change so long as we do not go into +a moving coordinate system. Is it possible, then, to associate with some of our +known “three-vectors” a fourth object, that we could call the “time component,” +in such a manner that the four obJects together would “rotate” the same wawy +--- Trang 327 --- +as position and time in space-time? We shall now show that there is, indeed, at +least one such thing (there are many of them, in fact): the three componenfs oƒ +momentum, ơnd the cnergụ œs the từne component, transform together to make +what we call a “four-vector.” In demonstrating this, since it is quite inconvenient +to have to write cs everywhere, we shall use the same trick concerning units of +the energy, the mass, and the momentum, that we used in Eq. (17.4). Energy +and mass, for example, difer only by a factor c2 which is merely a question of +units, so we can say energy is the mass. Instead of having to write the c2, we +put # = mn, and then, of course, 1Ÿ there were any trouble we would put in the +right amounts of e so that the units would straighten out in the last equation, +but not in the intermediate ones. +Thus our equations for energy and momentum are +=m =mo/Wl1— 02, (176) +Ð =0 =1mo0/V1— 02. +Also in these units, we have +E3 — pˆ = mạ. (17.7) +For example, 1Ÿ we measure energy in electron volts, what does a mass of 1 electron +volt mean? It means the mass whose rest energy is 1 electron volt, that is, mọc? +is one electron volt. For example, the rest mass of an electron is 0.511 x 108 eV. +Now what would the momentum and energy look like in a new coordinate +system? To find out, we shall have to transform 4q. (17.6), which we can do +because we know how the velocity transforms. Suppose that, as we measure it, +an object has a velocity 0, bu we look upon the same object rom the point of +view Of a space ship which itself is moving with a velocity u, and in that system +we use a prime to designate the corresponding thing. In order to simplify things +at first, we shall take the case that the velocity ø is in the direction of u. (Later, +we can do the more general case.) What is 0”, the velocity as seen from the space +ship? It is the composite velocity, the “diference�� between 0 and u. By the law +which we worked out before, +g= —, (17.8) +1—0U0 +Now let us calculate the new energy F”, the energy as the fellow in the space +ship would see it. He would use the same rest mass, of course, but he would +--- Trang 328 --- +use ? for the velocity. What we have to do is square œ, subtract 1% from one, +take the square root, and take the reciprocal: +g2 — 02 — 2u + uŸ +— 1—9uu+u2u2) +— L— 20 +u202— 02+ 2u — u2 +1 — t) = — +1— 2u + u2u2 Í +— l—02—u2+u2u? +—— I—2u+u2u2 ` +_— (—?)(1—u?) +— (1-uo} ` +'Therefore +1 c— 1— 0 (17.9) +V1i—2 v1—u2V1—w2. +The energy #7 is then simply rmọ times the above expression. But we want +to express the energy in terms of the unprimed energy and momentum, and we +note that +Pmm..... (mo/V1— 02) — (mou/V1— 0?)u +V1—02vli—u2 V1—u2 l +E=——, 17.10 +TC (17.10) +which we recognize as being exactly of the same form as +; ‡— uz +Ÿ =———m. +Next we must fñnd the new mmomentum 7Ø. This is just the energy #7 times 0, +and is also simply expressed in terms of / and ø: +b— pyy rmo(1 — ưo) 0u— TU — Tnow += %2 — — — ———— - —————— ———— +P„ VI-—ø»2V1-u2 (1—-u°) VTI—w2V1—u2 +/ Đ„ — tUE += ——ễ,Ụ 17.11 +--- Trang 329 --- +which we recognize as being of precisely the same form as +; % — UuÈ +# =———n. +vV1—-u2 +Thus the transformations for the new energy and momentum in terms of the +old energy and momentum are exactly the same as the transformations for # in +terms oŸ £ and z, and zø“ in terms of #z and ứ: all we have to do is, every tỉme we +see £ in (17.4) substitute #, and every time we see # substitute ø„, and then the +cquations (17.4) will become the same as Eqs. (17.10) and (17.11). This would +imply, iƒ everything works right, an additional rule that ø, = ø„ and that ø = Ø¿. +To prove this would require our goïng back and studying the case of motion up +and down. Actually, we did study the case of motion up and down in the last +chapter. We analyzed a complicated collision and we noticed that, in fact, the +transverse momentum is øø‡ changed when viewed from a moving system; so we +have already verified that „ = ø„ and 7, = ø;. The complete transformation, +then, 1s +g — Đ„ — tu +% 1 — „2 hà +Ủy = Đụ; +gi (17.12) +Đy — Dz:; +E/ — b -~ uDx +v1—-u2 +In these transformations, therefore, we have discovered four quantities which +transform like z, , z, and , and which we call the ƒour-uector mmomentum. Since +the momentum is a four-vector, it can be represented on a space-time diapgram +of a moving particle as an “arrow” tangent to the path, as shown in Fig. 17-4. +'This arrow has a time component equal to the energy, and its space components +Fig. 17-4. The four-vector momentum of a particle. +--- Trang 330 --- +represent its three-vector momentum; this arrow is more “real” than either the +energy or the momentum, because those just depend on how we look at the +diagram. +17-5 Four-vector algebra +The notation for four-vectors ¡is diferent than it is for three-vectors. In +the case of three-vectors, if we were to talk about the ordinary three-vector +mmomentum we would write it ø. If we wanted to be more specifc, we could say +it has three components which are, for the axes in question, ø„, ø„, and Øø;, Or +we could simply refer to a general component as ø;, and say that ¿ could either +be z, , or z, and that these are the three components; that is, imagine that ¿ is +any one of three directions, ø, , or z. The notation that we use for four-vecftors +is analogous to this: we write ø„ for the four-vector, and / stands for the ƒour +possible directions Ý, ø, , Or Z. +W© could, of course, use any notation we want; do not laugh at notations; +Invent them, they are powerful. In fact, mathematics is, to a large extent, +Invention of better notations. “The whole idea of a four-vector, In fact, is an +improvement in notation so that the transformations can be remembered easily. +A„, then, is a general four-vector, but for the special case oŸ momentum, the ?¿ +is identified as the energy, ø„ is the momentum in the z-direction, øy is that in +the -direction, and ø; is that in the z-direction. To add four-vectors, we add +the corresponding componenfs. +TỶ there is an equation among four-vectors, then the equation is true for cœch +cơmponen‡. Eor instance, 1f the law of conservation of three-vector momentum is +to be true in particle collisions, i.e., if the sum of the momenta for a large number +of interacting or colliding particles is to be a constant, that must mean that the +sums of all momenta in the z-direction, in the -direction, and in the z-direction, +for all the particles, must each be constant. 'This law alone would be impossible +in relativity because it 1s Zncomjplete; it is like talking about only bwo of the +components of a three-vector. lt is incomplete because iŸ we rotate the axes, we +mix the various componentfs, so we must include all three components in our law. +Thus, in relativity, we must complete the law of conservation of momentum by +extending it to include the #ữne component. This is øbsolutel necessar to go +with the other three, or there cannot be relativistic invariance. 'Phe conseruation +öƒ energụ 1s the fourth equation which goes with the conservation of momentum +--- Trang 331 --- +to make a valid four-vector relationship in the geometry of space and time. Thus +the law of conservation of energy and momentum in four-dimensional notation is +» Pụ — » Đụ (17.13) +particles particles +in out +or, in a slightly diferent notation +À pụ, = À ` Địu, (17.14) +where ? = l, 2,... refers to the particles goïng into the collision, 7 = 1, 2,... +refers to the particles coming out of the collision, and / = ø, , z, or ứ. You say, +“In which axes?” It makes no diference. The law is true for each component, +USINE đn axes. +In vector analysis we discussed one other thing, the dot produect oŸ bwo vecbors. +Let us now consider the corresponding thing in space-time. In ordinary rotation +we discovered there was an unchanged quantity #2 + #2 + z2. In four dimensions, +we find that the corresponding quantity is £ — #7 — 2 — z2 (Eq. 17.3). How can +we write that? One way would be to write some kind of four-dimensional thing +with a square dot between, like A„ L-] „; one of the notations which is actually +used is +3) A,A,=A?— A?— A?— A2. (17.15) +The prime on 3` means that the first term, the “time” term, is positive, but the +other three terms have minus signs. This quantity, then, will be the same in any +coordinate system, and we may call it the square of the length of the four-vector. +For instance, what is the square of the length of the four-vector momentum of +a single particle? This will be equal to gøý — Ø2 — Ø2 — Ø2 or, in other words, +E2 — p2, because we know that p¿ is . What is #2 — p?? It must be something +which is the same in every coordinate system. In particular, it must be the same +for a coordinate system which is moving right along with the particle, in which +the particle is standing still. If the particle is standing still, it would have no +mmomentum. So in that coordinate system, iÈ is purely i%s energy, which is the +same as its rest mass. Thus #3 — p° = m. So we see that the square of the +length of this vector, the four-vector momentum, is equal to mã. +--- Trang 332 --- +From the square of a vector, we can go on to invent the “dot product,” or the +product which is a scalar: iŸ ø„ is one four-vector and b„ is another four-vector, +then the scalar product is +» dubu = đi — ayÙy — quby — g„by. (17.16) +Tt is the same in all coordinate systems. +Finally, we shall mention certain things whose rest mass mọ is zero. A photon +of light, for example. A photon is like a particle, in that ib carries an energy +and a momentum. The energy of a photon is a certain constant, called Planeck”s +constant, times the frequency of the photon: # = hz. Such a photon also carries +a momentum, and the momentum of a photon (or of any other particle, in fact) +is 5 divided by the wavelength: p = h/^A. But, for a photon, there is a delnite +relationship bebween the frequency and the wavelength: = c/A. (The number +of waves per second, times the wavelength of each, is the distance that the light +goes in one second, which, oŸ course, is c.) Thus we see immmediately that the +energy of a photon must be the momentum times c, or 1Í c=— 1, the energụ œnd +momentưm are cqual. That 1s to say, the rest mass 1s zero. Let us look at that +again; that is quite curious. lf it is a particle of zero rest mass, what happens +when it stops? Ï£ neuer stopsf Tt always goes at the speed c. The usual formula +for energy is mo/V 1 — 02. Ñow can we say that rmọ = 0Ö and 0 = 1, so the energy +is 0? We cannof say that it is zero; the photon really can (and does) have energy +even though it has no rest mass, but this it possesses by perpetually going at the +speed of lightl +W© also know that the momentum of any particle is equal to its total energy +tỉmes is velocity: iŸ e = 1, p = 0# or, in ordinary units, p = 0/c?. Eor any +particle moving at the speed of light, p = ifc = 1. The formulas for the energy +of a photon as seen from a moving system are, of course, given by Eq. (17.12), +but for the momentum we must substitute the energy times é (or times 1 in this +case). The different energies after transformation means that there are diferent +frequencies. 'Phis is called the Doppler eÑfect, and one can calculate it easily from +Eq. (17.12), using also = p and = hứ. +As Minkowski said, “Space of itself, and time of itself will sink into mere +shadows, and only a kind of union between them shall survive.” +--- Trang 333 --- +}Ổo((tffGrt tro tro ŸÌf110©OrtSf@reS +18-1 The center of mass +In the previous chapters we have been studying the mechanics oŸ points, or +small particles whose internal structure does not concern us. For the next few +chapters we shall study the application of NÑewton's laws to more complicated +things. When the world becomes more complicated, it also becomes more +interesting, and we shall ñnd that the phenomena associated with the mechanics +of a more complex object than just a point are really quite striking. Of course +these phenomena involve nothing but combinations of NÑewton”s laws, but it is +sometimes hard to believe that only #' = ma is at work. +The more complicated objects we deal with can be of several kinds: water +fowing, galaxies whirling, and so on. “The simplest “eomplicated” object to +analyze, at the start, is what we call a r/g/đ body, a solid object that is turning +as it moves about. However, even such a simple object may have a most complex +motion, and we shall therefore first consider the simplest aspects of such motion, +in which an extended body rotates about a fized azs. A given point on such a +body then moves in a plane perpendicular to this axis. Such rotation of a body +about a fñxed axis is called pÏiøne rotatfion or rotation in two dimensions. We +shall later generalize the results to three dimensions, but in doïng so we shall +fnd that, unlike the case of ordinary particle mechanics, rotations are subtle and +hard to understand unless we first get a solid grounding in two dimensions. +The first interesting theorem concerning the motion of complicated objects +can be observed at work if we throw an object made of a lot of blocks and spokes, +held together by strings, into the aïr. Of course we know it goes in a parabola, +because we studied that for a particle. But now our object is no‡ a particle; 1% +wobbles and it jiggles, and so on. I$ does go in a parabola though; one can see +that. Wha# goes in a parabola? Certainly not the point on the corner of the +block, because that is jiggling about; neither is i9 the end of the wooden stick, +--- Trang 334 --- +or the middle of the wooden stick, or the middle of the block. But something +goes In a parabola, there is an efective “center” which moves In a parabola. So +our first theorem about complicated objects is to demonstrate that there 7s a +mean position which is mathematically defñnable, but not necessarily a point of +the material itself, which goes in a parabola. That ¡is called the theorem of the +center of the mass, and the proof of it is as follows. +W©e may consider any object as beïing made of lots of little particles, the atoms, +with various forces among them. Let ¿ represent an index which defines one of +the particles. (There are millions of them, so ¿ goes to 102, or something.) Then +the force on the ?th particle is, of course, the mass times the acceleration of that +particle: +E¡ = m;(dŠr;/d12). (18.1) +In the next few chapters our moving objects will be ones in which all the +parts are moving at speeds very much slower than the speed of light, and we shall +use the nonrelativistic approximation for all quantities. In these circumstances +the mass is constant, so that +E¡ = dẦ(m¿r;)/dfẺ. (18.2) +T we now add the force on all the particles, that is, IÝ we take the sum of all +the #;'s for all the diferent indexes, we get the total force, #'. Ôn the other +side of the equation, we get the same thing as though we added before the +diferentiation: Z(S ) +¡ Th¿T; +» E;=P=—=“— (18.3) +Therefore the total force is the second derivative of the masses times their +positions, added together. +Now the total force on all the particles is the same as the ezternal force. Why? +Although there are all kinds of forces on the particles because of the strings, the +wigplings, the pullings and pushings, and the atomie forces, and who knows what, +and we have to add all these together, we are rescued by Newton's Third Law. +Between any two particles the action and reaction are equal, so that when we +add all the equations together, if any two particles have forces between them I1 +caneels out in the sum; therefore the net result is only those forces which arise +from other particles which are not included in whatever object we decide to sun +over. So if Eq. (18.3) is the sum over a certain number of the particles, which +--- Trang 335 --- +together are called “the object,” then the ez#ernal force on the total object 1s +equal to the sum of ai the forces on all its constituent particles. +NÑow it would be nice if we could write Bq. (18.3) as the total mass times +some acceleration. We can. Let us say /Mƒ is the sum of all the masses, i.e., the +total mass. Then if we deffne a certain vector i? to be +R=À mr//(M, (18.4) +Eq. (18.3) will be simply +FP= d (MR)/dt? = M(d°R/dt2), (18.5) +since # is a constant. Thus we fñnd that the external force is the tota]l mass +times the acceleration of an imaginary point whose location is l. This poïnt is +called the cen#er oƒ mmass of the body. It is a point somewhere in the “middle7 +of the object, a kind of average r in which the diferent ?;'s have weights or +Importances proportional to the masses. +W© shall discuss this important theorem in more detail in a later chapter, and +we shall therefore limit our remarks to two points: First, if the external Íorces are +zero, if the object were floating in empty space, it might whirl, and jiggle, and +twist, and do all kinds of things. But the center oƒ mass, thìs artiflcially invented, +calculated position, somewhere in the middle, +uiÏl moue u#th a constant 0elocitg. +In particular, if it is initially at rest, ít will stay at rest. So If we have some kind +of a box, perhaps a space ship, with people ín it, and we calculate the location +of the center of mass and fñnd it is standing still, then the center of mass will +continue to stand still if no external forces are acting on the box. Of course, the +space ship may move a little in space, but that is because the people are walking +back and forth inside; when one walks toward the front, the ship goes toward +the back so as to keep the average position of all the masses in exactly the same +place. +ls rocket propulsion therefore absolutely impossible because one cannot move +the center of mass? No; but of course we fñnd that to propel an interesting part +of the rocket, an uninteresting part must be thrown away. In other words, if we +start with a rocket at zero velocity and we spit some gas out the back end, then +this little blob of gas goes one way as the rocket ship goes the other, but the +center of mass is still exactly where it was before. So we simply move the part +that we are interested in against the part we are not interested in. +--- Trang 336 --- +The second point concerning the center of mass, which is the reason we +introduced ït into our discussion at this time, is that it may be treated separately +from the “internal” motions of an object, and may therefore be ignored in our +discussion oŸ rotation. +18-2 Rotation of a rigid body +Now let us discuss rotations. Of course an ordinary object does not simply +rotate, it wobbles, shakes, and bends, so to simplify matters we shall diseuss the +motion of a nonexistent ideal object which we call a rigid body. 'Phis means an +object in which the forces bebween the atoms are so strong, and of such character, +that the little forces that are needed to move it do not bend it. Its shape stays +essentially the same as iÿ moves about. If we wish to study the motion of such a +body, and agree to ignore the motion of is center oŸ mass, there is only one thing +left for it to do, and that 1s to furn. We have to describe that. How? Suppose +there is some line in the body which stays put (perhaps it includes the center oŸ +mass and perhaps not), and the body is rotating about this particular line as +an axis. How do we defne the rotation? 'Phat is easy enough, for if we mark a +point somewhere on the obJect, anywhere except on the axis, we can always tell +exactly where the object is, if we only know where this point has gone to. The +only thing needed to describe the position of that point is an øngle. So rotation +consists of a study of the variations of the angle with time. +In order to study rotation, we observe the angle through which a body has +turned. Of course, we are not referring to any particular angle #ns¿de the object +itself; ¡% is not that we draw some angle øn the object. We are talking about the +angular change oƒ the position of the whole thing, from one tỉme to another. +Jirst, let us study the kinematics of rotations. 'Phe angle will change with +time, and just as we talked about position and velocity in one đimension, we +may talk about angular position and angular velocity in plane rotation. In fact, +there is a very interesting relationship between rotation in two dimensions and +one-dimensional displacement, in which almost every quantity has its analog. +First, we have the angle Ø which defnes how far the body has gone around; this +replaces the distance +, which defnes how far ¡it has gone ølong. In the same +manner, we have a velocity oŸ turning, œ = đØ/di, which tells us how mụuch the +angle changes in a second, just as ø = đs/d£ describes how fast a thing moves, +or how far it moves in a second. I the angle is measured in radians, then the +angular velocity œ will be so and so many radians per second. “The greater the +--- Trang 337 --- +angular velocity, the faster the object is turning, the faster the angle changes. +W© can go on: we can diferentiate the angular velocity with respect to time, and +we can call œ = dư /dt = d20/df2 the angular acceleration. That would be the +analog of the ordinary acceleration. +Q—Y C\|Vx ộ +x$P(x, y) +Að đã y +Fig. 18-1. Kinematics of two-dimensional rotation. +Now oŸ course we shall have to relate the dynamics oŸ rotation to the laws oŸ +dynamies of the particles of which the obJect is made, so we must fnd out how +a particular particle moves when the angular velocity is such and such. To do +this, let us take a certain particle which is located at a distance z from the axis +and say ÍÊ is in a certain location ?{z, ø) at a given instant, in the usual manner +(Fig. 18-1). If at a moment Af later the angle of the whole object has turned +through A0, then this particle is carried with it. It is at the same radius away +from ) as it was before, but is carried to Q. The first thíng we would like to know +is how much the distance + changes and how much the distance changes. lf @?P +is called r, then the length P@) is z A0, because of the way angles are defined. +The change in z, then, is simply the projection of z A0 in the z-direction: +Az = —PQsinØ = —r A0 - (0/r) =—ụA0. (18.6) +Similarly, +Au = +z A0. (18.7) +TÍ the object is turning with a given angular velocity œ, we fñnd, by dividing both +sides of (18.6) and (18.7) by A¿, that the velocity oŸ the particle is +Uy —= — and Uy = +u#. (18.8) +Of course if we want to fnd the magnitude of the velocity, we Just write +0= (J0ệ + 02 = V202 + 1333 = 0V #2 + J2 = ê†. (18.9) +--- Trang 338 --- +Tt should not be mysterious that the value of the magnitude of this velocity is (07; +in fact, it should be self-evident, because the distance that it moves is z.AØ and +the distance it moves per second is r A0/Af, or rứ. +Let us now move on to consider the dựngmics of rotation. Here a new concept, +ƒorce, must be introduced. Let us inguire whether we can invent something which +we shall call the #orque (L. torquere, to twist) which bears the same relationship to +rotation as force does to linear movement. Á force is the thing that is needed to +make linear motion, and the thing that makes something rotate is a “rotary force” +or a “twisting force,” i.e., a torque. Qualitatively, a torque is a “twist”; what is a +torque quantitatively? We shall get to the theory of torques quantitatively by +studying the øork done in turning an object, for one very nice way of delning a +force is to say how much work it does when it acts through a given displacement. +W© are going to try to maintain the analogy between linear and angular quantities +by equating the work that we do when we turn something a little bit when there +are forces acting on it, to the £orgue tỉmes the øngle it turns through. In other +words, the deflnition of the torque is goïng to be so arranged that the theorem oŸ +work has an absolute analog: force times distance is work, and torque times angle +1s goïng to be work. That tells us what torque is. Consider, for instance, a rigid +body of some kind with various forces acting on it, and an axis about which the +body rotates. Let us at frst concentrate on one force and suppose that this force +is applied at a certain point (z,). How much work would be done iŸ we were to +turn the object through a very small angle? 'Phat is easy. he work done is +AW = F„ Az + lụ A. (18.10) +W©e need only to substitute Bqs. (18.6) and (18.7) for Az and A¿ to obtain +AW = (zty— uEF„)A0. (18.11) +That is, the amount of work that we have done is, in fact, equal to the angle +through which we have turned the object, multiplied by a strange-looking combi- +nation of the force and the distanece. 'Phis “strange combination” is what we call +the torque. So, defining the change in work as the torque times the angle, we +now have the formula for torque in terms oŸ the forces. (Obviously, torque is not +a completely new idea independent of Newtonian mechanics—torque must have +a defnite deflnition in terms of the force.) +'When there are several forces acting, the work that is done 1s, of course, the +sum of the works done by all the forces, so that AW will be a whole lot of terms, +--- Trang 339 --- +all added together, for all the forces, cach oƒ uhách is proportional, houeuer, +to A0. W© can take the AØ outside and therefore can say that the change in the +work is equal to the sum of all the torques due to all the diferent forces that are +acting, times A0. 'This sum we might call the total torque, 7. Thus torques add +by the ordinary laws of algebra, but we shall later see that this is only because +we are working in a plane. lt is like one-dimensional kinematics, where the forces +simply add algebraically, but only because they are all in the same direction. lt +1s more complicated in three dimensions. 'Phus, for two-dimensional rotation, +T=À T¡. (18.13) +lt must be emphasized that the torque is about a given axis. lfa different axis is +chosen, so that all the ø; and ¿ are changed, the value of the torque is (usually) +changed too. +Now we pause briefly to note that our foregoing introduction oŸtorque, through +the idea of work, gives us a most important result for an object in equilibrium: 1Ý +all the forces on an object are in balance both for translation and rotation, not +only is the net ƒorce zero, but the total of all the #orgues is also zero, because if an +object is in equilibrium, 0ö 0ork ?s done bụ the ƒorces [or a small đisplacement. +Therefore, since AW = r AØ =0, the sum of all the torques must be zero. So +there are two conditions for equilibrium: that the sum of the forces 1s zero, and +that the sum of the torques is zero. Prove that it suffices to be sure that the sum +of torques about any øne axis (in two dimensions) is zero. +Now let us consider a single force, and try to ñgure out, geometrically, what +this strange thing # — „ amounts to. In Eig. 18-2 we see a force ' acting at +a point r. When the object has rotated through a small angle A0, the work done, +of course, is the component of force in the direction of the displacement times the +Fig. 18-2. The torque produced by a force. +--- Trang 340 --- +displacement. In other words, ¡È is only the tangential component of the force +that counts, and this must be multiplied by the distance r A0. Therefore we see +that the torque is also equal to the tangential component oŸ force (perpendicular +to the radius) times the radius. That makes sense in terms of our ordinary idea +of the torque, because 1f the force were completely radial, ít would not put any +“twist” on the body; it is evident that the twisting efect should involve only the +part of the force which is not pulling out from the center, and that means the +tangential component. Eurthermore, it is clear that a given force is more effective +on a long arm than near the axis. In fact, if we take the case where we push right +ơn the axis, we are not twisting at alll So ¡it makes sense that the amount of +twist, or torque, is proportional both to the radial distance and to the tangential +component of the force. +There is still a third formula for the torque which is very interesting. We +have Jjust seen that the torque is the force times the radius times the sine of the +angle œ, in Fig. 18-2. But if we extend the line of action of the force and draw +the line Ø5, the perpendicular distance to the line of action of the force (the +lcuer œrm of the force) we notice that this lever arm is shorter than r in just the +same proportion as the tangential part of the force is less than the total force. +Therefore the formula for the torque can also be written as the magnitude of the +force times the length of the lever arm. +The torque is also often called the mmomen# oŸ the force. "The origin of this +term is obscure, but it may be related to the fact that “moment” is derived from +the Latin mouữnentum, and that the capability of a force to move an object +(using the force on a lever or crowbar) increases with the length of the lever arm. +In mathematics “moment” means weighted by how far away it is om an axis. +18-3 Angular momentum +Although we have so far considered only the special case of a rigid body, +the properties of torques and their mathematical relationships are interesting +also even when an object is not rigid. In fact, we can prove a very remarkable +theorem: just as external force 1s the rate of change of a quantity ø, which we +call the total momentum of a collection of particles, so the external torque is the +rate of change of a quantity Ù which we call the angular mmormnentum oŸ the group +of particles. +To prove this, we shall suppose that there is a system of particles on which +there are some forces acting and fñnd out what happens to the system as a result +--- Trang 341 --- +O7 +Fig. 18-3. A particle moves about an axis Ó. +of the torques due to these forces. First, of course, we should consider just øne +particle. In Fig. 18-3 is one particle of mass ?n, and an axis Ó; the particle is not +necessarily rotating in a cirele about Ó, it may be moving in an ellipse, like a +planet going around the sun, or in some other curve. Ït is moving somehow, and +there are Íorces on it, and it accelerates according to the usual formula that the +#-component of force is the mass times the z-component of acceleration, etc. But +let us see what the #orgue does. The torque equals ø„ — ;„, and the force in +the ø- or -direction is the mass times the acceleration in the z- or -direction: +T—#Èu T— Uy = += zm(dŠu/dt2) — m(d°+/d12). (18.14) +Now, although this does not appear to be the derivative of any simple quantity, +1E is in fact the derivative of the quantity zrm(dụ/đf) — ym(d+z/dÐ): +d dụ d+z dˆụ + d+z dụ +— |zm| —- | —m| — || —=zm| —> — |Jm| —- +dt dt) ”“Vi di? di dí +(18.15) +d2 dụ dœ d2 d2+z +—m| —c |— | TT |m| —| =zm| —— ]_—-m| —- |: +MAV di di d2) — ”“Ẳp +So ït is true that the torque is the rate of change of something with timel So we +pay attention to the “something,” we give it a name: we call it b, the angular +1mmomentum: += ~zm(dụ/đĐ) — ym(d+z/dt) += #Ðụ — Da. (18.16) +Although our present discussion is nonrelativistic, the second form for Ù, given +above is relativistically correct. So we have found that there is also a rotational +--- Trang 342 --- +analog for the momentum, and that this analog, the angular momentum, is given +by an expression in terms of the components of linear momentum that is jus$ +like the formula for torque in terms of the force componentsl Thus, iŸ we want +to know the angular momentum of a particle about an axis, we take only the +component of the momentum that is tangential, and multiply it by the radius. In +other words, what counts for angular momentum is not how fast it 1s goïng œa +from the origin, but how much it is going around the origin. Only the tangential +part of the momentum counts for angular momentum. Eurthermore, the farther +out the line of the momentum extends, the greater the angular momentum. And +also, because the geometrical facts are the same whether the quantity ¡s labeled +por F) it is true that there is a lever arm (nø the same as the lever arm of the +force on the particlel) which is obtained by extending the line of the zmomentưm +and fñnding the perpendicular distance to the axis. Thus the angular momentum +is the magnitude of the momentum tỉmes the momentum lever arm. So we have +three formulas for angular momentum, just as we have three formulas for the +torque: +Ù = tDụ — UPz +— Ttang += p- lever arm. (18.17) +Like torque, angular momentum depends upon the position of the axis about +which it is to be calculated. +Before proceeding to a treatment of more than one particle, let us apply the +above results to a planet going around the sun. In which direction is the force? +'The force is toward the sun. What, then, is the torque on the object? Of course, +this depends upon where we take the axis, but we get a very simple result if +we take i% at the sun itself, for the torque is the force times the lever arm, or +the component of force perpendicular to z, times z. But there is no tangential +force, so there is no torque about an axis at the sun! "Therefore, the angular +momentum of the planet goïng around the sun must remain constant. Let us see +what that means. The tangential ecomponent of velocity, times the mass, times +the radius, will be constant, because that is the angular momentum, and the +rate of change of the angular momentum ¡is the torque, and, in this problem, +the torque is zero. OÝ course since the mass is also a constant, this means that +the tangential velocity times the radius is a constant. But this is something we +already knew for the motion of a planet. Suppose we consider a smaill amount of +--- Trang 343 --- +time Af. How far will the planet move when it moves from ? to @ (Fig. 18-3)? +How mụuch ørea will it sweep through? Disregarding the very tiny area QQ“P +compared with the mụch larger area @P@), it is simply half the base P€) times +the height, Of. In other words, the area that is swept through in unit time will +be cqual to the velocity times the lever arm of the velocity (times one-half). 'Thus +the rate of change of area is proportional to the angular momentum, which 1s +constant. So Kepler”s law about equal areas in equal times is a word description +of the statement of the law of conservation of angular momentum, when there is +no torque produced by the force. +18-4 Conservation of angular momentum +Now we shall go on to consider what happens when there is a large number +of particles, when an object is made of many pieces with many Íorces acting +between them and on them from the outside. Of course, we already know that, +about any given fxed axis, the torque on the ¿th particle (which is the force on +the ¿th particle times the lever arm of that force) is equal to the rate oŸ change of +the angular momentum of that particle, and that the angular momentum of the +¿th particle is 10s momentum times its momentum lever arm. NÑow suppose we +add the torques 7; for all the particles and call ít the total torque 7. Then this will +be the rate of change of the sum of the angular momenta of all the particles L, +and that defnes a new quantity which we call the total angular momentum Ù. +Just as the total mornentum of an object is the sum of the momenta of all the +parts, so the angular momentum is the sum of the angular momenta of all the +parts. hen the rate of change of the total Ù is the total torque: +dLị dù +T=Ồ 7=” . (18.18) +Now it might seem that the total torque is a complicated thing. 'There are all +those internal forces and all the outside forces to be considered. But, if we +take Newton”s law of action and reaction to say, not simply that the action +and reaction are equal, but also that they are đirccted exactlU oppositelu along +the seme line (NÑewton may or may not actually have said this, but he tacitly +assumed it), then the Ewo £orgues on the reacting objects, due to their mutual +interaction, will be equal and opposite because the lever arms for any axis are +equal. Therefore the internal torques balance out pair by pair, and so we have +the remarkable theorem that the ra#e oƒ chưnge oƒ the totaÌ œngular mmomentum +--- Trang 344 --- +about am a#is is cqual to the external torque about that azis! +T= `T¡ = To„y = dL/dị. (18.19) +Thus we have a very powerful theorem concerning the motion of large collections +of particles, which permits us to study the over-all motion without having to +look at the detailed machinery inside. This theorem is true for any collection of +objects, whether they form a rigid body or not. +One extremely important case of the above theorem is the la+ oƒ conseruation +öƒ angular mmomentum: 1ƒ no external torques act upon a system of particles, the +angular momentum remains constant. +A special case of great importance is that of a rigid body, that is, an object +of a defñnite shape that is just turning around. Consider an object that is ñxed +in its geometrical dimensions, and which is rotating about a ñxed axis. Various +parts of the object bear the same relationship to one another at all times. Ñow +let us try to ñnd the total angular momentum of this object. If the mass of one +OÝ its particles is rn¿, and its position or location is at (#;, ¡), then the problem +1s to fnd the angular momentum of that particle, because the total angular +tmmomentum is the sum of the angular momenta of all such particles in the body. +For an object going around in a circle, the angular momentum, of course, is the +mass times the velocity times the distance from the axis, and the velocity is equal +to the angular velocity times the distanece from the axis: +L¡ = tmju¡r¿ = mạrỆằ), (18.20) +or, sunming over all the particles ?, we get +L= Tu, (18.21) +T=Ồ mịr?. (18.22) +This is the analog of the law that the momentum is mass times velocity. +Velocity is replaced by angular velocity, and we see that the mass is replaced +by a new thing which we call the rmornen‡ oƒ inertia T, which is analogous to +the mass. Equations (18.21) and (18.22) say that a body has inertia for turning +which depends, not just on the masses, but on hou ƒar a+UdU the are from the +axis. So, IÝ we have two objects of the same mass, when we put the masses +--- Trang 345 --- +lÝ 5) sĩ +Fig. 18-4. The “inertia for turning” depends upon the lever arm of the +masses. +farther away from the axis, the inertia for turning will be higher. 'This is easily +demonstrated by the apparatus shown in Fig. 18-4, where a weight Mƒ is kept +from falling very fast because it has to turn the large weighted rod. At first, the +masses ?w are close to the axis, and M⁄ speeds up at a certain rate. But when +we change the moment of inertia by putting the wo masses rn much farther +away from the axis, then we see that MỸ accelerates much less rapidly than it did +before, because the body has much more inertia against turning. The moment of +inertia is the inertia against turning, and is the sum of the contributions of all +the masses, times their distances sguared, from the axis. +'There is one important diference between mass and moment of inertia which +is very dramatic. The mass of an object never changes, but its moment of inertia, +can be changed. lf we stand on a frictionless rotatable stand with our arms +outstretched, and hold some weights in our hands as we rotate slowly, we may +change our moment of inertia by drawing our arms in, but our mass does not +change. When we do this, all kinds of wonderful things happen, because of the +law of the conservation of angular momentum: lf the external torque is zero, then +the angular momentum, the moment of inertia times omega, remains constant. +Initially, we were rotating with a large moment of inertia 1 at a low angular +velocity œ, and the angular momentum was ;ư. Then we changed our moment +oŸ inertia by pulling our arms in, say to a smaller value f¿. 'Then the product Tœ, +which has to stay the same because the total angular momentum has to stay the +same, was Ïaœs2. So hư = la. That 1s, IÍ we reduce the moment of inertia, we +have to #nwcrease the angular velocity. +--- Trang 346 --- +I9 +(orefor' of IWerss: /Wort©ortÉ @Ÿ Írt©rfier +19-1 Properties of the center of mass +In the previous chapter we found that iŸ a great many Íorces are acting on a +complicated mass of particles, whether the particles comprise a rigid or a nonrigid +body, or a cloud oŸ stars, or anything else, and we ñnd the sum of all the forces +(that is, oŸ course, the external forces, because the internal forces balance out), +then if we consider the body as a whole, and say it has a total mass jM, there is +a certain point “inside” the body, called the cenmter oƒ mass, suụch that the net +resulting external force produces an acceleration of this point, jus as though the +whole mass were concentrated there. Let us now discuss the center of mass in a +little more detail. +The location of the center of mass (abbreviated CM) is given by the equation +» THẠT¡ +Tcw Sâm (19.1) +This is, of course, a vector equation which is really three equations, one for +cach of the three directions. We shall consider only the z-direction, because +1ƒ we can understand that one, we can understand the other two. What does +Xe = È})m;+¡/S `m¿ mean? Suppose for a moment that the object is divided +into little pieces, all of which have the same mass ?n; then the total mass 1s +simply the number / of pieces times the mass oŸ one piece, say one gram, or any +unit. Then this equation simply says that we add all the z's, and then divide +by the number of things that we have added: Xe =?n})z;/mN =3 `z/;/N. +In other words, Xem is the average of all the +ˆs, If the masses are equal. But +Suppose one of them were twice as heavy as the others. 'Phen in the sum, that + +would come in twice. 'This is easy to understand, for we can think of this double +mass as being split into two equal ones, just like the others; then in taking the +average, of course, we have to count that + twice because there are two Immasses +--- Trang 347 --- +there. Thus X is the average position, in the z-direction, of all the masses, every +mass being counted a number of times proportional to the mass, as though 1$ +were divided into “little grams.” From this it is easy to prove that X must be +somewhere between the largest and the smallest z, and, therefore lies inside the +envelope including the entire body. It does not have to be in the mater?al of the +body, for the body could be a cirele, like a hoop, and the center of mass is in the +center of the hoop, not in the hoop itself. +Of course, if an object is symmetrical in some way, for instance, a rectangle, +so that it has a plane of symmetry, the center of mass lies somewhere on the +plane of symmetry. In the case of a rectangle there are two planes, and that +locates it uniquely. But ïŸ it is just any symmetrical object, then the center of +gravity lies somewhere on the axis of symmetry, because in those circumstances +there are as many positive as negative 4s. +: ` QCM +Fig. 19-1. The CM of a compound body lies on the line Joining the +CM's of the two composite parts. +Another interesting proposition is the following very curious one. Suppose +that we imagine an object to be made oŸ two pieces, 4 and Ö (Eig. 19-1). Then +the center oŸ mass of the whole object can be calculated as follows. First, ñnd the +center of mass of piece 4, and then of piece Ø. Also, fnd the total mass of each +pilece, ƒ4 and Míp. hen consider a new problem, in which a pø¿n‡ mass Ma 1s +at the center of mass of object 4, and another øø¿n‡ mass Míp is at the center +of mass of object #. 'The center of mass of these two point masses is then the +center of mass of the whole object. In other words, if the centers of mass of +various parts of an object have been worked out, we do not have to start all over +again to find the center of mass of the whole object; we Just have to put the +pleces together, treating each one as a point mass situated at the center oŸ mass +of that piece. Let us see why that is. Suppose that we wanted to calculate the +center of mass of a complete object, some of whose particles are considered to +be members of object A and some mermbers of object . The total sum À `?nm;#; +--- Trang 348 --- +can then be split into two pieces—the sum 3) ¿?m¿#¿ for the A object only, and +the sum 3 }„ m¿z; for object Ö only. Now if we were computing the center of +mass of object 4 alone, we would have exactly the first of these sums, and we +know that this by itselfis Ma Xa, the total mass of all the particles in A tỉmes +the position of the center of mass of 4, because that is the theorem of the center +of mass, applied to object A. In the same mamner, jus6 by looking at object Ö, +we get Míp Xp, and of course, adding the two yields ÄMƒ XeM: += MAXaA+ MpXn. (19.2) +NÑow since Ä⁄ is evidently the sum of Mu and ăpg, we see that Eq. (19.2) can +be interpreted as a special example of the center of mass formula for two point +objects, one of mass f4 located at Xa and the other of mass Mfp located at Xg. +The theorem concerning the motion of the center of mass is very interesting, +and has played an Important part in the development of our understanding of +physics. Suppose we assume that Newton”s law is right for the small component +parts of a much larger object. Then this theorem shows that Newton's law is also +correct for the larger object, even iŸ we do not study the details of the object, but +only the total force acting on it and its mass. In other words, Newton's law has +the peculiar property that If it is right on a certain small scale, then 1% will be +right on a larger scale. Ifwe do not consider a baseball as a tremendously complex +thing, made of myriads of interacting particles, but study only the motion of the +center of mass and the external forces on the ball, we fnd #' = ma, where F' +1s the external force on the baseball, m 1s Its mass, and ø is the acceleration of +1ts center of mass. 5o = rmœ is a law which reproduces itself on a larger scale. +(There ought to be a good word, out of the Greek, perhaps, to describe a law +which reproduces the same law on a larger scale.) +Of course, one might suspect that the first laws that would be discovered +by human beings would be those that would reproduce themselves on a larger +scale. Why? Because the actual scale of the fundamental gears and wheels of the +universe are of atomie dimensions, which are so much finer than our observations +that we are nowhere near that scale in our ordinary observations. So the first +things that we would discover must be true for objects of no special size relative to +an atomic scale. If the laws for small particles did not reproduce themselves on a, +larger scale, we would not discover those laws very easily. What about the reverse +--- Trang 349 --- +problem? Must the laws on a small scale be the same as those on a larger scale? +OŸÝ course it is not necessarily so in nature, that at an atomic level the laws have +to be the same as on a large scale. Suppose that the true laws of motion of atoms +were given by some strange equation which does œø# have the property that when +we øo to a larger scale we reproduce the same law, but instead has the property +that if we go to a larger scale, we can øpprozimeite ?t bụ a certain ezpression such +that, if we extend that expression up and up, ? keeps reproducing ïtself on a +larger and larger scale. 'That is possible, and in fact that is the way It works. +Newton's laws are the “tail end” of the atomic laws, extrapolated to a very large +size. The actual laws of motion of particles on a fine scale are very peculiar, but +1ƒ we take large numbers of them and compound them, they approximate, but +on approximate, Newton”s laws. Newton”s laws then permit us to go on to a +higher and higher scale, and it still seems to be the same law. In fact, it becomes +more and more accurate as the scale gets larger and larger. This self-reproducing +factor of Newton's laws is thus really not a fundamental feature of nature, but +1s an Iimportant historical feature. We would never discover the fundamental +laws of the atomic particles at first observation because the first observations +are much too crude. In fact, i% turns out that the fundamental atomic laws, +which we call quantum mechanics, are quite diferent from Newton”s laws, and +are difficult to understand because all our direct experiences are with large-scale +objects and the small-scale atoms behave like nothing we see on a large scale. 5o +we cannot say, “An atom ¡is just like a planet going around the sun,” or anything +like that. It is like nof#h#ng we are familiar with because there is noứh#ng like +z‡. As we apply quantum mechanics to larger and larger things, the laws about +the behavior of many atoms together do øoø reproduce themselves, but produce +neu laus, which are Ñewton'”s laws, which then continue to reproduce themselves +from, say, micro-microgram size, which still is billions and billions of atoms, on +up to the size of the earth, and above. +Let us now return to the center of mass. 'Phe center of mass is sometimes +called the center of gravity, for the reason that, in many cases, gravity may be +considered uniform. Let us suppose that we have small enough dimensions that the +gravitational foree is not only proportional to the mass, but is everywhere parallel +to some fñxed line. Then consider an object in which there are gravitational Íorces +on each of its constituent masses. Let ?m¿ be the mass of one part. Then the +gravitational force on that part 1s ?m¿ times g. Now the question is, where can we +apply a single force to balance the gravitational force on the whole thing, so that +the entire object, if it is a rigid body, will not turn? The answer is that this force +--- Trang 350 --- +mmust go through the center of mass, and we show this in the following way. In +order that the body will not turn, the torque produced by all the forces must add +up to zero, because if there is a torque, there is a change of angular momentum, +and thus a rotation. So we must calculate the total of all the torques on all the +particles, and see how much torque there is about any given axis; ¡§ should be +zero 1ƒ this axis is at the center of mass. Now, measuring z horizontally and +vortically, we know that the torques are the forces in the -direction, times the +lever arm ø# (that is to say, the force times the lever arm around which we want +to measure the torque). Now the total torque is the sum +T= Àmga; = gà” T¿, (19.3) +so if the total torque is to be zero, the sum À `7n¿#¿ must be zero. But È) m¿#¿ = +1M Xe, the total mass times the distance of the center of mass from the axis. +'Thus the z-distance of the center of mass from the axis is zero. +Of course, we have checked the result only for the z-distance, but IÝ we use +the true center of mass the object will balance in any position, because IÝ we +turned ï© 90 degrees, we would have zs instead of zøs. In other words, when an +object is supported at its center of mass, there is no torque on i§ because of a +parallel gravitational fñield. In case the object is so large that the nonparallelism +of the gravitational forces is significant, then the center where one must apply +the balancing force is not simple to describe, and ¡it departs slightly from the +center of mass. hat is why one must distinguish between the center oŸ mass and +the center of gravity. The fact that an object supported exactly at the center of +mass will balance in all positions has another interesting consequence. ÏÝ, instead +of gravitation, we have a pseudo force due to acceleration, we may use exactÌy +the same mathematical procedure to fnd the position to support it so that there +are no torques produeced by the inertial force of acceleration. Suppose that the +object is held in some mamner inside a box, and that the box, and everything +contained ïn it, is accelerating. We know that, from the point of view of someone +at rest relative to this accelerating box, there will be an effective force due to +inertia. That is, to make the object go along with the box, we have to push on it +to accelerate it, and this force is “balanced” by the “force of inertia,” which is a +pseudo force equal to the mass times the acceleration of the box. To the man in +the box, this is the same situation as ïif the object were in a uniform gravitational +fñeld whose “ø” value is equal to the acceleration ø. 'Phus the inertial force due +to accelerating an obJect has no torque about the center of mass. +--- Trang 351 --- +This fact has a very interesting consequence. Ín an inertial frame that is +not accelerating, the torque is always equal to the rate of change of the angular +momentum. However, about an axis through the center of mass of an object +which ¿s accelerating, 1t is sfil true that the torque is equal to the rate of change +of the angular momentum. ven ïf the center of mass is accelerating, we may +still choose one special axis, namely, one passing through the center of mass, such +that it will still be true that the torque is equal to the rate of change of angular +qmomentum around that axis. Thhus the theorem that torque equals the rate of +change of angular momentum is true in two general cases: (1) a ñxed axis in +inertial space, (2) an axis through the center oŸ mass, even though the object +may be accelerating. +19-2 Locating the center of mass +'The mathematical techniques for the calculation of centers oŸ mass are in the +province of a mathematics course, and such problems provide good exercise In +integral calculus. After one has learned calculus, however, and wants to know +how to locate centers of mass, It is nice to know certain tricks which can be used +to do so. Ône such trick makes use of what is called the theorem of Pappus. lt +works like this: if we take any closed area in a plane and generate a solid by +moving it through space such that each poïnt is always moved perpendicular to +the plane of the area, the resulting solid has a total volume equal to the area of +the cross section times the distance that the center of mass movedl Certainly +this is true If we move the area in a straight line perpendicular to itself, but 1f +we move it in a circle or in some other curve, then it generates a rather peculiar +volume. For a curved path, the outside goes around farther, and the inside goes +around less, and these efects balance out. So If we want to locate the center +of mass of a plane sheet oŸ uniform density, we can remember that the volune +generat©ed by spinning ¡i% about an axis is the distance that the center of mass +goes around, times the area oŸ the sheet. +For example, If we wish to fnd the center of mass of a right triangle of +base D and height (Eig. 19-2), we might solve the problem in the following +way. Imagine an axis along H, and rotate the triangle about that axis through +a full 360 degrees. This generates a cone. The distance that the #-coordinate +of the center of mass has moved is 2rz. The area which is beïing moved is the +area of the triangle, sH D. So the z-distance of the center of mass tỉimes the +area, of the triangle is the volume swept out, which is of course x!22//3. Thus +--- Trang 352 --- +2S XI» --` +“ mà ` +f ` D \ +N —_— - ` r +Fig. 19-2. A right triangle and a right circular cone generated by +rotating the triangle. +(2xz)(3HD) = xD?H/3, or z = D/3. In a similar manner, by rotating about +the other axis, or by symmetry, we lnd = H/3. In fact, the center oŸ mass +of any uniform triangular area is where the three medians, the lines from the +vertices through the centers of the opposite sides, all meet. That point is 1/3 +of the way along each median. (C?ue: Slice the triangle up into a lot of little +pleces, each parallel to a base. Note that the median line bisects every piece, and +therefore the center of mass must lie on this line. +Now let us try a more complicated figure. Suppose that ït is desired to fñnd +the position of the center of mass of a uniform semicircular disc—a disc sliced in +half. Where is the center of mass? Eor a full disc, i is at the center, of course, +but a half-dise is more difficult. Let r be the radius and z be the distance of the +center of mass from the straight edge of the disc. Spin it around this edge as axis +to generate a sphere. Khen the center of mass has gone around 27rz, the area +is Zr2/2 (because it is only half a circle). The volume generated is, of course, +4r3/3, from which we fnd that +(2xz)(šmr?) = 4mr3/3, ++ = Ar/3n. +'There is another theorem of Pappus which is a special case of the above one, +and therefore equally true. Suppose that, instead of the solid semicircular disc, +we have a semicircular piece of wire with uniform mass density along the wire, +and we want to find its center of mass. In this case there is no mass in the +Interior, only on the wire. 'Then it turns out that the area which is swept by +a plane curved line, when it moves as before, is the distance that the center of +mass moves times the iength of the line. (The line can be thought of as a very +narrow area, and the previous theorem can be applied to it.) +--- Trang 353 --- +19-3 Finding the moment of inertia +Now let us discuss the problem of finding the zmomen‡s oƒ inertia of various +objects. The formula for the moment of inertia about the z-axis of an object 1s +T=) mị(zỶ + tý) +I= Jú2+2) đm = J2+20sat (19.4) +That is, we must sum the masses, each one multiplied by the square of its +distance (z‡ + ÿ) from the axis. Note that it is not the three-dimensional +distance, only the two-dimensional distance squared, even for a three-dimensional +object. For the most part, we shall restrict ourselves to two-dimensional objeects, +but the formula for rotation about the z-axis is just the same in three dimensions. +L—————— +x ——| |-gx +Fig. 19-3. A straight rod of length L rotating about an axis through +one end. +As a simple example, consider a rod rotating about a perpendicular axis +through one end (Eig. 19-3). Now we must sum all the masses times the z- +distances squared (the s being all zero in this case). What we mean by “the +sum,” of course, is the integral of z2 times the little elements of mass. lÝ we +divide the rod into small elements of length dz, the corresponding elements of +mass are proportional to đz, and if dz were the length of the whole rod the mass +would be Mĩ. Therefore +đưmm = M dr/L +and so +¬- '“—=. q95 += øˆ—>———=— “dư = ———. : +0 TL Lo b) +The dimensions of moment of inertia are always mass times length squared, so +all we really had to work out was the factor 1/3. +--- Trang 354 --- +Now what is ƒ If the rotation axis is at the center of the rod? We could just +do the integral over again, letting # range from —šL to +§1. But let us notice +a few things about the moment of inertia. We can imagine the rod as two rods, +cach of mass ă/2 and length 7/2; the moments of inertia of the two small rods +are equal, and are both given by the formula (19.5). Thherefore the moment of +Inertia 1s 2 2 +]= 2(M/2)L/2)ˆ = HE (19.6) +'Thus it is much easier to turn a rod about its center, than to swing it around an +Of course, we could go on to compute the moments oŸ inertia of various other +bodies of interest. However, while such computations provide a certain amount of +important exercise in the calculus, they are not basically of interest to us as such. +'There is, however, an interesting theorem which is very useful. Suppose we have +an object, and we want to fnd its moment of inertia around some axis. hat +means we want the inertia needed to carry it by rotation about that axis. Now if +we support the object on pivots at the center of mass, so that the obJect does not +turn as iE rotates about the axis (because there is no torque on it from inertial +effects, and therefore it will not turn when we start moving it), then the forces +needed to swing it around are the same as though all the mass were concentrated +at the center of mass, and the moment of inertia would be simply 71 = MHệu,, +where em 1s the distance from the axis to the center of mass. But of course +that is not the right formula for the moment of inertia of an object which is really +beïng rotated as it revolves, because not only is the center of it moving ïn a circle, +which would contribute an amount 1¡ to the moment of inertia, but also we must +turn it about its center of mass. So it is not unreasonable that we must add to 1 +the moment oŸ inertia ?„ about the center of mass. So it is a good guess that the +total moment of inertia about any axis will be +I=I+ MRậu. (19.7) +This theorem ¡s called the parailel-azis theorem, and may be easily proved. +The moment of inertia about any axis is the mass times the sum of the z¿'s and +the z;'s, each squared: 7 = Y)(z‡ + ÿ)m¿. We shall concentrate on the #'s, but +of course the 's work the same way. Now z is the distance of a particular point +mass from the origin, but let us consider how it would look iŸ we measured z +trom the CM, instead of z from the origin. To get ready for this analysis, we +--- Trang 355 --- +HP 1 + XeM: +Then we just square this to fnd +3? =0 +2XecM#; + XêM: +So, when this is multiplied by rm¿ and summed over all ;, what happens? Taking +the constants outside the summation sign, we get +Ty — » ma? + 2X*eM » m4 + XếM » Tạ. +The third sum is easy; it is just Äf Xổ. In the second sum there are two pieces, +one of them is Ð }m;z;, which is the tobal mass times the #-coordinate of the +center oŸ mass. But this contributes nothing, because # is rmeasured from the +center of mass, and in these axes the average position of all the particles, weighted +by the masses, is zero. The first sum, of course, is the #ø part of ?„. Thhus we +arrive at Bd. (19.7), jusE as we guessed. +Let us check (19.7) for one example. Let us just see whether it works for the +rod. For an axis through one end, the moment of inertia should be Ä# 2/3, for we +calculated that. The center of mass of a rod, of course, is in the center of the rod, +at a distance L/2. Therefore we should find that MfL”/3 = ML2/12+ M(L/2). +Since one-quarter plus one-twelfth is one-third, we have made no fundamental +©TTOT. +Incidentally, we did not really need to use an integral to ñnd the moment of +inertia (19.5). If we simply assume that it is A2? times +y, an unknown coefficient, +and then use the argument about the two halves to get z+ for (19.6), then from +our argument about transferring the axes we could prove that + = 1 + + SO + +must be 1/3. There is always another way to do itl +In applying the parallel-axis theorem, it is oŸ course Important to remember +that the axis for Ï¿ musứ be parallel to the axis about which the moment of inertia +is wanbed. +One further property of the moment of inertia is worth mentioning because it +1s often helpful in ñnding the moment of inertia of certain kinds of objects. This +property is that if one has a pÏane figure and a set of coordinate axes with origin +in the plane and z-axis perpendicular to the plane, then the moment of inertia of +this fñgure about the z-axis is equal to the sum of the moments of inertia about +--- Trang 356 --- +the zø- and -axes. This is easily proved by noting that +1„ = m(wŸ + z2) = À ) mu +(since z¿ =0). Similarly, +lạ = À m,(x; +z7)= » m7, +1= m(x2 + 2) = À mịư? + ` mịy; += l¿ + ly. +As an example, the moment ofinertia ofa uniform rectangular plate of mass Ä, +width +, and length b, about an axis perpendicular to the plate and through +1ts center is simply +I= M(u + L”)/12, +because its moment of inertia about an axis in its plane and parallel to its length +is Mu2/12, i.e., just as for a rod of length +, and the moment of inertia about +the other axis in its plane is MƒL”/12, just as for a rod of length L. +To summarize, the moment of inertia of an object about a given axis, which +we shall call the z-axis, has the following properties: +(1) The moment of inertia is +1= 3 mi(eỆ + uệ) = [GẺ +) dm, +(2) T the object is made of a number of parts, each of whose moment of inertia +1s known, the total moment of inertia is the sum of the moments of inertia +of the pieces. +(3) The moment of inertia about any given axis is equal to the moment of +inertia about a parallel axis through the ƠM plus the total mass times the +square of the distance from the axis to the ƠM. +(4) LÝ the object is a plane fñgure, the moment of inertia about an axis perpen- +dicular to the plane is equal to the sum of the moments of inertia about +any two mutually perpendicular axes lying in the plane and intersecting at +the perpendicular axis. +--- Trang 357 --- +The moments of inertia of a number of elementary shapes having uniform +mass densities are given in Table 19-1, and the moments of inertia of some other +objects, which may be deduced from 'Table 19-1, using the above properties, are +given in Table 19-2. +Table 19-1 +Thin rod, length U | .L rod at center ML?/12 +Thin concentric +circular ring, radii | .L ring at center | Mr +r3)/2 +r1 and 7a +Sphere, radius r through center 2Mr2 /5 +Table 19-2 +Rect. sheet, sides ø, b | || b at center Ma?/12 +Rect. sheet, sides a, b | _L sheet at M(a2 + b2)/12 +center +Thin annular ring, any diameter Mf(r?+r3)/4 +radii r1, ra +Rect. parallelepiped, || c, through M(a2 + b2)/12 +sides a, Ù, € center +lRt. circ. cyL, radius || L, through Mr?/2 +r, length L center +lRt. circ. cyL, radius .L L, through | A(r?/4+ L2/12) +r, length L center +19-4 Rotational kinetic energy +Now let us go on to discuss dynamics further. In the analogy between linear +motion and angular motion that we discussed in Chapter 18, we used the work +theorem, but we did not talk about kinetic energy. What is the kinetic energy of +a rigid body, rotating about a certain axis with an angular velocity œ¿? We can +immediately guess the correct answer by using our analogies. The moment of +inertia corresponds to the mass, angular velocity corresponds to velocity, and so +--- Trang 358 --- +the kinetic energy ought to be j1 œ2, and indeed it is, as will now be demonstrated. +Suppose the objJect is rotating about some axis so that each point has a velocity +whose magnitude is œ¿r¿, where r¿ is the radius from the particular poin$ to the +axis. Thhen If rm¿ is the mass of that point, the total kinetic energy of the whole +thing is just the sum of the kinetic energies of all of the littÌe pieces: +T=j » TUỆ = 5 » m¿(r¿o)Ÿ. +Now ¿2 is a constant, the same for all points. Thus +T= 3® mịn? = 310. (19.8) +At the end of Chapter 1S we pointed out that there are some interesting +phenomena associated with an object which is not rigid, but which changes from +one rigid condition with a defnite moment of inertia, to another rigid condition. +Namely, in our example of the turntable, we had a certain moment of inertia Ï +with our arms stretched out, and a certain angular velocity œị. When we pulled +our arms in, we had a diferent moment of inertia, f¿, and a diferent angular +velocity, œ2, bu again we were “rigid.” The angular momentum remained constant, +since there was no torque about the vertical axis of the turntable. 'Phis means +that Tới = Taøa¿. Now what about the energy? 'That is an interesting question. +'With our arms pulled in, we turn faster, but our moment of inertia is less, and it +looks as though the energies might be equal. But they are not, because what +does balanee is Tự, not Tư?. So if we compare the kinetie energy before and +after, the kinetic energy before is shư? = s0, where Ù = lịư1 = Ï2ús is the +angular momentum. Afterward, by the same argument, we have 7 = 3a and +since œa > œ0 the kinetic energy of rotation is greater than it was before. So we +had a certain energy when our arms were out, and when we pulled them in, we +were turning faster and had more kinetic energy. What happened to the theorem +of the conservation of energy? Somebody must have done some work. We did +workl When did we do any work? When we move a weight horizontally, we do +not do any work. If we hold a thing out and pull it in, we do not do any work. +But that is when we are not rotatingl When we are rotating, there is centrifugal +force on the weights. 'Phey are trying to y out, so when we are going around +we have to pull the weights in against the centrifugal force. So, the work we +do against the centrifugal force ought to agree with the difference in rotational +energy, and of course i% does. That is where the extra kinetic energy comes Írom. +--- Trang 359 --- +There is still another interesting feature which we can treat only descriptively, +as a matter of general interest. This feature is a little more advanced, but is +worth pointing out because it is quite curious and produces many interesting +cfects. +Consider that turntable experiment again. Consider the body and the arms +separately, from the point of view of the man who is rotating. After the weights +are pulled in, the whole object is spinning faster, but observe, #he centrol part +0ƒ the bod is not changed, yet 1 1s turning faster after the event than before. +So, 1Ý we were to draw a circle around the inner body, and consider only obJects +inside the circle, /he#r angular momentum would chønge; they are going faster. +'Therefore there must be a torque exerted on the body while we pull in our arms. +No torque can be exerted by the centrifugal force, because that is radial. 5o that +means that among the forces that are developed in a rotating system, centrifugal +force is not the entire story, £here is œnother force. This other force is called +Coriolis [orce, and i9 has the very strange property that when we move something +in a rotating system, it seems to be pushed sidewise. Like the centrifugal force, +it is an apparent force. But IÝ we live in a system that is rotating, and move +something radially, we fnd that we must also push ït sidewise to move it radially. +'This sidewise push which we have to exert 1s what turned our body around. +Now let us develop a formula to show how this Coriolis force really works. +Suppose Moe is sitting on a carousel that appears to him to be stationary. But +trom the point of view of jJoe, who is standing on the ground and who knows +the right laws of mechanics, the carousel is going around. Suppose that we have +drawn a radial line on the carousel, and that Moe is moving some mass radially +along this line. We would like to demonstrate that a sidewise force is required to +do that. We can do this by paying attention to the angular momentum of the +mass. Ït is always going around with the same angular velocity œ, so that the +angular momentum is += ThU‡angT — THUT ©† = m2. +So when the mass is close to the center, it has relatively little angular momentum, +but if we move it to a new position farther out, if we increase z, rm has more +angular momentum, so a #orque rnust be ezerted in order to move it along the +radius. (To walk along the radius in a carousel, one has to lean over and push +sidewise. Try it sometime.) The torque that is required is the rate of change of Ù +with tỉme as ?w moves along the radius. If m moves only along the radius, omega +--- Trang 360 --- +stays constant, so that the torque is +T= F(r= n = —— ) = 2m¿ur m +where #4 is the Coriolis force. What we really want to know is what sidewise +ƒorce has to be exerted by Moe in order to move ?n out at speed „ = dr/df. Thịs +1s Fạ = TÍr = 2m0. +Now that we have a formula for the Coriolis force, let us look at the situation +a little more carefully, to see whether we can understand the origin of this force +from a more elementary point of view. We note that the Coriolis force is the same +at every radius, and is evidentÌy present even at the originl But it is especially +easy to understand it at the origin, just by looking at what happens from the +Inertial system of Joe, who is standing on the ground. Figure 19-4 shows three +Successive views of mm Just as it passes the origin at ý = 0. Because of the rotation +of the carousel, we see that rm does not move in a straight line, but in a curued +pa‡h tangent to a diameter of the carousel where z = 0. In order for ?nw to gO +in a curve, there must be a force to accelerate i% in absolute space. This is the +Coriolis force. +1 ¡ 3 3 +Fig. 19-4. Three successive views of a point moving radially on a +rotating turntable. +This is not the only case in which the Coriolis force occurs. We can also +show that if an object is moving with constant speed around the cireumference +of a circle, there is also a Coriolis force. Why? Moe sees a velocity 0a; around +the circle. On the other hand, .Joe sees rm going around the circle with the +velocitY 0 — 0; + œr, because m is also carried by the carousel. “Therefore +we know what the force really is, namely, the total centripetal force due to the +velocitV 0, Or mu} /r; that is the actual force. Now from Moe”s poinb oŸ view, +this centripetal force has three pieces. We may write it all out as follows: +hạ — "....... 21mUjd0 — ThuỶr. +--- Trang 361 --- +Now, #¿ is the force that Moe would see. Let us try to understand it. Would Moe +appreciate the first term? “Yes,” he would say, “even If Ï were not turning, there +would be a centripetal force if Ï were to run around a circle with velocity 0x.” +'This is simply the centripetal force that Moe would expect, having nothing to do +with rotation. In addition, Moe is quite aware that there is another centripetal +force that would act even on objects which are standing still on his carousel. “This +1s the third term. But there is another term in addition to these, namely the +second term, which is again 2n. 'Phe Coriolis force ¿ was tangential when +the velocity was radial, and now it is radial when the velocity is tangenmtial. In +fact, one expression has a minus sign relative to the other. The force is always +in the same direction, relative to the velocity, no matter in which direction the +velocity is. The force is at right angles to the velocity, and of magnitude 2m0. +--- Trang 362 --- +}ŸOof(tífOIt ít SJ06EC© +20-1 Torques ỉn three dimensions +In this chapter we shall discuss one of the most remarkable and amusing +consequences of mechanics, the behavior of a rotating wheel. In order to do +this we must first extend the mathematical formulation of rotational motion, +the principles of angular momentum, torque, and so on, to three-dimensional +space. We shall not se these equations in all their generality and study all theïr +consequences, because this would take many years, and we must soon turn to +other subjects. In an introductory course we can present only the fundamental +laws and apply them to a very few situations oŸ special interest. +Pirst, we notice that If we have a rotation in three dimensions, whether of a +rigid body or any other system, what we deduced for two dimensions is still right. +That is, it is still true that z#„ — 9F „ is the torque “in the #-plane,” or the +torque “around the z-axis.” It also turns out that this torque ¡s still equal to the +rate of change oÝ #Ðp„ — 1p„, for iÝ we go back over the derivation of Eq. (18.15) +from Newton's laws we see that we did not have to assume that the motion was In +a plane; when we diferentiate #py — px, we get øF — 9F+„, so thìs theorem ¡is still +right. The quantity #ø„ — px, then, we call the angular momentum belonging +to the z#-plane, or the angular momentum about the z-axis. This being true, we +can use any other pair of axes and get another equation. For instance, we can use +the z-plane, and it is clear from symmetry that iŸƒ we just substitute for z and z +for , we would find ¿/È; — zÈ;, for the torque and ø„ — zø„ would be the angular +tmmomentum associated with the z-plane. Of course we could have another plane, +the zz-plane, and for this we would ñnd zF„ — #EF¿ = d/dt (zp„ — #p;). +'That these three equations can be deduced for the motion of a single particle +is quite clear. Furthermore, if we added such things as #p„ — #p„ together for +many particles and called it the total angular momentum, we would have three +kinds for the three planes ø, z, and zz, and if we did the same with the forces, +--- Trang 363 --- +we would talk about the torque in the planes z, z, and zz also. Thus we would +have laws that the external torque associated with any plane is equal to the rate +of change of the angular momentum associated with that plane. 'This is Just a +generalization of what we wrote in two dimensions. +But now one may say, “Ah, but there are more planes; after all, can we not take +some other plane at some angle, and calculate the torque on that plane from the +forces? Since we would have to write another set of equations for every such plane, +we would have a lot of equationsl” Interestingly enouph, it turns out that iÝ we +were to work out the combination #2 — “F„; for another plane, measuring the ++", Fụ:, ec., in that plane, the result can be written as some cørmbination oŸ the +three expressions for the z-, z- and zz-planes. 'Phere is nothing new. In other +words, if we know what the three torques in the z-, z-, and zz-planes are, then +the torque in any other plane, and correspondingly the angular momentum also, +can be written as some combination of these: six percent of one and ninety-two +percent of another, and so on. 'This property we shall now analyze. +Suppose that in the zz-axes, Joe has worked out all his torques and his +angular momenta in his planes. But Moe has axes #, z/, z” in some other direction. +To make it a little easier, we shall suppose that only the z- and „-axes have +been turned. Moe's zø and + are new, but his z” happens to be the same. That +1s, he has new planes, let us say, for z and zz. He therefore has new torques +and angular momenta which he would work out. For example, his torque in the +z/-plane would be equal to #2 — F>/ and so forth. What we must now do +1s to find the relationship between the new torques and the old torques, so we +will be able to make a connection from one set oŸ axes to the other. 5omeone +may say, “Phat looks just like what we did with vectors.” And indeed, that is +exactly what we are intending to do. Then he may say, “Well, isnˆt torque jusE a +vector?” It does turn out to be a vector, but we do not know that right away +without making an analysis. 5o in the following steps we shall make the analysis. +'W© shall not discuss every step in detail, since we only want to illustrate how it +works. The torques calculated by Joe are +Tự„ụ = #Eụ — Uy, +Ty =UF; — zÈy, (20.1) +Tyy —= Zl„ — œF;. +W© digress at this point to note that in such cases as this one may get the wrong +sien for some quantity if the coordinates are not handled in the right way. Why not +--- Trang 364 --- +write 7z = 2 — F;? 'The problem arises from the fact that a coordinate system may +be either “right-handed” or “left-handed” Having chosen (arbitrarily) a sign for, say +Tx„, then the correct expressions for the other two quantities may always be found by +interchanging the letters zz in either order +# OT # +Z ~— Z —> +Moe now calculates the torques in his system: +Tag! — #' Fụ — V Fy, +Tụ!z! = '.Eà, — Z Fụ, (20.2) +Ty? — z' Fụ„ — x'EQ, * +Now we suppose that one coordinate system is rotated by a ñxed angle Ø, such +that the z- and z/-axes are the same. (This angle Ø has nothing to do with +rotating objects or what is goïng on inside the coordinate system. Ít is merely +the relationship between the axes used by one man and the axes used by the +other, and is supposedly constant.) Thus the coordinates of the two systems are +related by ++ = #øcos 0 + 1 sin 0, +ˆ = cos0 — #sin 0, (20.3) +z' =z. +Likewise, because force is a vector it transforms into the new system in the +same way as do zø, , and z, since a thing is a vector I1 and only if the various +components transform in the same way as z, , and z: +tạ = Fạ cos Ø + Fý, sin Ø, +Tàu = Fy cosØ — F„ sìn Ú, (20.4) +Hà. — F,. +Now we can fnd out how the torque transforms by merely substituting for ++, , and z” the expressions (20.3), and for F2, f;„, f;/ those given by (20.4), +all into (20.2). 5o, we have a rather long string of terms for 7x⁄„ and (rather +surprisingly at fñrst) it turns out that it comes right down to #„ — F„, which +--- Trang 365 --- +we recognize to be the torque in the z-plane: +Tạ: = (œcos 8 + sin Ø)(F„ cos Ø — F„ sỉn 6) +— (cos Ø — #zsin Ø)(F„ cos Ø + Ƒ+„ sin 0) += #F,(cos? Ø + sin? Ø) — F„(sin? Ø + cos? Ø) ++ #ƑF+(— sin Ø cos Ø - sin Ø cos 8) ++ #(sin Ø cos Ø — sin Ø cos 0) += ky — UF„ = Tụy. (20.5) +'That result is clear, for iŸ we only turn our axes ?n fhe pÏøœne, the twist around z +in that plane is no diferent than it was before, because it is the same planel +What will be more interesting is the expression for 7„;:, because that is a new +plane. We now do exactly the same thing with the z/z-plane, and it comes out +as follows: +Tựụ/z = (ucosØ — zsin Ø)F„ +— Z(E„ cos Ø — EF„ sin 6) += (0E; — zFu) cosØ + (zF„ — œF,) sin 0 += Tụz„ COs Ở -E 7„„ sỉn . (20.6) +Einally, we do it for Z4”: +Tz„: = Z(F„ cos 9 + Fy sin 8) +— (# cos 8 + sin 0); += (zF„ — #F,) cos Ø — (WEF+„ — zF) sin 9 += Tz„ cOs Ở — Tựz sỉn Ö. (20.7) +W©e wanted to get a rule for fñnding torques in new axes in terms oŸ torques In +old axes, and now we have the rule. How can we ever remember that rule? lf we +look carefully at (20.5), (20.6), and (20.7), we see that there is a close relationship +between these equations and the equations for #z, , and z. IÝ, somehow, we could +call 7„„ the z-componen# of something, let us call it the z-component of 7, then it +would be all right; we would understand (20.5) as a vector transformation, since +the z-component would be unchanged, as it should be. Likewise, if we associate +with the z-plane the #ø-component of our newly invented vector, and with the +--- Trang 366 --- +zz-plane, the -component, then these transformation expressions would read +Tự: — Tự, +T„/ = T„ cOS Ö + Tụ sỉn Ø, (20.8) +Tụ: = Tụ COS Ở — 7„ sỉn Ø, +which is just the rule for vectorsl +Therefore we have proved that we may identify the combination of #2, — 1F» +with what we ordinarily call the z-component of a certain artificially invented +vector. Although a torque is a twist on a plane, and it has no ø ør?or¿ vector +character, mathematically it does behave like a vector. 'This vector is at right +angles to the plane of the twist, and its length is proportional to the strength +of the twist. The three components of such a quantity will transform like a real +VCCfEOF. +So we represent torques by vectors; with each plane on which the torque is +supposed to be acting, we associate a line at right angles, by a rule. But “at +right angles” leaves the sign unspecifed. 'To get the sign right, we must adopt a +rule which will tell us that if the torque were in a certain sense on the z-plane, +then the axis that we want to associate with it is in the “up” z-direction. 'Phat is, +somebody has to defne “right” and “left” for us. Supposing that the coordinate +system is øz, , z In a ripght-hand system, then the rule wïll be the following: 1f +we think of the twist as If we were turning a screw having a right-hand thread, +then the direction of the vector that we will associate with that bwist is in the +direction that the screw would advanee. +'Why is torque a vector? It is a miracle of good luck that we can associate a +single axis with a plane, and therefore that we can associate a vector with the +torque; it is a special property of three-dimensional space. In two dimensions, the +torque is an ordinary scalar, and there need be no direction associated with it. In +three dimensions, it is a vector. If we had four dimensions, we would be in great +difficulty, because (ïf we had time, for exarmple, as the fourth dimension) we would +not only have planes like #ø, z, and zz, we would also have #z-, #-, and z- +planes. There would be s#z of them, and one cannot represent six quantities as +one vector in four dimensions. +WSe will be living in three dimensions for a long time, so iÈ is well to notice +that the foregoing mathematical treatment did not depend upon the fact that + +was position and # was force; it only depended on the transformation laws for +vectors. Therefore If, instead of z, we used the ø-component of some other vector, +--- Trang 367 --- +1E is not going to make any difference. In other words, iŸ we were to calculate +„bu — aub„, where œ and b are vectors, and call it the z-component of some new +quantity c, then these new quantities form a vector c. We need a mathematical +notation for the relationship of the new vector, with i0s three components, to +the vectors œ and b. The notation that has been devised for this is e = œ x b. +W© have then, in addition to the ordinary scalar produect in the theory of vector +analysis, a new kind of product, called the øector product. Thus, 1Í œ— œ x b, +this is the same as writing +C„ = quÖ; — dzb„, +đụ = „by — d„Ù„, (20.9) +cy = q„bu — dub„. +TỶ we reverse the order of ø and b, calling œ, b and b, œø, we would have the sign of e +reversed, because c„ would be b„ø„ — b„a„. Therefore the cross product is unlike +ordinary multiplication, where øÖ = ba; for the cross product, bx œ=— —ø x b. +tErom this, we can prove at once that if œ = b, the cross product 1s zero. Thus, +øxqœ=0. +'The cross product is very important for representing the features of rotation, +and it is important that we understand the geometrical relationship of the three +vectors ø, b, and e. Of course the relationship in components is given in Eq. (20.9) +and from that one can determine what the relationship is in geometry. “The +answer is, frst, that the vector e is perpendicular to both œ and b. (Try to +calculate e - œ, and see if it does not reduce to zero.) Second, the magnitude of e +turns out to be the magnitude oŸ ø times the magnitude of b times the sine of +the angle between the two. In which direction does e point? Imagine that we +turn ø into b through an angle less than 180”; a screw with a right-hand thread +turning in this way will advance in the direction of e. The fact that we say a +righi-hand screw instead of a /eff-hand screw is a convention, and is a perpetual +reminder that if ø and b are “honest” vectors in the ordinary sense, the new kind +OŸ “vector” which we have created by œ x b is artificial, or slightly diferent ïn its +character from ø and b, because it was made up with a special rule. lf œ and b +are called ordinary vectors, we have a special name for them, we call them polar +0ectors. Examples of such vectors are the coordinate ?, force #'", momentum 7ø, +velocity , electric fñeld #, etc.; these are ordinary polar vectors. Vectors which +involve just one cross product in their defnition are called a#al 0ectors or pseudo +uectors. Examples of pseudo vectors are, of course, torque 7 and the angular +--- Trang 368 --- +mmomentum E. It also turns out that the angular velocity œ is a pseudo vector, +as is the magnetic field Ö. +In order to complete the mathematical properties of vectors, we should know +all the rules for their multiplication, using dot and cross products. In our +applications at the moment, we will need very little of this, but for the sake of +completeness we shall write down all of the rules for vector multiplication so that +we can use the results later. These are +(a) œ<(b+c)=aœaxb+axe, +(b) (œa) x b= œ(œ x b), +e œ-(bxe)—=(axb)-c, +() (b xe) = (a x b) 6010) +(đ) œ < (b x e) = b(œ - c) — c(œ - b), +(e) axœ=0, +( œ-(œ x b) =0. +20-2 The rotation equations using cross products +Now let us ask whether any equations in physics can be written using the +cross product. The answer, of course, is that a great many equations can be so +written. For instance, we see immediately that the torque is equal to the position +vector cross the Íorce: +T—=rx Œ. (20.11) +This is a vector summary of the three equations 7x = 1; — zF¿y, etc. By the +same token, the angular momentum vector, if there is only one particle present, +1s the distanece from the origin multiplied by the vector momentum: +TL —rxp. (20.12) +For three-dimensional space rotation, the dynamical law analogous to the law #' = +dp/dt of NÑewton, is that the torque vector is the rate of change with time of the +angular momentum vector: +T = dL/dt. (20.13) +TÝ we sum (20.13) over many particles, the external torque on a system is the +rate of change of the total angular momentum: +Text — dL:oị /dt. (20.14) +--- Trang 369 --- +Another theorem: I the total external torque is zero, then the total vector +angular momentum of the system is a constant. Thịis is called the law of conser- +0ation oƒ angular momentum. TÝ there is no torque on a given system, its angular +mmomentum cannot change. +What about angular velocity? ls i a vector? We have already discussed +turning a solid object about a fñxed axis, but for a moment suppose that we are +turning i% simultaneously about #uo axes. It might be turning about an axis +inside a box, while the box is turning about some other axis. 'Phe net result of +such combined motions is that the object simply turns about some new axisl +The wonderful thing about this new axis is that it can be fgured out this way. +T the rate of turning in the z-plane is written as a vector in the z-direction +whose length is equal to the rate of rotation in the plane, and ïf another vector is +drawn in the -direction, say, which is the rate oŸ rotation in the zz-plane, then +1ƒ we add these together as a vector, the magnitude of the result tells us how +fast the object is turning, and the direction tells us in what plane, by the rule of +the parallelopgram. 'Phat is to say, simply, angular velocity is a vector, where we +draw the magnitudes of the rotations in the three planes as projections at right +angles to those planes.* +As a simple application of the use of the angular velocity vector, we may +evaluate the power being expended by the torque acting on a rigid body. The +pOwer, Of course, is the rate of change of work with time; in three dimensions, +the power turns out to be P =7 -ứ. +AII the formulas that we wrote for plane rotation can be generalized to three +dimensions. For example, If a rigid body is turning about a certain axis with +angular velocity œ, we might ask, “What is the velocity of a poïint at a certain +radial position r?” We shall leave it as a problem for the student to show that +the velocity of a particle in a rigid body is given by 0 = œ x?, where œ is +the angular velocity and z is the position. Also, as another example of cross +products, we had a formula for Coriolis force, which can also be written using +cross products: #2 = 2w x œ. That is, if a particle is moving with velocity 0 +in a coordinate system which is, in fact, rotating with angular velocity œ, and +we want to think in terms of the rotating coordinate system, then we have to +add the pseudo force #,. +— * That this is true can be đerived by compounding the displacements of the particles of +the body during an infinitesimal time Af. It is not self-evident, and is left to those who are +interested to try to fgure it out. +--- Trang 370 --- +20-3 The gyroscope +Let us now return to the law of conservation of angular momentum. 'Phis law +may be demonstrated with a rapidly spinning wheel, or gyroscope, as follows +(see Eig. 20-1). TỶ we sit on a swivel chair and hold the spinning wheel with +1ts axis horizontal, the wheel has an angular momentum about the horizontal +axis. Angular momentum around a 0erfical axis cannot change because of the +(frictionless) pivot of the chair, so iƒ we turn the axis of the wheel into the vertical, +then the wheel would have angular momentum about the vertical axis, because it +is now spinning about this axis. But the ss¿em (wheel, ourself, and chair) canwnof +have a vertical component, so we and the chaïr have to turn in the direction +opposite to the spin of the wheel, to balance ït. +` JÐ_k h 1) +BEFORE AFTER +Fig. 20-1. Before: axis is horlzontal; moment about vertical axis = 0. +After: axis Is vertical; momentum about vertical axis Is still zero; man +and chair spin in direction opposite to spin of the wheel. +First let us analyze in more detail the thing we have just described. What is +surprising, and what we must understand, is the origin of the forces which turn +us and the chaïr around as we turn the axis of the gyroscope toward the vertical. +Jigure 20-2 shows the wheel spinning rapidly about the -axis. 'Pherefore is +angular velocity is about that axis and, it turns out, its angular momentum is +likewise in that direction. NÑow suppose that we wish to rotate the wheel about +the z-axis at a small angular velocity ©; what forces are required? After a short +tỉme A¿, the axis has turned to a new position, tilted at an angle AØ with the +--- Trang 371 --- +/ =1 F AL +x ớy . lọ y +Fig. 20-2. A gyroscope. +horizontal. Since the major part of the angular momentum is due to the spin on +the axis (very little is contributed by the slow turning), we see that the angular +momentum vector has changed. What is the change in angular momentum? The +angular momentum does not change in rmagn#tude, but it does change in đứecclion +by an amount A0. The magnitude of the vector AE is thus AÙ, = bọ A0, so +that the torque, which is the time rate of change of the angular momentum, is +7= AL/At = Lạ A0/At = LạO. Taking the directions of the various quantities +into account, we see that +T =f) x Lạ. (20.15) +'Thus, if €) and ọ are both horizontal, as shown in the fgure, 7 is 0ertzcøl. To +produce such a torque, horizontal forces #" and —.F' must be applied at the ends +of the axle. How are these forces applied? By our hands, as we try to rotate the +axis of the wheel into the vertical direction. But NÑewton's Phird Law demands +that equal and opposite forces (and equal and opposite forqgues) act on 0s. This +causes us to rotate in the opposite sense about the vertical axis z. +This result can be generalized for a rapidly spinning top. In the familiar case +of a spinning top, gravity acting on its center of mass furnishes a torque about +the point of contact with the floor (see Fig. 20-3). 'This torque is in the horizontal +direction, and causes the top to precess with its axis moving in a circular cone +about the vertical. If ©J ¡is the (vertical) angular velocity of precession, we again +fnd that +T = dL/dt = © x Lạ. +Thus, when we apply a torque to a rapidly spinning top, the direction of the +precessional motion is in the direction of the torque, or at right angles to the +forces producing the torque. +We may now claim to understand the precession of gyroscopes, and indeed +we do, mathematically. However, this is a mathematical thing which, in a sense, +--- Trang 372 --- +Fig. 20-3. A rapidly spinning top. Note that the direction of the +torque vector ¡is the direction of the precession. +appears as a “miracle.” It will turn out, as we go to more and more advanced +physics, that many simple things can be deduced mathematically more rapidly +than they can be really understood in a fundamental or simple sense. This is a +strange characteristic, and as we get into more and more advanced work there are +circumstances in which mathematics will produce results which mo one has really +been able to understand in any direct fashion. An example is the Dirac equation, +which appears in a very simple and beautiful form, but whose consequences are +hard to understand. In our particular case, the precession of a top looks like some +kind of a miracle involving right angles and circles, and twists and right-hand +serews. What we should try to do is to understand it in a more physical way. +How can we explain the torque in terms of the real forces and the accelerations? +W© note that when the wheel is precessing, the particles that are going around +the wheel are not really moving in a plane because the wheel is precessing +(see Fig. 20-4). As we explained previously (Fig. 19-4), the particles which are +crossing through the precession axis are moving in curued paths, and this requires +application of a lateral force. This is supplied by our pushing on the axle, which +`v— /2[ ]⁄*, x/LATER +: hi si NOW +_ấ 4 = +m. VÀNG +-⁄ ` R⁄” `*EARLIER +Fig. 20-4. The motion of particles in the spinning wheel of Fig. 20-2, +whose axIs Is turning, ¡is in curved lines. +--- Trang 373 --- +then communicates the force to the rim through the spokes. “Wait,” someone +says, “what about the particles that are goïng back on the other side?” It does +not take long to decide that there must be a force in the opposite direclion on +that side. The net force that we have to apply is therefore zero. The ƒorces +balance out, but one of them must be applied at one side of the wheel, and the +other must be applied at the other side of the wheel. We could apply these forces +directly, but because the wheel is solid we are allowed to do it by pushing on the +axle, since forces can be carried up through the spokes. +'What we have so far proved is that if the wheel is precessing, it can balance +the torque due to gravity or some other applied torque. But all we have shown 1s +that this is ø solution of an equation. 'Phat is, 1f the torque is given, and Zƒ ue +get the spinning started right, then the wheel will precess smoothly and uniformly. +But we have not proved (and it is not true) that a uniform precession is the +tmos‡ general motion a spinning body can undergo as the result oŸ a given torque. +The general motion involves also a “wobbling” about the mean precession. 'This +“wobbling” is called nu‡ation. +Some people like to say that when one exers a ÿorque on a øyroscope, i% ©urns +and it precesses, and that the torque øroduces the precession. Ït is very sirange +that when one suddenly lets go of a gyroscope, it does not ƒœl! under the action +of gravity, but moves sidewise insteadl Why ¡is it that the dourmwuard force of the +gravity, which we knou and ƒeel, makes it go sideu#se? All the formulas in the +world like (20.15) are not going to tell us, because (20.15) is a special equation, +valid only after the gyroscope 1s precessing nicely. What really happens, in detail, +1s the following. lf we were to hold the axis absolutely fñxed, so that it cannot +precess in any manner (but the top is spinning) then there is no torque acting, +not even a torque from gravity, because it is balanced by our fñngers. But iŸ we +suddenly let go, then there will instantaneously be a torque from gravity. Anyone +in his right mind would think that the top would fall, and that is what it starts +to do, as can be seen If the top is not spinning too fast. +'The gyro actually does fall, as we would expect. But as soon as it falls, 1t is +then turning, and If this turning were to continue, a torque would be required. In +the absence of a torque in this direction, the gyro begins to “fall” in the direction +opposite that of the missing force. 'Phis gives the gyro a component of motion +around the vertical axis, as it would have in steady precession. But the actual +motion “overshoots” the steady precessional velocity, and the axis actually rises +again to the level from which it started. The path followed by the end of the +axle is a cycloid (the path followed by a pebble that is stuck in the tread of an +--- Trang 374 --- +automobile tire). Ordinarily, this motion is too quick for the eye to follow, and it +damps out quickly because of the friction in the gimbal bearings, leaving only +the steady precessional drift (Eig. 20-5). The slower the wheel spins, the more +obvious the nutation is. +` 7x7 =Z +Fig. 20-5. Actual motion of tip of axIs of gyroscope under gravity Just +after releasing axis previously held fixed. +'When the motion settles down, the axis of the gyro is a little bít lower than +it was at the start. Why? (These are the more complicated details, but we bring +them in because we do not want the reader to get the idea that the gyroscope 1s +an absolute miracle. It 7s a wonderful thing, but it is not a miracle.) IÝ we were +holding the axis absolutely horizontally, and suddenly let go, then the simple +precession equation would t$ell us that it precesses, that it goes around in a +horizontal plane. But that is impossiblel Although we neglected it before, it is +true that the wheel has sornme moment of inertia about the precession axis, and +1f it is moving about that axis, even slowly, it has a weak angular momentum +about the axis. Where did it come from? Tf the pivots are perfect, there is no +torque about the vertical axis. How then does it get to precess if there is no +change in the angular momentum? 'Phe answer is that the cycloidal motion of +the end of the axis damps down to the average, steady motion of the center of +the equivalent rolling circle. 'Phat is, it settles down a little bit low. Because it is +low, the spin angular momentum now has a small vertical component, which is +exactly what ¡is needed for the precession. So you see it has to go down a little, +in order to go around. It has to yield a little bít to the gravity; by turning is +axis down a little bit, it maintains the rotation about the vertical axis. That, +then, is the way a gyroscope WwOorks. +--- Trang 375 --- +20-4 Angular momentum of a solid body +Before we leave the subject of rotations in three dimensions, we shall discuss, +at least qualitatively, a few effects that occur in three-dimensional rotations that +are not self-evident. The main efect is that, in general, the angular momentum +of a rigid body is no necessari in the same direction as the angular velocity. +Consider a wheel that is fastened onto a shaft ïn a lopsided fashion, but with +the axis through the center of gravity, to be sure (Fig. 20-6). When we spin +the wheel around the axis, anybody knows that there will be shaking at the +bearings because of the lopsided way we have it mounted. Qualitatively, we +know that in the rotating system there is centrifugal force acting on the wheel, +trying to throw its mass as far as possible from the axis. 'This tends to line +up the plane of the wheel so that it is perpendicular to the axis. To resist this +tendenecy, a torque is exerted by the bearings. lf there is a torque exerted by the +bearings, there must be a rate of change of angular momentum. How can there +be a rate of change of angular momentum when we are simply turning the wheel +about the axis? Suppose we break the angular velocity œ into components œ1 +and œs perpendicular and parallel to the plane of the wheel. What is the angular +mmomentum? “The moments of inertia about these two axes are đjƒƒferent, so the +angular momenbum components, which (in these particular, special axes only) +are equal to the moments of inertia times the corresponding angular velocity +components, are in a đjfƒferent ratio than are the angular velocity components. +'Therefore the angular momentum vector is in a direction in space øø‡ along the +axis. When we turn the object, we have to turn the angular momentum vector +in space, so we must exert torques on the shaft. +Lìị = hư +N Ầ L +L U ^ +La = laua « +Fig. 20-6. The angular momentum of a rotating body Is not necessarily +parallel to the angular velocity. +Although it is much too complicated to prove here, there is a very important +and interesting property of the moment of inertia which is easy to describe and to +--- Trang 376 --- +use, and which is the basis of our above analysis. This property is the following: +Any rigid body, even an irregular one like a potato, possesses three mutually +perpendicular axes through the ƠM, such that the moment of inertia about one +of these axes has the greatest possible value for any axis through the ƠM, the +moment of inertia about another of the axes has the minwửnwm possible value, +and the moment of inertia about the third axis is intermediate between these two +(or equal to one of them). These axes are called the pr/ncipal azes of the body, +and they have the important property that If the body is rotating about one +of them, its angular momentum is in the same direction as the angular velocity. +For a body having axes of symmetry, the principal axes are along the symmetry +a%Xes. +z4@:~———_ +lổ JÄÌ t | +| J⁄ ⁄ s +F. dã ⁄ +Fig. 20-7. The angular velocity and angular momentum of a rigid +body (4> B> C). +TÝ we take the z-, -, and z-axes along the principal axes, and call the +corresponding principal moments of inertia A, Ö, and Œ, we may easily evaluate +the angular momentum and the kinetic energy of rotation of the body for any +angular velocity œ0. IÝ we resolve œ into componenfs œ„, (œ„, and œ; along the +Z-, -, z-axes, and use unit vectors ?, 7, k, also along zø, , z, we may write the +angular momentum as +TL Au„¿ + Buy 2 + Cu¿k. (20.16) +--- Trang 377 --- +The kinetic energy of rotation is +KE = š(Au2 + Bưu + C2) (20.17) +--- Trang 378 --- +Tho lÍtrrreorerc Ê)setÏletéor- +21-1 Linear diferential equations +In the study of physics, usually the course is divided into a series of subjects, +such as mechanics, electricity, optics, etbc., and one studies one subJect after the +other. Eor example, this course has so far dealt mostly with mechanics. But a +strange thing occurs again and again: the equations which appear in diferent +fields of physics, and even in other sciences, are often almost exactly the same, +so that many phenomena have analogs in these different fñelds. To take the +simplest example, the propagation oŸ sound waves is in many ways analogous to +the propagation of light waves. IÝ we study acoustics in great detail we discover +that much of the work is the same as it would be iŸ we were studying opties in +great detail. 5o the study of a phenomenon in one field may permit an extension +of our knowledge in another field. It is best to realize from the first that such +extensions are possible, for otherwise one might not understand the reason for +spending a great deal of time and energy on what appears to be only a small +part of mechanics. +The harmonic oscillator, which we are about to sbudy, has close analogs in +many other fields; although we star with a mechanical example oŸ a weight on a +spring, or a pendulum with a small swing, or certain other mechanical devices, we +are really studying a certain điƒƒeremtial cquation. This equation appears again +and again in physics and in other sciences, and in fact it is a part oŸ so many +phenomena that its close study is well worth our while. Some of the phenomena +involving this equation are the oscillations of a mass on a spring; the oscillations +of charge Ñowing back and forth in an electrical circuit; the vibrations oŸ a tuning +fork which is generating sound waves; the analogous vibrations of the electrons +in an atom, which generate light waves; the equations for the operation of a +servosystem, such as a thermostat trying to adjust a temperature; complicated +interactions in chemical reactions; the growth of a colony of bacteria in interaction +--- Trang 379 --- +with the food supply and the poisons the bacteria produece; foxes eating rabbits +eating grass, and so on; all these phenomena follow equations which are very +similar to one another, and this is the reason why we study the mechanical +oscillator in such detail. 'Phe equations are called znear djfƒerential cquations +tuïth constant coefficien#s. A linear diferential equation with constant coefficients +1s a diferential equation consisting of a sum of several terms, each term beïng a +derivative of the dependent variable with respect to the independent variable, +and multiplied by some constant. Thus +dụ đa (đP + aụ— d0 1a/dÉfCT + ccc + ai de dt + ag# = ƒ) — ð) +1s called a linear diferential equation of order ø with constant coefficients (each +đ¿ 1s constant). +21-2 The harmonic oscillator +Perhaps the simplest mechanical system whose motion follows a linear difer- +ential equation with constant coeflicients is a mass on a spring: frst the spring +stretches to balance the gravity; once it is balanced, we then discuss the vertical +displacement of the mass from its equilibrium position (Fig. 21-1). We shall +call this upward displacement z, and we shall also suppose that the spring 1s +perfectly linear, in which case the force pulling back when the spring is stretched +1s precisely proportional to the amount of stretch. "hat is, the force is —kø +(with a minus sign to remind us that it pulls back). Thus the mass times the +acceleration must equal —kz: +md°+/dt2 = —ka. (21.2) +L9 +Fig. 21-1. A mass on a spring: a simple example of a harmonic +oscillator. +--- Trang 380 --- +Eor simplicity, suppose it happens (or we change our unit of time measurement) +that the ratio k/n = 1. We shall fñrst study the equation +d°+/di? = —z. (21.3) +Later we shall come back to Bq. (21.2) with the & and rn explicitly present. +We have already analyzed Eq. (21.3) in detail numerically; when we first +introduced the subject of mechanics we solved this equation (see Eq. 9.12) to +fnd the motion. By numerical integration we found a curve (Eig. 9-4) which +showed that 1Í rm was initially displaced, but at rest, it would come down and go +through zero; we did not then follow it any farther, but of course we know that +1t just keepbs going up and down——It osc/lÏates. When we calculated the motion +numerically, we found that it went through the equilibrium poïnt at ý = 1.570. +The length of the whole cycle is four times this long, or #o = 6.28 “sec.” This +was found numerically, before we knew much calculus. We assume that in the +meantime the Mathematics Department has brought forth a function which, +when differentiated twice, is equal to itself with a minus sign. (There are, oŸ +course, ways of getting at this function in a direct fashion, but they are more +complicated than already knowing what the answer is.) The function is # = cosứ. +Tf we differentiate this we fnd đz/đt = — sin£ and d”z/đt? = — cost = —z. The +function # = cos£ starts, at ứ —= 0, with z = 1, and no initial velocity; that was +the situation with which we started when we did our numerical work. Now that +we know that # = cosứ, we can calculate a prec¿se value for the time at which it +should pass z = 0. The answer is ý = Z/2, or 1.57080. We were wrong in the lasb +figure because of the errors of numerical analysis, but it was very closel +Now to go further with the original problem, we restore the time units to real +seconds. What is the solution then? First ofall, we might think that we can get the +constants & and ?m in by multiplying cos ý by something. So let us try the equation +œ = Acosf; then we fnd dz/dt = — Asinf, and đ?z/d‡2 = —Acost = —z. Thus +we discover to our horror that we diỉd not succeed in solving Eq. (21.2), but we +gọt Eq. (21.3) again! That fact illustrates one of the most important properties +of linear diferential equations: ?ƒ e rmuliipl a solulion oƒ the equalion DỤ ang +constant, ft ís again œ solulion. The mathematical reason for this is clear. IÍ ø is +a solution, and we multiply both sides of the equation, say by 4, we see that all +derivatives are also multiplied by 4, and therefore 4z is just as good a solution +of the original equation as ø was. The physics of it is the following. If we have a +weight on a spring, and pull it down twice as far, the force is Ewice as much, the +--- Trang 381 --- +resulting acceleration is twice as great, the velocity it acquires in a given tỉme is +twice as preat, the distance covered in a given time is twice as great; but it has +to cover ÿwice as great a distanee in order to get back to the origin because 1 1s +pulled down twice as far. So 1% takes the sdrne tữne to get back to the origin, +irrespective of the initial displacement. In other words, with a linear equation, +the motion has the same f#ữne paitern, no matter how “strong” it is. +That was the wrong thing to do—it only taught us that we can multiply +the solution by anything, and it satisfes the same equation, but not a diferent +cquation. After a little cut and try to get to an equation with a diferent constant +multiplying z, we fnd that we must alter the scale of fzme. In other words, +Eq. (21.2) has a solution oŸ the form +% = COSUgÝ. (21.4) +(It is important to realize that in the present case, œo is not an angular velocity +of a spinning body, but we run out of letters if we are not allowed to use the same +letter for more than one thing.) The reason we put a subscript “0” on œ is that we +are going to have more omegas before long; let us remermber that œọ refers to the +natural motion of this oscillator. Now we try Eq. (21.4) and this time we are more +successful, because đ#/đf = —øg sin œo£ and d2z/đt? = —u cosuoŸ = —u§z. 8o +at last we have solved the equation that we really wanted to solve. 'The equation +d3z/dt? = —u§z is the same as Eq. (21.2) IŸ ø8 = k/m. +The next thing we must investigate is the physical signifcance oŸ œạ. We +know that the cosine function repeats itself when the angle it refers to is 2. So +% = cosug# will repeat its motion, ¡% will go through a complete cycle, when the +“angle” changes by 2z. The quantity œg is often called the phøse of the motion. +In order to change œgÝ by 27, the time must change by an amount íạ, called +the per7od of one complete oscillation; of course #o must be such that ¿go = 27. +That is, go must account for one cycle of the angle, and then everything will +repeat itself—If we Increase ý by #o, we add 2z to the phase. Thus +tọ = 2/œo = 2mm. (21.5) +Thus if we had a heavier mass, it would take longer to oscillate back and forth +on a spring. That is because it has more inertia, and so, while the forces are the +same, it takes longer to get the mass moving. Ôr, ïf the spring is stronger, it will +move more quickly, and that is right: the period is less if the spring is stronger. +Note that the period of oscillation of a mass on a spring does not depend +in any way on hou ¡it has been started, how far down we pull ít. The period +--- Trang 382 --- +1s determined, but the amplitude of the oscillation is no determined by the +cquation of motion (21.2). The amplitude 2s determined, in fact, by how we let +go of it, by what we call the zn#tial condiHions or starting conditions. +Actually, we have not quite found the most general possible solution of +Edq. (21.2). There are other solutions. It should be clear why: because all of the +cases covered by # = øcos /œg start with an initial displacement and no initial +velocity. But it is possible, for instance, for the mass to start at z = 0, and we +may then give it an impulsive kick, so that it has some speed at ¿ = 0. Such +a motion is not represented by a cosine——it is represented by a sine. 'o put 1§ +another way, iÝ — cosœg# 1s a solution, then is it no obvious that if we were +to happen to walk into the room at some từne (which we would call “¿ = 0”) +and saw the mass as it was passing z = 0, ¡it would keep on goïng just the same? +Therefore, ø = cosœo cannot be the most general solution; it must be possible +to shift the beginning of tỉme, so to speak. As an example, we could write the +solution this way: # = øœcosœg(# — tị), where íq is some constant. "This also +corresponds to shifting the origin of time to some new instant. Eurthermore, we +may expand +cos (œo# + A) = cosugf# eos Á — sin „g£ sỉn A, +and write +œ= Acosœg£ + Bsin œg#, +where 4 = øcos A and = —asin A. Any one of these forms is a possible way +to write the complete, general solution of (21.2): that is, every solution of the +differential equation đ?z/df? = —„ÿz that exists in the world can be written as +(a) % = acOS0g(È — #1), +(b) % = acos (0£ + A), (21.6) +(c) %= Acosoo£ + Bsìn uot. +Some of the quantities in (21.6) have names: œọ is called the angular [requencU; +it is the number of radians by which the phase changes in a second. “That 1s +determined by the diferential equation. The other constants are not determined +by the equation, but by how the motion is started. Of these constants, œ measures +the maximum displacement attained by the mass, and is called the ampiitude +--- Trang 383 --- +of oscillation. "The constant A is sometimes called the phase of the oscillation, +but that is a confusion, because other people call «¿o£ + A the phase, and say +the phase changes with time. We might say that A is a phase shúf† from some +defned zero. Let us put it diferently. Diferent A?s correspond to motions in +diferent phases. 'That ¡is true, but whether we want to call A £he phase, or not, +1s another question. +21-3 Harmonic motion and circular motion +The fact that cosines are involved in the solution of Eq. (21.2) suggests that +there might be some relationship to circles. 'This is artificial, of course, because +there is no circle acbually involved in the linear motion—i% just goes up and down. +W©e may point out that we have, in fact, already solved that diferential equation +when we were studying the mechanics of circular motion. lf a particle moves In +a circle with a constant speed 0, the radius vector from the center of the cirele +to the particle turns through an angle whose size is proportional to the time. lÝ +we call this angle Ø = œt/R (Fig. 21-2) then đØ/đf = œạ = 0/R.. We know that +there is an acceleration a = 02/ = uŸR toward the center. Now we also know +that the position z, at a given moment, is the radius of the cirele times cos ổ, +and that is the radius times sin 0: +z= Rcos0, ụ= Rsin0. +Now what about the acceleration? What is the z-component of acceleration, +d2z/dt?? We have already worked that out geometrically; it is the magnitude +of the acceleration times the cosine of the projection angle, with a minus sign +because it is toward the center. +đ„ = —acoS = —wg]#cos 0 = —u0%. (21.7) +Fig. 21-2. A particle moving ¡In a circular path at constant speed. +--- Trang 384 --- +In other words, when a particle is moving ín a cirele, the horizontal component of +10s motion has an acceleration which is proportional to the horizontal displacement +from the center. Of course we also have the solution for motion in a circle: +% = Rcosuot. Equation (21.7) does not depend upon the radius oŸ the circle, so +for a circle of any radius, one fnds the same equation for a given œọ. Thus, for +several reasons, we expect that the displacement of a mass on a spring will turn +out to be proportional to cosœg#, and will, in fact, be exactly the same motion +as we would see if we looked at the z-component of the position of an object +rotating in a circle with angular velocity œo. As a check on this, one can devise +an experiment to show that the up-and-down motion of a mass on a spring is the +same as that ofa poïnt goïing around in a cirele. In Eig. 21-3 an arc light projected +on a screen casts shadows of a crank pin on a shaft and of a vertically oscillating +mass, side by side. If we let go of the mass at the right time from the right place, +and ïf the shaft speed is carefully adjusted so that the frequencies match, each +should follow the other exactly. One can also check the numerical solution we +obtained earlier with the cosine function, and see whether that agrees very well. +Light 1 +Projector +Screen +Fig. 21-3. Demonstration of the equivalence between simple harmonIc +motion and uniform circular motion. +Here we may point out that because uniform motion in a cirele is so closeÌy +related mathematically to oscillatory up-and-down motion, we can analyze oscil- +latory motion in a simpler way if we imagine it to be a projection oŸ something +goïing in a circle. In other words, although the distance means nothing in the +oscillator problem, we may still artificially supplement Eq. (21.2) with another +--- Trang 385 --- +equation using , and put the two together. If we do this, we will be able to +analyze our one-dimensional oscillator with circular motions, which is a lot easier +than having to solve a diferential equation. The trick in doing this is to use +complex numbers, a procedure we shall introduce in the next chapter. +21-4 Initial conditions +NÑow let us consider what determines the constants A and ?Ö, or ø and A. Of +course these are determined by how we start the motion. If we start the motion +with just a small displacement, that is one type of oscillation; 1Ÿ we start with +an initial displacement and then push up when we let go, we get still a diferent +motion. The constants A and ?Ö, or a and A, or any other way of putting it, are +determined, of course, by the way the motion started, not by any other features +of the situation. Thhese are called the #miii@Ï conditions. We would like to connect +the initial conditions with the constants. Although this can be done using any +one of the forms (21.6), it turns out to be easiest if we use Eq. (21.6c). Suppose +that at ý —= 0 we have started with an initial displacement øo and a certain +velocity øọ. Thịs is the most general way we can sbart the motion. (We cannot +specify the acceleration with which it started, true, because that is determined +by the spring, once we speclfy #o.) Now let us calculate 4 and Ø. We start with +the equation for #, +z = Acosœg£ + Bsin œ0f. +Since we shall later need the velocity also, we differentiate z and obtain += —ứg Äsin œg£ + œ0. cos 0g. +'These expressions are valid for all ¿, but we have special knowledge about z and 0 +at £—=0. So 1ƒ we put ‡ = 0 into these equations, on the left we get #øo and 0o, +because that is what øz and 0 are at ý = 0; also, we know that the cosine oŸ zero +1s unity, and the sine of zero is zero. Therefore we get +#e=A-1+:0=A4A +0 — —œoA-0+œgB: 1 = g8. +So for this particular case we find that +A =zo, B = %o(ua. +trom these values of Á and Ö, we can get ø and A if we wish. +--- Trang 386 --- +'That is the end of our solution, but there is one physically Interesting thing +to check, and that is the conservation of energy. 5ince there are no frictional +losses, energy ought to be conserved. Let us use the formula += acos (0g# + A); +0 = —ưgøasin (@g£ + A). +Now let us ñnd out what the kinetic energy 7' is, and what the potential energy +is. The potential energy at any moment is skz”, where # is the displacement +and & is the constant of the spring. If we substitute for +, using our expression +above, we get +U = šk#? = $kaŸ cos” (œạt + A). +Of course the potential energy is not constant; the potential never becomes +negative, naturally——there is always some energy in the spring, but the amount +of energy Ñuctuates with z. The kinetic energy, on the other hand, is sinu, and +by substituting for 0 we get +T= ÿmwŸ = š mua“ sinŸ (wọt + A). +Now the kinetic energy is zero when zø is at the maximum, because then there +1s no velocity; on the other hand, it is maximal when z is passing through zero, +because then it is moving fastest. This variation of the kinetic energy is just +the opposite of that of the potential energy. But the total energy ought to be a +constant. IÝ we note that k = mi, we see that +T+U= smưufa”[cos” (œạt + A) + sin” (œạt + A)] = 3mafdŸ. +The energy is dependent on the square of the amplitude; 1ƒ we have twice the +amplitude, we get an oscillation which has four times the energy. The øuerøge +potential energy is half the maximum and, therefore, half the total, and the +average kinetic energy is likewise half the total energy. +21-5 Forced oscillations +Next we shall discuss the ƒorced harmonic oscdllator, i.e., one in which there +is an external driving force acting. The equation then is the following: +md2+z/df? = —kaz + F(). (21.8) +--- Trang 387 --- +We would like to fnd out what happens in these cirecumstances. The external +driving force can have various kinds of functional dependence on the time; the +first one that we shall analyze is very simple—we shall suppose that the force is +oscillating: +†{) = Focos úf. (21.9) +Notice, however, that this œ is not necessarily œạ: we have œ under our control; +the forcing may be done at diferent frequencies. So we try to solve Eq. (21.8) +with the special force (21.9). What is the solution of (21.8)? One special solution, +(we shall discuss the more general cases laber) is +% = Ccosưf, (21.10) +where the constant is to be determined. In other words, we might suppose that +1f we kept pushing back and forth, the mass would follow back and forth in step +with the force. We can try it anyway. So we put (21.10) and (21.9) into (21.8), +and get +— nu cosÈ = —muf cos w‡ + Fù cos 0F. (21.11) +We have also put in k = múa, so that we will understand the equation better at +the end. Now because the cosine appears everywhere, we can divide it out, and +that shows that (21.10) is, in fact, a solution, provided we pick Œ just right. The +answer is that Œ must be +Œ = Fụ/m(wạ — œ2). (21.12) +'That Is, mm oscillates at the same frequency as the force, but with an amplitude +which depends on the frequency of the force, and also upon the frequency of the +natural motion of the oscillator. It means, frst, that if œ is very small compared +with œọ, then the displacement and the force are in the same direction. Ôn the +other hand, if we shake it back and forth very fast, then (21.12) tells us that Ở is +negative iŸ œ is above the natural frequenecy œọ oŸ the harmonic oscillator. (We +will call œọ the natural frequency of the harmonic oscillator, and œ the applied +frequency.) At very hiph requency the denominator may become very large, and +there is then not much amplitude. +Of course the solution we have found is the solution only 1ƒ things are started +Just right, for otherwise there is a part which usually dies out after a while. This +other part is called the #rønsient response to Ƒ(£), while (21.10) and (21.12) are +called the s£eadu-state response. +--- Trang 388 --- +According to our formula (21.12), a very remarkable thing should also occur: +1Ý œ is almost exactly the same as œ, then Œ should approach infinity. So If we +adjust the Írequenecy of the force to be “in time” with the natural frequenecy, then +we should get an enormous displacement. 'This is well known to anybody who +has pushed a child on a swing. It does not work very well to elose our eyes and +push at a certain speed at random. lf we happen to get the right timing, then +the swing goes very high, but if we have the wrong timing, then sometimes we +may be pushing when we should be pulling, and so on, and it does not work. +Tf we make œ exactly equal to œọ, we fnd that ¡§ should oscillate at an +#nfinite amplitude, which is, of course, impossible. 'Phe reason it does not is that +something goes wrong with the equation, there are some other frictional terms, +and other forces, which are not in (21.S) but which occur in the real world. So +the amplitude does not reach infinity for some reason; it may be that the spring +breaksl +--- Trang 389 --- +Algeobr« +22-1 Addition and multiplication +In our study of oscillating systems we shall have occasion to use one of the +mmost remarkable, almost astounding, formulas in all of mathematics. EFrom the +physicist's point of view we could bring forth this formula in two minutes or +so, and be done with it. But science is as much for intellectual enjoyment as +for practical utility, so instead of just spending a few minutes on this amazing +jewel, we shall surround the jewel by its proper setting in the grand design of +that branch of mathematics which is called elementary algebra. +Now you may ask, “What is mathematics doïng in a physics lecbure?” We +have several possible excuses: first, of course, mathematics is an important tool, +but that would only excuse us for giving the formula in two minutes. Ơn the +other hand, in theoretical physics we discover that all our laws can be written in +mathematical form; and that this has a certain simplicity and beauty about it. +So, ultimately, in order to understand nature it may be necessary to have a deeper +understanding of mathematical relationships. But the real reason is that the +subject is enjoyable, and although we humans cut nature up in different ways, and +we have diferent courses in diferent departments, such compartmentalization 1s +really artifcial, and we should take our intellectual pleasures where we fnd them. +Another reason for looking more carefully at algebra now, even though most +of us studied algebra in high school, is that that was the first time we studied it; +all the equations were unfamiliar, and it was hard work, just as physics 1s now. +lvery so often it is a great pleasure to look back to see what territory has been +covered, and what the great map or plan of the whole thing is. Perhaps some day +somebody in the Mathematics Department will present a lecture on mechanics In +such a way as to show what it was we were trying to learn in the physics coursel +The subject of algebra will not be developed from the point of view of a +mathematician, exactly, because the mathematicians are mainly interested in how +various mathematical facts are demonstrated, and how many assumptions are +absolutely required, and what is not required. They are not so interested in the +--- Trang 390 --- +result oŸ what they prove. For example, we may fñnd the Pythagorean theorem +quite interesting, that the sum of the squares of the sides of a right triangle 1s +equal to the square of the hypotenuse; that is an interesting fact, a curiously +simple thing, which may be appreciated without discussing the question of how +to prove it, or what axioms are required. So, in the same spirit, we shall describe +qualitatively, if we may put it that way, the system of elementary algebra. We +say clementaru algebra because there is a branch of mathematics called rmodern +algebra in which some of the rules such as œb = ba, are abandoned, and ït ¡s still +called algebra, but we shall not discuss that. +To discuss this subJect we start in the middle. We suppose that we already +know what integers are, what zero is, and what it means to increase a number +by one unit. You may say, “That is not in the middlel” But it is the middle from +a mathematical standpoint, because we could go even further back and describe +the theory of sets in order to đerzue some of these properties of integers. But we +are not goïing in that direction, the direction of mathematical philosophy and +mathematical logic, but rather in the other direction, from the assumption that +we know what integers are and we know how to count. +Tf we start with a certain number ø, an integer, and we count successively +one unit b times, the number we arrive at we call ø + 0, and that defines øddiion +Of integers. +Once we have defned addition, then we can consider this: if we start with +nothing and add ø to it, b times in succession, we call the result rmultiplication oŸ +integers; we call it b tữmes a. +Now we can also have a swccession. o0 tmultiplicœtions: 1Ÿ we start with 1 and +multiply by ø, b tỉimes in succession, we call that raising to œ pouer: aP. +Now as a consequence of these definitions it can be easily shown that all of +the following relationships are true: +(a) œ+b=b+a (b) a+(b+c)=(a+Ù)+ec +(c) ab= ba (d) a(b+c)= ab+ ac +(e)_ (ab)c= a(bc) () (œb)“= a°° +be _ „(b+c) be —_ „(be) (22.1) +(g) a a°=ø (h) (a) =4 +(Œ) a+0=wø 0) a:l=a +(k) aøl=a +'These results are well known and we shall not belabor the point, we merely list +--- Trang 391 --- +them. Of course, 1 and 0 have special properties; for example, œ + 0 is ø, ø times +1= 4a, and ø to the frst power 1s ø. +In this discussion we must also assume a few other properties like continuity +and ordering, which are very hard to deñne; we will let the rigorous theory do it. +Purthermore, it is defnitely true that we have written down too many “rules”; +some of them may be deducible from the others, but we shall not worry about +such matters. +22-2 The inverse operations +In addition to the direct operations of addition, multiplication, and raising to +a power, we have also the Zøw0erse operations, which are defned as follows. Let us +assume that ø and é are given, and that we wish to fnd what values of b satisfy +such equations as ø-L = eœ, ab = c, b“ =c. Ifa+b = c, b1s defined as e— a, which +1s called subfraction. 'The operation called division is also clear: if ab = c, then +b = c/a defines division—a solution of the equation øb = e “backwards.” Now if +we have a power 0“ = cand we ask ourselves, “What is b?,” ït is called the ath roo‡ +of c: b= c. Eor instance, if we ask ourselves the following question, “What +Integer, raised to the third power, equals 8?,” then the answer is called the cube +root of 8; ït is 2. Because b“ and a are not equal, there are #wo inverse problems +associated with powers, and the other inverse problem would be, “To what power +must we raise 2 to get 8?” Thịis is called taking the logarithm. Tf a° = e, we write +b = log„c. The fact that it has a cumbersome notation relative to the others does +not mean that it is any less elementary, at least applied to integers, than the other +processes. Although logarithms come late in an algebra class, in practice they are, +Of course, just as simple as roots; they are just a diferent kind of solution of an +algebraic equation. The direct and inverse operations are summarized as follows: +(a) addition (a) subtraction +a+b=ec b=c-—-q +(b)_ multiplication (b) division +qb=ec b = c/a +22.2 +(c) power (c)_ root (22.2) +b“=ec b— ức +(d) power (d) logarithm +œ°=€ b =log„e +--- Trang 392 --- +Now here ¡is the idea. 'These relationships, or rules, are correct for integers, +since they follow from the definitions of addition, multiplication, and raising to a +power. IỨe are goïng ‡o điscuss tohether or noÈ tue can broaden the cÏass oƒ objects +thích a, Ð, and c represent‡ so that the uuilÏ obeu these sœme rules, although the +processes for ø + ð, and so on, will not be defnable in terms of the direct action +of adding 1, for instance, or successive multiplications by integers. +22-3 Abstraction and generalization +'When we try to solve simple algebraic equations using all these defnitions, +we soon discover some insoluble problems, such as the following. Suppose that +we try to solve the equation ö = 3— 5. That means, according to our def- +inition of subtraction, that we must fnd a number which, when added to 5ð, +gives 3. And of course there 2s no such number, because we consider only +positive Integers; this is an insoluble problem. However, the plan, the great +idea, 1s this: œbsfraclon and generalzalion. From the whole structure of al- +gebra, rules plus integers, we abstract the original defñnitions of addition and +multiplication, bu we leave the rules (22.1) and (22.2), and assume these to +be true ?w general on a wider class oŸ numbers, even though they are originally +derived on a smaller class. “Thus, rather than using integers symbolically to +defñne the rules, we use the rules as the defñnition of the symbols, which then +represent a more general kind of number. As an example, by working with +the rules alone we can show that 3 — 5 =0-—2. In facÿ we can show that +one can make øil subtractions, provided we defne a whole set of new num- +bers: 0— 1,0 —2,0—3,0— 4, and so on, called the megøf2ue ?mtegers. Then +we may use all the other rules, like ø(b + e) = øb + ac and so forth, to ñnd +what the rules are for multiplying negative numbers, and we will discover, in +fact, that all of the rules can be maintained with negative as well as positive +1ntegers. +So we have increased the range of objects over which the rules work, but the +mmeaning of the symbols is difÑerent. +One cannot say, for instance, that —2 times 5 really means to add ð together +successively —2 times. hat means nothing. But nevertheless everything will +work out all right according to the rules. +An interesting problem comes up in taking powers. Suppose that we wish +to discover what a(3~5) means. We know only that 3 — 5 is a solution of the +problem, (3 — 5) +5 = 3. Knowing that, we know that a(3~5)að = a3. Therefore +--- Trang 393 --- +a(—~5) = a3/a5, by the definition of division. With a little more work, this can +be reduced to 1/a2. So we find that the negative powers are the reciprocals of +the positive powers, but 1/42 is a meaningless symbol, because if ø is a positive +or negative integer, the square of it is greater than 1, and we do not yet know +what we mean by 1 divided by a number greater than 1l +Onwardl The great plan is to continue the process of generalization; whenever +we fnd another problem that we cannot solve we extend our realm of numbers. +Consider division: we cannot find a number which is an integer, even a negative +integer, which is equal to the result of dividing 3 by 5. But if we suppose that all +tractional numbers also satisfy the rules, then we can talk about multiplying and +adding fractions, and everything works as well as it did before. +Take another example of powers: what is a3/5? We know only that (3/5)5 = 3, +since that was the defnition of 3/5. So we know also that (a(3/5))5 = ạ(3/5)(5) = +a3, because this is one of the rules. Then by the defnition of roots we fñnd that +a\3/5) — a3, +In this way, then, we can delñne what we mean by putting fractions in the +various symbols, by using the rules themselves to help us determine the defnition—— +1t is not arbitrary. It is a remarkable fact that all the rules still work for positive +and negative integers, as well as for fractionsl +We go on in the process of generalization. Are there any other equations +we cannot solve? Yes, there are. EFor example, it is impossible to solve this +cquation: b= 21⁄2 = v2. It is impossible to ñnd a number which is rational (a +fraction) whose square is equal to 2. Ib is very easy Íor us in modern days to +answer this question. We know the decimal system, and so we have no difliculty +in appreciating the meaning of an unending decimal as a type of approximation +to the square root of 2. Historically, this idea presented great dificulty to the +Greeks. To really delne ørecisel what is meant here requires that we add some +substance of continuity and ordering, and it is, in fact, quite the most dificult +step In the processes of generalization Just at this point. It was made, formally +and rigorously, by Dedekind. However, without worrying about the mathematical +rigor of the thing, it is quite easy to understand that what we mean is that we are +going to find a whole sequence of approximate fractions, perfect fractions (because +any decimal, when stopped somewhere, is oŸ course rational), which Just keeps +on going, getting closer and closer to the desired result. That is good enough +for what we wish to discuss, and it permits us to involve ourselves in irrational +numbers, and to calculate things like the square root of 2 to any accuracy that +we desire, with enough work. +--- Trang 394 --- +22-4 Approximating irrational numbers +The next problem comes with what happens with the irrational powers. +Suppose that we want to defne, for instance, 10Y2. In principle, the answer 1s +simple enough. lIÝ we approximate the square root of 2 to a certain number of +decimal places, then the power is rational, and we can take the approximate root, +using the above method, and get an øpprozimation to 10Y2. Then we may run it +up a few more decimal places (it is again rational), take the appropriate root, +this time a much higher root because there is a much bigger denominator in the +fraction, and get a better approximation. OÝ course we are going to geÈ some +enormously high roots involved here, and the work is quite difcult. How can we +cope with this problem? +In the computations of square roots, cube roots, and other small roots, there +1s an arithmetical process available by which we can get one decimal place after +another. But the amount of labor needed to calculate irrational powers and +the logarithms that go with them (the inverse problem) is so great that there +1s no simple arithmetical process we can use. Therefore tables have been built +up which permit us to calculate these powers, and these are called the tables +of logarithms, or the tables of powers, depending on which way the table 1s set +up. Ít is merely a question of saving time; iŸ we must raise some number to an +Irrational power, we can look it up rather than having to compute it. Of course, +such a computation is Just a technical problem, but it is an interesting one, and +of great historical value. In the first place, not only do we have the problem of +solving # = 10Y2, but we also have the problem of solving 10 = 2, or # = logig 2. +This is not a problem where we have to defne a new kind of number for the +result, it is merely a computational problem. The answer is simply an irrational +number, an unending decimal, not a new kind of a number. +Let us now discuss the problem oŸ calculating solutions of such equations. +The general idea is really very simple. If we could caleulate 101, and 10, and +101/10 and 10/1900 and so on, and multiply them all together, we would get +10114: or 10Y2, and that is the general idea on which things work. But instead +of calculating 10119 and so on, we shall caleulate 101/2, 101/4, and so on. Before +we start, we should explain why we make so mụch work with 10, instead of some +other number. Of course, we realize that logarithm tables are of great practical +utility, quite aside from the mathematical problem of taking roots, since with +any base at all, +logg(ac) = logy ø + logy e. (22.3) +--- Trang 395 --- +W© are all familiar with the fact that one can use this fact in a practical way to +multiply numbers iŸ we have a table of logarithms. The only question is, with +what base ö shall we compute? It makes no diference what base is used; we +can use the same principle all the time, and iŸ we are using logarithms to any +particular base, we can find logarithms to any other base merely by a change +in scale, a multiplying factor. IÝ we multiply Eq. (22.3) by 61, ¡it is Just as true, +and ïif we had a table of logs with a base ö, and somebody else multiplied all of +our table by 61, there would be no essential diference. Suppose that we know +the logarithms of all the numbers to the base b. In other words, we can solve +the equation b# = c for any c because we have a table. 'Phe problem is to ñnd +the logarithm of the same number c to some other base, let us say the base #. +We would like to solve #° = e. It is easy to do, because we can always write +z = bÝ, which delnes , knowing z and b. As a matter of fact, £ = log,ø. Then +if we put that in and solve for a”, we see that (bf)%“ = b* = e. In other words, +ta! is the logarithm of ein base b. Thus ø' = ø/£. Thus logs to base # are just +1/f, which is a constant, tìmes the logs to the base, ð. Therefore any log table is +equivalent to any other log table iŸ we multiply by a constant, and the constant +is 1/log,ø. This permits us to choose a particular base, and for convenience we +take the base 10. (The question may arise as to whether there is any natural +base, any base in which things are somehow simpler, and we shall try to fnd an +answer to that later. At the moment we shall just use the base 10.) +Now let us see how to calculate logarithms. We begin by computing successive +square roots of 10, by cut and try. The results are shown in Table 22-1. The +powers of 10 are given in the first column, and the result, 10, is given in the +third column. Thus 10! = 10. The one-half power of 10 we can easily work out, +because that is the square root of 10, and there is a known, simple process for +taking square roots of any number.* Ủsing this process, we find the first square +root to be 3.16228. What good is that? It already tells us something, it tells +us how to take 1005, so we now know at least one logarithm, if we happen to +need the logarithm of 3.16228, we know the answer is close to 0.50000. But we +must do a little bit bet6er than that; we clearly need more information. 5o we +take the square root again, and find 101/4, which is 1.77828. Now we have the +logarithm of more numbers than we had before, 1.250 is the logarithm of 17.78 +* 'TThere is a definite arithmetic procedure, but the easiest way to fnd the square root oŸ any +number X is to choose some ø fairly close, find N/a, average q = sia + (N/a)], and use this +avcrage a” for the next choice for ø. The convergence is very rapid—the number of significant +figures doubles each time. +--- Trang 396 --- +Table 22-1 +Successive Square Roots of Ten +1 1024 10.00000 9.00 +1/2 512 3.16228 4.32 +1/4 256 1.77828 3.113 +1/8 128 1.33352 2.668 +1/16 64 1.15478 2.476 +1/32 32 1.074607 2.3874 +1/64 16 1.036633 2.3445 +1/128 8 1.018152 2.3234? +1/256 4 1.0090350 2.3130194 +1/512 2 1.0045073 2.3077 °3 +1/1024 1 1.0022511 2.3051 2° +A/1024 A 1 +0.0022486A 2.3025 +(A => 0) +and, incidentally, if it happens that somebody asks for 105, we can get it, +because that is 10(0-5+0:25): ït js therefore the produet of the second and third +numbers. lÝ we can get enough numbers in column s to be able to make up +almost any number, then by multiplying the proper things in column 3, we can +get 10 to any power; that is the plan. So we evaluate ten successive square roots +of 10, and that is the main work which is involved in the calculations. +'Why don”t we keep on going for more and more accuracy? Because we begin +to notice something. When we raise 10 to a very small power, we get 1 plus +a small amount. “The reason for this is clear, because we are going to have to +take the 1000th power of 101/190 to get back to 10, so we had better not sbart +with too big a number; it has to be close to 1. What we notice is that the small +numbers that are added to 1 begin to look as though we are merely dividing +by 2 cach time; we see 1815 becomes 903, then 450, 225; so it is clear that, to an +excellent approximation, if we take another root, we shall get 1.00112 something, +and rather than actually ¿øke all the square roots, we øwess at the ultimate +limit. When we take a small fraction A/1024 as A approaches zero, what will +the answer be? Of course it will be some number close to 1 + 0.0022511A. Not +--- Trang 397 --- +exactly 1 + 0.0022511A, however—we can get a better value by the following +tríck: we subtract the 1, and then divide by the power s. This ought to correc +all the excesses to the same value. We see that they are very closely equal. Ät +the top of the table they are not equal, but as they come down, they get cÌoser +and closer to a constant value. What is the value? Again we look to see how the +Series is going, how it has changed with s. It changed by 211, by 104, by 53, by +26. These changes are obviously half of each other, very closely, as we go down. +'Therefore, if we kept going, the changes would be 13, 7, 3, 2 and 1, more or less, +or a total of 26. 'Phus we have only 26 more to go, and so we fñnd that the true +number is 2.3025. (Actually, we shall later see that the ezøc£ number should +be 2.3026, but to keep it realistic, we shall not alter anything in the arithmetic.) +trom this table we can now calculate any power of 10, by compounding the power +out of 1024ths. +Let us now actually calculate a logarithm, because the process we sha]l use is +where logarithm tables actually come from. The procedure is shown in Table 22-2, +and the numerical values are shown in Table 22-1 (columns 2 and 3). +Table 22-2 +Calculation of a logarithm: log 2 +2~ 1.77828 = 1.124682 +1.124682 ~ 1.074607 = 1.046598, etc. +-2 = (1.77828)(1.074607)(1.036633)(1.0090350)(1.000573) +Ị 308.254 +S73 +— 1030103 = += 10 (mñ 0254) +.l0g+o 2 = 0.30103 +Suppose we want the logarithm of 2. That is, we want to know to what power +we Imust raise 10 to get 2. Can we raise 10 to the 1/2 power? No; that is too bịg. +In other words, we can see that the answer is goïng to be bigger than 1/4, and +less than 1/2. Let us take the factor 101/4 out; we divide 2 by 1.778..., and get +1.124..., and so on, and now we know that we have taken away 0.250000 from +the logarithm. The number 1.124..., is now the number whose logarithm we +necd. When we are finished we shall add back the 1/4, or 256/1024. Ñow we +--- Trang 398 --- +look in the table for the next number just below 1.124..., and that is 1.074607. +We© therefore divide by 1.074607 and get 1.046598. Erom that we discover that 2 +can be made up of a product of numbers that are in Table 22-1, as follows: +2 = (1.77828)(1.074607) (1.036633)(1.0090350)(1.000573). +There was one factor (1.000573) left over, naturally, which is beyond the range of +our table. To get the logarithm of this factor, we use our result that 10/1924 œ +1+ 2.3025A/1024. We fnd A = 0.254. Therefore our answer is 10 to the +following power: (256 + 32 + 16 + 4+ 0.254)/1024. Adding those together, we geb +308.254/1024. Dividing, we get 0.30108, so we know that the logo 2 = 0.30103, +which happens to be right to 5 ñguresl +This is how logarithms were originally computed by Mr. Briggs of Halifax, +in 1620. He said, “[ computed successively 54 square roots of 10” We know he +really computed only the first 27, because the rest of them can be obtained by this +trick with A. His work involved calculating the square root of 10 twenty-seven +times, which is not mụuch more than the ten times we did; however, it was more +work because he calculated to sixteen decimal places, and then reduced his answer +to fourteen when he published it, so that there were no rounding errors. He +made tables of logarithms to fourteen decimal places by this method, which 1s +quite tedious. But all logarithm tables for three hundred years were borrowed +trom Mr. Briggs' tables by reducing the number of decimal places. Only in +modern times, with the WPA and computing machines, have new tables been +independently computed. 'Phere are much more efficient methods of computing +logarithms today, using certain series expansions. +In the above process, we discovered something rather interesting, and that +1s that for very small powers e we can calculate 10“ easily; we have discovered +that 10° = 1 + 2.3025, by sheer numerical analysis. Of course this also means +that 10923925 — ] + n iŸ n is very small. Now logarithms to any other base +are merely multiples of logarithms to the base 10. The base 10 was used only +because we have 10 fngers, and the arithmetic of it is easy, but if we ask for a +mathematically natural base, one that has nothing to do with the number of +ñngers on human beings, we might try to change our scale of logarithms in some +convenient and natural manner, and the method which people have chosen 1s +to redefne the logarithms by multiplying all the logarithms to the base 10 by +2.3025... This then corresponds to using some other base, and this ¡is called the +nakural base, or base e. Note that log„(1 + n) 3m, or eƒ” + as n — 0. +--- Trang 399 --- +It is easy enough to ñnd out what e is: e = 101/23925 or 109434294. an +Irrational power. Our table of the successive square roots of 10 can be used +to compute, not just logarithms, but also 10 to any power, so let us use it tO +calculate this natural base e. Eor convenience we transform 0.434294... into +444.73/1024. Now, 444.73 is 256 + 128 + 32+ 16+ 8+ 4+ 0.73. Therefore e, +since it is an exponent of a sum, will be a product of the numbers +(1.77828X1.33352(1.0746071.036633(1.018152)1.009035X1.001643) = 2.7184. +(The only problem is the last one, which is 0.73, and which is not in the table, +but we know that if A is small enough, the answer is 1 + 2.3025 A.) When we +multiply all these togebher, we get 2.7184 (it should be 2.7183, but it is good +enough). The use of such tables, then, is the way in which irrational powers and +the logarithms of irrational numbers are all calculated. 'Phat takes care of the +irrationals. +22-5 Complex numbers +Now it turns out that after all that work we s2 cannot solve every equationl +Eor instance, what is the square root of —1? Suppose we have to fnd z2 = —1. +'The square of no rational, of no irrational, of nothøng that we have discovered so +far, is equal to —1. 5o we again have to generalize our numbers to a still wider +class. Let us suppose that a speeific solution of z2 = —1 is called something, we +shall call it ¿; ¿ has the property, by defnition, that is square is —1. Thhat is +about all we are going to say about it; oŸ course, there is more than one root +of the equation #? = —1. Someone could write ¡, but another could say, “No, +l prefer —¿. My ¿ is minus your 2.” lt is jus as good a solution, and since the +only defnition that ¿ has is that 72 = —1, it must be true that any equation we +can write is equally true if the sign of ¿ is changed everywhere. This ¡is called +taking the cormplez conƒugate. Ñow we are goïng to make up numbers by adding +successive 7's, and multiplying ?'s by numbers, and adding other numbers, and +So on, according to all of our rules. In this way we fnd that numbers can all be +written in the form ø-+ ?g, where ø and g are what we call real numbers, i.e., the +numbers we have been defning up until now. The number ¿ is called the n¿£ +#maginar number. Any real multiple of ¿ is called pure ïmaginaru. The most +general number, ø, is of the form ø + ;q and is called a complez number. 'Things +do not get any worse IÍ, for instance, we multiply two such numbers, let us say +--- Trang 400 --- +(r-+2s)(p+ 4). Then, using the rules, we get +(+ is) + 14) = rp + rũ) + (4s)p + (15)(44) += rp + i(rq) + (sp) + (1/)(s9) += p~ sq) + lírg + sp), (22.4) +sỉnce ở = ¡2 = —1. Therefore all the numbers that now belong in the rules (22.1) +have this mathematical form. +NÑow you say, “This can go on foreverl We have defined powers of imaginaries +and all the rest, and when we are all fñnished, somebody else will come along with +another equation which cannot be solved, like øŠ + 3z2 = —2. Then we have to +generalize all over again!” But it turns out that œU#th thás one more inueniion, just +the square root of —1, cuer algebraic cquation can be solued! 'This 1s a fantastic +fact, which we must leave to the Mathematics Department to prove. The proofs +are very beautiful and very interesting, but certainly not self-evident. In fact, +the most obvious supposition is that we are goïing to have to invent again and +again and again. But the greatest miracle of all is that we do not. 'Phis is the +last invention. After this invention of complex numbers, we fnd that the rules +still work with complex numbers, and we are fñnished inventing new things. We +can fnd the complex power of any complex number, we can solve any equation +that is written algebraically, in terms of a ñnite number of those symbols. We +do not fnd any new numbers. 'Phe square root oŸ ¿, for instance, has a definite +result, it is not something new; and ?' is something. We will điscuss that now. +W© have already discussed multiplication, and addition is also easy; if we add +©wo cormplex numbers, (p + 7g) + (r + 7s), the answer is (p + r) + ¿(q + s). Now +we can add and multiply complex numbers. But the real problem, of course, 1s +to compute cơomplÌez pouers oƑ complez numnbers. It turns out that the problem +1s actually no more dificult than computing complex powers of real numbers. So +let us concentrate now on the problem of calculating 10 to a complex power, not +just an irrational power, but 10†?%), Of course, we must at all tỉmes use our +rules (22.1) and (22.2). Thus +10ŒT7%) = 10710!%, (22.5) +But 10” we already know how to compute, and we can always multiply anything +by anything else; therefore the problem is to compute only 107%. Let us call it +some complex number, # + 2. Problem: given s, ñnd z, ñnd . Now ïf +108 =z +, +--- Trang 401 --- +then the complex conjugate of this equation must also be true, so that +10”? =„— 1g. +(Thus we see that we can deduce a number oŸ things without actually computing +anything, by using our rules.) We deduce another interesting thing by multiplying +these together: +108108 = 10 =1= (+ i9)( — iu) = z? + Ÿ. (22.6) +'Thus if we fnd z, we have + also. +Now the problem is ho to compute 10 to an imaginary power. What guide +1s there? We may work over our rules until we can go no further, but here is a +reasonable guide: if we can compute it for any particular s, we can get it for all +the rest. IÝ we know 10”° for any one s and then we want it for twice that s, we +can square the number, and so on. But how can we fnd 10/5 for even one special +value of øs? 'To do so we shall make one additional assumption, which is not quite +in the category of all the other rules, but which leads to reasonable results and +permits us to make progress: when the power is small, we shall suppose that the +“law” 10° = 1+ 2.3025c is right, as c gets very small, not only for real c, bu for +comjplex as uell. Therefore, we begin with the supposition that this law is true +in general, and that tells us that 10° = 1 + 2.3095 - is, for s —> 0. So we assume +that 1Í s is very small, say one part in 1024, we have a rather good approximation +to 107%. +Now we make a table by which we can compute ø/! the Imaginary DOW©rS +of 10, that is, compute + and . It ¡is done as follows. "The first power we start +with is the 1/1024 power, which we presume is very nearly 1 + 2.3025//1024. +'Thus we start with +107/192 — 1.00000 -+ 0.0022486¿, (22.7) +and ïfƒ we keep multiplying the number by itself, we can get to a higher imaginary +power. In fact, we may just reverse the procedure we used in making our logarithm +table, and calculate the square, 4th power, 8th power, etc., oŸ (22.7), and thus +buïld up the values shown in Table 22-3. We notice an interesting thing, that +the ø numbers are positive at frst, but then swing negative. We shall look into +that a little bit more in a moment. But first we may be curious to fnd for what +number s the real part of 10/5 is zero. The -value would be 1, and so we would +have 103 = 1¿, or js = logjg7. As an example of how to use this table, just as +we calculated logs 2 before, let us now use Table 22-3 to fnd log+g¿. +--- Trang 402 --- +Table 22-3 +Successive Squares of 10/1024 — 1 - 0.0022486¿ +z/1024 1 1.00000 + 0.00225¿* +2/512 2 1.00000 + 0.00450¿ +¿/256 4 0.99996 + 0.00900; +z/128 8 0.99984 + 0.01800; +¿/64 16 0.999386 + 0.03599; +7/32 32 0.99742 + 0.07193¿ +z/16 64 0.98967 + 0.14349/ +7/8 128 0.95885 + 0.28402; +¡/4 256 0.83872 + 0.54467: +¡z/2 512 0.40679 + 0.91365: +z/1 1024 | —0.66928 + 0.74332¡ +* Should be 0.0022486; +Which of the numbers in Table 22-3 do we have to multiply together to +get a pure imaginary result? After a little trial and error, we discover that to +reduce z the most, it is best to multiply “512” by “128” 'This gives 0.13056 + +0.99159/. "Then we discover that we should multiply this by a number whose +imaginary part is about equal to the size of the real part we are trying to remove. +Thus we choose “64” whose 2-value is 0.14349, since that is closest to 0.13056. +This then gives —0.01308 + 1.00008/. Now we have overshot, and must đ¿uide +by 0.99996 + 0.009007. How do we do that? By changing the sign of ? and +multiplying by 0.99996 — 0.00900/ (which works if z2 + 2 = 1). Continuing in +this way, we fnd that the entire power to which 10 must be raised to glve ¿ 1s +(512 + 128 + 64 — 4— 2+ 0.20)/1024, or 698.20//1024. Tf we raise 10 to that +power, we can get ¿. Therefore logo ¿ = 0.68184:. +22-6 Imaginary exponents +To further investigate the subject of taking complex imaginary powers, let +us look at the powers of 10 taking swccess¿ue pouers, not doubling the power +each time, im order to follow Table 22-3 further and to see what happens to those +mỉnus signs. This is shown in Table 22-4, in which we take 10”, and just keep +--- Trang 403 --- +Table 22-4 +Successive Powers of 107⁄8 +0 1.00000 + 0.00000¿ +1 0.95882 + 0.28402¿ +2 0.83867 + 0.54465: +3 0.64944 + 0.76042/ +4 0.40672 + 0.91356: +b) 0.13050 + 0.99146 +6 —0.15647 + 0.9877 +t —0.43055 + 0.90260% +8 —0.66917 + 0.74315 +9 —0.85268 + 0.52249; +10 —0.96596 + 0.25880; +11 —0.99969 — 0.02620¿ +12 —0.95104 — 0.30905 +14 —0.62928 — 0.7771 7 +16 —0.10447 — 0.99453¿ +18 +0.45454 — 0.89098¿ +20 +0.86648 — 0.49967/ +22 +0.99884 + 0.05287/ +24 +0.80890 + 0.58836/ +multiplying it. We see that + decreases, passes through zero, swings aÌlmost to —1 +(ïf we could get in between ø = 10 and p = I1 it would obviously swing to —T), +and swings back. 'Phe -value is going back and forth too. +In Eig. 22-1 the dots represent the numbers that appear in Table 22-4, and +the lines are Just drawn to help you visually. So we see that the numbers ø and +oscillate; 107% repea#s ifself, ït is a periodie thing, and as such, it is easy enough +to explain, because 1Í a certain power is ¿, then the fourth power of that would +be 72 squared. It would be +1 again, and therefore, since 100:588 ¡s equal to ¡, by +taking the fourth power we diseover that 10272? is equal to +1. Therefore, if we +wanted 103%, for instance, we could write it as 1027?! times 10:28, In other +words, it has a period, it repeats. Of course, we recognize what the curves look +liket They look like the sine and cosine, and we shall call them, for a while, the +algebraic sine and algebraic cosine. However, instead of using the base 10, we +--- Trang 404 --- +" 10® =x + íy +MÀ 15 20 25 /30 +Figure 22-1 +shall put them into our natural base, which only changes the horizontal scale; +so we denote 2.3025s by , and write 107% = e#, where # is a real number. NÑow +cï = ø-+iụ, and we shall write this as the algebraie cosine of plus 2 tỉimes the +algebraic sine of ý. Thus +©“ = cosf + isin f. (22.8) +What are the properties of cosf and sin? Eirst, we know, for instance, that +#2 + 2 must be 1; we have proved that before, and it is just as true for base e +as for base 10. 'Therefore cos2f + sin2£ = 1. We also know that, for small +t, e# = 1+ it, and therefore cos is nearly 1, and sin is nearly , and so it +goes, that øÏÏ oƒ the 0uarious propertlics oƒ these remarkable ƒunctions, which +come from taking imaginary powers, ør© the same œs the sine and costne oj +trigonometrg. +ls the period the same? Let us fnd out. e to what power is equal to 2? What +1s the logarithm of ¿ to the base c? We worked ¡% out before, in the base 10 +it was 0.68184/, but when we change our logarithmic scale to e, we have to +multiply by 2.3025, and if we do that it comes out 1.570. 5o this will be called +“algebraic z/2” But, we see, ¡9 differs from the regular z/2 by only one place +in the last point, and that, of course, is the result of errors in our arithmetiel +So we have created two new functions in a purely algebraic manner, the cosine +and the sine, which belong to algebra, and only to algebra. We wake up at the +end to discover the very functions that are natural to geometry. 5o there is a +connection, ultimately, between algebra and geometry. +We summarize with this, the most remarkable formula in mathematics: +c'? = cosØ + ¿ sỉn 6. (22.9) +'This is our jewel. +--- Trang 405 --- +W©e may relate the geometry to the algebra by representing complex numbers +in a plane; the horizontal position of a point is ø, the vertical position of a point +1s (Eig. 22-2). We represent every complex number, # + 2. Then ïf the radial +distance to this poïint is called z and the angle is called Ø, the algebraic law is that +øœ + # is written in the form re”, where the geometrical relationships between +z, , r, and Ø are as shown. This, then, is the unifñcation of algebra and geometry. +Fig. 22-2. x + iy = re. +'When we began this chapter, armed only with the basic notions oŸ integers +and counting, we had little idea of the power of the processes of abstraction +and generalization. sing the set of algebraic “laws,” or properties of numbers, +q. (22.1), and the definitions of inverse operations (22.2), we have been able +here, ourselves, to manufacture not only numbers but useful things like tables of +logarithms, powers, and trigonometric functions (for these are what the Imaginary +powers of real numbers are), all merely by extracting ten successive square roos +of tenl +--- Trang 406 --- +Tồosortdrree© +23-1 Complex numbers and harmonic motion +In the present chapter we shall continue our discussion of the harmonic +oscillator and, in particular, the forced harmonic oscillator, using a new technique +in the analysis. In the preceding chapter we introduced the idea of complex +numbers, which have real and imaginary parts and which can be represented +on a diagram in which the ordinate represents the imaginary part and the +abscissa represents the real part. lÝ ø is a complex number, we may write it as +gœ = đ; + ?d¿, where the subscript rz means the real part of ø, and the subscript +means the imaginary part of ø. Referring to Fig. 23-1, we see that we may +also write a complex number ø = # +? in the form z + i = re”, where +r2 = z2 + 9Ÿ = (# + i0)(+ — iụ) = aa*. (The complex conjugate of a, written +đ*, is obtained by reversing the sign of ? in a.) So we shall represent a complex +number in either of two forms, a real plus an imaginary part, or a magnitude z and +a phase angle Ø, so-called. Given z and Ø, z and are clearly r cos Ø and r sin 8 +and, in reverse, given a complex number # -Ƒ 2, = v⁄/#2 + 2 and tan 0 = 0/z, +the ratio of the imaginary to the real part. +IMAGINARY +x REAL AXIS +Fig. 23-1. A complex number may be represented by a point in the +“complex plane.” +--- Trang 407 --- +W© are goïng to apply complex numbers to our analysis of physical phenomena +by the following trick. We have examples of things that oscillate; the oscillation +may have a driving force which is a certain constant times cos œý. Now such a force, +E = Fpocosut, can be written as the real part of a complex number #' = Fpe”“t +because e?“ — cosuf + ?sin ý. The reason we do this is that it is easier to work +with an exponential function than with a cosine. So the whole trick is to represent +our oscillatory functions as the real parts of certain complex functions. “The +complex number #' that we have so defned is not a real physical force, because +no force in physies is really complex; actual forces have no imaginary part, only +a real part. We shall, however, speak of the “force” Fpe”“t, but of course the +actual force 1s the real par‡ of that expression. +Let us take another example. Suppose we want to represent a force which +is a cosine wave that is out of phase with a delayed phase A. 'This, of course, +would be the real part of Fuef=^), but exponentials being what they are, we +may wribe e/@~Ä) = e?“fe—/A, 'Thus we see that the algebra of exponentials is +much easier than that of sines and cosines; this is the reason we choose to use +complex numbers. We shall often write +E= Fục lêct = ft, (23.1) +We write a little caret (2) over the #' to remind ourselves that this quantity is a +complex number: here the number 1s +là — Fạc ?S, +Now let us solve an equation, using complex numbers, to see whether we can +work out a problem for some real case. For example, let us try to solve +da + kử = T = ro COS (UẺ, (23.2) +d2 ?m 1m 1n +where #! is the force which drives the oscillator and z is the displacement. Now, +absurd though it may seem, let us suppose that z and #! are actually complex +numbers, for a mathematical purpose only. That is to say, ø has a real part and +an Imaginary part times ?, and ?#! has a real part and an imaginary part times ¿. +Now ïf we had a solution of (23.2) with complex numbers, and substituted the +complex numbers in the equation, we would get +———— + —————~—=_————- +đị2 m m +--- Trang 408 --- +Ti nn. na _ đu. HỘ +d2 m d2 m m— 1n +Now, since if two complex numbers are equal, their real parts must be equal and +their imaginary parts must be equal, we deduce that #he redl part oƑ + satlisfies +the cquation tuïth the real part oƒ the forcc. We must emphasize, however, that +this separation into a real part and an imaginary part is not valid in general, +but is valid only for equations which are znear, that is, for equations in which + +appears in every term only in the frst power or the zeroth power. Eor instance, +if there were in the equation a term À#2, then when we substitute #„ -L 7;, we +would get A(z„ + ¿z;)2, but when separated into real and imaginary parts this +would yield A(#2 — z?) as the real part and 22Aø„ø; as the imaginary part. So we +see that the real part of the equation would not involve just A#2, but also —Àz‡. +In this case we get a diferent equation than the one we wanted to solve, with z¿, +the completely artificial thing we introduced in our analysis, mixed in. +Let us now try our new method for the problem of the forced oscillator, that +we already know how to solve. We want to solve Đq. (23.2) as before, but we say +that we are going to try to solve +d2x„ kxz Êc*“t +PP + mm (23.3) +where “is a complex number. Of course # will also be complex, but remember +the rule: take the real part to fnd out what is really going on. So we try %O +solve (23.3) for the forced solution; we shall discuss other solutions later. The +forced solution has the same frequency as the applied force, and has some +amplitude of oscillation and some phase, and so i§ can be represented also +by some complex number ê whose magnitude represents the swing of z and +whose phase represents the time delay in the same way as for the force. Now +a wonderful feature of an exponential function is that d(©c”2®)/dt = iuâc!*t, +'When we diferentiate an exponential function, we bring down the exponenft as +a simple multiplier. 'Phe second derivative does the same thing, it brings down +another ?œ, and so It is very simple to write Immediately, by inspection, what the +equation is for Ê: every time we see a diferentiation, we simply multiply by 2œ. +(Differentiation is now as easy as multiplication! 'This idea oŸ using exponentials +in linear diferential equations is almost as great as the invention of logarithms, +--- Trang 409 --- +in which multiplication is replaced by addition. Here diferentiation is replaced +by multiplication.) Thus our equation becomes +()22 + (kê/m) = Ê/m. (23.4) +(We have cancelled the common factor e”“f,) See how simple it is! Diferential +equations are immediately converted, by sight, into mere algebraic equations; we +virtually have the solution by sight, that +4= —_— ` — +(k/m) — 0°) +since ()” = —w#. This maybe slightly simplifed by substituting k/m = u, +which gives +â = ÊJ/m(uậ — œ`). (23.5) +This, of course, is the solution we had before; for since m(œổ — (2) is a real +number, the phase angles of P and of ê are the same (or perhaps 1802 apart, If +@2 > œ[), as advertised previously. The magnitude of ê, which measures how far +it oscillates, is related to the size of the by the faetor 1/m(œ[ — œ2), and this +factor becomes enormous when œ is nearly equal to œo. So we get a very strong +response when we apply the right frequency œ (ïf we hold a pendulum on the +end of a string and shake it at just the right Ífrequency, we can make it swing +very high). +23-2 The forced oscillator with damping +That, then, ¡is how we analyze oscillatory motion with the more elegant +mathematical technique. But the elegance of the technique is not at all exhibited +in such a problem that can be solved easily by other methods. It is only exhibited +when one applies it to more difficult problems. Let us therefore solve another, +more dificult problem, which furthermore adds a relatively realistic feature to +the previous one. Equation (23.5) tells us that ¡f the frequency œ were exactly +cqual to œọ, we would have an infinite response. Actually, of course, no such +Infinite response occurs because some other things, like friction, which we have +so far ignored, limits the response. Let us therefore add to Eq. (23.2) a friction +Ordinarily such a problem is very difcult because of the character and +complexity of the frictional term. There are, however, many circumstances In +--- Trang 410 --- +which the frictional force 1s proportional to the speed with which the object moves. +An example of such friction is the friction for slow motion of an object in oil or +a thick liquid. “PThere is no force when ï§ is just standing still, but the faster 1t +moves the faster the oil has to go past the object, and the greater is the resistance. +So we shall assume that there is, in addition to the terms in (23.2), another +term, a resistance force proportional to the velocity: = —cdz/di. It will be +convenient, in our mathematical analysis, to write the constant é as rm times + +to simplify the equation a little. Phis is Just the same trick we use with k when +we replace it by ma, just to simplify the algebra. Thus our equation will be +m(dÊ+/dt?) + e(daz/dt) + kz = F (23.6) +or, writing e = my and k = mổ and dividing out the mass rm, +(d2+/d1?) + +(dz/dÐ) + + = F/m. (23.6a) +Now we have the equation in the most convenient form to solve. lÍ y is very +small, that represents very little friction; if +y is very large, there is a tremendous +amount of friction. How do we solve this new linear diferential equation? Suppose +that the driving force is equal to #ọ cos (¿# + A); we could put this into (23.6a) +and try to solve it, but we shall instead solve it by our new method. Thus we +write ` as the real part of #'e”““f and z as the real part of êe”“!, and substitute +these into Eq. (23.6a). It is not even necessary to do the actual substituting, for +we can see by inspection that the equation would become +[(¿œ)22 -+ +(#ø)ê + äöẬ£]e**t = (Ê/m)e**t, (23.7) +[As a matter of fact, IÝ we tried to solve Eq. (23.6a) by our old straightforward +way, we would really appreciate the magic of the “complex” method.] IÝ we divide +by e?”“f on both sides, then we can obtain the response £ to the given force /) it +â@= ÊJ/m(uŸ — 0 + iu). (23.8) +Thus again ê is given by Ê tỉmes a certain faetor. There is no technical name +for this factor, no particular letter for it, but we may call ít for discussion +DUTPOSGS: +th = ———cc +m(u8 — 2 + i2) +--- Trang 411 --- +â= ÊR. (23.9) +(Although the letters and œọ are in very common use, this f‡ has no particular +name.) This factor ## can either be written as p-L ¿g, Or as a certain magnitude ø +times e', If it is written as a certain magnitude times e'”, let us see what it +means. NÑow /' = Fụe!^, and the actual force ` is the real part of Fụel2e'“t, that +1s, Fo cos (J@£ + A). Next, Bq. (23.9) tells us that £ is equal to #'?. So, writing +R= pể?® as another name for Ï, we get +@—= RÊ = øe?Fụef2 = pFucf+A), +Einally, going even further back, we see that the physical , which is the real part +of the complex £e”“f, is equal to the real part of øFuef®+A)e/“f, But ø and Fụ +are real, and the real part of e#+^Ã**®) is simply cos (w# + A +9). Thus +# = pFọ cos (u‡ + A +8). (23.10) +'This tells us that the amplitude of the response is the magnitude of the force † +multipHed by a certain magnifcation factor, /ø; this gives us the “amount” of +oscillation. It also tells us, however, that #ø is not oscillating in phase with the +force, which has the phase A, but is shifted by an extra amount Ø. 'Therefore ø +and Ø represent the size of the response and the phase shift of the response. +Now let us work out what ø is. IÝ we have a complex number, the square of +the magnitude is equal to the number times its complex conJugate; thus +ø”= : +mˆ2(uậ — . + 0)(„8 — 2 — iu) (23.11) +—_ m3[(w2 — œ8)? + +42] +In addition, the phase angle Ø is easy to find, for if we write +1/R= 1/pe'” = (1/p)e"”” = m(ưậ — Ÿ + i20), +we see that +tan 0 = —+w/(uä — œ). (23.12) +]t is minus because tan(—0) = — tan/. A negative value for Ø results for all œ, +and this corresponds to the displacement zø lagging the force #'. +--- Trang 412 --- +(Q9 ứ) +Fig. 23-2. Plot of øˆ versus ứ. +0° +~90° (0 œ +—180°4~~~~~~~~~~~~~~~~~~~~~~—~~~~-—~~~---==== +Fig. 23-3. Plot of Ø versus ứ. +Figure 23-2 shows how øŸ varies as a funetion of frequency (øZ is physically +more interesting than ø, because øŸ is proportional to the square of the amplitude, +or more or less to the energw that is developed in the oscillator by the force). We +see that iŸ + is very small, then 1/(œđ — @”)Ÿ is the most important term, and +the response tries to go up toward infũnity when œ equals œạ. Now the “infnity” +is not actually infũnite because if œ = œọ, then 1/22 is still there. The phase +shift varies as shown In Fig. 23-3. +In certain circumstances we get a slightly diferent formula than (23.8), also +called a “resonance” formula, and one might think that it represents a dierent +phenomenon, but it does not. The reason is that IŸ + is very small the most +interesting part oŸ the curve is near œ = œọ, and we may replace (23.8) by +an approximate formula which is very accurate iŸ + is small and œ is near œ0. +Since „8 — 2 = (œg — œ)(œ0 + œ@), l œ is near œọ this is nearly the same as +2(o(o — 0) and +0 is nearly the same as +œo. Dsing these in (23.8), we see that +tu — 0Ÿ + 20 200(0 — @ + 2/2), so that +â Ê/2mœg(œạ =@+ 11/9) lÍ +<øg and @®úg. (23.13) +--- Trang 413 --- +It is easy to fnd the corresponding formula for øŸ. It is +g2 % 1/4mŠ8|(ao — ø)Š + 42/4. +W©e shall leave it to the student to show the following: if we call the maximum +height of the curve of øŸ vs. œ one unit, and we ask for the width Aø of the curve, +at one half the maximum height, the full width at half the maximum height of +the curve is A¿ = +, supposing that + is small. The resonance is sharper and +sharper as the frictional efects are made smaller and smaller. +As another measure of the width, some people use a quantity @Q which is +defined as Q = œg/+. The narrower the resonance, the higher the Q: = 1000 +means a resonance whose width is only 1000th of the frequency scale. The @Q of +the resonance curve shown in Fig. 23-2 is ð. +'The importance of the resonance phenomenon is that it occurs in many other +circumstances, and so the rest of this chapter will describe some of these other +circumstances. +23-3 Electrical resonance +'The simplest and broadest technical applications of resonanece are in electricity. +In the electrical world there are a number of obJects which can be connected to +make electric circuits. These 0dss?ue circu#t clements, as they are often called, +are of three main types, although each one has a little bit of the other wo mixed +in. Before describing them in greater detail, let us note that the whole idea of our +mnechanical oscillator beïng a mass on the end oŸa spring is only an approximation. +All the mass is not actually at the “mass”; some of the mass is in the inertia of +the spring. Similarly, all of the spring is not at the “spring”; the mass itself has a +little elasticity, and although it may appear so, it is not øbsolu‡elu rigid, and as it +goes up and down, i% fexes ever so slightly under the action of the spring pulling +it. The same thing is true in electricity. Phere is an approximation in which we +can lump things into “circuit elements” which are assumed to have pure, ideal +characteristics. It is not the proper time to discuss that approximation here, we +shall simply assume that it is true in the cireumstances. +The three main kinds of cireuit elements are the following. “The first is called a +capacitor (EFig. 23-4); an example is 0wo plane metallic plates spaced a very small +distance apart by an insulating material. When the plates are charged there is +a certain voltage diference, that is, a certain diference in potential, between +--- Trang 414 --- +ẠA C E +B D F +CAPACITOR RESISTOR _ INDUCTOR +Fig. 23-4. The three passive circuit elements. +them. "The same diference of potential appears bebween the terminals A4 and Ö, +because if there were any diference along the connecting wire, electricity would +fow right away. So there is a certain voltage diference V between the plates If +there is a certain electric charge +g and —q on them, respectively. Between the +plates there will be a certain electric field; we have even found a formula for 1% +(Chapters 13 and 14): +V = ơd/sạ = qd/eoA, (23.14) +where đ is the spacing and A is the area of the plates. Note that the potential +diference is a linear function of the charge. If we do not have parallel plates, +but insulated electrodes which are of any other shape, the diference in potential +1s still precisely proportional to the charge, but the constant of proportionality +may not be so easy to compute. However, all we need to know is that the +potential difference across a capacitor 2s proportional to the charge: V = q/C: +the proportionality constant is 1/Œ, where Œ is the capacitance oŸ the object. +'The second kind of circuit element is called a reszstor; 1t offers resistance to +the Ñow of electrical current. It turns out that metallic wires and many other +substances resist the fÑow of electricity in this manner: if there is a voltage +diference across a piece of some substance, there exists an electric current Ï = +dq/đt that is proportional to the electric voltage difference: +V = RÏI = hdaq/dt (23.15) +'The proportionality coefficient is called the resis‡tønece Rì. Thĩs relationship may +already be familiar to you; i% is Ohm”s law. +Tf we think of the charge g on a capacitor as being analogous to the displace- +ment # of a mechanical system, we see that the current, Ï = dg/dt, is analogous +to velocity, L/Œ is analogous to a spring constant k, and ?# is analogous to the +resistive coefficlent e = zm+y in Eq. (23.6). Now it is very interesting that there +--- Trang 415 --- +exists another circuit element which is the analog of massl “This is a coil which +builds up a magnetic feld within itself when there is a current in it. A changing +magnetic feld develops in the coil a voltage that is proportional to đĨ/đf (this +is how a transformer works, in fact). The magnetic feld is proportional to a +current, and the induced voltage (so-called) in such a coil is proportional to the +rate of change of the current: +V = LdI/dt = Ld°q/dtẺ. (23.16) +The coeficlent Ù is the sejf-?nductfance, and is analogous to the mass in a +mmechanical oscillating circuit. +Fig. 23-5. An oscillatory electrical circuit with resistance, Iinductance, +and capacitance. +Suppose we make a circuit in which we have connected the three circuit +elements in series (Eig. 23-5); then the voltage across the whole thing from 1 +to 2 is the work done in carrying a charge through, and it consists of the sum +of several pieces: across the induetor, Vy„ = Ld2q/di2; across the resistance, +Vn = Tìdq/dt; across the capacitor, Vơ = g/C. The sum of these is equal to the +applied voltage, V: +Ld?q/di° + Rdq/dt + q/C = VỆ). (23.17) +Now we see that this equation is exactly the same as the mechanical equa- +tion (23.6), and oŸ course it can be solved in exactly the same manner. WWe +suppose that V(£) is oscillatory: we are driving the circuit with a generator with +a pure sine wave oscillation. Then we can write our V(£) as a complex V with +the understanding that it must be ultimately multiplied by e”“, and the real part +taken in order to fnd the true W. Likewise, the charge g can thus be analyzed, +and then in exactly the same manner as in Eq. (23.8) we write the corresponding +equation: the second derivative of ậ is (2)2â; the fñrst derivative is (2)ệ. Thus +--- Trang 416 --- +Eq. (23.17) translates to +9 . 1Ì, _« +L()“ + T() + cl?Z V +L1(0u)2 + R() + — +which we can write in the form +ậ= Ÿ/L(uỆ — w2 + i2), (23.18) +where œä = 1/EŒ and + = R/L. It is exactly the same denominator as we had +in the mechanical case, with exactly the same resonance propertiesl The corre- +spondence between the electrical and mechanical cases is outflined in Table 23-1. +Table 23-1 +General Mechanical Eilectrical +characteristic property property +indep. variable time (#) time (?) +dep. variable position (z) charge (g) +inertia mass (mm) inductance (L) +resistance drag coef. (c = +m) resistance (lề = +yL) +stifness stifness (k) (capacitance)” (1/Œ) +resonant frequency u = k/m uạ = 1/LƠ +period to = 2mav/Èm/È to = 2xV ÙŒ +fgure of merit Q=,u0/^2 Q=uoL/R +We must mention a small technical point. In the electrical literature, a +diferent notation is used. (From one field to another, the subject is not really +any diferent, but the way of writing the notations is often different.) Eirst, 7 +1s commonly used instead of ¿ in electrical engineering, to denote —1. (After +all, ¿ must be the currentl) Also, the engineers would rather have a relationship +bebween V and ï than between W and , just because they are more used to i% +that way. Thus, since Ï = đ@/đf# = iuậ, we can just substitute ƒ/26 for ậ and get +Ÿ = („L+ R+ 1/iu@)Ê = ÔÏ. (23.19) +--- Trang 417 --- +Another way is to rewrite Eq. (23.17), so that it looks more familiar; one often +sees it written this way: +Ld1/dt + RT + q/© ƒ T dt = V(t). (23.20) +At any rate, we ñnd the relation (23.19) between voltage Ÿ and current Í which +is Just the same as (23.18) except divided by 2œ, and that produces Ed. (23.19). +The quantity #8 + /œÙ + 1/2 is a complex number, and is used so mụch in +electrical engineering that it has a name: it is called the cornplez ứmpedance, +2. Thus we can write Ÿ = 2Ÿ. The reason that the engineers like to do this +1s that they learned something when they were young: V = ÏÏ for resistances, +when they only knew about resistances and DƠ. Now they have become more +educated and have AC circuits, so they want the equation to look the same. Thus +they write Ÿ =ÊŸ, the only diference being that the resistance is replaced by a +more complicated thing, a complex quantity. So they insist that they cannot use +what everyone else in the world uses for imaginary numbers, they have to use a 7 +for that; it is a miracle that they did not insist also that the letter Z be an ?#l +(Then they get into trouble when they talk about current densities, for which +they also use 7. 'The difficulties of science are to a large extent the dificulties of +notations, the units, and all the other artificialities which are invented by man, +not by nature.) +23-4 Resonance in nature +Although we have discussed the electrical case in detail, we could also bring +up case after case in many fields, and show exactly how the resonance equation +is the same. 'Phere are many circumstances in nature in which something is +“oscillating” and in which the resonance phenomenon occurs. We said that in +an earlier chapter; let us now demonstrate it. If we walk around our study, +pulling books of the shelves and simply looking through them to ñnd an example +of a curve that corresponds to Eig. 23-2 and comes from the same equation, +what do we fnd? Just to demonstrate the wide range obtained by taking the +smallest possible sample, it takes only five or six books to produce quite a series +of phenomena which show resonances. +The first two are from mechanics, the frst on a large scale: the atmosphere oŸ +the whole earth. If the atmosphere, which we suppose surrounds the earth evenly +--- Trang 418 --- +Cycles per day +4a Ị a +1aha2‡ tahoo, 1ohạo +Fig. 23-6. Response of the atmosphere to external excitation. a Is +the required response if the atmospheric Sa-tide is of gravitational origin; +peak amplification ¡is 100 : 1. b ¡s derived from observed magnification +and phase of M;-tide. [Munk and MacDonald, “Rotation of the Earth,” +Cambridge University Press (1960)] +on all sides, is pulled to one side by the moon or, rather, squashed prolate into a +double tide, and if we could then let it go, it would go sloshing up and down; it is +an oscillator. Thịs oscillator is đrZuen by the moon, which is eÑfectively revolving +about the earth; any one component of the force, say in the z-direction, has a +cosine component, and so the response of the earth's atmosphere to the tidal pull +of the moon is that of an oscillator. The expected response of the atmosphere is +shown in Eig. 23-6, curve Ö (curve ø is another theoretical curve under discussion +in the book from which this is taken out of context). Now one might think +that we only have one point on this resonance curve, since we only have the one +frequency, corresponding to the rotation of the earth under the moon, which +Occurs at a period of 12.42 hours——12 hours for the earth (the tide is a double +bump), plus a little more because the moon is goïing around. But rom the size oŸ +the atmospheric tides, and from the phase, the amount of delay, we can get both +p and Ø. From those we can get œạ and +, and thus draw the entire curvel 'This is +an example of very poor seience. From two numbers we obtain two numbers, and +from those two numbers we draw a beautiful curve, which of course goes through +the very point that determined the curvel Ït is of no use nÏess te can Tncasure +sơmethzng else, and in the case of geophysics that is often very difcult. But in +this particular case there is another thing which we can show theoretically must +--- Trang 419 --- +have the same timing as the natural frequency œọ: that is, if someone disturbed +the atmosphere, it would oscillate with the frequency œg. Now there +0øs such a +sharp disturbance in 1883; the Krakatoa volcano exploded and half the island +blew of, and ¡it made such a terrific explosion in the atmosphere that the period +of oscillation of the atmosphere could be measured. It came out to 105 hours. +'The œọ obtained from Eig. 23-6 comes out 10 hours and 20 minutes, so there we +have at least one check on the reality of our understanding of the atmospheric +tides. +Next we go to the small scale of mechanical oscillation. This time we take a +sodium chỉloride crystal, which has sodium ions and chlorine iIons next to each +other, as we described in an early chapter. 'Phese ions are electrically charged, +alternately plus and minus. Now there is an interesting oscillation possible. +uppose that we could drive all the plus charges to the right and all the negative +charges to the left, and let go; they would then oscillate back and forth, the +sodium lattice against the chlorine lattice. How can we ever drive such a thing? +'That is easy, for If we apply an electric fñeld on the crystal, it will push the plus +charge one way and the minus charge the other wayl So, by having an external +electric field we can perhaps get the crystal to oscillate. The frequency of the +electric fñeld needed is so high, however, that it corresponds to ?nƒfrared radiatiom +So we try to fnd a resonance curve by measuring the absorption of infrared light +by sodium chloride. Such a curve is shown in Fig. 23-7. "The abscissa is not +frequenecy, but is given in terms of wavelength, but that is just a technical matter, +Of course, since for a wave there is a definite relation bebween Írequency and +wavelength; so it is really a frequency scale, and a certain Írequency corresponds +to the resonant frequency. +But what about the width? What determines the width? There are many +cases in which the width that is seen on the curve is not really the natural width + +that one would have theoretically. There are two reasons why there can be a +wider curve than the theoretical curve. If the objects do not all have the same +frequency, as might happen ïf the crystal were strained in certain reglons, so that +in those regions the oscillation frequency were slightly diferent than in other +regions, then what we have is many resonance curves on top oŸ each other; so we +apparently get a wider curve. The other kind of width is simply this: perhaps +we cannot measure the frequency precisely enough-——if we open the slit of the +spectrometer fairly wide, so although we thought we had only one Írequency, +we actually had a certain range Aœ, then we may not have the resolving power +needed to see a narrow curve. Ofhand, we cannot say whether the width in +--- Trang 420 --- +° +40 45 50 55 60 65 70 +Wavelength in microns (10~4 cm) +Fig. 23-7. Transmission of infrared radiation through a thin (0.17 u}) +sodium chloride film. [After R. B. Barnes, Z. Phys¡ik 75, 723 (1932). +Kittel, Introduction to Solid State Physics, Wiley, 1956.] +Fig. 23-7 is natural, or whether it is due to inhomogeneities in the crystal or the +ñnite width of the slit of the spectrometer. +Now we turn to a more esoteric example, and that is the swinging of a magnet. +Tf we have a magnet, with north and south poles, in a constant magnetic field, +the N end of the magnet will be pulled one way and the S5 end the other way, and +there will in general be a torque on it, so ít will vibrate about its equilibrium +position, like a compass needle. However, the magnets we are talking about are +a‡oms. 'These atoms have an angular momentum, the torque does not produee a +simple motion in the direction of the field, but instead, of course, a precession. +Now, looked at from the side, any one component is “swinging,” and we can +disturb or drive that swinging and measure an absorption. The curve in Fig. 23-8 +represents a typical such resonance curve. What has been done here is slightly +diferent technically. "The frequency of the lateral feld that is used to drive +this swinging is always kept the same, while we would have expected that the +investigators would vary that and plot the curve. They could have done it that +way, but technically it was easier for them to leave the frequency œ fñxed, and +change the strength of the constant magnetic field, which corresponds to changing +œg in our formula. 'They have plotted the resonance curve against œ0. Anyway, +this is a typical resonance with a certain œọ and +. +Now we go still further. Our next example has to do with atomiec nuclei. "The +motions of protons and neutrons in nuclei are oscillatory in certain ways, and we +--- Trang 421 --- +2.0 +1.8 +SIŠ 1.4 +S812 +GÌE 42 +la OERSTEDS +»J2 1.0 ~ +S5 0ø +SE 0.6 +0.2 +8100 8200 8300 8400 8500 8600 8700 8800 +STATIC MAGNETIC FIELD IN OERSTEDS +Fig. 23-8. Magnetic energy loss in paramagnetic organic compound +as function of applied magnetic field intensity. [Holden et al., Phys. +Rev. 75, 1614 (1949)] +can demonstrate this by the following experiment. We bombard a lithium atom +with protons, and we discover that a certain reaction, producing +-rays, actually +has a very sharp maximum typical of resonance. We note in Eig. 23-9, however, +one difference from other cases: the horizontal scale is not a frequency, it is an +energ! The reason is that in quantum mechanics what we think of classically as +the energy will turn out to be really related to a frequency of a wave amplitude. +'When we analyze something which in simple large-scale physics has to do with a +frequency, we fnd that when we do quantum-mechanical experiments with atomic +matter, we get the corresponding curve as a function of energy. In fact, this +curve is a demonstration of this relationship, in a sense. It shows that frequency +and energy have some deep interrelationship, which of course they do. +Now we turn to another example which also involves a nuclear energy level, +but now a mụch, much narrower one. The œ in Fig. 23-10 corresponds to an +energy of 100,000 electron volts, while the width + is approximately 105 electron +--- Trang 422 --- +: L]| HN L | | +s 4 Í í b : N +2 Z5 l : }TEm==m —? +BEPE/PRIrienman +300 400 500 600 +PROTON ENERGY IN KEV +Fig. 23-9. The intensity of gamma-radiation from lithium as a func- +tion of the energy of the bombarding protons. The dashed curve Is a +theoretical one calculated for protons with an angular momentum £ = 0. +[Bonner and Evans, Phys. Rev. 73, 666 (1948)] +AI —5 —5 —5 +m= 0 2-10 410 56V, A +0 -4 0 +4 +8 cm/sec vV +~—0.8% +Fig. 23-10. [Courtesy of Dr. R. Mössbauer] +--- Trang 423 --- +volt; in other words, this has a Q of 101! When this curve was measured it was +the largest @Q of any oscillator that had ever been measured. It was measured by +Dr. Mössbauer, and it was the basis of his Nobel prize. The horizontal scale here +1s velocity, because the technique for obtaining the slightly diferent requencies +was to use the Doppler efect, by moving the source relative to the absorber. Ône +can see how delicate the experiment is when we realize that the speed involved is +a few centimeters per secondl Ơn the actual scale of the figure, zero frequency +would correspond to a point about 1010 em to the left—slightly of the paperl +c 2 \ 7 +Đ 3 Z += NN 4 +ĐS TH % +li G: +° SN +200 300 400 500 +P‹ (MeV/c) +Fig. 23-11. Momentum dependence of the cross section for the +reactions (a) K” +p + A+ xử +7 and (b) K” +p > K?+n. +The lower curves in (a) and (b) represent the presumed nonresonant +backgrounds, while the upper curves contain in addition the superposed +resonance. [Ferro-Luzzi et al., Phys. Rev. Lett. 8, 28 (1962)] +Jinally, ifƒ we look in an issue oŸ the Phsical Reuieu, say that of January 1, +1962, will we fñnd a resonance curve? Every issue has a resonance curve, and +Fig. 23-11 is the resonance curve for this one. 'Phis resonance curve turns out to +be very interesting. It is the resonance found in a certain reaction among strange +--- Trang 424 --- +particles, a reaction in which a K— and a proton interact. “The resonance 1s +detected by seeing how many of some kinds of particles come out, and depending +on what and how many come out, one gets diferent curves, but of the same +shape and with the peak at the same energy. We thus determine that there is a +resonance at a certain energy for the K— meson. That presumably means that +there is some kind oŸ a state, or condition, corresponding to this resonance, which +can be attained by putting together a K— and a proton. This is a new particle, +or resonance. Today we do not know whether to call a bump like this a “particle” +or simply a resonance. When there is a very shørp resonance, it corresponds to a +very đefinaiie energu, just as though there were a particle of that energy present +in nature. When the resonance gets wider, then we do not know whether to +say there is a particle which does not last very long, or simply a resonance in +the reaction probability. In the second chapter, this point is made about the +particles, but when the second chapter was written this resonance was not known, +so our chart should now have still another partiele in itl +--- Trang 425 --- +TT-drtSf©reÉs +24-1 The energy of an oscillator +Although this chapter is entitled “transients,” certain parts of i% are, in a +way, part of the last chapter on forced oscillation. One of the features of a forced +oscillation which we have not yet discussed is the energy in the oscillation. Let +us now consider that energy. +In a mechanical oscillator, how much kinetic energy is there? lt is proportional +to the square of the velocity. NÑow we come to an important point. Consider an +arbitrary quantity A, which may be the velocity or something else that we want +to discuss. When we write A = Âc 1t, a complex number, the true and honest +A, in the physical world, is only the real part; therefore if, for some reason, we +want to use the sguøre of A, i% is not right to square the complex number and +then take the real part, because the real part of the square of a complex number +1s not just the square of the real part, but also involves the Z#maginarw part. So +when we wish to fñnd the energy we have to get away from the complex notation +for a while to see what the inner workings are. +Now the true physical 4 is the real part of Agef«++^) that is, A = Ao cos (£+ +A), where Â, the complex number, is written as Aoe?^. Now the square of this +real physical quantity is 4? = 4ä cos” („#+ A). The square of the quantity, then, +goes up and down from a maximum to zero, like the square of the cosine. 'Phe +square of the cosine has a maximum of 1 and a minimum of 0, and is average +value is 1/2. +In many circumstances we are not interested in the energy at any specifc +moment during the oscillation; for a large number of applications we merely +want the average of 42, the mmean of the square of A over a period of time large +compared with the period of oscillation. In those circumstances, the average +of the cosine squared may be used, so we have the following theorem: if A is +represented by a complex number, then the mean of .4” is equal to 3.43. NÑow 4ã +--- Trang 426 --- +1s the square of the magnitude of the complex Â. (This can be written in many +ways—some people like to write |A|?; others write, 44”, 4 times its complex +conjugate.) We shall use this theorem several tỉmes. +Now let us consider the energy in a forced oscillator. The equation for the +forced oscillator is +m d”z/dtÊ + m dz/dt + mua = F1). (24.1) +In our problem, of course, #'(£) is a cosine function of ¿. NÑow let us analyze the +situation: how much work is done by the outside force †'? "The work done by the +force per second, ¡.e., the power, is the force times the velocity. (W© know that +the diferemtial work in a tỉme đý is f'dz+, and the power is F'dz+/di.) Thus +da: d+z d2z d+z dz\ +P=F-.= — g5 —— —|- 24.2 +dt m7) (mm) s2 m)|+an ( n) (242) +But the ñrst two terms on the right can also be written as d/df[Sm(dz/dt)? + +smœf+2], as is immediately verifled by differentiating. That is to say, the term in +brackets is a pure derivative oŸ ©wo terms that are easy to understand——one is the +kinetic energy of motion, and the other is the potential energy oŸ the spring. Let +us call this quantity the sứored energu, that is, the energy stored in the oscillation. +uppose that we want the average power over many cycles when the oscillator is +beiïng forced and has been running for a long time. In the long run, the stored +energy does not change——its derivative gives zero average efect. In other words, +1 we average the power in the long run, đÌÏ the energu ulttmatelU ends up ín the +resistiue term +m(dz/df)?. There is some energy stored in the oscillation, but +that does not change with time, if we average over many cycles. Therefore the +mean power (P) is +(P) = (wém(dz/d£)). (24.3) +Using our method of writing complex numbers, and our theorem that (42) = +34ã, we may find this mean power. Thus if z = êc”“”, then dœ/dt = iuêâc”*!, +'Therefore, in these circumstances, the average power could be written as +(?)= 3m đg. (24.4) +In the notation for electrical circuits, đø/đ# is replaced by the current ï (T is +dq/dt, where q corresponds to +), and zw⁄ corresponds to the resistance Ÿ. Thus +--- Trang 427 --- +the rate of the energy loss—the power used up by the forcing function——is the +resistance in the circuit times the average square of the current: +(P) = RỊ?) = R- 313. (24.5) +This energy, of course, goes into heating the resistor; i% is sometimes called the +heating loss or the Joule heating. +Another interesting feature to discuss is how much energy is sfored. 'That is +not the same as the power, because although power was at first used to store up +some energy, after that the system keeps on absorbing power, insofar as there +are any heating (resistive) losses. At any moment there is a certain amount of +sbored energy, so we would like to calculate the mean stored energy (2) also. We +have already calculated what the average of (dz/đ#)2 is, so we find +(E) = ‡m((de/4)®) + ÿmeŠ(v?) 016) += 3m(0Ÿ + 08) šzg. +Now, when an oscillator is very efficient, and IÝ œ is near œọ, so that |Ê{ is large, +the stored energy is very high—we can get a large stored energy from a relatively +smaill force. 'Phe force does a great deal of work in getting the oscillation goiïng, +but then to keep it steady, all it has to do is to fight the friction. The oscillator +can have a great deal of energy if the friction is very low, and even though it is +oscillating strongly, not much energy is being lost. The eficiency of an oscillator +can be measured by how much energy is stored, compared with how much work +the force does per oscillation. +How does the stored energy compare with the amount of work that is done in +one cycle? 'Phis is called the @ of the system, and @ is defined as 27 times the +mean stored energy, divided by the work done per cycle. (If we were to say the +work done per rœđan instead of per cycle, then the 2z disappears.) +1 2 2 2 2 2 +sm((Z + 08) - (œ +Q=2n2— sa o ý SẺ ` ĐỘ GÔ — Tổ, (24.7) +“+mú2(#2) - 2/0 2% +Q} ïs not a very useful number unless it 1s very large. When it is relatively large, +1E gives a measure of how good the oscillator is. People have tried to deñne Q +in the simplest and most useful way; various defnitions difer a bit from one +another, but if Q is very large, all deÑnitions are in agreement. 'Phe most generally +accepted defnition is Bq. (24.7), which depends on œ¿. Eor a good oscillator, close +--- Trang 428 --- +to resonance, we can simplify (24.7) a little by setting œ = œo, and we then have +Q = ¿0/+, which is the definition of Q that we used before. +'What is @Q for an electrical circuit? To ñnd out, we merely have to translate +L form, R for m+, and 1/C for mới (see Table 23-1). The Q at resonance is +T/R, where œ is the resonance frequency. IÝ we consider a circuit with a hiph +Q, that means that the amount of energy stored in the oscillation is very large +compared with the amount of work done per cycle by the machinery that drives +the oscillations. +24-2 Damped oscillations +W©e now turn to our main topic of discussion: transients. By a transient 1s +meant a solution of the diferential equation when there is no force present, but +when the system is not simply at rest. (Of course, 1Ÿ it is standing still at the +origin with no force acting, that is a nice problem—it stays therel) Suppose the +oscillation starts another way: say it was driven by a force for a while, and then +we turn of the force. What happens then? Let us first get a rough idea of what +will happen for a very high Q system. 5o long as a force is acting, the stored +energy stays the same, and there is a certain amount of work done to maintain +1t. NÑow suppose we turn of the force, and no more work is being done; then the +losses which are eating up the energy of the supply are no longer eating up is +energy——there 7s no more driver. 'Phe losses will have to consume, so to speak, +the energy that is stored. Let us suppose that Q/2z = 1000. Then the work +done per cycle is 1/1000 of the stored energy. Is it not reasonable, sinee iE is +oscillating with no driving force, that in one cycle the system will still lose a +thousandth of its energy #⁄, which ordinarily would have been supplied from the +outside, and that it will continue oscillating, always losing 1/1000 of its energy +per cycle? 5o, as a guess, for a relatively high @ system, we would suppose that +the following equation might be roughly right (we will later do it exactly, and it +will turn out that it œøs rightl): +dE/dt = —=uE/Q. (24.8) +Thịs is rough because iE is true only for large Q. In each radian the system loses a +fraction 1/Q of the stored energy #2. Thus in a given amount oŸ tỉme đý the energy +will change by an amount œ đ£/@Q, since the number of radians associated with +the time để is œ d. What is the frequency? Let us suppose that the system moves +--- Trang 429 --- +so nicely, with hardly any force, that if we let go i9 will oscillate at essentially the +same frequency all by itself. 5o we will guess that œ is the resonant Írequency œ. +Then we deduce rom Eaq. (24.8) that the stored energy will vary as +EB= Eụe 9/9 = Eue-Ðt, (24.9) +This would be the measure of the energu at any moment. What would the +formula be, roughly, for the amplitude of the oscillation as a function of the +time? The same? Nol "The amount of energy in a spring, say, goes as the sguare +of the displacement; the kinetic energy goes as the sợuare of the velocity; so +the total energy goes as the sguare of the displacement. 'Thus the displacement, +the amplitude of oscillation, will decrease half as fast because of the square. In +other words, we guess that the solution for the damped transient motion will +be an oscillation of frequeney close to the resonance frequency œọ, in which the +amplitude of the sine-wave motion will diminish as e~?⁄2: +œ = Aoe"?2 cosugt. (24.10) +This equation and Fig. 24-1 give us an idea of what we should expect; now let +us try to analyze the motion øreciselu by solving the diferential equation of the +motion itself. +` ` ⁄ e-1/2 +° e~71/2 cos wo t +_—— —— £ +Fig. 24-1. A damped cosine oscillation. +So, starting with Eq. (24.1), with no outside force, how do we solve it? Being +physicists, we do not have to worry about the rmethod as mụuch as we do about +what the solution 2s. Armed with our previous experience, let us try as a solution +an exponential curve, z = Ac”*f, (Why do we try this? It is the easiest thỉng to +diferentiate!) We put this into (24.1) (with Ƒ{) = 0), using the rule that each +--- Trang 430 --- +time we diferentiate + with respect to time, we multiply by ¿ơ. So it is really +quite simple to substitute. 'Thus our equation looks like this: +(Ta? +i>œ+u)Ac'*t =0. (24.11) +The net result must be zero for øÏl t#mes, which is impossible unless (a) A = 0, +which is no solution at all—it stands still, or (b) +=2 +ia+ + ư8 =0. (24.12) +Tf we can solve this and find an ơ, then we will have a solution in which 4 need +not be zerol +œ =11/2#+ Vu — 32/4. (24.13) +For a while we shall assume that + is fairly small compared with œọ, so that +uẩ — 22/4 is definitely positive, and there is nothing the matter with taking the +square root. The only bothersome thing is that we get #œo solutionsl Thus +œi =i2/2+ Vai — +2/4=1+/2+u+x (24.14) +da =i1/2~— vưi — +2/4= i+/2— ư+x. (24.15) +Let us consider the ñrst one, supposing that we had not noticed that the square +root has two possible values. Then we know that a solution for # is zị = Ac'e1t, +where A is any constant whatever. NÑow, in substituting ơ+, because it is goïng to +come so many tỉmes and it takes so long to write, we shall call /„§ — +2/4 = œx. +Thus iœ¡ = —+/2 + 7+, and we get ø = Ae(7/2†1“+)!, or what is the same, +because of the wonderful properties of an exponential, +đị = Ae 11/2/1241, (24.16) +First, we recognize this as an oscillation, an oscillation at a frequency ¿„, which +1s not ezacflu the frequency œọ, but 1s rather close to œọ 1Ý ït is a good system. +Second, the amplitude of the oscillation is decreasing exponentiallyl If we take, +for instance, the real part of (24.16), we get +gì = Ae~ 1/2 eosu,f. (24.17) +--- Trang 431 --- +This is very mụuch like our guessed-at solution (24.10), except that the frequency +really is «¡„. This is the only error, so i% is the same thing——we have the right +idea. But everything is no all rightl What is not all right is that £here ¡s another +solution. +The other solution is œ¿, and we see that the diference is only that the sign +OŸ ¿J„ 1s reversed: +#ạ = Be~7/2e~1sat, (24.18) +What does this mean? We shall soon prove that if z¡ and #a are each a possible +solution of Eq. (24.1) with ?' = 0, then z¡ + #a is also a solution of the same +cquation! So the general solution #+ is of the mathematical form +œ=e 2U2( Aelsst + Be~ +), (24.19) +Now we may wonder why we bother to give this other solution, since we were +so happy with the frst one all by itself. What is the extra one for, because +Of course we know we should only take the real part? We know that we must +take the real part, but how did the rmafhematics know that we only wanted the +real part? When we had a nonzero driving force #{f), we put in an artjicial +force to go with it, and the #naginarw part of the equation, so to speak, was +driven in a delnite way. But when we put #{£) = 0, our convention that # +should be only the real part of whatever we write down is purely our own, and +the mathematical equations do not know it yet. 'Phe physical world høs a real +solution, but the answer that we were so happy with before is not real, it 1s +cormmplez. 'The equation does not know that we are arbitrarily going to take the +real part, so 1t has to present us, so to speak, with a complex conjugate type of +solution, so that by putting them together we can maœke a truhụ real solution; +that is what œ¿ is doïng for us. In order for z to be real, Be~“»† will have to be +the complex conjugate of Ae*“+f that the imaginary parts disappear. So it turns +out that Ö is the complex conjugate of A, and our real solution is +a=e TU2( Aesst+ A*esst), (24.20) +So our real solution is an oscillation with a phase shft and a damping—just as +advertised. +24-3 Electrical transients +Now let us see If the above really works. We construct the electrical circuit +shown in Fig. 24-2, in which we apply to an oscilloscope the voltage across the +--- Trang 432 --- +Fig. 24-2. An electrical circuit for demonstrating transients. +inductance Ù after we suddenly turn on a voltage by closing the switch Š. lt is an +oscillatory circuit, and it generates a transient of some kind. It corresponds to a +circumstance in which we suddenly apply a force and the system starts to oscillate. +Tt is the electrical analog of a damped mechanical oscillator, and we watch the +oscillation on an oscilloscope, where we should see the curves that we were trying +to analyze. (The horizontal motion of the oscilloscope is driven at a uniform speed, +while the vertical motion is the voltage across the inductor. “The rest of the circuit +1s only a technical detail. We would like to repeat the experiment many, many +tỉimes, since the persistence of vision is not good enough to see onÌy one trace +on the screen. So we do the experiment again and again by closing the switch +60 times a second; each time we close the switch, we also start the oscilloscope +horizontal sweep, and it draws the curve over and over.) In Figs. 24-3 to 24-6 we +see examples of damped oscillations, actually photographed on an oscilloscope +sereen. Pigure 24-3 shows a damped oscillation in a circuit which has a high @, a +small y. It does not die out very fast; it oscillates many times on the way down. +But let us see what happens as we decrease @, so that the oscillation dies out +more rapidly. We can decrease @ by increasing the resistance # in the circuit. +When we increase the resistance in the circuit, i9 dies out faster (Eig. 24-4). +'Then ïf we increase the resistance in the circuit still more, it dies out faster still +(Fig. 24-5). But when we put in more than a certain amount, we cannot see any +oscillation at alll "The question is, is this because our eyes are not good enough? +TÍ we increase the resistance still more, we get a curve like that of Fig. 24-6, which +does not appear to have any oscillations, except perhaps one. Now, how can we +explain that by mathematics? +The resistance 1s, of course, proportional to the + term in the mechanical +device. Specifically, + is //E. NÑow iŸ we increase the + in the solutions (24.14) +and (24.15) that we were so happy with before, chaos sets in when +/2 exceeds +œg; we must write i% a different way, as +y~/2+iVW^2/4—,u$ and y/2— iv^22/4- uậ. +--- Trang 433 --- +Figure 24-3 +Figure 24-4 +Figure 24-5 +Figure 24-6 +--- Trang 434 --- +Those are now the two solutions and, following the same line of mathematical +reasoning as previously, we again fñnd two solutions: e'*“f and e?%2†, Tf we now +substitute for œ, we get +z— Ae—(/3†V3/4— s8): +a nice exponential decay with no oscillations. Likewise, the other solution is ++ — Be-(%/2~V^2/4-ui)t. +Note that the square root cannot exceed +/2, because even IŸ «o = 0, one term +jusi equals the other. But ø is taken away from +2/4, so the square root is less +than +/2, and the term in parentheses is, therefore, always a positive number. +Thank goodnessl Why? Because ïf it were negative, we would find e raised to a +postfiue factor tỉìmes ‡, which would mean it was explodingl In putting more and +more resistance into the cireuit, we know it is not going to explode—qulite the +contrary. So now we have ©wo solutions, each one by itself a dying exponential, +but one having a much faster “dying rate” than the other. 'Phe general solution is +of course a combination of the two; the coefficients in the combination depending +upon how the motion starts—what the initial conditions of the problem are. In +the particular way this circeuit happens to be starting, the A is negative and the +B 1s positive, so we get the diference of bwo exponential curves. +Now let us discuss how we can fnd the two coefficients A and Ö (or Aand 4Š), +1ƒ we know how the motion was started. +Suppose that at # = 0 we know that ø = zo, and đz/đ# = 0ọ. TÝ we put £= 0, +% = #ọ, and d+/đt = 0ọ into the expressions ++ — e~1⁄2(Aefsat + A*e a9, +da /dt = e~1⁄2[(—+x/2 +4) Aezf + (—+/2T— iax„) A*e a1, +we fnd, sinee e? = e9 =1, +#o= A+ A4” =2An, +uọ = =(/2)(A+ A*) +ix(A— A*) += —*#o/2 + i„x(2Ar), +where 4= Ag-+¿4;, and 4Ý = Ag — ¿Ar. Thus we ñnd +An — zo/2 +--- Trang 435 --- +Ar = —(0o + +#o/2)/2„. (24.21) +This completely determines 4 and 4Ý, and therefore the complete curve of the +transient solution, in terms of how it begins. Incidentally, we can write the +solution another way if we note that +c® +e~?” =2cos8 and c8 — e~?# = 2isin 6. +W©e may then write the complete solution as +z—e 2 la COS(„Ý + to + 120/2 sỉn ¬ : (24.22) +where œ„ = +/œ — 2/4. Thịis is the mathematical expression for the way an +oscillation dies out. We shall not make direct use of it, but there are a number +of poïints we should like to emphasize that are true in more general cases. +First of all the behavior of such a system with no external force is expressed by a +sum, or superposition, of pure exponentials in tỉme (which we wrote as e?%!), 'This +is a good solution to try in such circumstances. The values of œ may be complex +in general, the imaginary parts representing damping. Finally the intimate +mathematical relation of the sinusoidal and exponentfial function discussed In +Chapter 22 often appears physically as a change from oscillatory to exponential +behavior when some physical parameter (in this case resistance, +) exceeds some +critical value. +--- Trang 436 --- +X}irnoer Sggséormes cn«Ï lïotosr +25-1 Linear diferential equations +In this chapter we shall discuss certain aspects of oscillating systems that are +found somewhat more generally than just in the particular systems we have been +discussing. For our particular system, the diferential equation that we have been +solving is +dỀz da 2 +mu + m + Ta0+ = F). (25.1) +Now this particular combination of “operations” on the variable ø has the +interesting property that if we substitute (+) for z, then we get the sum of the +same operations on z and ø; or, if we multiply z by a, then we get just ø times +the same combination. This is easy to prove. Just as a “shorthand” notation, +because we get tired of writing down all those letters in (25.1), we shall use the +symbol (+) instead. When we see this, it means the left-hand side of (25.1), +with z substituted in. With this system oŸ writing, Ù(z + ) would mean the +following: +đˆ(x+ d(z + +L(x+ụ) =m “Œ T9) „mm đŒ $9) muà(z + g). (25.2) +(We underline the Ù so as to remind ourselves that it is not an ordinary function.) +W©e sometimes call this an operator no‡øtion, but 1t makes no diference what we +call it, it is just “shorthand” +Our frst statement was that +Lí +) = L(z) + LỤU): (25.3) +which of course follows from the fact that ø(# + ) = a# + aụ, đ(z + U) /dt —= +dz/dt + dụ/dt, etc. +--- Trang 437 --- +Our second statement was, for constant ø, +T(az) = aE(œ). (25.4) +[Actually, (25.3) and (25.4) are very closely related, because iŸ we put # + # +into (25.3), this is the same as setting ø = 2 in (25.4), and so on. +In more complicated problems, there may be more derivatives, and more +terms in Ù; the question of interest is whether the two equations (25.3) and (25.4) +are maintained or not. If they are, we call such a problem a iZ»eør problem. In +this chapter we shall discuss some of the properties that exist because the system +1s linear, to appreciate the generality of some of the results that we have obtained +in our special analysis of a special equation. +Now let us study some of the properties of linear differential equations, having +illustrated them already with the specific equation (25.1) that we have studied +so closely. "The first property of interest is this: suppose that we have to solve +the diferential equation for a transient, the free oscillation with no driving force. +'That is, we want to solve +L(z) =0. (25.5) +Suppose that, by some hook or crook, we have found a particular solution, which +we shall call z¡. That is, we have an #¡ for which L(z¡) =0. Now we notice +that øz, 1s also a solution to the same equation; we can multiply this special +solution by any constant whatever, and get a new solution. In other words, IŸ we +had a motion of a certain “size,” then a motion ©wice as “big” is again a solution. +Proof: L(a#1) = &E(#1) = a-0 =0. +Next, suppose that, by hook or by crook, we have not only found øøwe solu- +tion #, but also another solution, z¿. (Remember that when we substituted +œ = e?®† for finding the transients, we found f#+»o values for œ, that is, two solutions, +#¡ and #øa.) Now let us show that the combination (# + #a) is also a solution. In +other words, if we put #ø = #1 + #a, # is again a solution of the equation. Why? +Because, if U(z¡) = 0 and (4a) = 0, then E(zi+z2) = E(œi)+ E(z:) = 0+0 = 0. +So if we have found a number of solutions for the motion of a linear system we +can add them together. +Combining these two ideas, we see, of course, that we can also add six of +one and two of the other: IÝ ø is a solution, so is œ#. Therefore any sum of +these tEwo solutions, such as (œ#i + z2), is also a solution. If we happen to +be able to fnd three solutions, then we fñnd that any combination of the three +solutions is again a solution, and so on. Iỳ turns out that the number of what +--- Trang 438 --- +we call ?dependent solufions* that we have obtained for our oscillator problem +is only ưuo. The number of independent solutions that one finds in the general +case depends upon what is called the number of degrees oƒ freedom. We shall +not discuss this in detail now, but if we have a second-order difÑferential equation, +there are only two independent solutions, and we have found both of them; so +we have the most general solution. +Now let us go on to another proposition, which applies to the sibtuation in +which the system is subjected to an outside force. Suppose the equation 1s +L(z) = F(). (25.6) +and suppose that we have found a special solution of it. Let us say that Joe”s +solution is z;, and that E(z;) = Ƒ). Šuppose we want to find yet another +solution; suppose we add to Joe”s solution one of those that was a solution of the +free equation (25.5), say z¡. Then we see by (25.3) that +TE(z„ + #1) = L(z) + L(xì) = F() +0 = F0). (25.7) +Therefore, to the “forced” solution we can add any “free” solution, and we still +have a solution. 'Phe free solution is called a frøns¿en‡ solution. +'When we have no force acting, and suddenly turn one on, we do not imme- +diately get the steady solution that we solved for with the sine wave solution, +but for a while there is a transient which sooner or later dies out, IÝ we wait long +enough. 'Phe “forced” solution does not die out, since it keeps on being driven +by the force. Ultimately, for long periods of time, the solution 1s unique, but +initially the motions are diferent for different circumstances, depending on how +the system was started. +25-2 Superposition of solutions +Now we come to another interesting proposition. Suppose that we have a +certain particular driving force #4 (let us say an oscillatory one with a cerbain +œ = œ„, but our conclusions will be true for any functional form oŸ F2) and we +have solved for the forced motion (with or without the transients; it makes no +difference). NÑow suppose some other force is acting, let us say #}ÿ, and we solve +* Solutions which cannot be expressed as linear combinations of each other are called +independent. +--- Trang 439 --- +the same problem, but for this different force. hen suppose someone comes +along and says, “[ have a new problem for you to solve; I have the force „ + Fỳ.” +Can we do it? Of course we can do it, because the solution is the sum of the +two solutions #„ and zø; for the forces taken separately——a most remarkable +circumstance indeed. IÝ we use (25.3), we see that +L(#¿ + #ụ) = L(z4) + L(œy) = †„) + tịÚ). (25.8) +This is an example of what is called the prznciple oƒ superposition for linear +systems, and it is very important. It means the following: if we have a complicated +force which can be broken up in any convenient manner into a sum of separate +pieces, each of which is in some way simple, in the sense that for each special +piece into which we have divided the force we can solve the equation, then the +answer is available for the +0hole force, because we may simply add the pieces of +the solufion back together, in the same manner as the total ƒorce is compounded +out of pieces (Eig. 25-1). +Fạ + Fp +Xa -F Xpb +Fig. 25-1. An example of the principle of superposition for linear +Systems. +--- Trang 440 --- +Let us give another example of the prineiple of superposition. In Chapter 12 +we said that it was one of the great facts of the laws of electricity that if we have +a certain distribution of charges ga and calculate the electric fñield #Z„ arising from +these charges at a certain place , and ïf, on the other hand, we have another +set of charges q; and we calculate the feld #; due to these at the corresponding +place, then if both charge distributions are present at the same time, the field +at P is the sưm of E„ due to one set plus #; due to the other. In other words, +1ƒ we know the fñeld due to a certain charge, then the feld due to many charges +is merely the vector sum of the ñelds of these charges taken individually. 'This +is exactly analogous to the above proposition that if we know the result of two +given forces taken at one time, then if the force is considered as a sum of them, +the response is a sum of the corresponding individual responses. +` dụ ` F +Fig. 25-2. The principle of superposition in electrostatics. +'The reason why this is true in electricity is that the great laws of electricity, +Maxwell's equations, which determine the electric field, turn out to be diferential +cquations which are ineør, ¡.e., which have the property (25.3). What corresponds +to the force is the chørøe generating the electric fñeld, and the equation which +determines the electric ñeld in terms of the charge is linear. +As another interesting example of this proposition, let us ask how it is possible +to “tune in” to a particular radio station at the same time as all the radio stations +are broadcasting. 'Phe radio station transmits, fundamentally, an oscillating +electric fñield of very high frequency which acts on our radio antenna. Ït is true +that the amplitude of the oscillation of the field ¡is changed, modulated, to carry +the signal of the voice, but that is very slow, and we are not going to worry about +it. When one hears “'Phis station is broadcasting at a frequency of 780 kilocycles,” +this indicates that 780,000 oscillations per second is the frequency of the electric +field of the station antenna, and this drives the electrons up and down at that +frequency in our antenna. Now at the same time we may have another radio +station in the same town radiating at a diferent frequency, say 550 kilocycles per +second; then the electrons in our antenna are also being driven by that frequency. +Now the question is, how is it that we can separate the signals coming into the +--- Trang 441 --- +one radio at 780 kilocycles from those coming in at 550 kilocycles? We certainly +do not hear both stations at the same tỉme. +By the principle of superposition, the response of the electric circuit in the +radio, the first part of which is a linear circuit, to the forces that are acting due +to the electric field q + Fỳ, is z„ + øạ. It therefore looks as though we will never +disentangle them. In fact, the very proposition of superposition seems to insist +that we cannot øø0oø?d having both of them In our system. But remember, for a +resonanf circuit, the response curve, the amount oŸ z per unit ?, as a function of +the requency, looks like Fig. 25-3. If it were a very high @ circuit, the response +would show a very sharp maximum. Now suppose that the two stations are +comparable in strength, that is, the two ƒorces are of the same magnitude. 'Phe +response that we get 1s the sum of ø„ and ø;. But, in Eig. 25-3, #ø„ is tremendous, +while #; is small. 5o, in spite of the fact that the two signals are equal in strength, +when they go through the sharp resonant circuit of the radio tuned for œ¿, the +frequency of the transmission of one station, then the response to this station +1s mụuch greater than to the other. Therefore the complete response, with both +signals acting, is almost all made up oŸ œ„, and we have selected the station we +œp_ (Úc ta ø +Fig. 25-3. A sharply tuned resonance curve. +Now what about the tuning? How do we tune it? We change œọ by changing +the Ù or the Œ of the circuit, because the frequency of the circuit has to do with +the combination of Ù and Œ. In particular, most radios are built so that one can +change the capacitance. When we retune the radio, we can make a new setting +of the dial, so that the natural frequenecy of the circuit is shifted, say, to œe. In +those circumstances we hear neither one station nor the other; we get silence, +provided there is no other station at frequency œ„. lf we keep on changing the +--- Trang 442 --- +capacitance until the resonance curve 1s at œụ, then of course we hear the other +station. 'That is how radio tuning works; it is again the prineciple of superposition, +combined with a resonant response.* +To conclude this discussion, let us describe qualitatively what happens if we +proceed further in analyzing a linear problem with a given force, when the force is +quite complicated. Out of the many possible procedures, there are two especially +useful general ways that we can solve the problem. Ône is this: suppose that we +can solve it for special known forces, such as sine waves of different frequencies. +W©e know it is child”s play to solve it for sine waves. So we have the so-called +“child°s play” cases. Now the question is whether our very complicated force +can be represented as the sum oŸ two or more “child”s play” forces. In Fig. 25-1 +we already had a fairly complicated curve, and of course we can make i% more +complicated still if we add in more sine waves. So 1t is certainly possible to +obtain very complicated curves. And, in fact, the reverse is also true: practically +every curve can be obtained by adding together ?mfinite muwmbers oŸ sỉine waves of +điferent wavelengths (or frequencies) for each one of which we know the answer. +W© just have to know how mụch of each sine wave to put in to make the given #', +and then our answer, zø, is the corresponding sum of the # sine waves, each +multipled by its effective ratio of z to #'Ô This method of solution is called +the method of Fourier transƒforms or Fourier œnalusis. YNe are not going to +actually carry out such an analysis just now; we only wish to describe the idea +involved. +Another way in which our complicated problem can be solved is the following +very interesting one. Suppose that, by some tremendous mental efort, it were +possible to solve our problem for a special force, namely an Zmpuise. 'The Íorce is +quickly turned on and then of; ït ¡is all over. Actually we need only solve for an +impulse of some unit strength, any other strength can be gotten by multiplication +by an appropriate factor. We know that the response ø for an impulse is a +damped oscillation. NÑow what can we say about some other Íorce, for instance a +force like that of Fig. 25-4? +Such a force can be likened to a succession of blows with a hammer. First there +1s no force, and all of a sudden there is a steady force—impulse, impulse, impulse, +* In modern superheterodyne receivers the actual operation is more complex. The amplifers +are all tuned to a fixed frequency (called IEF frequency) and an oscillator of variable tunable +frequency is combined with the input signal in a n„onl¿near circuit to produce a new frequency +(the diference of signal and oscillator frequency) equal to the IE frequency, which is then +amplifed. 'Phis will be discussed in Chapter 50. +--- Trang 443 --- +Fig. 25-4. A complicated force may be treated as a succession of +sharp Iimpulses. +impulse,... and then it stops. In other words, we imagine the continuous force +to be a series of impulses, very close together. Now, we know the result for an +Impulse, so the result for a whole series of impulses will be a whole series of +damped oscillations: it will be the curve for the first impulse, and then (slightly +later) we add to that the curve for the second impulse, and the curve for the +third impulse, and so on. 'Phus we can represent, mathematically, the complete +solution for arbitrary functions if we know the answer for an impulse. We get +the answer for any other force simply by integrating. This method is called the +Greens ƒunction rmnethod. A Green?s function is a response to an impulse, and +the method of analyzing any force by putting together the response of impulses +1s called the Green's function method. +The physical prineiples involved in both of these schemes are so simple, +involving just the linear equation, that they can be readily understood, but the +mathemaftical problems that are involved, the complicated integrations and so on, +are a little too advanced for us to attack right now. You will most likely return +to this some day when you have had more practice in mathematics. But the ?dea +1s very simple indeed. +Finally, we make some remarks on why Ï?near systems are so important. The +answer is simple: because we can solve theml So most of the tỉme we solve linear +problems. Second (and most important), it turns out that the ƒundamental laus +0ƒ phụsics are oflen linear. The Maxwell equations for the laws of electricity are +linear, for example. "The great laws of quantum mechanics turn out, so far as +we know, to be linear equations. 7 hø‡ is why we spend so much time on linear +cequations: because if we understand linear equations, we are ready, in principle, +to understand a lot of things. +We mention another situation where linear equations are found. When +displacements are small, many functions can be øpprozzmaœtcd linearly. For +--- Trang 444 --- +example, if we have a simple pendulum, the correct equation for its motion is +d20/dt? = —(g/L) sin 0. (25.9) +This equation can be solved by elliptic funections, but the easiest way to solve 1W is +numerically, as was shown in Chapter 9 on Newton”s Laws of Motion. A nonlinear +equation cannot be solved, ordinarily, any other way 0u# numerically. Now for +small Ø, sinØ is practically equal to Ø, and we have a linear equation. It turns +out that there are many circumstances where small efects are linear: for the +example here the swing of a pendulum through small arcs. As another example, +1ƒ we pull a little bit on a spring, the force is proportional to the extension. lÝ we +pull hard, we break the spring, and the force is a completely diferent function of +the distancel Linear equations are important. In fact they are so important that +perhaps fifty percent of the time we are solving linear equations in physics and +1n engineering. +25-3 Oscillations ỉn linear systems +Let us now review the things we have been talking about in the past few +chapters. lt is very easy for the physics of oscillators to become obscured by +the mathematics. The physics is actually very simple, and if we may forget the +mathematics for a moment we shall see that we can understand almost everything +that happens in an oscillating system. First, ifƒ we have only the spring and the +weight, it is easy to understand why the system oscillates—it is a consequence +of inertia. We pull the mass down and the force pulls it back up; as 1È passes +zoro, which is the place it likes to be, it cannot Just suddenly stop; because of its +qmomentum it keeps on goïing and swings to the other side, and back and forth. +So, if there were no fÍriction, we would surely expect an oscillatory motion, and +indeed we get one. But if there is even a little bit of friction, then on the return +cycle, the swing will not be quite as high as it was the first time. +Now what happens, cycle by cycle? 'Phat depends on the kind and amount +of friction. Suppose that we could concoct a kind of friction force that always +remains in the same proportion to the other forces, of inertia and in the spring, +as the amplitude of oscillation varies. In other words, for smaller oscillations +the friction should be weaker than for big oscillations. Ordinary friction does +not have this property, so a special kind of friction must be carefully invented +for the very purpose of creating a friction that is directly proportional to the +--- Trang 445 --- +velocity——so that for big oscillations iỀ is stronger and for small oscillations it +1s weaker. lf we happen to have that kind of friction, then at the end of each +successive cycle the system is in the same condition as it was at the start, except +a little bit smaller. All the forces are smaller in the same proportion: the spring +force 1s reduced, the inertial efects are lower because the accelerations are now +weaker, and the friction is less too, by our careful design. When we actually +have that kind of fiction, we ñnd that each oscillation is exactly the same as the +first one, except reduced in amplitude. If the first cycle dropped the amplitude, +say, to 90 percent of what it was at the start, the next will drop it to 90 percent +of 90 percent, and so on: ứhe sizes öƒ the oscdllatlions are reduced bụ the same +fraclion oj themselues in cuerw cụcÌle. An exponential function is a curve which +does just that. It changes by the same factor in each equal interval of time. That +1s to say, 1ƒ the amplitude of one cycle, relative to the preceding one, is called ø, +then the amplitude of the next is a2, and of the next, ø”. So the amplitude is +some constant raised to a power equal to the number of cycles traversed: +A= Aoad". (25.10) +But of course m œ ứ, so it is perfectly clear that the general solution will be some +kind of an oscillation, sine or cosine œ#, tỉmes an amplitude which goes as ÙÍ +more or less. But ö can be written as e °, 1ƒ b is positive and less than 1. So this +is why the solution looks like e~“ cosœo#. It is very sỉmple. +'What happens ïf the friction is not so artificial; for example, ordinary rubbing +on a table, so that the friction force is a certain constant amount, and is indepen- +dent of the size of the oscillation that reverses its direction each hal£cycle? Then +the equation is no longer linear, i§ becomes hard to solve, and must be solved +by the numerical method given in Chapter 9, or by considering each half-cycle +separately. 'Phe numerical method is the most powerful method of all, and can +solve any equation. lt is only when we have a simple problem that we can use +mathematical analysis. +Mathematical analysis is not the grand thing it is said to be; it solves only +the simplest possible equations. Äs soon as the equations get a little more +complicated, just a shade—they cannot be solved analytically. But the numerical +method, which was advertised at the beginning of the course, can take care of +any equation of physical interest. +Next, what about the resonance curve? Why is there a resonance? First, +Imagine for a moment that there is no friction, and we have something which +--- Trang 446 --- +could oscillate by itself. If we tapped the pendulum just right each time it went +by, of course we could make it go like mad. But if we close our eyes and do +not watch ït, and tap at arbitrary equal intervals, what is going to happen? +Sometimes we will fnd ourselves tapping when it is goỉing the wrong way. When +we happen to have the timing just right, of course, each tap is given at just the +right time, and so i% goes higher and higher and higher. So without friction we +get a curve which looks like the solid curve in Eig. 25-5 for diferent frequencies. +Qualitatively, we understand the resonance curve; in order to get the exact shape +of the curve it is probably just as well to do the mathematics. The curve goes +toward infinity as œ —> œọ, where œọ is the natural frequency of the oscillator. +Fig. 25-5. Resonance curves with various amounts of friction present. +Now suppose there is a little bit of friction; then when the displacement of +the oscillator is small, the friction does not affect it much; the resonance curve +1s the same, except when we are near resonance. Instead of becoming infinite +near resonance, the curve is only going to get so hiph that the work done by +our tapping each time is enough to compensate for the loss of energy by friction +during the cycle. 5o the top oŸ the curve is rounded oÑ——it does not go to infnity. +lf there is more friction, the top of the curve is rounded off still more. Now +someone might say, “I thought the widths of the curves depended on the friction.” +'That is because the curve is usually plotted so that the top of the curve ¡s called +one unit. However, the mathematical expression is even simpler to understand 1ƒ +we just plot all the curves on the same scale; then all that happens is that the +friction cuts down the topl T there is less friction, we can go farther up into that +little pinnacle before the friction cuts it of, so it looks relatively narrow. That is, +--- Trang 447 --- +the higher the peak of the curve, the narrower the width at half the maximum +height. +Jinally, we take the case where there is an enormous amount of friction. lt +turns out that iŸ there is too much friction, the system does not oscillate at all. +The energy in the spring is barely able to move it against the frictional force, +and so it slowly oozes down to the equilibrium poiïnt. +25-4 Analogs in physics +'The next aspect of this review is to note that masses and springs are not the +only linear systems; there are others. In particular, there are electrical systems +called linear circuits, in which we fnd a complete analog to mechanical systems. +W© dịd not learn exactly hy each of the objects in an electrical cireuit works in +the way it does—that is not to be understood at the present moment; we may +assert 1% as an experimentally verifable fact that they behave as stated. +For example, let us take the sinplest possible cireumstance. We have a piece +of wire, which is just a resistance, and we have applied to it a difference In +potential, V. Now the V means this: if we carry a charge g through the wire +from one terminal to another terminal, the work done is gV. 'Phe higher the +voltage diference, the more work was done when the charge, as we say, “falls” +from the hiph potential end of the terminal to the low potential end. So charges +release energy in goiïng om one end to the other. Now the charges do not simply +fñy om one end straight to the other end; the atoms in the wire ofer some +resistance to the current, and this resistance obeys the following law for almost +all ordinary substances: If there is a current 7, that is, so and so many charges +per second tumbling down, the number per second that comes tumbling through +the wire is proportional to how hard we push them——in other words, proportional +to how much voltage there is: +V =1TR= R(dq/d). (25.11) +'The coefficient ?‡ ¡is called the resisfance, and the equation is called Ohm's Law. +'The unit of resistanee is the ohm; it is equal to one volt per ampere. In mechanical +situations, to get such a frictional force in proportion to the velocity is dificult; +in an electrical system it is very easy, and this law is extremely accurate for most +mnetals. +W© are often interested in how much work is done per second, the power Ìoss, +or the energy liberated by the charges as they tumble down the wire. When +--- Trang 448 --- +we carry a charge g through a voltage V, the work is gV, so the work done per +second would be V(dqg/đ£), which is the same as Vĩ, or also IR- I = I2R. This +is called the heøat#ng loss—this is how much heat is generated in the resistance +per second, by the conservation of energy. It is this heat that makes an ordinary +incandescent light bulb work. +Of course, there are other interesting properties of mechanical systems, such +as the mass (inertia), and it turns out that there is an electrical analog to inertia +also. It is possible to make something called an znmductor, having a property +called zmductance, such that a current, once started through the inductance, đoes +not tuant to stop. Tt requires a voltage in order to change the currentl TẾ the +curren£ is constant, there is no voltage across an inductance. DƠC circuits do not +know anything about inductance; it is only when we changøe the current that the +efects of inductance show up. The equation is +V = L(d1/dt) = L(d°q/dt), (25.12) +and the unit of inductance, called the henr, 1s such that one volt applied to +an inductance of one henry produces a change of one ampere per second in the +current. Equation (25.12) is the analog of Ñewton”s law for electricity, if we wish: +V corresponds to #', Ù corresponds to mm, and Ï corresponds to velocityl All of the +consequent equations for the two kinds of systems will have the same derivations +because, in all the equations, we can change any letter to its corresponding +analog letter and we get the same equation; everything we deduce will have a +correspondence in the two systems. +Now what electrical thing corresponds to the mechanical spring, in which +there was a force proportional to the stretch? If we start with # = kz and replace +†'— V and z — q, we get V = ag. lt turns out that there 7s such a thing, in fact +1t is the only one of the three circuit elements we can really understand, because +we did study a pair of parallel plates, and we found that If there were a charge of +certain equal, opposite amounts on each plate, the electric fñeld between them +would be proportional to the size of the charge. 5o the work done in moving a +unit charge across the gap from one plate to the other is precisely proportional to +the charge. This work is the definiiion of the voltage difference, and it is the line +integral of the electric field from one plate to another. It turns out, for historical +reasons, that the constant oŸ proportionality is not called Œ, but 1/Œ. It could +have been called Œ, but it was not. So we have +V =q/C. (25.13) +--- Trang 449 --- +'The unit of capacitance, Œ, is the farad; a charge of one coulomb on each plate +of a one-farad capacitor yields a voltage diference of one volt. +There are our analogies, and the equation corresponding to the oscillating +circuit becomes the following, by direct substitution of Ù for rn, q for ø, etc: +m(d®+/dt?) + ym(da/dE) + ka = F, (25.14) +L(d°q/dt2) + R(dq/dt) + q/C = V. (25.15) +Now everything we learned about (25.14) can be transformed to apply to (25.15). +lvery conseqguence is the same; so mụuch the same that there is a brilliant thing +we can do. +Suppose we have a mechanical system which is quite complicated, not Just +one mass on a spring, but several masses on several springs, all hooked together. +What do we do? Solve it? Perhaps; but look, we can make an clecfr?cal circuit +which will have the same equations as the thing we are trying to analyzel For +instance, iŸ we wanted to analyze a mass on a spring, why can we not build +an electrical circuit in which we use an inductance proportional to the mass, a +resistance proportional to the corresponding +, 1/C proportional to k, allin +the same ratio? 'Phen, of course, this electrical circuit will be the exact analog +of our mechanical one, in the sense that whatever g does, in response to V +(V also is made to correspond to the forces that are acting), so the # would +do in response to the forcel So if we have a complicated thing with a whole +lot of interconnecting elements, we can interconnect a whole lot of resistances, +inductaneces, and capacitances, to #mn‡øte the mechanically complicated system. +What is the advantage to that? One problem is jus6 as hard (or as easy) as +the other, because they are exactly equivalent. The advantage is not that it is +any easier to solve the rmmathematical equations after we discover that we have an +electrical circuit (although that ¿s the method used by electrical engineersl), but +instead, the real reason for looking at the analog is that it is easier to make the +electrical circuit, and to chønge something in the system. +Suppose we have desipgned an automobile, and want to know how much it +1s going to shake when iÈ goes over a certain kind of bumpy road. We build an +electrical eireuit with inductances to represent the inertia of the wheels, spring +constants as capacitances to represent the springs of the wheels, and resistors to +represent the shock absorbers, and so on for the other parts of the automobile. +Then we need a bumpy road. All right, we apply a 0ol#age from a generator, +which represents such and such a kind of bump, and then look at how the left +--- Trang 450 --- +wheel jiggles by measuring the charge on some capacitor. Having measured it +(it is easy to do), we fñnd that it is bumping too much. Do we need more shock +absorber, or less shock absorber? With a complicated thíng like an automobile, +do we actually change the shock absorber, and solve it all over again? Nol, we +simply turn a dial; dial number ten is shock absorber number three, so we put ín +more shock absorber. 'Phe bumps are worse—all right, we try less. The bumps +are sbill worse; we change the stifness of the spring (dial 17), and we adjust all +these things eleciricallu, with merely the turn of a knob. +This is called an ønalog compu£er. Tt is a device which imitates the problem +that we want to solve by making another problem, which has the same equation, +but in another circumstance of nature, and which is easier to build, to measure, +to adjust, and to destroyl +25-5 Series and parallel impedances +Finally, there is an important item which is not quite in the nature of review. +'This has to do with an electrical cireuit in which there is more than one circuit +element. Eor example, when we have an inductor, a resistor, and a capacitor +connected as in Eig. 24-2, we note that all the charge went through every one +of the three, so that the current in such a singly connected thing is the same at +all points along the wire. Since the current is the same in each one, the voltage +across Ÿ‡ is I, the voltage across Ù is E(đdTI/đf), and so on. So, the total voltage +drop is the sum of these, and this leads to Eq. (25.15). Using complex numbers, +we found that we could solve the equation for the steady-state motion in response +to a sinusoidal force. We thus found that Ÿ = ÊÂ. Now Z is called the ?mpcdance +of this particular circuit. It tells us that if we apply a sinusoidal voltage, £, Ww© +get a current Ỉ. +Now suppose we have a more complicated circuit which has two pieces, +which by themselves have certain impedances, ÊWŸ¡ and 22 and we put them in +1 [2] [Z:] 2 1 2 +(a) Series (b) Parallel +Fig. 25-6. Two impedances, connected in series and ¡n parallel. +--- Trang 451 --- +series (Eig. 25-6a) and apply a voltage. What happens? It is now a little more +complicated, but if Ÿ is the current through VN the voltage diference across ôi, +1s ñ =Ỉ 2: similarly, the voltage across 2; 1s llổ =Ï VN The same current goes +through both. Thherefore the total voltage is the sum of the voltages across the +two sections and is equal to Ÿ= ñ + Ññ = (2¡ + 2,)Ï. This means that the +voltage on the complete circuit can be written Ÿ=Í 2.. where the VÀ of the +combined system in series is the sum of the two 2s of the sepDarate pieces: +2, = 2¡+ôa. (25.16) +This is not the only way things may be connected. We may aÌso connect +them in another way, called a parailel connection (Fig. 25-6b). Now we see that a +given voltage across the terminals, if the connecting wires are perfect conductfors, +1s efectively applied to both of the impedances, and will cause currents in each +independently. Therefore the current through Ñ¡ is cqual to ñ = / 2¡. The +current in 2, 1s TP = Ÿ/2¿. Tt is the sœme 0oltage. Now the total current +which is supplied to the terminals is the sưzn of the currents in the two sections: +? =Ÿ/ô\ +Ÿ/2;¿. Thịs can be written as +(1/22)+ (1/22) +1/2„ = 1/2¡ + 1/2. (25.17) +More complicated circuits can sometimes be simplified by taking pieces of +them, working out the succession of Impedances of the pieces, and combining +the circuit together step by step, using the above rules. If we have any kind of +circuit with many impedances connected ín all kinds of ways, and if we include +the voltages in the form of little generators having no impedance (when we pass +charge through it, the generator adds a voltage WV}), then the following principles +apply: (1) At% any junction, the sum oŸ the currents into a junction is zero. +That is, all the current which comes in must come back out. (2) IÝ we carry a +charge around any loop, and back to where it started, the net work done is zero. +These rules are called zchhoff s laas for electrical circuits. Theïr systematic +application to complicated circuits often simplifies the analysis of such circuits. +We mention them here in conjunction with Eqs. (25.16) and (25.17), in case you +have already come across such circuits that you need to analyze in laboratory +work. They will be discussed again in more detail next year. +--- Trang 452 --- +€)pfics: To EPrirtcfpÏlo oŸ Loáist Time© +26-1 Light +This is the fñrst of a number of chapters on the subject of electromagnetic +radiation. Light, with which we see, is only one small part of a vast spectrum of +the same kind of thing, the various parts of this spectrum being distinguished by +diferent values oŸ a certain quantity which varies. 'Phis variable quantity could be +called the “wavelength” As it varies in the visible spectrum, the light apparently +changes color from red to violet. If we explore the spectrum systematically, from +long wavelengths toward shorter ones, we would begin with what are usually called +radiotues. Radiowaves are technically available in a wide range of wavelengths, +some even longer than those used in regular broadcasts; regular broadcasts have +wavelengths corresponding to about 500 meters. 'Phen there are the so-called +“short waves,” i.e., radar waves, millimeter waves, and so on. There are no actual +boundaries between one range of wavelengths and another, because nature did +not present us with sharp edges. The number associated with a given name for +the waves are only approximate and, of course, so are the names we give to the +diferent ranges. +Then, a long way down through the millimeter waves, we come to what +we call the ?mƒrared, and thence to the visible spectrum. Then going in the +other direction, we get into a region which is called the ui#rœoolet. Where the +ultraviolet stops, the x-rays begin, but we cannot defne precisely where this +is; it is roughly at 10” m, or 1072 ø. These are “soft” x-rays; then there are +ordinary x-rays and very hard x-rays; then +-rays, and so on, for smaller and +smaller values of this dimension called the wavelength. +Within this vast range of wavelengths, there are three or more regions of +approximation which are especially interesting. In one of these, a condition exists +in which the wavelengths involved are very small compared with the dimensions +of the equipment available for their study; furthermore, the phobon energies, +--- Trang 453 --- +using the quantum theory, are small compared with the energy sensitivity of the +equipment. nder these conditions we can make a rough frst approximation +by a method called geometrical opfics. TỶ, on the other hand, the wavelengths +are comparable to the dimensions of the equipment, which is difficult to arrange +with visible light but easier with radiowaves, and ïf the photon energies are still +negligibly small, then a very useful approximation can be made by studying the +behavior of the waves, still disregarding the quantum mechanics. This method is +based on the classical theor oƒ electromagnetic radiation, which will be discussed +in a later chapter. Next, If we go to very short wavelengths, where we can +disregard the wave character but the photons have a very Íarge energy compared +with the sensitivity of our equipment, things get simple again. 'This ¡is the simple +photon picture, which we will describe only very roughly. The complete picture, +which unifies the whole thing into one model, will not be available to us for a +long time. +In this chapter our discussion is limited to the geometrical optics region, in +which we forget about the wavelength and the photon character of the lght, +which will all be explained in due time. We do not even bother to say what +the light zs, but just fñnd out ho ?£ behœues on a large scale compared with +the dimensions of interest. All this must be said in order to emphasize the fact +that what we are going to talk about is only a very crude approximation; this +is one of the chapters that we shall have to “unlearn” again. But we shall very +quickly unlearn it, because we shall almost immediately go on to a more accurate +mnethod. +Although geometrical optics is just an approximation, it is of very great +importance technically and of great interest historically. We shall present this +subject more historically than some of the others in order to give some idea. of +the development oŸ a physical theory or physical idea. +tirst, light is, of course, familiar to everybody, and has been familiar since +time mmemorial. NÑow one problem is, by what process do we see light? There +have been many theories, but it finally settled down to one, which is that there +1s something which enters the eye—which bounces of objJects into the eye. We +have heard that idea so long that we accept it, and it is almost impossible for +us to realize that very intelligent men have proposed contrary theories—that +something comes out of the eye and feels for the obJect, for example. Some other +Important observations are that, as light goes from one place to another, it goes +in sứraight lines, 1Ÿ there 1s nothing in the way, and that the rays do not seem +to interfere with one another. hat is, light is crisscrossing in all directions in +--- Trang 454 --- +the room, but the light that is passing across our line of vision does not affect +the light that comes to us from some object. 'This was once a most powerful +argument against the corpuscular theory; it was used by Huygens. If light were +like a lot of arrows shooting along, how could other arrows go through them so +easily? Such philosophical arguments are not of mụch weight. One could always +say that light is made up of arrows which go through each otherl +26-2 Reflection and refraction +The discussion above gives enough of the basic iđeø of geometrical optics—now +we have to go a little further into the quantitative features. Thus far we have light +going only in straight lines bebween two points; now let us study the behavior +of light when it hits various materials. The simplest object is a mirror, and the +law for a mirror is that when the light hits the mirror, it does not continue in a +straight line, but bounces of the mirror into a new straight line, which changes +when we change the inclination of the mirror. 'Phe question for the aneients +was, what is the relation between the two angles involved? This is a very simple +relation, discovered long ago. 'Phe light striking a mirror travels in such a way +that the two angles, between each beam and the mirror, are equal. For some +reason iÈ is customary to measure the angles from the normal to the mirror +surface. Thus the so-called law of refection 1s +0; = Ú,. (26.1) +'That is a simple enough proposition, but a more dificult problem is encoun- +tered when light goes from one medium into another, for example from air into +water; here also, we see that it does not go in a straight line. In the water the +ray is a% an inclination to its path in the air; if we change the angle Ø; so that +1t comes down more nearly vertically, then the angle of “breakage” is not as +\ø, ro, Ế +Fig. 26-1. The angle of incidence ¡s equal to the angle of reflection. +--- Trang 455 --- +Fig. 26-2. A light ray Is refracted when It posses from one medium +Into another. +Table 26-1 Table 26-2 +Anglein air Angle in water Anglein air Angle in water +10° 8° 10° 7-1/2° +207 15-1/2 207 15° +30 22-1/2° 30 22” +400 29 40 29° +502 35° 502 35° +60° 40-1/2° 60° 40-1/2° +702 45-1/2° 702 45” +80 502 802 48” +great. But iŸ we tilt the beam of light at quite an angle, then the deviation angle +1s very large. The question is, what is the relation of one angle to the other? +This also puzzled the ancients for a long time, and here they never found the +answerl It is, however, one of the few places in all of Greek physics that one +may fñnd any experimental results listed. Claudius Ptolemy made a list of the +angle in water for each of a number of diferent angles in air. Table 26-1 shows +the angles In the air, in degrees, and the corresponding angle as measured In +the water. (Ordinarily it is said that Greek scientists never did any experiments. +But it would be impossible to obtain this table of values without knowing the +ripht law, except by experiment. It should be noted, however, that these do +not represent independent careful measurements for each angle but only some +numbers interpolated from a few measurements, for they all ft perfectly on a +parabola.) +--- Trang 456 --- +'This, then, is one of the important steps in the development of physical law: +frst we observe an efect, then we measure it and list it in a table; then we try +to fnd the ruie by which one thing can be connected with another. The above +numerical table was made in 140 A.D., but ít was not until 1621 that someone +finally found the rule connecting the two anglesl The rule, found by Willebrord +Snell, a Dutch mathematician, is as follows: if Ø; is the angle in air and Ø„ is the +angle in the water, then i% turns out that the sine of Ø; is equal to some constant +multiple of the sine of Ø,„: +sin Ø¿ = n0sin Ø„. (26.2) +For water the number ø is approximately 1.33. Equation (26.2) is called Šnells +la; 1 permits us to predict how the light is goïng to bend when it goes Írom air +into water. Table 26-2 shows the angles in air and in water according to Snells +law. Note the remarkable agreement with Ptolemy's list. +26-3 Eermat?s principle of least tỉme +Now in the further development of science, we want more than just a formula. +Pirst we have an observation, then we have numbers that we measure, then we +have a law which summarizes all the numbers. But the real gior of science 1s +that te can fnd a U0ay öƒ thinkứng súch that the law 1s cuident. +The fñrst way of thinking that made the law about the behavior of light evident +was discovered by Fermat in about 1650, and it is called ¿he pr¿inciple oƒ least +time, or Ferma†s principle. His idea 1s thịs: that out of all possible paths that it +might take to get from one point to another, light takes the path which requires +the shortest từmc. +Let us first show that this is true for the case of the mirror, that this simple +prineiple contains both the law of straight-line propagation and the law for the +mirror. So, we are growing in our understandingl Let us try to ñnd the solution +to the following problem. In Eig. 26-3 are shown two points, A and Ö, and a +plane mirror, 1ƒ“. What is the way to get rom A to in the shortest time? +The answer is to go straight from 4 to Bƒ But if we add the extra rule that the +light has to sfứrike the mưrror and come back in the shortest time, the answer is +not so easy. QÔne way would be to go as quickly as possible to the mirror and +then go to Ö, on the path AJD2B. Of course, we then have a long path 2Ö. If we +move over a little to the right, to 2, we slipghtly increase the first distance, but +we greatly decrease the second one, and so the total path length, and therefore +--- Trang 457 --- +A __===—— Z +ụ _+Z : +Ẫ ` ZZ ⁄ +LÀ.^ +có CIỦI ¡ +I I Mà. I I +Fig. 26-12. A paraboloidal mirror. +miles away; we would like to cause all the light that comes in to come to a Íocus. +Of course we cannot draw the rays that go all the way up to the star, but we +still want to check whether the times are equal. Of course we know that when +the various rays have arrived at some plane ##“”, perpendicular to the rays, all +the tỉimes in this plane are equal (Eig. 26-12). The rays must then come down +to the mirror and proceed toward ¡n equal times. That is, we must fnd a +curve which has the property that the sum of the distances XX” + X”P' is a +constant, no matter where X is chosen. An easy way to find it is to extend the +length of the line XX” down to a plane ÙE. Now 1ƒ we arrange our curve so that +.A“=AP, BE" =BT', CC” = C”P', and so on, we will have our curve, +because then of course, 4A” + AP' = AA' + A7A” will be constant. Thus our +curve is the locus of all points equidistant from a line and a point. Such a curve +1s called a parabola; the mirror is made in the shape of a parabola. +'The above examples illustrate the principle upon which such optical devices +can be designed. The exact curves can be calculated using the principle that, to +focus perfectly, the travel times must be exactly equal for all light rays, as well +as being less than for any other nearby path. +We shall discuss these focusing optical devices further in the next chapter; let +us now discuss the further development of the theory. When a new theoretical +principle ¡is developed, such as the principle of least time, our first inclination +might be to say, “Well, that is very pretty; it is delightful; but the question is, +does it help at all in understanding the physics?” Someone may say, “Yes, look +at how many things we can now understand!” Another says, “Very well, but ÏI +can understand mirrors, too. Ï need a curve such that every tangent plane makes +--- Trang 465 --- +equal angles with the bwo rays. I can figure out a lens, too, because every ray +that comes to it is bent through an angle given by Snells law.” Evidently the +statement of least time and the statement that angles are equal on refection, +and that the sines of the angles are proportional on refraction, are the same. So +1s Iÿ merely a philosophical question, or one of beauty? 'There can be arguments +on both sides. +However, the Importance of a powerful prineiple is that #‡ predicts neu things. +Tt is easy to show that there are a number oŸ new things predicted by Fermaf”s +principle. First, suppose that there are f£hree media, glass, water, and aïr, and +we perform a refraction experiment and measure the index ø for one medium +against another. Let us call ma the index of air (1) against water (2); ma the +index of air (1) against glass (3). IÝ we measured water against glass, we should +fnd another index, which we shall call nmạs. But there is no ø pr?or¿ reason why +there should be any connection between 01s, 01s, and 2s. Ôn the other hand, +according to the idea of least time, there 2s a defnite relationship. The index ?1a +1s the ratio of two things, the speed in air to the speed in water; 1s is the ratio +of the speed ín air to the speed in gÌlass; 23 is the ratio of the speed in water to +the speed in glass. 'herefore we cancel out the air, and get +¬......` (26.5) +U3 ĐỊ (0a T2 +In other words, we ørcd¡¿ct that the Index for a new pair of materials can be +obtained from the indexes of the individual materials, both against air or against +vacuum. 5o iŸ we measure the speed of light in all materials, and from this get a +single number for each material, namely its index relative to vacuum, called m¿ +(mị is the speed in air relative to the speed in vacuum, etc.), then our formula is +easy. The index for any two materials 2 and 7 is +ng= TƯ = 2, (26.6) +Uj Tt¿ +Using only Snells law, there is no basis for a prediction of this kind.* But of +course this prediction works. The relation (26.5) was known very early, and was +a very strong argument for the prineciple of least time. +— * Although it can be deduced if the additional assumption is made that adding a layer of +one substance to the surface of another does not change the eventual angle of refraction in the +latter material. +--- Trang 466 --- +Another argument for the principle of least time, another prediction, is that +1 we measure the speed of light in water, it will be lower than in air. This is a +prediction of a completely diferent type. It is a brilliant prediction, because all +we have so far measured are øngles; here we have a theoretical prediction which +1s quite diferent from the observations from which Eermat deduced the idea of +least time. It turns out, in fact, that the speed in water 2s slower than the speed +in air, by just the proportion that is needed to get the right indexl +26-5 A more precise statement of Fermat?s principle +Actually, we must make the statement of the principle of least time a little +more accurately. It was not stated correctly above. It is #øcorrectu called the +principle of least time and we have gone along with the incorrect description Íor +convenience, but we must now see what the correct statement is. Suppose we had +a mirror as in Fig. 26-3. What makes the light think it has to go to the mirror? +The path of ieast time is clearly 4P. So some people might say, “Sometimes it is +a maximum time.” Ít is no‡ a maximum time, because certainly a curved path +would take a stil longer timel The correct statement is the following: a ray going +in a certain particular path has the property that if we make a small change (say +a one percent shift) in the ray in any manner whatever, say in the location at +which it comes to the mirror, or the shape of the curve, or anything, there will +be mo first-order change in the time; there will be only a second-order change in +the time. In other words, the principle is that light takes a path such that there +are many other paths nearby which take almost exactly the sazne tỉme. +The following is another difculty with the principle of least time, and one +which people who do not like this kind of a theory could never stomach. With +Snells theory we can “understand” light. Light goes along, 1% sees a surface, 1% +bends because it does something at the surface. “The idea of causality, that it goes +from one point to another, and another, and so on, is easy to understand. But +the prineiple of least time is a completely diferent philosophical principle about +the way nature works. Instead oŸ saying it is a causal thing, that when we do one +thing, something else happens, and so on, it says this: we set up the situation, +and igh# decides which is the shortest time, or the extreme one, and chooses +that path. But uhø‡ does it do, ho does it nd out? Does 1t srneÏl the nearby +paths, and check them against each other? The answer is, yes, it does, in a way. +That is the feature which is, of course, not known in geometrical optics, and +which is involved ïn the idea of auelength; the wavelength tells us approximately +--- Trang 467 --- +—T^ \ |) |_— :91) --H[D] +Fig. 26-13. The passage of radiowaves through a narrow silit. +how far away the light must “smell” the path in order to check it. Tt is hard +to demonstrate this fact on a large scale with light, because the wavelengths +are so t6erribly short. But with radiowaves, say 3-cm waves, the distances over +which the radiowaves are checking are larger. lf we have a source of radiowawves, +a detector, and a slit, as in Eig. 26-13, the rays of course go from ,Š to J because +1t is a straight line, and if we close down the slit it is all right—they still go. But +now if we move the detector aside to J2, the waves will not go through the wide +slit from 6 to D/, because they check several paths nearby, and say, “No, my +friend, those all eorrespond to diferent times.” Ôn the other hand, if we preuen‡ +the radiation from checking the paths by closing the slit down to a very narrow +crack, then there is but one path available, and the radiation takes it! With a +narrow slit, more radiation reaches than reaches it with a wide slitl +One can do the same thing with light, but it is hard to demonstrate on a large +scale. The efect can be seen under the following simple conditions. Eind a small, +bright light, say an unfrosted bulb in a street light far away or the reflection of +the sun in a curved automobile bumper. Then put two fingers in front of one eye, +so as to look through the crack, and squeeze the light to zero very gently. You +will see that the image of the light, which was a little dot before, becomes quite +elongated, and even stretches into a long line. 'Phe reason is that the ñngers are +very close together, and the light which is supposed to come in a straight line +1s spread out at an angle, so that when 1% comes into the eye 1% comes in from +sevoral directions. Also you wïll notice, if you are very careful, side maxima, a lot +of fringes along the edges too. Furthermore, the whole thing is colored. All of this +will be explained ïn due time, but for the present it is a demonstration that light +does not always go in straight lines, and it is one that is very easily performed. +--- Trang 468 --- +20-6 How it works +Finally, we give a very crude view of what actually happens, how the whole +thing really works, from what we now believe is the correct, quantum-dynamically +accurate viewpoint, but of course only qualitatively described. In following the +light from A to in Eig. 26-3, we find that the light does not seem to be in the +form of waves at all. Instead the rays seem to be made up of photons, and they +actually produce clicks in a photon counter, if we are using one. The brightness +of the light is proportional to the average number of photons that come in per +second, and what we calculate is the chance that a photon gets from A4 to Ö, say +by hitting the mirror. The iau for that chance is the following very strange one. +Take any path and fnd the time for that path; then make a complex number, +or draw a little complex vector, øe?, œbose œngle 9 is proportional to the time. +The number o turns per second is the frequency of the light. Now take another +path; it has, for instance, a different time, so the vector for it is turned through +a diferent angle—the angle being always proportional to the time. Take all the +available paths and add on a little vector for each one; then the answer is that +the chance of arrival of the photon is proportional to the square of the length of +the final vector, from the beginning to the endl +Fig. 26-14. The summation of probability amplitudes for many neigh- +boring paths. +Now let us show how this implies the principle of least time for a mirror. WWe +consider all rays, all possible paths AD, AHB, AC, etc., in Eig. 26-3. The +path A4AJD2 makes a certain small contribution, but the next path, 1⁄5, takes +a quite diferent time, so its angle Ø is quite diferent. Let us say that point Œ +corresponds to minimum time, where 1Ý we change the paths the times do not +change. So for awhile the times do change, and then they begin to change less and +less as we geb near point Œ (Fig. 26-14). So the arrows which we have to add are +coming almost exactly at the same angle for awhile near Œ, and then gradually +the time begins to increase again, and the phases go around the other way, and so +--- Trang 469 --- +on. Eventually, we have quite a tight knot. The total probability is the distanece +from one end to the other, squared. 4ửmost ølÏ oƒ that accwmnulated probabilit +occurs ïn the region tthere qÏl the œrrotus are ïn the sarmme đireciion (or in the same +phase). All the contributions from the paths which have very đjƒƑferent tỉìmes as +we change the path, cancel themselves out by pointing in diferent directions. +That is why, if we hide the extreme parts of the mirror, it still relects almost +exactly the same, because all we did was to take out a piece of the diagram inside +the spiral ends, and that makes only a very small change in the light. So this is +the relationship between the ultimate picture of photons with a probability of +arrival depending on an accumulation of arrows, and the principle of least time. +--- Trang 470 --- +Ấoormeofr'rcerÏl Ê)jpp££€-s +27-1 Introduction +In this chapter we shall discuss some elementary applications of the ideas of +the previous chapter to a number of practical devices, using the approximation +called geometrical optfics. Thïs is a most usefu]l approximation in the practical +design of many optical systems and instruments. Geometrical optics 1s either +very simple or else it is very complicated. By that we mean that we can either +study it only superficially, so that we can design instruments roughly, using rules +that are so simple that we hardly need deal with them here at all, since they +are practically of hipgh school level, or else, if we want to know about the small +errors in lenses and similar details, the subject gets so cormplicated that it is too +advanced to discuss herel TẾ one has an actual, detailed problem in lens design, +including analysis of aberrations, then he is advised to read about the subject +or else simply to trace the rays through the various surfaces (which is what the +book tells how to do), using the law of refraction from one side to the other, +and to ñnd out where they come out and see iŸ they form a satisfactory image. +People have said that this is too tedious, but today, with computing machines, it +is the right way to do it. One can set up the problem and make the calculation +for one ray after another very easily. 5o the subject is really ultimately quite +simple, and involves no new prineciples. Furthermore, it turns out that the rules of +either elementary or advanced optics are seldom characteristic of other fields, so +that there is no special reason to follow the subject very far, with one important +exception. +The most advanced and abstract theory of geometrical optics was worked +out by Hamilton, and it turns out that this has very important applications in +mechanics. Ïlt is actually even more important in mechanies than it is in opties, +and so we leave Hamilton”s theory for the subject ofadvanced analytical mechanies, +which is studied in the senior year or in graduate school. 5o, appreciating that +--- Trang 471 --- +Figure 27-1 +geometrical optics contributes very little, except for its own sake, we now go on +to discuss the elementary properties of simple optical systems on the basis of the +principles outlined ïn the last chapter. +In order to go on, we must have one geometrical formula, which is the following: +1ƒ we have a triangle with a small altitude h and a long base đ, then the diagonal s +(we are goiïng to need it to fnd the difference in time between two different routes) +is longer than the base (Fig. 27-1). How mụuch longer? The diference A = s— đ +can be found in a number of ways. One way is this. W©e see that s2 — đ2 = h2, +or (s — đ)(s + đ) = h?. But s— đ= A, and s+d2s. Thus +A~ h2/2s. (27.1) +This is all the geometry we need to discuss the formation of images by curved +surfacesl +27-2 The focal length of a spherical surface +The first and simplest situation to discuss is a single refracting surface, +separating §wo media with diferent indices of refraction (Fig. 27-2). We leave +the case of arbitrary indices of refraction to the student, because deas are always +l6) vị €C ớ +AIR GLASS +Fig. 27-2. Focusing by a single refracting surface. +--- Trang 472 --- +the most Important thing, not the specifc situation, and the problem is easy +enough to do in any case. So we shall suppose that, on the left, the speed is 1 +and on the right i6 is 1/n, where ø is the index of refraction. The light travels +more slowly in the glass by a facbOr n. +Now suppose that we have a point at Ó, at a distance s from the front surface +of the glass, and another point Ó“ at a distance sf inside the glass, and we desire +to arrange the curved surface in such a manner that every ray from @ which hits +the surface, at any point , will be bent so as to proceed toward the point Ó”. +For that to be true, we have to shape the surface in such a way that the time it +takes for the light to go from Ó to Ð, that is, the distance ÓØ?P divided by the +speed of light (the speed here is unity), plus œ - Ó“P, which is the time it takes +to go from P to Ớƒ, is equal to a constant independent of the point P. Thịs +condition supplies us with an equation for determining the surface. The answer +1s that the surface is a very complicated fourth-degree curve, and the student +may entertain himself by trying to calculate it by analytic geometry. It is simpler +to try a special case that corresponds to s —> œo, because then the curve 1s a +second-degree curve and is more recognizable. lt is interesting to compare this +curve with the parabolic curve we found for a focusing mirror when the light is +coming from infnity. +So the proper surface cannot easily be made——to focus the light from one +point to another requires a rather complicated surface. Ït turns out in practice +that we do not try to make such complicated surfaces ordinarily, but instead we +make a compromise. Instead of trying to get aøÏl the rays to come to a Íocus, we +arrange it so that only the rays fairly close to the axis Ó” come to a focus. The +farther ones may deviate if they want to, unfortunately, because the ideal surface +1s complicated, and we use instead a spherical surface with the right curvature at +the axis. Ït is so much easier to fabricate a sphere than other surfaces that it 1s +proftable for us to fnd out what happens to rays striking a spherical surface, +supposing that only the rays near the axis are going to be focused perfectly. +'Those rays which are near the axis are sometimes called parazial røs, and what +we are analyzing are the conditions for the focusing of paraxial rays. We shall +discuss later the errors that are introduced by the fact that all rays are not aÌways +close to the axis. +'Thus, supposing ? is close to the axis, we drop a perpendicular P@) such that +the height P@) is h. For a moment, we imagine that the surface is a plane passing +through ?. In that case, the time needed to go from @ to would exceed the +time from Ó to @, and also, the time from to Ó” would exceed the time from +--- Trang 473 --- +Q to Ở But that is why the glass must be curved, because the total excess +time must be compensated by the delay in passing from V to Q! Ñow the ezcess +time along route ÓP is h2/2s, and the excess time on the other route is nh2/2s”. +This excess time, which must be matched by the delay in going along VQ, differs +from what it would have been in a vacuum, because there is a medium present. +In other words, the time to go from V to Q is not as 1Í it were straight in the +air, but 1E is slower by the factor ø%œ, so that the excess delay in this distance 1s +then (wT— 1)VQ. And now, how large is VQ? TÍ the point Ở is the center of the +sphere and if its radius is #, we see by the same formula that the distance V) is +cqual to h”/2ÿ. Therefore we discover that the law that connects the distances s +and s/, and that gives us the radius of curvature ? of the surface that we need, is +(h2/2s) + (nh2/2s!) = (n — 1)h”/2R (27.2) +(1/s) + (n/s) = (n — 1)/R. (27.3) +If we have a position Ó and another position ÓØ”, and want to focus light from @ +to Ø7, then we can calculate the required radius of curvature ## of the surface by +this formula. +Now it turns out, interestingly, that the same lens, with the same curvature †, +will focus for other distances, namely, for any pair of distances such that the sum +of the two reciprocals, one multiplied by m, is a constant. 'Phus a given lens will +(so long as we limit ourselves to paraxial rays) focus not only from Ó to Ở', but +between an infinite number of other pairs of points, so long as those pairs of +points bear the relationship that 1/s + œ/s” is a constant, characteristic of the +In particular, an interesting case is that in which s —> oo. W© can see from the +formula that as one s increases, the other decreases. In other words, if point Ó +goes out, poini Ó“ comes in, and vice versa. As point Ó goes toward infinity, +point @“ keeps moving in until it reaches a certain distance, called the ƒocal +length ƒ', inside the material. If parallel rays come in, they will meet the axis +at a disbance ƒ7. Likewise, we could imagine it the other way. (Remember the +reciprocity rule: if light will go rom Ó to Ó”, of course it will also go from Ớ/ +to Ó.) Therefore, if we had a light source inside the gÌass, we might want to know +where the focus is. In particular, if the light in the glass were at infinity (same +problem) where would it come to a focus outside? Thịis distance is called ƒ. Of +course, we can also put it the other way. If we had a light source at ƒ and the +--- Trang 474 --- +light went through the surface, then ¡% would go out as a parallel beam. We can +easily fnd out what ƒ and 7 are: +m/Ƒ) =(n— 1)/R Or ƒƑ = Rn/(n— 1), (27.4) +1/ƒ =(n—1)/R OF ƒ =R/(n- ]). (27.5) +W© see an interesting thing: iƒwe divide each focal length by the corresponding +index of refraction we get the same resultl 'Phis theorem, in fact, is general. lt is +true of any system oŸ lenses, no matter how complicated, so it is interesting to +remember. We did not prove here that it is general—we merely noted i% for a +single surface, but it happens to be true in general that the two focal lengths of +a system are related in this way. Sometimes q. (27.3) is written in the form +1/s-+m/s = 1/ƒ. (27.6) +Thịis is more useful than (27.3) because we can measure ƒ more easily than we +can measure the curvature and index of refraction of the lens: if we are not +interested in designing a lens or in knowing how it got that way, but simply liÑt +1t of a shelf, the interesting quantity is ƒ, not the œ and the 1 and the #l +Now an interesting situation occurs If s becomes less than ƒ. What happens +then? IÝ s < ƒ, then (1/s) > (1/ƒ), and therefore s” is negative; our equation says +that the light will focus only with a negative value of s”, whatever that meansl +It does mean something very interesting and very defñnite. It is still a useful +formula, in other words, even when the numbers are negative. What it means is +shown in Fig. 27-3. If we draw the rays which are diverging from Ó, they will be +bent, it is true, at the surface, and they will not come to a focus, because ) is so +close in that they are “beyond parallel” However, they diverge as if they had +come from a point Ó“ ou£side the glass. This is an apparent image, sometimes +— ===- +Fig. 27-3. A virtual image. +--- Trang 475 --- +called a ơrtual image. The image @' in Fig. 27-2 is called a real zœmage. TỶ the +light really comes to a point, it is a real image. But if the light appears to be +coming ƒrom a point, a fictitious point diferent from the original point, it is a +virtual image. So when s” comes out negative, it means that Ó” is on the other +side of the surface, and everything is all right. +| — ===> => +AIR GLASS +Fig. 27-4. A plane surface re-images the light from ' to OÓ. +Now consider the interesting case where ?## is equal to infnity; then we have +(1/5) + (n/s) =0. In other words, s” = —?ws, which means that if we look from a +dense medium into a rare medium and see a poïnt in the rare medium, it appears +to be deeper by a factor ø. Likewise, we can use the same equation backwards, so +that iƒ we look into a plane surface at an object that is at a certain distance inside +the dense medium, it will appear as though the light is coming from not as far +back (Fig. 27-4). When we look at the bottom of a swimming pool from above, +it does not look as deep as iÈ really is, by a factor 3/4, which is the reciprocal of +the index of refraction of water. +W© could go on, of course, to discuss the spherical mirror. But if one appreci- +ates the ideas involved, he should be able to work it out for himself. Therefore +we leave it to the student to work out the formula for the spherical mirror, but +we mention that it is well to adopt certain conventions concerning the distances +involved: +(1) The object distance s is positive if the point Ó is to the left of the surface. +(2) The image distance s” is positive if the point Ớf is to the right of the surface. +(3) The radius of curvature of the surface is positive iƒ the center is to the right +of the surface. +In Fig. 27-2, for example, s, s/, and ## are all positive; in Pig. 27-3, s and ## are +--- Trang 476 --- +positive, but s” is negative. IÝ we had used a concave surface, our formula (27.3) +would still give the correct result if we merely make a negative quantity. +In working out the corresponding formula for a mirror, using the above +conventions, you will fnd that if you put ø = —1 throughout the formula (27.3) +(as though the material behind the mirror had an index —1), the right formula +for a mirror resultsl +Although the derivation of formula (27.3) is simple and elegamt, using least +time, one can of course work out the same formula using 5nell's law, remembering +that the angles are so small that the sines of angles can be replaced by the angles +themselves. +27-3 The focal length of a lens +Now we go on to consider another situation, a very practical one. Most of the +lenses that we use have two surfaces, not just one. How does this affect matters? +Suppose that we have ÿwo surfaces of diferent curvature, with glass filling the +space bebween them (Eig. 27-5). We want to study the problem of focusing from +a point Ó to an alternate point Ớ”. How can we do that? The answer is this: +Eirst, use formula (27.3) for the first surface, forgetting about the second surface. +This will tell us that the light which was diverging from @Ø will appear to be +converging or diverging, depending on the sign, from some other point, say Ó”. +Now we consider a new problem. We have a diferent surface, between glass and +air, in which rays are converging toward a certain point Ó'. Where will they +actually converge? We use the same formula again!l We fnd that they converge +at Ø“”. Thus, if necessary, we can go through 7ð surfaces by just using the same +formula in succession, from one to the nextl +x —m `... +Fig. 27-5. lmage formation by a two-surface lens. +--- Trang 477 --- +There are some rather high-class formulas that would save us considerable +energy in the few times in our lives that we might have to chase the light through +fve surfaces, but it is easier just to chase it through fve surfaces when the +problem arises than it is to memorize a lot of formulas, because it may be we +will never have to chase it through any surfaces at alll +In any case, the principle is that when we go through one surface we find a +new position, a new focal point, and then take that point as the starting poiïnt +for the next surface, and so on. In order to actually do this, since on the second +surface we are going from %øw to 1 rather than from 1 to øœ, and since in many +systems there is more than one kind of glass, so that there are indices 0ø, na, +..., we really need a generalization of formula (27.3) for a case where there are +two diferent indices, + and nạ, rather than only nø. 'Phen ït is not dificult to +prove that the general form of (27.3) is +(m1/5) + (na/s)) = (na — mì)/R. (27.7) +Particularly simple is the special case in which the two surfaces are very close +together——so close that we may ignore small errors due to the thickness. lÝ we +draw the lens as shown in Fig. 27-6, we may ask this question: How must the +lens be built so as to focus light from Ó to Ø7? Suppose the light comes exactÌy +to the edge of the lens, at point P. Then the excess tỉme in going from Ó to +is (mịh2/2s) + (nịh2/25/), ignoring for a moment the presence of the thickness 7” +of glass of index nạ. Now, to make the time for the direct path equal to that for +the path ÓPŒỚ", we have to use a piece of glass whose thickness 7' at the center +1s such that the delay introduced in going through this thickness is enough to +compensate for the excess time above. Therefore the thickness of the lens at the +O le) œ +HỊ HỊ +Fig. 27-6. A thin lens with two positive radii. +--- Trang 478 --- +center must be given by the relationship +(nìh2/2s) + (nìh?/25)) = (nạ — mì)T. (27.8) +W© can also express 7 in terms of the radii f¡ and ñ¿ of the two surfaces. Paying +attention to our convention (3), we thus fñnd, for ?#ị < #2 (a convex lens), +7 = (hˆ/2RI) - (h°/2Ã). (27.9) +'Therefore, we fnally get +(mì/s) + (mị/s) — (mạ — mị)(1/ị — 1/R). (27.10) +Now we note again that if one of the points is at infinity, the other will be at a +point which we will call the focal length ƒ. The focal length ƒ is given by +1/ƒ =(nm— 1)(1/Tị — 1/1). (27.11) +where ø = nạ/m. +Now, 1ƒ we take the opposite case, where s goes to infinity, we see that sf 1s +at the focal length ƒ7. This time the focal lengths are equal. (This is another +special case of the general rule that the ratio of the two focal lengths is the ratio +of the indices of refraction in the two media in which the rays focus. In this +particular optical system, the initial and fnal indices are the same, so the two +focal lengths are equal.) +Forgetting for a moment about the actual formula for the focal length, if +we bought a lens that somebody designed with certain radii of curvature and a +certain index, we could measure the focal length, say, by seeing where a point at +inñnity focuses. Once we had the focal length, it would be better to write our +equation in terms of the focal length directly, and the formula then is +(1/s) + (1/5) = 1/. (27.12) +Now let us see how the formula works and what it implies in diferent circum- +siances. First, it implies that IÝ s or sf is inñnite the other one is ƒ. That means +that parallel light focuses at a distance ƒ, and this in efect defines ƒ. Another +interesting thing i% says is that both points move in the same direction. lf one +moves to the right, the other does also. Another thing it says is that s and s/ +are equal if they are both equal to 2ƒ. In other words, if we want a symmetrical +situation, we ñnd that they will both focus at a distance 2ƒ. +--- Trang 479 --- +27-4 Magnification +So far we have discussed the focusing action only for points on the axis. NÑow +let us discuss also the imaging of objects not exactly on the axis, but a little +bít of, so that we can understand the properties of mmagnification. When we set +up a lens so as to focus light from a smaill filament onto a “point” on a screen, +we notice that on the screen we get a “picture” of the same filament, except of +a larger or smaller size than the true fñlament. 'Phis must mean that the light +comes to a focus from cach poøzn‡ of the filament. In order to understand this a +little better, let us analyze the thin lens system shown schematically in Eig. 27-7. +W©e know the following facts: +(1) Any ray that comes in parallel on one side proceeds toward a certain +particular point called the focus on the other side, at a distance ƒ from the +(2) Any ray that arrives at the lens from the focus on one side comes out +parallel to the axis on the other side. +Thịs is all we need to establish formula (27.12) by geometry, as follows: Suppose +we have an object at some distance #ø from the focus; let the height of the object +be . Then we know that one of the rays, namely PQ, will be bent so as to pass +through the focus # on the other side. Now ïf the lens will focus point P at all, +we can fnd out where if we fnd out where just one other ray goes, because the +new focus will be where the two intersect again. We need only use our ingenulty +to fñnd the exact direction of øne other ray. But we remember that a parallel +ray goes through the focus and 0e 0ersa: a ray which goes through the focus +will come out parallel' So we draw ray 7 through . (It is true that the actual +rays which are doing the focusing may be much more limited than the two we +have drawn, but they are harder to fñgure, so we make believe that we can make +ÔN CƯ ƯỢNG cu +Fig. 27-7. The geometry of imaging by a thin lens. +--- Trang 480 --- +this ray.) Since it would come out parallel, we draw 79 parallel to XW. The +Intersection ®Š is the point we need. 'Phis will determine the correcE place and +the correct height. Let us call the height ˆ and the distance from the focus, z. +Now we may derive a lens formula. Ủsing the similar triangles PVU and 7'XU, +we fnd , +—==~. 27.13 +Th (713) +Similarly, from triangles SW . and QX, we get +Ụ _— 9 +—==_.. 27.14 +n=Ủ (2714) +Solving each for /, we ñnd that +U_—# (27.15) +Equation (27.15) is the famous lens formula; in it is everything we need to know +about lenses: Ib tells us the magnification, #“/, in terms of the distances and +the focal lengths. It also connects the bwo distances z and øˆ with ƒ: +ma! = Ƒ, (27.16) +which is a much neater form to work with than Eq. (27.12). We leave it to the +student to demonstrate that if we call s = #z + ƒ and s” = zø' + ƒ, Bq. (27.12) is +the same as Eq. (27.16). +27-5 Compound lenses +Without actually deriving it, we shall briefy describe the general result when +we have a number of lenses. If we have a system of several lenses, how can +we possibly analyze it? 'Phat is easy. We start with some object and calculate +where its image is for the first lens, using formula (27.16) or (27.12) or any other +equivalent formula, or by drawing diagrams. So we fñnd an image. Then we treat +this image as the source for the next lens, and use the second lens with whatever +1ts focal length is to again ñnd an image. We simply chase the thing through +the succession of lenses. 'That is all there is to it. It involves nothing new in +principle, so we shall not go into it. However, there is a very interesting net +--- Trang 481 --- +result of the efects of any sequence of lenses on light that starts and ends up in +the same medium, say air. Any optical instrument——a telescope or a microscope +with any number of lenses and mirrors—has the following property: There exist +two planes, called the prinecipal pÏøœnes of the system (these planes are often fairly +close to the first surface of the first lens and the last surface of the last lens), +which have the following properties: (1) If light comes into the system parallel +from the first side, it comes out at a certain focus, at a distance from the second +principal plane equal to the focal length, Just as though the system were a thin +lens situated at this plane. (2) Tf parallel light comes in the other way, i1 comes +to a focus at the same distance ƒ from the ƒrs‡ principal plane, again as If a thin +lens where situated there. (See Eig. 27-8.) +Fig. 27-8. lllustration of the principal planes of an optical system. +Of course, iŸ we measure the distances # and z', and ÿ and z as before, +the formula (27.16) that we have written for the thin lens is absolutely general, +provided that we measure the focal length from the principal planes and not from +the center of the lens. It so happens that for a thin lens the principal planes are +coincident. It is just as though we could take a thin lens, slice i2 down the middle, +and separate it, and not notice that it was separated. Every ray that comes in +pops out immediately on the other side of the second plane from the same point +as it went into the first planel 'The principal planes and the focal length may be +found either by experiment or by calculation, and then the whole set oŸ propertfies +of the optical system are described. lt is very Interesting that the result is not +complicated when we are all ñnished with such a big, complicated optical system. +27-6 Aberrations +Before we get too excited about how marvelous lenses are, we must hasten +to add that there are also serious limitations, because of the fact that we have +--- Trang 482 --- +limited ourselves, strictly speaking, to paraxial rays, the rays near the axis. A +real lens having a fñnite size will, in general, exhibit aberrations. For example, +a ray that is on the axis, of course, goes through the focus; a ray that is very +close to the axis will still come to the focus very well. But as we go farther out, +the ray begins to deviate from the focus, perhaps by falling short, and a ray +striking near the top edge comes down and misses the focus by quite a wide +margin. So, instead of getting a point image, we get a smear. This efect is called +spherical œberration, because it is a property of the spherical surfaces we use in +place of the ripght shape. This could be remedied, for any specifc obJect distance, +by re-forming the shape of the lens surface, or perhaps by using several lenses +arranged so that the aberrations of the individual lenses tend to cancel each other. +Lenses have another fault: light of diferent colors has diferent speeds, or +diferent indices of refraction, in the glass, and therefore the focal length of a +given lens is diferent for diferent colors. 5o iŸ we image a white spot, the image +will have colors, because when we focus for the red, the blue is out of focus, or +vice versa. This property is called chrormatic aberrotion. +'There are still other faults. If the object is of the axis, then the focus really +1snˆt perfect any more, when it gets far enough of the axis. 'Phe easiest way to +verify this is to focus a lens and then tilt it so that the rays are coming in at a +large angle from the axis. hen the image that is formed will usually be quite +crude, and there may be no place where it focuses well. There are thus several +kinds of errors in lenses that the optical designer tries to remedy by using many +lenses to compensate each other”s errors. +How careful do we have to be to eliminate aberrations? Is it possible to make +an absolutely perfect optical system? Suppose we had built an optical system +that is supposed to bring light exactly to a point. Now, arguing from the poïnt +of view of least time, can we fnd a condition on how perfect the system has to +be? The system will have some kind oŸ an entrance opening for the light. IÝ we +take the farthest ray from the axis that can come to the focus (ïf the system +is perfect, of course), the times for all rays are exactly equal. But nothing is +perfect, so the question is, how wrong can the time be for this ray and not be +worth correcting any further? That depends on how perfectb we want to make +the image. But suppose we want to make the image as perfect as it possibly can +be made. 'Then, of course, our impression is that we have to arrange that every +ray takes as nearly the same time as possible. But ¡% turns out that this is not +true, that beyond a certain point we are trying to do something that is too ñne, +because the theory of geometrical optics does not workl +--- Trang 483 --- +Remember that the principle of least time 1s not an accurate formulation, +unlike the principle of conservation of energy or the principle of conservation +of momentum. 'Phe principle of least time is only an approzimation, and 1§ +1s interesting to know how much error can be allowed and still not make any +apparent diference. The answer is that if we have arranged that between the +maximal ray—the worst ray, the ray that is farthest out—and the central ray, the +diferenee in time is less than about the period that corresponds to one oscillation +of the light, then there is no use improving it any further. Light is an oscillatory +thing with a defñnite frequenecy that is related to the wavelength, and if we have +arranged that the time diference for diferent rays is less than about a period, +there is no use going any further. +27-7 Resolving power +Another interesting question—a very important technical question with all +optical instruments—is how much resoluing pouer they have. IÝ we build a +microscope, we want to see the objects that we are looking at. That means, for +instance, that if we are looking at a bacterium with a spot on each end, we want +to see that there are two dots when we magnify them. One might think that all +we have to do is to get enough magnification——we can always add another lens, +and we can always magnify again and again, and with the cleverness of designers, +all the spherical aberrations and chromatic aberrations can be cancelled out, +and there is no reason why we cannot keep on magnifying the image. So the +limitations of a microscope are not that it is impossible to build a lens that +magnifes more than 2000 diameters. We can build a system of lenses that +magnifes 10,000 diameters, but we s#Z/ could not see two points that are too +close together because of the limitations of geometrical opties, because of the +fact that least time is not precise. +To discover the rule that determines how far apart® bwo points have to be +so that at the image they appear as separate points can be stated in a very +beautiful way associated with the time it takes for diferent rays. Suppose that +we disregard the aberrations now, and imagine that for a particular point ? +(Fig. 27-9) all the rays rom object to image 7' take exactly the same tỉme. (It +is not true, because it is not a perfect system, but that is another problem.) +NÑow take another nearby point, P, and ask whether its image will be distinct +trom 7” In other words, whether we can make out the diference between them. +Of course, according to geometrical optics, there should be two point images, +--- Trang 484 --- +frtZ| C————_ T +ầm =ứY +Fig. 27-9. The resolving power of an optical system. +but what we see may be rather smeared and we may not be able to make out +that there are two points. The condition that the second poïnt is focused ín a +distinctly diferent place from the first one is that the two times for the extreme +rays P'ST and PT on each side of the big opening of the lenses to go Írom +one end to the other, must sœø‡ be equal from the two possible obJect points to +a given image point. Why? Because, if the times were equal, of course both +would ƒocus at the same point. So the times are not going to be equal. But by +how much do they have to difer so that we can say that both do noø‡ come to a +common fÍocus, so that we can distinguish the 6wo image points? 'The general rule +for the resolution of any optical instrument is this: two diferent point sources +can be resolved only if one source is focused at such a point that the times for +the maximal rays from the other source to reach that point, as compared with +its own true image point, difer by more than one period. It is necessary that the +diference in time between the top ray and the bottom ray to the rong focus +shall exceed a certain amount, namely, approximately the period of oscillation of +the light: +ta — tị > 1/1, (27.17) +where 1 is the frequency of the light (number of oscillations per second; also speed +divided by wavelength). IÝ the distance of separation of the two points is called +D, and 1ƒ the opening angle of the lens is called Ø, then one can demonstrate +that (27.17) is exactly equivalent to the statement that 2 must exceed À/nsin 0, +where ø is the index of refraction at and À is the wavelength. 'Phe smallest +things that we can see are therefore approximately the wavelength of light. A +corresponding formula exists for telescopes, which tells us the smallest diference +in angle bebween two stars that can just be distinguished.* +* "The angle is about À/D, where D is the lens diameter. Can you see why? +--- Trang 485 --- +Mgiocfrorneigraofic Hồ (cÏfqf6fGrte +28-1 Electromagnetism +'The most dramatic moments in the development of physics are those in which +great syntheses take place, where phenomena which previously had appeared +to be diferent are suddenly discovered to be but diferent aspects of the same +thing. The history of physics is the history of such syntheses, and the basis of +the success of physical science is mainly that we are øble to synthesize. +Perhaps the most dramatie moment in the development of physics during the +19th century occurred to J. C. Maxwell one day in the 1860°s, when he combined +the laws of electricity and magnetism with the laws of the behavior of light. As +a result, the properties of light were partly unravelled—that old and subtle stuf +that is so important and mysterious that it was felt necessary to arrange a special +creation for it when writing Genesis. Maxwell could say, when he was ñnished +with his discovery, “Let there be electricity and magnetism, and there is lightl” +For this culminating moment there was a long preparation in the gradual +discovery and unfolding of the laws of electricity and magnetism. This story we +shall reserve for detailed study next year. However, the story is, briely, as follows. +The gradually discovered properties of electricity and magnetism, of electric Íorces +of attraction and repulsion, and of magnetie forces, showed that although these +forces were rather complex, they all fell off inversely as the square of the distance. +We know, for example, that the simple Coulomb law for stationary charges is +that the electric force field varies inversely as the square of the distance. As a +consequence, for sufficiently great distances there is very little inÑuence of one +system of charges on another. Maxwell noted that the equations or the laws that +had been discovered up to this tìme were mutually inconsistent when he tried to +put them all together, and in order for the whole system to be consistent, he had +to add another term to his equations. With this new term there came an amazing +prediction, which was that a part of the electric and magnetic fields would fall of +--- Trang 486 --- +tmmuch more slowly with the distance than the inverse square, namely, inversely as +the first power of the distancel And so he realized that electric currents in one +place can affect other charges far away, and he predicted the basic efects with +which we are familiar today—radio transmission, radar, and so on. +lt seems a miracle that someone talking in Europe can, with mere electrical +inñuences, be heard thousands of miles away in Los Angeles. How is it possible? +lt is because the fields do not vary as the inverse square, but only inversely as +the first power of the distance. Finally, then, even light itself was recognized +to be electric and magnetie inÑuences extending over vast distances, generated +by an almost incredibly rapid oscillation of the electrons in the atoms. All +these phenomena we summarize by the word rad¿øtion or, more specifically, +clectromagnetic radiation, there being one or two other kinds of radiation also. +Almost always, radiation means electromagnetic radiation. +And thus is the universe knit together. The atomic motions of a distant star +siiHl have sufficient inÑuence at this great distance to set the electrons in our eye +in motion, and so we know about the stars. If this law did not exist, we would +all be literally in the dark about the exterior worldl And the electric surgings in +a galaxy fñve billion light years away——which is the farthest object we have found +so far—can still inÑuenee in a signilcant and detectable way the currents in the +great “dish” in front of a radio telescope. And so it is that we see the stars and +the galaxies. +'This remarkable phenomenon is what we shall discuss In the present chapter. +At the beginning of this course in physics we outlined a broad picture of the +world, but we are now better prepared to understand some aspects of it, and +so we shall now go over some parts of it again in greater detail. We begin by +describing the position of physics at the end of the 19%0h century. All that was +then known about the fundamental laws can be summarized as follows. +First, there were laws of forces: one force was the law of gravitation, which +we have written down several times; the force on an object of mass mm, due to +another of mass j, is given by +FPƑ.=GmMe,/rŸ, (28.1) +where e; is a unit vector directed from rn to Mĩ, and r is the distance between +Next, the laws of electricity and magnetism, as known at the end of the +19th century, are these: the electrical forces acting on a charge g can be described +--- Trang 487 --- +by two fields, called # and ?Ö, and the velocity ø of the charge g, by the equation +P=q(E+ox Đ). (28.2) +To complete thịs law, we have to say what the formulas for E and Ö are in a +given circumstance: iŸ a number of charges are present, # and the #Ö are each +the sum of contributions, one from each individual charge. So if we can find the +2 and B produced by a single charge, we need only to add all the efects from +all the charges in the universe to get the total # and BI 'This is the principle of +SuperposIfion. +What ¡is the formula for the electric and magnetic field produced by one +individual charge? It turns out that this is very complicated, and it takes a +great deal of study and sophistication to appreciate it. But that is not the +point. We write down the law now only to impress the reader with the beauty +of nature, so to speak, i.e., that it is possible to sunmarize all the fundamental +knowledge on one page, with notations that he is now familiar with. 'This law for +the fields of an individual charge 1s complete and accurate, so far as we know +(except for quantum mechanics) but it looks rather complicated. We shall not +study all the pieces now; we only write it down to give an Impression, to show +that it can be written, and so that we can see ahead of time roughly what ¡it +looks like. As a matter of fact, the most wseƒful way to write the correct laws of +electricity and magnetism is not the way we shall now write them, but involves +what are called field equat¿ons, which we shall learn about next year. But the +mathematical notations for these are different and new, and so we write the law +in an inconvenient form for calculation, but in notations that we now know. +'The electric ñeld, #, is given by +—{ | €Cr: rrd Cự 1 d2 +E= 47€o l# + e đdí (#) + c2 đí2 si (28.3) +What do the various terms tell us? Take the frst term, = —qe„:/4meor2. +That, of course, is Coulomb°s law, which we already know: g is the charge that is +produecing the field; ez¿ is the unit vector in the direction from the point where +E2 is measured, z is the distance from ? to g. But, Coulomb's law is wrong. The +discoveries of the 19th century showed that inÑuences cannot travel faster than +a certain fundamental speed c, which we now call the speed of light. I% is not +correct that the first term is Coulomb'°s law, not only because it is not possible to +know where the charge is nøu and at what distance it is œøu, but also because +--- Trang 488 --- +the only thing that can affect the fñield at a given place and time is the behavior +of the charges in the øasứ. How ƒár in the past? "The time delay, or refarded +time, so-called, is the time it takes, at speed e, to get from the charge to the field +point ?P. The delay is ?//e. +So to allow for this time delay, we put a littÌe prime on r, meaning how far +away 1% uas when the information now arriving at left g. Just for a moment +suppose that the charge carried a light, and that the light could only come +to at the speed c. Then when we look at g, we would not see where ït 1s +now, of course, but where it œøs at some earlier time. What appears in our +formula is the apparen‡ direction ezz—the direction it used to be—the so-called +retarded direction——and at the retarded distance r7. That would be easy enough to +understand, too, but it is also wrong. The whole thing is much more complicated. +There are several more terms. The next term is as though nature were trying +to allow for the fact that the efect is retarded, ¡f we might put it very crudely. It +suggests that we should calculate the delayed Coulomb field and add a correction +to it, which is its rate of change times the time delay that we use. Nature seems +to be attempting to guess what the field at the present time is going to be, by +taking the rate of change and multiplying by the time that ¡is delayed. But we +are not yet through. 'Phere is a third term——the second derivative, with respect +to ứ, of the unit vector in the direction of the charge. Now the formula ¡s finished, +and that is all there is to the electric ñeld from an arbitrarily moving charge. +'The magnetic field is given by +B=-e,: x Eực. (28.4) +We have written these down only for the purpose of showing the beauty of nature +or, in a way, the power of mathematics. We do not pretend to understand :ø0hø it +is possible to write so much in such a small space, but (28.3) and (28.4) contain +the machinery by which electric generators work, how light operates, all the +phenomena of electricity and magnetism. Of course, to complete the story we +also need to know something about the behavior of the materials involved——the +properties of matter——which are not described properly by (28.3). +To fñnish with our description of the world of the 19th century we must +mention one other great synthesis which occurred in that century, one with which +Maxwell had a great deal to do also, and that was the synthesis of the phenomena, +of heat and mechanics. We shall study that subject soon. +'What had to be added in the 20th century was that the dynamical laws of +Newton were found to be all wrong, and quantum mechanies had to be introduced +--- Trang 489 --- +to correct them. Newton'”s laws are approximately valid when the scale of things +1s sufficiently large. These quantum-mechanical laws, combined with the laws of +electricity, have only recently been combined to form a set of laws called guantwm +clectrodunøam#cs. In addition, there were discovered a number of new phenomena, +of which the first was radioactivity, discovered by Becquerel in 1898——he just +sneaked 1t in under the 19th century. 'PThis phenomenon of radioactivity was +followed up to produce our knowledge of nuclei and new kinds of forces that are +not gravitational and not electrical, but new particles with diferent interactions, +a subJect which has still not been unravelled. +Eor those purists who know more (the professors who happen to be reading +this), we should add that when we say that (28.3) is a complete expression of the +knowledge of electrodynamies, we are not being entirely accurate. There was a +problem that was not quite solved at the end of the 19th century. When we try +to calculate the ñeld from all the charges ?ncluding the charge itselƒ that tue tuanứ +the ficld to ac† on, we get into trouble trying to fnd the distance, for example, of +a charge om itself, and dividing something by that distance, which is zero. The +problem of how to handle the part of this fñeld which ¡is generated by the very +charge on which we want the field to act is not yet solved today. So we leave 1t +there; we do not have a complete solution to that puzzle yet, and so we shall +avoid the puzzle for as long as we can. +28-2 Radiation +That, then, is a summary of the world picture. Now let us use it to discuss +the phenomena called radiation. To discuss these phenomena, we must select +from Eq. (28.3) only that piece which varies inversely as the distance and not as +the square of the distance. lt turns out that when we fñnally do fnd that piece, it +1s so simple in its form that it is legitimate to study optics and electrodynamics +in an elementary way by taking it as “the law” of the electric ñeld produced by a +moving charge far away. We shall take it temporarily as a given law which we +will learn about in detail next year. +Of the terms appearing in (28.3), the first one evidentÌy goes inversely as +the square of the distance, and the second is only a correction for delay, so 1E +1s easy to show that both of them vary inversely as the square of the distance. +All of the efects we are interested in come from the third term, which is not +very complicated, after all. What this term says 1s: look at the charge and note +the direction of the unit vector (we can project the end of it onto the surface of +--- Trang 490 --- +a unit sphere). As the charge moves around, the unit vector wiggles, and fhe +acceleration oƒ that ni 0ector is that tục are looking or. Phat is all. Thus +q d2e„› +⁄= 4mcạc2 d2. (285) +1s a statement of the laws of radiation, because that is the only important term +when we get far enough away that the fñelds are varying inversely as the distance. +(The parts that go as the square have fallen off so mụuch that we are not interested +in them.) +NÑow we can go a little bít further in studying (28.5) to see what it means. +Suppose a charge is moving in any manner whatsoever, and we are observing it +from a distance. We imagine for a moment that in a sense it is “li up” (although +1t is light that we are trying to explain); we imagine it as a little white dot. Then +we would see this white dot running around. But we don” see ezøcfg how it is +running around right =øu, because of the delay that we have been talking about. +What counts is how iÿ was moving earler. The unit vector e„; is pointed toward +the apparent position of the charge. Of course, the end of ez:; goes on a slipght +curve, so that its acceleration has two components. One is the transverse piece, +because the end of it goes up and down, and the other is a radial piece because +1t stays on a sphere. Ït is easy to demonstrate that the latter is much smaller +and varies as the inverse square of? when r7 is very great. This is easy to see, Íor +when we imagine that we move a given source farther and farther away, then the +wigplings of ez; look smaller and smaller, inversely as the distance, but the radial +component of acceleration is varying much more rapidly than inversely as the +distance. So for practical purposes all we have to do is project the motion on a +plane at unit distance. 'Therefore we fñnd the following rule: Imagine that we look +at the moving charge and that everything we see is delayed——like a painter trying +to paint a scene on a screen at a unit distance. Á real painter, oŸ course, does +not take into account the fact that light is goïing at a certain speed, but paints +the world as he sees it. We want to see what his picture would look like. So we +see a dot, representing the charge, moving about in the picture. 'Phe acceleration +of that dot is proportional to the electric field. 'That ¡s all—all we need. +Thus Eq. (28.5) is the complete and correct formula for radiation; even +relativity efects are all contained ín it. However, we often want to apply it to a +still simpler cireumstance in which the charges are moving only a small distanece +at a relatively slow rate. Since they are moving slowly, they do no move an +appreciable distance from where they start, so that the delay time is practically +--- Trang 491 --- +constant. 'Then the law ¡s still simpler, because the delay time is fñxed. 'Phus we +imagine that the charge is executing a very tiny motion at an efectively constant +distance. The delay at the distance r is r/c. Then our rule becomes the following: +TÍ the charged object is moving in a very small motion and it is laterally displaced +by the distance #(#), then the angle that the unit vector e¿; is displaced is #/?, +and sinece z is practically constant, the #-component of d2e, /đf? is simply the +acceleration of ø itself at an earlier time divided by z, and so fñnally we get the +law we want, which is +E,0)=— Ta, (: — ^): (28.6) +47coc2r e +Only the component of øx perpendicular to the line of sight is important. Let +us see why that is. Evidently, If the charge is moving in and out straight at us, +the unit vector in that direction does not wiggle at all, and it has no acceleration. +So 1È is only the sidewise motion which is important, only the acceleration that +we see projected on the screen. +28-3 The dipole radiator +As our fundamental “law” of electromagnetic radiation, we are goïing to assume +that (28.6) is true, i.e., that the electric ñeld produced by an accelerating charge +which is moving nonrelativistically at a very large distance ? approaches that +form. 'The electric field varies inversely as r and is proportional to the acceleration +of the charge, projected onto the “plane of sight,” and this acceleration is not +today”s acceleration, but the acceleration that ¡it had at an earlier time, the +amount of delay being a time, r/e. In the remainder of this chapter we shall +discuss this law so that we can understand it better physically, because we are +goïng to use it to understand all of the phenomena of light and radio propagation, +such as refection, refraction, interference, difraction, and scattering. It is the +central law, and is all we need. All the rest of Eq. (28.3) was written down only +to set the stage, so that we could appreciate where (28.6) fts and how i% comes +about. +We shall discuss (28.3) further next year. In the meantime, we shall accept +it as true, but not just on a theoretical basis. We may devise a number of +experiments which illustrate the character of the law. In order to do so, we need +an accelerating charge. It should be a single charge, but if we can make a great +many charges move together, all the same way, we know that the ñeld will be the +--- Trang 492 --- +Fig. 28-1. A high-frequency signal generator drives charges up and +down on two wires. +sum oÝ the efects of each of the individual charges; we just add them together. +As an example, consider two pieces oŸ wire connected to a generator, as shown in +Fig. 28-1. The idea is that the generator makes a potential diference, or a field, +which pulls electrons away from piece 4 and pushes them into Ö at one moment, +and then, an infinitesimal time later, it reverses the efect and pulls the electrons +out of and pumps them back into A/ So in these bwo wires charges, leb us say, +are accelerating upward in wire A and upward in wire Ö for one moment, and a +moment later they are accelerating downward in wire 4 and downward in wire Ö. +The fact that we need ÿwo wires and a generator is merely that this is a way of +doïng it. The net result is that we merely have a charge accelerating up and down +as though 4 and were one single wire. A wire that is very short compared +with the distance light travels in one oscillation period is called an elecfric đipolÌe +oscillator. 'hus we have the cireumstanece that we need to apply our law, which +tells us that this charge makes an electric feld, and so we need an instrument to +detect an electric ñeld, and the instrument we use is the same thing—a pair of +wires like A and / If an electric field is applied to such a device, it will produce +a force which will pull the electrons up on both wires or down on both wires. +Thịs signal is detected by means of a rectifier mounted bebween 4 and ?Ø, and +a tiny, ñne wire carries the information into an amplifier, where it is amplified so +we can hear the audiofrequency tone with which the radiofrequency is modulated. +'When this probe feels an electric field, there will be a loud noise coming out of the +loudspeaker, and when there is no electric fñeld driving it, there will be no noise. +Because the room in which the waves we are measuring has other objects in +1%, our electric ñeld will shake electrons in these other objects; the electric field +makes these other charges go up and down, and ín going up and down, these +also produce an efect on our probe. Thus for a successful experiment we must +hold things fairly close together, so that the inuences from the walls and from +--- Trang 493 --- +ourselves—the refected waves—are relatively small. 5o the phenomena will not +turn out to appear to be precisely and perfectly in accord with Eq. (28.6), but +will be close enough that we shall be able to appreciate the law. +~ K "° ` +`. | .* +»e. ."“ +Fig. 28-2. The instantaneous electric field on a sphere centered at a +localized, linearly oscillating charge. +Now we turn the generator on and hear the audio signal. We fnd a strong +fñeld when the detector is parallel to the generator G at point 1 (Eig. 28-2). +We fnd the same amount of fñeld also at any other azimuth angle about the axis +of Œ, because it has no directional efects. On the other hand, when the detector +1s at 3 the field is zero. 'Phat is all right, because our formula said that the field +should be the acceleration of the charge projected perpendicular to the line of +sipht. “Therefore when we look down on Œ, the charge is moving toward and +away from D, and there is no efect. So that checks the frst rule, that there is +no efect when the charge is moving directly toward us. Secondly, the formula +says that the electric fñeld should be perpendicular to z and in the plane of G +and 7; so if we put Ö at 1 but rotate it 90°, we should get no signal. And this is +Just what we fnd, the electric feld is indeed vertical, and not horizontal. When +we move to some intermediate angle, we see that the strongest sipnal occurs +when it is oriented as shown, because although Œ is vertical, it does not produce +a fñeld that is simply parallel to itself—it is the projeciton oƒ the acceleration +perpendicular to the line oƒ sight that counts. The signal is weaker at 2 than it is +at 1, because of the proJection efect. +--- Trang 494 --- +28-4 Interference +Next, we may test what happens when we have two sources side by side +sevoral wavelengths apart (Fig. 28-3). The law is that the two sources should +add theïr efects at point 1 when both of the sources are connected to the same +generator and are both moving up and down the same way, so that the total +electric ñeld is the sum of the two and is twice as strong as it was before. +D S2 +Fig. 28-3. lllustration of interference of sources. +Now comes an interesting possibility. Suppose we make the charges In S1 +and Š› both accelerate up and down, but delay the timing of Š5› so that they are +1802 out of phase. 'Phen the field produced by ŠS¡ will be in one direction and +the field produced by ŠS+ will be in the opposite direction at any instant, and +therefore we should get øoø efect at point 1. The phase of oscillation is neatly +adjustable by means of a pipe which is carrying the signal to S¿. By changing +the length of this pipe we change the time it takes the signal to arrive at 5s and +thus we change the phase of that oscillation. By adjusting this length, we can +indeed fnd a place where there is no more signal left, in spite of the fact that +both 5¡ and 52 are movingl The fact that they are both moving can be checked, +because if we cut one out, we can see the motion of the other. So the bwo of +them together can produce zero iŸ everything is adjusted correctly. +Now, i1 is very interesting to show that the addition of the two fields is in +fact a 0ector addition. We have Jjust checked it for up and down motion, bu§ +let us check two nonparallel directions. First, we restore 5 and S2 to the same +phase; that is, they are again moving together. But now we turn 5: through 902, +as shown in Fig. 28-4. Now we should have at point 1 the sum of two efects, +--- Trang 495 --- +2. /R +S2 +Fig. 28-4. lllustration of the vector character of the combination of +SOUFCeS. +one of which is vertical and the other horizontal. The electric fñeld is the vector +sum oŸ these two in-phase signals—they are both strong at the same time and go +through zero together; the total fñeld should be a signal at 45°. If we turn +to get the maximum noise, it should be at about 45°, and not vertical. And if +we turn i% at right angles to that direction, we should get zero, which is easy to +mmeasure. Indeed, we observe just such behaviorl +Now, how about the retardation? How can we demonstrate that the signal is +retarded? We could, with a great deal of equipment, measure the time at which +1t arrives, but there is another, very simple way. Referring again to Fig. 28-3, +suppose that 5¡ and S52 are in phase. 'Phey are both shaking together, and +they produce equal electric fields at point 1. But suppose we go to a certain +place 2 which is closer to S2 and farther from Š¡. Then, in accordance with the +principle that the acceleration should be retarded by an amount equal to r/e, +1f the retardations are not equal, the signals are no longer in phase. Thus it +should be possible to fnd a position at which the distances of 1 from 5% and S2 +difer by some amount A, in such a manner that there is no net signal. 'Phat +is, the distance A is to be the distance light goes in one-half an oscillation of +the generator. We may go still further, and fñnd a poïint where the diferenece is +greater by a whole cycle; that is to say, the signal from the first antenna reaches +point 3 with a delay in time that is greater than that of the second antenna, +by just the length of time it takes for the electric current to oscillate once, and +therefore the two electric fñelds produced at 3 are in phase again. At point 3 the +signal is strong again. +This completes our discussion of the experimental verifcation of some of +the important features of Eq. (28.6). Of course we have not really checked +the 1/r variation of the electric feld strength, or the fact that there is also a +--- Trang 496 --- +magnetic fñeld that goes along with the electric ñeld. To do so would require +rather sophisticated techniques and would hardly add to our understanding at +this point. In any case, we have checked those features that are of the greatest +Importance for our later applications, and we shall come back to study some of +the other properties of electromagnetic waves next year. +--- Trang 497 --- +Xrfor'for-orte© +29-1 Electromagnetic waves +In this chapter we shall discuss the subject of the preceding chapter more +mathematically. We have qualitatively demonstrated that there are maxima, +and minima in the radiation fñeld from two sources, and our problem now is tO +describe the field in mathematical detail, not just qualitatively. +We have already physically analyzed the meaning of formula (28.6) quite +satisfactorily, but there are a few points to be made about it mathematically. In +the frst place, IŸ a charge is accelerating up and down along a line, in a motion +of very small amplitude, the fñield at some angle Ø from the axis of the motion is +in a direction at right angles to the line of sight and in the plane containing both +the acceleration and the line of sight (Fig. 29-1). Iƒ the distance is called r, then +at time £ the electric fñeld has the magnitude +—qa{(‡ — r/e) sin 8 +#(0 = _—_^.. (29.1) +47cgc2r +Fig. 29-1. The electric field E due to a positive charge whose retarded +acceleration is a”. +--- Trang 498 --- +Fig. 29-2. The acceleration of a certain charge as a function of time. +Fig. 29-3. The electric field as a function of position at a later time. +(The 1/r variation is ignored.) +where a(# — r/e) is the acceleration at the time (£ — r/c), called the retarded +acceleration. +Now it would be interesting to draw a picture of the fñeld under different +conditions. The thing that is interesting, of course, is the factor ø(£ ��� r/c), +and to understand it we can take the simplest case, Ø = 90”, and plot the field +graphically. What we had been thinking of before is that we stand in one position +and ask how the fñeld there changes with time. But instead of that, we are now +goïing to see what the field looks like at diferent positions in space at a given +Instant. So what we want is a “snapshot” picture which tells us what the fñeld +1s In diÑerent places. Of course it depends upon the acceleration of the charge. +Suppose that the charge at first had some particular motion: it was Initially +standing still, and ¡it suddenly accelerated in some mamner, as shown in Fig. 29-2, +and then stopped. 'Then, a little bit later, we measure the field at a diferent +place. Then we may assert that the feld will appear as shown in Eig. 29-3. At +cach point the fñeld is determined by the acceleration of the charge at an earlier +time, the amount earlier being the delay r/c. The field at farther and farther +points is determined by the acceleration at earlier and earlier times. So the curve +in Eig. 29-3 is really, in a sense, a “reversed” plot of the acceleration as a function +of time; the distanece is related to time by a constant scale factor c, which we +--- Trang 499 --- +often take as unity. This is easily seen by considering the mathematical behavior +of ø(£ — r/c). Evidently, if we add a little time Af, we get the same value for +a(È — r/c) as we would have if we had subtracted a little distance: Ar = —c Ai. +Stated another way: if we add a little time A£, we can restore œ(£ — r/e) to +1ts former value by adding a little distance Az = cAf. Thhat is, as tỉme goes on +the fteld mmoues qs a U0aue outuUard jrom the source. Thhat is the reason why we +sometimes say light is propagated as waves. I% is equivalent to saying that the +field is delayed, or to saying that the electric feld is moving outward as time +ØO©s OH. +An interesting special case is that where the charge g is moving up and down +in an oscillatory manner. The case which we studied experimentally in the last +chapter was one in which the displacement ø at any time ý was equal to a certain +constant zọ, the magnitude of the oscillation, times cos/. 'Then the acceleration +d = —02#0 COS UÉ — đọ COS UÉ, (29.2) +where ứo is the maximum acceleration, —œ2#o. Putting this formula into (29.1), +we fñnd (t— r/ +. Œọ COS(U(È — rc +1 =-qsin8 _.~ (29.3) +Now, ignoring the angle Ø and the constant factors, let us see what that looks +like as a function of position or as a function of time. +29-2 Energy of radiation +First of all, at any particular moment or in any particular place, the strength +of the field varies inversely as the distance r, as we mentioned previously. NÑow +we must point out that the energu content of a wave, or the energy efects that +such an electrie field can have, are proportional to the sợuare of the field, because +1Ý, for instance, we have some kind of a charge or an oscillator in the electric field, +then I1f we let the field act on the oscillator, it makes it move. lf this is a linear +oscillator, the acceleration, velocity, and displacement produced by the electric +fñeld acting on the charge are all proportional to the feld. So the kinetic energy +which is developed in the charge is proportional to the square of the fñield. Š5o we +shall take it that the energy that a field can deliver to a system is proportional +somehow to the square of the field. +This means that the energy that the source can deliver decreases as we gøet +farther away; in fact, 1t varles ?nuersclU as the square oƒ the đistance. But that +--- Trang 500 --- +Fig. 29-4. The energy flowing within the cone ABC D is independent +of the distance r at which ït is measured. +has a very simple interpretation: If we wanted to pick up all the energy we could +from the wave in a certain cone at a distance ?¡ (Eig. 29-4), and we do the same +at another distance r›, we ñnd that the amount of energy per unit area at any +one place øoes inversely as the square of r, but the area of the surface intercepted +by the cone goes đ/recfu as the square of z. So the energy that we can take out +of the wave within a given conical angle is the same, no matter how far away +we arel In particular, the total energy that we could take out of the whole wave +by putting absorbing oscillators all around is a certain fñxed amount. So the +fact that the amplitude of E varies as 1/7 is the same as saying that there is an +energy fux which is never lost, an energy which goes on and on, spreading over a +greater and greater effective area. Thus we see that after a charge has oscillated, +1t has lost some energy which it can never recover; the energy keeps going farther +and farther away without diminution. So ïÝ we are far enough away that our +basic approximation is good enouph, the charge cannot recover the energy which +has been, as we say, radiated away. Of course the energy still exists somewhere, +and is available to be picked up by other systems. We shall study this energy +“loss” further in Chapter 32. +Let us now consider more carefully how the wave (29.3) varies as a function +of time at a given place, and as a function of position at a given time. Again we +ignore the 1/r variation and the constants. +29-3 Sinusoidal waves +Pirst let us ñx the position r, and watch the field as a function of time. Ït is +oscillatory at the angular frequency œ. The angular frequency œ can be defned +--- Trang 501 --- +as the ra£© oƒ chưnge öoƒ phase tuïth từme (radians per second). We have already +studied such a thing, so it should be quite familiar to us by now. 'Phe per?od is +the time needed for one oscillation, one complete cycle, and we have worked that +out too; it is 2#/œ, because œ times the period is one cycle of the cosine. +Now we introduce a new quantity which is used a great deal in physics. This +has to do with the opposite situation, in which we fix £ and look at the wave +as a function of distance r. Of course we notice that, as a function of r, the +wave (29.3) is also oscillatory. That is, aside from 1/r, which we are ignoring, we +see that # oscillates as we change the position. So, in analogy with œ, we can +defne a quantity called the 0œøe rruwmber, symbolized as k. 'This is deñned as ứhe +rake oƒ change oƒ phase tuïth distœnce (radians per meter). 'That 1s, as we move +in space at a fñxed time, the phase changes. +'There is another quantity that corresponds to the period, and we might call +1t the period in space, but it is usually called the wavelength, symbolized À. The +wavelength is the distance occupied by one complete cycle. Ït is easy tO see, +then, that the wavelength is 2Z/&k, because & times the wavelength would be the +number of radians that the whole thing changes, being the product o£ the rate of +change of the radians per meter, times the number of meters, and we must make +a 27 change for one cycle. So &À = 27 is exactly analogous to ¿fọ = 27. +Now in our particular wave there is a definite relationship between the fre- +quency and the wavelength, but the above definitions of k and œ are actually +quite general. 'Phat is, the wavelength and the frequency may not be related in +the same way in other physical circumstances. However, in our circumstance +the rate of change of phase with distance is easily determined, because if we call +Ó = w(t — r/c) the phase, and diferentiate (partially) with respect to distance r7, +the rate of change, Øj/Ôr, is +lg|=£=Š: (29.4) +There are many ways to represent the same thing, such as +À =cío (29.5) À/=ec (29.7) +œ = €k (29.6) œÀ = 27c (29.8) +'Why is the wavelength equal to e times the period? 'Phat”s very easy, Of course, +because if we sit still and wait for one period to elapse, the waves, travelling at +--- Trang 502 --- +the speed c, will move a distance cứo, and will of course have moved over just +one wavelength. +In a physical situation other than that of lght, & is not necessarily related +to œ in this simple way. TỶ we call the distance along an axis #, then the formula +for a cosine wave moving in a direction z with a wave number & and an angular +frequency œ will be written in general as cos (É — &z). +Now that we have introduced the idea of wavelength, we may say something +more about the cireumstances in which (29.1) is a legitimate formula. We recall +that the field is made up of several pieces, one of which varles inversely as r7, +another part which varies inversely as r2, and others which vary even faster. It +would be worth while to know in what circumstances the 1/z part of the field is +the most important part, and the other parts are relatively small. Naturally, the +answer is “if we go “far enoughˆ away,” because terms which vary inversely as +the square ultimately become negligible compared with the 1/z term. How Íar is +“far enough”? The answer is, qualitatively, that the other terms are of order À/r +smaller than the 1/z term. Thus, so long as we are beyond a few wavelengths, +(29.1) is an excellent approximation to the field. Sometimes the region beyond a +few wavelengths is called the “wave zone.” +29-4 Two dipole radiators +Next let us discuss the mathematics involved in combining the efects of two +oscillators to fnd the net fñeld at a given point. This is very easy in the Íew cases +that we considered in the previous chapter. We shall first describe the efects +qualitatively, and then more quantitatively. Let us take the simple case, where +the oscillators are situated with their centers in the same horizontal plane as the +detector, and the line of vibration is vertical. +Figure 29-5(a) represents the top view of 6wo such oscillators, and in this +particular example they are half a wavelength apart in a NÑ-S direction, and are +oscillating together in the same phase, which we call zero phase. NÑow we would +like to know the intensity of the radiation in various directions. By the intensity +we mean the amount of energy that the fñeld carries past us per second, which is +proportional to the square of the fñield, averaged ïn time. So the thing to look at, +when we want to know how bright the light is, is the square of the electric field, +not the electric field itself. (The electric field tells the strength of the force felt +by a stationary charge, but the amount of energy that is going past, in watts per +square meter, is proportional to the square of the electric field. We shall derive +--- Trang 503 --- +2 của 2 2 củ ? +4 À/2——4 0 À/2———0 +2z |NG z |ÌNG +œ=0 œ=1 +(a) (@b) +Fig. 29-5. The intensities In various directions from two dipole oscilla- +tors one-half wavelength apart. Left: in phase (œ = 0). Right: one-half +period out of phase (œ = 7). +the constant of proportionality in Chapter 31.) TÝ we look at the array rom the W +side, both oscillators contribute equally and in phase, so the electric feld is Ewice +as strong as it would be from a single oscillator. Therefore the ?mtensit ¡s [our +times as sfrong ús ?t tuould be tƒ there tuere onÏỤ one oscillator. (The numbers +in Eig. 29-5 represent how strong the intensity would be in this case, compared +with what it would be if there were only a single oscillator of unit strength.) Ñow, +in either the Ñ or S5 direction along the line of the oscillators, since they are half +a wavelength apart, the efect of one oscillator turns out to be out of phase by +exactly half an oscillation from the other, and therefore the fñelds add to zero. +At a certain particular intermediate angle (in fact, at 309) the intensity is 2, and +1t falls of, 4, 2, 0, and so forth. We have to learn how to fnd these numbers at +other angles. It is a question of adding two oscillations with diferent phases. +Let us quickly look at some other cases of interest. Suppose the oscillators +are again one-half a wavelength apart, but the phase œ of one is set half a period +behind the other in its oscillation (Fig. 29-5b). In the W direction the intensity +is now zero, because one oscillator is “pushing” when the other one is “pulling” +But in the N direction the signal from the near one comes at a certain time, and +that of the other comes half a period later. But the latter was originallu half a +period behind in timing, and therefore it is now exactly 7n tưne with the first one, +and so the intensity in this direction is 4 units. The intensity in the direction +at 30” is still 2, as we can prove later. +Now we come to an interesting case which shows up a possibly useful feature. +Let us remark that one of the reasons that phase relations of oscillators are +interesting is for beaming radio transmitters. For instance, if we build an antenna +--- Trang 504 --- +system and want to send a radio signal, say, to Hawaii, we set the antennas up +as in Fig. 29-5(a) and we broadcast with our 0wo antennas in phase, because +Hawall is to the west of us. Then we decide that tomorrow we are going %O +broadcast toward Alberta, Canada. Since that is north, not west, all we have +to do 1s to reverse the phase of one of our antennas, and we can broadcast to +the north. 5o we can build antenna systems with various arrangements. Ôurs is +one of the simplest possible ones; we can make them much more complicated, +and by changing the phases in the various antennas we can send the beams In +various directions and send most of the power in the direction in which we wish +to transmit, without ever moving the antennal In both of the preceding cases, +however, while we are broadcasting toward Alberta we are wasting a lot of power +on Easter Island, and it would be interesting to ask whether it is possible to +send it in only øwe direction. At frst sight we might think that with a pair of +antennas of this nature the result is always going to be symmetrical. So let us +consider a case that comes out unsymmetrical, to show the possible variety. +2 Ẫ /4—2 +Fig. 29-6. A palr of dipole antennas giving maximum power in one +direction. +Tf the antennas are separated by one-quarter wavelength, and ïf the NÑ one +is one-fourth period behind the S one in time, then what happens (Fig. 29-6)? +In the W direction we get 2, as we will see later. In the SŠ direction we get zero, +because the signal from SŠ comes at a certain time; that from Ñ comes 902 later +in #ữne, but it is already 90° behind in its built-in phase, therefore it arrives, +altogether, 180” out of phase, and there is no efect. On the other hand, in the +NÑ direction, the Ñ signal arrives earlier than the 5 signal by 90” in time, because +1t is a quarter wavelength closer. But its phase is set so, that it is oscillating 90° +behữnd In tìme, which Just compensates the delay diference, and therefore the +two sipnals appear ogether in phase, making the field strength twice as large, +and the energy four tỉmes as great. +--- Trang 505 --- +Thus, by using some cÌleverness in spacing and phasing our antennas, we +can send the power all in one direction. But still it is distributed over a great +range of angles. Can we arrange it so that it is focused still more sharply in +a particular direction? Let us consider the case of Hawali again, where we are +sending the beam east and west but it is spread over quite an angle, because +even at 30” we are still getting half the intensity—we are wasting the power. Can +we do better than that? Let us take a situation in which the separation 1s ben +wavelengths (Fig. 29-7), which is more nearly comparable to the situation in which +we experimented in the previous chapter, with separations of several wavelengths +rather than a small raction of a wavelength. Here the picture is quite diÑerent. +To distant point +Fig. 29-7. The intensity pattern for two dipoles separated by 10À. +TÍ the oscillators are ten wavelengths apart (we take the in-phase case to make +it easy), we see that in the E—W direction, they are in phase, and we get a strong +intensity, four times what we would get if one of them were there alone. On the +other hand, at a very small angle away, the arrival times difÑfer by 180” and the +intensity is zero. To be precise, iŸ we draw a line from each oscillator to a distant +point and the diference A in the two distances is À/2, half an oscillation, then +they will be out of phase. So this firsb nuÌl occurs when that happens. (The +fgure is not drawn to scale; it is only a rough sketch.) This means that we do +indeed have a very sharp beam in the direction we want, because If we just move +over a little bit we lose all our intensity. Ủnfortunately for practical purposes, +1ƒ we were thinking of making a radio broadcasting array and we doubled the +distance A, then we would be a whole cycle out of phase, which is the same as +being exactly #n phase againl Thus we get many successive maxima and minima, +just as we found with the 23A spacing in Chapter 28. +Now how can we arrange to get rid of all these extra maxima, or “lobes,” as +they are called? We could get rid of the unwanted lobes in a rather interesting +--- Trang 506 --- +6+ _A = 5ố +10A 2 0° +30° +Fig. 29-8. A six-dipole antenna array and part of its intensity pattern. +way. Suppose that we were to place another set of antennas between the bwo +that we already have. 'That is, the outside ones are still 10À apart, but between +them, say every 2À, we have put another antenna, and we drive them all in +phase. There are now six antennas, and if we looked at the intensity in the +E—W direction, it would, of course, be much higher with six antennas than with +one. The fñeld would be six times and the intensity thirty-six times as great (the +square of the feld). We get 36 units of intensity in that direction. Now if we +look at neighboring points, we fnd a zero as before, roughly, but If we go farther, +to where we used to get a big “bump,” we get a much smaller “bump” now. Let +us try to see why. +The reason is that although we might expect to get a big bump when the +distance A is exactly equal to the wavelength, it is true that dipoles 1 and 6 are +then in phase and are cooperating in trying to get some strength in that direction. +But numbers 3 and 4 are roughly 3 a wavelength out oŸ phase with 1 and 6, and +althoupgh 1 and 6 push together, 3 and 4 push together too, but in opposite phase. +'Therefore there is very little intensity in this direction——=but there is something; +it does not balance exactly. This kind of thing keeps on happening; we get very +little bumps, and we have the strong beam in the direction where we want it. +But in this particular example, something else will happen: namely, since the +distance between successive dipoles is 2À, it is possible to find an angle where +the distance ổ between successiue đipoles is exactly one wavelength, so that the +effects from all of them are in phase again. Each one is delayed relative to the +next one by 3607, so they all come back in phase, and we have another strong +beam in that direction! It is easy to avoid this in practice because it is possible +to put the dipoles closer than one wavelength apart. IÝ we put in more antennas, +--- Trang 507 --- +closer than one wavelength apart, then this cannot happen. But the fact that +this can happen at certain angles, if the spacing is bigger than one wavelength, +is a very interesting and useful phenomenon in other applications——not radio +broadcasting, but in đjffraction gratings. +29-5 The mathematics of interference +Now we have finished our analysis of the phenomena of dipole radiators +qualitatively, and we must learn how to analyze them quantitatively. To ñnd the +efect of two sources at some particular angle in the most general case, where +the two oscillators have some intrinsic relative phase œ from one another and +the strengths 4q and 4s are not equal, we fnd that we have to add two cosines +having the same frequenecy, but with diferent phases. Ït is very easy to find this +phase diference; it is made up of a delay due to the diference in distance, and +the intrinsic, built-in phase of the oscillation. Mathematically, we have to fnd +the sum ?# of two waves: Ï = Ái cos (2£ + ôi) + Áa cos (0É + j2). How do we do +Tt is really very easy, and we presume that we already know how t$o do it. +However, we shall outline the procedure in some detail. Eirst, we can, if we are +clever with mathematics and know enough about cosines and sines, simply work +it out. "The easiest such case is the one where 4 and 4a are cqual, let us say +they are both equal to A. In those cireumstances, for example (we could call this +the trigonometric method of solving the problem), we have +Tỳ = Alcos (‡ + ở1) + cos (ø‡ + óa)]. (29.9) +Once, in our trigonometry class, we may have learned the rule that +cos A + cos 8 = 2cos 5(A + B) cos 3(A — 8). (29.10) +Tf we know that, then we can Immediately write l as +l= 2Acos 3(di — 9a) cos (0É + Sới + 392). (29.11) +So we find that we have an oscillatory wave with a new phase and a new amplitude. +In general, the result œ2 be an oscillatory wave with a new amplitude Áp, which +we may call the resultant amplitude, oscillating at the same frequency but with +--- Trang 508 --- +a phase diference óp, called the resultant phase. In view of this, our particular +case has the following result: that the resultant amplitude 1s +An = 2Acos š (di — óa), (29.12) +and the resultant phase is the average of the two phases, and we have completely +solved our problem. +⁄⁄2——T +œ $n ° * +Fig. 29-9. A geometrical method for combining two cosine waves. +The entire diagram ¡is thought of as rotating counterclockwise with +angular frequency 0. +Now suppose that we cannot remember that the sum of Ewo cosines is twice +the cosine of half the sum times the cosine of half the diference. 'Phen we may +use another method of analysis which is more geometrical. Any cosine function +of „# can be considered as the horizontal projectlon of a rofating uector. Suppose +there were a vector ¡ of length 4 rotating with time, so that its angle with the +horizontal axis is œ‡ + ởị. (WS shall leave out the œ# in a minute, and see that it +makes no diference.) Suppose that we take a snapshot at the tìme £ = 0, although, +in fact, the picture is rotating with angular velocity œ (Fig. 29-9). The projection +of Ai along the horizontal axis is precisely Ai cos (ð£ + ở). Now at £ =0 the +second wave could be represented by another vector, 4a, of length 4a and at +an angle ós, and also rotating. Phey are both rotating with the same angular +velocity œ, and therefore the relafiue positions of the two are fxed. The system +goes around like a rigid body. The horizontal projection oŸ Áa is 4a cos (0£ + da). +But we know from the theory of vectors that if we add the bwo vectors in the +ordinary way, by the parallelogram rule, and draw the resultant vector Án, the +#-component of the resultant is the sum of the #z-components of the other two +vectors. hat solves our problem. It is easy to check that this gives the correct +--- Trang 509 --- +result for the special case we treated above, where Ái = 4a = A. In this case, +we see from Fig. 29-9 that Áp lies midway between 4+ and 4a and makes an +angle 3(Óa — ới) with each. Therefore we see that Áp = 2Ácos 3(s — ới), a8 +before. Also, as we see from the triangle, the phase of Ág, as it goes around, is +the average angle of Áq and 4s when the two amplitudes are equal. Clearly, we +can also solve for the case where the amplitudes are not equal, Just as easily. We +can call that the geometrical way oŸ solving the problem. +There is still another way of solving the problem, and that is the ønalfical +way. hat is, instead of having actually to draw a picture like Fig. 29-9, we +can write something down which says the same thing as the picture: instead of +drawing the vectors, we write a complez mxwmber to represent each of the vectors. +'The real parts of the complex numbers are the actual physical quantities. So in +our particular case the waves could be written in this way: Aieff†1) [the real +part of this is Ai cos (ø£ + ởi)| and Asef@f†22), Ñow we can add the two: +h = Aiei@etrói) + Aasei6et92) = (Aie2t + Aac192)c«t (29.13) +Ñ= Aic? + Aac!?2 = Ancl6n, (29.14) +'This solves the problem that we wanted to solve, because it represents the result +as a complex number of magnitude Áp and phase ón. +To see how this method works, let us ñnd the amplitude An which is the +“length” of f. To get the “length” of a complex quantity, we always multiply the +quantity by its complex conjugate, which gives the length squared. he complex +conjugate is the same expression, but with the sign of the 7's reversed. 'Phus we +A? = (Aic'? + Aac??2)(Aie"??! + Aae”12), (29.15) +In multiplying this out, we get 4ƒ + 443 (here the es caneel), and for the cross +terms we have +Ai4Aa(cft®i=4) + cit02~91)), +e9 + e~?? = cosØ + isỉn Ø + cos Ø — ?sỉn 6. +That is to say, e'? + e~? = 2cosØ. Qur fnal result is therefore +4a = 4? + A2 + 2AI4a COS (Óa — Ị). (29.16) +--- Trang 510 --- +As we seo, this agrees with the length of Áp in Eig. 29-9, using the rules of +trigonometry. +Thus the sum of the two efects has the intensity 4? we would get with one +of them alone, plus the intensity 43 we would get with the other one alone, +plus a correction. 'Phis correction we call the mterƒference effect. It is really +only the diference bebween what we get simply by adding the intensities, and +what actually happens. We call it interference whether it is positive or negative. +(Interference in ordinary language usually suggests opposition or hindranee, but +in physics we often do not use language the way it was originally designedl) TỶ the +Interference term is positive, we call that case construcfzue interference, horrible +though it may sound to anybody other than a physicistl The opposite case is +called des‡ructzue interference. +Now let us see how to apply our general formula (29.16) for the case of Ewo +oscillators to the special situations which we have discussed qualitatively. To +apply this general formula, it is only necessary to fnd what phase diference, +Ó1 — đa, ©exists between the signals arriving at a given point. (It depends only on +the phase difference, of course, and not on the phase itself.) So let us consider +the case where the two oscillators, of equal amplitude, are separated by some +distance đ and have an intrinsic relative phase œ. (When one is at phase zero, the +phase of the other is œ.) Then we ask what the intensity will be in some azimuth +direction Ø from the E—W line. [Note that this is mof the same Ø as appears +in (29.1). We are torn between using an unconventional symbol like lý, or the +conventional symbol Ø (Fig. 29-10).| The phase relationship is found by noting +that the diference in distance from ? to the two oscillators is đsin Ø, so that the +phase diference contribution from this is the number of wavelengths in đsin 6, +multiplied by 2z. (Those who are more sophisticated might want to multiply the +wave number k, which is the rate of change of phase with distance, by đsin; +Aell0tta) To Point P +AeetZ đsin80 +Fig. 29-10. 'Iwo oscillators of equal amplitude, with a phase differ- +ence œ between them. +--- Trang 511 --- +1b is exactly the same.) The phase diference due to the distance difference is +thus 2zdsin Ø/^À, but, due to the timing of the oscillators, there is an additional +phase œ. So the phase diference at arrival would be +Óa — Ôi = œ+ 2mdsin 0/À. (29.17) +'This takes care of all the cases. 'Thus all we have to do is substitute this expression +into (29.16) for the case 4 = 4a, and we can calculate all the various results for +two antennas of equal intensity. +Now let us see what happens in our various cases. The reason we know, for +example, that the intensity is 2 at 30° in Eig. 29-5 is the following: the two +oscillators are ¿À apart, so at 30°, dsin Ø = À/4. Thus ó¿ — ởị = 2mÀ/4ÀA = m/2, +and so the interference term is zero. (We are adding two vectors at 909.) The +result is the hypotenuse of a 45° right-angle triangle, which is v⁄2 times the unit +amplitude; squaring it, we get ©wice the intensity of one oscillator alone. All the +other cases can be worked out in this same way. +--- Trang 512 --- +})rffr-(rcff©ore +30-1 The resultant amplitude due to ?øw equal oscillators +'This chapter is a direct continuation of the previous one, although the name +has been changed om /n#erference to Diffraction. No one has ever been able to +defñne the diference between interference and difraction satisfactorily. It is just a +question of usage, and there is no specife, important physical diference between +them. The best we can do, roughly speaking, is to say that when there are only +a Ífew sources, say ©wo, interfering, then the result is usually called interference, +but if there is a large number of them, it seems that the word difraction is more +often used. 5o, we shall not worry about whether it is interference or difraction, +but continue directly from where we left off in the middle of the subject in the +last chapter. +Thus we shall now discuss the situation where there are + equally spaced +oscillators, all of equal amplitude but diferent from one another in phase, either +because they are driven diferently in phase, or because we are looking at them at +an angle such that there is a difference in time delay. Eor one reason or another, +we have to add something like this: +T = Alcos œ£ + cos (uÉ + ở) + cos (É + 29) + - - - + cos (2£ + (n — 1))], (50.1) +where ở is the phase diference between one oscillator and the next one, as seen +in a particular direction. Specifically, ¿ = œ + 2rdsinØ/A. Ñow we must add all +the terms together. We shall do this geometrically. The frst one is of length A, +and ít has zero phase. “The next is also of length 4 and it has a phase equal to ó. +The next one is again of length A and it has a phase equal to 2ø, and so on. So +we are evidently going around an equiangular polygon with ø sides (Eig. 30-1). +Now the vertices, of course, all lie on a circle, and we can fñnd the net amplitude +mmost easily if we fnd the radius of that cirele. Suppose that @ is the center of +--- Trang 513 --- +A6 +Q ° +O Ai S my x +Fig. 30-1. The resultant amplitude of n = 6 equally spaced sources +with net successive phase differences ý. +the circle. Thhen we know that the angle Q6 is just a phase angle . (Thịs is +because the radius Q9 bears the same geometrical relation to 4a as QO bears +to Ai, so they form an angle ó between them.) Therefore the radius r must +be such that A = 2rsin 2/2, which fixes r. But the large angle Ó@Q7' is equal +to mó, and we thus fnd that Ág = 2rsinno2/2. Combining these bwo results to +eliminate r, we get +sin n@/2 +An=A————. 30.2 +" sin @/2 (30.2) +The resultant intensity is thus +sinˆ „j/2 +T=lo——=_. 30.3 +" sin? ø/2 (80.3) +Now let us analyze this expression and study some of its consequences. Ïn +the first place, we can check it for ø = 1. It checks: σ = Tạ. Next, we check it +for ø= =2: writing sin @ = 2sin @/2cos @/2, we find that An = 2A cos @/2, which +agrees with (29.12). +Now the idea that led us to consider the addition of several sources was that +we might get a much stronger intensity in one direction than in another; that the +nearby maxima which would have been present if there were only ©wo sources will +have gone down in strength. In order to see this efect, we plot the curve that +comes rom (30.3), taking œ to be enormously large and plotting the region near +=0. In the first place, iŸ ở is exactly 0, we have 0/0, but iŸ ở is inũnitesimal, +the ratio of the two sines squared is simply n2, since the sine and the angle are +--- Trang 514 --- +approximately equal. 'Phus the intensity of the maximum of the curve is equal +to n2 times the intensity of one oseillator. That is easy to see, because if they +are all in phase, then the little vectors have no relative angle and all œ of them +add up so the amplitude is ø times, and the intensity n2 times, stronger. +As the phase ó increases, the ratio of the Ewo sines begins to fall of, and the +first time it reaches zero is when #d/2 = z, because sin 7 = 0. In other words, +@ = 2#/n corresponds to the first minimum in the curve (Fig. 30-2). In terms +of what is happening with the arrows in Fig. 30-1, the first minimum occurs +when all the arrows come back to the starting point; that means that the total +accumulated angle in all the arrows, the total phase diference between the first +and last oscillator, must be 2z to complete the circle. +1.0 +H Ñ = +; \ rN TT _ +Z———-`.ø⁄“.——`-ò-s.⁄ ~ ——=- +0 1 2 3 4 nộ/2m 5 +Fig. 30-2. The Intensity as a function of phase angle for a large +number of oscillators of equal strength. +Now we go to the next maximum, and we want to see that it is really much +smaller than the first one, as we had hoped. We shall not go precisely to the +maximum position, because both the numerator and the denominator of (30.3) +are variant, but sin 2/2 varies quite slowly compared with sinnø/2 when øw is +large, so when sinnd/2 = I we are very close to the maximum. “The next +maximum of sin2eở/2 comes at œÓ/2 = 37/2, or ó = 3Z/n. This corresponds +to the arrows having traversed the circle one and a half times. On putting +ó = 3z/n into the Íormula to fnd the size of the maximum, we fñnd that +sin” 3x/2 = 1 in the numerator (because that is why we picked this angle), and +in the denominator we have sin? 3z/2n. Now if ø is sufficiently large, then this +angle is very small and the sine is equal to the angle; so for all practical purposes, +we can put sỉin 3/2n = 3z/2n. Thus we find that the intensity at this maximum +--- Trang 515 --- +is l = Ia(dn2/9z?). But øØỞlạ was the maximum intensity, and so we have +4/92 tỉmes the maximum intensity, which is about 0.045, less than 5 percent, of +the maximum intensity! Of course there are decreasing intensities farther out. +So we have a very sharp central maximum with very weak subsidiary maxima on +the sides. +Tt is possible to prove that the area of the whole curve, including all the little +bumps, is equal to 2wïÏo, or bwice the area of the dotted rectangle in Eig. 30-2. +ð= A/n= dsin60 \ +L_——+zz.Ì ' +T2 3 s n +Fig. 30-3. A linear array of n equal oscillators, driven with phases œ; +Now let us consider further how we may apply Eq. (30.3) in diferent cireum- +stances, and try to understand what ¡is happening. Let us consider our sources +to be all on a line, as drawn in Fig. 30-3. There are ø of them, all spaced by a +distance đ, and we shall suppose that the intrinsic relative phase, one to the next, +is œ. Then if we are observing in a given direction Ø from the normal, there is an +additional phase 2xđsin Ø/À because of the time delay between each successive +two, which we talked about before. Thus += œ+ 2rdsin Ø/À +? : / (30.4) += œ+ kdsin 8. +First, we shall take the case œ = 0. 'That ïs, all oscillators are in phase, and we +want to know what the intensity is as a function of the angle Ø. In order to ñnd +out, we merely have to put @ = kdsin Ø into formula (30.3) and see what happens. +In the first place, there is a maximum when ở = 0. 'PThat means that when all +the oscilators are in phase there is a strong intensity in the direction Ø = 0. Ôn +the other hand, an interesting question is, where is the first minimum? “Phat +occurs when @ = 27/n. In other words, when 2zdsinØ/A = 2m/n, we get the +--- Trang 516 --- +ñrst minimum of the curve. lf we get rid of the 27ˆs so we can look at it a little +better, it says that +ndsin 8 = À. (30.5) +Now let us understand physically why we get a minimum at that position. nd +is the total length Ù of the array. Referring to Eig. 30-3, we see that nđsin Ø = +LsinØ = A. What (30.5) says is that when A is equal to one tuauelength, we +get a minimum. Now why do we get a minimum when A = À? Because the +contributions of the various oscillators are then uniformly distributed in phase +from 0° to 360°. The arrows (Fig. 30-1) are going around a whole circle—we are +adding equal vectors in all directions, and such a sum is zero. So when we have +an angle such that A = À, we get a minimum. That is the first minimum. +There is another important feature about formula (30.3), which is that if +the angle ø is increased by any multiple of 2z, it makes no diference to the +formula. So we will get other strong maxima at ở = 27, 4m, 6z, and so forth. +Near cach of these great maxima the pattern of Fig. 30-2 is repeated. We may +ask ourselves, what is the geometrical circumstance that leads to these other +great maxima? 'Phe condition is that ô = 2m, where m is any integer. That is, +2zdsin 8/À = 2m. Dividing by 27, we see that +đsỉin 8 = mÀ. (30.6) +Thịis looks like the other formula, (30.5). No, that formula was nđsin Ø = À. The +diferenee is that here we have to look at the 7md?uidual sources, and when we say +đsin Ø = mÀ, that means that we have an angle Ø such that ổ = ?mÀ. In other +words, each source is now contributing a certain amount, and successive ones +are out of phase by a whole multiple of 360”, and therefore are contributing 7n +phase, because out of phase by 360” is the same as being in phase. So they all +contribute in phase and produce just as good a maximum as the one for rn = Ö +that we discussed before. 'Phe subsidiary bumps, the whole shape of the pattern, +1s jus$ like the one near ¿ = 0, with exactly the same minima on each side, etc. +Thus such an array will send beams in various directions—each beam having a +strong central maximum and a certain number of weak “side lobes.” 'The various +strong beams are referred to as the zero-order beam, the first-order beam, etc., +according to the value of ?m. ?n is called the order of the beam. +W© call attention to the fact that if đ is less than À, Eq. (30.6) can have +no solution except rm = 0, so that If the spacing 1s too small there is only one +possible beam, the zero-order one centered at Ø = 0. (Of course, there is also +--- Trang 517 --- +a beam in the opposite direction.) In order to get subsidiary great maxima, we +mmust have the spacing đ oŸ the array greater than one wavelength. +30-2 The difraction grating +In technical work with antennas and wires it is possible to arrange that all the +phases of the little oscillators, or antennas, are equal. The question is whether +and how we can do a similar thing with light. We cannot at the present time +literally make little optical-frequency radio stations and hook them up with +Infnitesimal wires and drive them all with a given phase. But there is a very +easy way to do what amounts to the same thing. +Suppose that we had a lot of parallel wires, equally spaced at a spacing đ, and +a radiofrequency source very far away, practically at infñnity, which is generating +am electric fñeld which arrives at each one of the wires at the same phase (it is +so far away that the từìme delay is the same for all oŸ the wires). (One can work +out cases with curved arrays, but let us take a plane one.) Then the external +electric fñeld will drive the electrons up and down in each wire. 'That is, the +fñeld which is coming from the original source will shake the electrons up and +down, and in moving, these represent + genera‡ors. Thịs phenomenon is called +scattering: a light wave from some source can induce a motion of the electrons +in a piece of material, and these motions generate their own waves. Thherefore all +that is necessary is to set up a lot of wires, equally spaced, drive them with a +radiofrequency source far away, and we have the situation that we want, without +a whole lot of special wiring. Tf the incidence is normal, the phases will be equal, +and we will get exactly the cireumstance we have been discussing. Therefore, If +the wire spacing is greater than the wavelength, we will get a strong intensity of +scattering in the normal direction, and in certain other directions given by (30.6). +This can dalso be done tuíth líghH Instead oŸ wires, we use a flat piece of glass +and make notches in it such that each of the notches scatters a little diferently +than the rest of the glass. If we then shine light on the glass, each one of the +notches will represent a source, and if we space the lines very fnely, but not +closer than a wavelength (which is technically almost impossible anyway), then +we would expect a miraculous phenomenon: the light not only will pass straight +through, but there will also be a strong beam at a finite angle, depending on +the spacing of the notchesl Such objects have actually been made and are in +common use—they are called đjfƒraction gratings. +--- Trang 518 --- +In one of its forms, a difraction grating consists of nothing but a plane glass +sheet, transparent and colorless, with scratches on it. There are often several +hundred scratches to the millimeter, 0ery carefully arranged so as to be equally +spaced. 'Phe efect of such a grating can be seen by arranging a projectOr so as +6o throw a narrow, vertical line of light (the image of a slit) onto a screen. When +we put the grating into the beam, with its scratches vertical, we see that the +line is still there but, in addition, on each side we have ønother strong patch +of light which is colored. “This, of course, is the slit Image spread out over a +wide angular range, because the angle Ø in (30.6) depends upon À, and lights of +diferent colors, as we know, correspond to diferent frequencies, and therefore +diferent wavelengths. 'The longest visible wavelength is red, and since đsin Ø = À, +that requires a larger 0. And we do, in fact, fnd that red is at a greater angle +out from the central imagel 'There should also be a beam on the other side, and +indeed we see one on the sereen. Then, there might be another solution of (30.6) +when ?m = 2. We do see that there is something vaguely there—very weak——and +there are even other beams beyond. +We have just argued that all these beams ought to be of the same strength, +but we see that they actually are not and, in fact, not even the first ones on the +right and left are equall “The reason is that the grating has been carefully built to +do just this. How? If the grating consists of very fine notches, inÑnitesimally wide, +spaced evenly, then all the intensities would indeed be equal. But, as a matter +of fact, although we have taken the simplest case, we could also have considered +an array of øœ#rs of antennas, in which each member of the pair has a certain +strength and some relative phase. In this case, it is possible to get intensities +which are diferent in the diferent orders. A grating is often made with little +“sawtooth” cuts instead of little symmetrical notches. By carefully arranging the, +“sawteeth,” more light may be sent into one particular order of spectrum than +Into the others. In a practical grating, we would like to have as mụch light as +possible in one of the orders. This may seem a complicated point to bring in, +but it is a very clever thing to do, because it makes the grating more useful. +So far, we have taken the case where all the phases of the sources are equal. +But we also have a formula for ô when the phases difer from one to the nex$ +by an angle œ. 'Phat requires wiring up our antennas with a slight phase shift +between each one. Can we do that with light? Yes, we can do it very easily, +for suppose that there were a source of light at infnity, œ an angle such that +the light is coming in at an angle địn, and let us say that we wish to discuss the +scattered beam, which is leaving at an angle Øẹu¿ (Fig. 30-4). The Ø¿u¿ is the +--- Trang 519 --- +đsin sụt đsin Ôn +Fig. 30-4. The path difference for rays scattered from adjacent rulings +of a grating Is đsin đụ: — đsin Ôịn. +same Ø as we have had before, but the Ø¡ạ is merely a means for arranging that +the phase of each source is diferent: the light coming from the distant driving +source first hits one scratch, then the next, then the next, and so on, with a phase +shift rom one to the other, which, as we see, is œ = —2dsinØ¡„/À. Therefore +we have the formula for a grating in which light both comes in and goes out at +an angle: +@ = 2#dsin Øsu¿/À — 2xdsin Øịn /À. (30.7) +Let us try to fñnd out where we get strong intensity in these cireumstances. The +condition for strong intensities 1s, of course, that ø should be a multiple of 2z. +'There are several interesting points to be noted. +One case of rather great interest ¡is that which corresponds to m = 0, where +đ is less than À; in fact, this is the only solution. In this case we see that +sin Øsụy = sinØ¡n, which means that the light comes out in the sœmne điccfion as +the light which was exciting the grating. We might think that the light “goes +right through.” No, it 1s đjƒeren# light that we are talking about. The light that +goes right through is from the original source; what we are talking about is the +new light uhách ¡s generated bụ scaltering. It turns out that the scattered light +1s going in the same direction as the original light, in fact it can interfere with +it—a feature which we will study later. +There is another solution for this same case. For a given Øịn, Øou„¿ may be +the supplemen# oŸ Ø¡n. 5o not only do we get a beam in the same direction as +the incoming beam but also one in another direction, which, if we consider 1E +carefully, is such that the angle oƒ ïncidence ¡s equal to the angle oƒ scattering. +'This we call the refected beam. +--- Trang 520 --- +So we begin to understand the basic machinery of reflection: the light that +comes in generates motions of the atoms in the refector, and the refector then +regeneraftes a. n„eu œue, and one of the solutions for the direction of scattering, +the on solution if the spacing of the scatterers is small compared with one +wavelength, is that the angle at which the light comes out is equal to the angle +at which it comes inl +Next, we discuss the special case when đ —> 0. 'PThat is, we have just a solid +plece of material, so to speak, but of Ññnite length. In addition, we want the phase +shift from one scatterer to the next to go to zero. In other words, we put more +and more antennas between the other ones, so that each of the phase differences +1s getting smaller, but the number of antennas is increasing in such a way that +the total phase diference, between one end of the line and the other, is constant. +Let us see what happens to (30.3) iƒ we keep the diference in phase nộ from one +end to the other constant (say nó = ®), letting the number go to infnity and +the phase shift ý of each one go to zero. But now ở is so small that sin ộ = ở, +and if we also recognize n2fo as T„, the maximum intensity at the center of the +beam, we find +T= 4l2sin? ›®/8Ẻ. (30.8) +This limiting case is what is shown in EFig. 30-2. +In such cireumstances we fnd the same general kind of a picture as for ñnite +spacing with đ > À; all the side lobes are practically the same as before, but +there are no higher-order maxima. lf the scatterers are all in phase, we get a +mmaximum in the direction Øs„y = 0, and a minimum when the distance A is equal +to À, just as for finite đ and nø. 5o we can even analyze a con#nuous distribution +Of scatterers or oscillators, by using integrals instead of summing. +L † —— +! =E=—ˆ——— †}——” +Fig. 30-5. The intensity pattern of a continuous line of oscillators has +a single strong maxImum and many weak “side lobes.” +--- Trang 521 --- +As an example, suppose there were a long line of oscillators, with the charge +oscillating along the direction of the line (Eig. 30-5). Erom such an array the +greatest intensity is perpendicular to the line. 'There is a little bit of intensity up +and down from the equatorial plane, but it is very slipht. With this result, we can +handle a more complicated situation. Suppose we have a set of such lines, each +producing a beam only in a plane perpendicular to the line. To find the intensity +in various directions from a series of long wires, Instead of infinitesimal wires, +1s the same problem as it was for infñnitesimal wires, so long as we are in the +central plane perpendicular to the wires; we just add the contribution from each +of the long wires. hat is why, although we actually analyzed only tiny antennas, +we might as well have used a grating with long, narrow slots. Each of the long +slots produces an efect only in its own direction, not up and down, but they are +all set next to each other horizontally, so they produce interference that way. +Thus we can build up more complicated situations by having various distri- +butions of scatterers in lines, planes, or in space. The first thing we did was to +consider scatterers in a line, and we have just extended the analysis to strips; we +can work it out by just doïng the necessary summations, adding the contributions +from the individual scatterers. The principle is always the same. +30-3 Resolving power of a grating +W© are now in a position to understand a number of interesting phenomena. +For example, consider the use oŸ a grating for separating wavelengths. We noticed +that the whole spectrum was spread out on the screen, so a grating can be used +as an instrument for separating light into its diferent wavelengths. One of the +interesting questions is: supposing that there were t©wo sources of slightly diferent +frequency, or slightly diferent wavelength, how close together in wavelength could +they be such that the grating would be unable to tell that there were really +two diferent wavelengths there? 'The red and the blue were clearly separated. +But when one wave ¡is red and the other is slightly redder, very close, how close +can they be? Thịis is called the resolưing pouer of the grating, and one way of +analyzing the problem is as follows. Suppose that for light of a certain color +we happen to have the maximum of the difracted beam occurring at a certain +angle. TÝ we vary the wavelength the phase 2zđdsin Ø/À is different, so oÝ course +the maximum occurs at a different angle. 'Phat is why the red and blue are +spread out. How different in angle must i% be in order for us to be able to see +it? Tƒ the two maxima are exactly on top of each other, of course we cannot see +--- Trang 522 --- +them. Tf the maximum oŸ one is far enough away from the other, then we can see +that there ¡is a double bump in the distribution of light. In order to be able to +Jjust make out the double bump, the following simple criterion, called iayleigh s +criterion, is usually used (Eig. 30-6). It is that the first minimum from one bump +should sit at the maximum of the other. Now it is very easy to calculate, when +one minimum sits on the other maximum, how much the diference in wavelength +is. The best way to do it is geometrically. +"~. —— h ` —— - +Fig. 30-6. lllustration of the Rayleigh criterion. The maximum of one +pattern falls on the first minimum of the other. +In order to have a maximum for wavelength ÀJ, the distance A (Eig. 30-3) +must be A7, and if we are looking at the znth-order beam, it is nA/. In other +words, 2zdsinØ/ÀX' = 2m, so ndsin 9, which is A, is rmwÀ! tìmes m, or wwÀ/. For +the other beam, of wavelength À, we want to have a mứnữmum at thĩs angle. +That is, we want A to be exactly one wavelength À more than mnA. That is, +A = mnÀ + À = mmnÀ'. Thus iŸ Ä' = À + A^A, we find +AA/A= 1/mn. (30.9) +The ratio À/AA is called the resolung pouer oŸ a grating; we see that it is equal +to the total number of lines in the grating, times the order. lt is not hard to +prove that this formula is equivalent to the formula that the error in Ífrequency 1s +cqual to the reciprocal time diference between extreme paths that are allowed +to interfere: +Aw = 1/T. +In fact, that is the best way to remember i%, because the general formula works +not only for gratings, but for any other instrument whatsoever, while the special +formula (30.9) depends on the fact that we are using a grating. +* In our case 7'= A/c = mnÀ/c, where c is the speed of light. The frequency = c/À, so +Au=eAA/A2. +--- Trang 523 --- +30-4 The parabolic antenna +Now let us consider another problem in resolving power. 'Phis has to do +with the antenna of a radio telescope, used for determining the position of radio +sources in the sky, i.e., how large they are in angle. Of course if we use any old +antenna and fnd signals, we would not know tom what direction they came. VWWe +are very interested to know whether the source is in one place or another. Ône +way we can find out is to lay out a whole series of equally spaced dipole wires on +the Australian landscape. Then we take all the wires from these antennas and +feed them into the same receiver, in such a way that all the delays in the feed +lines are equal. Thus the receiver receives signals from all of the dipoles in phase. +That is, it adds all the waves from every one of the dipoles in the same phase. +Now what happens? If the source is directly above the array, at inÑnity or nearly +so, then its radiowaves will excite all the antennas in the same phase, so they all +feed the receiver together. +Now suppose that the radio source is at a slight angle Ø from the vertical. +Then the various antennas are receiving signals a little out of phase. The receiver +adds all these out-of-phase signals together, and so we get nothing, if the angle Ø +is too big. How bịg may the angle be? Ansuer: we get zero If the angle A/Ù = 9 +(Fig. 30-3) corresponds to a 360° phase shift, that is, if A is the wavelength À. +'This is because the vector contributions form together a complete polygon with +zero resultant. The smallest angle that can be resolved by an antenna array of +length Ù is Ø = À/L. Notice that the receiving pattern oŸ an antenna such as +this is exactly the same as the intensity distribution we would get if we turned +the receiver around and made it into a transmitter. 'This is an example of +what is called a reciprocitU principle. It turns out, in fact, to be generally true +for any arrangement of antennas, angles, and so on, that if we frst work out +what the relative intensities would be in various directions if the receiver were a +transmitter instead, then the relative directional sensitivity of a receiver with +the same external wiring, the same array of antennas, is the same as the relative +intensity of emission would be if it were a transmitter. +Some radio antennas are made in a diferent way. Instead of having a whole +lot of dipoles in a long line, with a lot of feed wires, we may arrange them not in +a line but in a curve, and put the receiver at a certain point where it can detect +the scattered waves. This curve is cleverly designed so that if the radiowaves +are coming down from above, and the wires scatter, making a new wave, the +wires are so arranged that the scattered waves reach the receiver all at the same +--- Trang 524 --- +time (Eig. 26-12). In other words, the curve is a øaraboïa, and when the source +is exactly on is axis, we get a very strong intensity at the focus. In this case we +understand very clearly what the resolving power of such an instrument is. The +arranging of the antennas on a parabolie curve is not an essential point. It is only +a convenient way to get all the signals to the same point with no relative delay +and without feed wires. The angle such an instrument can resolve is still Ø = À/1, +where Ù is the separation of the first and last antennas. I$ does not depend on +the spacing of the antennas and they may be very close together or in fact be +all one piece of metal. NÑow we are describing a telescope mirror, of course. We +have found the resolving power of a telescopel (Sometimes the resolving power is +written Ø = 1.22À/L, where Ƒ is the diameter of the telescope. The reason that it +is not exactly ÀA/ is this: when we worked out that Ø = À/Ù, we assumed that all +the lines of dipoles were equal in strength, but when we have a circular telescope, +which is the way we usually arrange a telescope, not as much signal comes from the +outside edges, because it is not like a square, where we get the same intensity all +along a side. We get somewhat less because we are using only part of the telescope +there; thus we can appreciate that the efective diameter ¡s a little shorter than +the true diameter, and that is what the 1.22 factor tells us. In any case, it seems +a little pedantic to put such precision into the resolving power formula.*) +30-5 Colored fÌms; crystals +The above, then, are some of the efects of interference obtained by adding +the various waves. But there are a number of other examples, and even though +we do not understand the fundamental mechanism yet, we will some day, and +we can understand even now how the interference occurs. For example, when +a light wave hits a surface of a material with an index ø, let us say at normal +incidence, some of the light is refected. The reasơn for the reflection we are not +in a position to understand right now; we shall discuss it later. But suppose we +know that some of the light is refected both on entering and leaving a refracting +medium. 'Phen, if we look at the refection of a light source in a thin film, we +see the sum of two waves; If the thicknesses are small enoupgh, these two waves +* 'This is because Rayleigh”s criterion is a rough idea in the frst place. It tells you where it +begins to get very hard to tell whether the image was made by one or by two stars. Actually, if +sufficiently careful measurements of the exact intensity distribution over the difracted image +spot can be made, the fact that two sources make the spot can be proved even ïif Ø is less +than À/L. +--- Trang 525 --- +will produce an interference, either constructive or destructive, depending on +the signs of the phases. It might be, for instance, that for red light, we get an +enhanced reflection, but for blue light, which has a diÑerent wavelength, perhaps +we get a destructively interfering reflection, so that we see a bright red reflection. +l we change the thickness, i.e., if we look at another place where the film 1s +thicker, it may be reversed, the red interfering and the blue not, so it is bright +blue, or green, or yellow, or whatnot. So we see colors when we look at thin flms +and the colors change if we look at diferent angles, because we can appreciate +that the timings are diferent at diferent angles. Thus we suddenly appreciate +another hundred thousand situations involving the colors that we see on oil ñlms, +soap bubbles, etc. at diferent angles. But the principle is all the same: we are +only adding waves at diferent phases. +As another important application of difraction, we may mention the following. +W© used a grating and we saw the difracted image on the sereen. If we had used +mmonochromatic light, it would have been at a certain specifc place. Then there +were various higher-order images also. Erom the positions of the images, we could +tell how far apart the lines on the grating were, if we knew the wavelength of +the light. Erom the difference in intensity of the various images, we could ñnd +out the shape of the grating scratches, whether the grating was made of wires, +sawtooth notches, or whatever, t#thout being ablÌe to see them. This principle 1s +used to discover the positions of the ø‡oms ?n a crustal. 'Phe only complication +1s that a crystal is three-dimensional; it is a repeating three-dimensional array +of atoms. We cannot use ordinary light, because we must use something whose +wavelength is less than the space between the atoms or we get no effect; so we +must use radiation of very short wavelength, i.e., x-rays. 5o, by shining x-rays +into a crystal and by noticing how intense is the refection in the various orders, +we can determine the arrangement of the atoms inside without ever being able +to see them with the eyel It is in this way that we know the arrangement of the +atoms in various substances, which permitted us to draw those pictures In the +ñrst chapter, showing the arrangement of atoms in salt, and so on. We shaÏll later +come back to this subject and discuss 1t in more detail, and therefore we say no +more about this most remarkable idea at present. +30-6 Diffraction by opaque screens +Now we come to a very interesting situation. Suppose that we have an opaque +sheet with holes in it, and a light on one side of it. We want to know what the +--- Trang 526 --- +Intensity Is on the other side. What most people say is that the light shines +through the holes, and produces an efect on the other side. It will turn out that +one gets the right answer, to an excellent approximation, if he assumes that there +are sources distributed with uniform density across the open holes, and that +the phases of these sources are the same as they would have been ïf the opaque +material were absent. Of course, actually there are øoø sources at the holes, In +fact that is the only place that there are certaznl no sources. Nevertheless, we +get the correct difraction patterns by considering the holes to be the only places +that there are sources; that 1s a rather peculiar fact. We shall explain later why +this is true, but for now let us just suppose that it 1s. +In the theory of difraction there is another kind of difraction that we should +briefly discuss. It is usually not discussed in an elementary course as early as this, +only because the mathematical formulas involved in adding these little vectors +are a little elaborate. Otherwise i% is exactly the same as we have been doïng all +along. AlI the interference phenomena are the same; there is nothing very much +more advanced involved, only the cireumstances are more complicated and it is +harder to add the vectors together, that is all. +Suppose that we have light coming in from infnity, casting a shadow of an +object. Pigure 30-7 shows a screen on which the shadow of an object 4? is made +by a light source very far away compared with one wavelength. NÑow we would +expect that outside the shadow, the intensity is all bright, and inside 1t, it 1s +all dark. As a matter of fact, if we plot the intensity as a function of position +— > E +—> 'h s +————*® +— > A +Opaque Screen +Object +Fig. 30-7. A distant light source casts a shadow of an opaque obJect +on a screen. +--- Trang 527 --- +near the shadow edge, the intensity rises and then overshoots, and wobbles, and +oscillates about in a very peculiar manner near this edge (Eig. 30-9). We now +shall discuss the reason for this. If we use the theorem that we have not yet +proved, then we can replace the actual problem by a set of efective sources +uniformly distributed over the open space beyond the object. +We imagine a large number of very closely spaced antennas, and we wan$ +the intensity at some point P. That looks just like what we have been doïng. +Not quite; because our screen is not at infnity. We do not want the intensity +at infnity, but at a fñnite point. To calculate the intensity at some particular +place, we have to add the contributions from all the antennas. Eirst there is an +antenna at D, exactly opposite ; ïf we go up a little bít in angle, let us say a +height h, then there is an increase in delay (there is also a change in amplitude +because of the change in distance, but this is a very small efect if we are at all +far away, and is much less important than the diference in the phases). NÑow the +path diference PP — DP is approximately h2/2s, so that the phase diferenee is +proportional to the sợuare of how far we go trom , while in our previous work +øs was infinite, and the phase difference was zneariu proportional to h. When +the phases are linearly proportional, each vector adds at a constant angle to the +next vector. What we now need is a curve which is made by adding a lot of +inÑnitesimal vectors with the requirement that the angle they make shall increase, +not linearly, but as the sợuøre of the length of the curve. To construect that +curve involves slightly advanced mathematics, but we can always construct it by +actually drawing the arrows and measuring the angles. In any case, we get the +marvelous curve (called Cornu's spiral) shown in Eig. 30-8. Now how do we use +this curve? +Tf we want the intensity, let us say, at point , we add a lot of contributions +of diferent phases from point Ï2 on up to infnity, and from D down only to +point Ởp. So we start at p ¡n Fig. 30-8, and draw a series Of arrows OÝ ©Ver- +increasing angle. 'Pherefore the total contribution above point p all goes along +the spiraling curve. If we were to stop integrating at some place, then the total +amplitude would be a vector from 7? to that point; in this particular problem we +are going to infñnity, so the total answer is the vector p¿. Now the position +on the curve which corresponds to point p on the object depends upon where +point ? ¡s located, since point D, the inflection point, always corresponds to +the position of point . 'Thus, depending upon where ? is located above , the +beginning point will fall at various positions on the lower left part of the curve, +and the resultant vector p.., will have many maxima and minima (Fig. 30-9). +--- Trang 528 --- +S2 +Fig. 30-8. The addition of amplitudes for many In-phase oscillators +whose phase delays vary as the square of the distance from point D of +the previous figure. +1.0 R +0.25F----=--—~z +Fig. 30-9. The Iintensity near the edge of a shadow. The geometrical +shadow edge Is at xo. +--- Trang 529 --- +Ôn the other hand, if we are at Q, on the other side of , then we are using +only one end of the spiral curve, and not the other end. In other words, we do not +even start at JD, but at Hạ, so on this side we get an intensity which continuously +falls of as Q goes farther into the shadow. +One point that we can immediately calculate with ease, to show that we really +understand it, is the intensity exactly opposite the edge. 'Phe intensity here 1s +1/4 that of the incident light. Reason: Pxactly at the edge (so the endpoint ÖØ of +the arrow is at D in Fig. 30-8) we have half the curve that we would have had iŸ +we were far into the bright region. If our point †## is far into the light we go from +one end of the curve to the other, that is, one full unit vector; but if we are at +the edge of the shadow, we have only half the amplitude——1/4 the intensity. +In this chapter we have been finding the intensity produced in various direc- +tỉons from various distributions of sources. As a fñnal example we shall derive +a formula which we shall need for the next chapter on the theory of the index +of refraction. p to this point relative intensities have been sufficient for our +purpose, but this time we shall fnd the complete formula for the field in the +following situation. +30-7 The field of a plane of oscillating charges +Suppose that we have a plane full of sources, all oscillating together, with +their motion in the plane and all having the same amplitude and phase. What is +the fñeld at a finite, but very large, distance away from the plane? (We cannot +get very close, of course, because we do not have the right formulas for the field +close to the sources.) IÝ we let the plane of the charges be the zz-plane, then +we want to fñnd the field at the point ? far out on the z-axis (Eig. 30-10). We +Oscillating charge +€' : +Sheet of oscillating charges +Fig. 30-10. Radiation field of a sheet of oscillating charges. +--- Trang 530 --- +suppose that there are ? charges per unit area of the plane, and that each one of +them has a charge g. All of the charges move with simple harmonic motion, with +the same direction, amplitude, and phase. We let the motion of each charge, ¿th +respect to is on querage postfion, be #øọ cos (‡. Ôr, using the complex notation +and remembering that the real part represents the actual motion, the motion +can be described by #zoe“t, +Now we fnd the fñield at the point from all of the charges by fñnding the +ñeld there rom each charge g, and then adding the contributions from all the +charges. We know that the radiation field is proportional to the acceleration of +the charge, which is —œ2zoe”“ (and is the same for every charge). The electric +fñeld that we want at the point due to a charge at the point @ is proportional +to the acceleration of the charge g, but we have to remember that the feld at the +point ?P at the instant £ is given by the acceleration of the charge at the earlier +time f“ = £— r/c, where r/e is the time i% takes the waves to travel the distance r +from @ to P. Therefore the field at is proportional to +— 02zgefe—r/6), (30.10) +Using this value for the acceleration as seen from in our formula for the electric +fñeld at large distances from a radiating charge, we get +Electric feld at ? q 2zge⁄2ữ—r/e) +lim charge at Q ) ¬. . Ặ—— (30.11) +Now this formula is not quite right, because we should have used øø‡ the +acceleration of the charge but ?s cormmponen‡ perpendicular to the line Q?P. We +shall suppose, however, that the point ? is so far away, compared with the +distance of the point Q from the axis (the distance ø in Eig. 30-10), for those +changes that we need to take into account, that we can leave out the cosine facbor +(which would be nearly equal to 1 anyway). +To get the total fñeld at , we now add the efects of all the charges in the +plane. We should, of course, make a øecfor sum. But since the direction of the +electric feld is nearly the same for all the charges, we may, in keeping with the +approximation we have already made, just add the magnitudes of the fñelds. 'lo +our approximation the field at depends only on the distance z, so all charges +at the same z produce equal fñelds. So we add, frst, the felds of those charges In +a ring of width đø and radius ø. hen, by taking the integral over all ø, we will +obtain the total fñeld. +--- Trang 531 --- +The number of charges in the ring is the product of the surface area of the +ring, 2ø đo, and ?, the number of charges per unit area. We have, then, +2 iœ(t—r/c) +Total ñeld at = J _—1 “I0 n.2mpdp, (30.12) +47coc2 T +We wish to evaluate this integral from ø = 0 to ø = œ. The variable ứ, of +course, is to be held fñxed while we do the integral, so the only varying quantities +are ø and r. Leaving out all the constant factors, ineluding the ƒactor e”*°t, for +the moment, the integral we wish is +0=œo eiaur/o +J ——— jpdịp. (30.13) +To do this integral we need to use the relation bebween r and ø: +r?ˆ= p?+ 2. (30.14) +Since z is independent of ø, when we take the diferential of this equation, we get +2r dr = 2p dp, +which is lucky, since in our integral we can replace øđø by r dr and the z will +cancel the one in the denominator. 'Phe integral we want is then the simpler one +?=CC - +J e~⁄/$ dự, (30.15) +To integrate an exponential is very easy. We divide by the coeflicient oŸ r in the +exponent and evaluate the exponential at the limits. But the limits of z are not +the same as the limits of . When ø = 0, we have r = z, so the limits oŸ z are z +to infinity. We get for the integral +— C —i¡œ __ „—(iœ/c)z 30.16 +¬"m.“nnh (30.16) +where we have written oo for (œ/c)oo, since they both just mean a very large +numberl - +NÑow e"??° is a mysterious quantity. Its real part, for example, is cos (—o©), +which, mathematically speaking, is completely indefnite (although we would +--- Trang 532 --- +expect i% to be somewhere—or everywhere (?)—between +1 and —1l). But in a +phụs?cal situation, 1t can mean something quite reasonable, and usually can Jjust +be taken to be zero. 'To see that this is so In our case, we go back to consider +again the original integral (30.15). +W©e can understand (30.15) as a sum of many small complex numbers, each +of magnitude Az, and with the angle Ø = —œr/c in the complex plane. We can +try to evaluate the sum by a graphical method. In Eig. 30-11 we have drawn the +first five pieces of the sum. Each segment of the curve has the length Az and is +placed at the angle AØ = —ưw Ar/c with respect to the preceding piece. The sum +for these first five pieces is represented by the arrow from the starting point to +the end of the fifth segment. As we continue to add pieces we shall trace out a +polygon until we get back to the starting point (approximately) and then start +around once more. Adding more pieces, we just go round and round, staying +close to a circle whose radius is easily shown to be c/œ. We can see now why the +integral does not give a definite answerl +ạ= —, lmaginary Axis +A0 = — âr +¬—- Real Axis +1" “¿0 +lã \-Aø +cụ Sum >¬A0 +Fig. 30-11. Graphical solution of J" e—ðr/€ qr, +But now we have to go back to the øñh#s¿cs of the situation. In any real +situation the plane of charges cœnno‡ be infnite in extent, but must sometime +stop. lfit stopped suddenly, and was exactly circular in shape, our integral would +have some value on the cirele in Fig. 30-11. If, however, we let the number of +charges in the plane gradually taper off at some large distance from the center +(or else stop suddenly but in an irregular shape so for larger ø the entire ring +--- Trang 533 --- +lmaginary Axis +_g—®>‹j¿Start;r =z_ Real Axis +Fig. 30-12. Graphical solution of J" re #/£ dự, +of width đø no longer contributes), then the coefficient r in the exact integral +would decrease toward zero. Since we are adding smaller pieces but still turning +through the same angle, the graph of our integral would then become a curve +which is a spiral. The spiral would eventually end up at the center of our original +circle, as drawn in Eig. 30-12. 'Phe ph¿#/s/call correct integral is the complex +number 4 in the figure represented by the interval from the starting point to the +center of the circle, which is just equal to +¬... (30.17) +as you can work out for yourself. This is the same result we would get from +E4q. (30.16) if we set e”?% = 0. +(There is also another reason why the contribution to the integral tapers of +for large values of z, and that is the factor we have omitted for the projection of +the acceleration on the plane perpendicular to the line P@Q.) +W© are, of course, interested only in physical situations, so we will take e—”% +cqual to zero. Returning to our original formula (30.12) for the ñeld and putting +back all of the factors that go with the integral, we have the result +Total fñeld at P= —- T zoe«ứ=#/2) (30.18) +(remembering that 1/2 = —)). +It is interesting to note that (2#oe”““) is just equal to the œelociy of the +charges, so that we can also write the equation for the fñeld as +Total fñeld at P = _¬ [veloeity of charges]a ¿ _ ;/e, (30.19) +--- Trang 534 --- +which is a little strange, because the retardation is just by the distance z, which +is the shortest distance from ? to the plane of charges. But that is the way +1t comes out—fortunately a rather simple formula. (We may add, by the way, +that although our derivation is valid only for distances far from the plane of +oscillatory charges, it turns out that the formula (30.18) or (30.19) is correct at +any distance z, even for z < À.) +--- Trang 535 --- +Tho €)riqgirt of tho lHofretcfftco InăiÏox +31-1 The index of refraction +WS have said before that light goes slower in water than in air, and slower, +slightly, in air than in vacuum. 'This effect is described by the index of refraction 0ø. +Now we would like to understand how such a slower velocity could come about. In +particular, we should try to see what the relation is to some physical assumptions, +or statements, we made earlier, which were the following: +(a) That the total electric field in any physical circumstance can always be +represented by the sum of the fñelds rom all the charges in the universe. +(b) That the fñeld from a single charge is given by its acceleration evaluated +with a retardation at the speed œ, aøa¿/s (for the radiafion feld). +But, for a piece of glass, you might think: “Oh, no, you should modify all +this. You should say it is retarded at the speed c/w” That, however, is not right, +and we have to understand why it is not. +lt 7s approximately true that light or any electrical wave đoes øppear to travel +at the speed c/n through a material whose index of refraction is nø, but the +fñelds are still produced by the motions oŸ øÏ/ the charges——including the charges +moving in the material—and with these basic contributions of the fñeld travelling +at the ultimate velocity c. Our problem ¡is to understand how the apparenth +slower velocity comes about. +We shall try to understand the efect in a very simple case. A source which we +shall call “the ez#ernal source” is placed a large distance away om a thin plate +of transparent material, say glass. We inquire about the fñeld at a large distance +on the opposite side of the plate. The situation is illustrated by the diagram +of FEig. 31-1, where Š and ? are imagined to be very far away from the plate. +According to the principles we have stated earlier, an electric ñeld anywhere +--- Trang 536 --- +Arriving wave ; “Transmitted” wave +s8) =/ +ì É What is me +SE Han th +“Reflected” +Wave / Glass plate +Fig. 31-1. Electric waves passing through a layer of transparent +material. +that is far from all moving charges is the (vector) sum of the felds produced by +the external source (at ,S) ønd the fields produced by cách of the charges in the +plate of glass, cuer one tuith is proper retardation at the 0elocitu c. Remember +that the contribution of each charge is not changed by the presence of the other +charges. These are our basic principles. The fñield at can be written thus: +Eb—= » J2cách charge (31.1) +all charges +b—= +1. + » đ2cách charge› (31.2) +all other charges +where #2, ¡is the feld due to the source alone and would be precisely the fñeld +at ÐP 7 there tuere no rmmatertal present. We expect the field at P to be diferent +trom #⁄, ïf there are any other moving charges. +'Why should there be charges moving in the glass? We know that all material +consists of atoms which contain electrons. When the electric fñeld oøƒ the source +acts on these atoms it drives the electrons up and down, because I1 exerts a +force on the electrons. And moving electrons generate a field—they constitute +new radiators. These new radiators are related to the source Š, because they +are driven by the fñeld of the source. "The total fñeld is not just the feld of the +source Š, but it is modifñed by the additional contribution from the other moving +charges. This means that the fñeld is not the same as the one which was there +before the glass was there, but is modified, and it turns out that it is modified +in such a way that the field inside the glass appears to be moving at a diferent +speed. 'Phat is the idea which we would like to work out quantitatively. +--- Trang 537 --- +Now this is, in the exact case, pretty complicated, because although we have +said that all the other moving charges are driven by the source field, that is not +quite true. If we think of a particular charge, it feels not only the source, but like +anything else in the world, it feels øi/ of the charges that are moving. I§ feels, +in particular, the charges that are moving somewhere else in the glass. So the +total feld which is acting on a particular chorge is a combination of the fields +from the other charges, +0„ose rmmotions depend on tuhat this particular charge 1s +đo”ng! You can see that it would take a complicated set of equations to get the +complete and exact formula. Ït is so complicated that we postpone this problem +until next year. +Instead we shall work out a very simple case in order to understand all the +physical principles very clearly. We take a cireumstance in which the efects from +the other atoms are very small relative to the efects from the source. In other +words, we take a material in which the total fñeld is not modifed very much +by the motion of the other charges. That corresponds to a material in which +the index of refraction is very close to 1, which will happen, for example, 1f the +density of the atoms 1s very low. Our calculation will be valid for any case In +which the index is for any reason very close to 1. In this way we shall avoid the +complications of the most general, complete solution. +Incidentally, you should notice that there is another efect caused by the +motion of the charges in the plate. These charges will also radiate waves back +toward the source Š. 'Phis backward-going fñeld is the light we see relected from +the surfaces of transparent materials. It does not come from just the surface. The +backward radiation comes from everywhere in the Interior, but it turns out that +the total efect is equivalent to a refection from the surfaces. These refection +efects are beyond our approximation at the moment because we shall be limited +to a calculation for a material with an index so close to 1 that very little light is +refected. +Before we proceed with our study of how the index of refraction comes about, +we should understand that all that is required to understand refraction is to +understand why the apparent wave 0elocitu is diferent in diferent materials. Thhe +bending of light rays comes about just 0ecause the efective speed of the waves is +difÑferent in the materials. To remind you how that comes about we have drawn +in Eig. 31-2 several successive crests of an electric wave which arrives from a +vacuum onto the surface of a block of glass. The arrow perpendicular to the +--- Trang 538 --- +⁄⁄ ⁄ / +VACUUM „⁄ˆ ,⁄ ˆ/⁄ GILASS +⁄ ⁄ „ ,⁄ +"4 ⁄ XS +crests `\⁄ ⁄ +Fig. 31-2. Relation between refraction and velocity change. +wave crests indicates the direction of travel of the wave. Now all oscillations in +the wave must have the same ƒreqguenec. (We have seen that driven oscillations +have the same frequency as the driving source.) This means, also, that the wave +crests for the waves on both sides of the surface must have the same spacïng +qlong the surƒace because they must travel together, so that a charge sitting +at the boundary will feel only one frequency. The shorfes‡ distance bebween +crests of the wave, however, ¡is the wavelength which is the velocity divided by +the requency. Ôn the vacuum side it is Ào = 2zc/œ, and on the other side it is +À = 270/u or 2#c/(œn, 1Ÿ 0 = cƒn is the velocity of the wave. From the fgure we +can see that the only way for the waves to “ft” properly at the boundary is for +the waves in the material to be travelling at a dilferent angle with respect to the +surface. From the geometry of the fñgure you can see that for a “ñt” we must +have Ào/sin Øo = À/sin 0, or sin Øo/sỉn Ø = ø=, which is Snell's law. We shall, for +the resi of our discussion, consider only why light has an efective speed oŸ c/n +in material of index m=, and no longer worry, in this chapter, about the bending +of the light direction. +W© go back now to the situation shown in Fig. 3Í-1. We see that what we +hawve to do 1s to calculate the fñeld produced at by all the oscillating charges +in the glass plate. We shall call this part of the feld 4, and ït is just the sum +written as the second term in Bq. (31.2). When we add it to the term #⁄¿, due to +the source, we will have the total feld at P. +--- Trang 539 --- +Thịs is probably the most complicated thing that we are going to do this +year, but i is complicated only in that there are many pieces that have to be put +together; each piece, however, is very simple. nlike other derivations where we +say, “Forget the derivation, just look at the answerl,” in this case we do not need +the answer so mụch as the derivation. In other words, the thing to understand +now is the physical machinery for the produection of the index. +To see where we are going, let us fñrst ñnd out what the “correction fñeld” „ +would have to be if the total fñeld at ? is going to look like radiation from the +source that is slowed down while passing through the thin plate. If the plate had +no effect on it, the feld of a wave travelling to the right (along the z-axis) would +12; —= Eo cosu(É — z/c) (31.3) +or, using the exponential notation, +E, = Eoele~z/©), (31.4) +Now what would happen If the wave travelled more slowly in going through +the plate? Let us call the thickness of the plate Az. If the plate were not there the +wave would travel the distance Az in the time Az/e. But ifit appears to travel +at the speed c/n then it should take the longer tìme œ Az/c or the add¿Hional +time A£ = (m — 1) Az/c. After that it would continue to travel at the speed e +again. We can take into account the extra delay in getting through the plate by +replacing £ in Eq. (31.4) by (# — Af) or by [£ — (m — 1) Az/c|. 5o the wave after +Insertion of the plate should be written +J2after plate —— EocfelF-Œ=1) Az/e—z/s * (31.5) +W© can also write this equation as +DÀNG plate — eTie(n=1) Az/e Enel2—=z/©), (31.6) +which says that the wave after the plate is obtained from the wave which could +exist without the plate, i.e., from #;, by multiplying by the factor e~?#Œ=1)Az/€, +Now we know that multiplying an oscillating funetion like e?“f by a factor c?# +Just says that we change the phase of the oscillation by the angle Ø, which is, of +course, what the extra delay in passing through the thickness Az has done. lt +has retarded the phase by the amount œ(nø — 1) Az/e (retarded, because of the +mỉnus sign in the exponent). +--- Trang 540 --- +W©e have said earlier that the plate should adđ a fñeld 4 to the original +ñeld #„ = Ese!2Œ~Z/*), bụt we have found instead that the efect of the plate is +to rmmuliiplụ the ñeld by a factor which shifts its phase. However, that is really all +right because we can get the same result by adding a suitable complex number. +lt is particularly easy to find the right number to add in the case that Az is +small, for you will remember that IÝ z is a small number then e” is nearly equal +to (1+z). We can write, therefore, +e~s(@=1)AZ/€ = 1 — ju(n — 1) AZ/e. (31.7) +Using this equality in Eq. (31.6), we have +ö(m„— 1A Ố +đ2atter plate — Epge20=z/2 — o§n ) : Ege-z/©) h (31.8) +"¬—— —“_——~Ö +The first term is just the fñeld om the source, and the second term must just be +cqual to !4, the fñeld produced to the right of the plate by the oscillating charges +of the plate—expressed here in terms of the index of refraction ø, and depending, +of course, on the strength of the wave from the source. +'What we have been doïng is easily visualized if we look at the complex number +diagram in Eig. 31-3. We first draw the number #; (we chose some values for +z and £ so that E2, comes out horizontal, but this is not necessary). The delay +due to slowing down in the plate would delay the phase of this number, that 1s, +it would rotate ; through a negative angle. But this is equivalent to adding +the small vector „ at roughly right angles to !2¿. But that is just what the +factor —¿ means in the second term of Eq. (31.8). It says that if #2; is real, then +„4 is negative imaginary or that, in general, #⁄⁄; and !4 make a right angle. +lmaginary Axis +Angle = u(n — 1)Az/c += Real Axis +_¬ ` Ea +Fig. 31-3. Diagram for the transmitted wave at a particular £ and z. +--- Trang 541 --- +31-2 The feld due to the material +W©e now have to ask: Is the field #„ obtained in the second term of Eq. (31.8) +the kind we would expect from oscillating charges in the plate? If we can show +that it is, we will then have calculated what the index ø should bel [Since øœ +is the only nonfundamental number in Eq. (31.8).| We turn now to calculating +what field F„ the charges in the material will produee. (To help you keep track +of the many symbols we have used up to now, and will be using in the rest of +our calculation, we have put them all together in Table 31-1.) +Table 31-1 +Symbols used in the calculations +FZ„ = field from the source +tạ = field produced by charges in the plate +Az = thickness of the plate +z = perpendicular distance from the plate +m = index of refraction +œ = frequency (angular) of the radiation +NÑ = number of charges per unit volume in the plate +rạ = number of charges per unit area of the plate +qe — charge on an electron +mm —= mass of an electron +œg — resonant Írequency of an electron bound in an atom +Tf the source Š (of Eig. 31-1) is far off to the left, then the field #⁄; will have +the same phase everywhere on the plate, so we can write that in the neighborhood +of the plate +E, = Eoele~z/©), (31.9) +Right at the plate, where z = 0, we will have += Eoe"°f (at the plate). (31.10) +Bach of the electrons in the atoms of the plate will feel this electric field +and will be driven up and down (we assume the direction oŸ 2o is vertical) by +the electrie force g#. To fnd what motion we expect for the electrons, we will +--- Trang 542 --- +assume that the atoms are little oscillators, that is, that the electrons are fastened +elastically to the atoms, which means that If a force is applied to an electron its +displacement from its normal position will be proportional to the force. +You may think that this is a funny model of an atom ïŸ you have heard about +electrons whirling around in orbits. But that is just an oversimplifed picture. +'The correct picture of an atom, which is given by the theory of wave mechanics, +says that, so far as problems imuolưing light are concerned, the electrons behave +as though they were held by springs. So we shall suppose that the electrons have +a linear restoring force which, together with their mass rm, makes them behave +like little oscillators, with a resonant frequency œọ. We have already studied such +oscillators, and we know that the equation of their motion is written this way: +where #' is the driving force. +For our problem, the driving force comes from the electric fñeld of the wave +from the source, so we should use +=q.B, = qeEoe"°°, (31.12) +where q¿ is the electric charge on the electron and for #⁄; we use the expres- +sion = Eoe”f from (31.10). Our equation of motion for the electron is +dŠz 2 In) +m dP + uậ# ] = qeEoe'”“”. (31.13) +W© have solved this equation before, and we know that the solution is +2 = øoc 1°, (31.14) +where, by substituting in (31.13), we fnd that +=————x 31.15 +#0 m(u — œ2) ) ( ) +so that +I7 - +ca (31.16) +m(uỗ — Ø3) +--- Trang 543 --- +We have what we needed to know——the motion of the electrons in the plate. And +1t is the same for every electron, except that the mean position (the “zero” of +the motion) is, oŸ course, diferent for each electron. +Now we are ready to fnd the fñeld !„ that these atoms produce at the +point , because we have already worked out (at the end of Chapter 30) what +ñeld is produced by a sheet of charges that all move together. Referring back to +Eq. (30.19), we see that the field + at ÐP is just a negative constant times the +velocity of the charges retarded in time by the amount z/c. Diferentiating z in +Eq. (31.16) to get the velocity, and sticking in the retardation |or just putting #o +from (31.15) into (30.18)] yields +Tdqe |. qeEo 2œ(—z/c) +#„=—=—— ———=—— : 31.17 +` 2co€ le mÁ(œ8 — (2) , ' ) +Just as we expected, the driven motion of the electrons produced an extra wave +which travels to the right (that is what the factor e“2ữ=Z/ says), and the +amplitude of this wave is proportional to the number of atoms per unit area +in the plate (the factor 7?) and also proportional to the strength of the source +fñeld (the factor #o). Then there are some factors which depend on the atomic +properties (qe, rm, and œạ), as we should expect. +The most important thing, however, is that this formula (31.17) for 2+ looks +very much like the expression for ⁄„ that we got in Bq. (31.8) by saying that +the original wave was delayed in passing through a material with an index of +refraction n. 'Phe bwo expressions will, in fact, be identical if +—1)Az=—————.. 31.18 +(x— À2 2com(uậ — œ2) ' ) +Notice that both sides are proportional to Az, since ?, which is the number of +atoms øer ni œrea, is equal to N Az, where is the number of atoms per unit +0olume of the plate. Substituting Az for and cancelling the Az, we get our +main result, a formula for the index of refraction in terms of the properties of +the atoms of the material—and of the frequency of the light: +=l+———.-. 31.19 +⁄ " 2cogm(„8 — œ2) ' ) +'This equation gives the “explanation” of the index of refraction that we wished +to obtain. +--- Trang 544 --- +31-3 Dispersion +Notice that in the above process we have obtained something very interesting. +For we have not only a number for the index of refraction which can be computed +from the basic atomic quantities, but we have also learned how the index of +refraction should vary with the frequency œ of the light. 'This is something we +would never understand from the simple statement that “light travels slower in +a transparent material” We still have the problem, of course, of knowing how +many atoms per unit volume there are, and what is their natural frequenecy œọ. +W©e do not know this just yet, because it is diferent for every diferent material, +and we cannot get a general theory of that now. Formulation oŸ a general theory +of the properties of diferent substances—their natural frequencies, and so on——1s +possible only with quantum atomic mechanics. Also, diferent materials have +diÑferent properties and diferent indexes, so we cannot expect, anyway, to get a +general formula for the index which will apply to all substanees. +However, we shall discuss the formula we have obtained, in various possible +circumstances. Pirst of all, for most ordinary gases (for instance, for air, most +colorless gases, hydrogen, helium, and so on) the natural frequencies of the +electron oscillators correspond to ultraviolet light. These requencies are higher +than the frequencies of visible light, that is, œọ is mụuch larger than œ of visible +light, and to a frst approximation, we can disregard œŠ in comparison with øÿ. +Then we fñnd that the index is nearly constant. So for a gas, the index is nearly +constant. This is also true for most other transparent substances, like glass. If +we look at our expression a little more closely, however, we notice that as œ Tis©s, +taking a little bit more away from the denominator, the index also rises. 5o ?t +rises slowly with frequency. The index is higher for blue light than for red light. +That is the reason why a prism bends the light more in the blue than in the red. +The phenomenon that the index depends upon the frequency is called the +phenomenon of đ/sperszon, because it is the basis of the fact that light is “dispersed” +by a prism into a spectrum. “The equation for the index of refraction as a function +of frequeney 1s called a d¿spersion equation. So we have obtained a dispersion +cquation. (In the past few years “dispersion equations” have been fnding a new +use in the theory of elementary particles.) +Our dispersion equation suggests other interesting efects. If we have a +natural requency œọ which lies in the visible region, or if we measure the index +of refraction of a material like glass in the ultraviolet, where œ gets near œọ, we +see that at Írequencies very close to the natural frequency the index can get +--- Trang 545 --- +enormously large, because the denominator can go to zero. Next, suppose that œ +1s preater than œọ. This would occur, for example, If we take a material like glass, +say, and shine x-ray radiation on it. In fact, since many materials which are +opaque to visible light, like graphite for instance, are transparent to x-rays, we +can also talk about the index of refraction of carbon for x-rays. All the natural +frequencies of the carbon atoms would be much lower than the frequency we are +using in the x-rays, since x-ray radiation has a very high fÍrequency. 'Phe index of +refraction is that given by our dispersion equation iŸ we set œ«ọ equal to zero (we +neglect œđ in comparison with œŸ). +A similar situation would occur if we beam radiowaves (or lighÈ) on a gas +of free electrons. In the upper atmosphere electrons are liberated from their +atoms by ultraviolet light rom the sun and they sỉt up there as free electrons. +Eor free electrons œo = 0 (there is no elastic restoring force). Setting «go =0 in +our dispersion equation yields the correct formula for the index of refraction for +radiowaves In the stratosphere, where / is now to represent the density of free +electrons (number per unit volume) in the stratosphere. But let us look again at +the equation, iŸ we beam x-rays on matter, or radiowaves (or any electric waves) +on free electrons the term (œä — œ2) becomes ø„ega7e, and we obtain the result +that tò is less than one. That means that the efective speed of the waves in the +substanee is ƒøsfer than cl Can that be correct? +Tt is correct. In spite of the fact that it is said that you cannot send signals any +faster than the speed of light, it is nevertheless true that the index of refraction of +materials at a particular Írequency can be either greater or less than 1. Thịis just +means that the phase shøff which is produced by the scattered light can be either +posifive or negative. It can be shown, however, that the speed at which you can +send a signadl is not determined by the index at one frequency, but depends on +what the index is at mømy frequencies. What the index tells us is the speed at +which the œodes (or crests) of the wave travel. The node of a wave is not a signal +by itself. In a perfect wave, which has no modulations of any kind, i.e., which is +a steady oscillation, you cannot really say when it “starts,” so you cannot use +1t for a timing signal. In order to send a siøgna/ you have to change the wave +somehow, make a notch in it, make it a little bit fatter or thinner. 'Phat means +that you have to have more than one frequenecy in the wave, and it can be shown +that the speed at which s¿ønais travel is not dependent upon the index alone, +but upon the way that the index changes with the frequency. 'Phis subject we +must also delay (until Chapter 48). Then we will calculate for you the acbual +speed of s7ønals through such a piece of glass, and you will see that ít will not +--- Trang 546 --- +be faster than the speed of light, although the nodes, which are mathematical +points, do travel faster than the speed of light. +Just to give a slipht hint as to how that happens, you will note that the real +dificulty has to do with the fact that the responses of the charges are opposite +to the field, i.e., the sign has gotten reversed. 'Thus in our expression Íor +(Eq. 31.16) the displacement of the charge is in the direction opposite to the +driving feld, because (œ — œ2) is negative for small œọ. The formula says that +when the electric fñeld is pulling in one direction, the charge is moving in the +opposite direction. +How does the charge happen to be going in the opposite direction? lt certainly +does not start of in the opposite direction when the fñeld is frst turned on. When +the motion first starts there is a transient, which settles down after awhile, and +only hen 1s the phase of the oscillation of the charge opposite to the driving +field. And it is then that the phase of the transmitted field can appear to be +aduanccd with respect to the source wave. l§ is this œduance ín phase which 1s +meant when we say that the “phase velocity” or velocity of the nodes is greater +than c. In Fig. 31-4 we give a schematic idea of how the waves might look for +a case where the wave is suddenly turned on (to make a signal). You will see +from the diagram that the signal (i.e., the sfar£ of the wave) is not earlier Íor +the wave which ends up with an advance in phase. +Let us now look again at our dispersion equation. We should remark that +our analysis of the refractive Index gives a result that is somewhat simpler than +(a) E /Strt | +Wave with no HA +material I I | I t +(b) (| pc «4 +Transmitted wave t +withn>1 ⁄ I t +delay of phase +I I 1 +Transmitted wave +with n< 1 ' ' h ! t +advance of phase +Fig. 31-4. Wave “signals.” +--- Trang 547 --- +you would actually ñnd in nature. To be completely accurate we must add some +refinements. First, we should expect that our model of the atomic oscillator +should have some damping force (otherwise once started it would oscillate forever, +and we do not expectE that to happen). We have worked out before (Eq. 23.8) +the motion of a damped oscillator and the result is that the denominator in +E4. (31.16), and therefore in (31.19), is changed from (uậT— œ2) to (uẩ —œ 2+7), +where + is the damping coeficient. +W©e need a second modification to take into account the fact that there are +several resonant frequencies for a particular kind of atom. Ït is easy to ñx up +our dispersion equation by imagining that there are several different kinds of +oscillators, but that each oscillator acts separately, and so we simply add the +contributions of all the oscillators. Let us say that there are j„ electrons per +unit of volume, whose natural frequeney is ¿„ and whose damping factOr is +. +We would then have for our dispersion equation +đề Ahụ +n=1+s —ÐÖ ` —s..—- (31.20) +2corn T § — Ố † 7k9 +W©e have, finally, a complete expression which describes the index of refraction +that is observed for many substances.* 'The index described by this formula varies +with frequency roughly like the curve shown in Pig. 3Í-5. +You will note that so long as œ is not too close to one of the resonant +frequencies, the slope of the curve is positive. Such a positive slope is called +1 “-‡-~ “-‡-> +0 ŒỊ (2 FC. 1) +Fig. 31-5. The index of refraction as a function of frequency. +* Actually, although in quantum mechanics Eq. (31.20) is still valid, its interpretation is +somewhat diferent. In quantum mechanics even an atom with one electron, like hydrogen, has +several resonant frequencies. 'Therefore jWy is not really the number of electrons having the +frequency œ4, but is replaced instead by ) ƒ„, where is the number of atoms per unit volume +and ƒ„ (called the oscillator strength) is a factor that tells how strongly the atom exhibits each +of its resonant frequencies œy;. +--- Trang 548 --- +“normail” dispersion (because it is clearly the most common occurrence). Very +near the resonant frequencies, however, there is a small range oŸ œ”s for which +the slope is negative. Such a negative slope is often referred to as “anomalous” +(meaning abnormal) dispersion, because i% seemed unusual when it was first +observed, long before anyone even knew there were such things as electrons. +ttrom our point of view both slopes are quite “normail”! +31-4 Absorption +Perhaps you have noticed something a little strange about the last form +(Eq. 31.20) we obtained for our dispersion equation. Because of the term 2y +we put in to take account of damping, the Index of refraction 1s now a complez +nưmber! What does that mean? By working out what the real and imaginary +parts of m are we could write +tr — TỦ — ?nẺ, (31.21) +where ø and ø” are real numbers. (We use the minus sign in front of the ¿nw” +because then ø” will turn out to be a positive number, as you can show Íor +yourself.) +W©e can see what such a complex index means by going back to Eq. (31.6), +which is the equation of the wave after it goes through a plate of material with +an index nø. IÝ we put our complex ø into this equation, and do some rearranging, +W© getE +Eatey plate — «n7 Az/c e— 1(m°—1) ^z/e macf2Œ—=2/5) : (31.22) +"¬——ễ_————— +The last factors, marked B in Eq. (31.22), are just the form we had before, and +again describe a wave whose phase has been delayed by the angle /j(m' — 1) Az/c +in traversing the material. 'The first term (A) is new and is an exponential +factor with a reøl exponent, because there were ©wo ?)s that cancelled. Also, the +exponent is negative, so the factor is a real number less than one. It describes a +đecrease 1n the magnitude of the field and, as we should expect, by an amount +which is more the larger Az is. As the wave goes through the material, it is +weakened. 'Phe material is “absorbing” part of the wave. 'Phe wave comes out +the other side with less energy. We should not be surprised at this, because +the damping we put in for the oscillators is indeed a friction force and must +--- Trang 549 --- +be expected to cause a loss of energy. We see that the imaginary part øé of a +complex index of refraction represents an absorption (or “attenuation”) of the +wave. In fact, ?ø7 is sometimes referred to as the “absorption Index.” +We may also point out that an imaginary part to the index ø corresponds +to bending the arrow „in Fig. 3Í-3 toward the origin. It is clear why the +transmitted fñeld is then decreased. +Normally, for instance as in glass, the absorption of light is very smaill. +This is to be expected from our Eq. (31.20), because the imaginary part oŸ the +denominator, 2œ, is much smaller than the term (œ£ — œ2). But ïf the light +frequency œ is very close to œ„ then the resonance term (œÿ — @”) can become +small compared with 7+„œ and the index becomes almost completely imaginary. +The absorption of the light becomes the dominant efect. It is just this efect +that gives the dark lines in the spectrum of light which we receive from the sun. +The light from the solar surface has passed through the sun's atmosphere (as +well as the earth's), and the light has been strongly absorbed at the resonant +frequencies of the atoms in the solar atmosphere. +The observation of such spectral lines in the sunlight allows us to tell the +resonant frequencies of the atoms and hence the chemical composition of the +sun's atmosphere. The same kind of observations tell us about the materials in +the stars. From such measurements we know that the chemical elements in the +sun and in the stars are the same as those we ñnd on the earth. +31-5 The energy carried by an electric wave +W©e have seen that the imaginary part of the index means absorption. We +shall now use this knowledge to fnd out how much energy is carried by a light +wave. We have given earlier an argument that the energy carried by light is +proportional to #2, the tỉme average of the square of the electrie feld in the +wave. The decrease in # due to absorption must mean a loss of energy, which +would go into some friction of the electrons and, we might guess, would end up +as heat in the material. +Tf we consider the light arriving on a unit area, say one square centimeter, +of our plate in Eig. 31-1, then we can write the following energy equation (ïŸ we +assume that energy is conserved, as we đo!): +lnergy in per sec = energy out per sec -- work done per sec. (31.23) +--- Trang 550 --- +For the fñrst term we can write œ22, where œ is the as yet unknown constant +of proportionality which relates the average value of #2 to the energy being +carried. For the second term we must include the part from the radiating +atoms oŸ the material, so we should use œ(#; + !4)2, or (evaluating the square) +a(EJ + 2E,l2„ + E2). +All of our calculations have been made for a thin layer of material whose +index is not too far from 1, so that #„ would always be much less than #2; (Just +to make the calculations easier). In keeping with our approximations, we should, +therefore, leave out the term E2, because it is much smaller than #;„. You +may say: “Then you should leave out #,„ also, because #£ is much smaller +than F2” It is true that ¿„ is much smaller than #2, but we must keep +2; F„ or our approximation will be the one that would apply if we neglected the +presence of the material completely! One way of checking that our calculations +are consistent is to see that we always keep terms which are proportional to Ñ Az, +the area density of atoms in the material, but we leave out terms which are +proportional to (W Az)2 or any higher power of Az. Ours is what should be +called a “low-density approximation.” +In the same spirit, we might remark that our energy equation has neglected +the energy in the reflected wave. But that is OK because this term, Èoo, is +proportional to (N A2), since the amplitude of the reflected wave is proportional +to N Az. +Eor the last term in Eq. (31.23) we wish to compute the rate at which the +incoming wave is doing work on the electrons. We know that work is Íorce tỉmes +distance, so the røứe of doïing work (also called power) is the force times the +velocity. It is really #!- ø, but we do not need to worry about the dot produect +when the velocity and force are along the same direction as they are here (except +for a possible minus sign). So for each atom we take ge2;ò for the average rate +of doing work. Since there are W Az atoms in a unit area, the last term in +Eaq. (31.23) should be ) AzqeE2,u. Our energy equation now looks like +ðă? = aE? + 2aE,E„ + N AzqeE,0. (31.24) +The #2 terms cancel, and we have +2ăl,E„ =—N Azqef.0. (31.25) +W©e now go back to Ed. (31.19), which tells us that for large z +đ„ = _XÂzứ 0(ret by z/c) (31.26) +--- Trang 551 --- +(recalling that ạ = W A2). Putting Eq. (31.26) into the left-hand side of (31.25), +W© getE +NWAzdqe————— +2v S—— E;(at 2) - 0(ret by z/c). +However, #2;(at z) is J;(at atoms) rebarded by z/c. Since the average is inde- +pendent of time, it is the same now as retarded by z/e, or is ;(at atom$) - 0, +the same average that appears on the right-hand side of (31.25). The two sides +are therefore equal if +Ý“= 1, Or œ = cục. (31.27) +We have discovered that 1Ý energy is to be conserved, the energy carried ïn an +electric wave per unit area and per unit time (or what we have called the imensify) +must be given by cocE2. IÝ we call the intensity Š, we have +s— 1ntensity = +Ss= Or = cạụcE2, (31.28) +energy/area/time +where the bar means the #ữne a0eragc. We have a nice bonus result om our +theory of the refractive indexl +31-6 Diffraction of light by a screen +lt is now a good time to take up a somewhat diferent matter which we can +handle with the machinery of this chapter. In the last chapter we said that +when you have an opaque screen and the light can come through some holes, the +distribution of intensity—the difraction pattern——could be obtained by imagining +instead that the holes are replaced by sources (oscillators) uniformly distributed +over the hole. In other words, the difracted wave is the same as though the hole +were a new source. We have to explain the reason for that, because the hole is, of +course, just where there are øø sources, where there are ?ø accelerating charges. +Let us first ask: “What 7s an opaque screen?” Suppose we have a completely +opaque screen bebween a source Š and an observer at P, as in Fig. 3I-6(a). Tf the +screen is “opaque” there is no field at P. Why is there no field there? According +to the basic principles we should obtain the field at as the field ; of the +source delayed, plus the field from all the other charges around. But, as we +have seen above, the charges in the screen will be set in motion by the field 2, +--- Trang 552 --- +X E=E, E=0 , +Opaque screen +s E=E:; E=E: + Euai PP +xzhole +J—wall +S ~—plug P +x ° +E=E. F =Es + Ej + EDi,g = Ô +Fig. 31-6. Diffraction by a screen. +and these motions generate a new field which, if the screen is opaque, must +czactlU cancel the field 2 on the back side of the screen. You say: “What a +miracle that it balances ezøctl Suppose it was not exactly right!” T it were +not exactly right (remember that this opaque screen has some thickness), the +field toward the rear part of the screen would not be exactly zero. So, not +being zero, it would set into motion some other charges in the material of +the screen, and thus make a little more field, trying to get the total balanced +out. So if we make the screen thick enough, there is no residual feld, because +there is enough opportunity to fñnally get the thing quieted down. In terms +of our formulas above we would say that the screen has a large and imaginary +Index, so the wawve is absorbed exponentially as it goes through. You know, +of course, that a thin enough sheet of the most opaque material, even gold, 1s +transparent. +Now let us see what happens with an opaque screen which has holes in it, as +in Eig. 3I-6(b). What do we expect for the fñeld at P? 'The field at P can be +represented as a sum of two parts—the field due to the source Š plus the field +due to the wall, i.e., due to the motions of the charges in the walls. We might +expect the motions of the charges in the walls to be complicated, but we can fnd +out :0ha‡ ftelds the produce in a rather simple way. +--- Trang 553 --- +Suppose that we were to take the same screen, but plug up the holes, as +indicated in part (c) of the fñgure. We imagine that the plugs are of exactly +the same material as the wall. Mind you, the plugs go where the holes were In +case (b). Now let us calculate the fñeld at P. The field at P is certainly zero in +case (©), but it is aiso equal to the ñeld from the source plus the feld due to +all the motions of the atoms in the walls and in the plugs. We can write the +following equations: +Case (h): đà p—= Hs + F2wall› +Case (c): FEÒp=0=E,+E\ạ+ Ebiug +where the primes refer to the case where the plugs are in place, but 2 1s, of +course, the same in both cases. Now if we subtract the two equations, we get +đạt p= (Evan - van) - EDnng: +Now ïf the holes are not too smaill (say many wavelengths across), we would not +expect the presence of the plugs to change the fields which arrive at the walls +except possibly for a little bit around the edges of the holes. Neglecting this +small efect, we can set #⁄van = # „¡ị and obtain that +đài p — —Ebiug +We have the result that the field at ÐP hen there œre holes ìn a sereen (case +b) is the same (except for sign) as the field that is produced by ứhat part of a +complete opaque wall which is located there the holes are! (The sign 1s not too +interesting, since we are usually interested in intensity which is proportional to +the square of the field.) It seems like an amazing backwards-forwards argument. +It is, however, not only true (approximately for not too small holes), but useful, +and is the justification for the usual theory of diÑraction. +'The field TỚNG 1s computed in any particular case by remembering that the +motion of the charges euerhere in the sereen is just that which will cancel out +the fñeld #⁄¿ on the back of the screen. OÔnece we know these motions, we add the +radiation fields at ? due just to the charges in the plugs. +W© remark again that this theory of difraction is only approximate, and will +be good only if the holes are not too smaill. Eor holes which are too small the +Tinng term will be small and then the diference between #2 „ and #¡ (which +diference we have taken to be zero) may be comparable to or larger than the +small Tung term, and our approximation will no longer be valid. +--- Trang 554 --- +Miqcli(frore I)crrtppirntgg. Lígphhí Secrffor-rrtg/ +32-1 Radiation resistance +In the last chapter we learned that when a system is oscillating, energy is +carried away, and we deduced a formula for the energy which is radiated by an +oscillating system. If we know the electric field, then the average of the square +of the field times cọc is the amount of energy that passes Der square meter Der +second through a surface normal to the direction in which the radiation is goïng: +8 = (ạc(E®*). (32.1) +Any oscillating charge radiates energy; for instance, a driven antenna radiates +energy. Ifthe system radiates energy, then in order to account for the conservation +of energy we must find that power is being delivered along the wires which lead +into the antenna. 'That ¡s, to the driving circuit the antenna acts like a resisfance, +or a place where energy can be “lost” (the energy is not really lost, it is really +radiated out, but so far as the circuit is concerned, the energy is lost). In an +ordinary resistance, the energy which is “lost” passes into heat; in this case the +energy which is “lost” goes out into space. But from the standpoint of circuit +theory, without considering +0here the energy goes, the net effect on the circuit is +the same——energy is “lost” from that circuit. Therefore the antenna appears to +the generator as having a resistance, even though it may be made with perfectly +good copper. In fact, 1ƒ it is well built ít will appear as almost a pure resistance, +with very little inductanee or capacitance, because we would like to radiate as +much energy as possible out of the antenna. 'Phis resistance that an antenna +shows is called the radiation resistance. +TÍ a current ƒ is going to the antenna, then the average rate at which power +is delivered to the antenna is the average of the square of the current times the +resistance. The rate at which power is rød¿atcd by the antenna is proportional +--- Trang 555 --- +to the square of the current in the antenna, of course, because all the fields are +proportional to the currents, and the energy liberated is proportional to the +square of the field. The coefficient of proportionality between radiated power +and (T2?) is the radiation resistance. +An interesting question is, what is this radiation resistance due to? Leb us +take a simple example: let us say that currents are driven up and down in an +antenna. W© find that we have to put work in, iŸ the antenna is to radiate energy. +TỶ we take a charged body and accelerate it up and down it radiates energy; 1Í +1E were not charged it would not radiate energy. Ïlt is one thing to calculate +from the conservation of energy that energy is lost, but another thing to answer +the question, øgœ#ns‡ t0uhøt ƒorce are we doïing the work? 'Phat is an interesting +and very dificult question which has never been completely and satisfactorily +answered for electrons, althouph it has been for antennas. What happens is this: +in an antenna, the fields produced by the moving charges in one part of the +antenna react on the moving charges in another part of the antenna. We can +calculate these forces and fnd out how much work they do, and so ñnd the right +rule for the radiation resistance. When we say “We can calculate—” that is not +quite right— cannot, because we have not yet studied the laws of electricity at +short distances; only at large distances do we know what the electric field is. We +saw the formula (28.3), but at present it is too complicated for ws to calculate +the fñelds inside the wave zone. Of course, since conservation of energy is valid, +we can calculate the result all right without knowing the fñelds at short distances. +(As a matter of fact, by using this argument backwards it turns out that one can +find the formula for the forces at short distances only by knowing the field at +very large distances, by using the laws of conservation of energy, but we shall not +go into that here.) +The problem in the case of a single electron is this: if there is only one charge, +what can the force act on? It has been proposed, in the old classical theory, that +the charge was a little ball, and that one part of the charge acted on the other +part. Because of the delay in the action across the tiny electron, the force is not +exactly in phase with the motion. 'Phat is, IÝ we have the electron standing still, +we know that “action equals reaction.” So the various internal forces are equal, +and there is no net force. But if the electron is accelerating, then because of the +time delay across it, the force which is acting on the front from the back is not +exactly the same as the force on the back from the front, because of the delay in +the efect. This delay in the timing makes for a lack of balance, so, as a net efect, +the thing holds itself back by its bootstrapsl 'This model of the origin of the +--- Trang 556 --- +resistance to acceleration, the radiation resistance of a moving charge, has run +into many difculties, because our present view of the electron 1s that it is nof a +“little ball”; this problem has never been solved. Nevertheless we can calculate +exactly, of course, what the net radiation resistance force must be, i.e., how much +loss there must be when we accelerate a charge, in spite of not knowing directly +the mechanism of how that force works. +32-2 The rate of radiation of energy +Now we shall calculate the total energy radiated by an accelerating charge. +'To keep the discussion general, we shall take the case of a charge accelerating any +which way, but nonrelativistically. A% a moment when the acceleration is, say, +vertical, we know that the electric ñeld that is generated is the charge multiplied +by the projection of the retarded acceleration, divided by the distance. So we +know the electric field at any point, and we therefore know the square of the +electric ñeld and thus the energy cocF2 leaving through a unit area per second. +The quantity cọc appears quite often in expressions involving radiowave +propagation. Its reciprocal is called the #npedønce oƒ a 0uacuwm, and 1t is an easy +number to remember: it has the value 1/eoc = 377 ohms. So the power in watts +per square meter ¡is equal to the average of the field squared, divided by 377. +Using our expression (29.1) for the electric field, we fnd that +g_— 04s” 6 (33.2) +16r2cgr2c3 +1s the power per square meter radiated in the direction Ø. We notice that it goes +inversely as the square of the distance, as we said before. NÑow suppose we wanted +the total energy radiated in all directions: then we must integrate (32.2) over all +directions. Eirst we multiply by the area, to ñnd the amount that flows within a +little angle đØ (Eig. 32-1). We need the area of a spherical section. The way to +think of it is this: 1Ý r is the radius, then the width of the annular segment is z đÓ, +and the cireumference is 277 sin Ø, because 7 sin Ø is the radius of the circle. So +the area of the little piece of the sphere is 2zz sin Ø times r đ6: +dA = 2m sin 0 d0. (32.3) +By multiplying the ñux [(32.2), the power per square meter| by the area in square +meters included in the small angle đØ, we fnd the amount of energy that is +--- Trang 557 --- +Lzsn2)S/9 +Fig. 32-1. The area of a spherical segment is 27r sin 6 - r d6. +liberated in this direction between Ø and Ø + đđ; then we integrate that over all +the angles Ø from 0 to 180P: +q22 (* +P= Jsaa = “! sin” Ø d0. (32.4) +87coc? /ọ +By writing sin” Ø = (1— cos2 Ø) sin Ø it is not hard to show that J sin3 Ø dØ = 4/3. +Using that fact, we finally get +P=_-—.. 32.5 +6zegc3 (325) +This expression deserves some remarks. Eirst of all, since the vector ø“ had a +certain đirection, the 2 in (32.5) would be the square of the vector a', that is, +d - da", the length of the vector, squared. Secondly, the ñux (32.2) was calculated +using the retarded acceleration; that is, the acceleration at the time at which the +energy now passing through the sphere was radiated. We might like to say that +this energy was in fact liberated at this earlier time. 'Phis is not exactly true; it +is only an approximate idea. The exact time when the energy is liberated cannot +be defined precisely. All we can really calculate precisely is what happens in a +complete motion, like an oscillation or something, where the acceleration ñnally +ccases. Then what we fnd is that the total energy fux per cycle is the average +of acceleration squared, for a complete cycle. 'Phis is what should really appear +in (32.5). Or, iŸit is a motion with an acceleration that is initially and fñnally +zero, then the total energy that has flown out is the time integral of (32.5). +To illustrate the consequences of formula (32.5) when we have an oscillating +system, let us see what happens if the displacement + of the charge is oscillating +so that the acceleration ø is —œ2zo€?“†, "The average of the acceleration squared +--- Trang 558 --- +over a cycle (remember that we have to be very careful when we square things +that are written in complex notation——it really is the cosine, and the average +Of cos2 œf is one-half) thus is +(a2) = 3 zã. +'Therefore +q2u1z? +Pp= 12cac3` (32.6) +The formulas we are now discussing are relatively advanced and more or less +modern; they date from the beginning of the twentieth century, and they are very +famous. Because of their historical value, it is important for us to be able to read +about them ¡in older books. In fact, the older books also used a system of units +diferent from our present mks system. However, all these complications can be +straightened out in the ñnal formulas dealing with electrons by the following +rule: The quantity gỆ/4zco, where q is the electronic charge (in coulombs), has, +historically, been written as e2. It is very casy to calculate that e in the mks +system is numerically equal to 1.5188 x 10~1*, because we know that, numerically, +qe = 1.60206 x 10~†12 and 1/4zco = 8.98748 x 10. Therefore we shall often use +the convenient abbreviation : +c2 = -®—, (32.7) +47m €0 +TỶ we use the above numerical value of e in the older formulas and treat them as +though they were written in mks units, we will get the right numerical results. +Eor example, the older form of (32.5) is P = 3c2a'2/c. Again, the potential +energy of a proton and an electron at distance r is qg2/4zcạr or e2/r, with +e = 1.5188 x 101 (mks). +32-3 Radiation damping +Now the fact that an oscillator loses a certain energy would mean that if we +had a charge on the end oŸ a spring (or an electron in an atom) which has a +natural frequency œạọ, and we start it oscillating and let it go, it will not oscillate +forever, even ï i is in empty space millions of miles from anything. There is no +oil, no resistance, in an ordinary sense; no “viscosity.” But nevertheless it will +not oscillate, as we might once have said, “forever,” because If it is charged it is +radiating energy, and therefore the oscillation will slowly die out. How slowly? +'What is the @Q of such an oscillator, caused by the electromagnetic efects, the +--- Trang 559 --- +so-called radiation resistance or radiation damping of the oscillator? The @Q of +any oscillating system is the total energy content of the oscillator at any time +divided by the energy loss per radian: +Q= uy à +Or (another way to write i0), since đW/dó = (dW/đt)/(do/dt) = (dW/dL) /e, += —: 32.8 +ẹ@ dW/dt (3238) +Tf for a given @ this tells us how the energy of the oscillation dies out, đW/dt = +—(u/Q)W, which has the solution W = Wse—*“1⁄® ¡f Wg is the initial energy +(at £ = 0). +To ñnd the Q for a radiator, we go back to (32.8) and use (32.6) for đdW/di. +Now what do we use for the energy W/ of the oscillator? 'Phe kinetic energy +of the oscillator is jn2?, and the mean kinetic energy is mœ2z2/4. But we +remember that for the total energy of an oscillator, on the average half is kinetic +and half is potential energy, and so we double our result, and fñnd for the total +energy of the oscillator +W = šmuŸzã. (32.9) +'What do we use for the frequency in our formulas? We use the natural frequency œọ +because, for all practical purposes, that is the frequency at which our atom is +radiating, and for rm we use the electron mass rm=;. hen, making the necessary +divisions and cancellations, the formula comes down to +1 4me2 +—= =>: 32.10 +@Q_ 3Am,c2 ( ) +(In order to see it better and in a more historical form we write iÈ using our +abbreviation g2/4zco = e2, and the factor œo/c which was left over has been +writben as 2/A.) Since Q is dimensionless, the combination e2/mn„c? must be +a property only of the electron charge and mass, an intrinsic property of the +electron, and i9 must be a lengfh. It has been given a name, the classical electron +radius, because the early atomic models, which were invented to explain the +radiation resistance on the basis of the force of one part oŸ the electron acting +on the other parts, all needed to have an electron whose dimensions were of this +--- Trang 560 --- +general order of magnitude. However, this quantity no longer has the signifcance +that we believe that the electron really has such a radius. Numerically, the +magnitude of the radius is +ro = ——s =3.82 x 10ˆ°”m, (32.11) +Now let us actually calculate the Q of an atom that is emitting light—let us +say a sodium atom. Eor a sodium atom, the wavelength is roughly 6000 angstroms, +in the yellow part of the visible spectrum, and this is a typical wavelength. Thus +=——#x~5x10 32.12 +9= | (32.12) +so the Q of an atom is of the order 10. 'This means that an atomic oscillator +will oscillate for 10 radians or about 107 oseillations, before its energy falls by a +factor 1/e. The Írequency of oscillation of light corresponding to 6000 angstroms, += c/À, is on the order of 1015 cyeles/sec, and therefore the lifetime, the time +1t takes for the energy oŸ a radiating atom to die out by a factor l/e, is on the +order of 10” sec. In ordinary cireumstances, freely emitting atoms usually take +about this long to radiate. 'This ¡is valid only for atoms which are in empty space, +not being disturbed in any way. If the electron is in a solid and it has to hit +other atoms or other electrons, then there are additional resistances and diferent +damping. +The efective resistance term + in the resistance law for the oscillator can +be found from the relation 1/Q = +/œo, and we remember that the size oŸ + +determines how wide the resonance curve is (Fig. 23-2). Thus we have just +computed the œidths oƒ spectral lines for freely radiating atomsl Since À = 27/0, +we fnd that +AA = 2me Au/uŸ = 2mcy/uạ = 2xc/Quo += À/Q = 4mro/3 = 1.18 x 10”! m. (32.13) +32-4 Independent sources +In preparation for our second topic, the scattering of light, we must now +discuss a certain feature of the phenomenon of interference that we neglected to +discuss previously. 'Phis is the question of when interference does øø occur. lf +we have two sources 5 and %2, with amplitudes 4¡ and 4a, and we make an +--- Trang 561 --- +observation in a certain direction in which the phases of arrival of the two signals +are ôi and óa (a combination oŸ the acbual tỉme oŸ oscillation and the delayed +tỉme, depending on the position of observation), then the energy that we receive +can be found by compounding the §wo complex number vectors 4 and 4a, one +at angle ở and the other at angle ó2 (as we did in Chapter 29) and we find that +the resultant energy is proportional to +A? = 4? + A +2AiAa cos (ới — 9a). (32.14) +Now If the cross term 24 4a cos (ở — da) were not there, then the total energy +that would be received in a given direction would simply be the sum of the +energies, 4Ý + 43, that would be liberated by each source separately, which is +what we usually expect. 'That is, the combined intensity of light shining on +something om two sources is the sum of the intensities of the two lights. On the +other hand, if we have things set just right and we have a cross term, it is not +such a sum, because there is also some interference. lf there are circumstances +in which this term is of no importance, then we would say the interference 1s +apparently lost. Of course, in nature it is always there, but we may not be able +to detect it. +Let us consider some examples. Suppose, first, that the two sources are +7,000,000,000 wavelengths apart, not an impossible arrangement. 'Then in a given +direction 1È is true that there is a very defñnite value of these phase differences. +But, on the other hand, if we move just a haïr in one direction, a few wavelengths, +which is no distance at all (our eye already has a hole in it that is so large that we +are averaging the efects over a range very wide compared with one wavelength) +then we change the relative phase, and the cosine changes very rapidly. If we +take the øuerage of the intensity over a little region, then the cosine, which øgoes +plus, minus, plus, minus, as we move around, averages tO zero. +So iÝ we average over regions where the phase varies very rapidly with position, +we get no interference. +Another example. Suppose that the Ewo sources are two independent radio +oscillators—not a single oscillator being fed by two wires, which guarantees that +the phases are kept together, but two independent sources—and that they are not +precisclu tuned at the same frequency (it is very hard to make them at exactly +the same frequency without actually wiring them together). In this case we have +what we call two zndependen‡ sources. Of course, since the frequencies are not +exactly equal, although they started in phase, one of them begins to get a little +--- Trang 562 --- +ahead of the other, and pretty soon they are out of phase, and then it gets still +further ahead, and pretty soon they are in phase again. So the phase diference +between the two is gradually drifting with time, but if our observation is so crude +that we cannot see that little time, if we average over a much longer time, then +althouph the intensity swells and falls like what we call “beats” in sound, if these +swellings and fallings are too rapid for our equipment to follow, then again this +term averages Out. +In other words, in any cireumstance in which the phase shift averages out, we +get no interferencel +One fnds many books which say that two distinct light sources never interfere. +This is not a statement of physics, but is merely a statement of the degree of +sensitivity of the technique of the experiments at the time the book was written. +'What happens in a light source is that first one atom radiates, then another atom +radiates, and so forth, and we have just seen that atoms radiate a train of waves +only for about 10~Š sec; after 10” sec, some atom has probably taken over, then +another atom takes over, and so on. So the phases can really only stay the same +for about 10~Ẻ sec. Therefore, if we average for very much more than 10” sec, +we do not see an interference from two diferent sources, because they cannot hold +their phases steady for longer than 10” sec. With photocells, very high-speed +detection is possible, and one can show that there is an interference which varies +with time, up and down, in about 10~Š sec. But most detection equipment, of +course, does not look at such fine time intervals, and thus sees no interference. +Certainly with the eye, which has a tenth-of-a-second averaging time, there is no +chance whatever of seeing an interference between two diferent ordinary sources. +Recently ít has become possible to make light sources which get around this +efect by making all the atoms emit fogether in time. The device which does this +1s a very complicated thing, and has to be understood in a quantum-mechanical +way. It is called a laser, and it is possible to produce from a laser a source in +which the time during which the phase is kept constant, is very much longer than +10 sec. It can be of the order of a hundredth, a tenth, or even one second, and +so, with ordinary photocells, one can pick up the frequency between ©wo diferent +lasers. One can easily detect the pulsing of the beats between two laser sources. +Soon, no doubt, someone will be able to demonstrate two sources shining on a +wall, in which the beats are so slow that one can see the wall get bright and darkl +Another case in which the interference averages out is that in which, instead +of having only #wo sources, we have nan. In this case, we would write the +expression for 4A? as the sum of a whole lot of amplitudes, complex numbers, +--- Trang 563 --- +squared, and we would get the square of each one, all added together, plus cross +terms bebween every pair, and if the cireumnstances are such that the latter average +out, then there will be no effects ofinterference. It may be that the various sources +are located in such random positions that, although the phase diference between +4s and 4s is also defnite, it is very different from that bebtween 4: and 4a, etc. +So we would get a whole lot of cosines, many plus, many minus, all averaging out. +So it is that in many circumstances we do not see the efects of interference, +but see only a collective, total intensity equal to the sum of all the intensities. +32-5 Scattering of light +The above leads us to an efect which occurs in air as a consequence of the +Irregular positions of the atoms. When we were discussing the index of refraction, +we saw that an incoming beam of light will make the atoms radiate again. The +electric fñeld of the incoming beam drives the electrons up and down, and they +radiate because of their acceleration. Phis scattered radiation combines to give +a beam in the same direction as the incoming beam, but of somewhat diferent +phase, and this is the origin of the index of refraction. +But what can we say about the amount of re-radiated light in some other +direction? Ordinarily, If the atoms are very beautifully located in a nice pattern, +1t is easy to show that we get nothing in other directions, because we are adding +a lot of vectors with their phases always changing, and the result comes to zero. +But ïf the objects are randomlụ located, then the total intensity in any direction +1s the sươn of the intensities that are scattered by each atom, as we have just +discussed. Eurthermore, the atoms in a gas are in actual motion, so that although +the relative phase of two atoms is a definite amount now, later the phase would be +quite diferent, and therefore eøch cosine term will average out. Therefore, to fnd +out how much light is scattered in a given direction by a gas, we merely study the +efects of one øtom and multiply the intensity it radiates by the number of atoms. +Barlier, we remarked that the phenomenon of scattering of light of this nature +1s the origin of the blue of the sky. 'Phe sunlight goes through the air, and when +we look to one side of the sun—say at 90° to the beam——we see blue light; what +we now have to calculate is hou rmụch light we see and 0h it 1s blue. +Tf the incident beam has the electric ñeld* E = oec”“f at the point where +the atom is located, we know that an electron in the atom will vibrate up and +- * When a Caret appears on a vector iÈ signifies that the componen‡s of the vector are complex: +#2 = (E„y, lụ, E„). +--- Trang 564 --- +lncident beam + Atom +(unpolarized) „ XS +Scattered ¬ +Fig. 32-2. A beam of radiation falls on an atom and causes the +charges (electrons) in the atom to move. The moving electrons in turn +radiate In varlous directions. +down in response to this (Fig. 32-2). Erom Eq. (23.8), the response will be +Ê=—> “—-: (32.15) +m(uổ — 3 + iu) +W© could include the damping and the possibility that the atom acts like several +oscillators of diferent frequency and sum over the various frequencies, but for +simplicity let us just take one oscillator and neglect the damping. Then the +response to the external electric fñield, which we have already used in the calculation +of the index of refraction, is simply +&=—S—. (32.16) +m(duỗ — œ3) +We could now easily calculate the intensity of light that is emitted in various +đirections, using formula (32.2) and the acceleration corresponding to the above Z. +Rather than do this, however, we shall simply calculate the #ofal amownt oŸ +light scattered in ai/ directions, just to save time. 'Phe total amount of light +energy per second, scattered in all directions by the single atom, is oÝ course +given by Eaq. (32.6). 5o, putting together the various pieces and regrouping them, +W© getE +P = [(q2ø*/12meoe))qễ Eỗ Jmà(Ÿ — œ8)” ] += (šeocE8)(8a/3)(q¿/16n2cm¿e")[j`/(ø2 — œ8)”] += (šcocE8)(Smr3/3)|°/(Ÿ — w)'] (32.17) +for the total scattered power, radiated ín all directions. +--- Trang 565 --- +We have written the result in the above form because it is then easy %O +remember: First, the total energy that is scattered is proportional to the square +of the incident fñeld. What does that mean? Obviously, the square of the ineident +field is proportional to the energy which is coming in per second. In fact, the +energy incident per square meter per second is cọc times the average (H2) of +the square of the electric field, and If lo is the maximum value of #, then +(F2?) = šEÿ. In other words, the total energy scabtered is proportional to the +energy per square meter that comes in; the brighter the sunlight that is shining +in the sky, the brighter the sky is going to look. +Next, what ƒfraction oŸ the incoming light is scattered? Let us imagine a +“target” with a certain area, let us say ø, in the beam (not a real, material target, +because this would difract light, and so on; we mean an imaginary area drawn +in space). The total amount of energy that would pass through this surface ø +in a given circumstance is proportional both to the incoming intensity and to ơ, +and the total power would be +P= (š‹ạcE))ø. (32.18) +Now we invent an idea: we say that the atom scatters a total amount of +intensity which is the amount which would fall on a certain geometrical area, +and we give the answer by giving that area. That answer, then, is independent +of the incident intensity; it gives the ratio of the energy scattered to the energy +Ineident per square meter. In other words, the ratio +total energy scattered per second : +—————— san ørea. +energy incident per square meter per second +The signifcance of this area is that, if all the energy that impinged on that area +were to be spewed in all directions, then that is the amount of energy that would +be scattered by the atom. +This area is called a cross seclion for scaftering; the idea OoŸ cross section 1s +used constantly, whenever some phenomenon occurs in proportion to the intensity +of a beam. In such cases one always describes the amount of the phenomenon by +saying what the efective area would have to be to pick up that mụuch of the beam. +lt does not mean in any way that this oscillator actually has such an area. If +there were nothing present but a free electron shaking up and down there would +be no area directly associated with it, physically. It is merely a way of expressing +the answer to a certain kind of problem; it tells us what area the incident beam +--- Trang 566 --- +would have to hit in order to account for that much energy coming off. Thus, for +OUT Ca§@, +8mrổ 3 +Ø = + (2—oŸ)? (32.19) +(the subscript s is for “scattering”). +Let us look at some examples. First, if we go to a very low natural frequency œ0, +or to completely unbound electrons, for which œọ = 0, then the frequency œ +cancels out and the cross section is a constant. 'This low-frequency limit, or the +free electron cross section, is known as the Thomson scatfering cross seclion. IW +is an area whose dimensions are approximately 10~!5 meter, more or less, on a +side, i.e., 10—9 square meter, which is rather smalll +Ôn the other hand, ïf we take the case of light in the air, we remember that for +aïr the natural frequencies of the oscillators are higher than the frequency of the +light that we use. This means that, to a frst approximation, we can disregard ¿2 +in the denominator, and we fñnd that the scattering is proportional to the ƒourth +pouer oÊ the frequency. hat is to say, light which is of higher frequency by, say, +a factor of two, is siz‡een tứmes more intensely scattered, which is a quite sizable +diference. This means that blue light, which has about twice the frequency of +the reddish end of the spectrum, is scattered to a far greater extent than red +light. Thus when we look at the sky it looks that glorious blue that we see all +the timel +There are several points to be made about the above results. One interesting +question is, why do we ever see the clowds? Where do the clouds come from? +tverybody knows it is the condensation of water vapor. But, of course, the +water vapor Is already in the atmosphere 0eƒfore it condenses, so why don” we +see it then? After it condenses it is perfectly obvious. It wasnt there, now it 2s +there. 5o the mystery of where the clouds come from is not really such a childish +mystery as “Where does the water come from, Daddy?,” but has to be explained. +W© have just explained that every atom scatters light, and of course the water +vapor will scatter light, too. The mystery is why, when the water is condensed +into clouds, does it scatter such a fremendouslu greater amownt of light? +Consider what would happen If, instead of a single atom, we had an agglom- +erate of atoms, say Ewo, very close together compared with the wavelength of the +light. Remember, atoms are only an angstrom or so across, while the wavelength +of light is some 5000 angstroms, so when they form a clump, a few atoms together, +they can be very close together compared with the wavelength of light. Then +--- Trang 567 --- +when the electric fñeld acts, bo#h, oƒ the atoms tuiÏH tnoue together. he electrie +fñeld that is scattered will then be the sum of the two electric fields in phase, ï.e., +double the amplitude that there was with a single atom, and the enerøgu which +is scattered is therefore ƒour: tưnes what it is with a single atom, not twicel So +lumps of atoms radiate or scatter more energy than they do as single atoms. Ôur +argument that the phases are independent is based on the assumption that there +is a real and large difference in phase bebween any ÿwo atoms, which is true only +1f they are several wavelengths apart and randomly spaced, or moving. But if +they are right next to each other, they necessarily scatter in phase, and they have +a coherent interference which produces an increase in the scattering. +Tf we have atoms in a lump, which is a tiny droplet of water, then each one +will be driven by the electric field in about the same way as before (the efect of +one atom on the other is not important; it is Just to get the idea anyway) and +the amplitude of scattering from each one is the same, so the total field which is +scatered is /-fold increased. The 7m#ensitu of the light which is scattered is then +the square, or WZ-fold, increased. We would have expected, if the atoms were +spread out in space, only Ñ times as much as 1, whereas we get W2 times as +much as 1l "That is to say, the scattering of water in lumps of ) molecules each +is / times more intense than the scattering of the single atoms. So as the water +agglomerates the scattering increases. Does it increase øở ?nfimitum2? Nol When +does this analysis begin to fail? How many atoms can we put together before +we cannot drive this argument any further? Ansuer: IÝ the water drop gets so +big that om one end to the other is a wavelength or so, then the atoms are +no longer all in phase because they are too far apart. So as we keep increasing +the size of the droplets we get more and more scattering, until such a time that +a drop gets about the size of a wavelength, and then the scattering does not +Increase anywhere nearly as rapidly as the drop gets bigger. Eurthermore, the +blue disappears, because for long wavelengths the drops can be bigger, before +this limit is reached, than they can be for short wavelengths. Although the short +waves scatter more per atom than the long waves, there is a bigger enhancement +for the red end of the spectrum than for the blue end when all the drops are +bigger than the wavelength, so the color is shifted from the blue toward the red. +Now we can make an experiment that demonstrates this. We can make +particles that are very small at frst, and then gradually grow in size. We use a +solution of sodium thiosulfate (hypo) with sulfuric acid, which precipitates very +fine grains of sulfur. As the sulfur precipitates, the grains frst start very small, +and the scattering is a little bluish. Äs it precipitates more it gets more intense, +--- Trang 568 --- +and then it will get whitish as the particles get bigger. In addition, the light +which goes straight through will have the blue taken out. hat is why the sunset +1s red, of course, because the light that comes through a lot of air, to the eye has +had a lot of blue light scattered out, so i% is yellow-red. +Finally, there is one other important feature which really belongs in the next +chapter, on polarization, but it is so Interesting that we point it out now. “This +1s that the electric fñeld of the scattered light tends to vibrate in a particular +direction. The electric feld in the incoming light is oscillating in some way, and +the driven oscillator goes in this same direction, and if we are situated about at +right angles to the beam, we will see polarzcởd light, that is to say, light in which +the electric feld is going only one way. In general, the atoms can vibrate in any +direction at right angles to the beam, but if they are driven directly toward or +away from us, we do not see it. 5o if the incoming light has an electric ñeld which +changes and oscillates in any direction, which we call unpolarized light, then the +light which is coming out at 909 to the beam vibrates in only one direction! (See +Eig. 32-3.) +—X Electron +moVe€S In +4“ plane L k +Incident beam + +(unpolarized) +-L k ¡is plane polarized +Fig. 32-3. lllustration of the origin of the polarization of radiation +scattered at right angles to the incident beam. +There is a substance called polaroid which has the property that when light +goes throuph it, only the piece of the electric fñeld which is along one particular +axis can get throupgh. We can use this to test for polarization, and indeed we fnd +the light scattered by the hypo solution to be strongly polarized. +--- Trang 569 --- +MPolqrr=crfiort +33-1 The electric vector of light +In this chapter we shall consider those phenomena which depend on the fact +that the electric fñeld that describes the light is a vector. In previous chapters +we have not been concerned with the direction of oscillation of the electric field, +except to note that the electric vector lies in a plane perpendicular to the direction +of propagation. The particular direction in this plane has not concerned us. We +now consider those phenomena whose central feature is the particular direction +of oscillation of the electric field. +In ideally monochromatic light, the electric ñeld must oscillate at a defnite +frequency, but since the z-component and the -component can oscillate indepen- +dently at a defnite frequency, we must first consider the resultant efect produced +by superposing two independent oscillations at right angles to each other. What +kind of electric field is made up of an zø-component and a -component which +oscillate at the same frequency? If one adds to an z-vibration a certain amount of +u-vibration at the same phase, the result is a vibration in a new direction in the +-plane. Figure 33-1 ïllustrates the superposition of diferent amplitudes for the +z-vibration and the g-vibration. But the resultants shown in Fig. 33-l are not +the only possibilities; in all of these cases we have assumed that the z-vibration +and the -vibration are ?w phøse, but it does not have to be that way. It could +be that the z-vibration and the z-vibration are out of phase. +'When the z-vibration and the z-vibration are not in phase, the electric field +vector moves around in an ellipse, and we can illustrate this in a familiar way. lÝ +we hang a ball from a support by a long string, so that it can swing freely in a +horizontal plane, it will execute sinusoidal oscillations. If we imagine horizontal +z- and -coordinates with their origin at the rest position of the ball, the ball +can swing in either the zø- or -direction with the same pendulum frequency. +By selecting the proper initial displacement and initial velocity, we can set the +--- Trang 570 --- +" x / x ⁄ x +Ey„ =1 Ey =1 Ey =1 +E,=0 E,=Ÿ E,=1 +Ey„ =0 E„= 1 Ey„ =—1 +E,=1 E,=—1 E,= 1 +Fig. 33-1. Superposition of x-vibrations and y-vibrations in phase. +ball mm oscillation along either the z-axis or the -axis, or along any straight +line in the zz-plane. 'Phese motions of the ball are analogous to the oscillations +of the electric field vector illustrated in Fig. 33-1. In each instance, since the +z-vibrations and the -vibrations reach their maxima and minima at the same +time, the z- and -oscillations are in phase. But we know that the most general +motion of the ball is motion in an ellipse, which corresponds to oscillations In +which the zø- and ¿-directions are øøf in the same phase. “The superposition of #- +and ø-vibrations which are not in phase is illustrated in Fig. 33-2 for a variety +of angles bebween the phase of the z-vibration and that of the g-vibration. he +general result is that the electric vector moves around an ellipse. The motion in +a straight line is a particular case corresponding to a phase difference of zero (or +an integral multiple of z); motion in a circle corresponds to equal amplitudes +with a phase diference of 90° (or any odd integral multiple of z/2). +In Eig. 33-2 we have labeled the electric field vectors in the ø- and z-directions +with complex numbers, which are a convenient representation in which to express +the phase diference. Do not confuse the real and imaginary components of the +complex electric vector in this notation with the z- and +-coordinates of the fñeld. +The z- and -coordinates plotted in Fig. 33-1 and Fig. 33-2 are actual electric +felds that we can measure. The real and imaginary components of a complex +--- Trang 571 --- +⁄ ⁄2 G3 +Ey = cosuf; 1 COS0f; 1 COS0f; 1 +Ey = cosuf; 1 cos (0£ + T); eix/4 —sinwt; ï +SN NY SN +Exy = COSUf; 1 COSUf, 1 COS UŸf; 1 +Ey = cos († + ei3x/4 — Cosưf; —1 — CoS (U£ + T); —e!⁄4 +E,—=cosuf; 1 COS f; 1 cos(uf; 1 +Ey„ =sinuwf; —i — Cos (f + Š*); —el3”/4 cosœf; 1 +Fig. 33-2. Superposition of x-vibrations and y-vibrations with equal +amplitudes but various relative phases. The components Ex and Ey are +expressed In both real and complex notations. +electric ñeld vector are only a mathematical convenience and have no physical +significance. +NÑow for some terminology. Light is ¿mearlJ polarized (sometimes called +plane polarized) when the electric feld oscillates on a straight line; Eig. 33-1 +iHustrates linear polarization. When the end of the electric field vector travels In +an ellipse, the light is ellticall polarizcd. When the end of the electric feld +vector travels around a cirele, we have c¿rcular polar?zation. TỶ the end of the +electric vector, when we look at it as the light comes straight toward us, goes +around in a counterelockwise direction, we call it right-hand cireular polarization. +Figure 33-2(ø) illustrates right-hand circular polarization, and Fig. 33-2(c) shows +--- Trang 572 --- +left-hand circular polarization. In both cases the light is coming out of the paper. +Our convention for labeling left-hand and right-hand circular polarization is +consistent with that which is used today for all the other particles in physics +which exhibit polarization (e.g., electrons). However, in some books on optics +the opposite conventions are used, so one must be careful. +W©e have considered linearly, cireularly, and elliptically polarized light, which +covers everything except for the case of wnpolarizcd light. NÑow how can the light +be unpolarized when we know that it must vibrate in one or another of these +ellipses? If the light is not absolutely monochromatie, or if the z- and -phases +are not kept perfectly together, so that the electric vector first vibrates in one +direction, then in another, the polarization is constantly changing. Remember +that one atom emits during 10~Ẻ sec, and if one atom emits a certain polarization, +and then another atom emits light with a diferent polarization, the polarizations +will change every 10” sec. TỶ the polarization changes more rapidly than we +can detect i%, then we call the light unpolarized, because all the efects of the +polarization average out. None of the interference effects of polarization would +show up with unpolarized light. But as we see from the defnition, light is +unpolarized only if we are unable ©o fñnd out whether the light is polarized or +33-2 Polarization of scattered light +The first example of the polarization efect that we have already discussed +1s the scattering of lipht. Consider a beam of light, for example from the sun, +shining on the air. The electric feld will produce oscillations of charges in the +air, and motion of these charges will radiate light with its maximum intensity +in a plane normal to the direction of vibration of the charges. The beam from +the sun is unpolarized, so the direction of polarization changes constantly, and +the direction of vibration of the charges in the air changes constantly. lÝ we +consider light scattered at 90, the vibration of the charged particles radiates +to the observer only when the vibration is perpendicular to the observerˆs line +of sight, and then light will be polarized along the direction of vibration. So +scattering is an example of one means of producing polarization. +33-3 Birefringence +Another interesting efect of polarization is the fact that there are substances +for which the index of refraction is diferent for light linearly polarized in one +--- Trang 573 --- +direction and linearly polarized in another. Suppose that we had some material +which consisted of long, nonspherical molecules, longer than they are wide, and +suppose that these molecules were arranged in the substance with their long axes +parallel. Then what happens when the oscillating electric ñeld passes through this +substance? Suppose that because of the structure of the molecule, the electrons +in the substanece respond more easily to oscillations in the direction parallel to +the axes of the molecules than they would respond I1 the electric ñeld tries to +push them at right angles to the molecular axis. In this way we expect a diferent +response for polarization in one direction than for polarization at right angles to +that direction. Let us call the direction of the axes of the molecules the opfic #3. +'When the polarization is in the direction of the optic axis the index of refraction +is diÑerent than it would be if the direction of polarization were at right angles +to it. Such a substance is called b#eƒringenmt. It has two refrangibilities, i.e., +two indexes of refraction, depending on the direction of the polarization inside +the substance. What kind of a substance can be birefingent? In a birefringent +substance there must be a certain amount of lining up, for one reason or another, +of unsymmetrical molecules. Certainly a cubic crystal, which has the symmetry of +a cube, cannot be birefringent. But long needlelike crystals undoubtedly contain +mmolecules that are asymmetric, and one observes this effect very easily. +Let us see what efects we would expect if we were to shine polarized light +through a plate of a birefringent substance. lf the polarization is parallel to +the optic axis, the light will go through with one velocity; 1f the polarization is +perpendicular to the axis, the light is transmitted with a diferent velocity. An +Interesting situation arises when, say, light is linearly polarized at 45° to the +optic axis. NÑow the 45° polarization, we have already noticed, can be represented +as a superposition of the z- and the ¿-polarizations of equal amplitude and in +phase, as shown in EFig. 33-2(a). Since the z- and z-polarizations travel with +diferent velocities, their phases change at a diferent rate as the light passes +through the substance. So, although at the start the z- and ø-vibrations are in +phase, inside the material the phase diference between z- and ø-vibrations 1s +proportional to the depth in the substance. As the light proceeds through the +material the polarization changes as shown in the series oŸ diagrams in Fig. 33-2. +Tf the thickness of the plate is just right to introduce a 90° phase shift between +the z- and g-polarizations, as in Fig. 33-2(c), the light will come out circularly +polarized. Such a thickness is called a quarter-wave plate, because it introduces a +quarter-cycle phase diference between the zø- and the -polarizations. If linearly +polarized light is sent through ©wo quarter-wave plates, it will come out plane- +--- Trang 574 --- +polarized again, but at right angles to the original direction, as we can see from +Eig. 33-2(e). +One can easily illustrate this phenomenon with a piece of cellophane. Cello- +phane is made of long, fbrous molecules, and is not isotropic, since the fibers +lie preferentially in a certain direction. 'To demonstrate birefringence we need a +beam of linearly polarized light, and we can obtain this conveniently by passing +unpolarized light through a sheet of polaroid. Polaroid, which we will discuss +later in more detail, has the useful property that it transmits light that is linearly +polarized parallel to the axis of the polaroid with very little absorption, but +light polarized in a direction perpendicular to the axis of the polaroid is strongly +absorbed. When we pass unpolarized light through a sheet of polaroid, only that +part of the unpolarized beam which is vibrating parallel to the axis of the polaroid +gets through, so that the transmitted beam is linearly polarized. 'This same +property of polaroid is also useful in detecting the direction of polarization of a +linearly polarized beam, or in determining whether a beam is linearly polarized or +not. Ône simply passes the beam of light through the polaroid sheet and rotates +the polaroid in the plane normal to the beam. lf the beam 1s linearly polarized, +it will not be transmitted through the sheet when the axis of the polaroid is +normal to the direction of polarization. The transmitted beam is only slightly +attenuated when the axis of the polaroid sheet is rotated through 90”. Tf the +transmitted intensity is independent of the orientation of the polaroid, the beam +is not linearly polarized. +To demonstrate the birefringence of cellophane, we use two sheets of polaroid, +as shown in Eig. 33-3. The frst gives us a linearly polarized beam which we pass +through the cellophane and then through the second polaroid sheet, which serves +to detect any efect the cellophane may have had on the polarized light passing +CELLOPHANE ++1 :HÉ“ +XS stasopf +Fig. 33-3. An experimental demonstration of the birefringence of +cellophane. The electric vectors ¡n the light are indicated by the dot- +ted lines. The pass axes of the polaroid sheets and optic axes of the +cellophane are indicated by arrows. 'The incident beam is unpolarized. +--- Trang 575 --- +through it. If we first set the axes of the two polaroid sheets perpendicular to each +other and remove the cellophane, no light will be transmitted through the second +polaroid. If we now introduce the cellophane between the two polaroid sheets, and +rotate the sheet about the beam axis, we observe that in general the cellophane +makes it possible for some light to pass through the second polaroid. However, +there are two orientations of the cellophane sheet, at right angles to each other, +which permit no light to pass through the second polaroid. 'Phese orientations in +which linearly polarized light is transmitted through the cellophane with no efect +on the direction of polarization must be the directions parallel and perpendicular +to the optic axis of the cellophane sheet. +W© suppose that the light passes through the cellophane with bwo diferent +velocities in these two diferent orientations, but it is transmitted without changing +the direction of polarization. When the cellophane is turned halfway bebween +these two orientations, as shown in Eig. 33-3, we see that the light transmitted +through the second polaroid is bright. +Tt just happens that ordinary cellophane used in commercial packaging is +very close to a halfwave thickness for most of the colors in white light. Such +a sheet will turn the axis of linearly polarized light through 90° if the incident +linearly polarized beam makes an angle of 45° with the optic axis, so that the +beam emerging from the cellophane is then vibrating in the right direction to +pass through the second polaroid sheet. +TÝ we use white light in our demonstration, the cellophane sheet will be of the +proper half-wave thickness only for a particular component of the white light, +and the transmitted beam will have the color of this component. “The color +transmitted depends on the thickness of the cellophane sheet, and we can vary +the efective thickness of the cellophane by tilting i% so that the light passes +throuph the cellophane at an angle, consequently through a longer path in the +cellophane. As the sheet is tilted the transmitted color changes. With cellophane +of diferent thicknesses one can construct filters that will transmit diferent colors. +These flters have the interesting property that they transmit one color when the +two polaroid sheets have their axes perpendicular, and the complementary color +when the axes of the bwo polaroid sheets are parallel. +Another interesting application of aligned molecules is quite practical. Certain +plastics are composed oŸ very long and complicated molecules all twisted together. +'When the plastic is solidified very carefully, the molecules are all twisted in a mass, +so that there are as many aligned in one direction as another, and so the plastic +is not particularly birefringent. Usually there are strains and stresses introduced +--- Trang 576 --- +when the material is solidifed, so the material is not perfectly homogeneous. +However, if we apply tension to a piece of this plastic material, it is as iÍ we were +pulling a whole tangle of strings, and there will be more strings preferentially +aligned parallel to the tension than in any other direction. So when a stress 1s +applied to certain plastics, they become birefringent, and one can see the efects +of the birefringence by passing polarized light through the plastic. If we examine +the transmitted light through a polaroid sheet, patterns of light and dark fringes +will be observed (in color, if white light is used). The patterns move as stress is +applied to the sample, and by counting the fringes and seeing where most of them +are, one can determine what the stress is. Engineers use this phenomenon as a +means of finding the stresses in odd-shaped pieces that are difficult to calculate. +Another interesting example of a way of obtaining birefringence is by means +of a liquid substance. Consider a liquid composed of long asymmetric molecules +which carry a plus or minus average charge near the ends of the molecule, so that +the molecule is an electric dipole. In the collisions in the liquid the molecules will +ordinarily be randomly oriented, with as many molecules pointed in one direction +as in another. If we apply an electric fñeld the molecules will tend to line up, +and the moment they line up the liquid becomes birefringent. With two polaroid +sheets and a transparent cell containing such a polar liquid, we can devise an +arrangement with the property that light is transmitted only when the electric +fñeld is applied. So we have an electrical switch for light, which is called a Kerr +ccli. 'This efect, that an electric field can produece birefringence in certain liquids, +is called the Kerr efect. +33-4 Polarizers +So far we have considered substances in which the refractive index is diferent +for light polarized in diferent directions. Of very practical value are those crystals +and other substances in which not only the index, but also the coefficient of +absorption, 1s diferent for light polarized in diferent directions. By the same +arguments which supported the idea of birefringence, it is understandable that +absorption can vary with the direction in which the charges are forced to vibrate +in an anisotropic substance. Tourmaline is an old, famous example and polaroid +is another. Polaroid consists of a thin layer oŸ small crystals of herapathite (a +salt oŸ iodine and quinine), all aligned with their axes parallel. 'These crystals +absorb light when the oscillations are in one direction, and they do not absorb +appreciably when the oscillations are in the other direction. +--- Trang 577 --- +Suppose that we send light into a polaroid sheet polarized linearly at an +angle Ø to the passing direction. What intensity will come through? This incident +light can be resolved into a component perpendicular to the pass direction which +1s proportional to sinØ, and a component along the pass direction which 1s +proportional to cosØ. “The amplitude which comes out of the polaroid is only +the cosine Ø part; the sin Ø component is absorbed. The amplitude which passes +through the polaroid is smaller than the amplitude which entered, by a factor cos Ø. +'The energy which passes through the polaroid, i.e., the intensity of the light, 1s +proportional to the square of cos Ø. Cos? 6, then, is the intensity transmitted +when the light enters polarized at an angle Ø to the pass direction. The absorbed +intensity, of course, is sỉn? 0. +An interesting paradox is presented by the following situation. We know that +1E is not possible to send a beam oŸ light through ©wo polaroid sheets with their +axes crossed at right angles. But if we place a third polaroid sheet betueen the +first two, with Its pass axis at 45° to the crossed axes, some light is transmitted. +W© know that polaroid absorbs light, it does not create anything. Nevertheless, +the addition of a third polaroid at 45° allows more light to get through. 'Phe +analysis of this phenomenon is left as an exercise for the student. +One of the most interesting examples of polarization is not in complicated +crystals or dificult substances, but in one of the simplest and most familiar of +situations—the refection of light from a surface. Believe it or not, when light is +refected om a glass surface it may be polarized, and the physical explanation of +this is very simple. It was discovered empirically by Brewster that light reflected +from a surface is completely polarized if the reflected beam and the beam refracted +into the material form a right angle. The situation is illustrated in Eig. 33-4. If +the ineident beam is polarized in the plane of incidence, there wiïll be no refection +at all. Only ïf the incident beam is polarized normal to the plane of ineidenece will +1E be refected. The reason is very easy to understand. In the reflecting material +the light is polarized transversely, and we know that it is the motion of the charges +in the material which generates the emergent beam, which we call the refected +beam. 'Phe source of this so-called reflected light is not simply that the incident +beam is reflected; our deeper understanding of this phenomenon tells us that the +ineident beam drives an oscillation of the charges in the material, which in turn +generates the reflected beam. FErom Eig. 33-4 it is clear that only oscillations +normal to the paper can radiate in the direction of refection, and consequently the +refected beam will be polarized normal to the plane of incidence. If the ineident +beam is polarized in the plane of ineidence, there will be no refected light. +--- Trang 578 --- +¡ -Q08 ⁄Z +I + # +Fig. 33-4. Reflection of linearly polarized light at Brewster's angle. +The polarization direction ¡s indicated by dashed arrows; round dots +Iindicate polarization normal to the paper. +This phenomenon 1s readily demonstrated by reflecting a linearly polarized +beam from a flat piece of glass. lf the glass is turned to present diferent angles +of incidenee to the polarized beam, sharp attenuation of the refected intensity +is observed when the angle of inecidence passes through Brewster s angle. This +attenuation is observed only if the plane of polarization lies in the plane of +Incidenee. Tf the plane of polarization is normal to the plane of incidence, the +usual refected intensity is observed at all angles. +33-5 Optical activity +Another most remarkable efect of polarization is observed in materials com- +posed of molecules which do not have refection symmetry: molecules shaped +something like a corkscrew, or like a gloved hand, or any shape which, if viewed +through a mirror, would be reversed in the same way that a left-hand glove +reflects as a right-hand glove. Suppose all of the molecules in the substance are +the same, I.e., none is a mirror image of any other. Such a substance may show +an interesting efect called optical activity, whereby as linearly polarized light +passes through the substance, the direction of polarization rotates about the +beam axis. +To understand the phenomenon of optical activity requires some calculation, +but we can see qualitatively how the efect might come about, without actually +carrying out the calculations. Consider an asymmetric molecule in the shape +of a spiral, as shown in Eig. 33-5. Molecules need not actually be shaped like a +corkscrew in order to exhibit optical activity, but this is a simple shape which +--- Trang 579 --- +ị Z2 ⁄ Ex +01 zZ ZI+A " +Fig. 33-5. A molecule with a shape that ¡is not symmetric when +reflected in a mirror. A beam of light, linearly polarized in the y-direction, +falls on the molecule. +we shall take as a typical example of those that do not have reflection symmetry. +'When a light beam linearly polarized along the z-direction falls on this molecule, +the electric field will drive charges up and down the helix, thereby generating +a current in the z-direction and radiating an electric field l„ polarized in the +u-direction. However, if the electrons are constrained to move along the spiral, +they must also move in the z-direction as they are driven up and down. When +a current is ñowing up the spiral, it is also Ñowing into the paper at z = z1 +and out of the paper at z = z¡ + A, if A ¡is the diameter of our molecular spiral. +One might suppose that the current in the z-direction would produce no net +radiation, since the currents are in opposite directions on opposite sides of the +spiral. However, if we consider the zø-components of the electric field arriving +at z = zs, we see that the field radiated by the current at z = z¡ + A and the +fñeld radiated from z = z¡ arrive at z¿ separated in time by the amount A/c, +and thus separated in phase by + œ4/c. Since the phase difference is not +exactly r, the two fields do not cancel exactly, and we are left with a small +#-component in the electric fñeld generated by the motion of the electrons in the +molecule, whereas the driving electric fñeld had only a -component. This small +#-component, added to the large -component, produces a resultant field that is +tilted slightly with respect to the -axis, the original direction of polarization. +As the light moves through the material, the direction of polarization rotates +about the beam axis. By drawing a few examples and considering the currents +that will be set in motion by an incident electric fñield, one can convince himself +that the existence of optical activity and the sign of the rotation are independent +of the orientation of the molecules. +Corn syrup is a common substance which possesses optical activity. The +phenomenon is easily demonstrated with a polaroid sheet to produee a linearly +polarized beam, a transmission cell containing corn syrup, and a second polaroid +--- Trang 580 --- +sheet to detect the rotation of the direction of polarization as the light passes +through the corn syrup. +33-6 The intensity of reflected light +Let us now consider quantitatively the relection coeffcient as a function of +angle. Pigure 33-6(a) shows a beam of light striking a glass surface, where it +1s partly reflected and partly refracted into the glass. Let us suppose that the +incident beam, of unit amplitude, is linearly polarized normal to the plane of the +paper. We will call the amplitude of the refected wave b, and the amplitude of +the refracted wave ø. The refracted and reflected waves will, of course, be linearly +polarized, and the electric feld vectors of the incident, reflected, and refracted +waves are all parallel to each other. Pigure 33-6(b) shows the same situation, but +now we suppose that the incident wave, of unit amplitude, is polarized in the +plane of the paper. Now let us call the amplitude of the refected and refracted +wave Ö and A, respectively. +We wish 6o calculate how strong the refection is in the two situations illus- +trated in Fig. 33-6(a) and 33-6(b). We already know that when the angle bebtween +the relected beam and refracted beam is a right angle, there will be no reflected +wave in Fig. 33-6(b), but leb us see if we cannot get a quantitative answer——an +exact formula for Ö and ö as a function of the angle of incidence, ¿. +The principle that we must understand is as follows. The currents that are +generated in the glass produce two waves. First, they produce the reflected wave. +b —1 B —1 +` a `v v \ A +_ _\*< | +1 Glass 1 Glass +(a) (@) +Fig. 33-6. An incident wave of unit amplitude ¡s reflected and refracted +at a glass surface. In (a) the incident wave is linearly polarized normal +to the plane of the paper. In (b) the incident wave is linearly polarized +In the direction shown by the dashed arrows. +--- Trang 581 --- +Moreover, we know that if there were no currents generated in the glass, the +incident wave would continue straight into the glass. Remember that all the +sources in the world make the net field. The source of the incident light beam +produces a field of unit amplitude, which would move into the glass along the +dotted line in the fgure. 'This field is not observed, and therefore the currents +generated in the glass must produce a fñeld of amplitude —1, which moves along +the dotted line. Ủsing this fact, we will caleulate the amplitude of the refracted +waves, ø and 4. +In Eig. 33-6(a) we see that the field of amplitude ð is radiated by the motion +of charges Inside the glass which are responding to a field ø inside the glass, and +that therefore b is proportional to a. We might suppose that since our two fñgures +are exactly the same, except for the direction of polarization, the ratio B/A +would be the same as the ratio b/a. 'This is not quite true, however, because in +Fig. 35-6(b) the polarization directions are not all parallel to each other, as they +are in Fig. 33-6(a). It is only the component of A which is perpendicular to Ö, +Acos (¿ + r), which is efective in producing Ø. The correct expression for the +proportionality is then +a Acos(i+r). 33.1) +Now we use a trick. We know that in both (a) and (b) of Fig. 33-6 the electric +field in the glass must produce oscillations of the charges, which generate a field +of amplitude —1, polarized parallel to the incident beam, and moving in the +direction of the dotted line. But we see from part (b) of the figure that only +the component of 4 that is normal to the dashed line has the right polarization +to produce this field, whereas in Eig. 33-6(a) the full amplitude ø is efective, +since the polarization of wave ø is parallel to the polarization of the wave of +amplitude —1. 'Therefore we can write +A cos (¿ — 7) — _1 (33.2) +since the two amplitudes on the left side of Eq. (33.2) each produce the wave of +amplitude —1. +Dividing Eq. (38.1) by Bq. (33.2), we obtain +B _ cos ứ + n: (33.3) +b — cos(—r} +--- Trang 582 --- +a result which we can check against what we already know. IÝ we set (¿-+z) = 909, +Edq. (33.3) gives =0, as Brewster says it should be, so our results so far are at +least not obviousÌly wrong. +We have assumed unit amplitudes for the incident waves, so that ||2/12 is +the reflection coefficient for waves polarized in the plane of ineidence, and |b|2/12 +1s the refection coefficient for waves polarized normal to the plane of incidenee. +The ratio of these two reflection coefficients is determined by Eq. (33.3). +Now we perform a miracle, and compute not just the ratio, but each coefficlent +||? and |b|? individually! We know from the conservation of energy that the +energy in the reracted wave must be equal to the incident energy minus the energy +in the refected wave, 1 — ||? in one case, 1 — |b|? in the other. Purthermore, +the energy which passes into the glass in Fig. 33-6(b) is to the energy which +passes into the glass in Fig. 33-6(a) as the ratio of the squares of the refracted +amplitudes, |A|?/|a|?. One might ask whether we really know how to compute +the energy inside the glass, because, after all, there are energies of motion of the +atoms in addition to the energy in the electric fñeld. But it is obvious that all of +the various contributions to the total energy will be proportional to the square +of the amplitude of the electric feld. 'Pherefore we can write +1—|BIÊ - |Al2 +T— BE T lajP” (33.4) +We now substitute Eq. (33.2) to eliminate A/ø from the expression above, +and express Ö in terms of b by means of Eq. (33.3): +. co ự +r) +coS (¿ — r) — b (33.5) +1— ||? cosZ (2 — r) +This equation contains only one unknown amplitude, b. Solving for |b|2, we +obtain +b2 = Z8 ữ—") (33.6) +sin“ (2 + r) +and, with the aid of (33.3), +IBẸ = PHUE—T), (33.7) +tan (2 + r) +--- Trang 583 --- +So we have found the reflection coefficient |b|? for an incident wave polarized +perpendicular to the plane of ineidenee, and also the reflection coefficient |B|2 +for an incident wave polarized in the plane of incidencel +Tt is possible to go on with arguments of this nature and deduce that 0 is real. +To prove this, one must consider a case where light is coming from both sides of +the glass surface at the same tỉme, a situation not easy to arrange experimentally, +but fun to analyze theoretically. If we analyze this general case, we can prove +that b must be real, and therefore, in fact, that b = +sin (¿ — r)/sin (¿ + r). It +is even possible to determine the sign by considering the case OŸ a very, Very +thin layer in which there is relection from the front and from the back surfaces, +and calculating how mụuch light is reflected. We know how much light should be +reflected by a thin layer, because we know how much current is generated, and +we have even worked out the fields produced by such currents. +One can show by these arguments that +,— SnU—r) u_ tan r). (33.8) +sin (¿+ r) tan (2+ 7) +These expressions for the relection coefficients as a function of the angles of +incidence and refraction are called Eresnel”s refection formulas. +Tf we consider the limit as the angles ? and z go to zero, we fñnd, for the case +of normal ineidenee, that 2 b2 (¿— r)2/(¡+r) for both polarizations, since +the sines are practically equal to the angles, as are also the tangents. But we +know that sin2/sin? = ø, and when the angles are small, ¿/r 2ø. It is thus easy +to show that the coefflicient of refection for normal incidenece is +B2ˆ—?— (n — Dã +(6+1)? +Tt is interesting to ñnd out how mụuch light is reflected at normal incidence +from the surface oŸ water, for example. Eor water, ø is 4/3, so that the reflection +coefficient is (1/7)2 ~ 2%. At normal incidence, only two percent of the light is +refected from the surface oŸ water. +33-7 Anomalous refraction +The last polarization elfect we shall consider was actually one of the first to be +discovered: anomalous refraction. Sailors visiting Iceland brought back to Burope +--- Trang 584 --- +crystals of Iceland spar (CaCOs) which had the amusing property of making +anything seen through the crystal appear doubled, I.e., as two images. 'Phis came +to the attention of Huygens, and played an important role in the discovery of +polarization. As is often the case, the phenomena which are discovered first are +the hardest, ultimately, to explain. It is only after we understand a physical +concept thoroughly that we can carefully select those phenomena which most +clearly and simply demonstrate the concept. +axis Ÿ H +Fig. 33-7. The upper diagram shows the path of the ordinary ray +through a doubly refracting crystal. The extraordinary ray Is shown In +the lower diagram. The optic axIs lies in the plane of the paper. +Anomalous refraction is a particular case of the same birefringence that we +considered earlier. Anomalous refraction comes about when the optic axis, the +long axis of our asymmetric molecules, is no parallel to the surface of the crystal. +In Eig. 33-7 are drawn two pieces of birefringent material, with the optic axis as +shown. In the upper figure, the inecident beam falling on the material is linearly +polarized in a direction perpendicular to the optic axis of the material. When +this beam strikes the surface of the material, each point on the surface acts as a +source oŸ a wave which travels into the crystal with velocity ø¡, the velocity of +light in the crystal when the plane of polarization is normal to the optic axis. +The wavefront is Just the envelope or locus of all these little spherical waves, and +this wavefront moves straight through the crystal and out the other side. 'This is +--- Trang 585 --- +Just the ordinary behavior we would expect, and this ray ¡s called the ordinar +In the lower fñgure the linearly polarized light falling on the crystal has its +direction of polarization turned through 907, so that the optic axis lies in the +plane of polarization. When we now consider the little waves originating at any +point on the surface of the crystal, we see that they do not spread out as spherical +waves. Light travelling along the optic axis travels with velocity ø¡ because +the polarization is perpendicular to the optic axis, whereas the light travelling +perpendicular to the optic axis travels with velocity 0i because the polarizatlon +is parallel to the optic axis. In a birefringent material 0 z# 0¡, and in the figure +0Ịị < 0L. Á more complete analysis will show that the waves spread out on the +surface of an ellipsoid, with the optic axis as major axis of the ellipsoid. “The +envelope of all these elliptical waves 1s the wavefront which proceeds through +the crystal in the đirection shown. Again, at the back surface the beam will be +defected just as it was at the front surface, so that the light emerges parallel +to the incident beam, but displaced from it. Clearly, this beam does not follow +Snells law, but goes in an extraordinary direction. It ¡is therefore called the +cztraordinar4J ray, +'When an unpolarized beam strikes an anomalously refracting crystal, i% is +separated into an ordinary ray, which travels straight through in the normal +mamner, and an extraordinary ray which is displaced as it passes through the +crystal. 'Phese two emergent rays are linearly polarized at right angles to each +other. 'Phat this is true can be readily demonstrated with a sheet of polaroid +to analyze the polarization of the emergent rays. We can also demonstrate that +our interpretation of this phenomenon 1s correct by sending linearly polarized +light into the crystal. By properly orienting the direction of polarization of the +incident beam, we can make this light go straight through without splitting, or +we can make it go through without splitting but with a displacement. +W© have represented all the various polarization cases in Figs. 33-I and 33-2 +as superpositions of two special polarization cases, namely + and ø in various +amounts and phases. Other pairs could equally well have been used. Polarization +along any two perpendicular axes 4, ˆ inclined to #z and would serve as well [for +example, any polarization can be made up of superpositions of cases (a) and (e) +of Eig. 33-2]. It is interesting, however, that this idea can be extended to other +cases also. For example, any neør polarization can be made up by superposing +suitable amounts at suitable phases of right and left c#cular polarizations [cases +(c) and (g) of Fig. 33-2], since two equal vectors rotating in opposite directions +--- Trang 586 --- +Fig. 33-8. Two oppositely rotating vectors of equal amplitude add to +produce a vector In a fixed direction, but with an oscillating amplitude. +add to give a single vector oscillating in a straight line (Eig. 33-6). TỶ the phase +of one is shifted relative to the other, the line is inclined. 'Thus all the pictures +of Eig. 33-1 could be labeled “the superposition of equal amounts of right and +left circularly polarized light at various relative phases.” As the left slips behind +the right in phase, the direction of the linear polarization changes. 'Therefore +optically active materials are, in a sense, birefringent. Their properties can be +described by saying that they have diferent indexes for right- and left-hand +circularly polarized light. Superposition of right and left circularly polarized light +of diferent intensities produces elliptically polarized light. +Circularly polarized light has another interesting property—it carries øngulœr +momentum (about the direction of propagation). To illustrate this, suppose that +such light falls on an atom represented by a harmonic oscillator that can be +displaced equally wellin any direction in the plane ø. Then the z-displacement of +the electron will respond to the #„ component of the feld, while the -component +responds, equally, to the equal 2 component of the fñield but 90 behind in phase. +That is, the responding electron goes around in a circle, with angular velocity œ, +in response to the rotating electric field of the light (Fig. 33-9). Depending on +the damping characteristics of the response of the oscillator, the direction of the +displacement œ of the electron, and the direction of the force q¿#⁄ on it need not +be the same but they rotate around together. The # may have a component at +right angles to ø, so work is done on the system and a torque 7 is exerted. The +work done per second is 7w. Ôver a period of time 7' the energy absorbed is 7uT,, +while 77' is the angular momentum delivered to the matter absorbing the energy. +We see therefore that œ bewm oƒ right círcularlụ polarized light contaimimng a total +--- Trang 587 --- +Fig. 33-9. A charge moving In a circle in response to circularly polarized +light. +energu Ê carries an œnguÏar mmormnentum (uuith 0ector dárected œlong the đireclion +oƒ propagation) Ê/œ. For when this beam is absorbed that angular momentum is +delivered to the absorber. Left-hand circular light carries angular momentum of +the opposite sign, —Ê/œ. +--- Trang 588 --- +Miolqfitrsffc ifocés rrẻ Haclf(ffort +34-1 Moving sources +In the present chapter we shall describe a number of miscellaneous efects +in connection with radiation, and then we shall be finished with the classical +theory of light propagation. In our analysis of light, we have gone rather far and +Into considerable detail. "The only phenomena of any consequence associated +with electromagnetic radiation that we have not discussed is what happens If +radiowaves are contained in a box with reflecting walls, the size of the box +being comparable to a wavelength, or are transmitted down a long tube. The +phenomena of so-called cauify resonators and uaueguzdes we shal] discuss later; +we shall frst use another physical example—sound——=and then we shall return to +this subject. Except for this, the present chapter is our last consideration of the +classical theory of light. +W© can summarize all the efects that we shall now discuss by remarking that +they have to do with the effects OŸ rmouing sources. We no longer assume that +the source is localized, with all its motion beïng at a relatively low speed near a +fñxed point. +We recall that the fundamental laws of electrodynamics say that, at large +distances from a moving charge, the electric field is given by the formula +q d2e R- +J— _.nn (34.1) +'The second derivative of the unit vector ep; which points in the apparent direction +of the charge, is the determining feature of the electric fñeld. 'This unit vector +does not point toward the presen£ position of the charge, of course, but rather in +the direction that the charge would seem to be, if the information travels only at +the fñnite speed c from the charge to the observer. +--- Trang 589 --- +Associated with the electric feld is a magnetic feld, always at right angles +to the electric fñeld and at right angles to the apparent direction of the source, +given by the formula +b= —€©hP:X Hực. (34.2) +Until now we have considered only the case in which motions are nonrelativistic +in speed, so that there is no appreciable motion in the direction of the source +to be considered. Now we shall be more general and study the case where the +motion is at an arbitrary velocity, and see what different efects may be expected +in those cireumstances. We shall let the motion be at an arbitrary speed, but of +course we shall still assumne that the detector is very far from the source. +y_ (xứ) +ep! z(r) +P — 6A x +Fig. 34-1. The path of a moving charge. The true position at the +time 7 Is at Ï, but the retarded position Is at A. +W© already know from our discussion in Chapter 28 that the only things that +count in d2ep/đ‡2 are the changes in the đirection of en;. Let the coordinates +of the charge be (z,,2z), with z measured along the direction of observation +(Fig. 34-1). At a given moment in tỉme, say the moment 7, the three components +of the position are #(7), (7), and z(7). The distance ?? is very nearly equal +to fr) = eo + z(r). Ñow the direction of the vector eq; depends mainly on # +and ø, but hardly at all upon z: the transverse components of the unit vector are +z/R and /R, and when we diferentiate these components we get things like #2 +in the denominator: +d(4/R) — dz/dL dz + +dc hR dị R2 ` +So, when we are far enough away the only terms we have to worry about are the +--- Trang 590 --- +variations of z and ø. Thus we take out the factor Ïọ and get +m—...., +4mcoc2Ro_ di2 +_ _ 4megc2Rg dị2 ` (343) +where #ọ is the distance, more or less, to g; let us take it as the distance @?P to the +origin of the coordinates (z,,z). Thus the electric field is a constant multiplied +by a very simple thing, the second derivatives oŸ the z- and -coordinates. (We +could put it more mathematically by calling z and the fransuerse components +of the position vector ? of the charge, but this would not add to the clarity.) +Of course, we realize that the coordinates must be measured at the retarded +tỉme. Here we find that z(7) does afect the retardation. What tỉme is the +retarded time? Tf the time of observation is called ¿ (the time at P) then the +tỉme 7 to which this corresponds at A is not the time ý, but ¡is delayed by the +total distance that the light has to go, divided by the speed of light. In the +frst approximation, this delay is ffo/c, a constant (an uninteresting feature), +but in the next approximation we must include the efects of the position in the +z-direction at the time 7, because 1Ý g is a little farther back, there is a little +more retardation. 'This is an efect that we have neglected before, and ït is the +only change needed in order to make our results valid for all speeds. +What we must now do is to choose a certain value of £ and calculate the value +of r from it, and thus fnd out where z and ø are at that 7. These are then the +retarded z and ø, which we call z“ and z⁄, whose second derivatives debermine +the fñeld. 'Thus 7 is determined by +t=T+ Ro + zữ) +z() =z(). — w()=(). (31.4) +Now these are complicated equations, but it is easy enough to make a geometrical +picture to describe theïr solution. 'Phis picture will give us a good qualitative +feeling for how things work, but ít still takes a lot of detailed mathematics to +deduce the precise results of a complicated problem. +--- Trang 591 --- +x x'(t) +¬— c TÊN +'TO OBSERVER 0 +Fig. 34-2. A geometrical solution of Eq. (34.5) to find x/(£). +34-2 Einding the “apparent” motion +The above equation has an interesting simplifcation. If we disregard the +uninteresting constant delay ?2o/c, which just means that we must change the +origin of ý by a constant, then i% says that +cÈ = œT + Z(T), z' = #(T), ự =9(1). (34.5) +NÑow we need to fnd zˆ and ø as functions of f, not 7, and we can do this in +the following way: Ed. (34.5) says that we should take the actual motion and +add a constant (the speed of light) times 7. What that turns out to mean is +shown in Fig. 34-2. We take the actual motion of the charge (shown at left) and +imagine that as it is going around it is being swept away from the point ? at +the speed e (there are no contractions from relativity or anything like that; this +is Just a mathematical addition of the cr). In this way we get a new motion, +in which the line-ofsight coordinate is cý, as shown at the right. (The figure +shows the result for a rather complicated motion ¡in a plane, but of course the +motion may not be in one plane—it may be even more complicated than motion +in a plane.) The point is that the horizontal (¡.e., line-of-sight) distance now is +no longer the old z, but is z + cr, and therefore is cứ. Thus we have found a +picture of the curve, #“ (and #') against ¿l All we have to do to fnd the field +1s to look at the acceleration of this curve, 1.e., to diferentiate I% twice. So the +fnal answer is: in order to find the electric feld for a moving charge, take the +motion of the charge and translate it back at the speed e to “open it out”; then +the curve, so drawn, is a curve of the #” and z positions of the function of¿. The +acceleration of this curve gives the electric field as a function of ý. Ôr, If we wish, +we can now imagine that this whole “rigid” curve moves forward at the speed +--- Trang 592 --- +x x'(t) +Fig. 34-3. The x/(f) curve for a particle moving at constant speed v = +0.94c ¡in a circle. +c throupgh the plane of sight, so that the point oŸ intersection with the plane of +sight has the coordinates zø“ and ø⁄. The acceleration of this point makes the +electric field. 'This solution is just as exact as the formula we started with—it is +simply a geometrical representation. +TÍ the motion is relatively slow, for instance if we have an oscillator just goïng +up and down slowly, then when we shoot that motion away at the speed of light, +we would get, of course, a simple cosine curve, and that gives a formula we have +been looking at for a long tỉme: it gives the ñeld produced by an oscillating charge. +A more interesting example is an electron moving rapidly, very nearly at the +speed of light, in a cirele. I we look in the plane of the cirele, the retarded z(£) +appears as shown in Fig. 34-3. What is this curve? If we imagine a radius vector +trom the center of the circle to the charge, and iŸ we extend this radial line a +little bit past the charge, just a shade If it is going fast, then we come to a point +on the line that goes at the speed of light. "Therefore, when we translate the +motion back at the speed of light, that corresponds to having a wheel with a +charge on it rolling backward (without slipping) at the speed é; thus we fnd a +curve which is very close to a cycloid—it is called a curfate cụcloid. TẾ the charge +1s going very nearly at the speed of light, the “cusps” are very sharp indeed; ïf it +went at exactly the speed of light, they would be actual cusps, infinitely sharp. +“Inũnitely sharp” is interesting; it means that near a cusp the second derivative +1s enormous. Ônce in each cycle we get a sharp pulse of electric field. 'This is +not at all what we would get from a nonrelativistic motion, where each tỉme the +charge goes around there is an oscillation which is of about the same “strength” +all the time. Instead, there are very sharp pulses of electric fñeld spaced at time +Intervals Tạ apart, where 7§ is the period of revolution. 'Phese strong electric +--- Trang 593 --- +fñelds are emitted ïn a narrow cone in the direction of motion oŸ the charge. When +the charge is moving away from ?, there is very little curvature and there is very +little radiated fñeld in the direction of P. +34-3 Synchrotron radiation +We have very fast electrons moving in circular paths in the synchrotron; they +are travelling at very nearly the speed e, and it is possible to see the above +radiation as actual l2ghfl Let us discuss this in more detail. +In the synchrotron we have electrons which go around in cireles in a uniform +magnetic feld. First, let us see why they go in circles. From Eq. (28.2), we know +that the force on a particle in a magnetic field is given by +F—=quxÖ, (34.6) +and ït is at right angles both to the ñeld and to the velocity. As usual, the force is +equal to the rate of change of momentum with time. If the field is directed upward +out of the paper, the momentum of the particle and the force on it are as shown In +Fig. 34-4. 5ince the force is at right angles to the velocity, the kinetic energy, and +therefore the speed, remains consfan. All the magnetic fñeld does is to change +the đirecHon oƒ motion. In a short tìme A#, the momentum vector changes at +right angles to itself by an amount Ấp = #' At, and therefore ø turns through +an angle AØ = App = quB At/p, since |F'| = quB. But in this same time the +particle has gone a distance As = 0u Af. Evidently, the two lines 4Ø and ŒD vill +intersect at a point Ó such that @A = OŒ = l, where As = l¿A0. Combining +Ap ” ¬¬ +A ..... `... +Fig. 34-4. A charged particle moves in a circular (or helical) path in a +uniform magnetic field. +--- Trang 594 --- +this with the previous expressions, we fnd /tA0/At = Rœ = 0u = quBRijp, trom +which we fnd +p=ụqBh (34.7) +œ = quBịjp. (34.8) +Since this same argument can be applied during the next instant, the next, and +so on, we conclude that the particle must be moving in a c#cle of radius #, with +angular velocity œ. +'The result that the momentum of the particle is equal to a charge times the +radius times the magnetic field is a very important law that is used a great deal. +lt is important for practical purposes because If we have elementary particles +which all have the same charge and we observe them in a magnetic fñeld, we +can measure the radii of curvature of their orbits and, knowing the magnetic +fñeld, thus determine the momenta. of the particles. If we multiply both sides of +Eq. (34.7) by e, and express g in terms of the electronic charge, we can measure +the momentum in units of the elecfron uoχ. In those units our formula 1s +pc(eV) = 3 x 10Ề(q/q.)BR, (34.9) +where 7Ø, #, and the speed of light are all expressed in the mks system, the latter +being 3 x 10Ẻ, numerically. +The mks unit of magnetic field is called a ueber per square mecter. There 1s +an older unit which is still in common use, called a øœuss. One weber/1m is +equal to 10 gauss. To give an idea of how big magnetie fields are, the strongest +magnetic feld that one can usually make in iron is about 1.5 x 10 gauss; beyond +that, the advantage of using iron disappears. Today, electromagnets wound +with superconducting wire are able to produee steady fields of over 105 gauss +strength—that is, 10 mks units. The field of the earth is a few tenths of a øauss +at the equator. +Returning to Bq. (34.9), we could imagine the synchrotron running at a billion +electron volts, so øe would be 102 for a billion electron volts. (We shall come +back to the energy in just a moment.) Then, if we had a Ö corresponding to, say, +10,000 gauss, which is a good substantial field, one mks unit, then we see that +would have to be 3.3 meters. The actual radius of the Caltech synchrotron 1s +3.7 meters, the field is a little bigger, and the energy is 1.5 billion, but i§ is the +same Iidea. 5o now we have a feeling for why the synchrotron has the size it has. +--- Trang 595 --- +W©e have calculated the momentum, but we know that the total energy, +including the rest energy, is given by W = 4⁄ø2c2 + rn2c, and for an electron +the rest energy corresponding to me? is 0.511 x 108 eV, so when øe is 109 eV +we can neglect me”, and so for all practical purposes W = pe when the speeds +are relativistic. It is practically the same to say the energy of an electron is a +bilion electron volts as to say the momentum times e is a billion electron volts. +If W = 10 eV, it is easy to show that the speed differs from the speed of light +by but one part in eight million! +W© turn now to the radiation emitted by such a particle. A particle moving +on a circle of radius 3.3 meters, or 20 meters circumference, øoes around once +in roughly the time it takes light to go 20 meters. So the wavelength that +should be emitted by such a particle would be 20 meters—in the shortwave +radio region. But because of the piling up efect that we have been discussing +(Fig. 34-3), and because the distance by which we must extend the radius to +reach the speed e is only one part in eight million of the radius, the cusps of the +curtate cyeloid are enormously sharp compared with the distance between them. +The acceleration, which involves a second derivative with respect to time, gets +twice the “compression factor” of 8 x 108 because the time scale is reduced by +eight million twice in the neighborhood of the cusp. 'hus we might expect the +efective wavelength to be much shorter, to the extent of 64 times 1012 smaller +than 20 meters, and that corresponds to the x-ray region. (Actually, the cusp +1tself is not the entire determining factor; one must also include a certain region +about the cusp. Thịs changes the factor to the 3/2 power instead of the square, +but still leaves us above the optical region.) Thus, even though a sÌlowly moving +electron would have radiated 20-meter radiowaves, the relativistic efect cuts +down the wavelength so much that we can see itl Clearly, the light should be +polarizcd, with the electric fñeld perpendieular to the uniform magnetic field. +To further appreciate what we would observe, suppose that we were to take +such light (to simplify things, because these pulses are so far apart in time, we +shall just take one pulse) and direct it onto a difraction grating, which is a lot +Of scattering wires. After this pulse comes away from the grating, what do we +see? (We should see red light, blue light, and so on, if we see any light at all.) +What đo we see? The pulse strikes the grating head-on, and all the oscillators +in the grating, together, are violently moved up and then back down again, just +once. 'Phey then produce efects in various directions, as shown in Eig. 34-5. But +the point ? ¡s closer to one end of the grating than to the other, so at this point +the electric field arrives frst from wire 4, next from ?Ö, and so on; finally, the +--- Trang 596 --- +° [“— Pulse from electron +A» ~„ Radiation scattered +by grating +Fig. 34-5. The light which strikes a grating as a single, sharp pulse is +scattered in various directions as different colors. +pulse from the last wire arrives. In short, the sum of the refections from all the +successive wires is as shown in Eig. 34-6(a); it is an electric field which is a series +oŸ pulses, and it is very like a sine wave whose wavelength is the distance bebween +the pulses, just as it would be for monochromatic light striking the gratingl So, +we get colored light all right. But, by the same argument, will we not get light +from any kind of a “pulse”? No. Suppose that the curve were much smoother; +then we would add all the scattered waves together, separated by a small time +between them (Eig. 34-6b). Then we see that the feld would not shake at all, +it would be a very smooth curve, because each pulse does not vary muụch in the +time interval between pulses. +NHA -— 7m +(a) (b) +Fig. 34-6. The total electric field due to a series of (a) sharp pulses +and (b) smooth pulses. +The electromagnetic radiation emitted by relativistic charged particles cir- +culating in a magnetic fñeld is called sựnchrotron radiation. Tt 1s so named for +obvious reasons, but it is not limited specifically to synchrotrons, or even to +earthbound laboratories. It is exciting and interesting that it also occurs in +naturel +--- Trang 597 --- +M mx~. : +` sa ' . ++ ° * +Fig. 34-7. The crab nebula as seen in all colors (no filter). +34-4 Cosmic synchrotron radiation +In the year 1054 the Chinese and Japanese civilizations were among the most +advanced in the world; they were conscious oŸ the external universe, and they +recorded, most remarkably, an explosive bright star in that year. (Ït is amazing +that none of the European monks, writing all the books of the middle ages, even +bothered to write that a star exploded in the sky, but they did not.) Today we +may take a picture of that star, and what we see is shown in Eig. . Ôn the +outside is a big mass of red fñlaments, which is produced by the atoms of the thin +gas “ringing” at their natural frequencies; this makes a bright line spectrum with +diferent frequencies in it. The red happens in this case to be due to nitrogen. +On the other hand, in the central region is a mysterious, fuzzy patch of light +in a contimuous distribution of frequency, 1.e., there are no special frequencies +associated with particular atoms. Yet this is not dust “lit up” by nearby stars, +which is one way by which one can get a continuous spectrum. We can see stars +through ït, so it is transparent, but it is emiiting light. +--- Trang 598 --- +# : "M bo +(a) (b) +Fig. 34-8. The crab nebula as seen through a blue filter and a polaroid. +(a) Electric vector vertical. (b) Electric vector horizontal. +In Eig. 34-8 we look at the same object, using light in a region of the spectrum +which has no bright spectral line, so that we see only the central region. But +in this case, also, polarizers have been put on the telescope, and the two views +correspond to two orientations 909 apart. We see that the pictures are diferentl +That is to say, the light is polarized. The reason, presumably, is that there is a +local magnetic field, and many very energetic electrons are going around ïn that +magnetic field. +W© have just illustrated how the electrons could go around the fñeld in a cirele. +W© can add to thís, of course, any uniform motion in the direction of the fñield, +since the force, gu x #Ö, has no component in this direction and, as we have +already remarked, the synchrotron radiation is evidentÌy polarized in a direction +at right angles to the projection of the magnetic feld onto the plane of sight. +Putting these two facts together, we see that in a region where one picEure is +bright and the other one is black, the light must have its electric fñeld completely +polarized in one direction. This means that there is a magnetic field at right +angles to this direction, while in other regions, where there is a strong emission In +the other picture, the magnetic feld must be the other way. If we look carefully +at Eig. 34-8, we may notice that there is, roughly speaking, a general set of +“lines” that go one way in one picture and at right angles to this in the other. +The pictures show a kind of fibrous structure. Presumably, the magnetic feld +lines will tend to extend relatively long distances in their own direction, and +--- Trang 599 --- +so, presumably, there are long regions of magnetic field with all the electrons +spiralling one way, while in another region the field is the other way and the +electrons are also spiralling that way. +What keeps the electron energy so high for so long a time? After all, it +is 900 years since the explosion—how can they keep going so fast? How they +maintain their energy and how this whole thing keeps goïng is still not thoroughly +understood. +34-5 Bremsstrahlung +W© shall next remark briefly on one other interesting efect of a very fast- +moving particle that radiates energy. The idea 1s very similar to the one we have +Just discussed. Suppose that there are charged particles in a piece of matter and +a very fast electron, say, comes by (EFig. 34-9). Then, because of the electric ñeld +around the atomic nucleus the electron is pulled, accelerated, so that the curve +of its motion has a slight kink or bend ïn it. If the electron is travelling at very +nearly the speed of light, what is the electric fñeld produced in the direction C7? +Remember our rule: we take the actual motion, translate it backwards at speed c, +and that gives us a curve whose curvature measures the electric feld. It was +coming toward us at the speed 0, so we get a backward motion, with the whole +picture compressed into a smaller distance in proportion as e— 0 is smaller than +é. So, i1 — 0u/e & 1, there is a very sharp and rapid curvature at ', and when +we take the second derivative of that we get a very high field in the direction +of the motion. 5o when very energetic electrons move through matter they spit +radiation in a forward direction. "This is called bremsstrahlung. As a matter of +fact, the synchrotron is used, not so mụch to make high-energy electrons (actually +1ƒ we could get them out of the machine more conveniently we would not say +this) as to make very energetic photons—gamma rays—by passing the energetic +c-------_„B A Ạ p +s Nucleus ; +D^~Z” \D +(a) (œ) +Fig. 34-9. A fast electron passing near a nucleus radiates energy In +the direction of Its motion. +--- Trang 600 --- +electrons through a solid tungsten “target,” and letting them radiate photons +from this bremsstrahlung efect. +34-6 The Doppler efect +Now we go on to consider some other examples of the efects of moving sources. +Let us suppose that the source is a stationary atom which is oscillating at one +of its natural frequencies, œọ. Then we know that the frequenecy of the light +we would observe is œọ. But now let us take another example, in which we +have a similar oscillator oscillating with a frequenecy ¿¡, and at the same time +the whole atom, the whole oscillator, is moving along in a direction toward the +observer at velocity ø. Then the actual motion in space, of course, is as shown In +Fig. 34-10(a). NÑow we play our usual game, we add er; that is to say, we translate +the whole curve backward and we find then that it oscillates as in Fig. 34-10(Ð). +In a given amount of time 7, when the oscillator would have gone a distance 07, +on the zø“ vs. cí diagram it goes a distance (c— )7. So all the oscillations of +frequency œ in the time A7 are now found in the interval Af = (1 — 0/c) A7; +they are squashed together, and as this curve comes by us at speed c, we will see +light of a higher ƒrequencu, higher by just the compression factor (1— ø/ec). Thus +we Observe ớ +@=———. 34.10 +1 — 1e ) +W© can, of course, analyze this situation in various other ways. Suppose that +the atom were emitting, instead of sine waves, a series of pulses, pip, DĨp, Dip, +pĨỊp, at a certain frequency œị. At what frequency would they be received by +us? The frst one that arrives has a certain delay, but the next one is delayed +less because in the meantime the atom moves closer to the receiver. 'Pherefore, +the time bebween the “pips” is decreased by the motion. If we analyze the +x Ƒ——w=——— x [€=9~1 +(a) (b) +Fig. 34-10. The x-z and x/-f curves of a moving oscillator. +--- Trang 601 --- +geometry of the situation, we fñnd that the frequency of the pips is increased by +the factor 1/(1 — 0/e). +ls œ = œo/(1 — 0/c), then, the frequency that would be observed if we took +an ordinary atom, which had a natural equenecy œọ, and moved it toward the +receiver at speed 0? No; as we well know, the natural frequenecy œJ¡ of a moving +atom is not the same as that measured when i1 is standing still, because of the +relativistic dilation in the rate of passage of time. Thus if œạ were the true +natural frequency, then the modified natural frequency œ¡ would be +œ@ị = 004/1 — 02/c2. (34.11) +'Therefore the observed frequency œ is +"-.—-.ˆ (34.12) +1— Địc +'The shift in frequency observed in the above situation is called the Doppler +cffcct: 1Ÿ something moves toward us the light it emits appears more violet, and +1Ý it moves away it appears more red. +We shall now give ©wo more derivations of this same interesting and important +result. Suppose, now, that the sowrce is standing still and 1s emitting waves +at Ífrequency œọ, while the obseruer is moving with speed 0 toward the source. +After a certain period of tỉìme £ the observer will have moved to a new position, a +distance 0ý tom where he was at ‡¿ = 0. How many radians of phase will he have +seen go by? Á certain number, œg, went past any fixed point, and in addition +the observer has swept past some more by his own motion, namely a number kg +(the number of radians per meter times the distance). So the total number oŸ +radians in the time , or the observed frequeney, would be dị = œọ + kọu. We +have made this analysis from the point of view of a man at rest; we would like +to know how it would look to the man who is moving. Here we have tO WOTry +again about the diference in clock rate for the two observers, and this time that +means that we have to đ/uide by 4/1 — 02/c2. So 1Ÿ kọ is the wave number, the +number of radians per meter ¡in the direction of motion, and œọ is the frequency, +then the observed frequency for a moving man is +j„.h. (34.13) +V1= u2/e +--- Trang 602 --- +Eor the case of light, we know that ko = œo/c. So, in this particular problem, +the equation would read +œo(1-+ += ¿00 +26), (34.14) +v1_— 02/c2 +which looks completely unlike formula (34.12)! Is the frequency that we would +observe iŸ we move toward a source difÑferent than the frequency that we would +see if the source moved toward us? Of course notl The theory of relativity says +that these bwo must be ezøctl cqual. IÝ we were expert enough mathematicians +we would probably recognize that these 6wo mathematical expressions øre exactly +cquall In fact, the øecessar equality of the two expressions is one of the ways +by which some people like to demonstrate that relativity requires a time dilation, +because if we did not put those square-root factors in, they would no longer be +equal. +Since we know about relativity, let us analyze it in still a third way, which +may appear a little more general. (It is really the same thing, since it makes no +diference ?ø+ we do it!) According to the relativity theory there is a relationship +between position and time as observed by one man and position and time as seen +by another who is moving relative to him. We wrote down those relationships +long ago (Chapter 16). Thịs is the Ùoren#z transƒformation and its inverse: +, % + UỶ 4 — 0f +4 =———, #=——————, +v1— 12/c2 v1— 12/c2 ( ) +34.15 +rà t+ 0uz/c2 " U — 0a'/c? +1— 02/c2` V1— 02/2 +Tf we were standing still on the ground, the form of a wave would be cos (œ — &4); +all the nodes and maxima and minima would follow this form. But what would +a man in motion, observing the same physical wave, see? Where the feld is +zero, the positions of all the nodes are the same (when the field is zero, euerone +measures the feld as zero); that is a relativistic invariant. So the form is the +same for the other man too, except that we must transform it into his frame of +reference: +U — 0a'/c2 ø' — 0È | +cos (‡ — k#) = cos|d———————— _—k —————|. +( ) | v1_— 02/c2 v1_— 032/c2 +--- Trang 603 --- +TỶ we regroup the terms inside the brackets, we get +œ -E kU k + uu/c? +cos (wÈ — kø) = cos|———————f —-_——————# +( ) In. v1— 02/2 +`"¬——c—> `"¬————— += coS [ œ" t— k z']. (34.16) +This is again a wave, a cosine wave, in which there is a certain frequency œ/, a +constant multiplying f', and some other constant, &', multiplying +“. We call kí +the wave number, or the number of waves per meter, for the other man. 'Pherefore +the other man will see a new frequency and a new wave number given by +œ' = _=“ (34.17) +v1— 02/2 +ƒ NT tt uUIC (34.18) +v1— 02/2 +Tf we look at (34.17), we see that it is the same formula (34.13), that we obtained +by a more physical argument. +34-7 The œ,k four-vector +The relationships indicated in Eqs. (34.17) and (344.1) are very interesting, +because these say that the new frequeney œ” is a combination of the old Írequeney œ +and the old wave number &, and that the new wave number is a combination +of the old wave number and frequency. Now the wave number is the rate of +change of phase with distance, and the frequency is the rate of change of phase +with time, and in these expressions we see a close analogy with the Lorent⁄z +transformation of the position and tỉme: if œ¿ is thought of as being like ¿, and k +is thought of as being like z divided by c2, then the new œ will be like #, and +the new &“ will be like ø'/c?. That is to say, wnder the Lorentz transƒformation œ +and k trans[form the same tua as do È and z. They constitute what we call a +ƒour-uector; when a quantity has four components transforming like time and +space, it is a four-vector. Everything seems all right, then, except for one little +thing: we said that a four-vector has to have ƒour cormnponen#s; where are the +other two components? We have seen that œ and & are like time and space in one +space direction, but not in all directions, and so we must next study the problem +--- Trang 604 --- +Fig. 34-11. A plane wave travelling in an oblique direction. +of the propagation of light in three space dimensions, not just in one direction, +as we have been doïng up until now. +Suppose that we have a coordinate system, z, , z, and a wave which is +travelling along and whose wavefronts are as shown in Fig. 34-11. The wavelength +of the wawe is À, but the direction of motion of the wave does not happen to be +in the direction of one of the axes. What is the formula for such a wave? "The +answer is clearly cos (œ# — ks), where & = 2Z/A and s is the distance along the +direction of motion of the wave—the component of the spatial position in the +direction of motion. Let us put it this way: 1Ý r is the vector position of a poïnt in +space, then s is 7 - e„, where e;, is a unit vector in the direction of motion. 'PThat +1s, ø is JusE 7cos (, e,), the component of distance in the direction of motion. +Therefore our wave is cos (£ — key - r). +Now ït turns out to be very convenient to defne a vector k, which is called +the 60øøe ector, which has a magnitude equal to the wave number, 2Z/À, and is +pointed in the direction of propagation of the waves: +k = 27e¿/À = key. (34.19) +Using this vector, our wave can be written as cos (ý — k - r), or as cOs (UÉ — +k„& — kụu — k„z). What is the signifcance of a component of k, say k„? Clearly, +k„ 1s the rate of change of phase with respect to z. Referring to Fig. 34-11, we +see that the phase changes as we change ø, just as 1ƒ there were a wave along +, bu‡ oƒ œ longer uuauelength. "The “wavelength in the z-direction” is longer +than a natural, true wavelength by the secant of the angle œ between the actual +--- Trang 605 --- +direction of propagation and the z-axis: +Àz =À/cosơ. (34.20) +Therefore the rate of change of phase, which 1s proportional to the reciprocal +Of À„, is smaller by the factor cos œ; that is just how k„ would vary——it would +be the magnitude of k, times the cosine of the angle between & and the z-axisl +'That, then, is the nature of the wave vector that we use to represent a wave +in three dimensions. The four quantities œ, k„, k„, k; transform in relativity as +a four-vector, where œ corresponds to the time, and &„, ky, ky correspond to the +Z-, -, and z-components of the four-vector. +In our previous discussion of special relativity (Chapter 17), we learned that +there are ways of making relativistic dot products with four-vectors. lÝ we use +the position vector z„, where stands for the four components (time and three +space ones), and iŸ we call the wave vector k„, where the index again has four +values, time and three space ones, then the dot product of ø„ and &„ is written +» k„z„ (see Chapter 17). Thịis dot product is an invariant, independent of the +coordinate system; what is it equal to? By the defnition of this dot product in +four dimensions, it 1s +» ku#u = (UÈ — k„# — kwU — k;„z. (34.21) +We know from our study of vecbors that $ k„#„ is invariant under the Lorentz +transformation, since &„ is a four-vector. But this quantity is precisely what +appears inside the cosine for a plane wave, and it øough# to be invariant under a +Lorentz transformation. We cannot have a formula with something that changes +inside the cosine, since we know that the phase of the wave cannot change when +we change the coordinate system. +34-8 Aberration +In deriving Eqs. (34.17) and (34.18), we have taken a simple exarmple where +k happened to be in a direction of motion, but of course we can generalize it +to other cases also. Eor example, suppose there is a source sending out light in +a certain direction from the point of view of a man at rest, but we are moving +along on the earth, say (Fig. 34-12). Erom which direction does the light appear +to come? To ñnd out, we will have to write down the four components of &„ and +--- Trang 606 --- +(a) (b) +Fig. 34-12. A distant source S is viewed by (a) a stationary telescope, +and (b) a laterally moving telescope. +apply the Lorentz transformation. 'Phe answer, however, can be found by the +following argument: we have to point our telescope at an angle to see the light. +Why? Because light is coming down at the speed c, and we are moving sidewise +at the speed ø, so the telescope has to be tilted forward so that as the light comes +down it goes “straight” down the tube. It is very easy to see that the horizontal +distance is øÉ when the vertical distance is œý, and therefore, iƒ 6” is the angle +of tilt, tan" = o/c. How nicel How nice, indeed—except for one little thing: +6° 1s no‡ the angle at which we would have to set the telescope relatie to the +carth, because we made our analysis from the point of view of a “ñxed” observer. +When we said the horizontal distance is ý, the man on the earth would have +found a different distance, since he measured with a “squashed” ruler. l% turns +out that, because of that contraction efect, +tan Ø —= "=ồ (34.22) +v1— 12/c2 +which is equivalent to +sin Ø = U/c. (34.23) +It will be instructive for the student to derive this result, using the Lorentz +transformation. +This efect, that a telescope has to be tilted, is called øberraton, and it has +been observed. Hø+ can we observe it? Who can say where a given star should +be? Suppose we đo have to look in the wrong direction to see a star; how do we +--- Trang 607 --- +know it is the wrong direction? Because the earth goes around the sun. Today +we have to point the telescope one way; six months later we have to tilt the +telescope the other way. That is how we can tell that there is such an efect. +34-9 The momentum of light +Now we turn to a diferent topic. We have never, in all our discussion of the +past few chapters, said anything about the efects of the zmagnectic field that 1s +associated with light. Ordinarily, the efects of the magnetic fñield are very small, +but there is one interesting and important efect which is a consequence of the +magnetic fñeld. Suppose that light is coming from a source and is acting on a +charge and driving that charge up and down. We will suppose that the electric +ñeld is in the z-direction, so the motion of the charge is also in the z-direction: +1t has a position ø and a velocity 0, as shown in Fig. 34-13. The magnetic fñeld +is at right angles to the electric ñeld. Now as the electric feld acts on the charge +and moves iÿ up and down, what does the magnetic ñeld do? The magnetic ñeld +acts on the charge (say an electron) only when it is moving; but the electron +¡s moving, 1% is driven by the electric field, so the bwo of them work together: +'While the thing is going up and down it has a velocity and there is a force on it, +B times 0 times g; but in which đ¿rection 1s thìs force? Tf ¡s ín the đirecEion oƒ +the propagation öoƒ líght. Therefore, when light is shining on a charge and it is +oscillating in response to that light, there is a driving force in the direction of +the light beam. This is called rød¿aton pressure or light pressure. +Let us determine how strong the radiation pressure is. Evidently it is = gu +or, since everything is oscillating, it is the tửme auerage of this, (F). Erom (34.2) +the strength of the magnetic fñeld is the same as the strength of the electric ñeld +divided by c, so we need to ñnd the average of the electric fñeld, times the velocity, +times the charge, times 1/c: (F} = q(0E)/c. But the charge g times the fñeld # +° V +y E4 +Fig. 34-13. The magnetic force on a charge which ¡s driven by the +electric field ¡s in the direction of the light beam. +--- Trang 608 --- +1s the electric force on a charge, and the force on the charge times the velocity +1s the work đW//đf being done on the chargel “Therefore the force, the “pushing +momentum,” that is delivered per second by the light, ¡is equal to 1/c tìmes the +cnergu absorbed from the light per secondl That is a general rule, since we did +not say how strong the oscillator was, or whether some of the charges cancel +out. Ïn an cứữcumstance tthere láght ¡s being absorbed, there is œ pressure. The +mmomentum that the light delivers is always equal to the energy that is absorbed, +divided by œ: +Œ)= HE (34.24) +That light carries energy we already know. We now understand that it also +carries mnomeniưum, and further, that the momentum carried is always 1/e tỉmes +the energy. +'When light is emitted from a source there is a recoil efect: the same thing in +reverse. lÝ an atom is emitting an energy W ¡in some direction, then there is a +recoil momentum ø = W//c. T light is reflected normally from a mirror, we get +twice the Íorce. +'That is as far as we shall go using the classical theory of light. Of course we +know that there is a quantum theory, and that in many respects light acts like a +particle. 'Phe energy of a light-particle is a constant times the frequency: +W = hu = hư. (34.25) +We now appreciate that light also carries a momentum equal to the energy +divided by e, so it is also true that these efective particles, these phofons, carry +a Immomentum +p= W/c= hujc = hh. (34.26) +The dưcction of the momentum is, of course, the direction of propagation of the +light. So, to put it in vector form, +W = hư, Ð — hÀ. (34.27) +W© also know, of course, that the energy and momentum of a particle should +form a four-vector. We have just discovered that œ and & form a Íour-vectfOr. +Therefore it is a good thing that (34.27) has the same constant in both cases; +it means that the quantum theory and the theory of relativity are mutually +consistent. +--- Trang 609 --- +Equation (34.27) can be written more elegantly as p„ = ñk„, a relativistic +cquation, for a particle associated with a wave. Although we have discussed this +only for photons, for which & (the magnitude of k) equals œ¿/c and p = W/c, +the relation is much more general. In quantum mechanies øÏÏ particles, not only +photons, exhibit wavelike properties, but the frequency and wave number of the +waves is related to the energy and momentum of particles by (34.27) (called the +de Broglie relations) even when ø is not equal to W//e. +In the last chapter we saw that a beam of right or left circularly polarized light +aÌlso carries angular mmomentwm in an amount proportional to the energy € of the +wave. In the quantum picture, a beam of circularly polarized light is regarded as +a stream of photons, each carrying an angular momentum :E, along the direction +of propagation. 'Phat is what becomes of polarization in the corpuscular point of +view—the photons carry angular momentum like spinning rifle bullets. But this +“bullet” picture is really as incomplete as the “wave” picture, and we shall have +to discuss these ideas more fully in a later chapter on Quantum Behavior. +--- Trang 610 --- +€olor- Vistore +35-1 The human eye +'The phenomenon of colors depends partly on the physical world. We discuss +the colors of soap films and so on as beïing produced by interference. But also, +of course, it depends on the eye, or what happens behind the eye, in the brain. +Physics characterizes the light that enters the eye, but after that, our sensations +are the result of photochemical-neural processes and psychological responses. +'There are many interesting phenomena associated with vision which involve +a mixture of physical phenomena and physiological processes, and the full ap- +preciation of natural phenomena, as we see them, must go beyond physies In +the usual sense. We make no apologies for making these excursions into other +fields, because the separation of felds, as we have emphasized, is merely a human +convenience, and an unnatural thing. Nature is not interested in our separations, +and many of the interesting phenomena bridge the gaps between fields. +In Chapter 3 we have already discussed the relation of physics to the other +Sclences in general terms, but now we are goïng to look in some detail at a specific +fñeld in which physics and other sciences are very, very closely interrelated. That +area is 0/sion. In particular, we shall discuss color 0¿sion. In the present chapter +we shall discuss mainly the observable phenomena. of human vision, and in the +next chapter we shall consider the physiological aspects of vision, both in man +and in other animals. +Tt all begins with the eye; so, in order to understand what phenomena we see, +some knowledge of the eye is required. In the next chapber we shall discuss in some +detail how the various parts of the eye work, and how they are interconnected +with the nervous system. For the present, we shall describe only briely how the +eye functions (EFig. 35-l). +Light enters the eye through the cornea; we have already discussed how it is +bent and is imaged on a layer called the refnaø in the back of the eye, so that +--- Trang 611 --- +Cornea +queous +Suspensory Ì Ciliary +ligament Ít muscle +"W - Vitreous h humor ; +\CChoroid | erne 2 +Sclera=—=< +/ẨNL—- +ạỪ ị Macula lutea +ptiC nerve +Fig. 35-1. The eye. +diferent parts of the retina receive light from diferent parts of the visual fñeld +outside. 'Phe retina is not absolutely uniform: there is a place, a spot, in the +center of our fñeld of view which we use when we are trying to see things very +carefully, and at which we have the greatest acuity oÝ vision; it is called the ƒouea +or rnacula. The side parts of the eye, as we can immediately appreciate from +our experience in looking at things, are not as effective for seeing detail as is the +center of the eye. There is also a spot in the retina where the nerves carrying all +the information run out; that is a blind spot. “There is no sensitive part of the +retina here, and it is possible to demonstrate that if we close, say, the left eye and +look straight at something, and then move a ñnger or another small object slowly +out of the fñeld of view it suddenly disappears somewhere. The only practical use +of this fact that we know of is that some physiologist became quite a favorite in +the court of a king of Franece by pointing this out to him; in the boring sessions +that he had with his courtiers, the king could amuse himself by “cutting of theïr +heads” by looking at one and watching anotherˆs head disappear. +Jigure 35-2 shows a magnified view of the inside of the retina in somewhat +schematie form. In different parts of the retina there are diferent kinds of +structures. The objects that occur more densely near the periphery of the retina +are called rods. Closer to the fovea, we fnd, besides these rod cells, also cone +cells. We shall describe the structure of these cells later. As we get close to the +--- Trang 612 --- +" - . . . . hố. tom dsờa taye +H1N1 HE +- ~3— LHỊ.:_. +I003iiiii\EiBi +° ĐH © CLỢY +|} TT >-; tả ii gio 7 Ù +| vÌ -..9) 44/4)... q1. +M : tựi He 4 +—=~=_- s~i0——— ~i0 +Fig. 35-2. The structure of the retina. (Light enters from below.) +fovea, the number of cones increases, and in the fovea itself there are in fact +nothing but cone cells, packed very tightly, so tightly that the cone cells are +much fner, or narrower here than anywhere else. So we must appreciate that we +see with the cones right in the middle of the feld of view, but as we go to the +periphery we have the other cells, the rods. Now the interesting thing is that +in the retina each of the cells which is sensitive to light is not connected by a +fiber directly to the optic nerve, but is connected to many other cells, which are +themselves connected to each other. 'Phere are several kinds of cells: there are +cells that carry the information toward the optic nerve, but there are others that +are mainly interconnected “horizontally.” 'There are essentially four kinds of cells, +but we shall not go into these details now. The main thing we emphasize 1s that +the light signal is already being “thought about.” 'That is to say, the information +from the various cells does not immediately go to the brain, spot for spot, but in +the retina a certain amount of the information has already been digested, by a +combining of the information from several visual receptors. Ïlt is Important to +understand that some brain-function phenomena occur in the eye itself. +35-2 Color depends on intensity +One of the most striking phenomena of vision is the dark adaptation of the +eye. lÝ we go into the dark from a brightly lighted room, we cannot see very well +for a while, but gradually things become more and more apparent, and eventually +--- Trang 613 --- +we can see something where we could see nothing before. lf the intensity of the +light is very low, the things that we see have 0ø color. It is known that this +dark-adapted vision is almost entirely due to the rods, while the vision in bright +light is due to the cones. As a result, there are a number of phenomena that we +can easily appreciate because of this transfer of function from the cones and rods +together, to just the rods. +There are many situations in which, If the light intensity were stronger, we +could see color, and we would fnd these things quite beautiful. One example +1s that through a telescope we nearly always see “black and white” images of +faint nebulae, but W. Œ. Miller of the Mt. Wilson and Palomar Observatories +had the patience to make color pictures of some of these obJects. Nobody has +ever really seen these colors with the eye, but they are not artificial colors, 1% +is merely that the light intensity is not strong enough for the cones in our eye +to see them. Among the more spectacular such objects are the ring nebula and +the Crab nebula. 'Phe former shows a beautiful blue inner part, with a bright +red outer halo, and the latter shows a general bluish haze permeated by bright +red-orange filaments. +In the bright light, apparently, the rods are at very low sensitivity but, in the +dark, as tỉme goes on they pick up their ability to see light. 'Phe variations in light +intensity for which one can adapt is over a million to one. Nature does not do +all this with just one kind of cell, but she passes her job from bright-light-seeing +cells, the color-seeing cells, the cones, to low-intensity, dark-adapted cells, the +rods. Among the interesting consequences of this shift is, frst, that there is no +color, and second, that there is a diference in the relative brightness of diferently +colored objJects. It turns out that the rods see better toward the blue than the +cones do, and the cones can see, for example, deep red light, while the rods fñnd +that absolutely impossible to see. So red light is black so far as the rods are +concerned. 'Phus two pieces of colored paper, say blue and red, in which the red +might be even brighter than the blue in good light, will, in the dark, appear +completely reversed. lt is a very striking efect. IÝ we are in the dark and can +ñnd a magazine or something that has colors and, before we know for sure what +the colors are, we judge the lighter and darker areas, and if we then carry the +magazine into the light, we may see this very remarkable shiẾt between which was +the brightest color and which was not. 'Phe phenomenon is called the Purkimje +cffcct. +In Eig. 35-3, the dashed curve represents the sensitivity of the eye in the dark, +1.e., using the rods, while the solid curve represents it in the light. We see that +--- Trang 614 --- +100 ° A +|| LÝ 1} %x |L | +TRNNNMNMERMNVEHRBMMNMRB +J]BHNNNMIINEAREESNHNMR +34L |} } L1]. hịỰ | +TU INHNEUNNSZNRNNHMWNHNNNRB +FIRRRNIIIERRYRRRNB +T IIHNNERNHHHMNNNMNNNN +TRMMFMEmMHMHMRVMRBREMR +JsNmãi — +TẾT 60 40 20 600 80 60 40 20 500 80 60 40 20 400 +Wavelength in mu +Fig. 35-3. The spectral sensitivity of the eye. Dashed curve, rods; +solid curve, cones. +the peak sensitivity of the rods is in the green region and that of the cones is +more in the yellow region. Tf there is a red-colored page (red is about 650 mu) +we can see iE if it is brightly lighted, but in the dark it is almost invisible. +Another efect of the fact that rods take over in the dark, and that there are +no rods in the fovea, is that when we look straight at something in the dark, our +vision is not quite as acute as when we look to one side. A faint star or nebula +can sometimes be seen better by looking a little to one side than directly at it, +because we do not have sensitive rods in the middle of the fovea. +Another interesting efect of the fact that the number of cones decreases as +we go farther to the side of the feld of view is that even in a bright light color +disappears as the object goes far to one side. The way to test that is to look in +some particular fñxed direction, let a friend walk in from one side with colored +cards, and try to decide what color they are before they are right in front of you. +One ñnds that he can see that the cards are there long before he can determine +the color. When doïng this, ¡it is advisable to come ïn from the side opposite the +blïnd spot, because it is otherwise rather confusing to almost see the color, then +not see anything, then to see the color again. +Another interesting phenomenon is that the periphery of the retina is very +sensitive to motion. Although we cannot see very well from the corner of our +eye, If a little bug moves and we do not expect anything to be moving over there, +--- Trang 615 --- +we are Immediately sensitive to it. We are all “wired up” to look for something +Jiggling to the side of the ñeld. +35-3 Measuring the color sensation +Now we go to the cone vision, to the brighter vision, and we come to the +question which is most characteristic of cone vision, and that is color. As we +know, white light can be split by a prism into a whole spectrum of wavelengths +which appear to us to have diferent colors; that is what colors are, Of cOUTsSe: +appearances. Any source of light can be analyzed by a grating or a prism, and one +can determine the spectral distribution, i.e., the “amount” of each wavelength. A +certain light may have a lot of blue, considerable red, very little yellow, and so on. +'That is all very precise in the sense of physics, but the question is, what color will +1t appear to be? It is evident that the diferent colors depend somehow upon the +spectral distribution of the light, but the problem is to fnd what characteristics +of the spectral distribution produce the various sensations. For example, what +do we have to do to get a green color? We all know that we can simply take a +piece of the spectrum which is green. But is that the on way to get green, or +orange, or any other color? +ls there more than one spectral distribution which produces the same apparent +visual efect? 'Phe answer is, defnitely wes. 'Phere is a very limited number of +visual efects, in fact just a three-dimensional manifold of them, as we shall +shortly see, but there is an infnite number of diferent curves that we can draw +for the light that comes from diferent sources. Now the question we have to +discuss is, under what conditions do diferent distributions of light appear as +exactly the same color to the eye? +The most powerful psycho-physical technique in color judgment is to use the +eye as a nuưÏÌ ímstrument. That 1s, we do not try to defne what constitutes a +green sensation, or to measure in what circumstances we get a green sensation, +because it turns out that this is extremely complicated. Instead, we study the +conditions under which 6wo stimuli are ¿nđistnguishable. Then we do not have +to decide whether two people see the same sensation in diferent circumstances, +but only whether, 1ƒ for one person ÿwo sensations are the same, they are also the +same for another. We do not have to decide whether, when one sees something +green, what it feels like inside is the same as what it feels like inside someone +else when he sees something green; we do not know anything about that. +--- Trang 616 --- +To illustrate the possibilities, we may use a series of four proJector lamps +which have filters on them, and whose brightnesses are continuously adjustable +over a wide range: one has a red filter and makes a spot of red light on the +sereen, the next one has a green filter and makes a green spot, the third one +has a blue filter, and the fourth one is a white circle with a black spot in the +middle of it. Now IÝ we turn on some red light, and next to it put some green, +we see that in the area of overlap it produces a sensation which is not what we +call reddish green, but a new color, yellow in this particular case. By changing +the proportions of the red and the green, we can go through various shades of +orange and so forth. If we have set it for a certain yellow, we can also obtain +that same yellow, not by mixing these two colors but by mixing some other ones, +perhaps a yellow filter with white light, or something like that, to get the same +sensation. In other words, it is possible to make various colors in more than one +way by mixing the lights from various filters. +What we have just discovered may be expressed analytically as follows. A +particular yellow, for example, can be represented by a certain symbol Y, which is +the “sum” oŸ certain amounts of red-filtered light (?) and green-filtered light (GŒ). +By using two numbers, say ? and ø, to describe how bright the ! and Œ are, we +can write a formula for this yellow: +Y=rR+qG. (35.1) +The question is, can we make ai! the diferent colors by adding together two +or three lights of diferent, ñxed colors? Let us see what can be done in that +connection. We certainly cannot get all the diferent colors by mixing only red +and green, because, for instance, blue never appears in such a mixture. However, +by putting in some blue the central region, where all three spots overlap, may +be made to appear to be a fairly nice white. By mixing the various colors and +looking at this central region, we ñnd that we can get a considerable range of +colors in that region by changing the proportions, and so i is not impossible +that ai the colors can be made by mixing these three colored lights. We shall +discuss to what extent this is true; it is in fact essentially correct, and we shall +shortly see how to defne the proposition better. +In order to illustrate our point, we move the spots on the screen so that they +all fall on top of each other, and then we try to match a particular color which +appears in the annular ring made by the fourth lamp. What we once thought +was “white” coming from the fourth lamp now appears yellowish. We may try to +--- Trang 617 --- +match that by adjusting the red and green and blue as best we can by a kind of +trial and error, and we fnd that we can approach rather closely this particular +shade of “cream” color. So it is not hard to believe that we can make all colors. +W© shall try to make yellow in a moment, but before we do that, there is one +color that might be very hard to make. People who give lectures on color make +all the “bright” colors, but they never make Öroưn, and it is hard to recall ever +having seen brown light. As a matter of fact, this color is never used for any +stage efect, one never sees a spotlight with brown light; so we think it might +be impossible to make brown. In order to fnd out whether it is possible to +make brown, we point out that brown light is merely something that we are not +used to seeing without its background. Ás a matter of fact, we can make it by +mixing some red and yellow. To prove that we are looking at brown light, we +merely increase the brightness of the annular background against which we see +the very same light, and we see that that is, in fact, what we call brownl Brown +1s always a dark color next to a lighter background. We can easily change the +character of the brown. Eor example, iŸ we take some green out we get a reddish +brown, apparently a chocolaty reddish brown, and iŸ we put more green info it, +in proportion, we get that horrible color which all the uniforms of the Army are +made of, but the light from that color is not so horrible by itself; it is of yellowish +green, but seen against a light background. +Now we put a yellow filter in front of the fourth light and try to match that. +(The intensity must oŸ course be within the range of the various lamps; we cannob +match something which is too bright, because we do not have enough power in +the lamp.) But we cøn match the yellow; we use a green and red mixture, and +put in a touch of blue bo make it even more perfect. Perhaps we are ready to +believe that, under good conditions, we can make a perfect match of any given +color. +Now let us discuss the laws of color mixture. In the first place, we found that +diferent spectral distributions can produce the same color; next, we saw that +“any” color can be made by adding together three special colors, red, blue, and +green. The most interesting feature of color mixing is this: if we have a certain +light, which we may call X, and I1f it appears indistinguishable from Y, to the +eye (it may be a different spectral distribution, but it øppears indistinguishable), +we call these colors “equal,” in the sense that the eye sees them as equal, and we +X =Y. (35.2) +--- Trang 618 --- +Here is one of the great laws of color: if two spectral distributions are indistin- +guishable, and we øđở fo cách one a certain light, say Z (ïfÍ we write X + Z, +this means that we shine both lights on the same patch), and then we take Y +and add the same amount of the same other light, Z, fhe neu mmiztures œre œÏlso +?ndistingutshable: +X+Z=Y+/. (35.3) +W© have just matched our yellow; if we now shine pink light on the whole thing, +it will still match. 5o adding any other light to the matched lights leaves a match. +In other words, we can summarize all these color phenomena by saying that once +we have a match bebween two colored lights, seen next to each other in the same +circumstances, then this match will remain, and one light can be substituted for +the other light ím any other color mixing situation. In fact, it turns out, and +1t 1s very Important and interesting, that this matching of the color of lights is +not dependent upon the characteristics of the eye at the moment of observation: +we know that if we look for a long time at a bright red surface, or a bright red +light, and then look at a white paper, it looks greenish, and other colors are aÌso +distorted by our having looked so long at the bright red. TỶ we now have a match +between, say, two yellows, and we look at them and make them match, then we +look at a bright red surface for a long time, and then turn back to the yellow, +1t may not look yellow any more; l do not know what color it will look, but i§ +will not look yellow. Nevertheless #he ellotus tuilH still look rmmaitched, and so, as +the eye adapts to various levels of intensity, the color match still works, with the +obvious exception of when we go into the region where the intensity of the light +gets so low that we have shifted from cones to rods; then the color match is no +longer a color match, because we are using a diferent system. +The second principle of color mixing oŸ lights is this: am color œÈ aÌÏ can +be made from threc different colors, in our case, red, green, and blue lights. +By suitably mixing the three together we can make anything at all, as we +demonstrated with our bwo examples. Further, these laws are very interesting +mathematically. For those who are interested in the mathematics of the thíng, it +turns out as follows. Suppose that we take our three colors, which were red, green, +and blue, but label them 4, Ö, and Œ, and call them our prửmar+ colors. Then +any color could be made by certain amounts of these three: say an amount œ of +color 4, an amount Ö of color Ö, and an amount e of color Œ makes X: +X=øuA+bB+‹cŒC. (35.4) +--- Trang 619 --- +Now suppose another color Y is made from the same three colors: +Y=ưzA+bB+ecC. (35.5) +Then it turns out that the mixture of the two lights (ï is one of the consequences +of the laws that we have already mentioned) is obtained by taking the sum of +the components of X and Y: +Z=X+Y=(a+z)A+(b+Ù)B+(c+e)C. (35.6) +It is Just like the mathematics of the addition of vectors, where (øœ,b,e) are the +components of one vector, and (œ”, ,e) are those of another vector, and the new +light Z is then the “sum” of the vectors. This subject has always appealed to +physicists and mathematicians. In fact, Schrödinger wrote a wonderful paper on +color vision In which he developed this theory of vector analysis as applied to +the mixing of colors. +Now a question is, what are the correct primary colors to use? There is no +such thing as “the” correct primary colors for the mixing of lights. There may be, +for practical purposes, three paints that are more useful than others for getting a +greater variety of mixed pigments, but we are not discussing that matter now. +Ang threc dierentlU colored lights tuhatsoeuer* can aÌways be mixed in the correcb +proportion to produce øng color tha‡socuer. Can we demonstrate this fantastic +fact? Instead of using red, green, and blue, let us use red, blue, and yellow In +our projector. Can we use red, blue, and yellow to make, say, green? +By mixing these three colors in various proportions, we get quite an array of +diferent colors, ranging over quite a spectrum. But as a matter of fact, after a +lot of trial and error, we fnd that nothing ever looks like green. 'Phe question +1s, cœn we make green? 'Phe answer is yes. How? Pụ projecling some red on‡o +the green, then we can make a match with a certain mixture of yellow and bluel +So we have matched them, except that we had to cheat by putting the red on +the other side. But since we have some mathematical sophistication, we can +appreciate that what we really showed was not that X could always be made, say, +of red, blue, and yellow, but by putting the red on the other side we found that +red plus X could be made out o£ blue and yellow. Putting it on the other side of +the equation, we can interpret that as a negatie œmnownt, so 1Ÿ we wIll allow that +the coefficients in equations like (35.4) can be both positive and negative, and IŸ +* Except, of course, if one of the three can be matched by mixing the other two. +--- Trang 620 --- +we interpret negative amounts to mean that we have to øđd those to the of#her +side, then any color can be matched by any three, and there is no such thing as +“the” fundamental primaries. +We may ask whether there are three colors that come only with positive +amounts for all mixings. The answer is no. Every set of three primaries requires +negative amounts for some colors, and therefore there is no unique way to defne +a primary. In elementary books they are said to be red, green, and blue, but that +is merely because with these a t0ider range of colors is available without minus +signs for some of the combinations. +35-4 The chromaticity diagram +Now let us discuss the combination of colors on a mathematical level as a +geometrical proposition. If any one color is represented by Eaq. (35.4), we can +plot it as a vector in space by plotting along three axes the amounts ø, b, and e, +and then a certain color is a point. lf another color is a', , e, that color is +located somewhere else. 'Phe sum of the two, as we know, is the color which +comes from adding these as vectors. We can simplify this diagram and represent +everything on a plane by the following observation: if we had a certain color light, +and merely doubled ø and b and e, that is, if we make them all stronger in the +same ratio, it is the same color, but brighter. 5o iŸ we agree to reduce everything +to the sưmne ligh‡ imtensitụ, then we can project everything onto a plane, and +this has been done in Fig. 35-4. It follows that any color obtained by mixing a +given two in some proportion will lie somewhere on a line drawn between the two +points. Eor instance, a fifty-fifty mixture would appear halfway between them, +and 1/4 of one and 3/4 of the other would appear 1/4 of the way from one point +to the other, and so on. lÝ we use a blue and a green and a red, as primaries, +we see that all the colors that we can make with positive coefficients are inside +the dotted triangle, which contains almost all of the colors that we can ever see, +because all the colors that we can ever see are enclosed in the oddly shaped area, +bounded by the curve. Where did this area come from? Once somebody made a +very careful match of all the colors that we can see against three special ones. +But we do not have to check øÏl colors that we can see, we only have to check the +pure spectral colors, the lines of the spectrum. Any light can be considered as a +sum of various positive amounts of various pure spectral colors——pure from the +physical standpoint. AÁ given light will have a certain amount of red, yellow, blue, +and so on—spectral colors. So if we know how much of each of our three chosen +--- Trang 621 --- +X 520 +0.8 +510 G® ' +07 ï `, N 550 +| Ị SN 560 +0.6 Ị ¬ +h ` N70 +500 Ị ` +0.5 Ị ` N80 +Ị ` 500 +0.4 Ị ` 600 +ị ` X 620 +0.3 \aoo ¡ ¬-: 630 +Ị —_. 700 +0.2 Ị _« +01 480 Ị _ +: 470 % +0 400) +01 02 03 04 05 06 07x +Fig. 35-4. The standard chromaticity diagram. +primaries is needed to make each of these pure components, we can calculate +how much of each is needed to make our given color. So, if we ñnd out what the +color coeffficienfs of all the spectral eolors are for any given three primary colors, +then we can work out the whole color mixing table. +An example of such experimental results for mixing three lights together +1s given in Eig. 35-5. This fñgure shows the amount of each of three diferent +particular primaries, red, green and blue, which is required to make each of the +spectral colors. Red is at the left end of the spectrum, yellow is next, and so on, all +the way to blue. Notice that at some points minus signs are necessary. Ït is Írom +such data that it is possible to locate the position of all of the colors on a chart, +where the z- and the #-coordinates are related to the amounts of the different +primaries that are used. 'Phat is the way that the curved boundary line has been +found. It§ ¡is the locus of the pure spectral colors. NÑow any other color can be +made by adding spectral lines, of course, and so we fnd that anything that can be +produced by connecting one part of this curve to another is a color that is available +in nature. "The straight line connects the extreme violet end of the spectrum with +the extreme red end. It is the locus of the purples. Inside the boundary are colors +that can be made with lights, and outside it are colors that cannot be made with +lights, and nobody has ever seen them (except, possibly, in after-imagesl). +--- Trang 622 --- +TIIRRSRRRRRERRIRR +zøLLL-LIN-⁄H\/--- +TP TIIIIRNY//ERNIIRRN +'ĐIIIIIII) 451111111 +TRIINIIPSVITIĐII +—'2 720 680 640 600 560 520 480 440 400 +WAVELENGTH mự +Fig. 35-5. The color coefficients of pure spectral colors in terms of a +certain set of standard primary colors. +35-5 The mechanism of color vision +Now the next aspect of the matter is the question, œh¿ do colors behave In +this way? 'The simplest theory, proposed by Young and Helmholtz, supposes that +in the eye there are three diferent pigments which receive the light and that these +have diferent absorption spectra, so that one pigment absorbs stronplÌy, say, in +the red, another absorbs strongly in the blue, another absorbs in the green. Then +when we shine a light on them we will get diferent amounts of absorptions in the +three regions, and these three pieces of information are somehow maneuvered in +the brain or in the eye, or somewhere, to decide what the color is. It is easy to +demonstrate that all of the rules of color mixing would be a consequence oŸ this +proposition. 'There has been considerable debate about the thíng because the +next problem, oŸ course, is to fnd the absorption characteristics of each of the +three pigments. Ït turns out, unfortunately, that because we can transform the +color coordinates in any manner we want to, we can only ñnd all kinds of linear +combinations of absorption curves by the color-mixing experiments, but not the +curves for the individual pigments. People have tried in various ways to obtain a +specifc curve which does describe some particular physical property of the eye. +One such curve is called a brightness curue, demonstrated in Fig. 35-3. In this +fñgure are bwo curves, one for eyes in the dark, the other for eyes in the light; +the latter is the cone brightness curve. This is measured by ñnding what is the +smallest amount of colored light we need in order to be able to just see it. Thịis +mneasures how sensitive the eye is in different spectral regions. There is another +--- Trang 623 --- +very interesting way to measure this. If we take ©wo colors and make them appear +in an area, by flickering back and forth from one to the other, we see a flicker +1f the frequency 1s too low. However, as the frequency increases, the ficker will +ultimately disappear at a certain frequency that depends on the brightness of +the light, let us say at 16 repetitions per second. Now if we adjust the brightness +or the intensity of one color against the other, there comes an intensity where +the flicker at 16 cycles disappears. To get flicker with the brightness so adjusted, +we have to go to a much lower Írequency in order to see a flicker of the color. +So, we get what we call a flicker of the brightness at a higher frequency and, +at a lower frequeney, a flicker of the color. It is possible to match two colors +for “equal brightness” by this flicker technique. “The results are almost, but not +exactly, the same as those obtained by measuring the threshold sensitivity of the +eye for seeing weak light by the cones. Most workers use the ficker system as a +defñnition of the brightness curve. +Now, 1ƒ there are three color-sensitive pigments in the eye, the problem is to +determine the shape of the absorption spectrum of each one. How? We know +there are people who are color blind——eight percent of the male population, and +one-half of one percent of the female population. Most of the people who are color +blind or abnormal in color vision have a different degree of sensitivity than others +to a variation of color, but they still need three colors to match. However, there +are some who are called dichromais, for whom any color can be matched using only +tuo primary colors. The obvious suggestion, then, is to say that they are missing +one of the three pigments. lf we can find three kinds of color-blind dichromats who +have diferent color-mixing rules, one kind should be missing the red, another the +green, and another the blue pigmentation. By measuring all these types we can +determine the three curvesl It turns out that there are three types of dichromatic +color blindness; there are bwo common types and a third very rare type, and +from these three it has been possible to deduce the pigment absorption spectra. +Pigure 35-6 shows the color mixing of a particular type of color-blind person +called a deuteranope. Eor him, the loci of constant colors are not points, but +certain lines, along each of which the color appears to him to be the same. Tf the +theory that he is missing one of the three pieces of information is right, all these +lines should intersect at a point. If we carefully measure on this graph, they do +intersect perfectly. Obviously, therefore, this has been made by a mathematician +and does not represent real datal As a matter of fact, if we look at the latest +paper with real data, it turns out that in the graph of EFig. 35-6, the poïnt of +focus of all the lines is not exactly at the right place. Using the lines in the +--- Trang 624 --- +515 530 +0.8 +0Ì 550 +0.6 +0.5 580 +0.4 +0.3 Lo 620 +° mang +01 CC +2` ——— +0 BỒN +0D 01 02 03 04 05 06 0.7 +Fig. 35-6. Loci of colors confused by deuteranopes. +above figure, we cannot find reasonable spectra; we need negative and positive +absorptions in difÑferent regions. But using the new data of Yustova, it turns out +that each of the absorption curves is everywhere posifive. +Figure 35-7 shows a diferent kind of color blindness, that of the protanope, +which has a focus near the red end o£ the boundary curve. Yustova gets approxi- +mately the same position in this case. Using the three diferent kinds of color +blindness, the three pigment response curves have finally been determined, and +are shown in Eig. 35-8. Finally? Perhaps. There 2s a question as to whether the +three-pigment idea is right, whether color blindness results from lack oŸ one pig- +ment, and even whether the color-mix data on color blindness are right. Diferent +workers get diferent results. This field ïs still very much under development. +35-6 Physiochemistry of color vision +Now, what about checking these curves against actual pigments in the eye? +The pigments that can be obtained from a retina consist mainly oŸ a pigment +called 0isual purple. "The most remarkable features of this are, frst, that it isin +--- Trang 625 --- +515 530 +0.8 +0Ì 550 +0.6 +0.5 580 +0.4 +0.3 À————= +0.1 +0Ì 450 +0D 01 02 03 04 05 06 07 +Fig. 35-7. Loci of colors confused by protanopes. +2.0 +1.0 +Ð 2000 5000 6000 7000 +^ (Ã) +Fig. 35-8. The spectral sensitivity curves of a normal trichromat's +receptors. +--- Trang 626 --- +1.0 L†T —] L_TTLT _ } — +RIRIVabVRIIIN +TRÍ ms +NI ANRI +MỸ LÀN +"i00 — 500 g0g TT +\Wavelength mu +Fig. 35-9. The sensitivity curve of the dark-adapted eye, compared +with the absorption curve of visual purple. +the eye of almost every vertebrate animal, and second, that its response curve +ñts beautifully with the sensitivity of the eye, as seen in Eig. 35-9, in which are +plotted on the same scale the absorption of visual purple and the sensitivity of +the dark-adapted eye. 'Phis pigment is evidently the pigment that we see with +in the dark: visual purple is the pigment for the rods, and it has nothing to do +with color vision. This fact was discovered in 1877. Even today it can be said +that the color pigments of the cones have never been obtained in a test tube. In +1958 ít could be said that the color pigments had never been seen at all. But +since that time, two of them have been detected by Rushton by a very simple +and beautiful technique. +The trouble is, presumably, that since the eye is so weakly sensitive to bright +light compared with light oŸ low intensity, i needs a lot of visual purple to see +with, but not much of the color pigments for seeing colors. Rushton”s idea is to +leaue the pigmen‡ ín the cuc, and measure it anyway. What he does is this. There +is an instrument called an ophthalmoscope for sending light into the eye through +the lens and then focusing the light that comes back out. With i9 one can measure +how much is refected. 5o one measures the reflection coefficient of light which +has gone £œce through the pigment (refected by a back layer in the eyeball, +and coming out through the pigment oŸ the cone again). Nature is not always +so beautifully designed. 'Phe cones are interestingly designed so that the light +that comes into the cone bounces around and works its way down into the little +sensitive points at the apex. The light goes right down into the sensitive poiïnt, +bounees at the bottom and comes back out again, having traversed a considerable +--- Trang 627 --- +amount of the color-vision pigment; also, by looking at the fovea, where there +are no rods, one is not confused by visual purple. But the color of the retina +has been seen a long time ago: it is a sort of orangey pink; then there are all the +blood vessels, and the color of the material at the back, and so on. How do we +know when we are looking at the pigment? Ansuer: Pirst we take a color-blind +person, who has fewer pigments and for whom it is therefore easier to make +the analysis. Second, the various pigments, like visual purple, have an intensity +change when they are bleached by light; when we shine light on them they change +their concentration. So, while looking at the absorption spectrum of the eye, +Rushton put ano#her beam in the whole eye, which changes the concentration of +the pigment, and he measured the chøngøe in the spectrum, and the diference, of +course, has nothing to do with the amount of blood or the color of the refecting +layers, and so on, but only the pigment, and in this manner Rushton obtained +a curve for the pigment of the protanope eye, which is given in Fig. 35-10. +Double Density Double Density +0.04- P-54 ¬ ° ° P-59 ¬0.03 +0.03 R — ^\ +h 0.02 +0.02 +0.01 NÓ T] ì - +" L—_] +0508 mụ 550 600 `= +Fig. 35-10. Absorption spectrum of the color pigment of a protanope +colorblind eye (squares) and a normal eye (dots). +The second curve in Fig. 35-10 is a curve obtained with a normal eye. This +was obtained by taking a normal eye and, having already determined what one +piegment was, bleaching the other one in the red where the frst one is insensitive. +Red light has no efect on the protanope eye, but does in the normal eye, and thus +one can obtain the curve for the missing pigment. 'Phe shape of one curve fts +beautifully with Yustova's green curve, but the red curve is a little bít displaced. +So perhaps we are getting on the right track. Ôr perhaps not——the latest work +with deuteranopes does not show any defnite pigment missing. +--- Trang 628 --- +Color is not a question of the physics of the light itself. Color is a sensafion, +and the sensation for different colors is diferent in diferent cireumstances. For +instance, if we have a pink light, made by superimposing crossing beams of white +light and red light (all we can make with white and red is pink, obviousÌy), we +may show that white light may appear blue. If we place an object in the beams, +1t casts 0wo shadows——one illuminated by the white light alone and the other +by the red. For most people the “white” shadow of an object looks blue, but 1ƒ +we keep expanding this shadow until it covers the entire screen, we see that it +suddenly appears white, not bluel We can get other effects of the same nature +by mixing red, yellow, and white light. Red, yellow, and white light can produce +only orangey yellows, and so on. So If we mix such lights roughly equally, we +get only orange lipht. Nevertheless, by casting diferent kinds of shadows in the +light, with various overlaps of colors, one gets quite a series of beautiful colors +which are not in the light themselves (that is only orange), but in our sensations. +W© clearly see many diferent colors that are quite unlike the “physical” ones in +the beam. Ït is very important to appreciate that a retina ¡is already “thinking” +about the light; it is comparing what it sees in one reglon with what it sees In +another, although not consciously. What we know of how it does that is the +subJect of the next chapter. +BIBLIOGRAPHY +Committee on Colorimetry, Optical Society of America, 7 he Science oŸ Color, +Thomas Y. Crowell Company, New York, 1953. +HECHT, S., S5. SHLAER, and M. H. PIRENNE, “Energy, Quanta, and Vision,” Journal +öoƒ General Phạụs¿ologu, 1942, 25, 819-840. +MORGAN, CLIFFORD and ELIOT STELLAR, PhụsiologicaL Psuchology, 2nd ed., McGraw- +Hill Book Company, Inc., 1950. +NUBERG, N. D. and E. N. YUSTOVA, “Researches on Dichromatic Vision and the +5pectral Sensitivity of the Receptors of 'Irichromats,” presented at Symposium No. 8, +Visual Problems oƒ Colour, Vol. II; Ñational Physical Laboratory, Teddington, England, +September 1957. Published by Her Majestyˆs Stationery Office, London, 1958. +RUSHTON, W. A., “The Cone Pigments of the Human Fovea in Colour Blind and +Normal,” presented at Symposium No. 8, W7sual Problems oƒ Colour, Vol. Ij Ñational +Physical Laboratory, Ieddington, England, September 1957. Published by Her Majesty's +Stationery Ofice, London, 1958. +WOODWORTH, ROBERT S., Ezperimental Psụchologu, Henry Holt and Company, New +York, 1938. Revised edition, 1954, by Robert 5. Woodworth and H. Schlosberg. +--- Trang 629 --- +IHMoclhi(rtfsits ©Ÿ Sooïngj +30-1 The sensation of color +In discussing the sense of sight, we have to realize that (outside of a gallery of +modern artl) one does not see random spots of color or spots of light. When we +look at an object we see a man or a fhứng; in other words, the brain interprets +what we see. How it does that, no one knows, and it does it, of course, at a very +high level. Although we evidently do learn to recognize what a man looks like +after much experience, there are a number of features of vision which are more +elementary but which also involve combining information from diferent parts +of what we see. To help us understand how we make an interpretation of an +entire image, it is worth while to study the earliest stages of the putting together +of information from the diferent retinal cells. In the present chapter we shall +concentrate mainly on that aspect of vision, although we shall also mention a +number oŸ side issues as we go along. +An example of the fact that we have an accumulation, at a very elementary +level, of information from several parts of the eye at the same time, beyond our +voluntary control or ability to learn, was that blue shadow which was produced +by white light when both white and red were shining on the same screen. 'This +effect at least involves the knowledge that the background of the screen is pink, +even though, when we are looking at the blue shadow, it is only “white” light +coming into a particular spot in the eye; somewhere, pieces of information have +been put together. The more complete and familiar the context is, the more +the eye will make corrections for peculiarities. In fact, Land has shown that +1ƒ we mix that apparent blue and the red in various proportions, by using two +photographic transparencies with absorption in front of the red and the white in +diÑferent proportions, it can be made to represent a real scene, with real objects, +rather faithfully. In this case we get a lot of intermediate apparent colors %oo, +analogous to what we would get by mixing red and blue-green; it seems to be an +--- Trang 630 --- +Fig. 36-1. When a disc like the above ¡is spun, colors appear in only +one of the two darker “rings.” lf the spin direction is reversed, the colors +appear In the other ring. +almost complete set of colors, but if we look very hard at them, they are not so +very good. Even so, it is surprising how much can be obtained from just red and +white. The more the scene looks like a real situation, the more one is able to +compensate for the fact that all the light is actually nothing but pinkl +Another example is the appearance of “colors” in a black-and-white rotating +disc, whose black and white areas are as shown in Fig. 36-1. When the disc is +rotated, the variations of light and dark at any one radius are exactly the same; +1E is only the background that is diÑferent for the two kinds oŸ “stripes.” Yet one +of the “rings” appears colored with one color and the other with another.* Ño +one yet understands the reason for those colors, but it is clear that information +1s being put together at a very elementary level, in the eye itself, most likely. +Almost all present-day theories of color vision agree that the color-mixing +data indicate that there are only three pigments in the cones oŸ the eye, and that +1t is the spectral absorption in those three pigments that fundamentally produces +the color sense. But the total sensation that is associated with the absorption +characteristics of the three pigments acting together is not necessarily the sum +of the individual sensations. We all agree that yellow simply does no‡ seem to be +reddish green; in fact it might be a tremendous surprise to most people to discover +that light is, in fact, a mixture of colors, because presumably the sensation of +light is due to some other process than a simple mixture like a chord in music, +where the three notes are there at the same time and If we listen hard we can +hear them individually. We cannot look hard and see the red and the green. +* The colors depend on speed of rotation, on the brightness of illumination, and to some +extent on who looks at them and how intently he stares at them. +--- Trang 631 --- +The earliest theories of vision said that there are three pigments and three +kinds of cones, each kind containing one pigment; that a nerve runs from each +cone to the brain, so that the three pieces of information are carried to the brain; +and then in the brain, anything can happen. This, of course, is an incomplete +idea: it does no good to discover that the information is carried along the optic +nerve to the brain, because we have not even started to solve the problem. We +must ask more basic questions: Does it make any diference œhere the information +1s put together? Is it important that it be carried right up into the brain in the +optic nerve, or could the retina do some analysis frst? We have seen a picture +of the retina as an extremely complicated thing with lots of interconnections +(Fig. 35-2) and it might make some analyses. +As a matter of fact, people who study anatomy and the development of the +eye have shown that the retina is, in fact, the brain: in the development of +the embryo, a piece of the brain comes out in front, and long fibers grow back, +connecting the eyes to the brain. The retina is organized in just the way the brain +1s organized and, as someone has beautifully put it, “The brain has developed +a way to look out upon the world.” "The eye is a piece of brain that is touching +light, so to speak, on the outside. So ït is not at all unlikely that some analysis +of the color has already been made in the retina. +This gives us a very interesting opportunity. None of the other senses involves +such a large amount of caleulation, so to speak, before the sipnal gets into a nerve +that one can make measurements on. The calculations for all the rest of the +senses usually happen in the brain itself, where it is very dificult to get at specific +places to make measurements, because there are so many interconnections. Here, +with the visual sense, we have the light, three layers of cells making calculations, +and the results of the calculations being transmitted through the optic nerve. 5o +we have the first chance to observe physiologically how, perhaps, the first layers +of the brain work in their first steps. It is thus of double interest, not simply +Interesting for vision, but interesting to the whole problem of physiology. +The fact that there are three pigments does not mean that there must be +three kinds of sensations. One of the other theories of color vision has it that +there are really opposing color schemes (Eig. 36-2). That is, one of the nerve +fibers carries a lot of impulses if there is yellow being seen, and less than usual +for blue. Another nerve fiber carries green and red information in the same +way, and another, white and black. In other words, in this theory someone +has already started to make a guess as to the system of wiring, the method of +calculation. +--- Trang 632 --- +Neural Responses +_— + — + + — +ĐA TÌ +Oấ0O]§ +ˆ Photochenical Absorptions ˆ +y—b = k(Ø0++-22) +r—g = k(œ+^x—2Ø) +w—bk = ka(œ+++)—ka(œ+-+) +Fig. 36-2. Neural connections according to an “opponent” theory of +color vision. +The problems we are trying to solve by guessing at these first calculations +are questions about the apparent colors that are seen on a pink background, +what happens when the eye is adapted to diferent colors, and also the so-called +psychological phenomena. “The psychological phenomena are of the nature, for +Instance, that white does not “feel” like red and yellow and blue, and this +theory was advanced because the psychologists say that there are ƒour apparent +pure colors: “Thhere are four stimuli which have a remarkable capacity to evoke +psychologically simple blue, yellow, green, and red hues respectively. Unlike +sienna, magenta, purple, or most$ of the discriminable colors, these simple hues +are unmixed in the sense that none partakes of the nature of the other; specifically, +blue 1s not yellowish, reddish, or greenish, and so on; these are psychologically +primary hues.” That is a psychological fact, so-called. To fñnd out from what +evidence this psychological fact was deduced, we must search very hard indeed +through all the literature: In the modern literature all we fñnd on the subject are +repeats of the same statement, or of one by a German psychologist, who uses +as one of his authorities Leonardo da Vinei, who, of course, we all know was a +great artist. He says, “Leonardo thought there were five colors.” Then, looking +still further, we fñnd, ïn a stïill older book, the evidence for the subject. The book +says something like this: “Purple is reddish-blue, orange is reddish-yellow, but +can red be seen as purplish-orange? Are not red and yellow more unitary than +purple or orange? 'Phe average person, asked to state which colors are unitary, +names red, yellow, and blue, these three, and some observers add a fourth, green. +--- Trang 633 --- +Psychologists are accustomed to accept the four as salient hues.” So that is the +situation in the psychological analysis of this matter: if everybody says there are +three, and somebody says there are four, and they want it to be four, it will be +four. That shows the difficulty with psychological researches. It is clear that we +have such feelings, but it is very diffcult to obtain much information about them. +So the other direction to go is the physiological direction, to fñnd out experi- +mentally what actually happens in the brain, the eye, the retina, or wherever, +and perhaps to discover that some combinations of impulses from various cells +move along certain nerve fibers. Incidentally, primary pigments do not have to +be in separate cells; one could have cells in which are mixtures of the various +pigments, cells with the red and the green pigments, cells with all three (the +information of all three is then white information), and so on. There are many +ways of hooking the system up, and we have to ñnd out which way nature has +used. It would be hoped, ultimately, that when we understand the physiological +connections we will have a little bit of understanding of some of those aspects of +the psychology, so we look in that direction. +36-2 The physiology of the eye +W© begin by talking not only about color vision, but about vision in general, +just to remind ourselves about the interconnections in the retina, shown in +lig. 35-2. The retina is really like the surface of the brain. Although the +actual picture through a microscope 1s a little more complicated looking than +this somewhat schematized drawing, by careful analysis one can see all these +Interconnections. 'There is no question that one part of the surface of the retina +1s connected to other parts, and that the information that comes out on the +long axons, which produce the optic nerve, are combinations of information from +many cells. There are three layers of cells in the succession of function: there +are retinal cells, which are the ones that the light affects, an intermediate cell +which takes information from a single or a few retinal cells and gives i% out again +to several cells in a third layer of cells and carries it to the brain. There are all +kinds of cross connections between cells in the layers. +We now turn to some aspects of the structure and performance of the eye +(see Eig. 35-1). The focusing of the light is accormmplished mainly by the cornea, +by the fact that it has a curved surface which “bends” the light. This is why we +cannot see clearly under water, because we then do not have enough diference +between the index of the cornea, which is 1.37, and that of the water, which +--- Trang 634 --- +1s 1.33. Behind the cornea is water, practically, with an index of 1.33, and behind +that is a lens which has a very interesting structure: ït is a series of layers, like an +onion, except that it is all transparent, and ït has an index of 1.40 in the middle +and 1.38 at the outside. (It would be nice if we could make optical gÌass in which +we could adjust the index throughout, for then we would not have to curve it as +much as we do when we have a uniform index.) Furthermore, the shape of the +cornea is not that of a sphere. Ä spherical lens has a certain amount of spherical +aberration. The cornea is “flatter” at the outside than is a sphere, in Just such a +mamner that the spherical aberration is less for the cornea than it would be 1f +we put a spherical lens in therel 'Phe light is focused by the cornea-lens system +onto the retina. As we look at things that are closer and farther away, the lens +tightens and loosens and changes the focus to adjust for the diferent distances. +To adJjust for the total amount of light there is the iris, which is what we call the +color of the eye, brown or blue, depending on who it is; as the amount of light +increases and decreases, the iris moves in and out. +Let us now look at the neural machinery for controlling the accommodation oŸ +the lens, the motion of the eye, the muscles which turn the eye in the socket, and +the iris, shown schematically in Eig. 36-3. Ofall the information that comes out of +the optic nerve 4, the great majority is divided into one of two bundles (which we +will talk about later) and thence to the brain. But there are a few fibers, of interest +to us now, which do not run directly to the visual cortex of the brain where we “see” +the images, but instead go into the mid-brain HH. These are the fbers which mea- +sure the average light and make adjustment for the iris; or, if the image looks foggy, +they try to correct the lens; or, if there is a double image, they try to adjust the +eye for binocular vision. At any rate, they go through the mid-brain and feed back +into the eye. At !{ are the muscles which run the accommodation of the lens, and +at Ù another one that runs into the iris. "The iris has ?wo muscle systems. Ône is a +circular muscle which, when ït is excited, pulls in and eloses down the iris; it acts +very rapidly and the nerves are directly connected from the brain through short +axons Iinto the iris. The opposite muscles are radial museles, so that, when the +things get dark and the circular muscle relaxes, these radial muscles pull out. Here +we have, as in many places in the body, a pair of muscles which work in opposite +directions, and in almost every such case the nerve systems which control the two +are very delicately adjusted, so that when signals are sent in to tighten one, signals +are automatically sent in to loosen the other. "The iris is a peculiar exception: the +nerves which make the iris contract are the ones we have already described, but +the nerves which make the iris ezpønd come out from no one knows exactly where, +--- Trang 635 --- +fứ) (` +“) Ằ¬ R +ồ B \, +M (2 ⁄2 +NG ° +Fig. 36-3. The neural interconnections for the mechanical operation +of the eyes. +'Temporal +2 SÀ` 2m. +N F Nasal_ + ⁄ +đ[# ` | "CDC2 +si cc, +cW° / sCDđĐ5 +Fig. 36-4. The neural connections from the eyes to the visual cortex. +--- Trang 636 --- +go down into the spinal cord back of the chest, into the thoracic sections, out of the +spinal cord, up through the neck ganglia, and all the way around and back up into +the head in order to run the other end of the iris. In fact, the signal goes through +a completely diferent nervous system, not the central nervous system at all, but +the sympathetic nervous system, so it is a very strange way of making things go. +W© have already emphasized another strange thing about the eye, that the +light-sensitive cells are on the wrong side, so that the light has to go through +several layers of other cells before it gets to the receptors—it is built inside outl +So some of the features are wonderful and some are apparently stupid. +Figure 36-4 shows the connections of the eye to the part of the brain which is +most directly concerned with the visual process. The optic nerve fbers run into a +certain area just beyond D, called the lateral geniculate, whereupon they run out +to a section of the brain called the visual cortex. Notice that some of the fñbers +from each eye are sent over to the other side of the brain, so the picture formed is +incomplete. “The optic nerves from the left side of the right eye run across the optic +chiasma #, while the ones on the left side of the left eye come around and go this +same way. So the left side of the brain receives all the information which comes +from the left side of the eyeball of each eye, ¡.e., on the right side of the visual +feld, while the right side of the brain sees the left side of the visual feld. 'This +is the manner in which the information from each of the two eyes is put together +in order to tell how far away things are. This is the system of binocular vision. +The connections bebween the retina and the visual cortex are interesting. If a +spot in the retina is excised or destroyed in any way, then the whole fñber will die, +and we can thereby find out where it is connected. It turns out that, essentially, +the connections are one to one——for each spot in the retina there is one spot in +the visual cortex—and spots that are very close together in the retina are very +close together in the visual cortex. So the visual cortex still represents the spatial +arrangement of the rods and cones, but of course much distorted. Things which +are in the center of the field, which occupy a very small part of the retina, are +expanded over many, many cells in the visual cortex. It is clear that it is useful +to have things which are originally close together, still close together. The most +remarkable aspect of the matter, however, is the following. The place where one +would think it would be most important to have things close together would be +right in the middle of the visual feld. Believe it or not, the up-and-down line in +our visual ñeld as we look at something is of such a nature that the information +from all the points on the right side of that line is going Into the left side of the +brain, and information from the points on the left side is going into the right side +--- Trang 637 --- +of the brain, and the way this area is made, there is a cut right down through +the middle, so that the things that are very close together right in the middle are +very far apart in the brain! Somehow, the information has to go from one side of +the brain to the other through some other channels, which is quite surprising. +The question of how this network ever gets “wired” together is very interesting. +The problem of how mụuch is already wired and how much is learned is an old one. +Tt used to be thought long ago that perhaps it does not have to be wired carefully +at all, it is only just roughly interconnected, and then, by experience, the young +child learns that when a thing is “up there” it produces some sensation in the +brain. (Doctors always tell us what the young child “feels,” but how do #hey know +what a child feels at the age of one?) The chỉld, at the age oŸ one, supposedÌy sees +that an object is “up there,” gets a certain sensation, and learns to reach “there,” +because when he reaches “here,” it does not work. 'That approach probably is not +correct, because we already see that in many cases there are these special detailed +interconnections. More illuminating are some most remarkable experiments done +with a salamander. (Incidentally, with the salamander there is a direct crossover +connection, without the optic chiasma, because the eyes are on each side of the +head and have no common area. Salamanders do not have binocular vision.) +The experiment is this. We can cut the optic nerve in a salamander and the +nerve will grow out from the eyes again. Thousands and thousands of cell ñbers +will thus re-establish themselves. Now, in the optic nerve the fbers do not stay +adjacent to each other——it is like a great, sloppily made telephone cable, all the +fbers twisting and turning, but when it gets to the brain they are all sorted out +again. When we cut the optie nerve of the salamander, the interesting question +1s, will it ever get straightened out? The answer is remarkable: yes. lÝ we cut +the optic nerve of the salamander and it grows back, the salamander has good +visual acuity again. However, iŸ we cut the optic nerve and turn the cục upside +đoưn and let 1t grow back again, it has good visual acuity all right, but ít has a +terrible error: when the salamander sees a ñy “up here,” i% Jumps at it “down +there,” and it never learns. Thherefore there is some mysterious way by which the +thousands and thousands of fñbers ñnd theïir right places in the brain. +'This problem of how much is wired in, and how much is not, is an important +problem in the theory of the development of creatures. 'Phe answer is not known, +but is being studied intensively. +The same experiment in the case of a goldfish shows that there is a terrible +knot, like a great scar or complication, in the optic nerve where we cut it, but in +spite of all this the fibers grow back to theïr right places in the brain. +--- Trang 638 --- +In order to do this, as they grow into the old channels of the optic nerve they +mmust make several decisions about the direction in which they should grow. How +do they do this? 'Phere seem to be chemical clues that diferent fbers respond to +diferently. Think of the enormous number of growing fbers, each of which is an +individual difering in some way from its neighbors; in responding to whatever +the chemical clues are, it responds in a unique enough way to find its proper +place for ultimate connection in the brain! “This is an interesting——a fantastic— +thing. It is one of the great recently discovered phenomena of biology and is +undoubtedly connected to many older unsolved problems of growth, organization, +and development of organisms, and particularly of embryos. +One other interesting phenomenon has to do with the motion of the eye. +The eyes must be moved in order to make the two images coincide in diferent +circumstances. 'These motions are of diferent kinds: one is to follow something, +which requires that both eyes must go in the same direction, right or left, and the +other is to poïint them toward the same place at various distances away, which +requires that they must move oppositely. The nerves going into the muscles of +the eye are already wired up for just such purposes. 'Phere is one set of nerves +which will pull the muscles on the inside of one eye and the outside oŸ the other, +and relax the opposite museles, so that the two eyes move together. There is +another center where an excitation will cause the eyes to move in toward each +other from parallel. Either eye can be turned out to the corner if the other eye +moves toward the nose, but it is impossible consciously or unconsciously to turn +both eyes øu£ at the same time, not because there are no ?wwscles, but because +there is no way to send a signal to turn both eyes out, unless we have had an +accident or there is something the matter, for instance iŸ a nerve has been cut. +Although the museles of one eye can certainly steer that eye about, not even a +Yogi is able to move bo¿h eyes out freely under voluntary control, because there +does not seem to be any way to do it. We are already wired to a certain extent. +This is an Important point, because most of the earlier books on anatomy and +psychology, and so on, do not appreciate or do not emphasize the fact that we +are so completely wired already—they say that everything is just learned. +30-3 The rod cells +Let us now examine in more detail what happens in the rod cells. Pigure 36-5 +shows an electron micrograph of the middle of a rod cell (the rod cell keeps +going up out of the field). There are layer after layer of plane structures, shown +--- Trang 639 --- +¡ asss —==Đ:.. +Si AniE==OÍ- +OS —mm¬¬aể "—-] +Ị =—== ai +I m=====s rs +¡ GGHHỊI là +cc : +\ \ II bó v*95 +‡ .. —> C1 “ạt +I er =Đ +IS + ST +' P ~ +Fig. 36-5. Electron micrograph of a rod cell. +magnified at the right, which contain the substance rhodopsin (visual purple), the +dye, or pigment, which produces the efects of vision in the rods. The rhodopsin, +which is the pigment, is a big protein which contains a special group called +retinene, which can be taken of the protein, and which is, undoubtedly, the +main cause of the absorption of light. We do not understand the reason for the +planes, but it is very likely that there is some reason for holding all the rhodopsin +molecules parallel. 'The chemistry of the thing has been worked out to a large +extent, but there might be some physics to it. It may be that all of the molecules +are arranged in some kind of a row so that when one is excited an electron which +1s generated, say, may run all the way down to some place at the end to get +the signal out, or something. 'This subject is very important, and has not been +worked out. It is a field in which both biochemistry and solid state physics, or +something like it, will ultimately be used. +'This kind of a structure, with layers, appears in other circumstances where +light is important, for example in the chloroplast in plants, where the light causes +photosynthesis. IÝ we magnify those, we fñnd the same thing with almost the +same kind of layers, but there we have chlorophyll, of course, instead of retinene. +--- Trang 640 --- +CHạ CHạ CHạ +CS ĐÁ +đ NN⁄ N⁄ Ñ⁄ ÑẶ⁄ Ñ⁄ N +Fig. 36-6. The structure of retinene. +The chemical form of retinene is shown in Eig. 36-6. It has a series of alternate +double bonds along the side chaïin, which is characteristic of nearly all strongly +absorbing organic substaneces, like chlorophyll, blood, and so on. 'This substanece +is Impossible for human beings to manufacture in their own cells—we have to +eat it. So we eat it in the form of a special substance, which is exactly the same +as retinene except that there is a hydrogen tied on the right end; ¡it ¡is called +vitamin A, and if we do not eat enough of it, we do not get a supply of retinene, +and the eye becomes what we call mógh# blnd, because there is then not enough +pigment in the rhodopsin to see with the rods at night. +The reason why such a series of double bonds absorbs light very strongly is +also known. We may just give a hint: The alternating series of double bonds +is called a con7ugated double bond; a double bond means that there is an extra +electron there, and this extra electron is easily shifted to the right or left. When +light strikes this molecule, the electron of each double bond is shifted over by one +step. All the electrons in the whole chai¡n shift, like a string of dominoes falling +over, and though each one moves only a little distance (we would expect that, in a +single atom, we could move the electron only a little distance), the net effect is the +same as though the one at the end was moved over to the other endl Tt is the same +as though one electron went the whole distance back and forth, and so, in this +manner, we get a much stronger absorption under the influence of the electric field, +than if we could only move the electron a distance which is associated with one +atom. 5o, sỉnce iÈ is easy to move the electrons back and forth, retinene absorbs +light very strongly; that is the machinery of the physical-chemical end of it. +36-4 The compound (insect) eye +Let us now return to biology. The human eye is not the only kind oŸ eye. +In the vertebrates, almost all eyes are essentially like the human eye. However, +in the lower animals there are many other kinds of eyes: eye spots, various eye +--- Trang 641 --- +cups, and other less sensitive things, whiích we have no time to discuss. But +there is one other highly developed eye among the invertebrates, the cornpownd +eye of the insect. (Most insects having large compound eyes also have various +additional simpler eyes as well.) A bee is an insect whose vision has been studied +very carefully. It is easy to sbudy the properties of the vision of bees because +they are attracted to honey, and we can make experiments in which we identify +the honey by putting ¡it on blue paper or red paper, and see which one they come +to. By this method some very interesting things have been discovered about the +vision of the bee. +In the first place, in trying to measure how acutely bees could see the color +diference between two pieces of “white” paper, some researchers found they +were not very good, and others found they were fantastically good. Even if the +two pieces of white paper were almost exactly the same, the bees could still tell +the diference. The experimenters used zinc white for one piece of paper and +lead white for the other, and although these look exactly the same to us, the +bee could easily distinguish them, because they refect a diferent amount in the +ultraviolet. In this way it was discovered that the bee”s eye is sensitive over a wider +range of the spectrum than is our own. Our eye works from 7000 angstroms to +4000 angstroms, from red to violet, but the bee°s can see down to 3000 angstroms +into the ultravioletl “This makes for a number of diferent interesting efects. +In the first place, bees can distinguish between many flowers which to us look +alike. Of course, we must realize that the colors of owers are not designed +for our eyes, but for the bee; they are signals to attract the bees to a specifc +flower. We all know that there are many “white” fowers. Apparently white is +not very interesting to the bees, because it turns out that all of the white fowers +have difÑferent proportions of reflection in the u#rœ0iolet; they do not reflect one +hundred percent of the ultraviolet as would a true white. All the light is not +coming back, the ultraviolet is missing, and that is a color, jusÈ as, for us, 1Í +the blue is missing, it comes out yellow. 5o, all the fiowers are colored for the +bees. However, we also know that red cannot be seen by bees. Thus we might +expect that all red fowers should look black to the bee. Ñot sol Ä careful study +of red fowers shows, frst, that even with our own eye we can see that a great +majority of red fowers have a bluish tinge because they are mainly refecting an +additional amount in the blue, which is the part that the bee sees. Purthermore, +experiments also show that fowers vary in their refection of the ultraviolet over +difÑferent parts of the petals, and so on. So if we could see the Ñowers as bees see +them they would be even more beautiful and varied! +--- Trang 642 --- +lt has been shown, however, that there are a few red flowers which do no# +refect in the blue or in the ultraviolet, and uould, therefore, appear black to the +beel This was of quite some concern to the people who worry about this matter, +because black does not seem like an interesting color, since iE is hard to tell from +a dirty old shadow. It actually turned out that these owers were øøf visited by +bees, these are the owers that are visited by hummingbirds, and hummingbirds +can see the redl +Another interesting aspect of the vision of the bee is that bees can apparently +tell the direction of the sun by looking at a patch of blue sky, without seeing +the sun itself. We cannot easily do this. If we look out the window at the sky +and see that it is blue, in which direction is the sun? The bee can tell, because +the bee is quite sensitive to the polarization of light, and the scattered light +of the sky is polarized.* There is still some debate about how this sensitivity +operates. Whether it is because the relections of the light are diferent in diferent +circumstances, or the bee”s eye is directly sensitive, is not yet known.† +Tt is also said that the bee can notice ficker up to 200 oscillations per second, +while we see it only up to 20. 'The motions of bees in the hives are very quick; the +feet move and the wings vibrate, but it is very hard for us 6o see these motions +with our eye. However, IÝ we could see more rapidly we would be able to see the +motfion. lt is probably very Important to the bee that its eye has such a rapid +T©eSDOHNSG. +Now let us discuss the visual acuity we could expect from the bee. “The +eye of a bee is a compound eye, and it is made of a large number of special +cells called ormmnatzdia, which are arranged conically on the surface of a sphere +(roughly) on the outside of the bee°s head. Eigure 36-7 shows a picture of one +such ommatidium. At the top there is a transparent area, a kind of “lens,” but +actually it is more like a filter or light pipe to make the light come down along +the narrow fiber, which is where the absorption presumnably occurs. Out of the +other end of it comes the nerve fiber. The central fber is surrounded on ïts sides +by six cells which, in fact, have secreted the fiber. hat is enough description +* "The human eye also has a slight sensitivity to the polarization of light, and one can learn +to tell the direction of the sunl "The phenomenon that is involved here is called Haidznger”s +brush; ït 1s a faint, yellowish hourglass-like pattern seen at the center of the visual feld when +one looks at a broad, featureless expanse using polarizing glasses. It can also be seen in the +blue sky without polarizing glasses if one rotates his head back and forth about the axis of +'VISI1ON. +† Evidence obtained since this lecture was given indicates that the eye is directly sensitive. +--- Trang 643 --- +Fig. 36-7. The structure of an ommatidium (a single cell of a com- +pound eye). +for our purposes; the point is that it is a conical thing and many can ft next to +cach other all over the surface of the eye of the bee. +Now let us discuss the resolution of the eye of the bee. If we draw lines +(Fig. 36-8) to represent the ommatidia on the surface, which we suppose is a +sphere of radius r, we may actually calculate how wide each ommatidium is +by using our brains, and assuming that evolution is as clever as we arel lÝ we +have a very large ommatidium we do not have much resolution. 'That is, one +cell gets a piece of information from one direction, and the adjacent cell gets a +plece of information from another direction, and so on, and the bee cannot see +things in bebween very well. 5o the uncertainty of visual acuity in the eye will +surely correspond to an angle, the angle of the end of the ommatidium relative +to the center of curvature of the eye. (The eye cells, of course, exist only at the +surface of the sphere; inside that is the head of the bee.) This angle, from one +--- Trang 644 --- +Fig. 36-8. Schematic view of packing of ommatidia in the eye of a bee. +ommatidium to the next, is, of course, the diameter of the ommatidia divided by +the radius of the eye surface: +Afy = ð/r. (36.1) +So, we may say, “The ñner we make the ở, the more the visual acuity. So why +doesn”t the bee just use very, very fine ommatidia?” Ansuer: We know enouph +physics to realize that if we are trying to get light down into a narrow sÌo, we +cannot see accurately in a given direction because of the difraction efect. The +light that comes from several directions can enter and, due to difraction, we will +get light coming in at angle A0; such that +A0x = A/ð. (36.2) +NÑow we see that if we make the ổ too small, then each ommatidium does not +look in only one direction, because of difractionl If we make them too big, each +one sees in a definite direction, but there are not enough of them to get a good +view of the scene. So we adjust the distance ổ in order to make minimal the total +efect of these two. IÝ we add the two together, and fñnd the place where the sum +has a minimum (Fig. 36-9), we ñnd that +đ(AØ, + A6) 1 À +——————=Ù=_-—= 36.3 +đỗ r ð2) (36.5) +which gives us a distance +ỗ = VÀ. (36.4) +TÍ we guess that r is about 3 millimeters, take the light that the bee sees as +4000 angstroms, and put the two together and take the square root, we find +ổ= (3x10 x4x10"”)12m += 3.5 x 107” m = 3ð ". (36.5) +--- Trang 645 --- +A0 +n8 x pôa +/ >`AØy =ð/r +„8u =A/ö +Fig. 36-9. The optimum size for an ommatidium Is ổm. +The book says the diameter is 30 , so that is rather good agreementl So, +apparently, it really works, and we can understand what determines the size of +the bee?s eyel It is also easy to put the above number back in and find out how +good the bee”s eye actually is in angular resolution; it is very poor relative to +our own. We can see things that are thirty times smaller in apparent size than +the bee; the bee has a rather fuzzy out-of-focus image relative to what we can +see. Nevertheless it is all right, and it is the best they can do. We might ask +why the bees do not develop a good eye like our own, with a lens and so on. +'There are several interesting reasons. In the frst place, the bee is too small; ïf it +had an eye like ours, but on his scale, the opening would be about 30 in size +and diÑfraction would be so important that it would not be able to see very well +anyway. 'Phe eye is not good ïÍ it is too small. Secondly, if it were as big as the +bee's head, then the eye would occupy the whole head of the bee. The beauty of +the compound eye is that it takes up no space, it is just a very thin layer on the +surface of the bee. So when we argue that they should have done it our way, we +must remember that they had their own problemsl +36-5 Other eyes +Besides the bees, many other animals can see color. Eish, butterfies, birds, +and reptiles can see color, but ït is believed that most mammals cannot. "The +primates can see color. 'Phe birds certainly see color, and that accounts for the +colors of birds. Thhere would be no point in having such brilliantly colored males +1f the females could not notice itl "That is, the evolution of the sexual “whatever +1t 1s” that the birds have is a result of the female being able to see color. 5o next +time we look at a peacock and think of what a brilliant display of gorgeous color +1t is, and how delicate all the colors are, and what a wonderful aesthetic sense it +takes to appreciate all that, we should not compliment the peacock, but should +--- Trang 646 --- +compliment the visual acuity and aesthetic sense of the peahen, because that is +what has generated the beautiful scenel +All invertebrates have poorly developed eyes or compound eyes, but all the +verbebrates have eyes very similar to our own, with one exception. If we consider +the highest form of animal, we usually say, “Here we arel,” but if we take a less +prejudiced point of view and restrict ourselves to the invertebrates, so that we +cannot inelude ourselves, and ask what ¡is the highest invertebrate animal, most +zoologists agree that the ocfopus is the highest animaill It is very interesting +that, besides the development of its brain and its reactions and so on, which are +rather good for an invertebrate, it has also developed, independently, a different +eye. Ït is not a compound eye or an eye spot—it has a cornea, i§ has lids, it has +an iris, it has a lens, it has two regions of water, it has a retina behind. It is +essentially the same as the eye of the vertebratesl It is a remarkable example of +a coincidence in evolution where nature has twice discovered the same solution +to a problem, with one slipht improvement. In the octopus it also turns out, +amazingly, that the retina is a piece of the brain that has come out in the same +way in its embryonic development as is true for vertebrates, but the interesting +thing which is diferent is that the cells which are sensitive to light are on the +¿nside, and the cells which do the calculation are in back of them, rather than +“inside out,” as in our eye. So we see, at least, that there is no good reason for +its being inside out. The other time nature tried it, she got i% straightened outl +⁄. —_ —= ~ +L ⁄⁄⁄⁄ —— — +// ⁄ . Nà +ầ šƑ:. +\ x< s +C ^^ Ề L1 Lý +NN z.. +NN k +—3 «fffE---[[-~-~~~- +ELECTRON ` 2 P +WALL BACKSTOP Pị =|d¡|2 Pa =lới +ớa|2 +P; = |ớa|Ÿ +(a) (b) (c) +Fig. 37-3. lnterference experiment with electrons. +--- Trang 661 --- +In front of the backstop we place a movable detector. The detector might be a +geiger counter or, perhaps better, an electron multiplier, which is connected to +a loudspeaker. +We should say right away that you should not try to set up this experiment (as +you could have done with the two we have already described). 'This experiment +has never been done in just this way. 'Phe trouble is that the apparatus would +have to be made on an impossibly small scale to show the effects we are interested +in. We are doïng a “thought experiment,” which we have chosen because it is easy +to think about. We know the results that ouwld be obtained because there are +many experiments that have been done, in which the scale and the proportions +have been chosen to show the efects we shall describe. +'The first thing we notice with our electron experiment is that we hear sharp +“clicks” from the detector (that is, from the loudspeaker). And all “clicks” are +the same. 'Phere are no “half-clicks.” +W©e would also notice that the “clicks” come very erratically. Something like: +just as you have, no doubt, heard a geiger counter operating. If we count the +clicks which arrive in a sufficiently long time—say for many minutes—and then +count again for another equal period, we fnd that the two numbers are very +nearly the same. 5o we can speak of the auerage rơ‡e at which the clicks are +heard (so-and-so-many clicks per minute on the average). +As we move the detector around, the rø£e at which the clicks appear is faster +or slower, but the size (loudness) of each click is always the same. TÍ we lower +the temperature of the wire in the gun the rate of clicking slows down, but still +each click sounds the same. We would notice also that if we put two separate +detectors at the backstop, one ør the other would click, but never both at onece. +(Except that once in a while, if there were two clicks very close together in tỉme, +our ear might not sense the separation.) We conclude, therefore, that whatever +arrives at the backstop arrives in “lumps.” ATI the “lumps” are the same size: +only whole “lumps” arrive, and they arrive one at a time at the backstop. We +shall say: “Electrons always arrive in identical lumps.” +Just as for our experiment with bullets, we can now proceed to find exper- +Imentally the answer to the question: “What is the relative probability that +an electron “lumpˆ will arrive at the backstop at various distances ø from the +center?” As before, we obtain the relative probability by observing the rate of +clicks, holding the operation of the gun constant. “The probability that lumps +will arrive at a particular + is proportional to the average rate of clicks at that z. +--- Trang 662 --- +The result of our experiment is the interesting curve marked 1; in part (c) +Of Eig. 37-3. Yesl That is the way electrons go. +37-5 The interference of electron waves +Now let us try to analyze the curve of Fig. 37-3 %o see whether we can +understand the behavior of the electrons. 'Phe first thing we would say is that +since they come in lumps, each lump, which we may as well call an electron, has +come either through hole 1 or through hole 2. Let us write this in the form of a +“Proposition”: +ProposiHon A: Pach electron either goes through hole 1 ør i9 goes through +hole 2. +Assuming Proposition A, all electrons that arrive at the backstop can be +divided into two classes: (1) those that come through hole 1, and (2) those that +come through hole 2. 5o our observed curve must be the sum oŸ the effects of +the electrons which come through hole 1 and the electrons which come through +hole 2. Let us check this idea by experiment. First, we will make a measurement +for those electrons that come through hole 1. We block of hole 2 and make +our counts of the clicks from the detector. Erom the clicking rate, we get ¡. +The result of the measurement is shown by the curve marked ) in part (b) of +Fig. 37-3. The result seems quite reasonable. In a similar way, we measure , +the probability distribution for the electrons that come through hole 2. 'Phe +result of this measurement is also drawn in the fñgure. +The result ¿ obtained with boø¿h holes open is clearly not the sum of ị +and , the probabilities for each hole alone. In analogy with our water-wave +experiment, we say: “'Phere is interference.” +or clectrons: ha # h + Đ. (37.5) +How can such an interference come about? Perhaps we should say: “Well, +that means, presumably, that i% is no# true that the lumps go either through +hole 1 or hole 2, because if they did, the probabilities should add. Perhaps they +go in a more complicated way. They split in half and...” But nol They cannot, +they always arrive in lumps.... “Well, perhaps some of them go through 1, and +then they go around through 2, and then around a few more times, or by some +other complicated path... then by closing hole 2, we changed the chance that +an electron that sfarfed out through hole 1 would ñnally get to the backstop.... ” +--- Trang 663 --- +But noticel There are some points at which very few electrons arrive when bo£h +holes are open, but which receive many electrons If we close one hole, so cÏos?ng +one hole zncreased the number from the other. Notice, however, that at the +center of the pattern, 1a is more than twice as large as ị + ạ. It ¡is as though +closing one hole decreased the number of electrons which come through the other +hole. It seems hard to explain bo#h efects by proposing that the electrons travel +in complicated paths. +lt is all quite mysterious. And the more you look at it the more mmysterious +1t seems. Many ideas have been concocted to try to explain the curve for 1a +in terms of individual electrons goïng around in complicated ways throupgh the +holes. None of them has succeeded. None of them can get the right curve for Pa +in terms of ¡ and H. +Yet, surprisingly enough, the rmathemaftics for relating P?¡ and › to Địa 1s +extremely simple. For Ọa¿ is jusê like the curve Ïq¿ of Eig. 37-2, and f#hø was +simple. What is going on at the backstop can be described by two complex +numbers that we can call ói and óa (they are functions oŸ #, of course). The +absolute square of ôi gives the efect with only hole 1 open. That is, Pị = NG +The efect with only hole 2 open is given by ôa in the same way. That is, +Đ = lộa|?. And the combined efect of the two holes is just Địa = lôi + ôa|Ê. +The mafhemaiics is the same as that we had for the water wavesl (It is hard to +see how one could get such a simple result from a complicated game of electrons +going back and forth through the plate on some strange trajectory.) +W©e conclude the following: 'Phe electrons arrive in lumps, like particles, and +the probability of arrival of these lumps is distributed like the distribution of +intensity of a wave. It is in this sense that an electron behaves “sometimes like a, +particle and sometimes like a wave.” +Tncidentally, when we were dealing with classical waves we defned the intensity +as the mean over time of the square of the wave amplitude, and we used complex +numbers as a mathematical trick to simplify the analysis. But in quantum +mechanics it turns out that the amplitudes zmws be represented by complex +numbers. The real parts alone wïll not do. 'Phat is a technical point, for the +mmoment, because the formulas look just the same. +Since the probability of arrival through both holes is given so simply, although +1E 1s not equal to (ị + P)), that is really all there is to say. But there are a +large number oŸ subtleties involved in the fact that nature does work this way. +W©e would like to illustrate some of these subtleties for you now. Eirst, since the +--- Trang 664 --- +number that arrives at a particular point is not equal to the number that arrives +through 1 plus the number that arrives through 2, as we would have concluded +from Proposition A, undoubtedly we should conclude that Proposition A ¡s false. +Tt is no£ truc that the electrons go e#her through hole 1 or hole 2. But that +conclusion can be tested by another experiment. +37-6 Watching the electrons +We shall now try the following experiment. To our electron apparatus we add +a very strong light source, placed behind the wall and between the two holes, +as shown in Fig. 37-4. We know that electric charges scatter light. So when an +electron passes, however it does pass, on its way to the detector, it will scatter +some light to our eye, and we can see where the electron goes. Ïlf, for instance, +an electron were to take the path via hole 2 that is sketched in Eig. 37-4, we +should see a fash of light coming from the vicinity of the place marked A in the +fñgure. If an electron passes through hole 1 we would expect to see a flash om +the vicinity of the upper hole. If ít should happen that we get light from both +places at the same time, because the electron divides in half... Let us Just do +the experimentl +Ì IISHT +TT... +—¬ ¬——.« +ELECTRON 2 P, +PỊ, = Pj + P2 +(a) (b) (c) +Fig. 37-4. A different electron experiment. +Here is what we see: c0erw time that we hear a “click” from our electron +detector (at the backstop), we aÏso see a flash of light e#her near hole l ør near +hole 2, but neuer both at oncel And we observe the same result no matter where +--- Trang 665 --- +we put the detector. Erom this observation we conclude that when we look at +the electrons we fñnd that the electrons go either through one hole or the other. +JExperimentally, Proposition AÁ is necessarily true. +'What, then, is wrong with our argument øgø#ns‡ Proposition A? Why isnt Đa +Jjust equal to ị + f2? Back to experimentl Let us keep track of the electrons +and fnd out what they are doing. Eor each position (z-location) of the detector +we will count the electrons that arrive and aiso keep track of which hole they +went through, by watching for the ñashes. We can keep track of things this +way: whenever we hear a “click” we will put a count in Colummn 1 if we see the +fash near hole 1, and if we see the flash near hole 2, we will record a count +in Column 2. Every electron which arrives is recorded in one oŸ two cÌasses: +those which come through 1 and those which come through 2. Erom the number +recorded in Column 1 we get the probability Pj that an electron will arrive at +the detector via hole 1; and from the number recorded in Column 2 we get 2, +the probability that an electron will arrive at the detector via hole 2. If we now +repeat such a measurement for many values of z, we get the curves for fƒ and +shown in part (b) of Fig. 37-4. +Well, that is not too surprisingl We get for j something quite similar to +what we got before for P¡ by blocking off hole 2; and Độ is similar to what we got +by blocking hole 1. So there is no any complicated business like going through +both holes. When we watch them, the electrons come through just as we would +expect them to come through. Whether the holes are closed or open, those which +we see come through hole 1 are distributed in the same way whether hole 2 is +open or closed. +But waitl What do we have no for the £o£øl probability, the probability +that an electron will arrive at the detector by any route? We already have that +Information. We just pretend that we never looked at the light ñashes, and we +lump together the detector clicks which we have separated into the two columns. +W© must just ødd the numbers. Eor the probability that an electron will arrive +at the backstop by passing through e#her hole, we do fnd P1; = P{ + F¿. That +is, although we succeeded in watching which hole our electrons come through, +we no longer get the old interference curve sa, but a new one, Pí„ showing no +interferencel If we turn out the light Ta is restored. +W©e must conclude that t0hen te look at the electrons the distribution of them +on the sereen ¡is diferent than when we do not look. Perhaps it is turning on +our light source that disturbs things? It must be that the electrons are very +delicate, and the light, when it scatters of the electrons, gives them a jolt that +--- Trang 666 --- +changes their motion. We know that the electric fñeld of the light acting on a +charge will exert a force on it. So perhaps we shøouldở expect the motion to be +changcd. Anyway, the light exerts a big iniuence on the electrons. By trying to +“watch” the electrons we have changed their motions. 'That is, the Jolt given to +the electron when the photon is scattered by it is such as to change the electronˆs +motion enough so that if it rm2gh# have gone to where P¿ was at a maximum ï1§ +will instead land where Pa was a minimum; that is why we no longer see the +wavy interference effects. +You may be thinking: “Don”t use such a bright sourcel Turn the brightness +down! The light waves will then be weaker and will not disturb the electrons so +much. Surely, by making the light dimmer and dimmer, eventually the wave will +be weak enough that it will have a negligible efect.” O.K. Let's try it. The frst +thing we observe is that the fashes of light scattered from the electrons as they +pass by does no get weaker. lý ¿s alt0aUs the same-sizcd flash. The only thing +that happens as the light is made dimmer is that sometimes we hear a “click” +from the detector but see mo fÏlash at all. The electron has gone by without being +“seen.” What we are observing is that light aiso acts like electrons, we kneu that +it was “wavy,” but now we find that it is also “lumpy.” It always arrives—or is +scattered——in lumps that we call “photons.” As we turn down the ?mtensit of +the light source we do not change the s2ze of the photons, only the ra#e at which +they are emitted. 75ø explains why, when our source is dim, some electrons get +by without being seen. 'Phere did not happen to be a photon around at the time +the electron went through. +This is all a little discouraging. TỶ it ïs true that whenever we “see” the electron +we see the same-sized Ñash, then those electrons we see are ø/øs the disturbed +ones. Let us try the experiment with a dim light anyway. Now whenever we +hear a click in the debector we will keep a count in three columns: in Column (1) +those electrons seen by hole 1, in Column (2) those electrons seen by hole 2, +and in Column (3) those electrons not seen at all. When we work up our data, +(computing the probabilities) we fnd these results: Those “seen by hole 1” have +a distribution like P{; those “seen by hole 2” have a distribution like Đÿ (so that +those “seen by either hole 1 or 2” have a distribution like f{¿); and those “not +seen at all” have a “wavy” distribution just like 1a of Eig. 37-3Ì ]ƒ the electrons +đre not scen, tue hœue interferencel +'That is understandable. When we do not see the electron, no photon disturbs +it, and when we do see it, a photon has disturbed it. There is always the same +amount of disturbance because the light photons all produce the same-sized +--- Trang 667 --- +effects and the efect of the photons being scattered is enough to smear out any +interference effect. +1s there not søme way we can see the electrons without disturbing them? +W© learned in an earlier chapter that the momentum carried by a “photon” is +inversely proportional to its wavelength (p = h/A). Certainly the jolt given +to the electron when the photon is scattered toward our eye depends on the +momentum that photon carries. Ahal If we want to disturb the electrons onÌy +sliphtly we should not have lowered the 7m£ensit of the light, we should have +lowered its ƒreqguencw (the same as increasing its wavelength). Let us use light of +a redder color. We could even use infrared light, or radiowaves (like radar), and +“see” where the electron went with the help of some equipment that can “see” +light of these longer wavelengths. If we use “gentler” light perhaps we can avoid +disturbing the electrons so much. +Let us try the experiment with longer waves. We shall keep repeating our +experiment, each time with light of a longer wavelength. At first, nothing seems to +change. The results are the same. Then a terrible thing happens. You remember +that when we discussed the microscope we pointed out that, due to the œøe +na‡ure oŸ the light, there is a limitation on how close Ewo spots can be and still +be seen as two separate spots. This distance is of the order of the wavelength of +light. So now, when we make the wavelength longer than the distance between +our holes, we see a ÙZø fuzzy fash when the light is scattered by the electrons. +We can no longer tell which hole the electron went throughl We just know it went +somewherel And it is just with light of this color that we ñnd that the jolts given +to the elecbron are small enough so that Pí; begins to look like PĐịa—that we +begin to get some interference efect. And it is only for wavelengths much longer +than the separation of the two holes (when we have no chance at all of telling +where the electron went) that the disturbance due to the light gets sufficiently +small that we again get the curve 1a shown ín Eig. 37-3. +In our experiment we fnd that it is impossible to arrange the light in such +a way that one can tell which hole the electron went through, and at the same +time not disturb the pattern. It was suggested by Heisenberg that the then +new laws of nature could only be consistent iŸ there were some basic limitation +on our experimental capabilities not previously recognized. He proposed, as a +general principle, his wwcertaintU pr¿nciple, which we can state in terms of our +experiment as follows: “l% is Impossible to design an apparatus to determine +which hole the electron passes throuph, that will not at the same time disturb the +electrons enough to destroy the interference pattern.” lf an apparatus is capable +--- Trang 668 --- +of determining which hole the electron goes through, it cønnot be so delicate +that it does not disturb the pattern in an essential way. No one has ever found +(or even thought of) a way around the uncertainty principle. Šo we must assume +that it describes a basic characteristic of nature. +The complete theory of quantum mechanics which we now use to describe +atoms and, in fact, all matter depends on the correctness of the uncertainty +principle. Since quantum mechanics is such a successful theory, our belief in +the uncertainty principle is reinforced. But IÝ a way to “beat” the uncertainty +principle were ever discovered, quantum mechanics would give inconsistent results +and would have to be discarded as a valid theory of nature. +“Well,” you say, “what about Proposition A? It is true, or is it noø£ true, that +the electron either goes through hole 1 or i% goes through hole 2?” 'Phe only +answer that can be given is that we have found from experiment that there is +a certain special way that we have to think in order that we do not get into +inconsistencies. What we must say (to avoid making wrong predictions) is the +following. lf one looks at the holes or, more accurately, if one has a piece of +apparatus which is capable of determining whether the electrons go through +hole 1 or hole 2, then one cøn say that it goes either through hole 1 or hole 2. +but, when one does øø£ try to tell whích way the electron goes, when there 1s +nothing in the experiment to disturb the electrons, then one may øœø‡ say that +an electron goes either through hole 1 or hole 2. If one does say that, and starts +to make any deductions from the statement, he will make errors in the analysis. +Thịs is the logical tightrope on which we must walk if we wish to describe nature +successfully. +T the motion of all matter——as well as electrons—must be described in terms +of waves, what about the bullets in our first experiment? Why didn't we see +an interference pattern there? It turns out that for the bullets the wavelengths +were so tiny that the interference patterns became very fine. So fine, ¡in fact, that +with any detector of fñnite size one could not distinguish the separate maxima +and minima. What we saw was only a kind of average, which is the classical +curve. In Fig. 37-5 we have tried to indicate schematically what happens with +large-scale objects. Part (a) of the figure shows the probability distribution +one might predict for bullets, using quantum mechanics. The rapid wiggles are +supposed to represent the interference pattern one gets for waves of very short +wavcleneth. Any physical detector, however, straddles several wiggles of the +--- Trang 669 --- +Ỉ ha : (smoothed) +(a) (b) +Fig. 37-5. Interference pattern with bullets: (a) actual (schematic), +(b) observed. +probability curve, so that the measurements show the smooth curve drawn In +part (b) of the fgure. +37-7 First principles of quantum mechanics +We will now write a summary of the main conclusions of our experiments. We +will, however, put the results in a form which makes them true for a general class +of such experiments. We can write our sunmary more simply iŸ we fñrst deflne +an “ideal experiment” as one in which there are no uncertain external infÑuences, +1.e., no jigsling or other things going on that we cannot take into account. WWe +would be quite precise if we said: “An ideal experiment is one in which all of the +initial and fñnal conditions of the experiment are completely specified” What we +will call “an event” is, in general, just a specifc set of initial and fñnal conditions. +(For example: “an electron leaves the gun, arrives at the detector, and nothing +else happens.”) Ñow for our summary. +ĐUMMARY +(1) The probability of an event in an ideal experiment is given by the square +of the absolute value of a complex number @ which is called the probability +amplitude: +P = probability, +ở = probability amplitude, (37.6) +P= li. +--- Trang 670 --- +(2) When an event can occur in several alternative ways, the probability +amplitude for the event is the sum of the probability amplitudes for each +way considered separately. 'There is interference: +Ó = 0i TÓ2, +P=lói + ó2. (37.7) +(3) lf an experiment is performed which is capable of determining whether one +or another alternative is actually taken, the probability of the event is the +sum of the probabilities for each alternative. 'Phe interference 1s lost: +P=h+h. (37.8) +One might still like to ask: “How does it work? What is the machinery +behind the law?” No one has found any machinery behind the law. Ño one can +“explain” any more than we have just “explained.” Ño one will give you any deeper +representation of the situation. We have no ideas about a more basic mechanism +from which these results can be dedueed. +We uould like to emphasize a 0eru ïmportant difƒerence betueen cÏlassical œnd +quantum mmechanics. We have been talking about the probability that an electron +will arrive in a given circumstance. We have implied that in our experimental +arrangement (or even in the best possible one) it would be impossible to predict +exactly what would happen. We can only predict the oddsl 'This would mean, if +1t were true, that physics has given up on the problem of trying to predict exactly +what will happen in a defnite circumstance. Yesl physics høs given up. We do +not‡ knou hou to predict thất t0ould happen ín a giuen circwmuns‡ance, and we +believe now that it is Impossible, that the only thing that can be predicted is the +probability of diferent events. It must be recognized that this is a retrenchment +in our earlier ideal of understanding nature. It may be a backward step, but no +one has seen a way to avoid ït. +We make now a few remarks on a suggestion that has sometimes been made +to try to avoid the description we have given: “Perhaps the electron has some +kind of internal works—some inner variables—that we do not yet know about. +Perhaps that is why we cannot predict what will happen. If we could look more +closely at the electron we could be able to tell where iÿ would end up.” 5o far as +we know, that is Impossible. We would still be ín difficulty. Suppose we were to +assume that inside the electron there is some kind of machinery that determines +--- Trang 671 --- +where it is going to end up. hat machine must øiso determine which hole it 1s +going to go through on its way. But we must not forget that what ïs inside the +electron should not be dependent on what we do, and in particular upon whether +we open or close one of the holes. So 1ƒ an electron, before It starts, has already +made up its mind (a) which hole it is going to use, and (b) where it is goïing to +land, we should fnd ¡ for those electrons that have chosen hole 1, ¿ for those +that have chosen hole 2, awd necessarifu the sum PP + › for those that arrive +through the two holes. 'Phere seems to be no way around this. But we have +verified experimentally that that is not the case. And no one has figured a way +out of this puzzle. 5o at the present time we must limit ourselves to computing +probabilities. We say “at the present time,” but we suspect very strongly that +1E is something that will be with us forever—that it is impossible to beat that +puzzle—that this is the way nature really ¿s. +37-8 The uncertainty principle +'This is the way Heisenberg stated the uncertainty principle originally: If you +make the measurement on any object, and you can determine the #-component +Of its momentum with an uncertainty Ấø, you cannot, at the same time, know its +z-position more accurately than Az > h/2Ap. The uncertainties in the position +and momentum at any instant must have their product greater than half the +reduced Planck constant. 'This is a special case of the uncertainty principle that +was siated above more generally. 'Phe more general statement was that one +cannot design equipment in any way to determine which of two alternatives is +taken, without, at the same time, destroying the pattern of interference. +Let us show for one particular case that the kind ofrelation given by Heisenberg +must be true in order to keep from getting into trouble. We imagine a modification +of the experiment of Fig. 37-3, in which the wall with the holes consists of a plate +mounted on rollers so that it can move freely up and down (ïn the z-direction), +as shown in Eig. 37-6. By watching the motion of the plate carefully we can +try to tell which hole an electron goes through. Imagine what happens when +the detector is placed at z =0. We would expect that an electron which passes +through hole 1 must be defected downward by the plate to reach the detector. +Since the vertical component of the electron momentum is changed, the plate +must recoil with an equal momentum in the opposite direction. The plate will +get an upward kick. If the electron goes through the lower hole, the plate should +feel a downward kick. It is clear that for every position of the detector, the +--- Trang 672 --- +ROLLERS +v 1 [92 +xzZY- ~=~~“Tq ~ ~~DETECTOR +ELECTRON ` "TW p, +GUN ^ “lap, +MOTION FREE|l| +ROLLERS +WALL BACKSTOP +Fig. 37-6. An experiment in which the recoil of the wall is measured. +mmomentum received by the plate will have a different value for a traversal vỉa, +hole 1 than for a traversal via hole 2. Sol Without disturbing the electrons ø£ +all, but just by watching the piø/e, we can tell which path the electron used. +Now in order to do this it is necessary to know what the momentum of the +screen is, before the electron goes through. 5o when we measure the momentun +after the electron goes by, we can fñgure out how much the plate's momentum has +changed. But remember, according to the uncertainty principle we cannot at the +same time know the position of the plate with an arbitrary accuracy. But if we do +not know exactly uhere the plate is we cannot say precisely where the two holes +are. They will be in a difÑferent place for every electron that goes through. “This +means that the center of our interference pattern will have a diferent location for +cach electron. The wiggles of the interference pattern will be smeared out. We +shall show quantitatively in the next chapter that If we determine the momentum +of the plate sufficiently accurately to determine from the recoil measurement +which hole was used, then the uncertainty in the z-position of the plate will, +according to the uncertainty principle, be enough to shift the pattern observed at +the detector up and down in the z-direction about the distance from a maximum +to its nearest minimum. Such a random shift is just enough to smear out the +pattern so that no interference is observed. +The uncertainty principle “protects” quantum mechanics. Heisenberg rec- +ognized that if it were possible to measure the momentum and the position +simultaneously with a greater accuracy, the quantum mechanics would collapse. +So he proposed that it must be impossible. 'Phen people sat down and tried to +--- Trang 673 --- +figure out ways of doïng it, and nobody could fgure out a way to measure the +position and the momentum of anything——a screen, an electron, a billiard ball, +anything—with any greater accuracy. Quanbum mechanics maintains is perilous +but accurate existence. +--- Trang 674 --- +Tho Holqfforn oŸ WW@œto (rteÏ +MParticlo WiosrjppoirÉs +38-1 Probability wave amplitudes +In this chapter we shall discuss the relationship of the wave and particle +viewpoints. We already know, from the last chapter, that neither the wave +viewpoint nor the particle viewpoint is correct. sually we have tried to present +things accurately, or at least precisely enough that they will not have to be +changed when we learn more—it may be extended, but it will not be changedl +But when we try to talk about the wave picture or the particle picture, both are +approximate, and both will change. 'Pherefore what we learn in this chapter will +not be accurate in a certain sense; it is a kind of halEintuitive argument that will +be made more precise later, but certain things will be changed a little bit when +we interpret them correctly in quantum mechanics. The reason for doïng such a +thing, of course, is that we are not goïng to go directly into quantum mechanics, +but we want to have at least some idea of the kinds of efects that we will fñnd. +Purthermore, all our experiences are with waves and with particles, and so I£ is +rather handy to use the wave and particle ideas to get some understanding of +what happens in given circumstances before we know the complete mathematics +of the quantum-mechanical amplitudes. We shall try to ïllustrate the weakest +places as we go along, but most of it is very nearly correct——it is just a matter of +1nterpretation. +First of all, we know that the new way of representing the world in quantum +mechanics—the new Íframework—is to give an amplitude for every event that can +occur, and ïf the event involves the reception of one particle then we can give the +amplitude to ñnd that one particle at diferent places and at diferent times. The +probability of ñnding the particle is then proportional to the absolute square of +the amplitude. In general, the amplitude to fnd a particle in diferent places at +diferent times varies with position and time. +--- Trang 675 --- +In a special case the amplitude varies sinusoidally in space and time like +ci(6t—krr) (do not forget that these amplitudes are complex numbers, not real +numbers) and involves a defnite frequency œ and wave number &. Then it turns +out that this corresponds to a classical limiting situation where we would have +believed that we have a particle whose energy #/ was known and is related to the +frequency by +ý = hư, (38.1) +and whose momentum ø is also known and ¡ïs related to the wave number by +p= hk. (38.2) +This means that the idea of a particle is limited. "The idea of a particle— +1ts location, Its momentum, etc.—which we use so mụch, is in certain ways +unsatisfactory. For instance, If an amplitude to fnd a particle at diferent places +is given by eff—*)whose absolute square is a constant, that would mean that +the probability of ñnding a partiele is the same at all points. That means we do +not know t+0here 1 is—it can be anywhere—there is a great uncertainty in its +location. +Ôn the other hand, if the position of a particle is more or less well known and +we can predict it fairly accurately, then the probability of ñnding ï% in diferent +places must be confined to a certain region, whose length we call Az. Outside +this region, the probability is zero. Now this probability is the absolute square of +an amplitude, and if the absolute square is zero, the amplitude is also zero, so +that we have a wave train whose length is Az (Fig. 38-1), and the wavelength +(the distance bebween nodes oŸ the waves in the train) of that wave train is what +corresponds to the particle momentum. +Here we encounter a strange thing about waves; a very simple thing which +has nothing to do with quantum mechanies strictly. It is something that anybody +who works with waves, even if he knows no quantum mechanics, knows: namely, +tue cœnnot‡ define a unique tu0uauelength [or a shorÈ tuaue train. Sụch a wave traïn +^^ TT TnaAÀx— +Fig. 38-1. A wave packet of length Ax. +--- Trang 676 --- +does not haue a defnite wavelength; there is an indefniteness in the wave number +that is related to the fñnite length of the train, and thus there is an indefniteness +in the momentum. +38-2 Measurement of position and momentum +Let us consider two examples of this idea—to see the reason why there is an +uncertainty in the position and/or the momentum, if quantum mechanics is right. +W© have also seen before that ¡f there were not such a thing—If it were possible +to measure the position and the momentum of anything simultaneously——we +would have a paradox; It is fortunate that we do not have such a paradox, and +the fact that such an uncertainty comes naturally from the wave picture shows +that everything is mutually consistent. +—> {B “" +Fig. 38-2. Diffraction of particles passing through a slit. +Here is one example which shows the relationship bebween the position and +the momentum ïn a circumstance that is easy to understand. Suppose we have a +single slit, and particles are coming om very far away with a certain energy——sO +that they are all coming essentially horizontally (Eig. 38-2). We are going to +concentrate on the vertical components of momentum. All of these particles have +a certain horizontal momentum øạ, say, In a classical sense. So, in the classical +sense, the vertical momentum ø„, before the particle goes through the hole, is +defñnitely known. 'Phe particle is moving neither up nor down, because it came +from a source that is far away—and so the vertical momentum is OŸ cOurse zero. +But now let us suppose that it goes through a hole whose width is Ø. Then +after it has come out through the hole, we know the position vertically——the +--- Trang 677 --- +position—with considerable accuracy—namely +.* That is, the uncertainty +in position, Aø#, is of order . NÑow we might also want to say, since we know the +momentum is absolutely horizontal, that Apy is 2ero; but that is wrong. We once +knew the momentum was horizontal, but we do not know it any more. Before the +particles passed through the hole, we did not know their vertical positions. Now +that we have found the vertical position by having the particle come through the +hole, we have lost our information on the vertical momentuml Why? According +to the wave theory, there is a spreading out, or difÑfraction, of the waves after +they go through the slit, just as for light. Therefore there is a certain probability +that particles coming out of the slit are not coming exactly straight. The pattern +1s spread out by the difraction efect, and the angle of spread, which we can +defñne as the angle of the first minimum, is a measure of the uncertainty in the +ñnal angile. +How does the pattern become spread? 'To say ït is spread means that there +1s some chance for the particle to be moving up or down, that is, to have a +component of momentum up or down. We say chance and particle because we +can detect this difraction pattern with a particle counter, and when the counter +receives the particle, say at in Eig. 38-2, it receives the enfire particle, so that, +in a classical sense, the particle has a vertical momentum, in order to get from +the slit up to Œ. +To get a rough idea of the spread of the momentum, the vertical momentum 7„ +has a spread which is equal to øọ A0, where øo is the horizontal momentum. +And how bịg is AØ in the spread-out pattern? We know that the first minimum +Occurs at an angle AØ such that the waves from one edge of the slit have to travel +one wavelength farther than the waves from the other side—we worked that +out before (Chapter 30). Therefore AØ is À/, and so Az„ ïn this experiment +1s poÀ/B. Note that iƒ we make smaller and make a more accurate measurement +of the position of the particle, the difÑfraction pattern gets wider. Remember, +when we closed the slits on the experiment with the microwawves, we had more +Intensity farther out. So the narrower we make the slit, the wider the pattern +gets, and the more is the likelihood that we would fñnd that the particle has +sidewise momentum. Thus the uncertainty in the vertical momentum is inversely +proportional to the uncertainty oŸ . In fact, we see that the product of the +two is equal to øọAÀ. But À is the wavelength and øọ is the momentum, and in +* More precisely, the error in our knowledge of is +/2. But we are now only interested +in the general idea, so we won”t worry about factors of 2. +--- Trang 678 --- +accordance with quantum mechanics, the wavelength times the momentum is +Planck”s constant h. So we obtain the rule that the uncertainties in the vertical +qmomentum and ïn the vertical position have a product of the order h: +AwApy > h/2. (38.3) +W©e cannot prepare a system in which we know the vertical position of a particle +and can predict how ¡it will move vertically with greater certainty than given +by (38.3). That is, the uncertainty in the vertical momentum must exceed ñh/2A#, +where A¿# is the uncertainty in our knowledge of the position. +Sometimes people say quantum mechanics is all wrong. When the particle +arrived from the left, its vertical momentum was zero. And now that it has gone +through the slit, its position is known. Both position and momentum seem to be +known with arbitrary accuracy. It is quite true that we can receive a particle, and +on reception determine what its position is and what its momentum would have +had to have been to have gotten there. That is true, but that is not what the +uncertainty relation (38.3) refers to. Equation (38.3) refers to the predictabilitu +Of a situation, not remarks about the øøsý. It does no good to say “I knew what +the momentum was before it went throuph the slit, and now I know the position,” +because now the momentum knowledge is lost. The fact that i5 went through +the slit no longer permits us to predict the vertical momentum. We are talking +about a predictive theory, not just measurements after the fact. 5o we must talk +about what we can predict. +Now let us take the thing the other way around. Let us take another example +of the same phenomenon, a little more quantitatively. In the previous example +we measured the momentum by a classical method. Namely, we considered the +direction and the velocity and the angles, etc., so we got the momentum by +classical analysis. But since momentum is related to wave number, there exists +in nature still another way to measure the momentum of a particle—photon or +otherwise—which has no classical analog, because it uses q. (38.2). We measure +the :0auelengths oƒ the tuaues. Let us try to measure momentum in this way. +Suppose we have a grating with a large number of lines (Eig. 38-3), and send +a beam of particles at the grating. We have often discussed this problem: if +the particles have a defnite momentum, then we get a very sharp pattern in a +certain direction, because of the interference. And we have also talked about how +accurately we can determine that momentum, that is to say, what the resolving +power of such a grating is. Rather than derive it again, we refer to Chapter 30, +--- Trang 679 --- +h ~-~— +Fig. 38-3. Determination of momentum by using a diffraction grating. +where we found that the relative uncertainty in the wavelength that can be +measured with a given grating is 1/Nơn, where Ñ is the number of lines on the +grating and mm is the order of the diÑraction pattern. That is, +AA/A = 1/Nm. (38.4) +Now formula (38.4) can be rewritten as +AA/A3= 1/NmÀ = 1/L, (38.5) +where ÈÙ is the distance shown in Eig. 38-3. This distance is the difference bebween +the total distance that the particle or wave or whatever it is has to travel If +1t is refected from the bottom of the grating, and the distance that it has to +travel if it is relected from the top of the grating. That is, the waves which form +the difÑraction pattern are waves which come from different parts of the grating. +The first ones that arrive come from the bottom end of the grating, from the +beginning of the wave train, and the rest of them come from later parts of the +wave train, coming from diferent parts of the grating, until the last one ñnally +arrives, and that involves a point in the wave train a distance Ù behind the frst +point. So in order that we shall have a sharp line in our spectrum corresponding +to a delnite momentum, with an uncertainty given by (38.4), we have to have a +wave train oŸ at least length E. IÝ the wave train is too short we are not using +the entire grating. The waves which form the spectrum are being refected from +only a very short sector of the grating if the wave train is too short, and the +grating will not work right—we will ñnd a big angular spread. In order to get a +narrower one, we need to use the whole grating, so that at least a% some moment +the whole wave train is scattering simultaneously from all parts of the grating. +--- Trang 680 --- +Thus the wave train must be of length Ù in order to have an uncertainty in the +wavelength less than that given by (38.5). Incidentally, +AA/A? = A(1/A) = Ak/2m. (38.6) +'Therefore +Ak = 2n/L. (38.7) +where Ù is the length of the wave train. +This means that ifƒ we have a wave train whose length is less than b, the +uncertainty in the wave number must exceed 2Z/L. Or the uncertainty in a wave +number times the length of the wave train—we will call that for a moment Az—— +exceeds 2z. We call it Az because that is the uncertainty ¡in the location of the +particle. If the wave train exists only in a fñnite length, then that is where we +could fnd the particle, within an uncertainty Az. Now this property of waves, +that the length of the wave train times the uncertainty of the wave number +associated with 1t is at least 7, 1s a property that is known to everyone who +studies them. It has nothing to do with quantum mechanics. It is simply that if +we have a fñnite train, we cannot count the waves in it very precisely. Let us try +another way to see the reason for that. +Suppose that we have a fnite train of length L; then because of the way iE +has to decrease at the ends, as in Fig. 38-1, the number of waves in the length +is uncertain by something like +1. But the number of waves in Ù is kL/2z. Thus +k is uncertain, and we again get the result (38.7), a property merely oŸ waves. +The same thing works whether the waves are in space and & is the number of +radians per centimeter and b is the length of the train, or the waves are in time +and œ is the number of oscillations per second and 7' is the “length” in time +that the wave train comes in. That is, if we have a wave train lasting only for a +certain fñnite time 7, then the uncertainty in the frequency is given by +Au = 2m/T. (38.8) +W© have tried to emphasize that these are properties of waves alone, and they +are well known, for example, in the theory of sound. +The point is that in quantum mechanics we interpret the wave number as +being a measure of the momentum of a particle, with the rule that p = ñk, so +that relation (38.7) tells us that Ap h/Az. Thịs, then, is a limitation of the +classical idea of momentum. (Naturally, ¡it has to be limited in some ways iÝ we +--- Trang 681 --- +are goïng to represent particles by wavesl) It is nice that we have found a rule +that gives us some idea. of when there is a failure of classical ideas. +38-3 Crystal difraction +Next let us consider the reflection of particle waves from a crystal. A crystal +is a thick thing which has a whole lot of similar atoms—we will include some +complications later——in a nice array. The question ¡is how to set the array so that +we get a strong refected maximum in a given direction for a given beam oÏ, say, +light (x-rays), electrons, neutrons, or anything else. In order to obtain a strong +reflection, the scattering from all of the atoms must be in phase. 'There cannot +be equal numbers in phase and out of phase, or the waves will cancel out. 'Phe +way to arrange things is to ñnd the regions of constant phase, as we have already +explained; they are planes which make equal angles with the initial and fnal +directions (Eig. 38-4). +— dsin8 +Fig. 38-4. Scattering of waves by crystal planes. +Tf we consider two parallel planes, as in Fig. 38-4, the waves scattered from +the two planes will be in phase provided the diference in distance travelled by a +wavefront is an integral number of wavelengths. 'This diference can be seen to +be 2dsin Ø, where đ is the perpendicular distance between the planes. 'hus the +condition for coherent reflection is +2đsin Ø = nÀ (m = 1,2,...). (38.9) +--- Trang 682 --- +T, for example, the crystal is such that the atoms happen to lie on planes +obeying condition (38.9) with ø = 1, then there will be a strong reflection. T, +on the other hand, there are other atoms oŸ the same nature (equal in density) +halfway between, then the intermediate planes will also scatter equally strongly +and will interfere with the others and produce no efect. So đ in (38.9) must refer +to øđjacent planes; we cannot take a plane five layers farther back and use this +formulal +As a matter of interest, actual crystals are not usually as simple as a single +kind of atom repeated in a certain way. Instead; If we make a two-dimensional +analog, they are much like wallpaper, in which there is some kind of fñgure +which repeats all over the wallpaper. By “ñgure” we mean, in the case of atoms, +some arrangement——calcium and a carbon and three oxygens, etc., for calcium +carbonate, and so on—which may involve a relatively large number of atoms. +But whatever ït is, the fñgure is repeated in a pattern. 'Phis basic fñgure is called +a tn2t cell +'The basic pattern of repetition defines what we call the /œf#2ce tụpe; the lattice +type can be immediately determined by looking at the refections and seeing +what their symmetry is. In other words, where we fnd any reflections œý all +determines the lattice type, but in order to determine what is in each of the +elements oŸ the lattice one must take into account the ?mtensity of the scattering +at the various directions. Whách directions scatter depends on the type of lattice, +but hoa stronglu each scatters is determined by what is inside each unit cell, and +in that way the structure of crystals is worked out. +'Two photographs of x-ray difraction patterns are shown in Pigs. 38-5 and 38-6; +they illustrate scattering from rock salt and myoglobin, respectively. +—.... tật : +Sai ch lP:i: Ạ +.:.H::NN:..- - +`. Thu : +Figure 38-5 Eigure 38-6 +--- Trang 683 --- +Incidentally, an interesting thing happens ïf the spacings of the nearest planes +are less than A/2. In this case (38.9) has no solution for ø. Thus iŸ À is bigger +than twice the distance between adjacent planes then there is no side diÑraction +pattern, and the light—or whatever i% is—will go right through the material +without bouncing of or getting lost. So in the case of light, where À is mụch +bigger than the spacing, of course it does go through and there is no pattern of +reflection from the planes of the crystal. +77 NEUTRONS +—> —> _ +PILE-E GRAPHITE — NEU TRÒNS +SHORT-A NEUTRONS +Fig. 38-7. Diffusion of pile neutrons through graphite block. +'This fact also has an interesting consequence in the case of piles which make +neutrons (these are obviously particles, for anybody”s money!). IỶ we take these +neutrons and let them into a long block of graphite, the neutrons difuse and +work their way along (Eig. 3§-7). They difuse because they are bounced by the +atoms, but strictly, in the wave theory, they are bounced by the atoms because of +diÑfraction from the crystal planes. It turns out that if we take a very long piece +of graphite, the neutrons that come out the far end are all of long wavelengthl In +fact, 1ƒ one plots the intensity as a function of wavelength, we get nothing except +Fig. 38-8. Intensity of neutrons out of graphite rod as function of +wavelength. +--- Trang 684 --- +for wavelengths longer than a certain minimum (Eig. 38-8). In other words, we +can get very slow neutrons that way. Ônly the slowest neutrons come through; +they are not difracted or scattered by the crystal planes of the graphite, but +keep going right through like light through glass, and are not scattered out the +sides. 'Phere are many other demonstrations of the reality of neutron waves and +waves of other particles. +38-4 The size of an atom +We now consider another application of the uncertainty relation, Eq. (38.3). +lt must not be taken too seriously; the idea is right but the analysis is not very +accurate. The idea has to do with the determination of the size of atoms, and +the fact that, classically, the electrons would radiate light and spiral in until +they settle down right on top of the nucleus. But that cannot be right quantum- +mechanically because then we would know where each electron was and how fast +1Ù WaS IIOVInE. +3uppose we have a hydrogen atom, and measure the position of the electron; we +must not be able to predict exactly where the electron will be, or the momentum +spread will then turn out to be infnite. Every time we look at the electron, +1t 1s somewhere, but it has an amplitude to be in diferent places so there is a +probability of it being found in diferent places. Thhese places cannot all be at the +nucleus; we shall suppose there is a spread in position of order ø. 'That is, the +distance of the electron from the nucleus is usually about ø. We shall determine +ø by minimizing the total energy of the atom. +The spread in momentum is roughly 5/a because of the uncertainty relation, +so that IÝ we try to measure the momentum of the electron in some mamner, +such as by scattering x-rays of ¡it and looking for the Doppler efect from a +moving scatterer, we would expect not to get zero every time—the electron 1s +not sianding still—but the momenta must be oŸ the order p + /a. Then the +kinetic energy is roughly sm? = p2/2m = h2/2ma?. (In a sense, this is a kind +of dimensional analysis to ñnd out in what way the kinetic energy depends upon +the reduced Planck constant, upon mm, and upon the size of the atom. We need +not trust our answer to within factors like 2, z, etc. We have not even defned ø +very precisely.) NÑow the potential energy is minus eŸ over the distance from the +center, say —c2/a, where, we remember, e2 is the charge of an electron squared, +divided by 4zeog. Now the point is that the potential energy is reduced If ø gets +smaller, but the smaller ø is, the higher the momentum required, because of +--- Trang 685 --- +the uncertainty principle, and therefore the higher the kinetic energy. The total +©n©rgy 1s +E = hˆ/2ma? — c°/a. (38.10) +W© do not know what ø is, but we know that the atom is goïng to arrange itself +to make some kind oŸ compromise so that the energy is as little as possible. In +order to minimize #2, we difÑerentiate with respect to ø, set the derivative equal +to zero, and solve for ø. The derivative of F/ is +dE/da = —hŠ /maŠ + e2/a3, (38.11) +and setting đE/da = 0 gives for ø the value +dạ = h2/me? = 0.528 angstrom, += 0.528 x 1019 meter. (38.12) +This particular distance ¡is called the Eohr radzus, and we have thus learned +that atomic dimensions are of the order of angstroms, which is right: This is +pretty good——in fact, it is amazing, since until now we have had no basis for +understanding the size of atomsl Atoms are completely impossible from the +classical point of view, since the electrons would spiral into the nucleus. +Now if we put the value (38.12) for ao into (38.10) to fnd the energy, it comes +Eo = —€?/2ao = —me?/2h2 = —13.6 eV. (38.13) +'What does a negative energy mean? It means that the electron has less energy +when ï£ is in the atom than when ït is free. It means it is bound. It means it takes +energy to kick the electron out; i% takes energy of the order of 13.6 eV to ionize +a hydrogen atom. We have no reason to think that iE is not two or three times +this—or half of this—or (1/) times this, because we have used such a sÌopDy +argument. However, we have cheated, we have used all the constants in such a +way that it happens to come out the right numberl 'This number, 13.6 electron +volts, is called a Rydberg of energy; it is the ionization energy of hydrogen. +So we now understand why we do not fall through the Hoor. As we walk, our +shoes with their masses of atoms push against the ñoor with 2s mass of atoms. +In order to squash the atoms closer together, the electrons would be confned to +a smaller space and, by the uncertainty principle, their momenta would have to +be higher on the average, and that means high energy; the resistance to atomic +--- Trang 686 --- +compression is a quantum-mechanical efect and not a classical efect. Classically, +we would expect that if we were to draw all the electrons and protons cÌoser +together, the energy would be reduced still further, and the best arrangement of +positive and negative charges in classical physics is all on top of each other. 'This +was well known in classical physics and was a puzzle because of the existence of the +atom. Of course, the early scientists invented some ways out oŸ the trouble—but +never mỉnd, we have the r2gh‡ way out, nowl (Maybe.) +Incidentally, although we have no reason to understand it at the moment, in +a situation where there are many electrons it turns out that they try to keep +away from each other. IÝ one electron is occupying a certain space, then another +does not occupy the same space. More precisely, there are two spin cases, so that +two can sỉt on top of each other, one spinning one way and one the other way. +But after that we cannot put any more there. We have to put others in another +place, and that is the real reason that matter has strength. lf we could put all +the electrons in the same place it would condense even more than it does. Ït 1s +the fact that the electrons cannot all get on top of each other that makes tables +and everything else solid. +Obviously, in order to understand the properties of matter, we will have to +use quantum mechanics and not be satisfed with classical mechanics. +38-5 Energy levels +W© have talked about the atom in its lowest possible energy condition, but +1t turns out that the electron can do other things. It can jiggle and wiggle in a +more energetic manner, and so there are many difÑferent possible motions for the +atom. According to quantum mechanics, in a stationary condition there can only +be definite energies for an atom. We make a diagram (Fig. 38-9) in which we +ñị E= Mi c +—T Ỷ— Ƒọ +Fig. 38-9. Energy diagram for an atom, showing several possible +†ransitions. +--- Trang 687 --- +plot the energy vertically, and we make a horizontal line for each allowed value of +the energy. When the electron is Íree, i.e., when is energy is positive, it can have +any energy; it can be moving at any speed. But bound energies are not arbitrary. +'The atom must have one or another out of a set of allowed values, such as those +in Fig. 38-9. +Now let us call the allowed values of the energy ọ, F\, Hạ, Pz. lf an atom +1s initially in one of these “excited states,” F\, hạ, cức., 1t does not remain In +that state forever. Sooner or later it drops to a lower state and radiates energy +in the form of light. The frequency of the light that is emitted is determined +by conservation of energy plus the quantum-mechanical understanding that the +frequency of the light is related to the energy of the light by (38.1). Therefore +the frequency of the light which is liberated in a transition from energy 3 to +energy #⁄¡ (for example) is +'This, then, is a characteristic frequency of the atom and defñnes a spectral emission +line. Another possible transition would be from #2 to o. That would have a +diferent frequency +Another possibility is that if the atom were excited to the state ¡ it could drop +to the ground state lo, emitting a photon of frequency +(10 —= (Eì¡ — Eo)/h. (38.16) +'The reason we bring up three transitions is to poïnt out an interesting relationship. +It is easy to see rom (38.14), (38.15), and (3§.16) that +030 = 031 10. (38.17) +In general, ¡if we fnd two spectral lines, we shall expect to fnd another line at the +sum of the frequencies (or the diference in the frequencies), and that all the lines +can be understood by fñnding a series of levels such that every line corresponds +to the diference in energy oŸ some pair of levels. This remarkable coincidence in +spectral frequencies was noted before quantum mechanics was discovered, and it +1s called the J#z combination principle. Thịs is again a mystery from the point of +view of classical mechanics. Let us not belabor the point that classical mechanies +1s a failure in the atomic domain; we seem to have demonstrated that pretty well. +--- Trang 688 --- +W© have already talked about quantum mechanics as being represented by +amplitudes which behave like waves, with certain frequencies and wave numbers. +Let us observe how it comes about from the point of view of amplitudes that the +atom has definite energy states. This is something we cannot understand from +what has been said so far, but we are all familiar with the fact that confned +waves have definite frequencies. Eor instance, if sound is confined to an organ +pipe, or anything like that, then there is more than one way that the sound can +vibrate, but for each such way there is a defnite frequency. Thus an object in +which the waves are confined has certain resonance frequencies. It is therefore a +property of waves in a confned space—a subject which we will discuss in detail +with formulas later on—that they exist only at defnite frequencies. And since the +general relation exists between frequencies of the amplitude and energy, we are +not surprised to ñnd defnite energies associated with electrons bound in atoms. +38-6 Philosophical implications +Let us consider briefy some philosophical implications of quantum mechanics. +As always, there are bwo aspects of the problem: one is the philosophical implica- +tion for physics, and the other is the extrapolation of philosophical matters to +other fields. When philosophical ideas associated with science are dragged into +another field, they are usually completely distorted. 'Therefore we shall confine +our remarks as much as possible to physics itself. +First of all, the most interesting aspect is the idea of the uncertainty principle; +making an observation afects the phenomenon. lt has always been known +that making observations afects a phenomenon, but the poïnt is that the efect +cannot be disregarded or minimized or decreased arbitrarily by rearranging the +apparatus. When we look for a certain phenomenon we cannot help but disturb +1t in a certain minimum way, and (he disturbance ¡s necessarU ƒor the consistenc +0ƒ the 0ieupozni. The observer was sometimes Important in prequantum physics, +but only in a rather trivial sense. The problem has been raised: ïf a tree falls in +a forest and there is nobody there to hear it, does it make a noise? À real tree +falling in a reøl forest makes a sound, oŸ course, even if nobody is there. ven If +no one is present to hear it, there are other traces left. 'The sound will shake some +leaves, and if we were careful enough we might find somewhere that some thorn +had rubbed against a leaf and made a tỉny scratch that could not be explained +unless we assumed the leaf were vibrating. So in a certain sense we would have +to admit that there is a sound made. We might ask: was there a sensafion of +--- Trang 689 --- +sound? No, sensations have to do, presumably, with consciousness. And whether +anfs are conscious and whether there were ants in the forest, or whether the tree +was conscious, we do not know. Let us leave the problem in that form. +Another thing that people have emphasized since quantum mechanics was +developed is the idea that we should not speak about those things which we +cannot measure. (Actually relativity theory also said this.) Unless a thing can be +defined by measurement, it has no place in a theory. And since an accurate value +of the momentum of a localized particle cannot be defned by measurement it +therefore has no place in the theory. The idea that this is what was the matter +with classical theory 2s ø ƒalse position. IV is a careless analysis of the situation. +Just because we cannot ?neøsure position and momentum precisely does not ø +prior¿ mean that we cannot talk about them. It only means that we øeđ not talk +about them. "The situation in the sciences is this: A concept or an idea which +cannot be measured or cannot be referred directly to experiment may or may not +be useful. It need not exist in a theory. In other words, suppose we compare the +classical theory of the world with the quantum theory of the world, and suppose +that it is true experimentally that we can measure position and momentum only +imprecisely. The question is whether the 7deøs of the exact position of a partiele +and the exact momentum of a particle are valid or not. 'Phe classical theory +admits the ideas; the quantum theory does not. This does not ïn itself mean that +classical physics is wrong. When the new quantum mechanics was discovered, +the classical people—which included everybody except Heisenberg, Schrödinger, +and Born—said: “Look, your theory is not any good because you cannot answer +certain questions like: what is the exact position of a particle?, which hole does +1t go through?, and some others.” Heisenberg's answer was: “Í[ do not need to +answer such questions because you cannot ask such a question experimentally.” It +is that we do not bøoe to. Consider 6wo theories (a) and (b); (a) contains an idea +that cannot be checked directly but which is used in the analysis, and the other, +(b), does not contain the idea. TỶ they disagree in their predictions, one could not +claim that (b) is false because it cannot explain this idea that is in (a), because +that idea is one of the things that cannot be checked directly. It is always good +to know which ideas cannot be checked directly, but ¡it is not necessary to remove +them all. It is not true that we can pursue science completely by using only those +concepts which are directly subject to experiment. +In quantum mechanics itself there is a wave function amplitude, there 1s a +potential, and there are many constructs that we cannot measure directly. The +basis of a sclence is its ability to predct. "To predict means to tell what will +--- Trang 690 --- +happen in an experiment that has never been done. How can we do that? By +assuming that we know what is there, independent of the experiment. We must +extrapolate the experiments to a regilon where they have not been done. We +must take our concepts and extend them to places where they have not yet +been checked. If we do not do that, we have no prediction. So it was perfectly +sensible for the classical physicists to go happily along and suppose that the +position—which obviously means something for a baseball—meant something +also for an electron. It was not stupidity. It was a sensible procedure. Today we +say that the law of relativity is supposed to be true at all energies, but someday +somebody may come along and say how stupid we were. We do not know where +we are “stupid” until we “stick our neck out,” and so the whole idea is to put our +neck out. And the only way to ñnd out that we are wrong is to find out 0hø‡ +our predictions are. lt is absolutely necessary to make construets. +W©e have already made a few remarks about the indeterminacy of quantum +mechanics. That is, that we are unable now to predict what will happen In +physics in a given physical circumstance which is arranged as carefully as possible. +Tf we have an atom that is in an excited state and so is goiïng to emit a photon, +we cannot say œhen it will emit the photon. It has a certain amplitude to emit +the photon at any time, and we can predict only a probability for emission; we +cannot predict the future exactly. 'Phis has given rise to all kinds of nonsense +and questions on the meaning of freedom of will, and of the idea that the world +1s uncertain. +Of course we must emphasize that classical physics is also indeterminate, in a +sense. It is usually thought that this indeterminacy, that we cannot predict the +future, is an important quantum-mechanical thing, and this is said to explain the +behavior of the mind, feelings of free will, etc. But If the world +0ere classical——1f +the laws of mechanics were classical—it is not quite obvious that the mind would +not feel more or less the same. Ït is true classically that if we knew the position +and the velocity of every particle in the world, or in a box of gas, we could prediect +exactly what would happen. And therefore the classical world is deterministic. +Suppose, however, that we have a fnite accuracy and do not know ezact where +Just one atom is, say to one part in a billion. hen as it goes along it hits another +atom, and because we đid not know the position better than to one part in a +billion, we fnd an even larger error in the position after the collision. And that +1s amplifed, of course, in the next collision, so that 1ƒ we start with only a tiny +error i% rapidly magnifes to a very great uncertainty. To give an example: if +water falls over a dam, it splashes. If we stand nearby, every now and then a +--- Trang 691 --- +drop will land on our nose. 'Phis appears to be completely random, yet such a +behavior would be predicted by purely classical laws. The exact position of all +the drops depends upon the precise wigglings of the water before it goes Over +the dam. How? 'Phe tiniest irregularities are magnified in falling, so that we get +complete randomness. Obviously, we cannot really predict the position of the +drops unless we know the motion of the water absolutclu exzactlg. +Speaking more precisely, given an arbitrary accuracy, no matter how precise, +one can fnd a time long enough that we cannot make predictions valid for that +long a time. Now the poïnt is that this length oŸ time is not very large. It is not +that the time is millions of years if the accuracy is one part$ in a billion. "The +time goes, in fact, only logarithmically with the error, and i% turns out that in +only a very, very tiny time we lose all our information. If the accuracy is taken +to be one part in billions and billions and billions—no matter how many billions +we wish, provided we do stop somewhere—then we can fnd a time less than the +time it took to state the accuracy——after which we can no longer predict what is +going to happenl It is therefore not fair to say that from the apparent freedom +and indeterminacy of the human mỉnd, we should have realized that classical +“deterministic” physics could not ever hope to understand ït, and to welcome +quantum mechanics as a release from a “completely mechanistic” universe. For +already in classical mechanics there was indeterminability from a practical point +of view. +--- Trang 692 --- +Tho Minotic Thoorgg ©Ÿ Ấ(sos +39-1 Properties of matter +With this chapter we begin a new subject which will occupy us Íor some +time. It is the first part of the analysis of the properties of matter from the +physical point of view, in which, recognizing that matter is made out of a great +many atoms, or elementary parts, which interact electrically and obey the laws +of mechanics, we try to understand why various aggregates of atoms behave the +way they do. +Tt is obvious that this is a dificult subject, and we emphasize at the beginning +that it is in facE an eøremelu difficult subject, and that we have to deal with +it diferently than we have dealt with the other subjects so far. In the case of +mechanics and in the case oŸ light, we were able to begin with a precise statement +of some laws, like Newton”s laws, or the formula for the field produced by an +accelerating charge, from which a whole host of phenomena. could be essentially +understood, and which would produce a basis for our understanding of mechanics +and of light from that time on. 'That is, we may learn more later, but we do not +learn diferent physics, we only learn better methods of mathematical analysis to +deal with the situation. +W©e cannot use this approach efectively in studying the properties of matter. +W© can discuss matter only in a most elementary way; it is much too complicated +a subject to analyze directly ữom its specifc basic laws, which are none other +than the laws of mechanics and electricity. But these are a bit too far away Írom +the properties we wish to study; 1 takes too many steps to get from Newton”s laws +to the properties of matter, and these steps are, in themselves, fairly complicated. +We will now start to take some of these steps, but while many of our analyses +will be quite accurate, they will eventually get less and less accurate. We will +have only a rough understanding of the properties of matter. +--- Trang 693 --- +One of the reasons that we have to perform the analysis so imperfectly is that +the mathematics of it requires a deep understanding of the theory of probability; +we are not going to want to know where every atom is actually moving, but +rather, how many move here and there on the average, and what the odds are for +diferent efects. So this subject involves a knowledge of the theory of probability, +and our mathematics is not yet quite ready and we do not want to strain it too +Secondly, and more important from a physical standpoint, the actual behavior +of the atoms is not according to classical mechanics, but according to quantum +mechanics, and a correct understanding of the subject cannot be attained until +we understand quantum mechanics. Here, unlike the case of billiard balls and +automobiles, the diference between the classical mechanical laws and the quantum- +mmechanical laws is very important and very significant, so that many things that +we will deduce by classical physics will be fundamentally incorrect. Therefore +there will be certain things to be partially unlearned; however, we shall indicate In +every case when a result is incorrect, so that we will know just where the “edges” +are. One of the reasons for discussing quantum mechanics in the preceding +chapters was to give an idea as to why, more or less, classical mechanics 1s +incorrect in the various directions. +'Why do we deal with the subject now at all? Why not wait half a year, or a +year, until we know the mathematics of probability better, and we learn a little +quantum mechanies, and then we can do it in a more fundamental way? "The +answer is that it is a dificult subject, and the best way to learn is to do it slowlyl +'The first thing to do is to get some idea, more or less, of what ought to happen +in diferent circumstances, and then, later, when we know the laws better, we +will formulate them better. +Anyone who wants to analyze the properties of matter in a real problem +might want to start by writing down the fundamental equations and then try +%o solve them mathematically. Although there are people who try to use such +an approach, these people are the failures in this field; the real successes come +to those who start from a phụs¿caÏ point of view, people who have a rouph idea +where they are goïng and then begin by making the right kind of approximations, +knowing what is big and what is small in a given complicated situation. These +problems are so complicated that even an elementary understanding, although +Inaccurate and incomplete, is worthwhile having, and so the subject will be one +that we shall go over again and again, each time with more and more accuracy, +as we øo through our course in physics. +--- Trang 694 --- +Another reason for beginning the subject right now is that we have alreadly +used many of these ideas in, for example, chemistry, and we have even heard of +some of them in high school. It is interesting to know the physical basis for these +things. +As an interesting example, we all know that equal volumes of gases, at the +same pressure and temperature, contain the same number of molecules. "The +law of multiple proportions, that when two gases combine in a chemical reaction +the volumes needed always stand in simple integral proportions, was understood +ultimately by Avogadro to mean that equal volumes have equal numbers of atoms. +Now œhụ do they have equal numbers of atoms? Can we deduce from Newton's +laws that the number of atoms should be equal? We shall address ourselves to +that specifc matter in this chapter. In succeeding chapters, we shall discuss +varlous other phenomena involving pressures, volumes, temperature, and heat. +W©e shall also ñnd that the subject can be attacked om a nonatomic point oŸ +view, and that there are many interrelationships of the properties of substances. +For instance, when we compress something, it heats; If we heat it, i expands. +There is a relationship between these two facts which can be deduced indepen- +dently of the machinery underneath. This subject is called /hermodynamics. The +deepest understanding of thermodynamics comes, of course, from understanding +the actual machinery underneath, and that is what we shall do: we shall take +the atomic viewpoint from the beginning and use it to understand the various +properties of matter and the laws of thermodynamics. +Let us, then, discuss the properties of gases from the standpoint of Ñewton's +laws of mechanics. +39-2 The pressure of a gas +First, we know that a gas exerts a pressure, and we must clearly understand +what this is due to. If our ears were a few times more sensitive, we would hear +a perpetual rushing noise. Evolution has not developed the ear to that point, +because it would be useless iŸ it were so much more sensitive—we would hear +a perpetual racket. "The reason is that the eardrum is in contact with the air, +and air is a lot of molecules in perpetual motion and these bang against the +eardrums. In banging against the eardrums they make an irregular tattoo—boom, +boom, boom——which we do not hear because the atoms are so small, and the +sensitivity of the ear is not quite enough to notice it. The result of this perpetual +bombardment is to push the drum away, but of course there is an equal perpetual +--- Trang 695 --- +bombardment of atoms on the other side of the eardrum, so the net force on it +1s 2ero. lf we were to take the air away Írom one side, or change the relative +amounts of air on the ©wo sides, the eardrum would then be pushed one way or +the other, because the amount of bombardment on one side would be greater +than on the other. We sometimes feel this uncomfortable efect when we go up +too fast in an elevator or an airplane, especially if we also have a bad cold (when +we have a cold, inlammation closes the tube which connects the air on the inside +of the eardrum with the outside air through the throat, so that the Ewo pressures +cannot readily equalize). +F1 +ˆ V_ sŸ +Fig. 39-1. Atoms of a gas In a box with a frictionless piston. +In considering how to analyze the situation quantitatively, we imagine that +we have a volume oŸ gas in a box, at one end of which is a piston which can be +moved (Fig. 39-1). We would like to nd out what force on the piston results +from the fact that there are atoms in this box. “The volume of the box 1s V, +and as the atoms move around inside the box with various velocities they bang +against the piston. Suppose there is nothing, a vacuum, on the outside of the +piston. What of it? Tf the piston were left alone, and nobody held onto it, each +tỉme it got banged it would pick up a little momentum and it would gradually +get pushed out of the box. So in order to keep 1% from being pushed out of the +box, we have to hold it with a force #'. The problem is, how much force? Ône +way of expressing the force is to talk about the force per unit area: if A is the +area of the piston, then the force on the piston will be written as a number times +the area. We define the pressure, then, as equal to the force that we have to +apply on a piston, divided by the area of the piston: +D=FJ/A. (39.1) +To make sure we understand the idea (we have to derive i for another purpose +anyway), the diferential øork đW done on the gas in compressing it by moving +the piston in a diferential amount —d+ would be the force times the distance +that we compress it, which, according to (39.1), would be the pressure times the +--- Trang 696 --- +area, times the distance, which is equal to minus the pressure times the change +in the volume: +đW = F(—d+z) = —PAd+z = —PdV. (39.2) +(The area, A times the distance đz is the volume change.) The minus sign is there +because, as we compress it, we đecrease the volume; if we think about it we can +see that IÝ a gas 1s compressed, work is done øn ït. +How mụuch force do we have to apply to balance the banging of the molecules? +The piston receives from each collision a certain amount of momentum. A certain +amountf of momentum per second will pour into the piston, and it will start to move. +'To keep it from moving, we must pour back into it the same amount oŸ momentum +per second from our force. Of course, the force 7s the amount of momentum per +second that we must pour in. 'Phere is another way to put it: IŸ we let go of the +piston it will pick up speed because of the bombardments; with each collision +we get a little more speed, and the speed thus accelerates. The rate at which the +piston picks up speed, or accelerates, is proportional to the force on it. Šo we see +that the force, which we already have said is the pressure times the area, is equal +to the momentum per second delivered to the piston by the colliding molecules. +To calculate the momentum per second is easy——we can do it in two parts: +first, we fnd the momentum delivered to the piston by one particular atom in a +collision with the piston, then we have to multiply by the number of collisions +per second that the atoms have with the wall. The force will be the product of +these two factors. Now let us see what the two factors are: In the first place, +we shall suppose that the piston is a perfect “reflector” for the atoms. lÝ it is +not, the whole theory is wrong, and the piston will start to heat up and things +will change, but eventually, when equilibrium has set in, the net result is that +the collisions are efectively perfectly elastic. Ôn the average, every particle that +comes in leaves with the same energy. 5o we shall imagine that the gas is in +a sbeady condition, and we lose no energy to the piston because the piston 1s +standing still. In those circumstances, ïÝ a particle comes in with a certain speed, +1t comes out with the same speed and, we will say, with the same mass. +TÝ is the velocity of an atom, and „ is the #ø-component of 0, then 0z is +the z-component of momentum “in”; but we also have an equal component of +mmomentum “out,” and so the total momentum delivered to the piston by the +particle, in one collision, 1s 2m, because ït is “refected.” +Now, we need the number oŸ collisions made by the atoms in a second, or in +a certain amount of time đý; then we divide by đ¿. How many atoms are hitting? +--- Trang 697 --- +Let us suppose that there are atoms in the volume V, or „ = N/V in each unit +volume. 'To fnd how many atoms hit the piston, we note that, given a certain +amount oŸ time ứ, 1Ý a particle has a certain velocity toward the piston it will +hit during the time ý, provided ït is close enough. Tf i§ is too far away, it goes +only part way toward the piston in the time ý, but does not reach the piston. +Therefore it is clear that only those molecules which are within a distance 0„Ý +from the piston are going to hit the piston in the time ý. Thus the number of +collisions in a time £ is equal to the number of atoms which are in the region +within a distance 0„, and since the area of the piston is A, the øolzzxme occupied +by the atoms which are going to hit the piston is 0„¿A. But the mưmber of atoms +that are going to hit the piston is that volume tỉimes the number of atoms per +unit volume, ø„‡¿A. Of course we do not want the number that hit in a tỉme f, +we want the number that hit per second, so we divide by the time ứ, to get œ»zA. +(This time # could be made very short; iŸ we feel we want to be more elegant, we +call it đ, then diferentiate, but it is the same thing.) +So we fnd that the force is +` =nu¿„A - 2m0. (39.3) +See, the Íorce 7s proportional to the area, if we keep the particle density fxed as +we change the areal The pressure is then +P=2nm%}. (39.4) +Now we notice a little trouble with this analysis: First, all the molecules do +not have the same velocity, and they do not move in the same direction. So, all +the 02 s are diferent! So what we must do, of course, is to take an øerage of +the 02s, since each one makes its own contribution. What we want is the square +Of 0x, averaged over all the molecules: +P=nm(). (39.5) +Did we forget to include the factor 2? No; of all the atoms, only half are headed +toward the piston. 'Phe other half are headed the other way, so the number of +atoms per unit volume that are hitng the piston is only 0/2. +Now as the atoms bounce around, it is clear that there is nothing special +about the “z-direction”; the atoms may also be moving up and down, back and +forth, in and out. Therefore it is going to be true that (02), the average motion +--- Trang 698 --- +of the atoms in one direction, and the average in the other two directions, are all +goïing to be equal: +(02) = (toà) = (0ì). (39.6) +Tt is only a matter of rather tricky mathematics to notice, therefore, that they +are each equal to one-third of theïr sum, which is of course the square of the +magnitude of the velocity: +(02) = š (02 + 02 + 02) = (02)/3. (39.7) +This has the advantage that we do not have to worry about any particular +direction, and so we write our pressure formula again in this form: +P=(3)n(m%2/2). (39.8) +The reason we wrote the last factor as (w2/2) is that this is the kinetie energu +of the center-of-mass motion of the molecule. We fnd, therefore, that +PV = N(§)(mù°/2). (39.9) +With this equation we can calculate how much the pressure 1s, iŸ we know the +speeds. +As a very simple example let us take helium gas, or any other gas, like mercury +vapor, or potassium vapor of high enough temperature, or argon, in which all the +molecules are single atoms, for which we may suppose that there is no internal +motion in the atom. If we had a complex molecule, there might be some internal +motfion, mutual vibrations, or something. We suppose that we may disregard +that; this is actually a serious matter that we will have to come back to, but it +turns out to be all right. We suppose that the internal motion of the atoms can +be disregarded, and therefore, for this purpose, that the kinetic energy of the +center-of-mass motion is all the energy there is. So for a monatomic gas, the +kinetic energy is the total energy. In general, we are going to call Ư the total +energy (it is sometimes called the total zw„ternaÏ energy—we may wonder why, +since there is no ezfernal energy to a gas), i.e., all the energy of all the molecules +in the gas, or the object, whatever 1 1s. +For a monatomic gas we will suppose that the total energy is equal to +a number of atoms times the average kinetic energy of each, because we are +disregarding any possibility of excitation or motion inside the atoms themselves. +'Then, in these circumstances, we would have +PV =§U. (39.10) +--- Trang 699 --- +Incidentally, we can stop here and find the answer to the following question: +Suppose that we take a can of gas and compress the gas slowly, how much pressure +do we need to squeeze the volume down? Ït is easy to find out, since the pressure +1S Ỹ the energy divided by V. As we squeeze it down, we do work on the gas +and we thereby increase the energy . So we are going to have some kind of a +diferential equation: If we start out in a given cireumstance with a certain energy +and a certain volume, we then know the pressure. Now we start to squeeze, but +the moment we do, the energy increases and the volume W decreases, so the +Dr€SSUr€ ØO©S up. +So, we have to solve a diferential equation, and we will solve it in a moment. +We must first emphasize, however, that as we are compressing this gas, we are +supposing that all the work goes into increasing the energy of the atoms inside. +W©e may ask, “lsn't that necessary? Where else could it go?” It turns out that it +can go another place. Thhere are what we call “heat leaks” through the walls: the +hot (i.e., fast-moving) atoms that bombard the walls, heat the walls, and energy +goes away. We shall suppose for the present that this is not the case. +For somewhat wider generality, although we are still making some very special +assumptions about our gas, we shall write, not PW = 4U ;„ but +PV = (+y_- 1)U. (39.11) +It is written (+ — 1) times for conventional reasons, because we will deal with +a few other cases later where the number in front of Ứ will not be Ÿ› but will +be a diÑerent number. So, in order to do the thing in general, we call it + — 1, +because people have been calling it that for almost one hundred years. This +, +then, is Ỹ for a monatomic gas like helium, because Ỹ —l1l1s - +We© have already noticed that when we compress a gas the work done is —P? đV. +A compression in which there is no heat energy added or removed is called an +adiabatic compression, from the Greek ø (not) + điø (through) + ba¿ncin (to +go). (The word adiabatic is used in physics in several ways, and it is sometimes +hard to see what is common about them.) That is, for an adiabatic compression +all the work done goes into changing the internal energy. hat is the key—that +there are no other losses oŸ energy——for then we have ?DđVW = —đdƯ. But since +U = PV/(x— 1), we may write +đU = (PdV + VdP)/(x+- 1). (39.12) +So we have PđV = —(P.dV +V đP)/(+— 1), or, rearranging the terms, +? đV = +--- Trang 700 --- +—VdP,or +(+dV/V) + (dP/P) = 0. (39.13) +Fortunately, assuming that + is constant, as it is for a monatomic gas, we +can integrate this: it gives yln V + ln?? = lnCŒ, where ln C is the constant of +integration. lf we take the exponential of both sides, we get the law +PV? = C (a constant). (39.14) +In other words, under adiabatic conditions, where the temperature rises as we +compress because no heat is being lost, the pressure times the volume to the +Ỹ power is a constant for a monatomic gasl Although we derived it theoretically, +this 2s, in fact, the way monatomic gases behave experimentally. +39-3 Compressibility of radiation +We may give one other example of the kinetic theory of a gas, one which +1s not used in chemistry so mụuch, but is used in astronomy. We have a large +number of photons in a box in which the temperature is very high. (The box +1s, Of course, the gas in a very hot star. The sun is not hot enough; there are +still too many atoms, but at still higher temperatures in certain very hot stars, +we may neglect the atoms and suppose that the only objects that we have in +the box are photons.) Now then, a photon has a certain momentum ø. (Woe +always fnd that we are in terrible trouble when we do kinetic theory: ø is the +pressure, but ø is the momentum; 0 is the volume, but 0œ is the velocity; 7' is the +temperature, but 7' is the kinetic energy or the time or the torque; one must keep +one?s wits about onel) 'This ø is momentum, it is a vector. Going through the +same analysis as before, it is the ø-component of the vector ø which generates +the “kick,” and twice the z-component of the vector ø is the momentum which +1s given In the kick. Thus 2p„ replaces 2m„, and in evaluating the number of +collisions, „ 1s sfill 0„, so when we get all the way through, we fnd that the +pressure in Eq. (39.4) is, instead, +P=2npzu. (39.15) +Then, in the averaging, it becomes ø times the average of p„u„ (the same factor +of 2) and, fnally, putting in the other two directions, we fnd +PV = Nịp:-0)/3. (39.16) +--- Trang 701 --- +This checks with the formula, (39.9), because the momentum is ?zmo; it is a little +more general, that is all. The pressure times the volume is the total number of +atoms tỉmes 3(p - 0), averaged. +Now, for photons, what 1s p - 0? 'Phe momentum and the velocity are in the +same direction, and the velocity is the speed of light, so this is the momentum of +cach of the objects, times the speed of light. The momentum times the speed of +light of every photon is its energy: # = pc, so these terms are the enwergies of +cach of the photons, and we should, of course, take an average energy, times the +number of photons. So we have 3 of the energy inside the gas: +PV = U/3 (photon gas). (39.17) +For photons, then, since we have Š in front, (+ — 1) in (39.11) is ẩ› OTr Y= l +and we have discovered that radiation in a box obeys the law +PV*⁄3 = Ơ. (39.18) +So we know the compressibility of radiation! “That is what is used in an analysis +of the contribution of radiation pressure in a star, that is how we calculate it, +and how it changes when we compress it. What wonderful things are already +within our powerl +39-4 Temperature and kinetic energy +So far we have not dealt with £emperafure; we have purposely been avoiding +the temperature. Âs we compress a gas, we know that the energy of the molecules +increases, and we are used to saying that the gas gets hotter; we would like +to understand what this has to do with the temperature. If we try to do the +experiment, not adiabatically but at what we call constan‡ temperature, what are +we doing? We know that if we take two boxes of gas and let them sit next to +cach other long enough, even ïf at the start they were at what we call diÑferent +temperatures, they will in the end come to the same temperature. NÑow what +does that mean? 'That means that they get to a condition that they would get +%o If we left them alone long enoughl What we mean by equal temperature is +Jjust that—the fñinal condition when things have been sitting around interacting +with each other long enough. +Let us consider, now, what happens if we have two gases in containers +separated by a movable piston as in Eig. 39-2 (just for simplicity we shall take +--- Trang 702 --- +Œ) (2) +Fig. 39-2. Atoms of two different monatomic gases are separated by +a movable piston. +two monatomic gases, say helium and neon). In container (1) the atoms have +mass ?mị, velocity 0, and there are m per unit volume, and in the other container +the atoms have mass rnạ, velocity 0a, there are øạ atoms per unit volume. What +are the conditions for equilibrium? +Obviously, the bombardment from the left side must be such that it moves +the piston to the right and compresses the other gas until its pressure builds up, +and the thing will thus slosh back and forth, and will gradually come to rest at a +place where the pressures are equal on both sides. So we can arrange that the +pressures are equal; that just means that the internal energies per unit volume are +cqual, or that the numbers øw times the average kinetic energies on each side are +cqual. What we have to try to prove, eventually, is that the nươnbers themselues +are equal. So far, all we know is that the numbers times the kinetic energies are +equal, +mm (mi0/2) = na(ma02/2), +from (39.8), because the pressures are equal. We must realize that this is not the +only condition over the long run, but something else must happen more sÌowly +as the true complete equilibrium corresponding to equal temperatures sets in. +'To see the idea, suppose that the pressure on the left side were developed by +having a very high density but a low velocity. By having a large mœ and a small 0, +we can get the same pressure as by having a small œ and a large 0. The atoms +may be moving slowly but be packed nearly solidly, or there may be fewer but +they are hitting harder. WIHI it stay like that forever? At first we might think so, +but then we think again and find we have forgotten one important point. 'Phat is, +that the intermediate piston does not receive a steady pressure; it wiggles, Jus$ +like the eardrum that we were first talking about, because the hangings are not +absolutely uniform. There is not a perpetual, steady pressure, but a tattoo——the +pressure varies, and so the thing jiggles. Suppose that the atoms on the right +side are not jiggling much, but those on the left are few and far between and very +energetic. The piston will, now and then, get a big impulse from the left, and will +--- Trang 703 --- +U2 +Fig. 39-3. A collision between unequal atoms, viewed in the CM +system. tị = |VWị — VcM|, Uạ = |Va — VCM|. +be driven against the slow atoms on the right, giving them more speed. (As each +atom collides with the piston, it either gains or loses energy, depending upon +whether the piston is moving one way or the other when the atom strikes it.) So, +as a result of the collisions, the piston fnds itself Jjiggling, jiggling, jiggling, and +this shakes the other gas—it gïves energy to the other atoms, and they build up +faster motions, until they balance the jiggling that the piston is giving to them. +The system comes to some equilibrium where the piston is moving at such a +mean square speed that it picks up energy from the atoms at about the same +rate as it puts energy back into them. So the piston picks up a certain mean +Irregularity in speed, and ït is our problem to ñnd it. When we do fñnd ït, we can +solve our problem better, because the gases will adjust their velocities until the +rate at which they are trying to pour energy into each other through the piston +will become equal. +lt is quite dificult to fñgure out the details of the piston in this particular +circumstance; although it is ideally simple to understand, it turns out to be a +little harder to analyze. Before we analyze that, let us analyze another problem +in which we have a box of gas but now we have two diferent kinds of molecules +in it, having masses mm and mạ, velocitiles 0 and 0a, and so forth; there is now +a much more intimate relationship. Tf all of the No. 2 molecules are standing +stil, that condition is not going to last, because they get kicked by the No. 1 +mmolecules and so pick up speed. Tf they are all going much faster than the No. 1 +mmolecules, then maybe that will not last either—they will pass the energy back +to the No. 1 molecules. So when both gases are in the same box, the problem is +to fnd the rule that determines the relative speeds of the two. +'This 1s still a very dificult problem, but we will solve it as follows. First we +consider the following sub-problem (again this is one of those cases where—never +miỉnd the derivation——in the end the result is very simple to remember, but the +--- Trang 704 --- +derivation is Just ingenmious). Let us suppose that we have two molecules, of +diferent mass, colliding, and that the collision is viewed in the center-of-mass +(CM) system. In order to remove a complication, we look at the collision in the +CM. AÀs we know from the laws of collision, by the conservation of momentum +and energy, after the molecules collide the only way they can move is such that +cach maintains its own original speed—and they just change their đireciion. So +we have an average collision that looks like that in Fig. 39-3. Suppose, Íor a +moment, that we watch all the collisions with the CM at rest. Suppose we +imagine that they are all initially moving horizontally. Of course, after the fñrst +collision some of them are moving at an angle. In other words, ¡f they were all +going horizontally, then at least some would later be moving vertically. Ñow in +some other collision, they would be coming ïn from another direction, and then +they would be twisted at still another angle. So even if they were completely +organized in the beginning, they would get sprayed around at all angles, and +then the sprayed ones would get sprayed some more, and sprayed some more, +and sprayed some more. Ultimately, what will be the distribution? Ansuer: ït +tuiiÏ be cquallụ likelụ to [ind œng padr mmouứng ín an đirecHion ?n space. After that +further collisions could not change the distribution. +They are equally likely to go in all directions, bu how do we say that? There +1s Of course øoø likelihood that they will go in any specifc direction, because a +specifc direction is too exact, so we have to talk about per unit “something.” +'The idea is that any area on a sphere centered at a collision point will have just +as many molecules going through ¡% as go through any other equal area on the +sphere. So the result of the collisions wïll be to distribute the directions so that +equal areas on a sphere will have equal probabilities. +Incidentally, ¡if we just want to discuss the original direction and some other +direction an angle Ø from ït, it is an interesting property that the difÑferenmtial +area of a sphere oŸ unit radius is sin Ø đØ times 2Z (see Eig. 32-1). And sin 0 d9 +is the same as the diferential of — cosØ. 5o what it means is that the cosine of +the angle Ø between any two directions ¡is equally likely to be anything from —1 +to +1. +Next, we have to worry about the actual case, where we do not have the +collision in the CM system, but we have bwo atoms which are coming together +with vector velocities 0¡ and ø¿. What happens now? W©e can analyze this +collision with the vector velocities 0 and 0a in the following way: We first say +that there is a certain CM; the velocity of the CM is given by the “average” +velocity, with weights proportional to the masses, so the velocity of the CM +--- Trang 705 --- +1S ĐGM = (mị01 + nạ0a)/(mì + na). T we watch thịs collision in the CM +system, then we see a collision Just like that in Eig. 39-3, with a certain relative +velocity œ coming in. The relative velocity is just 0 — 0a. Now the idea is that, +first, the whole CM is moving, and in the CM there is a relative velocity +, and +the molecules collide and come of in some new direction. All this happens while +the CM keeps right on moving, without any change. +Now then, what is the distribution resulting from this? From our previous +argument we conclude this: that at equilibrium, øÏl direclions ƒor t0 are equall +ljkclụ, relalioe to the đirecHon oƒ the motlion öƑ the CM.* There will be no +particular correlation, in the end, between the direction of the motion of the +relative velocity and that of the motion of the CM. Of course, If there were, the +collisions would spray i% about, so it is all sprayed around. So the cosine of the +angle between +0 and ®CM 1s zero on the average. Phat is, +But 00 - 0cw¡ can be expressed in terms of 0 and 0a as well: +— (ĐịT— 9a) - (mịU+ + m202) +4Ð + ỦCM = ———————————— +T11 + †12 +— (miUỆ — mạu)) + (mạ — 1m) (0 - 02). (39.20) +THỊ + T2 +First, let us look at the Ø) - 0s; what is the average of 0 - 0a? That is, what +1s the average of the component of velocity of one molecule in the direction of +another? Surely there is just as much likelihood of ñnding any given molecule +moving one way as another. 7e querage oƒ the 0clocitU 0a ïn ang direclion f5 +zero. Certainly, then, in the direction of 0, 0a has zero average. So, the average +OŸ Øy - 0a is zerol Therefore, we conclude that the average of ru must be equal +to the average of maø2. That is, the auerage kinetic energu oƒ the tuo rnust be +cqual: +(3m01) = (5ma0)). (39.21) +TÝ we have two kinds of atoms in a gas, it can be shown, and we presume to have +shown it, that the average of the kinetic energy of one is the same as the average +* 'TPhis argument, which was the one used by Maxwell, involves some subtleties. Although +the conclusion is correct, the result does noø# follow purely from the considerations of symmetry +that we used before, since, by going to a reference frame moving through the gas, we may find +a distorted velocity distribution. We have not found a simple proof of this result. +--- Trang 706 --- +of the kinetic energy of the other, when they are both in the same gas in the +same box in equilibrium. That means that the heavy ones will move slower than +the light ones; this is easily shown by experimentation with “atoms” of diferent +mmasses in an air trough. +k *Ö|,o P + +Fig. 39-4. Two gases in a box with a semipermeable membrane. +Now we would like to go one step further, and say that if we have two different +gases separafed in a box, they will also have equal average kinetic energy when +they have ñnally come to equilibrium, even though they are not in the same box. +W©e can make the argument in a number of ways. One way is to argue that ïf +we have a fixed partition with a tiny hole in it (Eig. 39-4) so that one gas could +leak out through the holes while the other could not, because the molecules are +too big, and these had attained equilibrium, then we know that in one part, +where they are mixed, they have the same average kinetic energy, but some come +through the hole without loss oŸ kinetic energy, so the average kinetic energy in +the pure gas and in the mixture must be the same. That is not too satisfactory, +because maybe there are no holes, for this kind of molecule, that separate one +kind from the other. +Let us now go back to the piston problem. We can give an argument which +shows that the kinetic energy of this piston must also be simazu]. Actually, that +would be the kinetic energy due to the purely horizontal motion of the piston, +so, forgetfing its up and down motion, it will have to be the same as 3in20),.. +Likewise, from the equilibrium on the other side, we can prove that the kinetic +energy of the piston 1s 31m1 0`... Although this is not in the middle of the gas, +but is on one side oŸ the gas, we can still make the argument, although it is a +little more difficult, that the average kinetic energy of the piston and of the gas +mmolecules are equal as a result of all the collisions. +TÍ this still does not satisfy us, we may make an artificial example by which +the equilibrium is generated by an object which can be hit on all sides. Suppose +that we have a short rod with a ball on each end sticking through the piston, on +a frictionless sliding universal joint. Each baill is round, like one of the molecules, +and can be hit on all sides. 'Phis whole obJect has a certain total mass, ?m. Now, +--- Trang 707 --- +we have the gas molecules with mass rm+ and mass rmạ as before. The result of +the collisions, by the analysis that was made before, is that the kinetic energy +Of mm because of collisions with the molecules on one side must be 31m, on +the average. Likewise, because of the collisions with molecules on the other side, +1t has to be simazuỆ on the average. So, therefore, both sides have to have the +sœme kinetie energy when they are in thermal equilibrium. So, although we only +proved it for a mixture of gases, it is easily extended to the case where there are +two diferent, separate gases at the same temperature. +Thus hen ue hœue tuo gases aÈ the sarmne temperature, the mean kinctic +cnergu oƒ the CM motions œre cqual. +The mean molecular kinetic energy is a property only of the “temperature.” +Being a property of the “temperature,” and not oƒ the gas, we can use it as a +defnition of the temperature. The mean kinetic energy of a molecule is thus +some function of the temperature. But who ïs to tell us what scale to use for +the temperature? We may arbitrarily define the scale of temperature so that +the mean energy is linearly proportional to the temperature. The best way to +do it would be to call the mean energy itself “the temperature.” 'Phat would +be the simplest possible function. Ủnfortunately, the scale of temperature has +been chosen differently, so instead of calling it temperature directly we use a +constant conversion factor between the energy of a molecule and a degree of +absolute temperature called a degree Kelvin. The constant of proportionality is +k = 1.38 x 10”? joule for every degree Kelvin.* So if 7' is absolute temperature, +our deflnition says that the mean molecular kinetic energy is 3kT . (The Ỷ 1s put +in as a matter of convenience, so as to get rid of it somewhere else.) +W© point out that the kinetic energy associated with the component of motion +in any particular direction is only 3k1. The three independent directions that +are involved make 1 3kT. +39-5 The ideal gas law +Now, oŸ course, we can put our definition oŸ temperature into Eq. (39.9) and +so fnd the law for the pressure of gases as a function of the temperature: 1È 1s +that the pressure times the volume is equal to the total number of atoms times +the universal constant k, times the temperature: +PV = NRT. (39.22) +* The centigrade scale is just this Kelvin scale with a zero chosen at 273.16 °K, so 7'= +273.16 + centigrade temperature. +--- Trang 708 --- +Purthermore, at the same temperature and pressure and volume, the ø6%wmnber oƒ +atoms is determined; it too is a universal constantl So equal volumes of different +gases, at the same pressure and temperature, have the same number of molecules, +because of Newton”s laws. 'That is an amazing conclusionl +In practice, when dealing with molecules, because the numbers are so large, +the chemists have artificially chosen a specifc number, a very large number, and +called it something else. They have a number which they call a mole. A mole is +merely a handy number. Why they did not choose 102 objects, so it would come +out even, is a historical question. 'Phey happened to choose, for the convenient +number of objects on which they standardize, ẤWọ = 6.02 x 102” objects, and this +is called a mole of objects. 5o instead of measuring the number of molecules in +units, they measure in terms of numbers of moles.* In terms of Wo we can write +the number of moles, times the number of atoms in a mole, times &?, and if we +want to, we can take the number of atoms in a mole times &, which is a mole”s +worth of k, and call it something else, and we do—we call it f. A mole's worth +of k is 8.317 joules: !? = Nọk = 8.317 J- mole~! -°K~!, Thus we also fnd the +gas law written as the number of moles (also called ý) times #7, or the number +of atoms, times k?: +PV = NÌRT. (39.23) +Tt is the same thing, just a difÑferent scale for measuring numbers. We use l as a +unit, and chemists use 6 x 1023 as a unitl +We now make one more remark about our gas law, and that has to do with +the law for objects other than monatomie molecules. We have dealt only with the +CM motion of the atoms of a monatomic gas. What happens i1f there are forces +present? First, consider the case that the piston is held by a horizontal spring, +and there are forces on it. The exchange of jiggling motion between atoms and +piston at any moment does not depend on where the piston is at that moment, +of course. The equilibrium conditions are the same. NÑo matter where the piston +1s, is sbeed of motion must be such that it passes energy to the molecules in just +the right way. 5o i% makes no diference about the spring. 'Phe speed at which +the piston has to move, on the average, 1s the same. So our theorem, that the +mean value of the kinetic energy in one direction is skT, #s truc t”hether there +đre [orces present or no†. +* What the chemists call molecular weights are the masses in grams of a mole of a molecule. +The mole is defined so that the mass of a mole of carbon atoms of isotope 12 (i.e., having +6 protons and 6 neutrons in the nucleus) is exactly 12 grams. +--- Trang 709 --- +Consider, for example, a diatomic molecule composed of atoms rnx and rng. +'What we have proved is that the motion of the ƠM of part A and that of part +are such that (2m03) = (šmpu?) = ŠkT. How can this be, if they are held +together? Although they are held together, when they are spinning and turning +in there, when something hits them, exchanging energy with them, 5e onhụ +thứng that counts 1s hou [ast theU are mong. That alone determines how fast +they exchange energy in collisions. At the particular instant, the force is not an +essential point. Therefore the same principle is right, even when there are Íorces. +Let us prove, fnally, that the gas law is consistent also with a disregard of +the internal motion. We did not really include the internal motions before; we +just treated a monatomic gas. But we shall now show that an entire object, +considered as a single body of total mass jM, has a velocity of the CM such that +(šMuễỗm) = $kT. (39.24) +In other words, we can consider either the separate pieces or the whole thing! Let +us see the reason for that: The mass of the diatomic molecule 1s MỸ = ma + mp, +and the velocity of the cenber oŸ mass is equal to ®ew = (maAUaA +?ngop)/M. +Now we need (øổ„,). lÝ we square 0cM, We geb +2 mÄ 0Ä + 2mATnpUA - 0p +1 20% +ĐỒM E TT TT: +NÑow we multiply 3M and take the average, and thus we get +h 3 mA5kT + mAmp(0A - 0p) +1ngŠkT +QM»ễm)=——”————nqg— +— 3 TnA1np(ĐA - Đp) +(We have used the fact that (ma + mnpg)/M = 1.) Now what is (0a: 0p)? (I +had better be zerol) To fnd out, leb us use our assumption that the relative +velocity, 0 — ĐA — 0g 1s not any more likely to poiïnt in one direction than in +another—that is, that its average component in any direction is zero. TÌhus we +assume that +kh * ĐCM) =0. +--- Trang 710 --- +But what 1s ® - eM? ÏIt 1s +— (ĐA— 9B): (mAUÐA +1np0B) ++Ð ' ỦCM — TT +— mAU0Ậ + (mp — 1nA)(ĐA - ĐB) — TpUỆ +=—————nr — " +Therefore, since n3) = (mp2), the first and last terms cancel out on the +average, and we are left with +(mp — TA)(ĐA , ĐbB) =0. +Thus iŸ mau # ng, we find that (0a - 0s) = 0, and therefore that the bodily +motion of the entire molecule, regarded as a single particle of mass Mƒ, has a +kinetic energy, on the average, equal to 3kT. +Incidentally, we have also proved at the same time that the average kinetic +energy of the ?mternal motions of the diatomic molecule, disregarding the bodily +motion of the CM, is 3k7! For, the total kinetic energy of the parts of the +molecule is SInAĐA + 3InpU$, whose avorage is 3kT + 3T, or 3k7'. The kinetic +energy of the center-of-mass motion is 3T, so the average kinetic energy of the +rotational and vibratory motions of the bwo atoms inside the molecule is the +diference, 3kT : +The theorem concerning the average energy of the CM motion is general: for +any object considered as a whole, with forces present or no, for every independent +direction oŸ motion that there is, the average kinetic energy in that motion is skT - +These “independent directions of motion” are sometimes called the đegrees oƒ +reedom oŸ the system. The number of degrees oŸ freedom of a molecule composed +of rz atoms is 3r, since each atom needs three coordinates to defne its position. +The entire kinetic energy of the molecule can be expressed either as the sum of +the kinetic energies of the separate atoms, or as the sum of the kinetic energy of +the CM motion plus the kinetic energy of the internal motions. The latter can +sometimes be expressed as a sum of rotational kinetic energy of the molecule and +vibrational energy, but this is an approximation. Our theorem, applied to the +r-atom molecule, says that the molecule will have, on the average, 3rk7'/2 joules +of kinetic energy, of which 3kT is kinetic energy of the center-of-mass motion of +the entire molecule, and the rest, 3(r— 1)kT, is Internal vibrational and rotational +kinetic energy. +--- Trang 711 --- +Tho Prinerplos of S£(ffsffcerl Wĩoclerrefes +40-1 The exponential atmosphere +W©e have discussed some of the properties of large numbers of intercolliding +atoms. “The subject is called kinetic theory, a description of matter from the point +of view of collisions bebween the atoms. Eundamentally, we assert that the gross +properties of matter should be explainable in terms of the motion of its parts. +We limit ourselves for the present to conditions of thermail equilibrium, that is, +to a subclass of all the phenomena of nature. The laws of mechanics which apply +Jjust to thermal equilibrium are called s‡a#isf¿cal mmechœnics, and 1n this section +we want to become acquainted with some of the central theorems oÊ this subject. +W©e already have one of the theorems of statistical mechanies, namely, the +mmean value of the kinetie energy for any motion at the absolute temperature 7 +1s 3k7 for each independent motion, i.e., for each degree of reedom. That tells +us something about the mean square velocities of the atoms. Our objective now +1s to learn more about the positions of the atoms, to discover how many of them +are going to be in diferent places at thermal equilibrium, and also to go into +a little more detail on the distribution of the velocities. Although we have the +mean square velocity, we do not know how to answer a question such as how +many of them are going three times faster than the root mean square, or how +many of them are goïng one-quarter oŸ the root mean square speed. Or have they +all the same speed exactly? +So, these are the two questions that we shall try to answer: How are the +molecules distributed in space when there are forces acting on them, and how +are they distributed in velocity? +lt turns out that the two questions are completely independent, and that +the distribution of velocities is always the same. We already received a hint +of the latter fact when we found that the average kinetic energy is the same, +skT per degree of freedom, no matter what forces are acting on the molecules. +--- Trang 712 --- +The distribution of the velocities of the molecules is independent of the forces, +because the collision rates do not depend upon the forces. +Let us begin with an example: the distribution of the molecules in an at- +mosphere like our own, but without the winds and other kinds of disturbance. +3uppose that we have a column of gas extending to a great height, and at thermal +equilibrium——=unlike our atmosphere, which as we know gets colder as we go up. +W© could remark that if the temperature difered at diferent heights, we could +demonstrate lack oŸ equilibrium by connecting a rod to some balls at the bottom +(Fig. 40-1), where they would pick up skT from the molecules there and would +shake, via the rod, the balls at the top and those would shake the molecules +at the top. So, ultimately, of course, the temperature becomes the same at all +heights in a gravitational ñeld. +j h+dh +Mechanism / h +for equalizing / +temperature Ũ +Fig. 40-1. The pressure at height h must exceed that at h + dh by +the weight of the intervening gas. +Tf the temperature is the same at all heights, the problem 1s to discover by +what law the atmosphere becomes tenuous as we øo up. lf N is the total number +of molecules in a volume W' of gas at pressure ?, then we know PỰ = Nk1, +or P= nÈkT, where nœ = N/V is the number of molecules per unit volume. In +other words, if we know the number of molecules per unit volume, we know +the pressure, and vice versa: they are proportional to each other, since the +--- Trang 713 --- +temperature is constant in this problem. But the pressure is not constant, 1% +must increase as the altitude is reduced, because it has to hold, so to speak, the +weight of all the gas above it. That is the clue by which we may determine how +the pressure changes with height. If we take a unit area at height 5, then the +vertical force from below, on this unit area, is the pressure . The vertical force +per unit area pushing down at a height h + dh would be the same, in the absence +of gravity, but here it is not, because the force from below must exceed the force +from above by the weight of gas in the section bebween h and h + dh. Now mg is +the force of gravity on each molecule, where ø is the acceleration due to gravlty, +and + dh is the total number of molecules in the unit section. So this gives us +the diferential equation „+, — Đụ = đP = —mgn dh. Since P = nÈT, and T' +1s constant, we can eliminate either or øœ, say ?, and get +dh KT” +for the diferential equation, which tells us how the density goes down as we go +UP 1n energy. +We thus have an equation for the particle density ø, which varies with height, +but which has a derivative which is proportional to itself. Now a function which +has a derivative proportional to itself is an exponential, and the solution of this +diferential equation 1s +n = nạc— møh/RT. (40.1) +Here the constant oŸ integration, mo, is obviously the density at h = 0 (which +can be chosen anywhere), and the density goes down exponentially with height. +Note that If we have diferent kinds of molecules with diferent masses, they +go down with diferent exponentials. The ones which were heavier would decrease +with altitude faster than the light ones. Therefore we would expect that because +oxygen is heavier than nitrogen, as we go higher and higher in an atmosphere +with nitrogen and oxygen the proportion of nitrogen would increase. 'Phis does +not really happen in our own atmosphere, at least at reasonable heights, because +there is so much agitation which mixes the gases back together again. It is not +am isothermal atmosphere. Nevertheless, there 7s a tendency for lighter materials, +like hydrogen, to dominate at very great heights in the atmosphere, because the +lowest masses continue to exist, while the other exponentials have all đied out +(Fig. 40-2). +--- Trang 714 --- +1.0 +0.8 +0.6 +0.4 +0.2 +20 40 60 80 +HEIGHT (Kilometer) +Fig. 40-2. The normalized density as a function of height in the earth's +gravitational field for oxygen and for hydrogen, at constant temperature. +40-2 The Boltzmamn law +Here we note the interesting fact that the numerator in the exponent of +Eq. (40.1) is the pofen#ial energụ oŸ an atom. 'Therefore we can also state this +particular law as: the density at any point is proportional to +c_the potential energy of each atom/kT: +That may be an accident, i.e., may be true only for this particular case of +a uniform gravitational fñeld. However, we can show that it is a more general +proposition. Suppose that there were some kind of force other than gravity acting +on the molecules in a gas. For example, the molecules may be charged electrically, +and may be acted on by an electric field or another charge that attracts them. +Ór, because of the mutual attractions of the atoms for each other, or for the wall, +or for a solid, or something, there is some force of attraction which varies with +position and which aects on all the molecules. NÑow suppose, for simplicity, that +the molecules are all the same, and that the force acts on each individual one, so +that the total force on a piece of gas would be simply the number of molecules +--- Trang 715 --- +times the force on each one. To avoid unnecessary complication, let us choose a +coordinate system with the z-axis in the direction of the force, È'. +In the same manner as before, if we take two parallel planes in the gas, +separated by a distance đz, then the force on each atom, times the ø atoms per +cmở (the generalization of the previous øzng), tìmes đz+, must be balanced by the +pressure change: ïn du — đP —= kT'dn. Ôr, to put thís law in a form which will +be useful to us later, +t`=kT “ã (nn). (40.2) +For the present, observe that —Ƒ' đz is the work we would do in taking a molecule +from z to # + đz, and ïf ' comes from a potential, ¡.e., if the work done can be +represented by a potential energy at all, then this would also be the difference in +the potential energy (P.E.). The negative diferential oŸ potential energy is the +work done, f'đz, and we find that đ(Inw) = —d(P.E.)/kT, or, after integrating, +n = (constant)e_—P.E./*?, (40.3) +Therefore what we noticed in a special case turns out to be true in general. +(What if ` does not come from a potential? Then (40.2) has no solution at all. +tEnergy can be generated, or lost by the atoms running around in cyclic paths +for which the work done is not zero, and no equilibrium can be maintained at +all. Thermail equilibrium cannot exist if the external forces on the atoms are +not conservative.) Equation (40.3), known as ollzmann/s lau, 1s another of the +principles of statistical mechanics: that the probability of ñnding molecules in a +given spatial arrangement varies exponentially with the negative of the potential +energy of that arrangement, divided by k7. +'This, then, could tell us the distribution of molecules: Suppose that we had a +positive ion in a liquid, attracting negative ions around it, how many oŸ them +would be at diÑferent distances? If the potential energy is known as a function oŸ +distance, then the proportion of them at difÑferent distances is given by this law, +and so on, through many applications. +40-3 Evaporation of a liquid +In more advanced statistical mechanics one tries to solve the following impor- +tant problem. Consider an assembly of molecules which attract each other, and +suppose that the force between any two, say ? and 7, depends only on their sepa- +ration r;;, and can be represented as the derivative oŸ a potential function W(z;;). +--- Trang 716 --- +P.E. vự) +Fig. 40-3. A potential-energy function for two molecules, which +depends only on their separation. +Jigure 40-3 shows a form such a function might have. For z > rọ, the energy +decreases as the molecules come together, because they attract, and then the +energy increases very sharply as they come still closer together, because they repel +strongly, which is characteristic of the way molecules behave, roughly speaking. +Now suppose we have a whole box full of such molecules, and we would like +to know how they arrange themselves on the average. The answer is e—P-E./FT, +The total potential energy in this case would be the sum over all the pairs, +supposing that the forces are all in pairs (there may be three-body forces in more +complicated things, but in electricity, for example, the potential energy 1s all in +pairs). Then the probability for ñnding molecules in any particular combination +OÝ ?¿;'s will be proportional to +exp|— ` V(n)/kT]. +Now, if the temperature is very high, so that k7' 3 |V(ro)|, the exponent is +relatively small almost everywhere, and the probability of ñnding a molecule is +almost independent of position. Let us take the case of Jjust two molecules: the +e—P.P/FT would be the probability of ñnding them at various mutual distances z. +Clearly, where the potential goes most negative, the probability 1s largest, and +where the potential goes toward infinity, the probability is almost zero, which +occurs for very small distances. hat means that for such atoms in a gas, there +is no chance that they are on top of each other, since they repel so stronglÌy. +But there is a greater chance of ñnding them per un#t 0olwme at the poïnt ro +than at any other point. How much greater, depends on the temperature. lf +the temperature is very large compared with the diference in energy between +--- Trang 717 --- +r =rọ and r = œ, the exponential is always nearly unity. In this case, where the +mean kinetic energy (about k7) greatly exceeds the potential energy, the forces +do not make mụuch difÑference. But as the temperature falls, the probability of +fñnding the molecules at the preferred distance rọ gradually increases relative to +the probability of fñnding them elsewhere and, in fact, if k?' is much less than +JV(ro)|, we have a relatively large positive exponent in that neighborhood. In +other words, in a given volume they are mwch more likely to be at the distance +of minimum energy than far apart. As the temperature falls, the atoms fall +together, clump in lumps, and reduce to liquids, and solids, and molecules, and +as you heat them up they evaporate. +The requirements for the determination of exactly how things evaporate, +exactly how things should happen ín a given circumstance, involve the following. +Eirst, to discover the correct molecular-force law V{(r), which must come from +something else, quantum mechanics, say, or experiment. But, given the law of +force between the molecules, to discover what a billion molecules are goïng to do +merely consists of studying the function e- 2, V/KT, Surprisingly enough, since +1t is such a simple function and such an easy idea, given the potential, the labor +1s enormouslU complicated; the dificulty is the tremendous number of variables. +In spite of such difficulties, the subJect is quite exciting and interesting. Ït is +often called an example of a “many-body problem,” and it really has been a very +interesting thing. In that single formula must be contained all the details, for +example, about the solidification of gas, or the forms of the crystals that the solid +can take, and people have been trying to squeeze it out, but the mathematical +difculties are very great, not in writing the law, but in dealing with so enorrmmous +a number of variables. +'That then, ¡is the distribution of particles in space. 'That is the end of classical +statistical mechaniecs, practically speaking, because if we know the Íorces, we +can, in principle, fnd the distribution ïn space, and the distribution of velocities +1s something that we can work out once and for all, and is not something that +is diÑerent for the different cases. The great problems are in getting particular +Information out of our formal solution, and that is the main subject of classical +statistical mechanics. +40-4 The distribution of molecular speeds +Now we go on to discuss the distribution of velocities, because sometimes +1t is interesting or useful to know how many of them are moving at diferent +--- Trang 718 --- +speeds. In order to do that, we may make use of the facts which we discovered +with regard to the gas in the atmosphere. We take it to be a perfect gas, as we +have already assumed in writing the potential energy, disregarding the energy of +mutual attraction of the atoms. “The only potential energy that we included in our +first example was gravity. We would, of course, have something more complicated +1f there were forces between the atoms. Thus we assume that there are no forces +bebween the atoms and, for a moment, disregard collisions also, returning later +to the justifcation of this. Now we saw that there are fewer molecules at the +height h than there are at the height 0; according to formula (40.1), they decrease +exponentially with height. How can there be fewer at greater heights? After +all, do not all the molecules which are moving up at height 0 arrive at h? Nol, +because some of those which are moving up at 0 are going too slowly, and cannot +climb the potential hill to h. With that clue, we can calculate how many must +be moving at various speeds, because from (40.1) we know Ø rnamw/ are moving +with less than enough speed to climb a given distance h. Those are just the ones +that account for the fact that the density at h is lower than at 0. +Now let us put that idea a little more precisely: let us count how many +molecules are passing from below to above the plane h = 0 (by calling it +height = 0, we do not mean that there is a Ñoor there; it is Just a convenient label, +and there is gas at negative 5). These gas molecules are moving around in every +direction, but some of them are moving through the plane, and at any momentf a +certain number per second of them are passing through the plane from below to +Fig. 40-4. Only those molecules moving up at h = 0 with sufficient +velocity can arrive at height h. +--- Trang 719 --- +above with diferent velocities. Now we note the following: if we call œ the velocity +which is jusi needed to get up to the height Đ (kinetic energy rwu2/2 = rmgh), +then the number of molecules per second which are passing upward through +the lower plane in a vertical direction with velocity component greater than +1s exactly the same as the number which pass through the upper plane with +am upward velocity. Thhose molecules whose vertical velocity does not exceed +cannot get through the upper plane. So therefore we see that +Number passing h = 0 with 0; > œ = number passing h = h with 0; > 0. +But the number which pass through h with any velocity greater than 0 is less than +the number which pass through the lower height with any velocity greater than 0, +because the number of atoms is greater; that is all we need. We know already +that the distribution of velocities is the same, after the argument we made earlier +about the temperature being constant all the way through the atmosphere. So, +since the velocity distributions are the same, and ï$ is Jjust that there are more +atorms Ìower down, clearly the number ?ø+>o(h), passing with positive velocity at +height h, and the number ø-.o(0), passing with positive velocity at height 0, are +in the same ratio as the densities at the two heights, which is e~””9*⁄'T, But +m>o(h) = n>„(0), and therefore we find that +m>„(0) —c-mgh/KT — ,—mu”/2kT +m>o(0) : +since 3mu2 = mịạgh. Thus, in words, the number of molecules per unit area +per second passing the height 0 with a z-component of velocity greater than u +is e—942/2#” tìmes the total number that are passing through the plane with +velocity greater than zero. +Now this is not only true at the arbitrarily chosen height 0, but of course 1 +is true at any other height, and thus the distributions of velocities are all the +samel (The fñnal statement does not involve the height h, which appeared only +in the intermediate argument.) The result is a general proposition that gives us +the distribution oŸ velocities. It tells us that if we drill a little hole in the side +of a gas pipe, a very tiny hole, so that the collisions are few and far bebween, +1.e., are farther apart than the diameter of the hole, then the particles which are +coming out will have different velocities, but the fraction of particles which come +out at a velocity greater than œ is e—”®92/2RT, +--- Trang 720 --- +Now we return to the question about the neglect of collisions: Why does 1t +not make any diference? We could have pursued the same argument, not with a +ñnite height h, but with an infũnitesimal height h, which is so small that there +would be no room for collisions between 0 and 5h. But that was not necessary: +the argument is evidently based on an analysis of the energies involved, the +conservation of energy, and in the collisions that occur there is an exchange of +energies among the molecules. However, we do not really care whether we follow +the same molecule iŸ energy is merely exchanged with another molecule. So i% +turns out that even ïŸ the problem is analyzed more carefully (and it is more +diffiecult, naturally, to do a rigorous job), it still makes no diference in the result. +Tt is interesting that the velocity distribution we have found is just +m%„ % e_ kimetic energy/kT" (40.4) +This way of describing the distribution of velocities, by giving the number +of molecules that pass a given area with a certain minimum z-component, 1s +not the most convenient way oŸ giving the velocity distribution. Eor instance, +inside the gas, one more often wants to know how many molecules are moving +with a z-component of velocity between two given values, and that, of course, 1s +not directly given by Eaq. (40.4). We would like to state our result in the more +conventional form, even though what we already have written is quite general. +Note that tt is not possible to sau that ang tmmolecule has eœactlU sormne stated +0elocitu; none of them has a velocity ezacflu cqual to 1.7962899173 meters per +second. So in order to make a meaningful statement, we have to ask how many are +to be found in some ranøe of velocities. We have to say how many have velocities +between 1.796 and 1.797, and so on. On mathematical terms, let ƒ(u) du be +the fraction of all the molecules which have velocities between and +œ + du or, +what is the same thing (ïf du is infnitesimal), all that have a velocity w with +aà range du. Pigure 40-5 shows a possible form for the function ƒ(u), and the +shaded part, of width đu and mean height ƒ(u), represents this fraction ƒ(u) du. +'That is, the ratio of the shaded area to the total area of the curve is the relative +proportion of molecules with velocity within du. If we define ƒ(u) so that the +traction having a velocity in this range is given directly by the shaded area, then +the total area must be 100 percent of them, that 1s, +J ƒ(u) du = 1. (40.5) +--- Trang 721 --- +Fig. 40-5. A velocity distribution function. The shaded area is f(u) du, +the fraction of particles having velocities within a range đu about u. +Now we have only to get this distribution by comparing it with the theorem +we derived before. Pirst we ask, what is the number of molecules passing through +an area per second with a velocity greater than ö, expressed in terms of ƒ(u)? At +first we might think it is merely the integral of T ƒ(u) du, but it 1s not, because +we want the number that are passing the area per second. The faster ones pass +more often, so to speak, than the slower ones, and in order to express how many +pass, you have to multiply by the velocity. (We discussed that in the previous +chapter when we talked about the number of collisions.) In a given tỉme £ the +total number which pass through the surface is all of those which have been able +to arrive at the surface, and the number which arrive come from a distance uứ. +So the number of molecules which arrive is not simply the number which are +there, but the number that are there per unit volume, multiplied by the distance +that they sweep through in racing for the area through which they are supposed +to go, and that distance is proportional to w. Thus we need the integral of u +times ƒ{(u) du, an infinite integral with a lower limit u, and this must be the same +as we found before, namely e~muÖ/ 2F? with a proportionality constant which +we will get later: +J tƒ(u) du = const - c— mu)/2RT, (40.6) +Now ïf we diferentiate the integral with respect to u, we get the thing that +is inside the integral, ¡.e., the integrand (with a minus sign, since is the lower +limit), and if we diferentiate the other side, we get œ times the same exponential +--- Trang 722 --- +(and some constants). The 's cancel and we find +ƒ(u) du = Cc— mw)/2kT gu, (40.7) +We retain the dư on both sides as a reminder that it is a đ¿s‡r¿bution, and 1t tells +what the proportion is for velocity between and -+L du. +The constant Œ must be so determined that the integral is unity, according +to Eq. (40.5). Now we can prove* that +J e~*” dạ = Vn. +Using this fact, iE is easy to fnd that Ở = v/1mm/2mkT'. +Since velocity and momentum are proportional, we may say that the distribu- +tion oŸ momenta is also proportional to e—K:E⁄⁄#” per unit momentum range. lt +turns out that this theorem is true in relativity too, ïÝ it is in terms of momentum, +while iŸ it is in velocity it is not, so it is best to learn i% in momentum instead oŸ +in velocity: +ƒ(p) dp = Ce~-P-/*T đạp, (40.8) +So we find that the probabilities of diferent conditions of energy, kinetic and +potential, are both given by e—°"e'sv/F` a very easy thing to remember and a +rather beautiful proposition. +So far we have, of course, only the distribution of the velocities “vertically.” +We might want to ask, what is the probability that a molecule is moving in +another direction? Of course these distributions are connected, and one can +obtain the complete distribution from the one we have, because the complete +distribution depends only on the square of the magnitude of the velocity, not +upon the z-component. It must be something that is independent of direction, +* 'To get the value of the integral, let +1= ƑS c*” da, +2= ƑS ct2 đái [S cty = [S, ƑS c9) đáp, +which is a double integral over the whole z-plane. But this can also be written in polar +coordinates as 3 +12—= la e~T“ . 9r dry = xJo © tát =1. +--- Trang 723 --- +and there 1s only one function involved, the probability of diferent magnitudes. +We have the distribution of the z-component, and therefore we can get the +distribution of the other components from it. The result ¡is that the probability +is sbill proportional to e—-E/'? but now the kinetie energy involves three parts, +muy/2, mu /2, and nu 2/2, sumnmed ïn the exponent. Or we can write it as a +product: +ƒ(Uz, 0y, 0y) duy duy duy +œ €— mo2/2ET,c—may/2kT, ma 2/2ET duy duy duy. (40.9) +You can see that this formula must be right because, frst, it is a function only +Of 2, as required, and second, the probabilities of various values of 0; obtained +by integrating over all u„ and œy is just (40.7). But this one function (40.9) can +do both those thingsl +40-5 The speciũc heats of gases +Now we shall look at some ways to test the theory, and to see how successful +1s the classical theory of gases. We saw earlier that iŸ Ư is the internal energy +of N molecules, then PV = Nk7' = (+ — 1)U holds, sometimes, for some gases, +maybe. If it is a monatomic gas, we know this is also equal to Ỹ of the kinetic +energy of the center-of-mass motion of the atoms. lỶ it is a monatomic gas, then +the kinetic energy is equal to the internal energy, and therefore + — l = '- But +Suppose I1 is, say, a more complicated molecule, that can spin and vibrate, and +let us suppose (it turns out to be true according to classical mechanics) that +the energies of the internal motions are also proportional to k7? 'Phen at a +given temperature, in addition to kinetic energy 3kT, 1t has internal vibrational +and rotational energies. So the total U includes not just the kinetic energy, but +also the rotational and vibrational energies, and we get a diferent value of +. +'lechnically, the best way to measure + is by measuring the specifc heat, which +is the change in energy with temperature. We will return to that approach later. +For our present purposes, we may suppose + 1s found experimentally from the +PV” curve for adiabatic compression. +Let us make a calculation of + for some cases. First, for a monatomiec gas +is the total energy, the same as the kinetic energy, and we know already that + +should be Š- For a diatomic gas, we may take, as an example, oxygen, hydrogen +1odide, hydrogen, etc., and suppose that the diatomic gas can be represented +--- Trang 724 --- +as two atoms held together by some kind of force like the one of Fig. 40-3. We +may also suppose, and it turns out to be quite true, that at the temperatures +that are of interest for the diatomic gas, the pairs of atoms tend strongly to be +separated by rọ, the distance of potential minimum. Tf this were not true, If the +probability were not strongly varying enough to make the great majority sit near +the bottom, we would have to remermber that oxygen gas is a mixture oŸ Os and +single oxygen atoms in a nontrivial ratio. We know that there are, in fact, very +few single oxygen atoms, which means that the potential energy minimum is very +much greater in magnitude than k7, as we have seen. Since they are clustered +strongly around rọ, the only part of the curve that is needed is the part near +the minimum, which may be approximated by a parabola. A parabolie potential +Implies a harmonie oscillator, and in fact, to an excellent approximation, the +oxygen molecule can be represented as two atoms connected by a spring. +Now what is the total energy of this molecule at temperature 7? We know +that for each of the two atoms, each of the kinetic energies should be 3SkT, SO +the kinetic energy of both of them is 3k7 + Šk7. We can also put this in a +diferent way: the same Ỷ plus Ỷ can also be looked at as kinetic energy of the +center of mass (ở), kinetic energy of rotation (f), and kinetic energy of vibration +(5). We know that the kinetic energy of vibration is 3, since there is just one +dimension involved and each degree of freedom has 5k1. Regarding the rotation, +it can turn about either of two axes, so there are two independent motions. We +assume that the atoms are some kind of points, and cannot spin about the line +Joining them; this is something to bear in mỉnd, because iÝ we get a disagreement, +maybe that is where the trouble is. But we have one more thing, which is the +potential energy of vibration; how much is that? In a harmonic oscillator the +average kinetic energy and average potential energy are equal, and therefore the +potential energy of vibration is 3kT, also. The grand total of energy is Ư = SkT, +or kĩ is ‡U per atom. 'Phat means, then, that + is Ỹ instead of Ÿ› 1.e., y = 1.286. +We may compare these numbers with the relevant measured values shown +in Table 40-1. Looking frst at helium, which is a monatomic gas, we fnd +very nearÌy Ÿ› and the error is probably experimental, although at such a low +temperature there may be some forces between the atoms. Krypton and argon, +both monatomic, agree also within the accuracy of the experiment. +We© turn to the diatomiec gases and fñnd hydrogen with 1.404, which does not +agree with the theory, 1.286. Oxygen, 1.399, is very similar, but again not in +agreement. Hydrogen iodide again is similar at 1.40. It begins to look as though +the right answer is 1.40, but ït is not, because If we look further at bromine +--- Trang 725 --- +Table 40-1 +Values of the specifc heat ratio, +, for various gases +He —180 1.660 +lér 19 1.68 +Ar lỗ 1.668 +Ha 100 1.404 +O2 100 1.399 +HI 100 1.40 +Đra 300 1.32 +lạ 185 1.30 +C©aHs 15 1.22 +we see 1.32, and at iodine we see 1.30. Since 1.30 is reasonably close to 1.286, +iodine may be said to agree rather well, but oxygen is far of. 5o here we have a +dilemnma. We have it right for one molecule, we do not have it right for another +mmolecule, and we may need to be pretty ingenious in order to explain both. +Let us look further at a still more complicated molecule with large numbers +of parts, for example, C2Hạ, which is ethane. It has eight diferent atoms, and +they are all vibrating and rotating in various combinations, so the total amount +OŸ internal energy must be an enormous number of kT”s, at least 12k7' for kinetic +energy alone, and y— 1 must be very close to zero, or +y almost exactly 1. In fact, +1t 2s lower, but 1.22 is not so mụuch lower, and is higher than the lẰ calculated +from the kinetic energy alone, and it is just not understandablel +Purthermore, the whole mystery is deep, because the diatomic molecule +cannot be made rigid by a limit. Even if we made the couplings stifer indefnitely, +although it might not vibrate much, it would nevertheless keep vibrating. The +vibrational energy inside is still k7, since it does not depend on the strength of +the coupling. But if we could imagine øsolute rigidity, stopping all vibration to +eliminate a variable, then we would get Ứ = 5kT and + = 1.40 for the diatomic +case. This looks good for Hạ or Ó¿. Ôn the other hand, we would still have +problems, because + for either hydrogen or oxygen varies with temperaturel From +the measured values shown in PFig. 40-6, we see that for Hạ, + varies from about +1.6 at —185°Ở to 1.3 at 2000°G. The variation is more substantial in the case of +--- Trang 726 --- +1/6 Hạ +1.4E *% ¬ +"`. dd +1.2 +1.0 +0 500 1000 1500 2000 +TEMPERATURE (°C) +Fig. 40-6. Experimental values of + as a function of temperature for +hydrogen and oxygen. Classical theory predicts y = 1.286, independent +of temperature. +hydrogen than for oxygen, but nevertheless, even in oxygen, + tends defñnitely to +go up as we go down in temperature. +40-6 The failure of classical physics +So, all in all, we might say that we have some difficulty. We might try some +force law other than a spring, but it turns out that anything else will only make ++ higher. If we include more forms of energy, y approaches unity more closely, +contradicting the facts. All the classical theoretical things that one can think of +will only make it worse. 'Phe fact is that there are electrons in each atom, and +we know from theïir spectra that there are internal motions; each of the electrons +should have at least skT of kinetic energy, and something for the potential energy, +so when these are added in, + gets still smaller. It is ridiculous. It is wrong. +The first great paper on the dynamical theory of gases was by Maxwell in +1859. On the basis of ideas we have been discussing, he was able accurately to +explain a great many known relations, such as Boyle”s law, the difusion theory, +the viscosity of gases, and things we shall talk about in the next chapter. He listed +all these great successes in a fñnal summary, and at the end he said, “Eïnally, by +establishing a necessary relation between the motions of translation and rotation +(he is talking about the skT theorem) of all particles not spherical, we proved +that a system of such particles could not possibly satisfy the known relation +--- Trang 727 --- +between the two specific heats.” He is referring to + (which we shall see later is +related to two ways of measuring specifc heat), and he says we know we cannot +get the right answer. +Ten years later, in a lecture, he said, “I have now put before you what I +consider to be the greatest dificulty yet encountered by the molecular theory.” +'These words represent the first discovery that the laws of classical physics were +wrong. This was the frst indication that there was something fundamentally +Impossible, because a rigorously proved theorem did not agree with experiment. +About 1890, Jeans was to talk about this puzzÌe again. One often hears it said +that physicists at the latter part of the nineteenth century thought they knew all +the signifcant physical laws and that all they had to do was to calculate more +decimal places. Someone may have said that once, and others copied it. But a +thorough reading of the literature of the time shows they were all worrying about +something. Jeans said about this puzzle that it is a very mysterious phenomenon, +and it seems as though as the temperature falls, certain kinds of motions “freeze +T we could assume that the vibrational motion, say, did not exist at low +temperature and did exist at high temperature, then we could imagine that a gas +might exist at a temperature sufficiently low that vibrational motion does not +occur, so + = 1.40, or a higher temperature at which it begins to come in, so + +falls. The same might be argued for the rotation. If we can eliminate the rotation, +say 1t “freezes out” at suficiently low temperature, then we can understand the +fact that the + of hydrogen approaches 1.66 as we go down in temperature. How +can we understand such a phenomenon? Of course that these motions “freeze +out” cannot be understood by classical mechanics. It was only understood when +quantum mechanics was discovered. +Without proof, we may state the results for statistical mechanics of the +quantum-mechanical theory. We recall that according to quantum mechanics, a +system which is bound by a potential, for the vibrations, for example, will have +a discrete set of energy levels, i.e., states of diferent energy. Now the question +is: how is statistical mechanics to be modified according to quantum-mechanical +theory? It turns out, interestingly enough, that although most problems are +more dificult in quantum mechaniecs than in classical mechanics, problems In +statistical mechanics are much easier in quantum theoryl "The simple result we +have in classical mechanies, that ø = nọc—°"e'sy/FT_ becomes the following very +Important theorem: Tf the energies of the set of molecular states are called, say, +đọ, Eị, Eạ,..., đụ, ..., then in thermal equilibrium the probability of ñnding a +--- Trang 728 --- +molecule in the particular state of having energy #¿ is proportional to e—#:/FT, +That gives the probability of beïing in various states. In other words, the relative +chance, the probability, of being in state #¡ relative to the chance of being in +state 2o, 1s +—Ei/kT +mì (40.10) +Pạ_— c-Eu/RT +which, of course, is the same as +mị = nọc” ị—Eu)/ET. (40.11) +since ị = mị/N and Pụ = nọ/N. So ït is less likely to be in a higher energy state +than in a lower one. The ratio of the number of atoms in the upper state to the +number in the lower state is e raised to the power (minus the energy diference, +over #7')—a very simple proposition. +Now it turns out that for a harmonic oscillator the energy levels are evenly +spaced. Calling the lowest energy #g = 0 (it actually is not zero, it is a little +diferent, but it does not matter If we shift all energies by a constant), the first +one is then = ñœ, and the second one is 2œ, and the third one is 3ñ¿, and +SO OI. +Now let us see what happens. We suppose we are studying the vibrations +of a diatomic molecule, which we approximate as a harmonic oscillator. Let +us ask what is the relative chance of finding a molecule in state # instead of +in state ọ. The answer is that the chance of fñnding it in state F, relative +to that of fnding it in state Eo, goes down as e—”⁄“/'T_ NÑow suppose that k7 +1s much less than ñ¿, and we have a low-temperatfure cireumstance. 'Then the +probability of its beïng in state #¡ is extremely small. Practically all the atoms +are in state họ. IÝ we change the temperature but still keep it very small, then +the chance of its being in state #¡ = ñœ remains infinitesimal—the energy of the +oscillator remains nearly zero; it does not change with temperature so long as the +temperature is much less than hớ. All oscillators are in the bottom state, and +their motion is efectively “frozen”; ¿here ?s no contribution oj tt to the specific +heœt. We can judge, then, from Table 40-1, that at 100°C, which is 373 degrees +absolute, k7' is much less than the vibrational energy in the oxygen or hydrogen +molecules, but not so in the iodine molecule. "The reason for the diference 1s +that an iodine atom is very heavy, compared with hydrogen, and although the +forces may be comparable in iodine and hydrogen, the iodine molecule is so +heavy that the natural frequency of vibration is very low compared with the +--- Trang 729 --- +natural frequency of hydrogen. With ñœ higher than k7” at room temperature +for hydrogen, but lower for iodine, only the latter, iodine, exhibits the classical +vibrational energy. As we increase the temperature of a gas, starting from a very +low value of 7', with the molecules almost all in theïr lowest state, they gradually +begin to have an appreciable probability to be in the second state, and then in +the next state, and so on. When the probability is appreciable for many states, +the behavior of the gas approaches that given by classical physics, because the +quantized states become nearly indistinguishable from a continuum oŸ energies, +and the system can have almost any energy. Thus, as the temperature rises, +we should again get the results of classical physics, as indeed seems to be the +case in Pig. 40-6. Ib is possible to show in the same way that the rotational +states of atoms are also quantized, but the states are so much closer together +that in ordinary circumstances &?' is bigger than the spacing. 'Phen many levels +are excited, and the rotational kinetic energy in the system participates in the +classical way. The one example where this is not quite true at room temperafure +is for hydrogen. +This is the first time that we have really deduced, by comparison with +experiment, that there was something wrong with classical physics, and we have +looked for a resolution of the difficulty in quantum mechaniecs in mụch the same +way as it was done originally. It took 30 or 40 years before the next difficulty was +discovered, and that had to do again with statistical mechanies, but this time +the mechanics of a photon gas. That problem was solved by Planck, in the early +years of the 20th century. +--- Trang 730 --- +Tho l?rotrrtrcrrt WOtOrrrortế +41-1 Equipartition of energy +The Brownian movement was discovered in 1827 by Robert Brown, a botanist. +'While he was studying miecroscopic life, he noticed little particles of plant pollens +Jiggling around in the liquid he was looking at in the microscope, and he was +wise enough to realize that these were not living, but were just little pieces of +dirt moving around in the water. In fact he helped to demonstrate that this +had nothing to do with life by getting from the ground an old piece of quartz in +which there was some water trapped. lt must have been trapped for millions and +millions of years, but inside he could see the same motion. What one sees is that +very tiny particles are jiggling all the time. +This was later proved to be one of the efects of molecular moton, and we +can understand it qualitatively by thinking of a great push ball on a playing field, +seen from a great distance, with a lot of people underneath, all pushing the ball +in various directions. We cannot see the people because we imagine that we are +too far away, but we can see the ball, and we notice that it moves around rather +irregularly. We also know, from the theorems that we have discussed in previous +chapters, that the mean kinetic energy of a small particle suspended in a liquid +or a gas will be 3kT even though it is very heavy compared with a molecule. lf +1t 1s very heavy, that means that the speeds are relatively slow, but it turns out, +actually, that the speed is not really so slow. In fact, we cannot see the speed +of such a particle very easily because although the mean kinetic energy is 3T, +which represents a speed of a millimeter or so per second for an objJect a micron +or ©wo in diameter, this is very hard to see even in a microscope, because the +particle continuously reverses its direction and does not get anywhere. How far +it does get we will discuss at the end of the present chapter. 'Phis problem was +ñrst solved by Binstein at the beginning of the 20th century. +--- Trang 731 --- +Incidentally, when we say that the mean kinetic energy of this particle is 3kT, +we claim to have derived this result from the kinetic theory, that is, from Newton”s +laws. We shall ñnd that we can derive all kinds of things—marvelous thỉings—from +the kinetic theory, and it is most interesting that we can apparently get so much +from so little. Of course we do not mean that Newton”s laws are “little”——they +are enough to do it, really—=what we mean is that e did not do very much. How +do we get so much out? The answer is that we have been perpetually making +a certain immportant assumption, which is that if a given system is in thermal +equilibrium at some temperature, it will also be In thermal equilibrium with +amthing else at the same temperature. For instance, iŸ we wanted to see how +a particle would move if it was really colliding with water, we could imagine +that there was a gas present, composed of another kind of particle, little fñne +pellets that (we suppose) do not interact with water, but only hit the particle +with “hard” collisions. Suppose the particle has a prong sticking out of ït; all +our pellets have to do is hit the prong. We know all about this imaginary gas of +pellets at temperature 7——it is an ideal gas. Water is complicated, but an ideal +gas is simple. Now, owr particle has to be in cquilibrtum uuíth the gas oƒ pellets. +Therefore, the mean motion of the particle must be what we get Íor gaseous +collisions, because 1ƒ it were not moving at the right speed relative to the water +but, say, was moving faster, that would mean that the pellets would pick up +energy from it and get hotter than the water. But we had started them at the +same temperature, and we assume that if a thing is once in equilibrium, it stays In +equilibrium——parts of it do not get hotter and other parts colder, spontaneously. +This proposition is true and can be proved from the laws of mechanics, but +the proof is very complicated and can be established only by using advanced +mechanics. ]t is much easier to prove in quantum mechanics than it is in classical +mechanics. Ït was proved first by Boltzmamn, but for now we simply take it to +be true, and then we can argue that our particle has to have 3kT OŸ energy lf it +1s hit with artifcial pellets, so it also must have 3kT when I1 is being hit with +water at the same temperature and we take away the pellets; so it 1s 3k1. ltis a +strange line of argument, but perfectly valid. +In addition to the motion of colloidal particles for which the Brownian move- +ment was first discovered, there are a number of other phenomena, both in the +laboratory and in other situations, where one can see Brownian movement. lÝ we +are trying to buïld the most delicate possible equipment, say a very small mirror +on a thin quartz fiber for a very sensitive ballistic galvanometer (Eig. 41-1), the +mirror does not stay put, but jiggles all the time——all the time—so that when we +--- Trang 732 --- +MR L ñ +(a) (b) +Fig. 41-1. (a) A sensitive light-beam galvanometer. Light from a +source L ¡is reflected from a small mirror onto a scale. (b) A schematic +record of the reading of the scale as a function of the time. +shine a light on it and look at the position of the spot, we do not have a perfect +instrument because the mirror is always jiggling. Why? Because the average +kinetic energy of rotation of this mirror has to be, on the average, 3kT : +'What is the mean-square angle over which the mirror will wobble? Suppose we +ñnd the natural vibration period of the mirror by tapping on one side and seeing +how long ít takes to oscillate back and forth, and we also know the moment of +inertia, ƒ. We know the formula for the kinetic energy oŸ rotation—it is given by +Eq. (19.8): 7= šTø?. That is the kinetic energy, and the potential energy that +goes with it will be proportional to the square of the angle—it is W = sa03. But, +1ƒ we know the period £o and calculate from that the natural frequenecy œo = 27/1o, +then the potential energy is V = jIœÿØ?. Now we know that the average kinetic +©h©rgy is 2k1, but sỉnee it is a harmonic oscillator the average potential energy +1s alsO 3kT. 'Thus +31u0(0”) = $kT, +(02) = kT/Tu§. (41.1) +In this way we can calculate the oscillations of a galvanometer mirror, and thereby +fnd what the limitations of our instrument wiïll be. lf we want to have smaller +oscillations, we have to cool the mirror. An interesting question is, 0ere to cool +it. This depends upon where ït is getting its “kicks” from. Tf i§ is through the +fber, we cool it at the top—ïf the mirror is surrounded by a gas and is getting hit +mmostly by collisions in the gas, it is better to cool the gas. As a matter of fact, if +we know where the damnpzng of the oscillations comes from, it turns out that that +1s always the source of the ñuctuations also, a point which we will come back to. +--- Trang 733 --- +(a) (b) +Fig. 41-2. A high-Q resonant circuit. (a) Actual circuit, at tempera- +ture T. (b) Artificial circuit, with an ideal (noiseless) resistance and a +“noise generator” G. +The same thing works, amazingly enough, in elecfrical circu#ts. Suppose that +we are building a very sensitive, accurate amplifier for a defñnite frequeney and +have a resonant circuit (Fig. 41-2) in the input so as to make it very sensitive to +this certain frequency, like a radio receiver, but a really good one. Suppose we +wish to go down to the very lowest limit of things, so we take the voltage, say +of the inductance, and send ït into the rest of the amplifier. Of course, in any +circuit like this, there is a certain amount of loss. ÏIt is not a perfect resonant +circuit, but it is a very good one and there is a littÏe resistance, say (we put +the resistor in so we can see it, but i is supposed to be small). NÑow we would +like to ñnd out: How much does the voltage across the inductance Ñuctuate? +Ansuer: We know that s1 2 is the “kinetic energy”—the energy associated with +a coil in a resonant circuit (Chapter 25). Therefore the mean value of 3,1? is +cequal to 3kT——that tells us what the rms current is and we can find out what +the rms voltage is from the rms current. For if we want the voltage across the +inductance the formula is Ứy = ¿¿L , and the mean absolute square voltage on +the inductanee is (Wÿ) = L?œ8(1?), and putting in šE(1?) = škT, we obtain +(Vỷ) = LukT. (41.2) +Đo now we can design circuits and tell when we are going to get what is called +Johnson noise, the noise associated with thermal fuctuationsl +Where do the Ñuctuations come from this time? 'They come again from the +resistor——they come from the fact that the electrons in the resistor are jiggling +around because they are in thermal equilibrium with the matter in the resistor, +and they make fuctuations in the density of electrons. 'Phey thus make tiny +electric ñelds which drive the resonant circuit. +tElectrical engineers represent the answer in another way. Physically, the +resistor is efectively the source of noise. However, we may replace the real circuit +--- Trang 734 --- +having an honest, true physical resistor which is making noise, by an artifcial +circuit which contains a little generator that is going to represent the noise, and +now the resistor is otherwise ideal—no noise comes from it. All the noise is +in the artificial generator. And so if we knew the characteristics of the noise +generated by a resistor, Iƒ we had the formula for that, then we could calculate +what the circuit is going to do in response to that noise. So, we need a formula +for the noise Ñuctuations. Now the noise that is generated by the resistor is at +all frequencies, since the resistor by itself is not resonant. Of course the resonant +circuit only “listens” to the part that is near the right frequency, but the resisbor +has many diferent frequencies in it. We may describe how strong the generator +1s, as follows: The mean power that the resistor would absorb if it were connected +directly across the noise generator would be (2)/ñR, if E were the voltage from +the generator. But we would like to know in more detail how much power there +is aW every Írequency. “There is very little power in any one Írequency; it is a +distribution. Let P(0) dư be the power that the generator would deliver in the +Írequency range đ« into the very same resistor. Then we can prove (we shall +prove it for another case, but the mathematics is exactly the same) that the +DOWeT comes out +P() dự = (2/)kT du, (41.3) +and is mdependent oƒ the resistance when put this way. +41-2 Thermal equilibrium of radiation +Now we go on to consider a still more advanced and interesting proposition +that is as follows. S5uppose we have a charged oscillator like those we were talking +about when we were discussing light, let us say an electron oscillating up and +down in an atom. lfit oscillates up and down, it radiates light. Now suppose that +this oscillator is in a very thin gas of other atoms, and that from tỉme to time +the atoms collide with it. 'Then in equilibrium, after a long time, this oscillator +will pick up energy such that is kinetic energy of oscillation is skT , and since +1t 1s a harmonic oscillator, its entire energy will become &?'. That is, oÝ course, +a wrong description so far, because the oscillator carries clecfr/c charge, and 1ƒ +it has an energy kT' it is shaking up and down and radiating light. 'Pherefore +1t 1s impossible to have equilibrium of real matter alone without the charges +in it emitting light, and as light is emitted, energy fows away, the oscillator +loses its & as time goes on, and thus the whole gas which is colliding with the +--- Trang 735 --- +oscillator gradually cools of. And that is, of course, the way a warm sÈove cools, +by radiating the light into the sky, because the atoms are jiggling their charge +and they continually radiate, and slowly, because of this radiation, the jiggling +motion slows down. +On the other hand, if we enclose the whole thing in a box so that the light +does not go away to infnity, then we can eventually get thermail equilibrium. We +may either put the gas in a box where we can say that there are other radiators +in the box walls sending light back or, to take a nicer example, we may sSuppose +the box has mirror walls. It is easier to think about that case. Thus we assume +that all the radiation that goes out from the oscillator keeps running around in +the box. Thhen, of course, it is true that the oscillator starts to radiate, but pretty +sSoon it can maintain its &Ƒ' of energy in spite of the fact that ¡% is radiating, +because it is being illuminated, we may say, by its own light reflected from the +walls of the box. That is, after a while there is a great deal of light rushing +around in the box, and although the oscillator is radiating some, the light comes +back and returns some of the energy that was radiated. +We shall now determine how much light there must be in such a box at +temperature 7 in order that the shining of the light on this oscillator will +generate just enough energy to account for the light it radiated. +Let the gas atoms be very few and far between, so that we have an ideal +oscillator with no resistance except radiation resistance. hen we consider that +at thermal equilibrium the oscillator is doïng two things at the same time. First, +1t has a mean energy &?, and we calculate how much radiation it emits. Second, +this radiation should be exactly the amount that would result because of the fact +that the light shining on the oscillator is scattered. Since there is nowhere else +the energy can go, this efective radiation is really just scattered light from the +light that is in there. +Thus we frst calculate the energy that is radiated by the oscillator per second, +1ƒ the oscilator has a certain energy. (We borrow from Chapter 32 on radiation +resisbance a number of equations without going back over their derivation.) The +energy radiated per radian divided by the energy of the oscillator is called 1/Q +(Eq. 32.8): 1/Q = (ÄW/đf)/œạW. ỦUsing the quantity +, the damping constant, +this can also be written as 1/Q = +/@ọ, where œc is the natural frequency of +the oscillator—if gamma is very small, @ is very large. 'Phe energy radiated per +second is then n +W_ œgW_ u¿ạW+ +--- Trang 736 --- +The energy radiated per second is thus simply gamma times the energy of the +oscillator. Now the oscillator should have an average energy k7”, so we see that +gamma k7 is the average amount oŸ energy radiated per second: +(dW/dt) = +kT. (41.5) +Now we only have to know what gamma is. Gamma is easily found from +Eq. (32.12). It is +œọ — 2 Tguậ +mm. e7 (41.6) +where ?o = e2/mc? is the classical electron radius, and we have set À = 2/00. +Our ñnal result for the average rate of radiation of light near the Írequency œ0 +1s therefore 2 +dị _ 2T0e0NP (41.7) +dt b) C +Next we ask how much light must be shining on the oscillator. It must be +enough that the energy absorbed from the light (and thereupon scattered) is just +exactly this much. In other words, the emitted light is accounted for as scaftered +light rom the light that is shining on the oscillator in the cavity. 5o we must +now calculate how much light is scattered from the oscillator if there is a certain +axmount—=unknown——oŸ radiation ineident on it. Let 7(6) đà be the amount of +light energy there is at the frequenecy œ, within a certain range đœ (because there +is no light at ezacflu a certain Írequenecy; it is spread all over the spectrum). So +T(() is a certain spectral đistribution which we are now goïng to find—it is the +color of a furnace at temperature 7' that we see when we open the door and +look in the hole. NÑow how much light is absorbed? We worked out the amount +of radiation absorbed from a given incident light beam, and we calculated it in +terms OŸ a cross secfion. It is just as though we said that all of the light that falls +on a certain cross section is absorbed. So the total amount that is re-radiated +(scattered) is the incident intensity 7(œ) dœ multiplied by the cross section ø. +The formula for the cross section that we derived (Eq. 32.19) did not have +the damping included. It is not hard to go through the derivation again and put +in the resistance term, which we neglected. If we do that, and calculate the cross +section the same way, we get +8rrÿ „4 +n Í= — Ä)2 + 12/2 ) (4L8) +--- Trang 737 --- +Now, as a function of equenecy, ø; is Of significant size only Íor œ very near +to the natural frequency œo. (Remember that the @ for a radiating oscillator +is about 108.) The oscillator scatters very strongly when œ is equal to œạ, and +very weakly for other values of œ. Therefore we can replace œ by œạ and œŠ — uậ +by 2¿o(d — œạ), and we get +Øs= 3[(— œ0)2+ 22/41 (41.9) +Now the whole curve is localized near œ = œọ. (We do not really have to make +any approximations, but it is much easier to do the integrals if we simplify the +cquation a bit.) Ñow we multiply the intensity in a given frequency range by the +cross section oŸ scattering, to get the amount of energy scattered in the range đi. +'The £o£al energy scattered is then the integral of this for all ¿. Phus +=ÍJ 1(()Øs() do +' (41.10) +—— là 2arqu8 I(œ) dư +› 3|@@—sp)+29/4] +Now we set đW; /dt = 3ykT. Why three? Because when we made our analysis +of the cross section in Chapter 32, we assumed that the polarization was such +that the light could drive the oscillator. lf we had used an oscillator which could +move only in one direction, and the light, say, was polarized in the wrong way, +iÿ would not give any scattering. So we must either average the cross section of +an oscillator which can go only in one direction, over all directions of incidence +and polarization of the light or, more easily, we can imagine an oscillator which +will follow the fñeld no matter which way the fñeld is pointing. Such an oscillator, +which can oscillate equally in three directions, would have 3k7 average energy +because there are 3 degrees of freedom in that oscillator. 5o we should use 3+k7" +because of the 3 degrees of freedom. +Now we have to do the integral. Let us suppose that the unknown spectral +distribution 7(œ) of the light is a smooth curve and does not vary very much +across the very narrow Írequency region where ø; is peaked (Eig. 41-3). Then the +only signifcant contribution comes when œ is very close to œọ, within an amount +gamma, which is very small. So therefore, although 7ƒ(œ) may be an unknown +and complicated function, the only place where 1t is Important is near œ = œ0, +--- Trang 738 --- +!I(0) +I{do)==—=———¬ mm ———— +(0o — 3 (go -E 3 +Fig. 41-3. The factors in the integrand (41.10). The peak is the +resonance curve 1/[(œ — øo)° + 47/4]. To a good approximation the +factor /(œ) can be replaced by (œ0). +and there we may replace the smooth curve by a fñat one—a “constant”——at +the same height. In other words, we simply take 7(œ) outside the integral sign +and call it J(óo). We may also take the rest of the constants out in front of the +integral, and what we have left is +> DIẾU +§r᧠1 (œ0 J ————sax —=3*kT. 41.11 +3 001( ) 0 (œ — œạ)2 + +2/4 ( ) +Now, the integral should go from 0Ö to oo, bu§ 0 is so far from œọ that the curve is +all ñnished by that time, so we go instead to minus oo—i% makes no diference and +1E is much easier to do the integral. The integral is an inverse tangent function +of the form ƒ dz/(z” + a?). If we look it up in a book we see that it is equal +to Z/a. So what it comes to for our case is 2r/y. Thherefore we get, with some +rearranging, +T(œ0) = ————. 41.12 +(60) 4n2rqu ( ) +Then we substitute the formula (41.6) for gamma (do not worry about writing +œọ; since it is true of any œọ, we may just call i9 œ¿) and the formula for 7(0) +then comes out +14) KT (41.13) +(U} = h * +And that gives us the distribution of light in a hot furnace. It is called the +blackbodw radiation. Black, because the hole in the furnace that we look at is +black when the temperatfure is zero. +--- Trang 739 --- +Inside a closed box at temperature 7, (41.13) is the distribution of energy +of the radiation, according to classical theory. Eirst, let us notice a remarkable +feature of that expression. The charge of the oscillator, the mass of the oscillator, +all properties specifc to the oscillator, caøncel out, because once we have reached +equilibrium with one oscillator, we must be at equilibrium with any other oscillator +of a diferent mass, or we will be in trouble. So this is an important kind of +check on the proposition that equilibrium does not depend on what we are In +cquilibrium with, but onh on the temjperature. NÑow let us draw a picture of the +T(() curve (Eig. 41-4). It tells us how much light we have at diferent frequencies. +I() 2Tọ +RADIO | IR | VISIBLE UV X-RAYS +Fig. 41-4. The blackbody intensity distribution at two temperatures, +according to classical physics (solid curves). The dashed curves show +the actual distribution. +The amount of intensity that there is in our box, per unit frequency range, +goes, as we see, as the square of the frequency, which means that if we have a +box at any temperature at all, and If we look at the x-rays that are coming out, +there will be a lot of theml +Of course we know this is false. When we open the furnace and take a look +at i9, we do not burn our eyes out from x-rays at all. It is completely false. +Purthermore, the #o‡øÏ energy in the box, the total of all this intensity summed +over all frequencies, would be the area under this infinite curve. 'Therefore, +something is fundamentally, powerfully, and absolutely wrong. +Thus was the classical theory øabsolutel ?ncaœpable of correctly describing +the distribution of light from a blackbody, just as it was incapable of correctly +describing the specifc heats of gases. Physicists went back and forth over this +derivation from many diferent points of view, and there is no escape. This 7s +the prediction of classical physics. Equation (41.13) is called Pagleigh s lau, and +1E is the prediction of classical physics, and is obviously absurd. +--- Trang 740 --- +41-3 Equipartition and the quantum oscillator +The dificulty above was another part of the continual problem of classical +physics, which started with the dificulty of the specifc heat of gases, and now has +been focused on the distribution of light in a blackbody. Now, of course, at the +time that theoreticians studied this thing, there were also many rneasurements +of the actual curve. And ït turned out that the correct curve looked like the +dashed curves in Fig. 4l-4. That is, the x-rays were not there. If we lower the +temperature, the whole curve goes down in proportion to 7”, according to the +classical theory, but the observed curve also cuts of sooner at a lower temperatfure. +Thus the low-frequency end of the curve is right, but the high-frequency end +is wrong. Why? When Bïr James Jeans was worrying about the specifc heats +Of gases, he noted that motions which have high frequency are “frozen out” as +the temperature goes too low. That is, if the temperature is too low, 1ƒ the +frequency is too high, the oscillators đo no# haue kT' of energy on the average. +Now recall how our derivation of (41.13) worked: It all depends on the energy of +an oscillator at thermal equilibrium. What the &7' of (41.5) was, and what the +same #7" in (41.13) is, is the mean energy of a harmonic oscillator oŸ Írequency œ +at temperature 7. Classically, this is k7, but experimentally, nol—not when the +temperature is too low or the oscillator frequency is too high. And so the reason +that the curve falls off is the same reason that the specifc heats of gases fail. +Tt is easier to s6udy the blackbody curve than it is the specilc heats of gases, +which are so complicated, therefore our attention is focused on determining the +true blackbody curve, because this curve is a curve which correctly tells us, a% +every Írequency, what the average energy of harmonic oscillators actually is as a +function of temperature. +Planck studied this curve. He first determined the answer empirically, by +ftting the observed curve with a nice function that fñtted very well. 'Phus he +had an empirical formula for the average energy of a harmonic oscillator as a +function of frequency. In other words, he had the r2øgh formula instead of k7, +and then by ñddling around he found a simple derivation for it which involved a +very peculiar assumption. That assumption was that the harmomic oscillator can +take up energies onlU ñœ a a time. The idea that they can have an energu œ‡ +gÌl is false. Of course, that was the beginning of the end of classical mechanics. +The very frst correctly determined quantum-mechanical formula will now be +derived. Suppose that the permitted energy levels of a harmonic oscillator were +equally spaced at ñư apart, so that the oscillator could take on only these diferent +--- Trang 741 --- +—.— E¿ = 4hu Pị = Aexp(—4hu/kt) +—”? E,—3ñu P›= Aexp(-3ñu/kt) +—M— E,—2nu P›— Aexp(—2ñu/kt) +—" — Eìi = ñœw = Aexp( —ñœ/kt) +—" m=0 P=A +Fig. 41-5. The energy levels of a harmonic oscillator are equally +spaced: En = như. +energies (Eig. 41-5). Planck made a somewhat more complicated argument than +the one that is beïng given here, because that was the very beginning of quantun +mnechanics and he had to prove some things. But we are goïing to take it as a fact +(which he demonstrated in this case) that the probability of occupying a level of +energy #2 is P(E) = ae—/*”, Tf we go along with that, we will obtain the right +result. +Suppose now that we have a lot of oscillators, and each is a vibrator of +frequency œạg. Some of these vibrators will be in the bottom quantum state, +some will be in the next one, and so forth. What we would like to know is the +average energy of all these oscillators. To ñnd out, let us calculate the total +energy of all the oscillators and divide by the number of oscillators. 'Phat will +be the average energy per oscillator in thermal equilibrium, and will also be the +energy that is in equilibrium with the blackbody radiation and that should go +in Eq. (41.13) in place of k7. Thus we let go be the number of oscillators that +are in the ground state (the lowest energy state); ¡ the number of oscillators +in the state Eq; ÑN¿ the number that are in state #2; and so on. According to +the hypothesis (which we have not proved) that in quantum mechanics the law +that replaced the probability e~P-E-/*T or e~K-E./*T ịn classical mechanics is that +the probability goes down as e—=^#/“T, where A is the excess energy, we shall +assume that the number Ñ) that are in the first state will be the number Nụ that +are in the ground state, times e—”“/*T, Similarly, N;, the number of oseillators +in the second state, is W¿ = Woe~?“/*T_ 'To simplify the algebra, let us call +e—hœ/RT — „. Then we simply have Ñ¡ = Nogz, Na = Ngz2,..., N„ = Ngz”. +The total energy of all the oscillators must fñrst be worked out. lan oscillator +is in the ground state, there is no energy. lf it is in the first state, the energy +is ñœ, and there are ÄÑ¡ of them. So Nqñú, or ñằNoz is how múch energy we get +--- Trang 742 --- +from those. 'Phose that are in the second state have 2œ, and there are Ms of +them, so Ms - 2hœ = 2hœNoz2 is how much energy we get, and so on. Then we +add it all together to get Eto¿ = Nohằœ(0 + # + 2z2 + 3x9 + ---). +And now, how many oscillators are there? Of course, j\ọ is the number that +are in the ground state, 1 in the first state, and so on, and we add them together: +Na = No(1 + # + z2 + z + ---). Thus the average energy is +() = Je — Nghe(D + + 2p + 3p go) (41.14) +Nhẹt NWMo(l+z+z2+z3+---) +Now the two sums which appear here we shall leave for the reader to play with +and have some fun with. When we are all ñnished summing and substituting +for ø in the sum, we should get—if we make no mistakes in the sum—— +(E) = chu/ET—T (41.15) +This, then, was the first quantum-mechanical formula ever known, or ever dis- +cussed, and it was the beautiful culmination of decades of puzzlement. Maxwell +knew that there was something wrong, and the problem was, what was r/gh#? +Here is the quantitative answer of what is right instead of k7'. 'Phis expression +should, of course, approach k7” as œ —> 0 or as 7 —> œ. See If you can prove that +1 does—learn how to do the mathematics. +Thịis 1s the famous cutof factor that Jeans was looking for, and if we use it +instead of k7'in (41.13), we obtain for the distribution of light in a black box +hư) dụ +T(œ) dự = x2c5(cho/Ef — 1)' (41.16) +We© see that for a large œ, even though we have œỞ in the numerator, there is +an e raised to a tremendous power in the denominator, so the curve comes down +again and does not “blow up”——we do not get ultraviolet light and x-rays where +we do not expect theml +One might complain that in our đerivation of (41.16) we used the quantum +theory for the energy levels of the harmonic oscillator, but the classical theory in +determining the cross section ơ;. But the quantum theory of light interacting +with a harmonic oscillator gives exactly the same result as that given by the +classical theory. 'That, in fact, is why we were justified in spending so much +--- Trang 743 --- +tỉme on our analysis of the index of refaction and the scattering of light, using +a model of atoms like little oscillators—the quantum formulas are substantially +the same. +Now let us return to the Johnson noise in a resistor. We have already remarked +that the theory of this noise power is really the same theory as that of the classical +blackbody distribution. In fact, rather amusingly, we have already said that if the +resistance in a circuit were not a real resisbtance, but were an antenna (an antenna +acts like a resistance because i radiates energy), a radiation resistance, it would +be easy for us to caleulate what the power would be. It would be just the power +that runs into the antenna from the light that is all around, and we would get the +same distribution, changed by only one or two factors. We can suppose that the +resistor is a generator with an unknown power spectrum (0). The spectrum is +determined by the fact that this same generator, connected to a resonant circuit +OŸ ơng [requenc, as in Eig. 41-2(b), generates in the inductance a voltage of the +magnitude given in Eq. (41.2). One is thus led to the same integral as in (41.10), +and the same method works to give Bq. (41.3). Eor low temperatures the k7 +in (41.3) must of course be replaced by (41.15). The two theories (blackbody +radiation and Johnson noise) are also closely related physically, for we may of +course connect a resonant circuit to an ønÉennae, so the resistance Ïl is a Dure +radialion resistance. Since (41.2) does not depend on the physical origin of +the resistance, we know the generator Œ for a real resistance and for radiation +resistance is the same. What is the origin of the generated power (4) if the +resistance # is only an ideal antenna in equilibrium with its environment at +temperature 7? It is the radiation ï(œ) in the space at temperature 7' which +impinges on the antenna and, as “received signals,” makes an efective generator. +Therefore one can deduce a direct relation of P() and 7(0), leading then from +(41.13) to (41.3). +AII the things we have been talking about——the so-called Johnson noise and +Planck's distribution, and the correct theory of the Brownian movement which +we are about to describe—are developments of the first decade or so of the 20th +century. Now with those points and that history in mind, we return to the +Brownlan movement. +41-4 The random walk +Let us consider how the position of a jiggling particle should change with tỉme, +for very long tỉmes compared with the time bebtween “kicks.” Consider a little +--- Trang 744 --- +S36 +Fig. 41-6. A random walk of 36 steps of length !. How far is Szø from +B? Ans: about 6l on the average. +Brownian movement particle which is jiggling about because it is bombarded +on all sides by irregularly jiggling water molecules. Query: After a given length +of time, how far away is it likely to be from where it began? 'This problem was +solved by Einstein and Smoluchowski. lÝ we imagine that we divide the time +imo little intervals, let us say a hundredth of a second or so, then after the first +hundredth of a second it moves here, and in the next hundredth it moves some +more, in the next hundredth of a second it moves somewhere else, and so on. Ïn +terms oŸ the rate of bombardment, a hundredth of a second is a very long tỉme. +The reader may easily verify that the number of collisions a single molecule of +water receives in a second is about 1012, so in a hundredth of a second it has +1012 eollisions, which is a lot! Therefore, after a hundredth of a second it is not +going to remember what happened before. In other words, the collisions are +all random, so that one “step” is not related to the previous “step.” lt is like +the famous drunken sailor problem: the sailor comes out of the bar and takes +a sequence of steps, but each step is chosen at an arbitrary angle, at random +(Fig. 41-6). The question is: After a long time, where is the sailor? Of course we +do not knowl It is impossible to say. What do we mean——he is just somewhere +more or less random. Well then, on the average, where is he? (n the œuerage, +hoa far ad from the bar has he gone? We have already answered this question, +because once we were discussing the superposition of light from a whole lot of +diferent sources at diferent phases, and that meant adding a lot of arrows at +diferent angles (Chapter 32). There we discovered that the mean square of the +distance from one end to the other of the chain of random steps, which was the +intensity of the light, is the sum of the intensities of the separate pieces. Ảnd so, +by the same kind of mathematics, we can prove immediately that if lầy is the +vector distance from the origin after / steps, the mean square of the distance +from the origin is proportional to the number of steps. That is, (Wy) = NL, +where Ù is the length of each step. Since the number of steps is proportional to +the time in our present problem, he rmeøn square đistance ?s proportional to the +--- Trang 745 --- +từne: +(R?) = at. (41.17) +This does not mean that the mmean distance is proportional to the time. Tf the +mean distance were proportional to the time it would mean that the drifting is +a% a nice uniform velocity. The sailor 7s making some relatively sensible headway, +but only such that his mean square distance is proportional to time. “That is the +characteristic oŸ a random walk. +'W©e may show very easily that in each successive step the square of the distance +increases, on the average, by L2. Eor if we write Ry = Rx„_¡ + L, we ñnd that +HẠy šs +RN : RN = HẠ = HÀ _¡+2RAN_ 1: L+ L3, +and averaging over many trials, we have (Hy) = (N$,_¡)+ LẺ, since (Rw_1 - L) = +0. Thus, by induction, +(RA) = NH3. (41.18) +NÑow we would like to calculate the coefficient œ in Eq. (41.17), and to do so +we must add a feature. W© are going to suppose that iŸ we were to put a force on +this particle (having nothing to do with the Brownian movement—we are taking +a side issue for the moment), then i9 would react in the following way against +the force. First, there would be inertia. Let m be the coefficient of inertia, the +efective mass of the object (not necessarily the same as the real mass of the +real particle, because the water has to move around the particle if we pull on it). +Thus if we talk about motion in one direction, there is a term like m(d2z/đt?) on +one side. And next, we want also to assume that if we kept a steady pull on the +object, there would be a drag on i§ from the Ñuid, proportional to its velocity. +Besides the inertia of the fuid, there is a resistance to fow due to the viscosity +and the complexity of the fuid. It is absolutely essential that there 0e some +irreversible losses, something like resistance, in order that there be Ñuctuations. +There is no way to produce the k7' unless there are also losses. The source of the +ñuctuations is very closely related to these losses. What the mechanism of this +drag is, we will discuss soon——we shall talk about forces that are proportional to +the velocity and where they come from. But let us suppose for now that there +1s such a resistance. Then the formula for the motion under an external force, +when we are pulling on it in a normal manner, is +m TT = Rau (41.19) +--- Trang 746 --- +The quantity can be determined directly from experiment. For example, we +can watch the drop fall under gravity. Then we know that the fÍorce is mø, and +is ng divided by the speed of fall the drop ultimately acquires. OÔr we could put +the drop in a centrifuge and see how fast it sediments. Or if it is charged, we can +put an electric feld on it. So # is a measurable thing, not an artifcial thing, and +1E is known for many types of colloidal particles, etc. +Now let us use the same formula in the case where the force is not external, +but is equal to the irregular forces of the Brownian movement. We shall then +try to determine the mean square distance that the object goes. Instead of +taking the distances in three dimensions, let us take just one dimension, and +fñnd the mean of #Ÿ, just to prepare ourselves. (Obviously the mean of #2 is +the same as the mean of # is the same as the mean of zŸ, and therefore the +mean square of the distance is just 3 times what we are going to calculate.) +The z-component of the irregular forces is, of course, jusÈ as irregular as any +other component. What is the rate of change of #”? It is đ(z2)/dt = 2z(dz/dĐ), +so what we have to ñnd is the average of the position tỉimes the velocity. We +shall show that this is a constant, and that therefore the mean square radius +will increase proportionally to the time, and at what rate. Now 1Ý we multiply +Eq. (41.19) by z, maz(d2z/đt?) + na(d+/đt) = zF}„. We want the tỉme average +oŸ z(dz/đ#), so let us take the average of the whole equation, and study the three +terms. Now what about z times the force? If the particle happens to have gone +a certain distance z, then, since the irregular force is comjpletel irregular and +does not know where the particle started from, the next impulse can be in any +direction relative to ø. lÝ ø is positive, there is no reason why the average force +should also be in that direction. It is just as likely to be one way as the other. +The bombardment forces are not driving it in a deflnite direction. So the average +value of z tỉmes #' is zero. On the other hand, for the term mz(d2z/đt?) we will +have to be a little fancy, and write this as +d2 d[z(dz/đÐ)] dz\Ÿ +TA nà SN HT _m() : +Thus we put in these two terms and take the average of both. 5o let us see how +much the first term should be. NÑow z times the velocity has a mean that does not +change with time, because when it gets to some position it has no remembrance of +where it was before, so things are no longer changing with time. So this quantity, +on the average, is zero. We have left the quantity m2, and that is the only thing +--- Trang 747 --- +we know: ?nw02/2 has a mean value 3kT. Therefore we fñnd that +d2z da: +implies +2 hủ, 2 +— �� — 0 +0m63) + ST (38) =0, +d(? kT +dự) AT. (41.0) +Therefore the object has a mean square distance (2), at the end of a certain +amount of , equal to +(R?) = 6kT—. (41.21) +And so we can actually determine ho ƒar the particles gol We first must +determine how they react to a steady force, how fast they drift under a known +force (to fnd #), and then we can determine how far they go in their random +motions. This equation was of considerable importance historically, because +it was one of the first ways by which the constant k was determined. After +all, we can measure /, the time, how far the particles go, and we can take an +average. 'Phe reason that the determination oŸ k was important is that in the +law PV = ]tT' for a mole, we know that , which can also be measured, is equal +to the number of atoms in a mole times &. A mole was originally defned as +so and so many ørømns of oxygen-16 (now carbon is used), so the number of +a‡øms in a mole was not known, originally. It is, of course, a very interesting +and important problem. How bịg are atoms? How many are there? So one of +the earliest determinations of the number of atoms was by the determination +of how far a dirty little particle would move if we watched it patiently under a +microscope for a certain length of time. And thus Boltzmann”s constant k and +the Avogadro number /ọ were determined because #‡ had already been measured. +--- Trang 748 --- +Applic(rfiores œŸ Nireofic Thoorgg +42-1 Evaporation +In this chapter we shall discuss some further applications of kinetic theory. +In the previous chapter we emphasized one particular aspect of kinetic theory, +namely, that the average kinetic energy in any degree of freedom of a molecule +or other object is 3kT. The central feature of what we shall now discuss, on +the other hand, is the fact that the probability of fnding a particle in diferent +places, per unit volume, varies as e—P9tential enersy/kT: wo sha]l make a number of +applications of this. +'The phenomena which we want to study are relatively complicated: a liquid +evaporating, or electrons in a metal coming out of the surface, or a chemical +reaction in which there are a large number o£atoms involved. In such cases it is no +longer possible to make rom the kinetic theory any simple and correct statements, +because the situation is too complicated. 'Therefore, this chapter, except where +otherwise emphasized, is quite inexact. The idea to be emphasized is only that +we can understand, from the kinetic theory, znøre or Íess how things ought to +behave. By using thermodynamic arguments, or some empirical measurements +of certain critical quantities, we can get a more accurate representation of the +phenomena. +However, it is very useful to know even only more or less why something +behaves as it does, so that when the situation is a new one, or one that we have +not yet started to analyze, we can say, more or less, what ought to happen. So +this discussion is highly inaccurate but essentially right—right in idea, but a +little bit simplifed, let us say, in the specifc details. +'The first example that we shall consider is the evaporation of a liquid. Suppose +we have a box with a large volume, partially flled with liquid in equilibrium and +with the vapor at a certain temperature. We shall suppose that the molecules of +the vapor are relatively far apart, and that inside the liquid, the molecules are +--- Trang 749 --- +packed close together. The problem is to ñnd out how many molecules there are +in the vapor phase, compared with the number there are in the liquid. How dense +1s the vapor at a given temperature, and how does it depend on the temperature? +Let us say that ?ø+ equals the number of molecules per unit volume in the +vapor. That number, of course, varies with the temperature. lf we add heat, +we get more evaporation. Now let another quantity, 1/V4, equal the number oŸ +atoms per unit volume ïn the liquid: We suppose that each molecule in the liquid +occupies a certain volume, so that if there are more molecules of liquid, then all +together they occupy a bigger volume. 'Phus if W„ is the volume occupied by one +mmolecule, the number oŸ molecules in a unit volume is a unit volume divided by +the volume of each molecule. Furthermore, we suppose that there is a force of +attraction between the molecules to hold them together in the liquid. Otherwise +we cannot understand why ¡it condenses. Thus suppose that there is such a force +and that there is an energy of binding of the molecules in the liquid which is lost +when they go into the vapor. 'That is, we are going %o suppose that, in order +to take a single molecule out of the liquid into the vapor, a certain amount of +work W has to be done. There is a certain diference, WỨ, in the energy of a +molecule in the liquid from what it would have ïf it were in the vapor, because +we have to pull it away from the other molecules which attraet ït. +Now we use the general principle that the number of atoms per unit volume in +two diferent regions Ìs 2 /m = e~(E2~F1)/RT, So the number ø per unit volume +in the vapor, divided by the number 1/M+ per unit volume in the liquid, is equal +nVy =e VU (42.1) +because that is the general rule. It is like the atmosphere in equilibrium under +gravity, where the gas at the bottom is denser than that at the top because of +the work møh needed to liẾt the gas molecules to the height h. In the liquid, the +mmolecules are denser than in the vapor because we have to pull them out through +the energy “hill” W, and the ratio of the densities is e—W/*T, +'This is what we wanted to deduce—that the vapor density varies as e to the +minus some energy or other over k7'. The factors in front are not really interesting +to us, because in most cases the vapor density is very much lower than the liquid +density. In those circumstances, where we are not near the critical point where +they are almost the same, but where the vapor density is much lower than the +liquid density, then the fact that nø is very much less than 1/Vạ is occasioned +by the facb that W/ is very much greater than k7. So formulas such as (42.1) +--- Trang 750 --- +are interesting only when Wƒ is very much bigger than &?Ƒ, because in those +circumstances, since we are raising e to minus a tremendous amount, iŸ we change +T'a little bít, that tremendous power changes a bit, and the change produced in +the exponential factor is very much more important than any change that might +occur in the factors out in front. Why should there be any changes in such factors +as Vạ? Because ours was an approximate analysis. After all, there is not really a +defnite volume for each molecule; as we change the temperature, the volume W4 +does not stay constant—the liquid expands. 'Phere are other little features like +that, and so the actual situation is more complicated. There are slowly varying +temperature-dependent factors all over the place. In fact, we might say that WZ +1tself varies sliphtly with temperature, because at a higher temperature, at a +diferent molecular volume, there would be diferent average attractions, and so +on. 5o, while we might think that iŸ we have a formula in which everything varies +in an unknown way with temperature then we have no formula at all, IÝ we realize +that the exponent W//kT7 is, in general, very large, we see that in the curve oŸ the +vapor density as a function of temperature most of the variation is occasioned by +the exponential factor, and iŸ we take W as a constant and the coefficient 1/Vạ +as nearly constant, it is a good approximation for short intervals along the curve. +Most of the variation, in other words, is of the general nature e~ W/*T, +lt turns out that there are many, many phenomena in nature which are +characterized by having to borrow an energy from somewhere, and in which +the central feature of the temperature variation is e to the minus the energy +over kĩ. This is a useful fact only when the energy is large compared with kĩ, +so that most of the variation is contained in the variation of the k7' and not in +the constant and in other factors. +Now let us consider another way of obtaining a somewhat similar result for +the evaporation, but looking at it in more detail. To arrive at (42.1), we simply +applied a rule which is valid at equilibrium, but in order to understand things +better, there is no harm in trying to look at the details of what is going on. We +may also describe what is going on in the following way: the molecules that +are in the vapor continually bombard the surface of the liquid; when they hit +it, they may bounce of or they may get stuck. 'There is an unknown factor for +that —maybe 50-50, maybe 10 to 90——we do not know. Let us say they always +get stuck—we can analyze it over again later on the assumption that they do +not always get stuck. Then at a given moment there will be a certain number +of atoms which are condensing onto the surface of the liquid. 'Phe number of +condensing molecules, the number that arrive on a unit area per unit time, is the +--- Trang 751 --- +number ø per unit volume times the velocity 0. 'This velocity of the molecules +is relabed to the temperature, because we know that šznø is equal to $7 on +the average. So 0 is some kind oŸ a mean velocity. Of course we should integrate +over the angles and get some kind of an average, but it is roughly proportional +to the root-mean-square velocity, within some factor. Thus +NW.=nu (42.2) +is the rate at which the molecules arrive per unit area and are condensing. +At the same time, however, the atoms in the liquid are jiggling about, and +from time to time one of them gets kicked out. Now we have to estimate how +fast they get kicked out. 'The idea will be that at equilibrium the number that +are kicked out per second and the number that arrive per second are equal. +How many get kicked out? In order to get kicked out, a particular molecule has +to have acquired by accident an excess energy over i%s neighbors—a considerable +excess energy, because it is attracted very strongly by the other molecules in +the liquid. Ordinarily it does not leave because 1£ 1s so strongly attracted, but +in the collisions sometimes one of them gets an extra energy by accident. And +the chance that it gets the extra energy W which it needs in our case is very +small if W » k7. In fact, e~W/FT ịs the chanee that an atom has picked up +more than this much energy. That is the general prineiple in kinetic theory: in +order to borrow an excess energy Wƒ over the average, the odds are e to the +minus the energy that we have to borrow, over k7. Now suppose that some +mnolecules have borrowed this energy. We now have to estimate how many leave +the surface per second. Of course, just because a molecule has the necessary +energy does not mean that it will actually evaporate, since it may be buried too +deeply inside the liquid or, even I1 it is near the surface, it may be travelling +in the wrong direction. 'Phe number that are going to leave a unit area per +second 1s going to be something like this: the number of atoms there are near +the surface, per unit area, divided by the time it takes one to escape, multiplied +by the probability e~W/*” that they are ready to escape in the sense that they +have enough energy. +W© shall suppose that each molecule at the surface of the liquid oceupies +a certain cross-sectional area A. Then the number of molecules per unit area +of liquid surface will be 1/A. And now, how long does i9 take a molecule to +escape? If the molecules have a certain average speed œ, and have to move, say, +one molecular diameter D, the thickness of the frst layer, then the time it takes +--- Trang 752 --- +to get across that thickness is the time needed to escape, if the molecule has +enough energy. The time will be 2/0. Thus the number evaporating should be +approximately +ÁN, =(1/A)(0/D)c -WT, (42.3) +Now the area of each atom times the thickness of the layer is approximately +the same as the volume W+ occupied by a single atom. And so, in order to get +equilibrium, we must have /¿ = Ñ¿, or +nù = (0/V„)e W/ET, (42.4) +W©e may cancel the 0s, since they are equal; even though one is the velocity of a +molecule in the vapor and the other is the velocity of an evaporating molecule, +these are the same, because we know their mean kinetic energy (in one direction) +1S skT. But one may object, “NÑol Nol These are the especially fast-moving ones; +these are the ones that have picked up excess energy.” Not really, because the +moment they start to pull away from the liquid, they have to iose that excess +energy against the potential energy. So, as they come to the surface they are +slowed down to the velocity øl It is the same as it was in our discussion of +the distribution of molecular velocities in the atmosphere—at the bottom, the +molecules had a certain distribution of energy. 'Phe ones that arrive at the top +have the sœne distribution of energy, because the slow ones did not arrive at +all, and the fast ones were slowed down. “The molecules that are evaporating +have the same distribution of energy as the ones inside—a rather remarkable +fact. Anyway, it is useless to try to argue so closely about our formula because of +other inaccuracies, such as the probability of bouncing back rather than entering +the liquid, and so on. Thus we have a rough idea of the rate of evaporation and +condensation, and we see, of course, that the vapor density mø varies in the same +way as before, but now we have understood it in some detail rather than just as +an arbitrary formula. +'This deeper understanding permits us to analyze some things. For example, +suppose that we were to pump away the vapor at such a great rate that we +removed the vapor as fast as it formed (ïf we had very good pumps and the liquid +was evaporating very slowly), how fast would evaporation occur iŸ we maintained +a liquid temperature 7? Suppose that we have already experimentally measured +the equilibrium vapor density, so that we know, at the given temperature, how +many molecules per unit volume are in equilibrium with the lquid. Now we +would like to know ho ƒast it will evaporate. Even though we have used only a +--- Trang 753 --- +rough analysis so far as the evaporation part oŸ i is concerned, the number of +vapor molecules ørr?u¿ng was not done so badly, aside from the unknown factor oŸ +refection coefficient. 5o therefore we may use the fact that the number that are +leaving, at equilibrium, is the same as the number that arrive. True, the vapor +1s being swept away and so the molecules are only coming out, but ïf the vapor +were left alone, it would attain the equilibrium density at which the number that +come back would equal the number that are evaporating. 'Pherefore, we can +easily see that the number that are coming of the surface per second is equal to +the unknown refection coeficient # times the number that would come down +to the surface per second were the vapor still there, because that is how many +would balance the evaporation at equilibrium: +N,=nuR = (0R/V,)ce WIST, (42.5) +Of course, the number of molecules that hít the liquid from the vapor is easy to +calculate, since we do not need to know as much about the forces as we do when +we are worrying about how they get to escape through the liquid surface; iW is +much easier to make the argument the other way. +42-2 Thermionic emission +W©e may give another example of a very practical situation that is similar to +the evaporation of a liquid—so similar that it is not worth making a separate +analysis. It is essentially the same problem. In a radio tube there is a source of +electrons, namely a heated tungsten filament, and a positively charged plate to +attract the electrons. Any electron that escapes from the surface of the tungsten +1s Immediately swept away to the plate. 'That ¡is our ideal “pump,” which 1s +“pumping” the electrons away all the time. Now the question is: How many +electrons per second can we get out of a piece of tungsten, and how does that +number vary with temperature? The answer to that problem is the same as (42.5), +because it turns out that in a piece of metal, electrons are attracted to the lons, +or to atoms, of the metal. They are attracted, if we may say it crudely, to the +metal. In order to get an electron out of a piece of metal, it takes a certain +amountf of energy or work to pull it out. Thịis work varies with the diferent kinds +of metal. In fact, it varies even with the character of the surface of a given kind +of metal, but the total work may be a few electron volts, which, incidentally, is +typical of the energy involved in chemical reactions. We can remember the latter +--- Trang 754 --- +fact by remembering that the voltage in a chemical cell like a fashlight battery, +which is produced by chemical reactions, is about one volt. +How can we fnd out how many electrons come out per second? It would be +quite dificult to analyze the efects on the electrons going out; 1t is easier %O +analyze the situation the other way. So, we could start out by imagining that +we did not draw the electrons away, and that the electrons were like a gas, and +could come back to the metal. Thhen there would be a certain density of electrons +at equilibrium which would, of course, be given by exactly the same formula +as (42.1), where Wạ is the volume per electron in the metal, roughly, and VỬ is +cqual to gsó, where ø is the so-called t0uork ƒunection, or the voltage needed to pull +an electron off the surface. Thịs would tell us how many electrons would have to +be in the surrounding space and striking the metal in order to balance the ones +that are coming out. And thus it is easy to calculate how many are coming out +1Ý we sweep away all of them, because the number that are coming out is exactly +equal to the number that would be going in with the above density of electron +“vapor.” In other words, the answer is that the current of electricity that comes +in per unit area is equal to the charge on each times the number that arrive per +second per unit area, which is the number per unit volume times the velocity, as +we have seen many times: +l = qenu = (qe0/V„)e” 9/1, (42.6) +Now one electron volt eorresponds to k7' at a temperature of 11,600 degrees. The +filament of the tube may be operating at a temperature of, say, 1100 degrees, so +the exponential facbor is something like e~†Đ; when we change the temperature a +little bit, the exponential factor changes a lot. Thus, again, the central feature +of the formula is the e~%2/'T, As a matter of fact, the factor in front is quite +wrong—it turns out that the behavior of electrons in a metal is not correctly +described by the classical theory, but by quantum mechanics, but this only +changes the factor in front a little. Actually, no one has ever been able to get +the thing straightened out very well, even though many people have used the +high-class quantum-mechanical theory for their calculations. The big problem is, +does W/ change slightly with temperature? lf it does, one cannot distinguish a +W changing slowly with temperature from a diferent coefficient in front. That +1s, If W changed linearly, say, with temperature, so that W = Wo + œkT', then +we would have +c~W/ET — Q=(Wo+oET)/ET — ¿~«¿—Wb/KT. +--- Trang 755 --- +'Thus a linearly temperature-dependent W is equivalent to a shifted “constant.” +Tt is really quite difficult and usually fruitless to try to obtain the coeflcient in +the tont accurately. +42-3 Thermail ionization +Now we go on to another example of the same idea; always the same idea. +This has to do with ionization. Suppose that in a gas we have a whole lot of +atoms which are in the neutral state, say, but the gas is hot and the atoms can +become ionized. We would like to know how many ions there are in a given +circumstance if we have a certain density of atoms per unit volume at a certain +temperature. Again we consider a box in which there are / atoms which can +hold electrons. (If an electron has come of an atom, it is called an ?øøw, and 1Í +the atom is neutral, we simply call it an atom.) Then suppose that, at any given +mmoment, the number of neutral atoms is øœ„, the number of ions is n¿, and the +number of electrons is n¿, all per unit volume. “he problem is: What is the +relationship of these three numbers? +In the first place, we have two conditions or constraints on the numbers. Eor +instance, as we vary different conditions, like the temperature and so on, ?„ -Ƒ ¿ +would remain constant, because this would be simply the number Ý of atomie +nuclei that are in the box. If we keep the number of nuclei per unit volume +fñxed, and change, say, the temperature, then as the ionization proceeded some +atoms would turn to ions, but the total number of atoms plus ions would be +unchanged. "That is, n„ + n¿ = N. Another condition is that if the entire gas +1s to be electrically neutral (and if we neglect double or triple ionization), that +means that the number of ions is equal to the number of electrons at all times, +Or n¿ — ẹ. These are subsidiary equations that simply express the conservation +of charge and the conservation of atoms. +'These equations are true, and we ultimately will use them when we consider a +real problem. But we want to obtain another relationship between the quantities. +We can do this as follows. We again use the idea that it takes a certain amount +oŸ energy to lift the electron out of the atom, which we call the ion2zation energu, +and we would write it as W/, in order to make all of the formulas look the same. +So we let W/ equal the energy needed to pull an electron out of an atom and +make an ion. NÑow we again say that the number of free electrons per unit volume +in the “vapor” ¡is equal to the number of bound electrons per unit volume In +the atoms, times e to the minus the energy diference between being bound and +--- Trang 756 --- +being free, over k7'. That is the basic equation again. How can we write it? The +number of free electrons per unit volume would, of course, be ›;, because that is +the definition of n„. NÑow what about the number of electrons per unit volume +that are bound to atoms? The total number of places that we could put the +electrons is apparentÌy ø=„ + nm¿, and we will suppose that when they are bound +each one is bound within a certain volume W. So the total amount of volume +which is available to electrons which would be bound is (nạ + n;¿)M4, so we might +want to write our formula as +— —— Hạ — —W/RT +n„ = (nạ+m;)1 € . +The formula is wrong, however, in one essential feature, which is the following: +when an electron is already on an atom, another electron cannot come to that +volume anymorel In other words, all the volumes of all the possible sites are not +really available for the one electron which is trying to make up its mind whether +or not to be in the vapor or in the condensed position, because in this problem +there is an extra feature that when one electron is where another electron is, it is +not allowed to go—It 1s repelled. Eor that reason, it comes out that we should +count only that part of the volume which is available for an electron to sit on or +not. That is, those which are already occupied do not count in the total available +volume, but the only volume which is allowed is that of the 7øns, where there are +vacant places for the electron to go. Then, in those cireumstances, we fnd that a +nicer way to write our formula is +ngụ _ 1 —W/kT +n Ứ € . (42.7) +This formula ¡is called the Saha ?omization cquation. Now let us see 1Ÿ we can +understand qualitatively why a formula like this is right, by arguing about the +kinetic things that are happening. +First, every once in a while an electron comes to an ion and they combine +to make an atom. And also, every once in a while, an atom gets into a collision +and breaks up into an ion and an electron. Now those two rates must be equal. +How fast do electrons and ions fnd each other? 'The rate is certainly increased if +the number of electrons per unit volume is increased. It is also increased 1f the +number o ions per unit volume is increased. 'That is, the total rate at which +recombination is occurring is certainly proportional to the number of electrons +times the number of ions. Now the total rate at which ionization is occurring +--- Trang 757 --- +due to collisions must be dependent linearly on how many atoms there are tO +ionize. And so the rates will balance when there is some relationship bebween +the product n„1w and the number of atoms, øœ„. The fact that this relationship +happens to be given by this particular formula, where Wƒ is the ionization energy, +1s Of course a little bit more information, bu we can easily understand that +the formula would necessarily involve the concentrations of the electrons, ions, +and atoms in the combination n¿n¿/n„ to produce a constant independent of +the nˆ°s, and dependent only on temperature, the atomic cross sections, and other +constant factOTs. +We may also note that, since the equation involves the numbers per unt +0olưmne, 1Ÿ we were to do bwo experiments with a given total number Ý of +atoms plus ions, that is, a certain fñxed number of nuclei, but using boxes with +diferent volumes, the m s would all be smaller in the larger box. But since the +ratio nem¿/n„ sbays the same, the #ofal mumber oŸ electrons and ions must be +greater in the larger box. To see this, suppose that there are / nuclei inside a box +of volume V, and that a fraction ƒ of them are ionized. Then n¿ = ƒN/V = 1m, +and n„ = (1— ƒƑ)N/V. Then our equation becomes +2 —W/kT +j N_ c7. (42.8) +1—-ƑV l +In other words, if we take a smaller and smaller density of atoms, or make the +volume of the container bigger and bigger, the fraction ƒ of electrons and ions +must increase. That ionization, just from “expansion” as the density goes down, +is the reason why we believe that at very low densities, such as in the cold +space bebween the stars, there may be ions present, even though we might not +understand it from the point of view of the available energy. Althouph it takes +many, many k1? of energy to make them, there are ions present. +'Why can there be ions present when there is so much space around, while if +we increase the density, the ions tend to disappear? Ansuer: Consider an atom. +very onece in a while, light, or another atom, or an ion, or whatever it is that +maintains thermal equilibrium, strikes it. Very rarely, because it takes such a +terrilic amount of excess energy, an electron comes of and an ion is left. Now +that electron, if the space is enormous, wanders and wanders and does not come +near anything for years, perhaps. But once in a very great while, it does come +back to an ion and they combine to make an atom. 5o the rate at which electrons +are coming out from the atoms is very slow. But if the volume is enormous, an +--- Trang 758 --- +electron which has escaped takes so long to fñnd another ion to recombine with +that its probability of recombination is very, very small; thus, in spite oŸ the large +excess energy needed, there may be a reasonable number of electrons. +42-4 Chemical kinetics +The same situation that we have just called “ionization” is also found in +a chemical reaction. For instance, if two objects 4 and Ö combine into a +compound 4, then if we think about it for a while we see that Á ¡is what we +have called an atom, ?Ö is what we call an electron, and A is what we call an ion. +With these substitutions the equations of equilibrium are exactly the same in +form: hạng +_“=“ ={ VIKT, (42.9) +'This formula, of course, 1s not exact, since the “constant” e depends on how mụch +volume is allowed for the 4 and Ö to combine, and so on, but by thermodynamic +arguments one can identify what the meaning of the W7 in the exponential facbor +1s, and it turns out that it is very close to the energy needed in the reaction. +Suppose that we tried to understand this formula as a result of collisions, +much in the way that we understood the evaporation formula, by arguing about +how many electrons came of and how many of them came back per unit time. +Suppose that 4 and Ö combine in a collision every once in a while to form a +compound 4Ø. And suppose that the compound 4Ö is a complicated molecule +which jiggles around and is hit by other molecules, and from tỉme to time it gets +enough energy to explode and break up again into 4 and Ö. +Now it actually turns out, in chemical reactions, that if the atoms come +together with too small an energy, even though energy may be released in the +reaction A + — AB, the fact that A and may touch each other does not +necessarily make the reaction start€. I% usually is required that the collision be +A+B W +Fig. 42-1. The energy relationship for the reaction A + 8 AB. +--- Trang 759 --- +rather hard, in fact, to get the reaction to go at all—a “soft” collision between 4 +and may not do it, even though energy may be released in the process. So +let us suppose that it is very common in chemical reactions that, in order for A +and to form 4Ö, they cannot just hit each other, but they have to hit each +other th sufficient energụ. This energy is called the acfiation energu—the +energy needed to “activate” the reaction. Call 4 the activation energy, the +excess energy needed in a collision in order that the reaction may really occur. +Then the rate f; at which 4 and Ö produce 4 would involve the number of +atoms of Á times the number of atoms of Ö, times the rate at which a single +atom would strike a certain cross section ơap, times a facbor c—4/ÈT wbich is +the probability that they have enough energy: +lì = nAnpuơaApe" 2 AT, (42.10) +Now we have to fnd the opposite rate, #„. There is a certain chance that A4? +will ñy apart. In order to ñy apart, it not only must have the energy W which it +needs in order to get apart at all but, just as it was hard for A and Ö to come +together, so there is a kind of hill that A and Ö have to climb over to get apart +again; they must have not only enough energy just to get ready to pull apart, +but a certain excess. It is like climbing a hill to get into a deep valley; they have +to climb the hill coming in and they have to climb out oÊ the valley and then over +the hill coming back (Eig. 42-1). Thus the rate at which 4 goes to A and +will be proportional to the number øag that are present, tìmes c~(W+4")/kT,; +R„ = cnapge (W+A*)/ET, (42.11) +The đ will involve the volume of atoms and the rate of collisions, which we +can work out, as we did the case of evaporation, with areas and times and +thicknesses; but we shall not do this. 'Phe main feature of interest to us is that +when these two rates are equal, the ratio of them is equal to unity. 'This tells +us that nAng/nAp = cc—W/*” as before, where e involves the cross sections, +velocities, and other factors independent of the nˆs. +The interesting thing is that the rate of the reaction also varies as e—©0nst/#?: +although the constant is not the same as that which governs the concentrations; +the activation energy 4Ý is quite diferent from the energy W/. W gouerns the +proportions oƑ A, B, and AB that tue haue ?n cquilibrium, but ïŸ we want to know +how fast A + goes to A4, that is not a question of equilibrium, and here a +--- Trang 760 --- +diferent energy, the actiuation energụ, governs the rate of reaction through an +exponential factor. +Furthermore, 4Ý is not a fundamental constant like W. Suppose that at the +surface of the wall—or at some other place—4 and ?Ö could temporarily stick +there in such a way that they could combine more easily. In other words, we might +find a “tunnel” through the hill, or perhaps a lower hill. By the conservation +of energy, when we are all fñnished we have still made 4Ø out of A and Ö, +so the energy diference W/ will be quite independent of the way the reaction +occurred, but the activation energy 4” will depend 0erw much on the way the +reaction occurs. This is why the rates of chemical reactions are very sensitive to +outside conditions. We can change the rate by putting in a surface of a diferent +kind, we can put it ím a “diferent barrel” and it will go at a diferent rate, 1Í +it depends on the nature of the surface. Or if we put in a third kind of object +it may change the rate very much; some things produce enormous changes In +rate simply by changing the 4Ÿ a little bit—they are called cafalsts. A reaction +might practically not occur at all because 4Ÿ is too big at the given temperature, +but when we put in this special stuff, the catalyst, then the reaction øoes very +fast indeed, because 4Ÿ is reduced. +Incidentally, there is some trouble with such a reaction, A plus Ø, making +AĐ, because we cannot conserve both energy and momentum when we try to put +two objects together to make one that is more stable. 'Pherefore, we need at least +a third objJect Œ, so the actual reaction is mụch more complicated. The forward +rate would involve the product nam=pzwc, and it might seem that our formula is +going wrong, but nol When we look at the rate at which 4 goes the other way, +we fnd that it also needs to collide with Œ, so there is an nApgmœ 1n the reverse +rate; the øœs cancel out in the formula for the equilibrium concentrations. 'Phe +law of equilibrium, (42.9), which we first wrote down is absolutely guaranteed to +be true, no matter what the mechanism of the reaction may bel +42-5 binstein?s laws of radiation +W©e now turn to an interesting analogous situation having to do with the +blackbody radiation law. In the last chapter we worked out the distribution law +for the radiation in a cavity the way Planck did, considering the radiation from +an oscillator. The oscillator had to have a certain mean energy, and since it was +oscillating, ¡it would radiate and would keep pumping radiation into the cavity +until it piled up enough radiation to balance the absorption and emission. In +--- Trang 761 --- +that way we found that the intensity of radiation at frequency œ was given by +the formula +hư) dụ +T(œ) dự = x2c5(cho/Ef.— 1) (42.12) +This result involved the assumption that the oscillator which was generating +the radiation had defñnite, equally spaced energy levels. We did not say that +light had to be a photon or anything like that. There was no discussion about +how, when an atom goes from one level to another, the energy must come out +in one unit of energy, ñœ, in the form of light. Planek”s original idea was that +the matter was quantized but not the light: material oscillators cannot take up +Just any energy, but have to take it in lumps. EFurthermore, the trouble with the +derivation is that it was partially classical. We calculated the rate of radiation +trom an oscillator according to classical physics; then we turned around and +said, “No, this oscillator has a lot of energy levels.” So gradually, in order to +find the right result, the completely quantum-mechanical result, there was a +slow development which culminated in the quantum mechanics of 1927. But in +the meantime, there was an attempt by Einstein to convert Planeck's viewpoint +that only oscillators of matter were quantized, to the idea that light was really +photons and could be considered in a certain way as particles with energy ñơ. +Purthermore, Bohr had pointed out that an system of atoms has energy levels, +but they are not necessarily equally spaced like Planck's oscillator. And so ït +became necessary to rederive or at least rediscuss the radiation law from a more +completely quantum-mechanical viewpoint. +Binstein assumed that Planck's ñnal formula was right, and he used that +formula to obtain some new information, previously unknown, about the inter- +action of radiation with matter. His discussion went as follows: Consider any +two of the many energy levels of an atom, say the ?mth level and the ø%th level +(Fig. 42-2). NÑow Einstein proposed that when such an atom has light of the right +frequency shining on it, it can absorb that photon of light and make a transition +from state mø to state rm, and that the probability that this occurs per second +Spontaneous emission +Absorption E..— Induced emission +Fig. 42-2. Transitions between two energy levels of an atom. +--- Trang 762 --- +depends upon the two levels, of course, but 1s proportional to hot tntense the +light ¡s that is shining on it. Let us call the proportionality constant „„, merely +to remind us that this is not a universal constant of nature, but depends on the +particular pair of levels: some levels are easy to excite; some levels are hard to +excite. Now what is the formula goiïng to be for the rate oŸ emission from ?m to m0? +Einstein proposed that this must have two parts to it. Eirst, even if there were +no lipht present, there would be some chance that an atom in an excited state +would fall to a lower state, emitting a photon; this we call spon‡aneous emission. +Tt is analogous to the idea that an oscillator with a certain amount of energy, +even in classical physics, does not keep that energy, but loses it by radiation. +Thus the analog of spontaneous radiation of a classical system is that if the atom +is in an excited state there is a certain probability A„„, which depends on the +levels again, for it to go down from rn to øœ, and this probability is independent +of whether light is shining on the atom or not. But then Einstein went further, +and by comparison with the classical theory and by other arguments, concluded +that emission was also inÑuenced by the presence of light—that when light of +the right frequeney is shining on an atom, iÿ has an increased rate of emitting a +photon that is proportional to the intensity of the light, with a proportionality +constant „„. Later, if we deduce that this coefficient is zero, then we will have +found that Einstein was wrong. Of course we will ñnd he was right. +'Thus Hinstein assumed that there are three kinds of processes: an absorption +proportional to the intensity of light, an emission proportional to the inten- +sity of light, called #wduced em4ssion or sometimes stimulated emission, and a +spontaneous emission independent of light. +Now suppose that we have, in equilibrium at temperature 7”, a certain number +of atoms „ ¡in the state m and another number N„„ in the state rm. Then the +total number of atoms that are goïng from ?ø to ?m is the number that are in the +state times the rate per second that, iÝ one is in , iÿ goes up to ?m. So we have +a formula for the number that are going from ø to ?w per second: +Tu ym = NaBu„1(0). (42.13) +The number that will go from ?n to ? is expressed in the same manner, as the +number NMự„ that are in m, times the chance per second that each one goes down +toø. This time our expression 1s +Tìm sa —= Nu[Ama Ð BaT(6)]}. (42.14) +--- Trang 763 --- +Now we shall suppose that in thermail equilibrium the number of atoms goïng up +must equal the number coming down. That is one way, at least, in which the +number will be sure to stay constant in each level. So we take these bwo rates to +be equal at equilibrium. But we have one other piece of information: we know how +large /„„, is compared with N„——the ratio of those Ewo 1s e~(Em~n)/FT, NÑow +Binstein assumed that the only light which is efective in making the transition +from ? to rn is the light which has the frequency corresponding to the energy +diference, so Ủy — ly = hớ ïn all our formulas. Thus +Nụ = Nạe “e/FT, (42.15) +Thus if we set the two rates equal: WaÐz„z„1() = N„[Am«ø + B„øT(6)], and +divide by „„, we get +Bauu1(0)c “(FT = Am + B„„1(6). (42.16) +From this equation, we can calculate ƒ(œ). It is simply +T(¿) = —————m—- 42.17 +(5) B,.chs/r—B..„ (42.17) +But Planck has already told us that the formula must be (42.12). Therefore +we can deduce something: First, that ;„„ must equal „, since otherwise we +cannot get the (c°«⁄/*T — 1). So Einstein discovered some things that he did +not know how to calculate, namely thaf the ?nduced emission probabtlitụ and the +absorption probabilitụ must be cqual. Thịs is interesting. And furthermore, in +order for (42.17) and (42.12) to agree, +Amn/ Đma must be hư” /n?cẺ. (42.18) +So If we know, for instance, the absorption rate for a given level, we can deduce +the spontaneous emission rate and the induced emission rate, or any combination. +This 1s as far as Einstein or anyone else could go using such arguments. To +actually compute the absolute spontaneous emission rate or the other rates for +any specifc atomie transition, of course, requires a knowledge of the machinery +of the atom, called quantum electrodynamics, which was not discovered until +eleven years later. This work of Einstein was done in 1916. +_ * This is not the only way one can arrange to keep the numbers of atoms in the various levels +constant, but it is the way it actually works. That every process must, in thermal equilibrium, +be balanced by its exact opposite is called the prữnciple oƑ detaied baÌancing. +--- Trang 764 --- +Blue m +Red, laser light +Fig. 42-3. By exciting, say by blue light, a higher state h, which may +emit a photon leaving atoms in state m, the number in this state m +becomes sufficiently large to start laser action. +'The possibility of induced emission has, today, found interesting applications. +T there is light present, it will tend to induce the downward transition. "The +transition then adds its ñứ to the available light energy, if there were some atorms +sitting in the upper state. Now we can arrange, by some nonthermal method, to +have a gas in which the number in the state ?m is very much greater than the +number in the state øœ. 'This is far out of equilibrium, and so is not given by the +formula e~”“/*T_ which is for equilibrium. We can even arrange it so that the +number in the upper state is very large, while the number in the lower state is +practically zero. Then light which has the requency corresponding to the energy +diference l„ — ly will not be strongly absorbed, because there are not many +atoms in state e to absorb it. Ôn the other hand, when that light is present, 1% +will induce the emission from this upper statel So, if we had a lot of atoms in +the upper state, there would be a sort of chain reaction, in which, the moment +the atoms began to emit, more would be caused to emit, and the whole lot of +them would dump down together. This is what is called a iaser, or, in the case +of the far infrared, a maser. +Various tricks can be used to obtain the atoms in state ?m. 'There may be +higher levels to which the atoms can get if we shine in a strong beam of light of +high frequency. Erom these high levels, they may trickle down, emitting various +photons, until they all get stuck in the state m. Tf they tend to stay in the +state rnm without emitting, the sbate is called rmefastable. And then they are +all dumped down together by induced emissions. Ône more technical point——1f +we put this system in an ordinary box, iÿ would radiate in so many different +directions spontaneously, compared with the induced efect, that we would still +be ín trouble. But we can enhance the induced efect, increase I1ts efficiency, by +--- Trang 765 --- +putting nearly perfect mirrors on each side of the box, so that the light which is +emitted gets another chance, and another chance, and another chanece, to induce +more emission. Although the mirrors are almost one hundred percent reflecting, +there is a slight amount of transmission of the mirror, and a little light gets out. +In the end, of course, from the conservation of energy, all the light goes out in +a nice uniform straight direction which makes the strong light beams that are +possible today with lasers. +--- Trang 766 --- +})rff—irSsrore +43-1 Collisions between molecules +WSe have considered so far only the molecular motions in a gas which is in +thermal equilibrium. We want now to discuss what happens when things are +near, but not exactly in, equilibrium. In a situation far from equilibrium, things +are extremely complicated, but in a situation very close to equilibrium we can +easily work out what happens. To see what happens, we must, however, return +to the kinetic theory. Statistical mechanics and thermodynamics deal with the +equilibrium situation, but away from equilibrium we can only analyze what occurs +atom by atom, so to speak. +As a simple example of a nonequilibrium cireumstance, we shall consider +the difusion of ions in a gas. Suppose that in a gas there is a relatively small +concentration of ions——electrically charged molecules. If we put an electric feld +on the gas, then each ion will have a force on it which is diferent from the Íorces +on the neutral molecules of the gas. If there were no other molecules present, an +ion would have a constant acceleration until it reached the wall of the container. +But because of the presence of the other molecules, it cannot do that; its velocity +increases only until it collides with a molecule and loses its momentum. Ïlt starts +again to pick up more speed, but then it loses is momentum again. “The net +efect is that an ion works its way along an erratic path, but with a net motion in +the direction of the electric force. We shall see that the ion has an average “drift” +with a mean speed which is proportional to the electric fñeld——the stronger the +ñeld, the faster it goes. While the fñeld is on, and while the ion is moving along, it +1s, OŸ course, øø£ in thermal equilibrium, it is trying to get to equilibrium, which +1s to be sitting at the end of the container. By means of the kinetic theory we +can compute the drift velocity. +lt turns out that with our present mathematical abilities we cannot really +compute ørec¿scfy what will happen, but we can obtain approximate results +--- Trang 767 --- +which exhibit all the essential features. We can find out how things will vary with +pressure, with temperature, and so on, but it will not be possible to get precisely +the correct numerical factors in front of all the terms. We shall, therefore, in our +derivations, not worry about the precise value of numerical factors. They can be +obtained only by a very mụuch more sophisticated mathematical treatment. +Before we consider what happens in nonequilibrium situations, we shall need +to look a little closer at what goes on in a gas In thermail equilibrium. We shall +need to know, for example, what the average time between successive collisions +of a molecule is. +Any molecule experiences a sequence of collisions with other molecules—in a +random way, of course. A particular molecule will, in a long period of time 7, +have a certain number, /, of hits. If we double the length of time, there will be +twice as many hits. So the number of collisions is proportional ©o the time 7'. +W©e would like to write it this way: +N =Tịr. (43.1) +We have written the constant of proportionality as 1/7, where 7 will have the +dimensions of a time. The constant 7 is the average time between collisions. +Suppose, for example, that in an hour there are 60 collisions; then 7 is one minute. +We would say that 7 (one minute) is the œuerage tữme between the collisions. +W©e may often wish to ask the following question: “What is the chønce that a +molecule will experience a collision during the next sznall ¿mterual of tìme dự?” +The answer, we may intuitively understand, is đ#/7. But let us try to make a +more convincing argument. Suppose that there were a very large number of +molecules. How many will have collisions in the next interval of time đý? I there +is equilibrium, nothing is changing øw the œ0erage with tỉìme. So ) molecules +waiting the time đ¿ will have the same number of collisions as øwe molecule +waiting for the time Ñ d. That number we know is đ/7. So the number oŸ +hits of W molecules is Ñ đ//7 in a time đ¿, and the chance, or probability, of a +hit for any one molecule is just 1/N as large, or (1/N)(N đi/T) = đt/T, as We +guessed above. 'That is to say, the fraction of the molecules which will sufer a +collision in the time đi is đ/7. To take an example, iÝ 7 is one minute, then in +one second the raction of particles which will sufer collisions is 1/60. What this +means, of course, is that 1/60 of the molecules happen to be close enough to +what they are goïing to hit next that #heir collisions will occur in the next second. +When we say that 7, the mean time between collisions, is one minute, we +do not mean that all the collisions will occur at times separated by exactly one +--- Trang 768 --- +minute. À particular particle does not have a collision, wait one minute, and then +have another collision. The times between successive collisions are quite variable. +W© will not need it for our later work here, but we may make a small diversion +to answer the question: “What are the times bebween collisions?” We know that +for the case above, the øuerage time 1s one minute, but we might like to know, +for example, what is the chance that we get no collision for #¿o minutes? +We shall ñnd the answer to the general question: “What is the probability +that a molecule will go for a time ý without having a collision?” At some arbitrary +instant—that we call ¿ = 0—we begin to watch a particular molecule. What is +the chance that it gets by until ¿ without colliding with another molecule? 'To +compute the probability, we observe what is happening to all Nọ molecules in a +container. After we have waited a tỉme £, some of them will have had collisions. +We let N(£) be the number that have noøứ had collisions up to the time ý. W() +1s, Of course, less than Wọ. We can find W(f) because we know how it changes +with time. I we know that N(#) molecules have got by until £, then W( + đi), +the number which get by until £ + đứ, is iess than N(£) by the number that have +collisions in đ¿. The number that collide in đý we have written above in terms of +the mean tỉme 7 as dNÑ = N(£) đt/T. We have the equation +N(t+ đi) = N( — N(t —. (43.2) +The quantity on the left-hand side, N(£ + đứ), can be written, according to the +defnitions of calculus, as Ý(f) + (4N/đf) đt. Making this substitution, Eq. (43.2) +yields +dN( Nự +dNỤ) __ Nữ) (43.3) +'The number that are being lost in the interval để is proportional to the number +that are present, and inversely proportional to the mean life r. Equation (43.3) +is easily integrated iŸ we rewrite it as +dN(®) d‡ +—=x =——- 43.4 +N( T (43.9 +lach side is a perfect differential, so the integral is +In ÝN() = —£/7 + (a constant), (43.5) +--- Trang 769 --- +which says the same thỉng as +NŒ) = (constant)e_!⁄, (43.6) +We know that the constant must be just ÄÑọ, the total number of molecules +present, since all of them start at # = 0 to wait for their “next” collision. We can +write our result as +N() = Nge 1, (43.7) +If we wish the probabiitu of no collision, P(£), we can get it by dividing ÝŒ) +by Äụ, so +P() =1, (43.8) +Our result is: the probability that a particular molecule survives a tỉme ý without +a collision is e—!⁄”, where 7 is the mean time between collisions. The probability +starts out at 1 (or certainty) for ý = 0, and gets less as £ gets bigger and +bigger. The probability that the molecule avoids a collision for a time equal +to 7 is e~†! 0.37. The chanee is less than one-half that it will have a greater +than average time between collisions. That is all right, because there are enough +molecules which go collision-free for times much Íonger than the mean time before +colliding, so that the average time can still be 7. +W© originally defned 7 as the average time be£ueen collisions. "The result +we have obtained in Eaq. (43.7) also says that the mean tỉme from an arbiiraru +starting instant to the mez£ collision 1s also 7. W© can demonstrate this somewhat +surprising fact in the following way. The number of molecules which experience +their nœez collision in the interval df at the time £ after an arbitrarily chosen +starting time is W() dt/7. Their “time until the next collision” is, of course, +Just ý. The “average time until the next collision” is obtained in the usual way: +1 ƒ“, N)di +Average time until the next collision = —— J ‡ DẦU kia +No 0 + +Using ÝNŒ) obtained in (43.7) and evaluating the integral, we ñnd indeed that 7 +is the average time from a/ instant until the next collision. +43-2 The mean free path +Another way of describing the molecular collisions is to talk not about the ##me +between collisions, but about hou ƒar the particle moves bebween collisions. If +--- Trang 770 --- +we say that the average time between collisions is 7, and that the molecules have +a mean velocity 0, we can expect that the average đis‡œnce between collisions, +which we shall call †, is just the produect of 7 and 0ø. "This distance between +collisions is usually called the rmeøn free path: +Mean free path Ï = 70. (43.9) +In this chapter we shall be a little careless about +ø0ha£ kind oƒ auerage we +mean in any particular case. The various possible averages—the mean, the root- +mmean-square, etc.—are all nearly equal and difer by factors which are near to +one. Since a detailed analysis is required to obtain the correct numerical factors +anyway, we need not worry about which average is required at any particular +point. We may also warn the reader that the algebraic symbols we are using Íor +soơme of the physical quantities (e.g., Ï for the mean free path) do not follow a +generally accepted convention, mainly because there is no general agreement. +Just as the chance that a molecule will have a collision in a short từme để +is equal to đf/7, the chance that it will have a collision in goïng a distance đz +1s dœ/I. EFollowing the same line of argument used above, the reader can show +that the probability that a molecule will go at least the distance # before having +its next collision is e-#⁄t, +The average distance a molecule goes before colliding with another molecule—— +the mean free path /j——will depend on how many molecules there are around and +on the “size” of the molecules, i.e., how bịg a target they represent. The effective +“size” of a target in a collision we usually describe by a “collision cross section,” +the same idea that is used in nuclear physics, or in light-scattering problems. +Consider a moving particle which travels a distance đa through a gas which +has nọ scatterers (molecules) per unit volume (Fig. 43-1). IÝ we look at each unit +Collision area Is ơc +—dx unit area +9 e 9 2 +_a 2 2 +'Total number of Total area covered is ¿nọ dx +molecules is nọ dx +Fig. 43-1. Collision cross section. +--- Trang 771 --- +of area perpendicular to the direction of motion of our selected particle, we will +fnd there ọ đa molecules. If each one presents an efective collision area. or, as +1t 1s usually called, “collision cross section,” øơ¿, then the total area covered by +the scatterers 1s Z¿ng đz. +By “collision cross section” we mean the area within which the center of our +particle must be located ïf it is to collide with a particular molecule. TỶ molecules +were little spheres (a classical picture) we would expect that ø¿ = (71 +72)Ÿ, +where r¡ and rz:ạ are the radii of the two colliding objects. The chance that +our particle will have a collision ¡is the ratio of the area covered by scattering +mmolecules to the total area, which we have taken to be one. So the probability of +a collision in going a distance đz is just ơeno d4: +Chanece of a collision in đø = đ¿ng đø. (43.10) +W©e have seen above that the chance of a collision in đz can also be written +in terms oŸ the mean free path Ï as dz/Ï. Comparing this with (43.10), we can +relate the mean free path to the collision cross section: +T1 70; (43.11) +which 1s easier to remember If we write it as +Øcnol = 1. (43.12) +This formula can be thought of as saying that there should be one collision, on +the average, when the particle goes through a distance / in which the scattering +molecules cowld just cover the total area. In a cylindrical volume of length ï +and a base of unit area, there are øoÏ scatterers; if each one has an area øơ„ the +total area covered is ngÏơ„, which is just one unit of area. The whole area 1s +not covered, of course, because some molecules are partly hidden behind others. +That is why some molecules go farther than / before having a collision. It is +only on the œuerage that the molecules have a collision by the time they go the +distance ỉ. From measurements of the mean free path / we can determine the +scattering cross section ơ‹, and compare the result with calculations based on a +detailed theory of atomic structure. But that is a diferent subjectl So we return +to the problem of nonequilibrium states. +--- Trang 772 --- +43-3 The drift speed +W©e want to describe what happens to a molecule, or several molecules, which +are diferent in some way from the large majority of the molecules in a gas. +We shall refer to the “majority” molecules as the “background” molecules, and +we shall call the molecules which are diferent from the background molecules +“gpecial” molecules or, for short, the Š-molecules. A molecule could be special +for any number of reasons: lt might be heavier than the background molecules. +It might be a diferent chemical. It might have an electric charge—i.e., be an +ion in a background oŸ uncharged molecules. Because of their different masses +or charges the S-molecules may have forces on them which are diferent from +the forces on the background molecules. By considering what happens to these +S-molecules we can understand the basic efects which come into play in a similar +way in many diferent phenomena. To list a few: the difusion of gases, electric +currents in batteries, sedimentation, centrifugal separation, etc. +We begin by concentrating on the basic process: an ,S-molecule in a back- +ground gas is acbed on by some specific force #' (which might be, e.g., gravitational +or electrical) and ¿nø addion by the not-so-specifc forces due to collisions with +the background molecules. We would like to describe the general behavior of the +S-molecule. What happens to it, 7n detø#L is that it darts around hither and yon +as iÿ collides over and over again with other molecules. But ifwe watch it carefully +we see that it does make some net progress in the direction of the force #'. We +say that there is a đrÿf, superposed on its random motion. We would like to +know what the speed of its drift is—its drjft 0elocitu—due to the force È'. +TÍ we star to observe an S-molecule a% some instant we may expect that it is +somewhere bebween two collisions. In addition to the velocity it was leftƠ with +after Its last collision it is picking up some velocity component due to the force È'. +In a short time (on the average, in a tỉme 7) it will experience a collision and +start out on a new piece of is trajectory. I§ will have a new starting velocity, +but the same acceleration from #'. +To keep things simple for the moment, we shall suppose that after each +collision our S-molecule gets a cormmpletely “fresh” start. 'That is, that it keeps no +remembrance ofits past acceleration by #'. 'This might be a reasonable assumption +1f our S-molecule were much lighter than the background molecules, but it is +certainly not valid in general. We shall discuss later an improved assumption. +For the moment, then, our assumption is that the S-molecule leaves each +collision with a velocity which may be in any direction with equal likelihood. The +--- Trang 773 --- +starting velocity will take it equally in all directions and will not contribute to +any net motion, so we shall not worry further about its initial velocity after a +collision. In addition to its random motion, each S-molecule will have, at any +mmoment, an additional velocity in the direction of the force #', which it has picked +up s¿nce its last collision. What is the ø0erøage value oŸ fh¡s part of the velocity? +It is just the acceleration #'/m (where mm is the mass of the S-molecule) times +the auerage tỉme s?nce the last collision. Now the average tỉme s/nce the Ìasf +collision must be the same as the average time ưøn#l the mez£ collision, which +we have called 7, above. The øueraøe velocity from #', of course, is just what is +called the drift velocity, so we have the relation +drift — bu (43.13) +'This basic relation is the heart of our subject. There may be some complication +in determining what 7 is, but the basic process is defined by Eaq. (43.13). +You will notice that the drift velocity is proportional to the force. 'There +1s, unfortunately, no generally used name for the constant of proportionality. +Diferent names have been used for each diferent kind of force. lfin an electrical +problem the force is written as the charge times the electric field, E' = q, then +the constant of proportionality between the velocity and the electric fñeld # is +usually called the “mobility.” In spite of the possibility of some confusion, +shall use the term rnobzl# for the ratio of the drift velocity to the force Íor øng +force. We write +Đarift — LẺ: (43.14) +in general, and we shall call „ the mobility. We have from Eq. (43.13) that +u = T/m. (43.15) +The mobility is proportional to the mean time between collisions (there are Íewer +collisions to slow it down) and inversely proportional to the mass (more inertia +means less speed picked up between collisions). +To get the correct numerical coeflicient in Eq. (43.13), which is correct as +given, takes some care. Without intending to confuse, we should still point out +that the arguments have a subtlety which can be appreciated only by a careful +and detailed study. To illustrate that there are dificulties, in spite of appearances, +we shall make over again the argument which led to Eq. (43.13) in a reasonable +but erroneous way (and the way one will fñnd in many textbooksl). +--- Trang 774 --- +We might have said: The mean time between collisions is 7. After a collision +the particle starts out with a random velocity, but it picks up an additional +velocity bebween collisions, which is equal to the acceleration times the time. +Since it takes the time 7 to arrive at the øœeø£ collision it gets there with the +velocity (EF/m}7r. At the beginning of the collision it had zero velocity. So bebween +the two collisions it has, on the average, a velocity one-half of the fñnal velocity, so +the mean drift velocity is 3f'r/m. (Wrongl) This result is wrong and the result +in Eq. (43.13) is right, although the arguments may sound equally satisfactory. +The reason the second result is wrong is somewhat subtle, and has to do with the +following: The argument is made as though all collisions were separated by the +mean time 7. The fact is that some times are shorter and others are longer than +the mean. Short times occur ?møre often but make Íess contribution to the drift +velocity because they have less chance “to really get going.” If one takes proper +account of the d¿stribution of free times bebween collisions, one can show that +there should not be the factor 3 that was obtained from the second argument. +The error was made in trying to relate by a simple argument the øuerage fnal +velocity to the average velocity itself. This relationship is not simple, so i is best +to concentrate on what is wanted: the average velocity itself. 'Phe fñrst argument +we gave determines the average velocity directly—and correctlyl But we can +perhaps see now why we shall not in general try to get all of the correct numerical +coefficients in our elementary derivationsl +W©e return now to our simplifying assumption that each collision knoecks out +all memory of the past motion—that a fresh start is made after each collision. +uppose our S-molecule is a heavy object in a background of lighter molecules. +Then our S-molecule will not lose its “forward” momentum in each collision. +Tt would take several collisions before its motion was “randomized” again. We +should assume, instead, that at each collision—in each time 7 on the average—it +loses a certain fraction of its momentum. We shall not work out the details, but +Just state that the result is equivalent to replacing 7, the average collision time, +by a new——and longer—7 which corresponds to the average “forgetting time,” i.e., +the average time to forget its forward momentum. With such an interpretation +oŸ7 we can use our formula (43.15) for situations which are not quite as simple +as we Írs assumed. +43-4 Tonic conductivity +W©e now apply our results to a special case. Suppose we have a gas in a vessel +in which there are also some ions—atoms or molecules with a net electric charge. +--- Trang 775 --- +W©e show the situation schematically in Fig. 43-2. If two opposite walls of the +container are metallic plates, we can connect them to the terminals of a battery +and thereby produce an electric ñeld in the gas. The electric ñeld will result in a +force on the ions, so they will begin to drift toward one or the other of the plates. +An electric current will be induced, and the gas with its ions will behave like a +resistor. By computing the ion ow from the drift velocity we can compute the +resistance. We ask, specifically: How does the Ñow of electric current depend on +the voltage diference V that we apply across the two plates? +E———bn—— +metal " sợ 9° ° +_- s9 E Lo +Area A ° ° +Gas with n¡ lons : +° per unit volume ++ ° ° — +Insulator +To battery with voltage V +Fig. 43-2. Electric current from an Ionized gas. +W© consider the case that our container is a rectangular box of length b and +cross-sectional area A (Fig. 43-2). If the potential diference, or voltage, from +one plate to the other is V, the electric ñeld # between the plates is V/b. (The +electric potential is the work done in carrying a unit charge from one plate to +the other. The force on a unit charge is E. lf E is the same everywhere bebween +the plates, which is a good enough approximation for now, the work done on a +unit charge is just #b, so V = Eb.) The special force on an ion of the gas is g, +where g is the charge on the ion. The drift velocity of the ion is then / times +this force, or +Đariftt = #ÉF` = hạ = nạ ?' (43.16) +An electric current 7 is the fow of charge in a unit time. The electric current +to one oŸ the plates is given by the total charge of the ions which arrive at the +plate in a unit of time. If the ions drift toward the plate with the veloclfYy 0ari£y; +--- Trang 776 --- +then those which are within a distance (0azie - 7) will arrive at the plate in the +time 7'. H there are n¿ ions per unit volume, the number which reach the plate +in the tìme 7° is (m¿ - A- 0arie : 7). Each ion carries the charge g, so we have that +Charge collected in 7 = gn¿ Auayity1. (43.17) +The current 7 is the charge collected in 7' divided by 7), so +T = gn; Aoayttt. (43.18) +Substituting 0azie Írom (43.16), we have +I = tq`n; nà (43.19) +W© fnd that the current is proportional to the voltage, which is just the form of +Ohm's law, and the resistance is the inverse of the proportionality constant: +BẾ uq”n; T (43.20) +W© have a relation between the resistance and the molecular properties ?;, q, +and , which depends in turn on rn and 7. If we know 0ø¿ and q from atomic +mneasurements, a measurement of # could be used to determine /, and from +alsO 7. +43-5. Molecular difusion +W©e turn now to a diferent kind of problem, and a diferent kind of analysis: +the theory of difusion. Suppose that we have a container of gas in thermal +equilibrium, and that we introduce a small amount of a diferent kind of gas +at some place in the container. We shall call the original gas the “background” +gas and the new one the “special” gas. The special gas will start to spread out +through the whole container, but it wiïll spread slowly because of the presence +of the background gas. This slow spreading-out process is called đjfƒus¿on. The +difusion is controlled mainly by the molecules of the special gas getting knocked +about by the molecules of the background gas. After a large number of collisions, +the special molecules end up spread out more or less evenly throughout the +whole volume. We must be careful no to confuse difusion of a gas with the +--- Trang 777 --- +gross transport that may occur due to convection currents. Most commonly, the +mixing of two gases occurs by a combination of convection and difusion. We are +interested now only in the case that there are œo “2n” currents. The gas 1s +spreading only by molecular motions, by difusion. We wish to compute how fast +difusion takes place. +We now compute the ne£ fiou of molecules of the “special” gas due to the +molecular motions. There will be a net fow only when there is some nonuniform +distribution oÊ the molecules, otherwise all oŸ the molecular motions would average +to give no net fow. Let us consider first the fow in the z-direction. To fnd the +fow, we consider an imaginary plane surface perpendicular to the #-axis and +count the number of special molecules that cross this plane. To obtain the net +flow, we must count as positive those molecules which cross in the direction of +positive ø and swbfract from this number the number which cross in the negative +z-direction. Ás we have seen many times, the number which cross a surface area, +in a time A7 is given by the number which start the interval A7' in a volume +which extends the distance ø A7' from the plane. (Note that 0, here, is the actual +molecular velocity, not the drift velocity.) +W©e shall simplify our algebra by giving our surface one unit of area. Then the +number of special molecules which pass from left to right (taking the +z-direction +to the right) is ø_ AT, where ø_ is the number of special molecules per unit +volume to the left (within a factor of 2 or so, but we are ignoring such factorsl). +The number which cross from right to left is, similarly, »¡u AT, where ¡ is the +number density of special molecules on the right-hand side of the plane. IÝ we +call the molecular current .J, by which we mean the net Ñow of molecules per +unit area per unit time, we have +n_~bAT'—nuAT' +J= ———AT (43.21) +jJ =(n_ — n0. (43.22) +'What shall we use for ø_ and œ¡? When we say “the density on the left,” +how ƒár to the left do we mean? We should choose the density at the place from +which the molecules started their “fñight,” because the number which s¿ør£ such +trips is determined by the number present at that place. So by øœ_ we should +mean the density a distance to the left equal to the mean free path i, and by n+, +the density at the distance / to the right of our imaginary surface. +--- Trang 778 --- +lt is convenient to consider that the distribution of our special molecules +in space is described by a continuous function of z, , and z which we shall +call n„. By n„(z,,2z) we mean the number density of special molecules in a +small volume element centered on (#,,2). In terms oŸ %„ we can express the +diference (m —m_) as +(n—n )=“ “Ax==s.2J, (43.23) +Substituting this result in Eq. (43.22) and neglecting the factor of 2, we get +Jy = lu CS, (43.24) +da: +W© have found that the ñow of special molecules is proportional to the derivative +of the density, or to what is sometimes called the “gradient” of the density. +Tt is clear that we have made several rough approximations. Besides various +factors of two we have left out, we have used ø where we should have used œ„, and +we have assumed that ø+ and ø+_ refer to places at the perpendicular distance Ï +from our surface, whereas for those molecules which do not travel perpendicular +to the surface element, / should correspond to the san distance from the surface. +AlI of these refinements can be made; the result of a more careful analysis shows +that the right-hand side of Bq. (43.24) should be multiplied by 1/3. 5o a better +anSWer 1s lờ đ +J>==———. 43.25 +` Š d+z ) +Similar equations can be written for the currents in the - and z-directions. +The current J„ and the density gradient đn„/dø can be measured by macro- +scopic observations. 'Pheir experimentally determined ratio is called the “difusion +coeffcient,” D. 'That 1s, +J„==D s, (43.26) +W© have been able to show that for a gas we expect +D= ÿlo. (43.27) +So far in this chapter we have considered two distinct processes: rmob¿lt, the +drift of molecules due to “outside” forces; and đjƒƒus¿on, the spreading determined +only by the internal forces, the random collisions. 'There is, however, a relation +--- Trang 779 --- +between them, since they both depend basically on the thermal motions, and the +mean free path / appears in both calculations. +Tf, in Eq. (43.25), we substitute ỉ = 0r and 7 = ưn, we have +J„ = —gmuŸ”" an” (43.28) +But m2 depends only on the temperature. We recall that +3mu° = ŠKT, (43.29) +J„y = —HkT'——. 43.30 +We ñnd that D, the đjƒfusion coefficient, is just k7' times , the mobilöty coeflcient: +D= HT. (43.31) +And it turns out that the numerical coefficient in (43.31) is exactly right—no +extra factors have to be thrown in to adjust for our rough assumptions. We +can show, in fact, that (43.31) must a0øys be correct—even in complicated +situations (for example, the case of a suspension in a liquid) where the details of +our simple calculations would not apply at all. +To show that (43.31) must be correct in general, we shall derive it in a different +way, using only our basic principles of statistical mechanics. lmagine a situation +in which there is a gradient of “special” molecules, and we have a difusion current +proportional to the density gradient, according to Eq. (43.26). We now apply +a force field in the z-direction, so that each special molecule feels the force #'. +According to the đeffnition of the mobility there will be a drift velocity given +drift — uử: (43.32) +By our usual arguments, the dưới current (the net number of molecules which +pass a unit oŸ area in a unit of time) will be +đQrift —= TìaUdrift› (43.33) +--- Trang 780 --- +drift — nạ: (43.34) +W©e now øđ7ust the force #! so that the drift current due to #! just balances +the difusion, so that there is mo net fiou of our special molecules. We have +J„ + qrift — 0, OT +D—— =nạ„uF. 43. +TNG.. (43.35) +Under the “balance” conditions we fñnd a steady (with time) gradient of +density given by +đng — nạjpF +———=_——- 43.36 +d+z D ) +But noticel We are describing an egui¿brium condition, so our egu¿lbrzum +laws of statistical mechanics apply. According to these laws the probability of +ñnding a molecule at the coordinate z is proportional to e~U/*T, where Ù is the +potential energy. In terms of the number density n„, this means that +nạ = nục” U/T, (43.37) +If we diferentiate (43.37) with respect to #, we find +dn, —_ "¡J..nn h +"đạc =—= -Tìiọ€ * k7 mm (43.38) +dng mạụ dƯÙ +———=_->m_— 43.39 +d+z kí dư ) +In our situation, since the force #' is in the z-direction, the potential energy is +jusb —z, and —đỮ/dz = F'. Equation (43.39) then gives +dng — nạF +———= —- 43.4 +đa kT (43.40) +[This is just exactly Eq. (40.2), from which we deduced e~Ứ⁄*” ¡n the ñrst place, +so we have come in a circlel. Comparing (43.40) with (43.36), we get exactly +Eq. (43.31). We have shown that Eq. (43.31), which gives the đdifusion current +in terms of the mobility, has the correct coeflicient and is very generally true. +Mobility and difusion are intimately connected. 'This relation was fñrst deduced +by Einstein. +--- Trang 781 --- +43-6 Thermal conductivity +The methods of the kinetic theory that we have been using above can be +used also to compute the #hermal conducliitụ of a gas. TỶ the gas at the top oŸ a +container is hotter than the gas at the bottom, heat will fow from the top to the +bottom. (We think of the top being hotter because otherwise convection currents +would be set up and the problem would no longer be one oŸ heat conduection.) +The transfer of heat from the hotter gas to the colder gas 1s by the difusion +of the “hot” molecules—those with more energy——downward and the difusion +of the “cold” molecules upward. To compute the Ñow of thermal energy we +can ask about the energy carried downward across an element of area by the +downward-moving molecules, and about the energy carried upward across the +surface by the upward-moving molecules. "The difference will give us the net +downward flow of energy. +The thermail conductivity & is deÑned as the ratio of the rate at which thermal +energy is carried across a unit surface area, to the temperature gradient: +1 dQ đT +———=-E—. 43.41 +Adr cdz (43.44) +Since the details of the caleculations are quite similar to those we have done above +in considering molecular difusion, we shall leave it as an exercise for the reader +to show that Eml +R— TT, (43.42) +where &7 /(+ — 1) is the average energy of a molecule at the temperature 7'. +TÝ we use our relation øz¿ = 1, the heat conductivity can be written as +¬..... (43.43) ++— lØƠ, +W© have a rather surprising result. We know that the average velocity of gas +molecules depends on the termperature but no‡ ơn the densit. We expect ơ, +to depend only on the s2ze of the molecules. So our simple result says that the +thermal conduectivity œ (and therefore the zafe of fow of heat in any particular +circumstance) is independent of the đensitu of the gasl "The change in the number +OŸ “carriers” of energy with a change in density is just compensated by the larger +distance the “carriers” can øo between collisions. +--- Trang 782 --- +One may ask: “ls the heat fow independent of the gas density in the limit +as the density goes to zero? When there is no gas at all?” Certainly notl The +formula (43.43) was derived, as were all the others in this chapter, under the +assumption that the mean free path between collisions is mụch smaller than any +of the dimensions oŸ the container. Whenever the gas density is so low that a +mmolecule has a fair chance of crossing from one wall of its container to the other +without having a collision, none of the calculations of this chapter apply. We +mmust in such cases go back to kinetic theory and calculate again the details of +what will occur. +--- Trang 783 --- +Tĩìo L{átt-s of Titor-rrtoelÏggreerrttfcS +44-1 Heat engines; the first law +So far we have been discussing the properties of matter from the atomic +point of view, trying to understand roughly what will happen if we suppose that +things are made of atoms obeying certain laws. However, there are a number of +relationships among the properties of substances which can be worked out without +consideration of the detailed structure of the materials. The determination of the +relationships among the various properties of materials, without knowing their +internal structure, is the subject oŸ 0hermodnamwcs. Historically, thermodynamics +was developed before an understanding of the internal structure of matter was +achieved. +To give an example: we know from the kinetic theory that the pressure oŸ a gas +1s caused by molecular bombardment, and we know that if we heat a gas, so that +the bombardment increases, the pressure must increase. Conversely, ¡if the piston +in a container of the gas is moved inward against the force of bombardment, the +energy of the molecules bombarding the piston will increase, and consequently the +temperature will increase. So, on the one hand, if we increase the temperature at +a given volume, we increase the pressure. Ôn the other hand, if we compress the +gas, we will find that the temperature will rise. From the kinetic theory, one can +derive a quantitative relationship between these two effects, but instinctively one +might guess that they are related in some necessary fashion which is independent +of the details of the collisions. +Let us consider another example. Many people are familiar with this interesting +property of rubber: lf we take a rubber band and pull it, it gets warm. lf one +puts ít between his lips, for example, and pulls it out, he can feel a distinct +warming, and this warming is reversible in the sense that if he relaxes the rubber +band quickly while ¡it is between his lips, it is distinctly cooled. That means that +when we stretch a rubber band ït heats, and when we release the tension of the +--- Trang 784 --- +X2 +Fig. 44-1. The heated rubber band. +band it cools. Now our instincts might suggest that if we heated a band, it might +pull: that the fact that pulling a band heats it might imply that heating a band +should cause it to contract. And, in fact, if we apply a gas Ñame to a rubber +band holding a weight, we will see that the band contracts abruptly (EFig. 44-1). +So it is true that when we heat a rubber band ït pulls, and this fact ¡is defnitely +related to the fact that when we release the tension of it, it cools. +'The internal machinery of rubber that causes these efects is quite complicated. +W©e will describe it from a molecular point of view to some extent, although our +main purpose in this chapter is to understand the relationship of these efects +independently of the molecular model. Nevertheless, we can show from the +mmolecular model that the efects are closely related. One way to understand the +behavior of rubber is to recognize that this substance consists oŸ an enormous +tangle of long chains of molecules, a kind of “molecular spaghetti,” with one extra +complication: between the chains there are cross-links——like spaghetti that 1s +sometimes welded together where it crosses another piece of spaghetti—a grand +tangle. When we pull out such a tangle, some of the chains tend to line up along +the direction of the pull. At the same time, the chains are in thermal motion, so +they hit each other continually. It follows that such a chain, if stretched, would +not by Itself remain stretched, because it would be hit from the sides by the other +chains and other molecules, and would tend to kink up again. So the real reason +why a rubber band tends to contract is this: when one pulls it out, the chains are +lengthwise, and the thermal agitations of the molecules on the sides of the chains +tend to kink the chains up, and make them shorten. Ône can then appreciate +that 1f the chains are held stretched and the temperature is increased, so that +the vigor of the bombardment on the sides of the chaïns is also increased, the +chains tend to pull in, and they are able to pull a stronger weight when heated. +Tf, after being stretched for a time, a rubber band ¡s allowed to relax, each chain +--- Trang 785 --- +becomes soft, and the molecules striking it lose energy as they pound into the +relaxing chain. So the temperature falls. +We have seen how these two processes, contraction when heated and cooling +during relaxation, can be related by the kinetic theory, but i£ would be a tremen- +dous challenge to determine from the theory the precise relationship bebween the +two. We would have to know how many collisions there were each second and +what the chains look like, and we would have to take account of all kinds of other +complications. The detailed mechanism is so complex that we cannot, by kinetic +theory, really determine exactly what happens; still, a definite relation between +the bwo efects we observe can be worked out without knowing anything about +the internal machineryl +The whole subject of thermodynamics depends essentially upon the following +kind of consideration: because a rubber band is “stronger” at higher temperatures +than it is at lower temperatures, it ought to be possible to lift weights, and to +move them around, and thus to do work with heat. In fact, we have already seen +experimentally that a heated rubber band can lift a weight. 'Phe study of the way +that one does work with heat is the beginning of the science of thermodynamics. +Can we make an engine which uses the heating efect on a rubber band to do work? +One can make a silly looking engine that does just this. It consists of a bicycle +wheel in which all the spokes are rubber bands (Eig. 44-2). If one heats the rubber +bands on one side of the wheel with a païr of heat lamps, they become “stronger” +than the rubber bands on the other side. 'Phe center of gravity of the wheel will +šxv: +Fig. 44-2. The rubber band heat engine. +--- Trang 786 --- +be pulled to one side, away from the bearing, so that the wheel turns. Às it turns, +cool rubber bands move toward the heat, and the heated bands move away from +the heat and cool, so that the wheel turns slowly so long as the heat is applied. The +efficiency of this engine is extremely low. Four hundred watts of power pour into +the two lamps, but it is just possible to lift a fy with such an enginel Án interesting +question, however, is whether we can get heat to do the work in more efHicient ways. +In fact, the science of thermodynamics began with an analysis, by the great +engineer Sadi Carnot, of the problem of how to build the best and most eficient +engine, and this constitutes one of the few famous cases in which engineering +has contributed fundamentally to physical theory. Another example that comes +to mind is the more recent analysis of information theory by Claude Shannon. +'These two analyses, inecidentally, turn out to be closely related. +Now the way a steam engine ordinarily operates 1s that heat from a fire +boils some water, and the steam so formed expands and pushes on a piston +which makes a wheel go around. So the steam pushes the piston——what then? +One has to fnish the Job: a stupid way to complete the cycle would be to let +the steam escape into the air, for then one has to keep supplying water. Ï% +is cheaper—more efficient—to let the steam go into another box, where it is +condensed by cool water, and then pump the water back into the boiler, so that +1t circulates continuously. Heat is thus supplied to the engine and converted into +work. Now would it be better to use alcohol? What property should a substance +have so that it makes the best possible engine? 'Phat was the question to which +Carnot addressed himself, and one of the by-products was the discovery of the +type of relationship that we have just explained above. +'The results of thermodynamies are all contained implicitly in certain appar- +ently simple statements called the iœus oƒ thermodunamics. At the tìme when +Carnot lived, the fñrst law of thermodynamics, the conservation Of energy, Was +not known. Carnot's arguments were so carefully drawn, however, that they are +valid even though the first law was not known in his timel Some tỉme afterwards, +Clapeyron made a simpler derivation that could be understood more easily than +Carnot”s very subtle reasoning. But it turned out that Clapeyron assumed, not +the conservation of energy in general, but that heø£ was conserved according to +the caloric theory, which was later shown to be false. So it has often been said +that Carnots logic was wrong. But his logic was quite correct. Only Clapeyron's +simplifed version, that everybody read, was incorrect. +'The so-called second law of thermodynamiecs was thus discovered by Carnot +before the first law!l It would be interesting to give Carnot”s argument that did +--- Trang 787 --- +not use the first law, but we shall not do so because we want to learn physics, +not history. We shall use the first law from the start, in spite of the fact that a +great deal can be done without ït. +Let us begin by stating the first law, the conservation of energy: if one has a +system and puts heat into it, and does work on it, then its energy is increased by +the heat put in and the work done. We can write this as follows: The heat Q +put into the system, plus the work W/ done on the system, is the increase in the +energy of the system; the latter energy is sometimes called the internal energy: +Change in U = Q+ W. (44.1) +The change in Ứ can be represented as adding a little heat AQ and adding a +little work AM: +AU = AQ+ AW, (44.2) +which is a diferential form of the same law. We know that very well, from an +earlier chapter. +44-2 The second law +Now, what about the second law of thermodynamics? We know that iŸ we +do work against friction, say, the work lost to us is equal to the heat produeed. +T we do work In a room at temperature 7', and we do the work slowly enough, +the room temperature does not change much, and we have converted work into +heat at a given temperature. What about the reverse possibility? Is it possible +to convert the heat back into work at a given temperature? The second law of +thermodynamics asserts that it is not. It would be very convenient to be able to +convert heat into work merely by reversing a process like friction. If we consider +only the conservation of energy, we might think that heat energy, such as that in +the vibrational motions of molecules, might provide a goodly supply of useful +energy. But Carnot assumed that it is impossible to extract the energy of heat +at a single temperature. In other words, ¡f the whole world were at the same +temperature, one could not convert any of its heat energy into work: while the +process of making work go into heat can take place at a given temperature, one +cannot reverse it to get the work back again. Specifically, Carnot assumed that +heat cannot be taken in at a certain temperature and converted into work tu¿th +no other change in the system or the surroundings. +--- Trang 788 --- +That last phrase is very important. Suppose we have a can of compressed +air at a certain temperature, and we let the air expand. It can do work; it can +make hammers go, for example. It cools off a little in the expansion, but iÝ we +had a bïg sea, like the ocean, at a given temperature—a heat reservoir——we could +warm it up again. So we have taken the heat out of the sea, and we have done +work with the compressed air. But Carnot was not wrong, because 0œ đứd no£ +leœue cueruthing as ït uas. TÝ we recompress the air that we let expand, we will +ñnd we are doing extra work, and when we are finished we will discover that we +not only got no work out of the system at temperature 7, but we actually put +some in. We must talk only about situations in which the øeÝ result of the whole +process is to take heat away and convert it into work, just as the net result of the +process of doing work against friction is to take work and convert i% into heat. TÍ +we move in a circle, we can bring the system back precisely to its starting poïnt, +with the net result that we did work against friction and produced heat. Can we +reverse the process? Turn a switch, so that everything goes backwards, so the +friction does work against us, and cools the sea? According to Carnot: nol So +let us suppose that this is impossible. +T it were possible it would mean, among other things, that we could take +heat out of a cold body and put it into a hot body at no cost, as it were. Now +we know it is natural that a hot thing can warm up a cool thing; iŸ we simply +put a hot body and a cold one together, and change nothing else, our experlence +assures us that it is not going to happen that the hot one gets hotter, and the +cold one gets colderl But if we could obtain work by extracting the heat out of +the ocean, say, or from anything else at a single temperature, then that work +could be converted back into heat by friction at some other temperature. For +instance, the other arm of a working machine could be rubbing something that is +already hot. The net result would be to take heat from a “cold” body, the ocean, +and to put it into a hot body. Now, the hypothesis of Carnot, the second law +of thermodynamies, is sometimes stated as follows: heat cannot, of itself, low +from a cold to a hot object. But, as we have Just seen, these two statements are +equivalent: fñrst, that one cannot devise a process whose only result is to convert +heat to work at a single temperature, and second, that one cannot make heat +fow by itself from a cold to a hot place. We shall mostly use the fñrst form. +Carnot's analysis of heat engines is quite similar to the argument that we +gave about weight-lifting engines in our discussion of the conservation of energy +in Chapter 4. In fact, that argument was patterned after Carnot°s argument +about heat engines, and so the present treatment will sound very much the same. +--- Trang 789 --- += T1 ° +Fig. 44-3. Heat engine. +Suppose we build a heat engine that has a “boiler” somewhere at a temper- +ature 71. A certain heat Qạ is taken rom the boiler, the steam engine does +some work W/, and it then delivers some heat Qs into a “condenser” at another +temperature 7¿ (Eig. 44-3). Carnot did not say how mụuch heat, because he did +not know the frst law, and he did not use the law that Qs was cqual to + +because he did not believe it. Although everybody thought that, according to +the caloric theory, the heats Q¡ and Qs would have to be the same, Carnot did +not say they were the same——that is part of the cleverness of his argument. lÝ we +do use the first law, we fnd that the heat delivered, Qa, is the heat Q that was +put in minus the work W/ that was done: +Q; = Q¡ — (44.3) +(Tf we have some kind of cyclic process where water is pumped back into the boiler +after it is condensed, we will say that we have heat Q¡ absorbed and work WZ +done, during each cycle, for a certain amount of water that goes around the +cycle.) +Now we shall build another engine, and see if we cannot get more work +from the same amount of heat being delivered at the temperature 71, with the +condenser still at the temperature 7¿. We shall use the same amount of heat +from the boiler, and we shall try to get more work than we did out of the steam +engine, perhaps by using another ẨÑuid, such as aleohol. +44-3 Reversible engines +Now we must analyze our engines. Ône thing is clear: we will lose something +1f the engines contain devices in which there is friction. 'Phe best engine will be +a Ífrictionless engine. We assume, then, the same idealization that we did when +we studied the conservation of energy; that is, a perfectly frictionless engine. +We must also consider the analog of frictionless motion, “frictionless” heat +transfer. If we put a hot object at a hipgh temperature against a cold object, so +--- Trang 790 --- +Fig. 44-4. Reversible heat transfer. +that the heat fows, then it is not possible to make that heat fow In a reverse +direction by a very small change in the temperature of either object. But when +we have a practically frictionless machine, if we push it with a little force one +way, 1% goes that way, and If we push it with a little force the other way, it goes +the other way. We need to fnd the analog of frictionless motion: heat transfer +whose direction we can reverse with only a tiny change. Ilf the diference in +temperature is finite, that is impossible, but if one makes sure that heat fows +always between two things at essentially the same temperature, with just an +inñnitesimal diference to make it fow in the desired direction, the fow is said to +be reversible (Eig. 44-4). If we heat the object on the left a little, heat will ow to +the right; If we cool it a little, heat will fow to the left. So we fñnd that the ideal +engine is a so-called reuers¿ble engine, in which every process is reversible in the +sense that, by minor changes, infnitesimal changes, we can make the engine go +in the opposite direction. hat means that nowhere in the machine must there +be any appreciable friction, and nowhere in the machine must there be any place +where the heat of the reservoirs, or the ame of the boiler, is in direct contact +with something definitely cooler or warmer. +Let us now consider an idealized engine in which all the processes are reversible. +To show that such a thing 1s possible in prineciple, we will give an example of an +engine cycle which may or may not be practical, but which is at least reversible, +in the sense of Carnot's idea. Suppose that we have a gas in a cylinder equipped +with a frictionless piston. The gas is not necessarily a perfect gas. The Ñuid does +not even have to be a gas, but to be specifc let us say we do have a perfect gas. +Also, suppose that we have two heat pads, 71 and 72——great big things that have +defnite temperatures, 71 and 7¿. We will suppose in this case that 71 is higher +than 7¿. Let us first heat the gas and at the same time expand it, while it 1s +--- Trang 791 --- +Z⁄⁄⁄42 +2 ⁄Z⁄⁄⁄4 +Step (1) lIsothermal expansion at T¡, absorb heat Q¡ +⁄Z⁄⁄⁄4 +Step (2) Adiabatic expansion, temperature falls ffom Tị to Ta +Step (3) Isothermal compression at Tạ, deliver heat Qz +⁄Z⁄⁄4 +Step (4) Adiabatic compression, temperature rises from Tạ to Tị +Fig. 44-5. Steps in Carnot cycle. +in contact with the heat pad at 7. As we do this, pulling the piston out very +slowly as the heat fows into the gas, we will make sure that the temperature +of the gas never gets very far from 7. lÝ we pull the piston out too fast, the +temperature of the gas will fall too much below 71 and then the process will not +be quite reversible, but if we pull it out sÌlowly enough, the temperature of the gas +will never depart much from 71. Ôn the other hand, if we push the piston back +slowly, the temperature would be only infũnitesimally higher than 71, and the +--- Trang 792 --- +9 ì &2 @ À +Ũ Area= JH +T=T, \@ +' N «`. > +Volume +Fig. 44-6. The Carnot cycle. +heat would pour back. We see that such an isothermal (constant-temperature) +expansion, done slowly and gently enough, is a reversible process. +To understand what we are doing, we shall use a plot (Fig. 44-6) of the pressure +of the gas against its volume. As the gas expands, the pressure falls. The curve +marked (1) tells us how the pressure and volume change if the temperature is +kept fñxed at the value 71. For an ideal gas this curve would be PW = Nk1). +During an isothermal expansion the pressure falls as the volume increases until +we s6op at the point b. At the same time, a certain heat Q+ must fow into the gas +from the reservoir, for if the gas were expanded without being in contact with the +reservoir i would cool of, as we already know. Having completed the isothermal +expansion, stopping at the point 0, let us take the cylinder away from the reservoir +and continue the expansion. 'This time we permit no heat to enter the cylinder. +Again we perform the expansion slowly, so there is no reason why we cannot +reverse it, and we again assume there is no friction. The gas continues to expand +and the temperature falls, since there is no longer any heat entering the cylinder. +We let the gas expand, following the curve marked (2), until the temperature +falls to Tạ, at the point marked c. 'This kind of expansion, made without adding +heat, is called an adiabatic expansion. For an ideal gas, we already know that +curve (2) has the form V7 = constant, where + is a constant greater than 1, so +that the adiabatic curve has a more negative slope than the isothermal curve. +The gas cylinder has now reached the temperature 72, so that if we put it on the +heat pad at temperature 7¿ there will be no irreversible changes. Now we slowly +--- Trang 793 --- +Bi s†+¬ "- +Useful +Q¡—W Qị —W/ Wwork +Fig. 44-7. Reversible engine A being driven backwards by engine 8. +compress the gas while it is in contact with the reservoir at 7¿, following the +curve marked (3) (Fig. 44-5, Step 3). Because the cylinder is in contact with the +reservoir, the temperature does not rise, but heat Qs fows from the cylinder into +the reservoir at the temperature 7¿. Having compressed the gas isothermally +along curve (3) to the point d, we remove the cylinder from the heat pad at +temperature 7¿ and compress it still further, without letting any heat fow out. +The temperature will rise, and the pressure will follow the curve marked (4). lÝ +we carry out each step properly, we can return to the point œ at temperature 7] +where we started, and repeat the cycle. +We see that on this diagram we have carried the gas around a complete cycle, +and during one cycle we have put Q1 in at temperature 71, and have removed Qs +at temperature 72. Now the poïnt is that this cycle is reversible, so that we could +represent all the steps the other way around. We could have gone backwards +instead offorwards: we could have started at point ø, at temperature 71, expanded +along the curve (4), expanded further at the temperature 72, absorbing heat Qa, +and so on, going around the cycle backward. lf we go around the cycle in one +direction, we must do work on the gas; IÝ we go in the other direction, the gas +does work on us. +Incidentally, it is easy to fnd out what the total amount of work is, because +the work during any expansion is the pressure tỉimes the change in volume, ƒ PđV. +On this particular diagram, we have plotted ?P vertically and V horizontally. 5o +1Ÿ we call the vertical distance z and the horizontal distance ø, this is ƒda=—in +other words, the area under the curve. So the area under each of the numbered +curves is a measure of the work done by or on the gas in the corresponding step. +lt is easy to see that the net work done is the shaded area of the picture. +--- Trang 794 --- +Now that we have given a single example of a reversible machine, we shall +suppose that other such engines are also possible. Let us assume that we have +a reversible engine 4 which takes Qị at 71, does work M7, and delivers some +heat at 7¿. Now let us assume we have any other engine , made by man, +already designed or not yet invented, made of rubber bands, steam, or whatever, +reversible or not, which ¡is designed so that it takes in the same amount of heat +at T7, and rejects the heat at the lower temperature 75 (Eig. 44-7). Assume +that engine Ö does some work, W/“. NÑow we shall show that W7 is not greater +than W——that no engine can do more work than a reversible one. Why? Suppose +that, indeed, W7” were bigger than W/. 'Then we could take the heat Q+ out of +the reservoir at 71, and with engine we could do work W7 and deliver some +heat to the reservoir at 72; we do not care how much. That done, we could save +some of the work W”, which is supposed to be greater than W/; we could use +a part of it, W, and save the remainder, W“ — W, for useful work. With the +work W we could run engine A backwards because ?‡ ¡s a reuersible engine. It will +absorb some heat from the reservoir at 7s and deliver @\ back to the reservoir +at 7. After this double cycle, the net result would be that we would have put +everything back the way it was before, and we would have done some excess +work, namely W“ — W, and ai we would have done would be to extract energy +from the reservoir at 7¿! We were careful to restore the heat @ị to the reservoir +at 71. So that reservoir can be small and “inside” our combined machine A4 + Ö, +whose net effect is therefore bo extract a net heat W“ — W from the reservoir +at T2 and convert it into work. But to obtain uscful work from a reservoir at a +single temperature with no other changes is impossible according to Carnot”s +postulate; it cannot be done. Therefore no engine which absorbs a given amount +of heat from a higher temperature 71 and delivers it at the temperature 7¿ can +do more work than a reversible engine operating under the same temperature +conditions. +Now suppose that engine #Ö is also reversible. Then, of course, not only must +W7 be not greater than W/, but now we can reverse the argument and show that +W cannot be greater than W/”. So, if both engines are reversible they must both +do the same amount of work, and we thus come to Carnotˆs brilliant conclusion: +that if an engine is reversible, it makes no diference how it is designed, because +the amount oŸ work one will obtain if the engine absorbs a given amount of +heat at temperature 71 and delivers heat at some other temperature 75 đoes no‡ +depend on the design oƒ the engine. Ït 1s a property of the world, not a property +of a particular engine. +--- Trang 795 --- +Tf we could fnd out what the law is that determines how much work we obtain +when we absorb the heat Q at 71 and deliver heat at 7¿, this quantity would +be a universal thing, independent of the substance. Of course If we knew the +properties of a particular substance, we could work it out and then say that all +other substances must give the same amount oŸ work in a reversible engine. That +1s the key idea, the clue by which we can fnd the relationship between how much, +for Instance, a rubber band contracts when we heat it, and how much it eools +when we let it contract. Imagine that we put that rubber band in a reversible +machine, and that we make it go around a reversible cycle. "The net result, the +total amount of work done, is that universal function, that great function which +1s Iindependent of substance. So we see that a substance's properties must be +limited in a certain way; one cannot make up anything he wants, or he would be +able to invent a substance which he could use to produce more than the maximum +allowable work when he carried it around a reversible cycle. This principle, this +limitation, is the only real rule that comes out of the thermodynamics. +44-4 The efficiency of an ideal engine +Now we shall try to ñnd the law which determines the work W as a function +of Q, 7, and 7¿. It is clear that W is proportional to Q\, for if we consider two +reversible engines in parallel, both working together and both double engines, +the combination is also a reversible engine. If each one absorbed heat Q\, the +two together absorb 2@¡ and the work done is 2W, and so on. So it is not +unreasonable that W is proportional to 1. +Now the next Important step is 0o find this universal law. We can, and will, +do so by studying a reversible engine with the one particular substance whose +laws we know, a perfect gas. It is also possible to obtain the rule by a purely +logical argument, using no particular substance at all. 'Phis is one of the very +beautiful pieces of reasoning in physics and we are reluctant not to show it to you, +so for those who would like to see it we shall discuss it in just a moment. But +first we shall use the much less abstract and simpler method of direct calculation +for a perfect gas. +We need only obtain formulas for Q and Qs (for WÝ is Just Qì — Qa), the heats +exchanged with the reservoirs during the isothermal expansion or contraction. +For example, how much heat @ is absorbed from the reservoir at temperature T1 +during the isothermal expansion [marked (1) in Eig. 44-6] from point ø, at +pressure ø„, volume W⁄, temperature 71, to point b with pressure øạ, volume Vj, +--- Trang 796 --- +and the same temperature 71? Eor a perfect gas each molecule has an energy that +depends only on the temperature, and since the temperature and the number of +mmolecules are the same at ø and at b, the internal energy is the same. 7Öere 7s +no change n U; all the work done by the gas, +W= J pdV, +during the expansion is energy + taken from the reservoir. During the expansion, +øpV = NkT], or +Qị = J pdV = J NRT (44.4) +or a a +Q¡ = NEkTiÌn TA +1s the heat taken from the reservoir at 71. In the same way, for the compression +at T2 [curve (3) of Fig. 44-6] the heat delivered to the reservoir at 72 is +To finish our analysis we need only fñnd a relation between W„/Wạ and Vi/Vạ. +This we do by noting that (2) is an adiabatic expansion from ở to é, during which +ÐV7 is a constant. Since pV = W7, we can write this as (pV)V~! = const or, +in terms of 7' and WV, as TV~! = const, or +TP” =TpVT—}, (44.6) +Likewise, since (4), the compression rom đ to ø, is also adiabatic, we fnd +TIV2~1 = TpV?—”, (44.6a) +Tf we divide this equation by the previous one, we fnd that Vp/Vạ must equal V„/Va, +so the ÌIn)s in (41.4) and (44.5) are equal, and that +Q:i - Q2 +—=_—.- 44.7 +T Thọ (44.7) +--- Trang 797 --- +Q2 +————'': VWa +Q2 +W22 Q3 +Q3 | T; +Fig. 44-8. Engines 1 and 2 together are equivalent to engine 3. +Thịs is the relation we were seeking. Although proved for a perfect gas engine, +we know it must be true ƒor an reuersible engine at aÌ1L +Now we shall see how this universal law could also be obtained by logical +argument, without knowing the properties of any specifc substances, as follows. +Suppose that we have three engines and three temperatures, let us say 7], 1ạ, +and 7$. Let one engine absorb heat Q from the temperature 71 and do a certain +amount of work Wìa, and let it deliver heat Qs to the temperature 73 (Eig. 44-8). +Let another engine run backwards between 7¿ and 7. Suppose that we let the +second engine be of such a size that it will absorb the same heat @s, and deliver +the heat Qs. We will have to put a certain amount of work, W⁄sa, into it —negative +because the engine is running backwards. When the first machine goes through a +cycle, it absorbs heat Q¡ and delivers Qs at the temperature 75s; then the second +machine takes the same heat Q)z out of the reservoir at the temperature 7z and +delivers it into the reservoir at temperature 7¿. Thherefore the net result of the +two machines in tandem is to take the heat €Q) from 7, and deliver Qs at 15. +The two machines are thus equivalent to a third one, which absorbs Q at 7ì, +does work W⁄a, and delivers heat Qs at 7¿, because W/qa = W1s — W2a, as one +can immediately show from the frst law, as follows: +Ma — W2 = (đi — Q3) — (Q2 — Qš) = Qìị — đà = Ma. (44.8) +W© can now obtain the laws which relate the eficiencies of the engines, because +there clearly must be some kind of relationship between the efficiencies of engines +running between the temperatures 71 and 7, and between 7¿ and 7, and +between 7] and 7. +--- Trang 798 --- +W©e can make the argument very clear in the following way: We have just +seen that we can always relate the heat absorbed at 71 to the heat delivered +at 1¿ by fñnding the heat delivered at some other temperature 7s. 'Pherefore +we can get all the enginesˆ properties iŸ we introduce a standard temperature, +analyzing everything with that standard temperature. In other words, iŸ we +knew the eficiency of an engine running between a certain temperature 7' and a +certain arbitrary standard temperature, then we could work out the efficiency +for any other diference in temperature. Because we assume we are using only +reversible engines, we can work from the initial temperature down to the standard +temperature and back up to the final temperature again. We shall defñne the +standard temperature arbitrarily as one đegree. We shall also adopt a special +symbol for the heat which is delivered at this standard temperature: we shall +call it @s. In other words, when a reversible engine absorbs the heat @Q at +temperature 7, it will deliver, at the unit temperature, a heat Qs. lƒ one engine, +absorbing heat‡ Q\ œ‡ Tì, dcliuers the heat Qs at one degree, and ¡ƒ an cngine +absorbing heat Q3 œ‡ temperoture Tạ tuilH also deliuer the sưme hea‡ Qs a‡ one +degree, then tt ƒollous that an engine tuhích œbsorbs heat Q1 a£ the temperature Tì +tuiil deliuer heat Q3 ÿ ít runs betUeen Tì ơnd Tạ, as we have already proved by +considering engines running between three temperatures. So all we really have +to do is to fnd how mụch heat Q¡ we need to put in at the temperature 71 in +order to deliver a certain amount of heat Qs at the unit temperature. lf we +discover that, we have everything. The heat Q, of course, is a function of the +temperature 7”. It is easy to see that the heat must increase as the temperature +increases, for we know that it takes work to run an engine backwards and deliver +heat at a higher temperature. ÏIt is also easy to see that the heat @Qị must be +proportional to Qs. So the great law ¡is something like this: for a given amount +of heat Qs delivered at one degree from an engine running at temperature 7 +degrees, the heat @ absorbed must be that amount Qs times some increasing +function of the temperature: +Q@=QsƒŒ). (44.9) +44-5 The thermodynamic temperature +At this stage we are not going to try to ñnd the formula for the above increasing +function of the temperature in terms of our familiar mercury temperature scale, +but instead te shall define temperature bụ œa neu scale. At one tỉme “the +temperature” was defned arbitrarily by dividing the expansion oŸ water into +--- Trang 799 --- +even degrees of a certain size. But when one then measures temperature with a +mmercury thermometer, one ñnds that the degrees are no longer even. But 2 +tue cøn rmake a definzlion oƒ temperature thích ¡is índependen‡ oƒ an particular +substance. We can use that function ƒ(7), which does not depend on what device +we use, because the efficiency of these reversible engines is independent of their +working substances. Since the function we found is rising with temperature, +we will defne the ƒfunction itselƒ as the temperature, measured in units of the +standard one-degree temperature, as follows: +Q= ST, (44.10) +Qs= 5:1. (44.11) +This means that we can tell how hot an object is by ñnding out how much heat +is absorbed by a reversible engine working between the temperature of the object +and the unit temperature (Fig. 44-9). IÝ seven times more heat is taken out +of a boiler than is delivered at a one-degree condenser, the temperature of the +boiler will be called seven degrees, and so forth. So, by measuring how much +heat is absorbed at diferent temperatures, we determine the temperature. 'Phe +temperature defñned in this way is called the aøbsolute thermodnamic temperature, +and ï§ is independent of the substance. We shall use this defñnition exclusively +from now on. +TTi.nnnnl +Reversible ° +"¬ š ¬" +@Qs=S-1° +22222221zzzz0\ +Fig. 44-9. Absolute thermodynamic temperature. +*W©e have previously defned our scale of temperature in a diferent way, namely by stating +that the mean kinetic energy of a molecule in a perfect gas is proportional to the temperature, +or that the perfect gas law says pV is proportional to 7. Is this new defnition equivalent? Yes, +since the final result (444.7) derived from the gas law is the same as that derived here. We shall +discuss this point again in the next chapter. +--- Trang 800 --- +Now we see that when we have two engines, one working between 71 and one +degree, the other working between 72 and one degree, delivering the same heat +at unit temperature, then the heats absorbed must be related by +II Ss= Tp” (44.12) +But that means that if we have a single engine running between 71 and 7, then +the result of the whole analysis, the grand fñnale, is that Q) 1s to 71 as Q2 1s +to Tạ, 1ƒ the engine absorbs energy Qị at temperature 71 and delivers heat Q2 at +temperature 7¿. Whenever the engine is reversible, this relationship between the +heats must follow. 'Phat is all there is to it: that is the center of the universe of +thermodynamics. +TÍ this is all there is to thermodynamies, why is it considered such a dificult +subject? In doïng a problem involving a given mass of some substance, the +condition of the substance at any moment can be described by telling what its +temperature is and what its volume is. IÝ we know the temperature and volume of +a substanee, and that the pressure is some function of the temperature and volume, +then we know the internal energy. One could say, “[ do not want to do it that way. +'Tell me the temperature and the pressure, and I will tell you the volume. Ï can +think of the volume as a function of temperature and pressure, and the internal +energy as a function of temperature and pressure, and so on.” 'Phat is why thermo- +dynamics is hard, because everyone uses a dilerent approach. TỶ we could only sit +down once and decide on our variables, and stick to them, it would be fairly easy. +Now we start to make deductions. Just as # = ma is the center of the +universe in mechanies, and it goes on and on and on after that, in the same way +the principle just found is all there is to thermodynamies. But can one make +conclusions out of it? +We begin. 'To obtain our firs§ conclusion, we shall combine both laws, the +law of conservation of energy and this law which relates the heats Qa and }), +and we can easily obtain the effficiencw oƒ a reuersible engine. From the ñrst law, +we have W = Qq — Qs. According to our new principle, +Q›= _ Gì, +so the work becomes +W =@\) (:- T.) = Gì TT (44.13) +--- Trang 801 --- +which tells us the eficiency of the engine—how much work we get out oŸ so +much heat. 'Phe efficiency of an engine is proportional to the diference in the +temperatures between which the engine runs, divided by the higher temperature: +Efficiency = T5, (44.14) +The eficiency cannot be greater than unity and the absolute temperature cannot +be less than zero, absolute zero. So, since 72 must be positive, the efficiency is +always less than unity. That is our frst conclusion. +44-6 Entropy +Equation (44.7) or (44.12) can be interpreted in a special way. Working always +with reversible engines, a heat Q at temperature 71 is “equivalent” to Qs at 72 1Ÿ +Q1/Tì = Q2/a, in the sense that as one is absorbed the other is delivered. This +suggests that if we call Q/7' something, we can say: in a reversible process as +mụch Q/ is absorbed as is liberated; there is no gain or loss of Q/7' This Q/T +1s called enfrop, and we say “there is no net change in entropy in a reversible +cycle” Tf 7 is 1°, then the entropy is Qs/1° or, as we symbolized it, Qs/1° = ®. +Actually, Š is the letter usually used for entropy, and it is numerically equal to +the heat (which we have called Qs) delivered to a 19-reservoir (entropy is not +1tself a heat, ¡ít ¡is heat divided by a temperature, hence it is measured in 7oulÌes +per' degree). +Now it is interesting that besides the pressure, which is a function of the +temperature and the volume, and the internal energy, which is a function of +temperature and volume, we have found another quantity which is a function +of the condition, 1.e., the entropy of the substance. Let us try to explain how +we compute i%, and what we mean when we call it a “function of the condition.” +Consider the system in t©wo diferent conditions, mụch as we had in the experiment +where we did the adiabatic and isothermal expansions. (Incidentally, there is +no need that a heat engine have only two reservoirs, it could have three or four +diferent temperatures at which it takes in and delivers heats, and so on.) We +can move around on a pV diagram all over the place, and go from one condition +to another. In other words, we could say the gas is in a certain condition ø, +and then it goes over to some other condition, ð, and we will require that this +transition, made from ø to b, be reversible. Now suppose that all along the path +from ø to b we have little reservoirs at diferent temperatures, so that the heat đQ +--- Trang 802 --- +removed from the substance at each little step is delivered to each reservoir at +the temperature corresponding to that point on the path. 'Then let us connect +all these reservoirs, by reversible heat engines, to a single reservoir at the unit +temperature. When we are fñnished carrying the substance from œø to b, we shall +bring all the reservoirs back to their original condition. Any heat đ@ that has +been absorbed from the substance at temperature 7' has now been converted by +a reversible machine, and a certain amount of entropy đŠ has been delivered at +the unit temperature as follows: +dđS = dQƒ/T. (44.15) +Let us compute the total amount of entropy which has been delivered. The +entropy difÑference, or the entropy needed to go from œ to b by this particular +reversible transformation, is the total entropy, the total of the entropy taken out +of the little reservoirs, and delivered at the unit temperature: +Sy — 8, = J do, (44.16) +The question is, does the entropy difÑference depend upon the path taken? 'There +is more than one way to go from ø to 0. Remember that in the Carnot cycle +we could go from œ to cin Eig. 44-6 by frst expanding isothermally and then +adiabatically; or we could first expand adiabatically and then isothermally. So +the question is whether the entropy change which occurs when we go from ø to b +in Eig. 44-10 is the same on one route as it is on another. Ï must be the sưme, +§ C Reservoirs +L. a © @) +ẹ : đ CG€) +aw~T[ ]L ]L ]LILIL ]|Engines +dS-1° +Volume +Fig. 44-10. Change in entropy during a reversible transformation. +--- Trang 803 --- +because if we went all the way around the cycle, goïng forward on one path and +backward on another, we would have a reversible engine, and there would be no +loss of heat to the reservoir at unit temperature. In a totally reversible cycle, no +heat must be taken from the reservoir at the unit temperature, so the entropy +needed to go from ø to b is the same over one path as it is over another. Ït is +tndependent oƒ pa‡h, and depends only on the endpoints. We can, therefore, say +that there is a certain function, which we call the entropy of the substance, that +depends only on the condition, ï.e., only on the volume and temperature. +We can fnd a function Š(V, 7) which has the property that iŸ we compute +the change in entropy, as the substance is moved along any reversible path, In +terms of the heat rejected at unit temperature, then +AS= | —,, 44.17 +l1 (1417) +where đ@) ¡is the heat removed from the substance at temperature 7'. 'Phis total +entropy change is the diference between the entropy calculated at the initial and +fnal points: +AS = 50W, Ti) — S(Wà, Tạ) = Ln (44.18) +This expression does not completely defñne the entropy, but rather only the +điference of entropy between two diferent conditions. Ônly if we can evaluate +the entropy for one special condition can we really deñne Š absolutely. +For a long time it was believed that absolute entropy meant nothing——that +only diferences could be deñned——but fñnally Nernst proposed what he called the +AS=S¿— S; +Ẹ AS = S— S, +° +Total Entropy Change = 0 +Volume +Fig. 44-11. Change in entropy in a completely reversible cycle. +--- Trang 804 --- +heat theorem, which 1s also called the third law of thermodynamics. Ït is very +simple. We will say what it is, but we will not explain why it is true. Nernst”s +postulate states simply that the entropy of any object at absolute zero is zero. +W©e know of one case of 7' and V, namely 7' = 0, where ®Š is zero; and so we can +get the entropy at any other point. +To give an illustration of these ideas, let us calculate the entropy of a perfect +gas. In an isothermal (and therefore reversible) expansion, ƒ đQ/7 is Q/T, since +T is constant. Therefore (rom 41.4) the change in entropy is +5(1„,7) — 5(W,.T) = Nkm +so S(VW, 7) = NklnV plus some function of 7 only. How does Š depend on 7? +We know that for a reversible adiabatic expansion, no hea‡ ¡s czchangcd. Thus +the entropy does not change even though W changes, provided that 7' changes +also, such that TV~! = constant. Can you see that this implies that +5(V,T) = Nk|nV.+ —. +a, +where ø is some constant independent of bo£h V and 7”? [a is called the chemical +constant. It depends on the gas in question, and may be determined experimen- +tally rom the Nernst theorem by measuring the heat liberated in cooling and +condensing the gas until it is brought to a solid (or for helium, a liquid) at 09, by +integrating Í dQ/T. It can also be determined theoretically by means of Planck's +constant and quantum mechanics, but we shall not study it in this course.] +Now we shall remark on some of the properties of the entropy of things. We +ñrst remember that if we go along a reversible cycle from ø to Ò, then the entropy +of the substance will change by 5;— %„. And we remember that as we go along the +path, the entropy——the heat delivered at unit temperature——increases according +to the rule để = dQ/T, where đ@ is the heat we remove from the substance +when its temperature is 7. +W©e already know that if we have a reversible cwcle, the total entropy of +everything is not changed, because the heat Q+ absorbed at 71 and the heat Qs +delivered at 72 correspond to equal and opposite changes in entropy, so that the +net change in the entropy is zero. So for a reversible cycle there is no change +in the entropy of anything, including the reservoirs. This rule may look like the +conservation of energy again, but it is not; it applies only to reversible cycles. lỶ +we include irreversible cycles there is no law of conservation of entropy. +--- Trang 805 --- +W© shall give two examples. Eirst, suppose that we do irreversible work on +an object by friction, generating a heat @ on some object at temperature 7'. The +entropy is increased by Q/7'. The heat Q is equal to the work, and thus when +we do a certain amount of work by fiction against an object whose temperature +1s 7, the entropy of the whole world increases by W/T. +Another example of irreversibility is this: If we put together two objects that +are at diferent temperatures, say 71 and 7ÿ, a certain amount of heat will ow +from one to the other by itself. Suppose, for instance, we put a hot stone in cold +water. Then when a certain heat AQ) is transferred from 71 to 7¿, how much +does the entropy of the hot stone change? It decreases by AQ/71. How much +does the water entropy change? It increases by AQ/7;. The heat will, of course, +fow only from the higher temperature 71 to the lower temperature 72, so that +AC) is positive 1Ÿ T71 is greater than 7¿. So the change in entropy of the whole +world is positive, and ït is the diference of the two fractions: +A@ _ AQ +AS= mm xa (44.19) +So the following proposition 1s true: in any process that is irreversible, the +entropy of the whole world is increased. Only in reversible processes does the +entropy remain constant. Since no process is absolutely reversible, there is always +at least a small gain in the entropy; a reversible process is an idealization In +which we have made the gain oŸ entropy minimail. +Unfortunately, we are not going to enter into the ñeld of thermodynamics very +far. Qur purpose is only to illustrate the principal ideas involved and the reasons +why it is possible to make such arguments, but we will not use thermodynamics +very much in this course. Thermodynamics is used very often by engineers and, +particularly, by chemists. So we must learn our thermodynamics in practice in +chemistry or engineering. Because it is not worthwhile duplicating everything, +we shall ]ust give some discussion of the origin of the theory, rather than much +detail for special applications. +The two laws of thermodynamics are often stated this way: +kirst lau: — the energy oŸ the universe is always constant. +Second lau: the entropy of the universe is always increasing. +'That is not a very good statement of the second law; it does not say, for example, +that in a reversible cycle the entropy stays the same, and i% does not say exactly +--- Trang 806 --- +what the entropy is. It is just a clever way of remembering the two laws, but it +does not really tell us exactly where we stand. We have summarized the laws +discussed in this chapter in Table 44-1. In the next chapter we shall apply these +laws to discover the relationship between the heat generated in the expansion of +a rubber band, and the extra tension when it is heated. +Table 44-1 +Summary of the laws of thermodynamics +trst lau: +Heat put into a system + Work done on a system = Increase in internal energy of the +system: +đQ + ädW = dU. +Second lau: +A process whose on net result is to take heat from a reservoir and convert i% %o +work is impossible. +No heat engine taking heat Q1 from 7: and delivering heat €Qs at 75 can do more +work than a reversible engine, for which +Tì — 1: +w=oico=o( TP), +The entrop oƒ a sụstem ¡s defincd this t0ag: +(a) If heat AQ) ¡is added reversibly to a system at temperature 7', the increase in +entropy of the system is A9 = AQ/T. +(b) At T=0, S=0 (th¿rd lau). +In a reuersible change, the total entropy of all parts of the system (including reservoirs) +does not change. +In ?rreuersible changqe, the total entropy of the system always increases. +--- Trang 807 --- +Miltarsfrcrfforts ©Ÿ Thhor'rttodÏggrterrttfc-S +45-1 Internal energy +'Thermodynamies is a rather dificult and complex subJect when we come to +apply it, and it is not appropriate for us %o go very far into the applications +in this course. The subject is of very great importance, oŸ course, to engineers +and chemists, and those who are interested in the subject can learn about the +applications in physical chemistry or in engineering thermodynamics. 'Phere +are also good equation reference books, such as Zemansky's Heø‡ ønd Thermo- +đựngmics, where one can learn more about the subject. In the Eneyclopedia +Britannica, fourteenth edition, one can find excellent articles on thermodynamics +and thermochemistry, and in the article on chemistry, the sections on physical +chemistry, vaporization, liquefication of gases, and so on. +The subject of thermodynamics is complicated because there are so many +difÑferent ways of describing the same thing. If we wish to describe the behavior +of a gas, we can say that the pressure depends on the temperature and on the +volume, or we can say that the volume depends on the temperature and the +pressure. Ôr with respect to the internal energy Ứ, we might say that it depends +on the temperature and volume, If those are the variables we have chosen——but +we might also say that it depends on the temperature and the pressure, or the +pressure and the volume, and so on. In the last chapter we discussed another +function of temperature and volume, called the entropy Š, and we can of course +construct as many other functions of these variables as we like: U — 15 is a +function of temperature and volume. 5o we have a large number of diferent +quantities which can be functions of many diferent combinations of variables. +To keep the subject simple in this chapter, we shall decide at the start +to use #emperøture and 0uolưmne as the independent variables. Chemists use +temperature and pressure, because they are easier to measure and control in +chemical experiments, but we shall use temperature and volume throughout this +--- Trang 808 --- +chapter, except in one place where we shall see how to make the transformation +into the chemists” system of variables. +We shall frst, then, consider only one system of independent variables: tem- +perature and volume. Secondly, we shall diseuss only two dependent functions: +the internal energy and the pressure. All the other functions can be derived +from these, so it is no necessary to discuss them. With these limitations, +thermodynamics is still a fairly difficult subject, but it is not quite so impossiblel +Jirst we shall review some mathematics. IÝ a quantity is a function of two +variables, the idea of the derivative of the quantity requires a little more careful +thought than for the case where there is only one variable. What do we mean +by the derivative of the pressure with respect to the temperature? The pressure +change accompanying a change in the temperature depends partÌy, of course, on +what happens to the øolưmne while T' is changing. We must specify the change +in V before the concept of a derivative with respect to 7' has a precise meaning. +W©e might ask, for example, for the rate of change of with respect to 7' If V is +held constant. 'This ratio is just the ordinary derivative that we usually write +as đP/đT. We customarily use a special symbol, ØP/ØT, to remind us that P +depends on another variable V as well as on 7', and that this other variable 1s +held constant. We shall not only use the symbol Ø to call attention to the fact +that the other variable is held constant, but we shall also write the variable that +is held constant as a subscript, (ØP/ØT)v. Since we have only two independent +variables, this notation is redundant, but ¡§ will help us keep our wits about us +in the thermodynamie jungle of partial derivatives. +Let us suppose that the funection ƒ(z,) depends on the two independent +variables z and . By (؃/Øz)„ we mean simply the ordinary derivative, obtained +in the usual way, If we treat as a constant: +(7) "¬-...ˆ....J] +Øzjy„ Az>0 Am +Similarly, we defñne +(5) — mi đŒ‹9 + Â) - ƒŒ, 0). +Øy œ Au->0 Aw +For example, if ƒ(z,) = #” + z, then (؃/9z)„ = 2z + , and (0ƒ /Ø)„ = #. +W© can extend this idea to higher derivatives: Ø2ƒ/Ø2 or 92ƒ/Øwôz. The latter +symbol indicates that we frst diferentiate ƒ with respect to ø, treating as a +--- Trang 809 --- +constant, then diferentiate the result with respect to , treating ø as a constant. +The actual order of diferentiation is immaterial: Ø®ƒ/0zØy = 6?ƒ/0yÔz. +We will need to compute the change Aƒ in ƒ(z,) when + changes to # + Az +ơnd changes to + Ay. WSe assume throughout the following that Az and A +are infñnitesimally small: +Aƒ =ƒ(+ Az,w+ Aw) — ƒ(œ,w) += ƒ( + Az, + Auw) ~ ƒ(,w + Au) + ƒ(œ,w + Au) — ƒ(œ,9) +"————_———- ——— += Azl== Av| == 45.1 +đu, to SỂn, 080 +The last equation is the fundamental relation that expresses A ƒ in terms of Az +and A#. +As an example of the use of this relation, let us calculate the change in the +internal energy U(7, V) when the temperature changes from 7' to 7+ AT and +the volume changes from V to V + AV. Using Eq. (45.1), we write +AU=A7| = AV[—]- 45.2 +l6), *^Y Ấn), 2 +In our last chapter we found another expression for the change AU in the internal +energy when a quantity of heat AQ) was added to the gas: +AU = AQ— PAY. (45.3) +In comparing Eqs. (45.2) and (45.3) one might at fñrst be inclined to think that +?P=—(0U/9V}r, but this is not correct. 'To obtain the correct relation, let us frst +suppose that we add a quantity of heat AQ to the gas while keeping the volume +constant, so that AV =0. With AV =0, Ea. (45.3) tells us that AU = AQ@Q, +and Eaq. (45.2) tells us that AU = (9U/ØT)v AT, so that (9U/9T)v = AQ/AT. +The ratio AQ/A 7, the amount of heat one must put into a substance in order +to change its temperature by one degree with the volume held constant, is called +the specific heut‡ aL constant 0olưme and is designated by the symbol Cự. By this +argument we have shown that +—=| =tw. 45.4 +(ốm), = " +--- Trang 810 --- +VOLUME +Fig. 45-1. Pressure-volume diagram for a Carnot cycle. The curves +marked T and 7 — AT are Isothermail lines; the steeper curves are +adiabatic lines. AV ¡is the volume change as heat AQ ¡is added to the +gas at constant temperature 7. AP is the pressure change at constant +volume as the gas temperature is changed from ï to Ï — AT. +Now let us again add a quantity of heat AQ) to the gas, but this time we +will hold 7' constant and allow the volume to change by AV. The analysis in +this case is more complex, but we can calculate AU by the argument of Carnot, +mmaking use of the Carnot cycle we introduced in the last chapter. +The pressure-volume diagram for the Carnot cycle is shown in EFig. 45-1. As +we have already shown, the total amount of work done by the gas in a reversible +cycle is AQ(A7/7), where AQ) is the amount of heat energy added to the gas +as iÿ expands isothermally at temperature ?' from volume W to V + AV, and +T7 — AT is the fñnal temperature reached by the gas as it expands adiabatically +on the second leg of the cycle. Now we will show that this work done is also +given by the shaded area in Fig. 45-1. In any circumstances, the work done by +the gas is ƒ PđV, and is positive when the gas expands and negative when the +gas is compressed. If we plot vs. V, the variation of P and V is represented +by a curve which gives the value of corresponding to a particular value of W. +As the volume changes from one value to another, the work done by the gas, +the imtegral ƒ PdV, is the area under the curve connecting the imnitial and final +values of V. When we apply this idea to the Carnot cycle, we see that as we go +around the cycle, paying attention to the sign of the work done by the gas, the +net work done by the gas is just the shaded area in Eig. 45-1. +Now we want to evaluate the shaded area geometrically. “The cycle we +have used in Fig. 45-1 difers from that used in the previous chapter in that +we now suppose that A7 and AÁQ) are infinitesimally small. We are working +--- Trang 811 --- +Fig. 45-2. Shaded area = area enclosed by dashed lines = area of +rectangle = AP AV. +between adiabatic lines and isothermail lines that are very close together, and +the fñgure described by the heavy lines in FEig. 45-1 will approach a parallelogram +as the increments A7' and A@ approach zero. The area of this parallelogram is +just AV AP, where AV ¡ïs the change in volume as energy A@) ¡s added to the gas +at constant temperature, and ẤP ïs the change in pressure as the temperature +changes by AT at constant volume. Ône can easily show that the shaded area +in Eig. 45-1 is given by AV AP by recognizing that the shaded area is equal to +the area enelosed by the dotted lines in Eig. 45-2, which in turn differs from the +rectangle bounded by A?P and AV only by the addition and subtraction of the +cqual triangular areas in Fig. 45-2. +Now let us summarize the results of the arguments we have developed so Íar: +Work done by the gas = shaded area = AV AP = AQ TT +ra (heat needed to change V by AV}kenstant +(45.5) += AV - (change in P when 7 changes by AT }eonstant V +AV (heat needed to change V by AV)+ = T(0P/ØT)v. +Equation (45.5) expresses the essential result of Carnot's argument. The whole +of thermodynamics can be deduced from Ea. (45.5) and the Eirst Law, which is +siated in Eq. (45.3). Equation (45.5) is essentially the Second Law, although it +--- Trang 812 --- +was originally deduced by Carnot in a slightly diferent form, since he did not +use our defñnition of temperature. +Now we can proceed to calculate (ØU/9V)+. By how much would the internal +energy change if we changed the volume by AV? Eirst, U changes because +heat is put in, and second, Ứ changes because work is done. 'Phe heat put ín is +AQ=7| — | AV, +9=r(ñn) vo +according to Eq. (45.5), and the work done on the substance is —? AVW. Therefore +the change AU ïn internal energy has two pieces: +AU=T| — ) AV_- PAYV. (45.6) +Dividing both sides by AVW, we fnd for the rate of change of Ư with V at +constant T7" aU ạP +—_ =7T| — —P. 45.7 +(ấn), = TẲm), “ +In our thermodynamics, in which 7' and W are the only variables and and U +are the only functions, Eqs. (45.3) and (45.7) are the basic equations from which +all the results of the subject can be deduced. +45-2 Applications +Now let us discuss the meaning of Eq. (45.7) and see why i% answers the +questions which we proposed in our last chapter. We considered the following +problem: in kinetie theory it is obvious that an increase in temperature leads +to an increase in pressure, because of the bombardments of the atoms on a +piston. For the same physical reason, when we let the piston move back, heat +1s taken out of the gas and, in order to keep the temperature constant, heat +will have to be put back in. The gas cools when it expands, and the pressure +rises when iÈ is heated. “There must be some connection between these two +phenomena, and this connection is given explicitly in Bq. (45.7). IÝ we hold the +volume fxed and increase the temperature, the pressure rises at a rate (ØP/ØT)v. +Related to that fact is this: iŸ we increase the volume, the gas wiïll cool unless we +pour some heat in to maintain the temperature constant, and (ØU/ØV)+ tells +us the amount oŸ heat needed to maintain the temperature. Equation (45.7) +--- Trang 813 --- +expresses the fundamental interrelationship between these two efects. 'Phat is +what we promised we would fnd when we came to the laws of thermodynamics. +'Without knowing the internal mechanism of the gas, and knowing only that we +cannot make perpetual motion of the second type, we can deduce the relationship +between the amount of heat needed to maintain a constant temperature when +the gas expands, and the pressure change when the gas is heated at constant +volumel +Now that we have the result we wanted for a gas, let us consider the rubber +band. When we stretch a rubber band, we find that i%s temperature rises, and +when we heat a rubber band, we find that it pulls itselfin. What is the equation +that gives the same relation for a rubber band as Eq. (45.3) gives for gas? Eor a +rubber band the situation will be something like this: when heat A@) is put in, +the internal energy is changed by AU and some work is done. The only diference +will be that the work done by the rubber band is —#` AL instead of P AV, where +t' is the force on the band, and E is the length of the band. The force #! is a +function oŸ temperature and of length of the band. Replacing P AV in Ea. (45.3) +by —FAL, we get +AU = AQ+ FAL. (45.8) +Comparing Eqs. (45.3) and (45.8), we see that the rubber band equation is +obtained by a mere substitution of one letter for another. Furthermore, IŸ we +substitute Ù for V, and —F' for ?, all of our discussion of the Carnot cycle +applies to the rubber band. We can immediately deduce, for instance, that the +heat AQ) needed to change the length by AT is given by the analog to Eq. (45.5): +AQ=-—7T(ð0F/ØT)„AL. Thịs equation tells us that iŸ we keep the length of a +rubber band fxed and heat the band, we can calculate how much the force will +increase in terms of the heat needed to keep the temperature constant when the +band is relaxed a little bít. So we see that the same equation applies to both gas +and a rubber band. In fact, if one can write AU = AQ+ AAĐ?, where A and +represent diferent quantities, force and length, pressure and volume, etc., one can +apply the results obtained for a gas by substituting 4 and Ö for —P and V. Eor +example, consider the electric potential diference, or “voltage,” # in a battery +and the charge AZ that moves through the battery. We know that the work done +in a reversible electric cell, like a storage battery, is AZ. (Since we include +no PAV term in the work, we require that our battery maintain a constant +volume.) Let us see what thermodynamics can tell us about the performance of +--- Trang 814 --- +a battery. IÝ we substitute for P and Z for V in Eq. (45.6), we obtain +AU 8E +^z= Ty), 1. (45.9) +Equation (45.9) says that the internal energy is changed when a charge AZ +moves through the cell. Why is AU/AZ not simply the voltage of the battery? +(The answer is that a real battery gets warm when charge moves through the +cell. The internal energy of the battery is changed, frst, because the battery did +some work on the outside circuit, and second, because the battery is heated.) +The remarkable thing is that the second part can again be expressed in terms +of the way in which the battery voltage changes with temperature. Incidentally, +when the charge moves through the cell, chemical reactions occur, and Eq. (45.9) +suggests a nifty way of measuring the amount oŸ energy required to produce +a chemical reaction. All we need to do is construct a cell that works on the +reaction, measure the voltage, and measure how much the voltage changes with +temperature when we draw no charge from the batteryl +Now we have assumed that the volume of the battery can be maintained +constant, since we have omitted the P AV term when we set the work done by the +battery equal to #ZAZ. It turns out that it is technically quite dificult to keep +the volume constant. It is much easier to keep the cell at constant atmospheric +pressure. Eor that reason, the chemists do not like any of the equations we +have written above: they prefer equations which describe performance under +constant pressure. We chose at the beginning of this chapter to use V and 7 as +independent variables. 'Phe chemists prefer and 7, and we will now consider +how the results we have obtained so far can be transformed into the chemists' +system of variables. Remember that in the following treatment confusion can +easily set in because we are shifting gears from 7' and V to 7' and P. +We started in Eq. (45.3) with AU = AQ— PAV; PAV may be replaced by +EAZ or AAPB. T we could somehow replace the last term, PĐAV, by VAP, +then we would have interchanged W and ?, and the chemists would be happy. +Well, a clever man noticed that the diferential of the product PV is d(PV) = +PdV +VäảP, and ïf he added this equality to Ba. (45.3), he obtained +A(PV)=PAV+VAP +AU=AQ —- PAV +A(U+PV)=AQ ~+VAP +--- Trang 815 --- +In order that the result look like Eq. (45.3), we define Ứ + PV to be something +new, called the en£halpụ, H, and we write AH = AQ + V AP. +Now we are ready to transform our results into chemists' language with the +following rules: —›> H, P — —V, V P. For example, the fundamental +relationship that chemists would use instead of Bq. (45.7) is +9H ØV +—c= | =-T|—] +. +(õn),— TẤm), +Tt should now be clear how one transforms to the chemists' variables 7' and P. +We now go back to our original variables: for the remainder of this chapter, 7 +and V are the independent variables. +Now let us apply the results we have obtained to a number of physical +situations. Consider frst the ideal gas. From kinetic theory we know that the +internal energy of a gas depends only on the motion of the molecules and the +number oŸ molecules. 'Phe internal energy depends on 7', but not on V. If we +change W, but keep 7' constant, Ứ is not changed. Therefore (9U/ØV)+ =0, and +Eq. (45.7) tells us that for an ideal gas +7T] —-P-=(0. 45.10 +Vốn), ga +Equation (45.10) is a diferential equation that can tell us something about P. +W© take account oÊ the partial derivatives in the following way: 5ince the partial +derivative is at constant V, we will replace the partial derivative by an ordinary +derivative and write explicitly, to remind us, “constant V7” Equation (45.10) then +becomes AP +T AT” P=\0; const V, (45.11) +which we can integrate to get +ln? = ln7' + const; const V, +P = const x 7) const V, (45.12) +W©e know that for an ideal gas the pressure per mole is equal to +P=— 45.13 +m (45.13) +--- Trang 816 --- +which is consistent with (45.12), since V and ?? are constants. Why did we +bother to go through this calculation 1ƒ we already knew the results? Because +we have been using #o ?ndependent defnitions oƒ temperaturel At one stage +we assumed that the kinetic energy of the molecules was proportional to the +temperature, an assumption that defñnes one scale of temperature which we will +call the ideal gas scale. The 7'in Eq. (45.13) is based on the gas scale. WWe +also call temperatures measured on the gas scale k¿netic temperatures. Later, +we deñned the temperature in a second way which was completely independent +of any substance. From arguments based on the Second Law we defined what +we might call the “grand thermodynamie absolute temperature” 7', the T' that +appears in Eq. (45.12). What we proved here is that the pressure of an ideal gas +(defined as one for which the internal energy does not depend on the volume) is +proportional to the grand thermodynamic absolute temperature. We also know +that the pressure is proportional to the temperature measured on the gas scale. +Therefore we can deduce that the kinetic temperature is proportional to the +“grand thermodynamie absolute temperature.” 'Phat means, of course, that iŸ we +were sensible we could make two scales agree. In this instance, at least, the two +scales hœue been chosen so that they coincide; the proportionality constant has +been chosen to be 1. Most of the time man chooses trouble for himself, but in +this case he made them equall +45-3 The Clausius-Clapeyron equation +The vaporization of a liquid is another application of the results we have +derived. Suppose we have some liquid in a cylinder, such that we can compTress +it by pushing on the piston, and we ask ourselves, “If we keep the temperature +constant, how does the pressure vary with volume?” In other words, we want +to draw an isothermail line on the P-V diagram. The substance in the cylinder +1s not the ideal gas that we considered earlier; now it may be in the liquid or +the vapor phase, or both may be present. If we apply suficient pressure, the +substance will eondense to a liquid. Now 1Ý we squeeze still harder, the volume +changes very little, and our isothermal line rises rapidly with decreasing volume, +as shown at the left in Eig. 45-3. +TÝ we increase the volume by pulling the piston out, the pressure drops until +we reach the point at which the liquid starts to boïl, and then vapor starts to form. +TÝ we pull the piston out farther, all that happens is that more liquid vaporizes. +'When there is part liquid and part vapor in the cylinder, the two phases are in +--- Trang 817 --- +5 LIQUID ` +ỡ AND VAPOR T - AT +¬ VAPOR +VOLUME +Fig. 45-3. lsothermal lines for a condensable vapor compressed in a +cylinder. At the left, the substance ¡s in the liquid phase. At the right, +the substance ¡is vaporized. In the center, both liquid, and vapor are +present in the cylinder. +ứ AE T +S5 T—ÁT +VOLUME +Fig. 45-4. Pressure-volume diagram for a Carnot cycle with a con- +densable vapor ¡in the cylinder. At the left, the substance ¡s in the liquid +state. A quantity of heat L ¡s added at temperature 7 to vaporize the +liquid. The vapor expands adiabatically as changes to ï — AT. +--- Trang 818 --- +equilibriunm——liquid is evaporating and vapor is condensing at the same rate. If +we make more room for the vapor, more vapor is needed to maintain the pressure, +so a little more liquid evaporates, but the pressure remains constant. Ôn the at +part of the curve in Fig. 45-3 the pressure does not change, and the value of the +pressure here is called the 0apor pressure d‡ temperature T'. As we continue to +increase the volume, there comes a time when there is no more liquid to evaporate. +At this juncture, if we expand the volume further, the pressure will fall as for +an ordinary gas, as shown at the right of the P-W diagram. The lower curve in +Jig. 45-3 is the isothermal line at a slightly lower temperature 7'— A7. The +pressure in the liquid phase ¡is slightly reduced because liquid expands with an +increase in temperature (for most substances, but not for water near the Íreezing +point) and, of course, the vapor pressure is lower at the lower temperature. +We will now make a cycle out of the ©wo isothermal lines by connecting +them (say by adiabatic lines) at both ends of the upper flat section, as shown in +Hig. 45-4. We are going to use the argument of Carnot, which tells us that the +heat added to the substance in changing it from a liquid to a vapor is related +to the work done by the substance as it goes around the cycle. Let us call U +the heat needed to vaporize the substance in the cylinder. Äs in the argument +immediately preceding Eq. (45.5), we know that L(A7/T) = work done by the +substance. As before, the work done by the substance is the shaded area, which is +approximately AP(W&q — Vạ), where AP ïs the diference in vapor pressure at the +two temperatures 7' and 7'— A7", V@œ is the volume of the gas, and V}, is the volume +of the liquid, both volumes measured at the vapor pressure at temperature 7'. +Setting these t©wo expressions for the area equal, we get AT '/T = AP(V&œ— Vp), +ữ ØĐa,/ØT 45.14 +T(Ves — Vr) ~= ( vap/ )- ( , ) +Equation (45.14) gives the relationship between the rate of change of vapor +pressure with temperature and the amount of heat required to evaporate the +liquid. Thịis relationship was deduced by Carnot, but ït is called the Clausius- +Clapeyron equation. +Now let us compare Eq. (45.14) with the results deduced from kinetic theory. +Usually V@œ is very mụuch larger than Vạ,. So Vœ — Vụ Veœ = RT/P per mole. lỶ +we further assume that Ù is a constant, independent of temperature—notf a very +øood approximation——then we would have ÔP/Ø7' = L/(RT2/P). The solution +--- Trang 819 --- +of this diferential equation is +P= conste-1/RT, (45.15) +Let us compare this with the pressure variation with temperature that we deduced +earlier from kinetic theory. Kinetic theory indicated the possibility, at least +roughly, that the number of molecules per unit volume of vapor above a liquid +would be +n = ly} menu (45.16) +where ỨỮc — y, is the internal energy per mole in the gas minus the internal +energy per mole in the liquid, i.e., the energy needed to vaporize a mole of +liquid. Equation (45.15) from thermodynamics and Bq. (45.16) from kinetie +theory are very closely related because the pressure is nk”', but they are not +exactly the same. However, they will turn out to be exactly the same if we +assume Ữa — , = const, instead of Ù = const. If we assume Ứœ — y = const, +independent of temperature, then the argument leading to Eq. (45.15) will +produce Eq. (45.16). Since the pressure is constant while the volume is changing, +the change in internal energy Ưœ — y, is equal to the heat Ù put in minus the +work done P(W&q — Vạ), so b = (Ueœ + PVS) — (Úr + PV¡). +This comparison shows the advantages and disadvantages of thermodynamics +over kinetic theory: Eirst of all, Eq. (45.14) obtained by thermodynamiecs is +exact, while Eq. (45.16) can only be approximated, for instance, iŸ U is nearly +constant, and if the model is right. Second, we may not understand correctly +how the gas goes into the liquid; nevertheless, Eq. (45.14) is right, while (45.16) +1s only approximate. Third, although our treatment applies to a gas condensing +into a liquid, the argument is true for any other change of state. Eor instance, +the solid-to-liquid transition has the same kind of curve as that shown in Figs. +45-3 and 45-4. Introducing the latent heat for melting, ÄMƒ/mole, the formula +analogous to Eq. (45.14) then is (Ømex/Ø1)v = Mf/[T(Via — Vsona)]. Although +we may not understand the kinetic theory of the melting process, we nevertheless +have a correct equation. However, when we cønw understand the kinetic theory, +we have another advantage. Equation (45.14) is only a diferential relationship, +and we have no way of obtaining the constants of integration. In the kinetic +theory we can obtain the constants also if we have a good model that describes +the phenomenon completely. So there are advantages and disadvantages to +cach. When knowledge is weak and the situation is complicated, thermodynamic +--- Trang 820 --- +relations are really the most powerful. When the situation is very simple and a +theoretical analysis can be made, then it is better to try to get more inÍormation +from theoretical analysis. +One more example: blackbody radiation. We have discussed a box containing +radiation and nothing else. We have talked about the equilibrium bebween the +oscillator and the radiation. We also found that the photons hitting the wall of +the box would exert the pressure , and we found P?V = U/3, where Ù is the +total energy of all the photons and V is the volume of the box. If we substitute +U =3äPV in the basic Eq. (45.7), we fndÝ +9U ØP +tt =.." P. (45.17) +Since the volume of our box is constant, we can replace (ØP/ØT)v by đP/đT' to +obtain an ordinary diferential equation we can integrate: ln = 4ln7 + const, +or P = const x 7. The pressure of radiation varies as the fourth power of the +temperature, and the total energy density of the radiation, U/V = 3P, also +varies as 7%, It is usual to write U/W = (4ø/e)T'*, where e is the speed of light +and ø is called the S0efan-Bollzmann constant. TW is not possible to get ơ from +thermodynamies alone. Here is a good example of its power, and its limitations. +To know that U/V goes as 7 is a great deal, but to know how big U/V actually +1s at any temperature requires that we go into the kind of detail that only a +complete theory can supply. Eor blackbody radiation we have such a theory and +we can derive an expression for the constant øơ in the following manner. +Let I(œ) d¿ be the intensity distribution, the energy fow through 1 m2 +in one second with fequency between œ and œ + dư. The energy density +distribution = energy/volume = Ï(œ) dư/c is +U : +VỀ total energy density += J energy density between œ and œ + đư +* In this case (ØP/Ø8V)z~ = 0, because in order to keep the oscillator in equilibrium at a +given temperature, the radiation in the neighborhood of the oscillator has to be the same, +regardless of the volume of the box. 'Phe total quantity of photons inside the box must therefore +be proportional to i%s volume, so the internal energy per unit volume, and thus the pressure, +đdepends only on the temperature. +--- Trang 821 --- +— II ® T(œ) dụ +=Í —- +trom our earlier discussions, we know that +1(U) = -ss>.rm: +T2c2(ch2/ET — 1) +Substituting this expression for Ï(œ) in our equation for U/V, we get +U — 1 ®% he dụ +V_ x2cồ .-...m +If we substitute œ = hư/kT, the expression becomes +U _ (k7)? Ẻ + dạ +V_ h3m?2 js c®—1' +This integral is just some number that we can get, approximately, by drawing +a curve and taking the area by counting squares. lt is roughly 6.5. The math- +ematicians among us can show that the integral is exactly z?/15.* Comparing +this expression with U/V = (4ơ/e)7, we find +k*n? watts +Z=———==56ï x10 —————i +60h3c3 (meter)2(degree) +T we make a small hole in our box, how muụch energy will ow per second +through the hole of unit area? To go from energy density to energy flow, we +multiply the energy density U/V by c. We also multiply by „ which arises +as follows: frst, a factor of 3ì because only the energy which is Ñowing ou# +* Since (e° — 1)! =e~*"+e—?* +..., the integral is +»xỊ e— „3 dạy, +But la e~” dạ = 1/n, and differentiating with respect to ?› three tỉmes gives la z3e~”# dạ = +6/n*, so the integral is 6(1 + 18 + 5T +---) and a good estimate comes from adding the frst +few terms. In Chapter 50 we will ñnd a way to show that the sum of the reciprocal fourth +powers of the integers is, in fact, x^/90. +--- Trang 822 --- +escapes; and second, another factor $3 because energy which approaches the +hole at an angle to the normal is less efective in getting through the hole by a +cosine factor. The average value of the cosine is 3 Tt is clear now why we write +U/V = (4ø/e)T®: so that we can ultimately say that the ñux om a small hole +is ơ7® per unit area. +--- Trang 823 --- +MHatchot anéeÏl peaerfF” +46-1 How a ratchet works +In this chapter we discuss the ratchet and paw]l, a very simple device which +allows a shaft to turn only one way. The possibility of having something turn +only one way requires some detailed and careful analysis, and there are some +Very Interesting consequences. +The plan of the discussion came about in attempting to devise an elementary +explanation, from the molecular or kinetic point of view, for the fact that there +1s a maximum amount of work which can be extracted from a heat engine. Of +course we have seen the essence oŸ Carnot's argument, but it would be nice to +fñnd an explanation which is elementary in the sense that we can see what is +happening physically. Now, there are complicated mathematical demonstrations +which follow from Newton”s laws to demonstrate that we can get only a certain +amount of work out when heat fows from one place to another, but there is great +difculty in converting this into an elementary demonstration. In short, we do +not understand it, although we can follow the mathematics. +In Carnot's argument, the fact that more than a certain amount of work +cannot be extracted in goïng from one temperature to another is deduced from +another axiom, which is that if everything is at the same temperature, heat cannot +be converted to work by means of a cyclic process. Eirst, let us back up and try +to see, in at least one elementary example, why this simpler statement is true. +Let us try to invent a device which will violate the Second Law of 'Thermo- +dynamics, that is, a gadget which will generate work from a heat reservoir with +everything at the same temperature. Let us say we have a box of gas at a certain +temperature, and inside there is an axle with vanes in it. (See Fig. 46-1 but take +Tì = Tạ =T, say.) Đecause of the bombardments of gas molecules on the vane, +* See Parrando and Espanol, Am. J. Phys. 64, 1125 (1996) for a critical analysis of this +chapter. +--- Trang 824 --- +ẢNN, ) nHÍ +Fig. 46-1. The ratchet and pawl machine. +the vane oscillates and jiggles. All we have to do is to hook onto the other end +of the axle a wheel which can turn only one way—the ratchet and pawl. 'Phen +when the shaft tries to jiggle one way, it will not turn, and when it jiggles the +other, it will turn. Then the wheel will slowly turn, and perhaps we might even +te a fea onto a string hanging from a drum on the shaft, and lift the Real Now +let us ask ïf this is possible. According to Carnot”s hypothesis, i% is impossible. +But ïf we just look at it, we see, prữna ƒacie, that 1% seems quite possible. So +we must look more closely. Indeed, if we look at the ratchet and pawÌl, we see a +number of complications. +First, our idealized ratchet is as simple as possible, but even so, there 1s a +pawl, and there must be a spring in the pawl. The pawl must return after coming +off a tooth, so the spring is necessary. +Another feature of this ratchet and pawl, not shown in the figure, is quite +essential. Suppose the device were made of perfectly elastic parts. After the pawl +1s lifted of the end of the tooth and is pushed back by the spring, it will bounce +against the wheel and continue to bounece. 'Phen, when another fÑuctuation came, +the wheel could turn the other way, because the tooth could get underneath +during the moment when the pawl was upl 'Therefore an essential part of the +irreversibility of our wheel is a damping or deadening mechanism which stops +the bouncing. When the damping happens, of course, the energy that was in the +pawl goes into the wheel and shows up as heat. So, as it turns, the wheel will +get hotter and hotter. To make the thing simpler, we can put a gas around the +wheel to take up some of the heat. Anyway, let us say the gas keeps rising in +temperature, along with the wheel. WIlI it go on forever? Nol The pawl and +wheel, both at some temperature 7', also have Brownian motion. 'Phis motion is +--- Trang 825 --- +such that, every once in a while, by accident, the pawl lifts itself up and over a +tooth just at the moment when the Brownian motion on the vanes is trying to +turn the axle backwards. And as things get hotter, this happens more often. +So, this is the reason this device does not work in perpetual motion. When +the vanes get kicked, sometimes the pawl lifts up and goes over the end. But +sometimes, when it tries to turn the other way, the pawl has already lifted due +to the fuctuations of the motions on the wheel side, and the wheel goes back the +other wayl The net result is nothing. It is not hard to demonstrate that when +the temperature on both sides is equal, there will be no net average motion of +the wheel. Of course the wheel will do a lot of jiggling this way and that way, +but it will not do what we would like, which is to turn jus one way. +Let us look at the reason. Ït is necessary to do work against the spring in +order to lift the pawl to the top of a tooth. Let us call this energy c, and let Ø +be the angle between the teeth. 'Phe chance that the system can accumulate +enough energy, c, to get the pawl over the top of the tooth, is e~*/*T, But the +probability that the pawl will aceidentally be up is also e—*“/““, So the number +of times that the paw] is up and the wheel can turn backwards freely is equal to +the number of times that we have enough energy to turn it forward when the +pawl is down. We thus get a “balance,” and the wheel will not go around. +46-2 The ratchet as an engine +Let us now go further. Take the example where the temperature of the vanes +1s 7 and the temperature of the wheel, or ratchet, is 75, and 75 is less than 71. +Because the wheel is cold and the ñuctuations of the paw] are relatively infrequent, +it will be very hard for the pawl to attain an energy c. Because of the high +temperature 71, the vanes will often attain the energy c, so our gadget will go in +one direction, as designed. +W©e would now like to see ïŸ it can lift weights. Onto the drum in the middle +we tie a string, and put a weight, such as our fea, on the string. We let be the +torque due to the weight. If Ù is not too great, our machine will lift the weight +because the Brownian fuctuations make it more likely to move in one direction +than the other. We want to fnd how much weight it can lift, how fast it goes +around, and so on. +First we consider a forward motion, the usual way one designs a ratchet to +run. In order to make one step forward, how much energy has to be borrowed +trom the vane end? We must borrow an energy < ©o liẾt the pawl. The wheel +--- Trang 826 --- +turns through an angle Ø against a torque , so we also need the energy Ø. The +total amount of energy that we have to borrow is thus + b0. 'PThe probability +that we get this energy is proportional to e~(Œ+9)/`”:. Actually, it is not only +a question of getting the energy, but we also would like to know the number of +times per second it has this energy. The probability per second is proportional +to e—(+£9)/T: and we shall call the proportionality consbant 1 /T. Tt will cancel +out in the end anyway. When a forward step happens, the work done on the +weight is ÙØ. The energy taken from the vane is e-+ bØ. The spring gets wound +up with energy c, then I% goes clatter, clatter, bang, and this energy øgoes intO +heat. All the energy taken out goes to lift the weight and to drive the paw], +which then falls back and gives heat to the other side. +Now we look at the opposite case, which is backward motion. What happens +here? To get the wheel to go backwards all we have to do is supply the energy to +litt the pawl high enough so that the ratchet will slip. This is still energy c. Ôur +probability per second for the pawl to lift this high is now (1/r)e ““*, Our +proportionality constant is the same, but this time k7; shows up because of the +diferent temperature. When this happens, the work is released because the wheel +slips backward. It loses one notch, so it releases work 6Ø. 'The energy taken from +the ratchet system is c, and the energy given to the gas at 7 on the vane side +1s ÙØ +c. It takes a little thinking to see the reason for that. Suppose the pawl +has lifted itself up accidentally by a Ñuctuation. Then when it falls back and the +spring pushes it down against the tooth, there is a force trying to turn the wheel, +because the tooth is pushing on an inclined plane. 'Phis force is doïing work, and +So is the force due to the weights. So both together make up the total force, and +all the energy which is sÌlowly released appears at the vane end as heat. (Of course +it must, by conservation of energy, but one must be careful to think the thing +throughl) We notice that all these energies are exactly the same, but reversed. +So, depending upon which of these bwo rates is greater, the weight is either slowly +lifted or slowly released. Of course, it is constantly jiggling around, going up for +a while and down for a while, but we are talking about the average behavior. +Suppose that for a particular weight the rates happen to be equal. hen we +add an infñnitesimal weight to the string. The weight will slowly go down, and +work will be done on the machine. Energy will be taken from the wheel and +given to the vanes. If instead we take of a little bit of weight, then the imbalance +is the other way. The weight is lited, and heat is taken from the vane and put +into the wheel. So we have the conditions of Carnot”s reversible cycle, provided +that the weight is just such that these ©wo are equal. This condition is evidently +--- Trang 827 --- +Table 46-1 +Summary of operation of ratchet and paw]. +toruard: Needs energy c+L9 from vane. .'.Rate = 1 e- 0+) +'Takes from vane L0 + +Does work L0 +Gives to ratchet € +1 —€/KT' +Backuard: Needs energy € for pawl. ...Rate= _—e 2 +Takes from ratchet ec +Releases work L0 same as above with sign reversed. +Gives to vane L0 + +L0 +TÝ system is reversible, rates are equal, hence c}ỳ 1£ =. +Heat toratchet Hence Q2 _ T2 +Heat rom vane LØ+c” QL TL +that (c+ E8)/Tì = é/T¿. Let us say that the machine is slowly lifting the weiglt. +lnergy ¡ is taken from the vanes and energy Qs is delivered to the wheel, and +these energies are in the ratio (e + 1Ø)/c. IÝ we are lowering the weight, we also +have Q1/Qs = (e+ LØ)/c. Thus (Table 46-1) we have +Q1/Qa = 11/1. +Furthermore, the work we get out is to the energy taken from the vane as LØ is +to ÙØ + , hence as (7T — 75)/T\. We see that our device cannot extract more +work than this, operating reversibly. 'This ¡is the result that we expected from +Carnot's aregument, and the main result of this lecbure. However, we can use +our device to understand a few other phenomena, even out of equilibrium, and +therefore beyond the range of thermodynamics. +Let us now calculate hou ƒas‡ our one-way device would turn if everything +were at the same temperature and we hung a weight on the drum. If we pull very, +very hard, of course, there are all kinds of complications. 'Phe pawl slips over the +ratchet, or the spring breaks, or something. But suppose we pull gently enough +--- Trang 828 --- +that everything works nicely. In those circumstances, the above analysis is right +for the probability of the wheel going forward and backward, iŸ we remember +that the two temperatures are equal. In each step an angle Ø is obtained, so the +angular velocity is Ø times the probability of one of these Jumps per second. Ïl§ +øoes forward with probability (1/r)e~(+#9/*T and backward with probability +(1/r)e~*/**, so that for the angular velocity we have +tụ (0/7)(e+19)/#T _— e~€/FT) += (0/r)e °/FT(e~19/RT — 1), (46.1) +l we plot œ against L, we get the curve shown in EFig. 46-2. We see that it +makes a great diference whether Ù, is positive or negative. lf Ù increases in the +positive range, which happens when we try to drive the wheel backward, the +backward velocity approaches a constant. As Ù becomes negative, œ really “takes +off” forward, since e to a tremendous power is very greatl +Fig. 46-2. Angular velocIty of the ratchet as a function of torque. +The angular velocity that was obtained from diferent forces is thus very +unsymmetrical. Going one way iÈ is easy: we get a lot of angular velocity for a +little force. Going the other way, we can put on a lot of force, and yet the wheel +hardly goes around. +We ñnd the same thing in an clectrical rectiffer. Instead of the force, we have +the electric ñeld, and instead of the angular velocity, we have the electric current. +In the case of a rectifier, the voltage is not proportional to resistance, and the +--- Trang 829 --- +situation is unsymmetrical. 'Phe same analysis that we made for the mechanical +rectifer will also work for an electrical rectifier. In fact, the kind of formula +we obtained above is typical of the current-carrying capacities of rectifers as a +function of their voltages. +Now let us take all the weights away, and look at the original machine. Tf +1T; were less than 71, the ratchet would go forward, as anybody would believe. +But what ¡is hard to believe, at first sight, is the opposite. If 72 is greater than +Tì, the ratchet goes around the opposite wayl A dynamic ratchet with lots of +heat in it runs itself backwards, because the ratchet pawl is bouncing. If the +pawl, for a moment, is on the incline somewhere, it pushes the inclined plane +sideways. But it is a”aøs pushing on an inclined plane, because if it happens to +lift up high enough to get past the point of a tooth, then the inelined plane slides +by, and it comes down again on an inclined plane. So a hot ratchet and paw] is +ideally built to go around in a direction exactly opposite to that for which it was +originally designedl +In spite of all our cleverness of lopsided design, ïf the bwo temperatures are +exactly equal there is no more propensity to turn one way than the other. The +moment we look at it, it may be turning one way or the other, but in the long +run, it gets nowhere. The fact that it gets nowhere is really the fundamental +deep principle on which all of thermodynamies is based. +46-3 Reversibility in mechanics +'What deeper mechanical principle tells us that, in the long run, 1f the tem- +perature is kept the same everywhere, our gadget will turn neither to the right +nor to the left? We evidently have a fundamental proposition that there is no +way to design a machine which, left to itself, will be more likely to be turning +one way than the other after a long enough time. We must try to see how this +follows from the laws of mechanics. +The laws of mechanics go something like this: the mass times the acceleration +1s the force, and the force on each partiele is some complicated function of the +positions of all the other particles. 'There are other situations in which forces +depend on velocity, such as in magnetism, but let us not consider that now. +W© take a simpler case, such as gravity, where forces depend onlÌy on position. +Now suppose that we have solved our set of equations and we have a certain +motion #(£) for each particle. In a complicated enough system, the solutions are +very complicated, and what happens with time turns out to be very surprising. +--- Trang 830 --- +TÍ we write down any arrangement we please for the particles, we will see this +arrangement actually occur if we wait long enough† Tf we follow our solution for +a long enoupgh time, it tries everything that it can do, so to speak. “This is not +absolutely necessary in the simplest devices, but when systems get complicated +enough, with enough atoms, it happens. Now there is something else the solution +can do. If we solve the equations of motion, we may get certain functions such +as £ + †? +. We claim that another solution would be —# + ¿2 — f3. In other +words, iŸ we substitute — everywhere for ý throughout the entire solution, we +will once again get a solution of the same equation. This follows from the fact +that if we substitute —ý for ‡ in the original diferential equation, nothing is +changed, since only second derivatives with respect to # appear. This means that +1ƒ we have a certain motion, then the exact opposite motion is also possible. In +the complete confusion which comes if we wait long enough, it finds itself going +one way sometimes, and it fñnds itself going the other way sometimes. 'There is +nothing more beautiful about one of the motions than about the other. 5o iE is +impossible to design a machine which, in the long run, is more likely to be going +one way than the other, if the machine is sufficiently complicated. +One might think up an example for which this is obviously untrue. IÝ we take +a wheel, for instance, and spin it in empty space, it will go the same waxy Íorever. +So there are some conditions, like the conservation of angular momentum, which +violate the above argument. 'Phis just requires that the argument be made with a +little more care. Perhaps the walls take up the angular momentum, or something +like that, so that we have no special conservation laws. Then, If the system 1s +complicated enouph, the argument is true. Ïlt is based on the fact that the laws +of mechanics are reversible. +For historical interest, we would like to remark on a device invented by +Maxwell, who first worked out the dynamical theory of gases. He supposed the +following situation: We have two boxes of gas at the same temperature, with a +little hole bebween them. At the hole sits a little demon (who may be a machine +of coursel). 'There is a door on the hole, which can be opened or closed by the +demon. He watches the molecules coming from the left. Whenever he sees a fast +molecule, he opens the door. When he sees a slow one, he leaves it closed. lf +we want him to be an extra special demon, he can have eyes at the back of his +head, and do the opposite to the molecules from the other side. He lets the slow +ones through to the left, and the fast through to the right. Pretty soon the left +side will get cold and the right side hot. Then, are the ideas of thermodynamics +violated because we could have such a demon? +--- Trang 831 --- +lt turns out, 1Ÿ we build a fñnite-sized demon, that the demon himself gets so +warm that he cannot see very well after a while. The simplest possible demon, as +an example, would be a trap door held over the hole by a spring. A fast molecule +comes through, because it is able to lift the trap door. 'Phe slow molecule cannot +get through, and bounces back. But this thing is nothing but our ratchet and +pawl in another form, and ultimately the mechanism will heat up. If we assume +that the specifc heat of the demon is not infinite, it mus$ heat up. It has but +a fñnite number of internal gears and wheels, so it cannot get rid of the extra +heat that it gets from observing the molecules. Soon it is shaking from Brownian +motion so mụuch that it cannot tell whether 1% is coming or going, much less +whether the molecules are coming or going, so it does not work. +46-4 Irreversibility +Are all the laws of physics reversible? Evidently notl Just try to unscramble +an egsl Run a moving picture backwards, and it takes only a few minutes for +everybody to start laughing. The most natural characteristic of all phenomena is +their obvious irreversibility. +'Where does irreversibility come from? It does not come from Newton”s laws. +Tf we claim that the behavior of everything is ultimately to be understood in +terms of the laws of physics, and I1f it also turns out that all the equations have +the fantastic property that if we put ý = — we have another solution, then every +phenomenon is reversible. How then does it come about in nature on a large scale +that things are not reversible? Obviously there must be some law, some obscure +but fundamental equation, perhaps in electricity, maybe in neutrino physics, in +which it does matter which way tỉme øoes. +Let us discuss that question now. We already know one of those laws, which +says that the entropy is always increasing. Ifwe have a hot thing and a cold thing, +the heat goes from hot to cold. So the law of entropy is one such law. But we +expect to understand the law of entropy from the point of view of mechanies. In +fact, we have just been successful in obtaining all the consequences of the argument +that heat cannot fow backwards by itself from just mechanical arguments, and +we thereby obtained an understanding of the Second Law. Apparently we can get +Irreversibility from reversible equations. But +0øs it on a mechanical argument +that we used? Let us look into it more closely. +Since our question has to do with the entropy, our problem 1s to try to ñnd a +mieroscopic description of entropy. lÝ we say we have a certain amount oŸ energy +--- Trang 832 --- +in something, like a gas, then we can get a microscopic picture of it, and say +that every atom has a certain energy. All these energies added together give us +the total energy. Similarly, maybe every atom has a certain entropy. lf we add +everything up, we would have the total entropy. It does not work so well, but let +us see what happens. +As an example, we calculate the entropy diference bebween a gas at a certain +temperature at one volume, and a gas at the same temperature at another volume. +W©e remember, from Chapter 44, that we had, for the change in entropy, +AS= | —.. +In the present case, the energy of the gas is the same before and after expansion, +since the temperature does not change. So we have to add enough heat to equal +the work done by the gas or, for each little change in volume, +dQ = PdV. +Putting this in for đQ, we get +W2 dV — f2 NET dV +AS= P—-= —— — +Vị T w V T += NkÌn —< +as we obtained in Chapter 44. For instance, iŸ we expand the volume by a factor +of 2, the entropy change is VNkln2. +Let us now consider another interesting example. Suppose we have a box +with a barrier in the middle. Ôn one side is neon (“black” molecules), and on +the other, argon (“white” molecules). NÑow we take out the barrier, and let them +mix. How much has the entropy changed? It is possible to imagine that instead +of the barrier we have a piston, with holes in it that let the whites through but +not the blacks, and another kind of piston which is the other way around. lf we +move one piston to each end, we see that, for each gas, the problem is like the +one we just solved. So we get an entropy change of Nkln2, which means that +the entropy has increased by kln2 per molecule. "The 2 has to do with the extra +room that the molecule has, which is rather peculiar. It is not a property of the +molecule itself, but of ho+ much room the molecule has to run around ín. This is +--- Trang 833 --- +a strange situation, where entropy increases but where everything has the same +temperature and the same energyl 'Phe only thing that is changed is that the +mmolecules are distributed diferently. +We well know that if we just pull the barrier out, everything will get mixed +up after a long time, due to the collisions, the jiggling, the banging, and so on. +tvery once in a while a white molecule goes toward a black, and a black one +goes toward a white, and maybe they pass. Gradually the whites worm their +way, by accident, across into the space of blacks, and the blacks worm their way, +by accident, into the space of whites. If we wait long enough we get a mixture. +Clearly, this is an irreversible process in the real world, and ought to involve an +increase in the entropy. +Here we have a simple example of an irreversible process which is completely +composed of reversible events. Every time there is a collision between any two +molecules, they go of in certain directions. lÝ we took a moving picture oŸ a +collision in reverse, there would be nothing wrong with the picture. Ín fact, one +kind oÊ collision is just as likely as another. 5o the mixing is completely reversible, +and yet it is irreversible. Everyone knows that iŸ we started with white and with +black, separated, we would get a mixture within a few minutes. Ïf we sat and +looked at it for several more minutes, it would not separate again but would stay +mixed. So we have an irreversibility which is based on reversible situations. But +we also see the reason now. We started with an arrangement which is, in some +sense, ordered. Due to the chaos of the collisions, it becomes disordered. Ïf ¡s the +change from an ordered arrangement to a disordered arrangement tuhïch 1s the +source oƒ the irreuersiblitg. +lt is true that if we took a motion picture of this, and showed it backwards, +we would see it gradually become ordered. Someone would say, “That is against +the laws of physics!” So we would run the fiÌm over again, and we would look +at every collision. Every one would be perfect, and every one would be obeying +the laws of physics. The reason, of course, 1s that every molecule°s velocities are +just right, so iŸ the paths are all followed back, they get back to their original +condition. But that is a very unlikely cireumstance to have. IÝ we start with the +gas in no special arrangement, just whites and blacks, it will never get back. +46-5 Order and entropy +So we now have to talk about what we mean by disorder and what we mean +by order. It is not a question of pleasant order or unpleasant disorder. What is +--- Trang 834 --- +diferent in our mixed and unmixed cases is the following. Suppose we divide +the space into little volume elements. If we have white and black molecules, how +many ways could we distribute them among the volume elements so that white is +on one side, and black on the other? On the other hand, how many ways could +we distribute them with no restriction on which goes where? Clearly, there are +many more ways to arrange them in the latter case. We measure “disorder” by +the number of ways that the insides can be arranged, so that from the outside it +looks the same. The logarithm oƒ that number öƒ U0ags f3 the entropy. The number +of ways in the separated case is less, so the entropy 1s less, or the “disorder” 1s +So with the above technical deñnition of disorder we can understand the +proposition. Eirst, the entropy measures the disorder. Second, the universe +always goes from “order” to “disorder,” so entropy always increases. Order is +not order in the sense that we like the arrangement, but in the sense that the +number of diferent ways we can hook ï up, and still have it look the same from +the outside, is relatively restricted. In the case where we reversed our motion +picture of the gas mixing, there was not as mụch disorder as we thought. Every +single atom had exactly the correct speed and direction to come out rightl 'The +entropy was not hiph after all, even though it appeared so. +What about the reversibility of the other physical laws? When we talked +about the electric ñeld which comes from an accelerating charge, it was said that +we must take the retarded field. At a time # and at a distance z from the charge, +we take the field due to the acceleration a® a tỉme £— r/c, not £+r/e. So it looks, +at first, as iƒ the law of electricity is not reversible. Very strangely, however, the +laws we used come from a set of equations called Maxwells equations, which +are, In fact, reversible. Eurthermore, it is possible to argue that IfÍ we were %O +use only the advanced field, the field due to the state of affairs at £ + r/c, and +do it absolutely consistently in a completely enelosed space, everything happens +exactly the same way as if we use retarded fieldsl “This apparent irreversibility in +electricity, at least in an enclosure, is thus not an irreversibility at all. We have +some feeling for that already, because we know that when we have an oscillating +charge which generates fñelds which are bounced from the walls of an enclosure +we ultimately get to an equilibrium in which there is no one-sidedness. The +retarded field approach is only a convenienee in the method of solution. +So far as we know, all the fundamental laws of physics, like NÑewton”s equations, +are reversible. Thhen where does irreversibility come from? It comes from order +going to disorder, but we do not understand this until we know the origin of the +--- Trang 835 --- +order. Why is it that the situations we find ourselves in every day are always +out of equilibriun? One possible explanation is the following. Look again at +our box of mixed white and black molecules. Now ï§ is possible, if we wait long +enough, by sheer, grossly improbable, but possible, accident, that the distribution +of molecules gets to be mostly white on one side and mostly black on the other. +After that, as times goes on and accidents continue, they geb more mixed up +again. +Thus one possible explanation oŸ the high degree of order in the present-day +world 1s that it is just a question of luck. Perhaps our universe happened to +have had a Ñuctuation of some kind in the past, in which things got somewhat +separatwed, and now they are running back together again. This kind of theory is +not unsymmetrical, because we can ask what the separated gas looks like either +a little in the future or a little in the past. In either case, we see a ørey smear at +the interface, because the molecules are mixing again. No matter which way we +run time, the gas mixes. So this theory would say the irreversibility is just one +of the accidents of life. +We would like to argue that this is not the case. Suppose we do not look at +the whole box at once, but only at a piece of the box. Then, at a certain moment, +Suppose we discover a certain amount of order. In this little piece, white and +black are separate. What should we deduce about the condition in places where +we have not yet looked? If we really believe that the order arose from complete +disorder by a ñuctuation, we must surely take the most likely Ñuctuation which +could produce it, and the most likely condition is so that the rest of it has +also become disentangledl 'Therefore, from the hypothesis that the world is a +fuctuation, all of the predictions are that if we look at a part of the world we +have never seen before, we will fnd it mixed up, and not like the piece we just +looked at. TIf our order were due to a Ñuctuation, we would not expect order +anywhere but where we have just noticed ït. +Now we assume the separation is because the past of the universe was really +ordered. It is not due to a Ñucbuation, but the whole thing used to be white and +black. 'This theory now predicts that there will be order in other places—the order +is not due to a fuctuation, but due to a much higher ordering at the beginning +of time. 'Phen we would expect to fnd order in places where we have not yet +looked. +'The astronomers, for example, have only looked at some of the stars. E/very +day they turn their telescopes to other stars, and the new stars are doing the +same thing as the other stars. We therefore conclude that the universe 1s noÝ a +--- Trang 836 --- +fuctuation, and that the order is a memory of conditions when things started. +Thịs is not to say that we understand the logic ofit. For some reason, the universe +at one time had a very low entropy for its energy content, and since then the +entropy has increased. So that is the way toward the future. That is the origin +of all irreversibility, that is what makes the processes of growth and decay, that +makes us remermber the past and not the future, remember the things which are +closer to that moment in the history of the universe when the order was higher +than now, and why we are not able to remember things where the disorder 1s +higher than now, which we call the future. So, as we commented ín an earlier +chapter, the entire universe is in a glass of wine, if we look at it closely enough. +In this case the gÌlass of wine is complex, because there is water and glass and +light and everything else. +Another delight of our subject of physics is that even simple and idealized +things, like the ratchet and pawl, work only because they are part of the universe. +The ratchet and pawl works in only one direction because i% has some ultimate +contact with the rest of the universe. lf the ratchet and pawl were in a box and +isolated for some sufficient time, the wheel would be no more likely to go one way +than the other. But because we pull up the shades and let the light out, because +we cool of on the earth and get heat from the sun, the ratchets and pawls that +we make can turn one way. This one-wayness is interrelated with the fact that +the ratchet is part of the universe. It is part of the universe not only in the sense +that it obeys the physical laws of the universe, but its one-way behavior is tied to +the one-way behavior of the entire universe. It cannot be completely understood +until the mystery of the beginnings of the history of the universe are reduced +still further from speculation to scientifc understanding. +--- Trang 837 --- +Seorrreel. TÌĨ:© trđrt© ©cjfrcrffOre +47-1 Waves +In this chapter we shall discuss the phenomenon of 0øøes. 'Phis is a phe- +nomenon which appears in many contexts throughout physics, and therefore our +attention should be concentrated on it not only because of the particular example +considered here, which is sound, but also because of the mụch wider application +of the ideas in all branches of physics. +lt was pointed out when we studied the harmonic oscillator that there are +not only mechanical examples of oscillating systems but electrical ones as well. +Waves are related to oscillating systems, except that wave oscillations appear not +only as time-oscillations at one place, but propagate in space as well. +W©e have really already studied waves. When we studied light, in learning +about the properties of waves in that subjJect, we paid particular attention to the +Interference in space of waves from several sources at diferent locations and all +at the same frequency. 'Phere are two Important wave phenomena that we have +not yet discussed which occur ïn light, i.e., electromagnetic waves, as well as in +any other form of waves. The frst of these is the phenomenon of ?n#erƒerence +ín từme rather than interference in space. lf we have two sources of sound which +have slightly diferent frequencies and if we listen to both at the same tỉme, +then sometimes the waves come with the crests together and sometimes with +the crest and trough together (see Fig. 47-1). The rising and falling of the sound +that results is the phenomenon of Öeøa#s or, in other words, of interference In +time. “The second phenomenon involves the wave patterns which result when +the waves are confned within a given volume and reflect back and forth from +walls. +'These efects could have been discussed, of course, for the case of electromag- +netic waves. 'Phe reason for not having done this is that by using one example we +would not generate the feeling that we are actually learning about many different +--- Trang 838 --- +Fig. 47-1. lnterference in time of two sound sources with slightly +different frequencies, resulting ¡in beats. +subjects at the same time. In order to emphasize the general applicability of +waves beyond electrodynamies, we consider here a different example, in particular +sound_waves. +Other examples of waves are water waves consisting oŸ long swells that we see +coming in to the shore, or the smaller water waves consisting of surface tension +ripples. As another example, there are two kinds of elastic waves in solids; a +cormpressional (or longitudinal) wave in which the particles of the solid oscillate +back and forth along the direction oŸ propagation of the wave (sound waves in a +gas are of this kind), and a transverse wave in which the particles of the solid +oscillate in a direction perpendicular to the direction of propagation. Earthquake +waves contain elastic waves of both kinds, generated by a motion at some place +in the earth”s crust. +Still another example of waves is found in modern physics. These are waves +which give the probability amplitude of ñnding a particle at a given place—the +“matter waves” which we have already discussed. 'Pheir frequency is proportional +--- Trang 839 --- +to the energy and their wave number is proportional to the momentum. 'They +are the waves of quantum mechanics. +In this chapter we shall consider only waves for which the velocity is inde- +pendent of the wavelength. 'This is, for example, the case for light in a vacuum. +The speed of light is then the same for radiowaves, blue light, green light, or +for any other wavelength. Because of this behavior, when we began to describe +the wave phenomenon we did not notice at first that we had wave propagation. +Instead, we said that If a charge is moved at one place, the electric fñeld at a +distance + was proportional to the acceleration, not at the time #, but at the +carlier time ¿ — #ø/c. Therefore if we were to picture the electric fñeld in space +at some instant of time, as in Eig. 47-2, the electric fñeld at a time £ later would +have moved the distance cứ, as indicated in the fgure. Mathematically, we can +say that in the one-dimensional example we are taking, the electric field is a +function of z — cứ. We soe that at £ =0, it is some function of z. lIf we consider +a later time, we need only to increase ø somewhat to get the same value of the +electric feld. Eor example, if the maximum field occurred at z = 3 at time zero, +then to fnd the new position of the maximum field at time # we need +# — CÈ = Ồ OF œ ==ä+ đi. +W© see that this kind of function represents the propagation oŸ a wave. +Such a function, ƒ(œ — c£), then represents a wave. We may summarize thìs +description of a wave by saying simply that +ƒ(œ — œ8) = ƒ(œ + Az — c(t+ At)), +——«t——+ +1 1 P.4 +Fig. 47-2. The solid curve shows what the electric field might be like +at some instant of time and the dashed curve shows what the electric +field ¡is at a time £ later. +--- Trang 840 --- +when Az = cAứ. There is, of course, another possibility, i.e., that instead of +a source to the left as indicated in Fig. 47-2, we have a source on the right, so +that the wave propagates toward negative z. hen the wave would be described +by ø(z + cŸ). +There is the additional possibility that more than one wave exists in space +at the same time, and so the electric field is the sum of the two fields, each one +propagating independently. "This behavior of electric fñelds may be described +by saying that if ƒfi(œ — c£#) is a wave, and IÍ ƒ2(œ — c£) is another wave, then +their sum is also a wave. This is called the principle of superposition. 'Phe same +prineiple is valid in sound. +W© are familiar with the fact that If a sound is produced, we hear with +complete fñdelity the same sequence of sounds as was generated. IÝ we had high +frequencies travelling faster than low frequencies, a short, sharp noise would be +heard as a succession of musical sounds. 5imilarly, ¡f red light travelled faster +than blue light, a flash of white light would be seen first as red, then as white, +and ñnally as blue. We are familiar with the fact that this is not the case. Both +sound and light travel with a speed in air which is very nearly independent of +frequency. Examples of wave propagation-for which this independence is not true +will be considered in Chapter 48. +In the case of light (electromagnetic waves) we gave a rule which determined +the electric field at a point as a result of the acceleration of a charge. One might +expect now that what we should do is give a rule whereby some quality of the +air, say the pressure, is determined at a given distance from a source in terms +of the source motion, delayed by the travel time of the sound. In the case of +light this procedure was acceptable because all that we knew was that a charge +at one place exerts a force on another charge at another place. The details of +propagation from the one place to the other were not absolutely essential. In +the case of sound, however, we know that i propagates through the air between +the source and the hearer, and it is certainly a natural question to ask what, +at any given moment, the pressure of the air is. We would like, in addition, to +know exactly how the air moves. In the case of electricity we could accept a +rule, since we could say that we do not yet know the laws of electricity, but we +cannot make the same remark with regard to sound. We would not be satisled +with a rule stating how the sound pressure moves through the air, because the +process ought to be understandable as a consequence of the laws of mechanics. +In short, sound is a branch of mechanies, and so ït is to be understood in terms +of NÑewton's laws. The propagation oŸ sound from one place to another is merely +--- Trang 841 --- +a consequence of mechanics and the properties of gases, 1Í it propagates in a gas, +or of the properties of liquids or solids, 1Ý it propagates through such mediums. +Later we shall derive the properties of light and its wave propagation in a similar +way from the laws of electrodynamics. +47-2 The propagation of sound +W©e shall give a derivation of the properties of the propagation of sound +bettueen the source and the receiver as a consequence of NÑewton”s laws, and we +shall not consider the interaction with the source and the receiver. Ordinarily +we emphasize a result rather than a particular derivation of it. In this chapter +we take the opposite view. The point here, in a certain sense, is the derivation +itself. This problem of explaining new phenomena in terms of old ones, when +we know the laws of the old ones, is perhaps the greatest art of mathematical +physics. The mathematical physicist has two problems: one is to ñnd solutions, +given the equations, and the other is to fnd the equations which describe a new +phenomenon. 'Phe derivation here is an example of the second kind of problem. +W©e shall take the simplest example here—the propagation of sound in one +dimension. To carry out such a derivation i% is necessary frst to have some +kind of understanding of what is going on. Eundamentally what is involved is +that If an object is moved at one place in the air, we observe that there is a +disturbance which travels through the air. IÝ we ask what kind of disturbance, +we would say that we would expect that the motion of the object produces a +change of pressure. Of course, if the object is moved gently, the air merely flows +around it, but what we are concerned with is a rapid motion, so that there is not +sufficient time for such a fow. 'Phen, with the motion, the air is compressed and a +change of pressure is produced which pushes on additional air. 'Phis air is in turn +compressed, which leads again to an extra pressure, and a wave is propagated. +W©e now want to formulate such a process. We have to decide what variables +we need. In our particular problem we would need to know how much the air has +moved, so that the air đisplacemen#‡ in the sound wave is certainly one relevant +variable. In addition we would like to describe how the air đens¿zfy changes as it +is displaced. 'Phe air pressure also changes, so this is another variable of interest. +Then, of course, the air has a 0elocztu, so that we shall have to describe the +velocity of the air particles. The air particles also have øcceleraiions——but as we +list these many variables we soon realize that the velocity and acceleration would +be known ïif we knew how the air đ¿splacemen‡ varies with time. +--- Trang 842 --- +As we said, we shall consider the wave in one dimension. We can do this +1ƒ we are sufliciently far from the source that what we call the eœuefronts are +very nearly planes. We thus make our argument simpler by taking the least +complicated example. We shall then be able to say that the displacement, x, +depends only on z and ¿, and not on # and z. Therefore the description of the +air is given by x(z, £). +ls this description complete? It would appear to be far from complete, for we +know none of the details of how the air molecules are moving. They are moving +in all directions, and this state of afairs is certainly not described by means of +this function x(z,£). From the point of view of kinetic theory, If we have a higher +density of molecules at one place and a lower density adjacent to that place, +the molecules would move away from the region of higher density to the one of +lower density, so as to equalize this diference. Apparently we would not get an +oscillation and there would be no sound. What is necessary to get the sound +wave is this situation: as the molecules rush out of the reglon of higher density +and higher pressure, they give momentum to the molecules in the adjacent region +of lower density. For sound to be generated, the regions over which the density +and pressure change must be much larger than the distance the molecules travel +before colliding with other molecules. 'Phis distance is the mean free path and +the distance between pressure crests and troughs must be much larger than this. +Otherwise the molecules would move freely from the crest to the trough and +immediately smear out the wave. +Tt is clear that we are going to describe the gas behavior on a scale large +compared with the mean free path, and so the properties of the gas will not be +described in terms of the individual molecules. The displacement, for example, +will be the displacement of the center of mass of a small element of the gas, and +the pressure or density will be the pressure or density in this region. We shall +call the pressure and the density ø, and they will be functions oŸ z and ứ. We +must keep in mind that this description is an approximation which is valid only +when these gas properties do not vary too rapidly with distanee. +47-3 The wave equation +'The physics of the phenomenon of sound waves thus involves three features: +I. The gas moves and changes the density. +TL. The change in density corresponds to a change in pressure. +--- Trang 843 --- +THI. Pressure inequalities generate gas motion. +Let us consider II first. For a gas, a liquid, or a solid, the pressure 1s some +function of the density. Before the sound wave arrives, we have equilibrium, +with a pressure b and a corresponding density øo. Á pressure in the medium +is connected to the density by some characteristic relation = ƒ(ø) and, in +particular, the equilibrium pressure ạ is given by ạ = ƒ(øo). The changes of +pressure in sound from the equilibrium value are extremely small. Á convenient +unit for measuring pressure is the bar, where 1 bar = 105 N/m2. The pressure of +1 standard atmosphere is very nearly 1 bar: 1 atm = 1.0133 bars. In sound we +use a logarithmic scale oŸ intensities since the sensitivity of the ear is roughly +logarithmic. 'This scale is the decibel scale, in which the acoustic pressure level +for the pressure amplitude ? ¡is defñned as +T (acoustic pressure level) = 20 logio(P/Đx«:) in dB, (47.1) +where the reference pressure ›e¿ = 2 x 10710 bar. A pressure amplitude of +P = 10?P,¿ = 2 x 10” bar* corresponds to a moderately intense sound of +60 decibels. We see that the pressure changes in sound are extremely small +compared with the equilibrium, or mean, pressure of 1 atm. 'Phe displacements +and the density changes are correspondingly extremely small. In explosions we +do not have such small changes; the excess pressures produced can be greater +than 1 atm. These large pressure changes lead to new efects which we shall +consider later. In sound we do not often consider acoustic Intensity levels over +100 đB; 120 dB ¡s a level which is painful to the ear. Therefore, for sound, if we +P=h+F. 0= P0 + Ø; (47.2) +we shall always have the pressure change 2 very small compared with ọ and +the density change ø¿ very small compared with øo. hen +Tạ + Đ, = ƒ(po + 0e) = ƒ(0o) + 0e (po): (47.3) +where Pụ = ƒ(0øo) and ƒ7(øo) stands for the derivative of ƒ(ø) evaluated at ø = /øo. +W© can take the second step in this equality only because ø¿ is very small. We fnd +in this way that the excess pressure #2 is proportional to the excess density øe, +* With this choice of fz;er, the P is not the peak pressure in the sound wave but the +“root-mean-square” pressure, which is 1/(2)1⁄2 times the peak pressure. +--- Trang 844 --- +and we may call the proportionality factor &: +Đụ = Kpe; where = ƒf(øo) = (đP/dp)o. (47.4) +'The relation we needed for II is this very simple one. +———— x(x, £) ——— +{— EOLD VOLUME h NEW VOLUME +I l 1 I +x x+Ax x+x(x,t†)_ (x+Ax)+x(x+Ax, t) +ma... x(X+Ax, t) — Mr +Fig. 47-3. The displacement of the air at x is x(x, f), and at x + Ax +Ít is x(x + Ax,£). The original volume of the air for a unit area of the +plane wave is Ax; the new volume is Ax + x(x + Ax, †) — x(x, t). +Let us now consider I. We shall suppose that the position of a portion of +aïir undisturbed by the sound wawe is ø and the displacement at the time £ due +to the sound is x(z,£), so that its new position is # + x(z,£), as in Eig. 47-3. +NÑow the undisturbed position of a nearby portion of air is ø + Äz, and its new +position is # + Az + x(œ + Az,f). We can now fñnd the density changes in the +following way. Since we are limiting ourselves to plane waves, we can take a unit +area perpendicular to the z-direction, which ¡is the direction of propagation of +the sound wave. The amount of air, per unit area, in Az is then øg Az, where /øo +1s the undisturbed, or equilibrium, air density. 'Phis air, when displaced by the +sound wave, now lies between ø + x(z,#) and #ø + Az + x(z + Az,f), so that we +have the same matter in this interval that was in Az when undisturbed. TỶ ø is +the new density, then +Ø0o Â# = p[# + Az + x(œ + Az,t) — z — x(#, Đ. (47.5) +Since Az is smaill, we can write x(œ + Az,f) — x(%,f) = (Ôx/Øz) Az. Thịs +derivative is a partial derivative, since x depends on the time as well as on z. +Our cquation then is +po Â#z = ñ Az+ Az) (47.6) +--- Trang 845 --- +Øo = (Øo + P2) +/o + Ø‹- (47.7) +Now in sound waves all changes are small so that ø¿ is small, x is small, and +Øx/Øx is also small. Therefore in the relation that we have just found, += —0Ø0 <— — Đe 47.8 +fe = —f0 ốc — Ðe ST (47.8) +we can neglect ø« Øx/Øz compared with øoØx/ÔØz. Thus we get the relation we +needed for I: : += —/0 =~. I 47.9 +Đe Ø0 Ôz ( ) ( ) +'This equation is what we would expect physically. If the displacements vary with +z, then there will be density changes. The sign is also right: if the displacement x +increases with #ø, so that the air is stretched out, the density must go down. +W© now need the third equation, which is the equation of the motion produced +by the pressure. If we know the relation between the force and the pressure, we +can then get the equation of motion. If we take a thin slab of air of length Az +and oŸ unit area perpendicular to #, then the mass oÝ air in this slab is øg Az and +it has the acceleration Ø”x/Ôf, so the mass tỉmes the acceleration for this slab +of matter is øo Az(02x/Ø12). (It makes no diference for small Az whether the +acceleration Ø”x/Ø/2 is evaluated at an edge of the slab or at some intermediate +position.) IÝ now we ñnd the force on this matter for a unit area perpendicular to #, +it will then be equal to øo Az(Ø”x/ô12). We have the force in the -+z-direction, +at #, of amount P{z,f) per unit area, and we have the force in the opposite +direction, at ø + Az, of amount P{+ + Az,£) per unit area (Eig. 47-4): +gP 9P, +P(z,t)—P Az,f)=———Az=—-=A 47.10 +TỶ... ẽn..ẽn.. (47.10) +P(&x,t)_—~ ~ P(x+Ax,t) +_—> Ax ~— +Fig. 47-4. The net force in the positive x-direction produced by the +pressure acting on unit area perpendicular to x is —(ÔP/ôx) Ax. +--- Trang 846 --- +since Aø is small and since the only part of P which changes is the excess +pressure #,. We now have TIT: +92x 9P, +=2 —=T— IH 47.11 +Ø0 PT Ôz (HD ( ) +and so we have enough equations to interconnect things and reduce down to one +variable, say to x. We can eliminate 2 from TII by using II, so that we get +ØŠx Ôp; +=2 =TR an; 47.12 +f0 y2 h 9z ) +and then we can use I to eliminate ø¿. In this way we fnd that øo cancels out +and that we are left with : : +CX VU X, (47.13) +Ø2 8z2 +We shall call c2 = ,, so that we can write +9? 1 Ø2 +“X=-, (47.14) +z2 c2 Ø12 +'This is the wave equation which describes the behavior of sound in matter. +47-4 Solutions of the wave equation +We now can see whether this equation really does describe the essential +properties of sound waves in matter. We want to deduce that a sound pulse, +or disturbance, will move with a constant speed. We want to verify that bwo +difÑerent pulses can move through each other—the principle of superposition. We +also want to verify that sound can go either to the right or to the left. All these +properties should be contained in this one equation. +We have remarked that any plane-wave disturbance which moves with a +constant velocity 0 has the form ƒ(œ — 0É). Ñow we have to see whether Xx(%, É) = +ƒ(œ — 9Ê) is a solution of the wave equation. When we calculate Øx/Ôz, we get +the derivative of the function, Øx/Ø = ƒ7( — o‡). Differentiating once more, +we fnd : +--- Trang 847 --- +The diferentiation of this same function with respect to £ gives —u times +the derivative of the function, or Øx/Ø# = —0ƒf(œ — 0), and the second time +derivative 1s +93x _— „2 +Pướnn: ƒ {œ -— 0t). (47.16) +It is evident that ƒ(œ — 9£) will satisfy the wave equation provided the wave +velocity ø is equal to ca. +W©e fnd, therefore, from the iaus oƒ mmechanics that any sound disturbance +propagates with the velocity c;, and in addition we fnd that +c =R}? = (4P/dp)g ”, +and so e haue related the tuaue 0elocftU to a propert oƒ the mmedium. +TỶ we consider a wave travelling in the opposite direction, so that x(#, £) = +g(œ + 0É), it is easy to see that such a disturbance also satisfes the wave equation. +The only diference bebween such a wave and one travelling from left to right +1s in the sign of 0, but whether we have # + 0Ý or ø — 0Ý as the variable in the +function does not affect the sign of 02x/Ø2, since it involves only 2. It follows +that we have a solution for waves propagating ¡in either direction with speed es. +An extremely interesting question is that of superposition. Suppose one +solution of the wave equation has been found, say xi. This means that the second +derivative of xị with respect to z# is equal to 1/cŸ tỉimes the second derivative +oŸ xi with respect to ý. Now any other solution xa has this same property. If we +superpose these two solutions, we have +x(, £) = XI(#,f) + Xa(, É), (47.17) +and we wish to verify that Xx(%,É) is also a wave, i.e., that x satisfies the wave +cequation. We can easily prove this result, since we have +ĐA _ ĐẠI, ĐA¿ (47.18) +8z2 9z2 9z2 +and, in addition, +ĐA _ ĐA ĐA, (47.19) +Ø2 Ø12 912 +It follows that 02x/Øz2 = (1/c$) 02x/Ø12, so we have verified the prineiple of +superposition. 'Phe proof of the principle of superposition follows from the fact +that the wave equation is ineør in X. +--- Trang 848 --- +W©e can now expect that a plane light wave propagating in the z-direction, +polarized so that the electric fñeld is in the ¿-direction, will satisfy the wave +equation +08B, _ 108B, #23 +9z2 c2 0Ÿ2 +where c is the speed of light. "This wave equation is one of the consequences of +Maxwells equations. "The equations of electrodynamics will lead to the wave +equation for light just as the equations of mechanics lead to the wave equation +for sound. +47-5 The speed of sound +Our deduction of the wave equation for sound has given us a ƒormula which +connects the wave speed with the rate of change oŸ pressure with the density at +the normal pressure: +c= (2) . (47.21) +In evaluating this rate of change, it is essential to know how the temperature +varles. In a sound wave, we would expect that in the region of compression the +temperature would be raised, and that in the reglon oŸ rarefaction the temperature +would be lowered. Newton was the first to caleulate the rate of change of pressure +with density, and he supposed that the temperature remained unchanged. He +argued that the heat was conducted from one region to the other so rapidly that +the temperature could not rise or fall. This argument gives the isothermal speed +of sound, and it is wrong. “The correcÿ deduction was given later by Laplace, +who put forward the opposite idea—that the pressure and temperature change +adiabatically in a sound wave. 'Phe heat flow om the compressed region to the +rarefed region is negligible so long as the wavelength is long compared with the +mean free path. Under this condition the slight amount of heat fow in a sound +wave does not affect the speed, although it gives a small absorption of the sound +energy. VWe can expect correctly that this absorption increases as the wavelength +approaches the mean free path, but these wavelengths are smaller by factors of +about a million than the wavelengths of audible sound. +The actual variation of pressure with density in a sound wave is the one that +allows no heat ñow. 'Phis corresponds to the adiabatic variation, which we found +--- Trang 849 --- +to be PV7 = const, where V was the volume. 5ince the density ø varies inversely +with V, the adiabatic connection between ? and ø is +P = const Øø , (47.22) +from which we get đP/do = +P/p. W© then have for the speed oŸ sound the +relation P +cẶ= J~, (47.23) +W© can also write c2 = yPV/øV and make use of the relation PV = NET. +Purther, we see that øW is the mass of gas, which can also be expressed as mm, +or as / per mole, where rn is the mass of a molecule and / is the molecular +weight. In this way we fnd that +... .—= (47.24) +from which ït is evident that the speed of sound depends only on the gas temper- +ature and not on the pressure or the density. We also have observed that +kT = 3m(02), (47.25) +where (22) is the mean square of the speed of the molecules. It follows that +cả = (/3)(0”), or +c; = Œ) Dạy: (47.26) +This equation states that the speed oŸ sound is some number which is roughly +1/(3)!⁄2 times some average speed, øay, of the molecules (the square root of the +mean square velocity). In other words, the speed oŸ sound is of the same order of +magnitude as the speed of the molecules, and is actually somewhat less than this +average speed. +Of course we could expect such a result, because a disturbance like a change +in pressure is, after all, propagated by the motion of the molecules. However, +such an argument does not $ell us the precise propagation speed; it could have +turned out that sound was carried primarily by the fastest molecules, or by the +slowest molecules. lt is reasonable and satisfying that the speed of sound 1s +roughly 3 of the average molecular speed 0ạy. +--- Trang 850 --- +}?o(fÉs +48-1 Adding two waves +Some time ago we discussed in considerable detail the properties of light +waves and theïir interference—that is, the efects of the superposition of two waves +trom different sources. In all these analyses we assumed that the frequencies +of the sources were all the same. In this chapter we shall discuss some of the +phenomena which result from the interference of two sources which have đjfƒerent +frequencies. +Tt is easy to guess what is going to happen. Proceeding in the same way as we +have done previously, suppose we have two equal oscillating sources of the same +frequency whose phases are so adjusted, say, that the signals arrive in phase at +some point . At that point, if it is light, the light is very strong; if it is sound, +1t is very loud; or iŸ i is electrons, many of them arrive. Ôn the other hand, if +the arriving signals were 180 out of phase, we would get no signal at , because +the net amplitude there is then a minimum. Now suppose that someone twists +the “phase knob” of one of the sources and changes the phase at back and +forth, say, ñrst making ¡it 0° and then 180”, and so on. Of course, we would then +ñnd variations in the net signal strength. Now we also see that 1Ÿ the phase of +one source is slowly changing relative to that of the other in a gradual, uniform +mamner, starting at zero, going up to ten, twenty, thirty, forty degrees, and so +on, then what we would measure at would be a series of strong and weak +“pulsations,” because when the phase shifts through 360° the amplitude returns +to a maximum. Of course, to say that one source is shifting its phase relative to +another at a uniform rate is the same as saying that the number of oscillations +per second ¡s slightly diferent for the bwo. +So we know the answer: iŸ we have two sources at slightly diferent frequencies +we should fnd, as a net result, an oscillation with a slowly pulsating intensity. +That is all there really is to the subjectl +--- Trang 851 --- +cos 107£ +cos 87£ +¬ Z " s. ` +V r⁄ ^* ⁄ V +/⁄ ¬ ¿_ ⁄ ¬ ¿_⁄ +_Z ^° ° _Z ¬ ° _Z +Fig. 48-1. The superposition of two cosine waves with frequencies In +the ratio 8 : 10. The precise repetition of the pattern within each “beat” +is not typical of the general case. +lt is very easy to formulate this result mathematically also. Suppose, Íor +example, that we have two waves, and that we do not worry for the moment +about all the spatial relations, but simply analyze what arrives at P. FErom one +source, let us say, we would have cosư#, and from the other source, cos œs#, +where the ÿwo œ's are not exactly the same. Of course the amplitudes may not +be the same, either, but we can solve the general problem later; let us fñrst take +the case where the amplitudes are equal. 'Phen the total amplitude at ? is the +sum of these Ewo cosines. If we plot the amplitudes of the waves against the time, +as in Fig. 48-1, we see that where the crests coincide we get a strong wave, and +where a trough and crest coincide we get practically zero, and then when the +crests coincide again we get a strong wave again. +Mathematically, we need only to add ÿwo cosines and rearrange the result +somehow. 'There exist a number of useful relations among cosines which are not +dificult to derive. Of course we know that +ci(a+tD) — dat, (48.1) +and that e?“ has a real part, cosø, and an imaginary part, sinaø. If we take the +--- Trang 852 --- +real part of e+°),we get cos (a + b). IÝ we multiply out: +c'“e?" = (cos ø + isỉn ø)(eos b + ¿ sỉn b), +we get cos ø cOs 0 — sin øsin b, plus some imaginary parts. But we now need only +the real part, so we have +cos (ø -L Ù) = cos acos Ò — sin ø sin b. (48.2) +Now if we change the sign oŸ b, since the cosine does not change sign while the +sine does, the same equation, for negative Ù, 1s +cos (œ — Ù) = cos acos Ö + sin ø sin b. (48.3) +Tf we add these t©wo equations together, we lose the sines and we learn that the +produect of bwo cosines is half the cosine of the sum, plus half the cosine of the +diferenee: +COS # cos b = 5 cos (ø + b) + 3 cos (ø — Ù). (48.4) +Now we can also reverse the formula and find a formula for cos œ + cos Ø iÝ we +simply let œ =ø+band 8= a— b. That is, a = š(œ+ Ø) and b= š(œT— đ), so +cos œ -Ƒ cos ở = 2cos 2(œ + đ) cos s(œ — đ). (48.5) +Now we can analyze our problem. 'Phe sum of cosœ+# and cos (aÝ 1s +COS 1Ý + COS 2É = 2coOS s(01 + 2)f cos s(01 — 0a)t. (48.6) +Now let us suppose that the two frequencies are nearly the same, so that 3(01 +2) +1s the average frequency, and is more or less the same as either. But œ — œ2 +1s much smailler than tị or (2 because, as We suppose, ¿ị and ¿¿ are nearly +cqual. 'Phat means that we can represent the solution by saying that there 1s +a high-frequency cosine wave more or less like the ones we started with, but +that its “size” is slowly changing——its “size” is pulsating with a frequency which +appears to be š(œ — œ2). But is this the frequency at which the beats are +heard? Although (48.6) says that the amplitude goes as cos 3(0I — 02)t, what it +is really telling us is that the high-frequency oscillations are contained between +two opposed cosine curves (shown dotted in Pig. 48-1). Ôn this basis one could +say that the amplitude varies at the frequency 3 (01 — 02), but iŸ we are talking +--- Trang 853 --- +about the ?m#ensiu oŸ the wave we must think of it as having ©wice this requency. +That ¡s, the modulation of the amplitude, in the sense of the strength of its +intensity, is at Írequency œ0 — ¿œ¿, although the formula tells us that we multiply +by a cosine wave at half that frequency. The technical basis for the diference +1s that the high frequency-wave has a little diferent phase relationship in the +second half-cycle. +lgnoring this small complication, we may conclude that if we add two waves oŸ +Írequency œ¡ and œ¿, we will get a net resulting wave oŸ average Írequency s(01 + +œ2) which oscillates in strength with a frequency œ1 — 0a. +TÍ the two amplitudes are diferent, we can do it all over again by multiplying +the cosines by diferent amplitudes 4i and 4¿, and do a lot of mathematics, +rearranging, and so on, using equations like (48.2)-(48.5). However, there are +other, easier ways of doing the same analysis. For example, we know that 1% +1s mụuch easier to work with exponentials than with sines and cosines and that +we can represent Ái cos¿# as the real part of Aie?“, The other wave would +similarly be the real part of 4se'“2!, Tf we add the two, we get Aief“1t + Ase2t, +Tf we then factor out the average frequency, we have +Aiest + Aae?ezt — crti+2)1/2( Ai —e2)1/2 + Aae 161—42)1/2, (48.7) +Again we have the high-frequency wave with a modulation at the lower frequency. +48-2 Beat notes and modulation +T we are now asked for the intensity of the wave of Eq. (48.7), we can either +take the absolute square of the left side, or of the right side. Let us take the left +side. The intensity then is +T= A?+ A?+2Ai4a cos (wị — 62). (48.8) +W© see that the intensity swells and falls at a Írequency ¿¡ — œ2, varying between +the limits (Ai + 4s)2 and (4¡ — 4a)2. If Ai # 4a, the minimum intensity is not +One more way to represent this idea is by means of a drawing, like Fig. 48-2. +W© draw a vector of length 4i, rotating at a frequency œ0, ©o represent one of +the waves in the complex plane. We draw another vector of length 4a, going +around at a frequency œ¿s, to represent the second wave. lf the bwo frequencies +are exactly equal, their resultant is of ñxed length as it keeps revolving, and we +--- Trang 854 --- +01 — (J2 —= (J +Fig. 48-2. The resultant of two complex vectors of equal frequency. +Fig. 48-3. The resultant of two complex vectors of unequal frequency, +as seen In the rotating frame of reference of one vector. Nine successive +positions of the slowly rotating vector are shown. +get a definite, fñxed intensity from the two. But if the frequencies are slightly +diferent, the bwo complex vectors go around at diferent speeds. Figure 48-3 +shows what the situation looks like relative to the vector Aie*“+, We see that +4s is turning slowly away from 44, and so the amplitude that we get by adding +the two is first strong, and then, as it opens out, when it gets to the 180° relative +position the resultant gets particularly weak, and so on. Äs the vectors øo around, +the amplitude of the sum vector gets bigger and smaller, and the intensity thus +pulsates. It is a relatively simple idea, and there are many difÑferent ways of +representing the same thing. +The efect is very easy to observe experimentally. In the case of acoustics, we +may arrange two loudspeakers driven by two separate oscillators, one for each +loudspeaker, so that they each make a tone. We thus receive one note from one +source and a diferent note from the other source. If we make the frequencies +exactly the same, the resulting efect will have a defnite strength at a given space +--- Trang 855 --- +location. If we then de-bune them a little bit, we hear some variations in the +intensity. 'Phe farther they are de-buned, the more rapid are the variations of sound. +'The ear has some trouble following variations more rapid than ten or so per second. +We may also see the efect on an oscilloscope which simply displays the sum +of the currents to the two speakers. If the frequency of pulsing is relatively low, +we simply see a sinusoidal wave train whose amplitude pulsates, but as we make +the pulsations more rapid we see the kind of wave shown in Eig. 48-1. As we go +to greater frequency diferences, the “bumps” move closer together. Also, ¡f the +amplitudes are not equal and we make one signal stronger than the other, then +we get a wave whose amplitude does not ever become zero, Just as we expect. +tverything works the way it should, both acoustically and electrically. +The opposite phenomenon occurs tool In radio transmission using so-called +aqmplitude modulalion (AM), the sound is broadcast by the radio station as +follows: the radio transmitter has an AC electric oscillation which is at a very +high frequency, for example 800 kilocycles per second, in the broadcast band. +T this carr¿er signal is turned on, the radio station emits a wave which is of +uniform amplitude at 800,000 oscillations a second. 'Phe way the “information” +1s transmitted, the useless kind of information about what kind of car to buy, is +that when somebody talks into a microphone the amplitude of the carrier signal +1s changed in step with the vibrations of sound entering the microphone. +Tf we take as the simplest mathematical case the situation where a soprano +1s singing a perfect note, with perfect sinusoidal oscillations of her vocal cords, +then we get a signal whose strength is alternating as shown in Fig. 48-4. 'Phe +audiofrequency alternation is then recovered in the receiver; we get rid of the +carrier wave and Just look at the envelope which represents the oscillations of the +vocal cords, or the sound of the singer. The loudspeaker then makes corresponding +vibrations at the same frequency in the aïir, and the listener is then essentially +Fig. 48-4. A modulated carrier wave. In this schematic sketch, +(c/(0m = 5. ln an actual radiowave, ¿c/œm ~ 100. +--- Trang 856 --- +unable to tell the diÑerence, so they say. Because of a number of distortions and +other subtle effects, it is, in fact, possible to tell whether we are listening to a +radio or to a real soprano; otherwise the idea is as indicated above. +48-3 Side bands +Mathematically, the modulated wave described above would be expressed as +5 = (1+ bCOS50„È) cos 0¿È, (48.9) +where ¿J¿ represents the frequency of the carrier and œư„ 1s the frequency of the +audio tone. Again we use all those theorems about the cosines, or we can use c!; +it makes no difference—it is easier with e', but it is the same thing. We then get +8 = coswf + 2b cos (0e + @„)Ê + 2b€OS (0e — œm)Ê. (48.10) +So, from another point of view, we can say that the output wave of the system +consists of three waves added in superposition: first, the regular wave at the +frequency œ;, that is, at the carrier frequenecy, and then ?wo new waves aE ÿWO +new frequencies. One is the carrier frequenecy plus the modulation frequency, and +the other is the carrier frequency minus the modulation frequency. lf, therefore, +we make some kind of plot of the intensity being generated by the generator as +a function of frequency, we would fnd a lot of intensity at the frequency of the +carrier, naturally, but when a singer started to sing, we would suddenly also ñnd +intensity proportional to the strength of the singer, ð, at frequeney ứ; + (dự +and œ — œ„, as shown in Eig. 48-5. These are called s¿de bønds; when there is +a modulated signal from the transmitter, there are side bands. If there is more +than one note at the same time, say œ„ and œ„„/, there are bwo instruments +ức — tUm Úc (Úc -E 0m œ +Fig. 48-5. The frequency spectrum of a carrier wave ¿c modulated +by a single cosine Wave (m. +--- Trang 857 --- +playing; or if there is any other complicated cosine wave, then, oÝ course, we can +see from the mathematics that we geÈ some more waves that correspond to the +frequencies œ¿ + ư„y¿. +'Therefore, when there is a complicated modulation that can be represented as +the sum of many cosines,* we fnd that the actual transmitter is transmitting over +a range of frequencies, namely the carrier frequency plus or minus the maximum +frequency that the modulation signal contains. +Although at frst we might believe that a radio transmitter transmits only +at the nominal frequenecy of the carrier, since there are big, superstable crystal +oscillators in there, and everything ¡is adjusted to be at precisely 800 kilocycles, +the moment someone øwnowwces that they are at 800 kilocyeles, he modulates +the 800 kilocycles, and so they are no longer precisely at 800 kilocyclesl Suppose +that the amplifers are so built that they are able to transmit over a øgood range +of the ear”s sensitivity (the ear can hear up to 20,000 cycles per second, but +usually radio transmitters and receivers do not work beyond 10,000, so we do +not hear the highest parts), then, when the man speaks, his voice may contain +frequencies ranging up, say, to 10,000 cycles, so the transmitter is transmitting +Írequencies which may range from 790 to 810 kilocycles per second. Now ïf there +were another station at 795 kc/sec, there would be a lot oŸ confusion. Also, if we +made our receiver so sensitive that it picked up only 800, and did not pick up the +10 kilocycles on either side, we would not hear what the man was saying, because +the information would be on these other frequenciesl 'Therefore it is absolutely +essential to keep the stations a certain distance apart, so that their side bands +do not overlap and, also, the receiver must not be so selective that it does not +permit reception of the side bands as well as of the main nominal frequenecy. In +the case of sound, this problem does not really cause much trouble. We can hear +over a +20 kc/sec range, and we have usually from 500 to 1500 ke/sec in the +broadcast band, so there is plenty of room for lots of stations. +The television problem is more difficult. As the electron beam goes across +the face of the picture tube, there are various little spots of light and dark. hat +* A slight side remark: In what circumstances can a curve be represented as a sum of a lot +of cosines? Ansuer: In all ordinary circumstances, except for certain cases the mathematicians +can dream up. OÝ course, the curve must have only one value at a given point, and it must +not be a crazy curve which jumps an infinite number of times in an infinitesimal distance, or +something like that. But aside from such restrictions any reasonable curve (one that a singer is +going to be able to make by shaking her vocal cords) can always be compounded by adding +cosine waves together. +--- Trang 858 --- +“lght” and “dark” is the “signal” Now ordinarily the beam scans over the whole +picture, 500 lines, approximately, in a thirtieth of a second. Let us consider +that the resolution oŸ the picture vertically and horizontally is more or less the +same, so that there are the same number of spots per inch along a scan line. +We want to be able to distinguish dark from light, dark from light, dark from +light, over, say, 500 lines. In order to be able to do this with cosine waves, the +shortest wavelength needed thus corresponds to a wavelength, from maximum +to maximum, of one 250th of the screen size. So we have 250 x 500 x 30 pieces +of information per second. 'Phe highest frequency that we are going %O CaTTY, +therefore, is close to 4 megacycles per second. Actually, to keep the television +stations apart, we have to use a little bit more than thịs, about 6 mc/sec; part +of it is used to carry the sound signal, and other information. So, television +channels are 6 megacycles per second wide. It certainly would not be possible +to transmit TV on an 800 kc/sec carrier, since we cannot modulate at a higher +frequency than the carrier. +At any rate, the television band starts at 54 megacycles. The first transmission +chamnel, which is channel 2 (!), has a frequency range from 54 to 60 me/sec, +which is 6 mec/sec wide. “But,” one might say, “we have just proved that there +were side bands on both sides, and therefore it should be bwice that wide.” lt +turns out that the radio engineers are rather clever. If we analyze the modulation +signal using not jusÈ cosine terms, but cosine and sine terms, to allow for phase +diferences, we then see that there is a defnite, invariant relationship between +the side band on the high-frequency side and the side band on the low-frequency +side. What we mean is that there is no new information on that other side band. +So what is done is to suppress one side band, and the receiver is wired inside +such that the information which is missing is reconstituted by looking at the +single side band and the carrier. Single side-band transmission is a clever scheme +for decreasing the band widths needed to transmit information. +48-4 Localized wave traïns +The next subject we shall discuss is the interference of waves in both space +and time. Suppose that we have bwo waves travelling in space. We know, of +course, that we can represent a wave travelling in space by ef©f—*#), 'This might +be, for example, the displacement in a sound wave. “This is a solution of the +wave equation provided that ¿2 = k2c2, where e is the speed of propagation of +the wave. In this case we can write it as e_?#Œ—©which is of the general form +--- Trang 859 --- +ƒ(œ — cf). Therefore this must be a wave which is travelling at this velocity, œ/È, +and that is c and everything is all right. +Now we want to add t©wo such waves together. Suppose we have a wave that is +travelling with one requency, and another wave travelling with another frequency. +We leave to the reader to consider the case where the amplitudes are diferent; it +makes no real diference. Thus we want to add cf61#—E1#) + c162t—2#),WWe can +add these by the same kind of mathematics we used when we added signal waves. +Of course, if c is the same for both, this is easy, since ït is the same as what we +did before: +c?i1(—z/e) + c?a2(—z/e) — của + củ, (48.11) +except that f = £— #/e is the variable instead of ¿. So we get the same kind of +mmodulations, naturally, but we see, of course, that those modulations are moving +along with the wave. In other words, ¡if we added two waves, but these waves +were not just oscillating, but also moving in space, then the resultant wave would +move along also, at the same speed. +Now we would like to generalize this to the case of waves in which the +relationship between the frequency and the wave number & is not so simple. +Example: material having an index of refraction. We have already studied the +theory of the index oŸ refraction in Chapter 31, where we found that we could +write k = nœ/c, where #w is the index of refraction. As an interesting example, +for x-rays we Íound that the index m is +mn==1— 3como°” (48.12) +W© actually derived a more complicated formula in Chapter 31, but this one is +as good as any, as an example. +Incidentally, we know that even when œ and &k are not linearly proportional, +the ratio œ/& is certainly the speed of propagation for the particular requency +and wave number. We call this ratio the phase 0elocifu; 1t is the speed at which +the phase, or the nodes of a single wave, would move along: +Up =T‹ (48.13) +This phase velocity, for the case of x-rays In gÌass, is greater than the speed of +light in vacuum (since ø in 48.12 is less than 1), and that is a bit bothersome, +because we do not think we can send signals faster than the speed of lightl +--- Trang 860 --- +'What we are going to discuss now is the interference of two waves in which œ +and & have a defnite formula relating them. The above formula for øœ says that +& is given as a defnite function of œ. To be specifc, in this particular problem, +the formula for & in terms OŸ œ iS +k=“-, (48.14) +where ø = W42/2comn, a constant. At any rate, for each frequeney there is a +defñnite wave number, and we want to add ©wo such waves together. +Let us do it just as we did in Eq. (48.7): +cf(61t—kiz) + cf(2t—kaz) — cilŒite2)£— (ị +ka)#] /2 +x {eflket—ez)t—~(Ri—ka)x]/2 + T6 —62)— (ai —ka)e]/21, (48.15) +So we have a modulated wave again, a wave which travels with the mean frequency +and the mean wave number, but whose strength is varying with a form which +depends on the diference frequency and the diference wave number. +Now let us take the case that the diference between the two waves is relatively +small. Let us suppose that we are adding two waves whose Írequencies are nearly +cqual; then (œ -Ƒ œ2)/2 is practically the same as either one of the œs, and +similarly for (k¡ + &a)/2. Thus the speed of the wave, the fast oscillations, +the nodes, is still essentially œ/k. But look, the speed oŸ propagation of the +mmodulation is not the samel How much do we have to change # to account for a +certain amount of ý? 'The speed of this modulation wawve is the ratio +(1 — 2 +UM —= T—~- 48.16 +¬ (48.16) +The speed of modulation is sometimes called the group 0elocitu. Tf we take the +case that the difference in frequency is relatively small, and the diference in +wave number is then also relatively small, then this expression approaches, in +the limit, +=—. 48.17 +Đụ dẸ ( ) +In other words, for the slowest modulation, the slowest beats, there is a defnite +speed at which they travel which is not the same as the phase speed of the +waves—what a mysterious thingl +--- Trang 861 --- +The group 0clocitỤ 1s the dertueliue oƒ ( títh respect to k, and the phase +0elocit is (/k. +Let us see if we can understand why. Consider two waves, again of slightly +diferent wavelength, as In Fig. 48-1. They are out of phase, in phase, out of +phase, and so on. Now these waves represent, really, the waves in space travelling +with slightly diferent frequencies also. Now because the phase velocity, the +velocity of the nodes of these wo waves, is not precisely the same, something +new happens. Suppose we ride along with one of the waves and look at the other +one; 1ƒ they both went at the same speed, then the other wave would stay right +where it was relative to us, as we ride along on this crest. We ride on that crest +and right opposite us we see a crest; if the two velocities are equal the crests stay +on top of each other. But it is no so that the bwo velocities are really equal. +There is only a small diference in frequency and therefore only a small dilerence +in velocity, but because of that diference in velocity, as we ride along the other +wave moves slowly forward, say, or behind, relative to our wave. So as time goes +on, what happens to the node? IÝ we move one wave train Just a shade forward, +the node moves forward (or backward) a considerable distance. That is, the sum +of these two waves has an envelope, and as the waves travel along, the envelope +rides on them at a diferent speed. “The growp 0elocitu is the speed at which +modulated signals would be transmitted. +T we made a signal, i.e., some kind of change in the wave that one could +recognize when he listened to it, a kind of modulation, then that modulation +would travel at the group velocity, provided that the modulations were relatively +slow. (When they are fast, it is mụch more difficult to analyze.) +Now we may show (at long last), that the speed oŸ propagation oŸ x-rays in a +block of carbon is no greater than the speed of light, although the phase velocity +7s greater than the speed of light. In order to do that, we must find dư /dk, which +we get by diferentiating (48.14): dk/dœ = 1/c + a/œ°c. The group velocity, +therefore, is the reciprocal of this, namely, +Dạ — + +a/ø5) (48.18) +which is smaller than cl So although the phases can travel faster than the speed +of light, the modulation signals travel slower, and that is the resolution of the +apparent paradoxl Of course, if we have the simple case that œ = ke, then dư /dk +1s also c. So when all the phases have the same velocity, naturally the group has +the same velocity. +--- Trang 862 --- +48-5 Probability amplitudes for particles +Let us now consider one more example of the phase velocity which is extremely +interesting. It has to do with quantum mechanics. We know that the amplitude +to fnd a particle at a place can, in some circumstances, vary in space and time, +let us say in one dimension, in this manner: += Ac6-kz), (48.19) +where œ is the frequency, which is related to the classical idea of the energy +through # = ñư, and & is the wave number, which is related to the momentum +through p = ñk. We would say the particle had a defnite momentum ? ïf the +wave number were exactly k&, that is, a perfect wave which goes on with the +same amplitude everywhere. Equation (48.19) gives the amplitude, and if we +take the absolute square, we get the relative probability for fñnding the partiecle +as a function of position and time. This is a cons‡øn‡, which means that the +probability ¡is the same to find a particle anywhere. Now suppose, instead, that +we have a situation where we know that the particle is more likely to be at one +place than at another. We would represent such a situation by a wave which has +a maximum and dies out on either side (Fig. 48-6). (Tt is not quite the same as a +wave like (48.1) which has a series oŸ maxima, but it is possible, by adding several +waves of nearly the same œ and & together, to get rid of all but one maximum.) +_ÍÍ lẦh,. +_ NV x +Fig. 48-6. A localized wave train. +Now in those circumstances, since the square of (48.19) represents the chance +of finding a particle somewhere, we know that at a given instant the particle +1s most likely to be near the center of the “lump,” where the amplitude of the +wave is maximum. If now we wait a few moments, the waves will move, and +after some time the “lump” will be somewhere else. IÝ we knew that the partiele +originally was situated somewhere, classically, we would ezpec£ that it would +later be elsewhere as a matter of fact, because it has a specd, after all, and a +--- Trang 863 --- +mmomentum. 'Phe quantum theory, then, will go into the correct classical theory +for the relationship of momentum, energy, and velocity only if the group velocity, +the velocity of the modulation, is equal to the velocity that we would obtain +classically for a particle of the same mmomentum. +Ït is now necessary to demonstrate that this is, or is not, the case. According +to the classical theory, the energy is related to the velocity through an equation +E=———__—. (48.20) +v1_— 02/2 +Similarly, the momentum is +?=—————. 48.21 +v1_— 02/2 ( ) +That ¡is the classical theory, and as a consequence of the classical theory, by +eliminating 0, we can show that +E2 — pc? = m2c*, +That is the four-dimensional grand result that we have talked and talked about, +that 0„Є = rm?; that is the relation between energy and momentum in the +classical theory. Now that means, since these #⁄'s and ø's are going to become +œ's and k's, by substitution of # = ñœ and p = ñk, that for quantum mechanics +1t is necessary that +—— — hŸk = mỶc. (48.22) +'This, then, is the relationship between the frequency and the wave number of a +quantum-mechanical amplitude wave representing a particle of mass mm. From +this equation we can deduce that œ is +œ = €VW k2 + mm2c2/h2. +The phase velocity, œ/k, is here again faster than the speed of lightl +Now let us look at the group velocity. The group velocity should be dư/dk, +the speed at which the modulations move. We have to diferentiate a square root, +which is not very dificult. The derivative is +dụ —- ké +dE - /k2 + m2c2/h2` +--- Trang 864 --- +Now the square root is, after all, ¿/c, so we could write this as dư/dk = c2k/ú. +Further, k/œ is p/H, so +Uy = cêp +But from (48.20) and (48.21), c2p/E = 9, the velocity of the particle, according to +classical mechanics. So we see that whereas the fundamental quantum-mechanical +relationship # = hư and p = ñk, for the identification of œ and & with the classical +E and p, only produees the equation œj2—k?c2 = m2c1/h2, now we also understand +the relationships (48.20) and (48.21) which connected #7 and ø to the velocity. Of +course the group velocity must be the velocity of the particle if the interpretation +1s going to make any sense. If we think the particle is over here at one tỉme, +and then ten minutes later we think it is over there, as the quantum mechanics +said, the distance traversed by the “lump,” divided by the time interval, must +be, classically, the velocity of the particle. +48-6 Waves in three dimensions +We shall now bring our discussion of waves to a close with a few general +remarks about the wave equation. 'These remarks are intended to give some +view of the future—not that we can understand everything exactly just now, but +rather to see what things are going to look like when we study waves a little +more. Eirst of all, the wave equation for sound in one dimension was +93x 1 63x +9x2 c2 0/2) +where c is the speed of whatever the wave is—in the case of sound, it is the sound +speed; in the case of lipght, ¡it is the speed of light. We showed that for a sound +wave the displacements would propagate themselves at a certain speed. But +the excess pressure also propagates at a certain speed, and so does the excess +density. So we should expect that the pressure would satisfy the same equation, +as indeed it does. We shall leave it to the reader to prove that it does. Hữn‡: +Øc 1s proportional to the rate of change of x with respect to z. Therefore if we +diferentiate the wave equation with respect to z, we will Immediately discover +that Øx/Øz satisfies the same equation. That is to say, øc satisfies the same +cequation. But , is proportional to ø;, and therefore , does too. So the pressure, +the displacements, everything, satisfy the same wave equation. +--- Trang 865 --- +sually one sees the wave equation for sound written in terms of pressure +instead of in terms of displacement, because the pressure is a scalar and has no +direction. But the displacement is a vector and has direction, and it is thus easier +to analyze the pressure. +The next matter we discuss has to do with the wave equation in three +dimensions. We know that the sound wave solution in one dimension is cf=*2), +with œ = kc;, but we also know that in three dimensions a wave would be +represented by e/«f~kzz—kuw—kz?)_ where, in this case, œ2 = k?c;, which is, of +course, (k2 + k2 + k2)c¿. Now what we want to do is to guess what the correct +wave equation in three dimensions is. Naturally, for the case of sound this can be +deduced by going through the same dynamiec argument in three dimensions that +we made in one dimension. But we shall not do that; instead we just write down +what comes out: the equation for the pressure (or displacement, or anything) is +2 2 2 2 +0E OP ĐH 101. (48.23) +9z2 Ø2 9z2 đc Ø12 +That this is true can be verifed by substituting in e/£='*)_ Clearly, every time +we diferentiate with respect to z, we multiply by —¿k„. If we diferentiate twice, +it is equivalent to multiplying by —k‡, so the first term would become —k?P,, +for that wave. Similarly, the second term becomes —kjƑ,, and the third term +becomes —kŸP,. On the right, we get —(œ”/c2)P„. Then, if we take away the P,'s +and change the sign, we see that the relationship between & and œ is the one +that we want. +'Working backwards again, we cannot resist writing down the grand equation +which corresponds to the dispersion equation (48.22) for quantum-mechanical +waves. lÝ ý represents the amplitude for fñnding a particle at position z#, ,z, at +the time £, then the great equation of quantum mechanics for free particles 1s +¬ 9$ 09324 02 1 Ø0 m°c +=a+t.as+*đ.ssa—-anas=e r0 (48.24) +9x2 ð0ụ 2? ôÔz2 c? Ø12 h2 +Jirst of all, the relativity character of this expression is suggested by the ap- +pearance of z, , z and # ¡in the nice combination relativity usually involves. +Second, it is a wave equation which, if we try a plane wave, would produce as +a consequence that —kŠ + œ2/c2 = mm2c2/h, which is the right relationship for +quantum mechanics. 'Phere is still another great thing contained in the wave +equation: the fact that any superposition of waves is also a solution. So this +--- Trang 866 --- +cequation contains all of the quantum mechaniecs and the relativity that we have +been discussing so far, at least so long as it deals with a single particle in empty +space with no external potentials or forces on itl +48-7 Normal modes +Now we turn to another example of the phenomenon of beats which is rather +curious and a little diferent. Imagine two equal pendulums which have, between +them, a rather weak spring connection. They are made as nearly as possible the +same length. HÝÍ we pull one aside and let go, it moves back and forth, and it pulls +on the connecting spring as it moves back and forth, and so i really is a machine +for generating a force which has the natural frequency of the other pendulum. +'Therefore, as a consequence of the theory of resonance, which we studied before, +when we put a force on something at just the right frequeney, it will drive it. 5o, +sure enough, one pendulum moving back and forth drives the other. However, In +this cireumstance there is a new thing happening, because the total energy of the +system 1s fnite, so when one pendulum pours its energy into the other to drive +1t, i ñnds itself gradually losing energy, until, if the timing is just right along +with the speed, it loses all its energy and is reduced to a stationary conditionl +Then, of course, it is the other pendulum ball that has all the energy and the +frst one which has none, and as time goes on we see that it works also in the +opposite direction, and that the energy is passed back into the first ball; this is a +very interesting and amusing phenomenon. We said, however, that this is related +to the theory of beats, and we must now explain how we can analyze this motion +from the point of view of the theory of beats. +W© note that the motion of either of the two balls is an oscillation which has +an amplitude which changes cyclically. 'herefore the motion of one of the balls is +presumably analyzable in a diferent way, in that it is the sum of ©wo oscillations, +present at the same time but having two slightly diferent frequencies. Therefore +it ought to be possible to fnd two other motions in this system, and to claim +that what we saw was a superposition of the bwo solutions, because this is of +course a linear system. Indeed, ¡it is easy to fnd two ways that we could start +the motion, each one of which is a perfect, single-frequeney motion——absolutely +periodic. The motion that we started with before was not strictly periodic, since +1t did not last; soon one ball was passing energy to the other and so changing its +amplitude; but there are ways of starting the motion so that nothing changes and, +OÝ cOurse, as soon as we see it we understand why. Eor example, If we made both +--- Trang 867 --- +pendulums go together, then, since they are of the same length and the spring is +not then doïng anything, they will of course continue to swing like that for all +time, assuming no iction and that everything is perfect. Ôn the other hand, +there is another possible motion which also has a definite frequency: that 1s, 1Ý +we move the pendulums oppositely, pulling them aside exactly equal distances, +then again they would be in absolutely periodic motion. We can appreciate that +the spring just adds a little to the restoring force that the gravity supplies, that +is all, and the system just keeps oscillating at a slightly higher frequency than +in the first case. Why higher? Because the spring is pulling, in addition to the +gravitation, and it makes the system a little “stifer,” so that the frequency of +this motion is just a shade higher than that of the other. +Thus this system has two ways in which ¡i% can oscillate with unchanging +amplitude: it can either oscillate in a manner in which both pendulums go the +same way and oscillate all the time at one frequency, or they could go in opposite +directions at a slightly higher frequency. +Now the actual motion of the thing, because the system is linear, can be +represented as a superposition of the two. (The subject of this chapter, remember, +is the efects of adding two motions with diferent frequenecies.) So think what +would happen If we combined these two solutions. IÝ at ¿ = 0 the two motions +are started with equal amplitude and in the same phase, the sum of the two +motions means that one ball, having been impressed one way by the fñrst motion +and the other way by the second motion, is at zero, while the other ball, having +been displaced the same way in both motions, has a large amplitude. Äs time +goes on, however, the bwo basic rno#ions proceed independently, so the phase +of one relative to the other is slowly shifting. That means, then, that after a +sufciently long time, when the tỉme is enough that one motion could have gone +“9005” oscillations, while the other went only “900,” the relative phase would be +Jjust reversed with respect to what it was before. 'Phat ¡s, the large-amplitude +motion will have fallen to zero, and in the meantime, of course, the initially +motionless ball will have attained full strengthl +So we see that we could analyze this complicated motion either by the idea +that there is a resonance and that one passes energy to the other, or else by the +superposition of two constant-amplitude motions at two diferent frequencies. +--- Trang 868 --- +JModios +49-1 The reflection of waves +This chapter will consider some of the remarkable phenomena which are a +result oŸ confñning waves in some fñnite region. We will be led first to discover a Íew +particular facts about vibrating strings, for example, and then the generalization +of these facts will give us a principle which is probably the most far-reaching +principle of mathematical physics. +Our fñirst example of confining waves will be to confne a wave at one boundary. +Let us take the simple example of a one-dimensional wave on a string. Ône could +equally well consider sound in one dimension against a wall, or other situations +of a similar nature, but the example of a string will be sufficient for our present +purposes. Suppose that the string is held at one end, for example by fastening +1E to an “infñnitely solid” wall. This can be expressed mathematically by saying +that the displacement of the string at the position z = 0 must be zero, because +the end does not move. Now if it were not for the wall, we know that the general +solution for the motion is the sum of two functions, f{œ — c£) and G(z + cf), the +ñrst representing a wave travelling one way in the string, and the second a wave +travelling the other way ¡n the string: +ụ = ư — c£) + G(+ + cÈ) (49.1) +1s the general solution for any string. But we have next to satisfy the condition +that the string does not move at one end. IÝ we put z = 0in Eq. (49.1) and +examine g for any value of, we get = F(—c£) + G(+ct). Now ïf this is to be +zero for all tìmes, it means that the function G(c£) must be —F{(—c£). In other +words, G of anything must be —# of minus that same thing. Tf this result is put +back into Ed. (49.1), we fnd that the solution for the problem is += F(% — cÈ) — F(—z — ©et). (49.2) +Tlt is easy to check that we will get — 0 1Ý we set ø = 0. +--- Trang 869 --- +Fixed End K&+ c8) +—F(-x+ cỒð +Fig. 49-1. Reflection of a wave as a superposition of two travelling waves. +Jigure 49-1 shows a wave travelling in the negative z-direction near # = 0, +and a hypothetical wave travelling in the other direction reversed in sign and on +the other side of the origin. We say hypothetical because, of course, there is no +string to vibrate on that side of the origin. The total motion of the string is to be +regarded as the sum of these two waves in the region of positive ø. As they reach +the origin, they will always cancel at = 0, and fñnally the second (refected) +wave will be the only one to exist for positive ø and it will, of course, be travelling +in the opposite direction. 'These results are equivalent to the following statement: +1Ý a wave reaches the clamped end of a string, it will be refected with a change +in sign. Such a refection can always be understood by imagining that what is +coming to the end of the string comes out upside down from behind the wall. +In short, IÝ we assume that the string is inñnite and that whenever we have a +wave going one way we have another one going the other way with the stated +symmetry, the displacement at ø = 0 will always be zero and it would make no +diference if we clamped the string there. +'The next poïnt to be discussed is the reflection of a periodic wave. Suppose that +the wave represented by #'{+ — c#) is a sine wave and has been reflected; then the +reflected wave — Ƒ"(—#— œ) is also a sine wave of the same frequency, but travelling +in the opposite direction. 'Phis situation can be most simply described by using +the complex function notation: F(z—£) = c⁄2Œ~#/® and F(—z— ct) = cl0+†z/9), +--- Trang 870 --- +It can be seen that iŸ these are substibuted in (49.2) and ïŸ # is set equal to 0, +then z = 0 for all values of #, so it satisfes the necessary condition. Because of +the properties of exponentials, this can be written in a simpler form: +ụ= c*t(e~etJe — cie#/©) = —94e**f sìn (a/e). (49.3) +There is something interesting and new here, in that this solution tells us that +1ƒ we look at any fxed z, the string oscillates at frequency œ. NÑo matter where +this point is, the requenecy is the samel But there are some places, in particular +wherever sin (u#/c) = 0, where there is no displacement at all. Furthermore, iŸ +at any time ý we take a snapshot of the vibrating string, the picture will be a +sine wave. However, the displacement of this sine wave will depend upon the +tỉme ý. From inspection of Eq. (49.3) we can see that the length of one cycle of +the sine wave is equal to the wavelength of either of the superimposed waves: +À =2#c/u. (49.4) +The points where there is no motion satisfy the condition sỉn (¿#/c) = 0, which +means that (0#/c) = 0,7, 27,..., na, ... These points are called øodes. Between +any £wo successive nodes, every point moves up and down sinusoidally, but the +pattern of motion stays fñxed in space. 'Phis is the fundamental characteristic of +what we call a mode. TỶ one can find a pattern of motion which has the property +that at any point the object moves perfectly sinusoidally, and that all points +move at the same frequency (though some will move more than others), then we +have what is called a mode. +49-2 Confned waves, with natural frequencies +The next interesting problem is to consider what happens ïf the string is +held at both ends, say at z = 0 and z = L. We can begin with the idea of the +reflection of waves, starting with some kind of a bump moving ïn one direction. +As time goes on, we would expect the bump to get near one end, and as time +goes still further it will become a kíind oŸ little wobble, because it is combining +with the reversed-image bump which is coming from the other side. Einally the +original bump will disappear and the image bump will move in the other direction +to repeat the process at the other end. 'Phis problem has an easy solution, but +an interesting question is whether we can have a sinusoidal motion (the solution +Jusb described is øer2od¿e, but of course it is not s¿auso¿daliu periodie). Let us try +--- Trang 871 --- +to put a sinusoidally periodic wave on a string. lÝ the string is tied at one end, +we know iÈ must look like our earlier solution (49.3). If it is tied at the other +end, it has to look the same at the other end. So the only possibility for periodic +sinusoidal motion is that the sine wave must neatly ft into the string length. If +it does not fit into the string length, then it is not a natural frequeney at which +the string can continue to oscillate. In short, if the string is started with a sine +wave shape that just fñts in, then it will continue to keep that perfect shape of a +sine wave and will oscillate harmonically at some frequency. +Mathematically, we can write sin kz for the shape, where & is equal to the +factor (œ/c) in Eqs. (49.3) and (49.4), and this function will be zero a% ø = 0. +However, it must also be zero at the other end. 'Phe significance of this is that & +1s no longer arbitrary, as was the case for the half-open string. With the string +closed at both ends, the only possibility is that sin (k) = 0, because this is the +only condition that will keep both ends fñxed. NÑow in order for a sine to be zero, +the angle must be either 0, z, 2z, or some other integral multiple of x. “The +equation +kÙ = rr (49.5) +will, therefore, give any one of the possible &'s, debending on what integer is put +in. Eor each of the k's there is a certain frequenecy œ, which, according to (49.3), +1s simply +œ = ke = trrc/L. (49.6) +So we have found the following: that a string has a property that it can have +sinusoidal motions, Duứ onlụ œ£ certain [requencies. Phis 1s the most important +characteristic of confined waves. No matter how complicated the system is, 1t +always turns out that there are some patterns oŸ motion which have a perfect +sinusoidal time dependence, but with frequencies that are a property of the +particular system and the nature of its boundaries. In the case of the string we +have many diferent possible frequencies, each one, by defnition, corresponding +to a mode, because a mode is a pattern of motion which repeats itself sinusoidally. +Jigure 49-2 shows the first three modes for a string. Eor the fñrst mode the +wavelength À is 2L. 'Phis can be seen iŸ one continues the wave out to ø = 2b +to obtain one complete cycle of the sine wave. The angular Írequency œ 1s 27c +divided by the wavelength, in general, and in this case, since À is 2, the frequency +is /,b, which is in agreement with (49.6) with m = 1. Let us call the frst mode +frequeney œị. Now the next mode shows two loops with one node in the middle. +For this mode the wavelength, then, is simply E. 'The corresponding value of & is +--- Trang 872 --- +—-. —CỀL—* +l ~”T~ +S ⁄⁄ ` 2 x +Fig. 49-2. The first three modes of a vibrating string. +twice as great and the frequency is twice as large; It is 2œ. Eor the third mode +1t 1s 3ú, and so on. So all the diferent frequencies of the string are multiples, +1, 2, 3, 4, and so on, of the lowest frequenecy 01. +Returning now to the general motion of the string, it turns out that any +possible motion can always be analyzed by asserting that more than one mode +1s operating at the same time. In fact, for general motion an infnite number of +modes must be excited at the same time. To get some idea of this, let us illustrate +what happens when there are two modes oscillating at the same time: Suppose +that we have the first mode oscillating as shown by the sequence of pictures in +Hig. 49-3, which illustrates the defection of the string for equally spaced time +intervals extending through half a cycle of the lowest frequeney. +Now, at the same time, we suppose that there is an oscillation of the second +mode also. Eigure 49-3 also shows a sequence of pictures of this mode, which +at the start is 90° out of phase with the first mode. “This means that at the +start it has no displacement, but the t©wo halves of the string have oppositely +directed velocities. NÑow we recall a general principle relating to linear systems: +1ƒ there are any two solutions, then their sum is also a solution. 'Pherefore a third +possible motion of the string would be a displacement obtained by adding the +two solutions shown in Fig. 49-3. "The result, also shown in the fgure, begins +to suggest the idea of a bump running back and forth between the ends of the +string, although with only 6wo modes we cannot make a very good picture oÝ it; +more modes are needed. 'This result is, in fact, a special case of a great principle +--- Trang 873 --- +^Nj„„ ư =^^ +——. bụng La ^^ +mm (1 t=5 má +mu tr} mw +——FIRST MODE —— COMPOSITE WAVE +—— SECOND MODE +Fig. 49-3. Two modes combine to give a travelling wave. +for linear systems: +Anụ motion at dÌlÌ can be analUzcd bụ assuming that it ¡s the sưm oƒ the +motions oƒ dÌÌ the difƒerent mmodes, combined uuith œppropriate amplitudes and +phases. +The importance of the principle derives from the fact that each mode is very +simple—it is nothing but a sinusoidal motion in time. lt is true that even the +general motion of a string is not really very complicated, but there are other +systems, for example the whipping of an airplane wing, in which the motion +1s much more complicated. Nevertheless, even with an airplane wing, we fnd +there is a certain particular way of twisting which has one Írequency and other +ways Of twisting that have other frequencies. If these modes can be found, then +the complete motion can always be analyzed as a superposition of harmonic +oscillations (except when the whipping is of such degree that the system can no +longer be considered as linear). +49-3 Modes in two dimensions +The next example to be considered is the interesting situation of modes in +two dimensions. p to this point we have talked only about one-dimensional +situations—a stretched string or sound waves in a tube. Ultimately we should +consider three dimensions, but an easier step will be that to two dimensions. +--- Trang 874 --- +⁄ Clamped Edges +b\š A2Nc +„Wave e*“t[e-lsstlwr] +JÊN | +lò ` 4x +Fig. 49-4. Vibrating rectangular plate. +Consider for deÑniteness a rectangular rubber drumhead which is conlned so as to +have no displacement anywhere on the rectangular edge, and let the dimensions of +the rectangle be ø and b, as shown ïn Fig. 49-4. Now the question is, what are the +characteristics of the possible motion? We can start with the same procedure used +for the string. If we had no confnement at all, we would expect waves travelling +along with some kind of wave motion. For example, (e“Đ(e~?#z##2*z⁄) would +represent a sine wave travelling in some direction which depends on the relative +values of k„ and k„. Now how can we make the z-axis, that is, the line = 0, a +node? sing the ideas developed for the one-dimensional string, we can imagine +another wave represented by the complex function (—e“f)(e~?*z#=i*”), The +superposition of these waves will give zero displacement at = 0 regardless of +the values of z and ý. (Although these functions are delned for negative where +there is no drumhead to vibrate, this can be ignored, since the displacement is +truly zero at =0.) In this case we can look upon the second function as the +refected wave. +However, we want a nodal line at = b as well as at —= 0. How do we +do that? 'Phe solution is related to something we did when studying refection +trom crystals. These waves which cancel each other at = 0 will do the same +a% = b only if 2bsin Ø is an integral multiple of À, where Ø is the angle shown in +Fig. 49-4: +mÀ = 2bsin 0, m =0, 1, 2,... (49.7) +Now in the same way we can make the -axis a nodal line by adding two +more functions —(e“®)(eTf#«#+2u) and +(e“)(e†2Rz#—i*9), cach representing +a refection of one of the other bwo waves from the ø = 0 line. The condition for +a nodal line at ø = œø is similar to the one for # = 0. It is that 2acos Ø must also +be an integral multiple of À: +?„\À = 2acos 0. (49.8) +--- Trang 875 --- +Then the fñnal result is that the waves bouncing about in the box produce a +standing-wave pattern, that is, a defnite mode. +So we must satisfy the above two conditions iŸ we are to have a mode. Let +us first ñnd the wavelength. 'Phis can be obtained by eliminating the angle Ø +from (49.7) and (49.8) to obtain the wavelength in terms of ø, b, œ and mm. +The easiest way to do that is to divide both sides of the respective equations +by 2b and 2a, square them, and add the two equations together. The result +is sin? Ø + cos2 Ø = 1 = (nÀ/2a) + (mA/2ð)2, which can be solved for À: +1 n m2 +¬== Ta +_a: 49.9 +À2 4a2 + 4i2 (49.9) +In this way we have determined the wavelength in terms of two integers, and +from the wavelength we immediately get the frequency œ, because, as we know, +the equency is equal to 2c divided by the wavelength. +'This result 1s interesting and important enough that we should deduce it by +a purely mathematical analysis instead of by an argument about the refections. +Let us represent the vibration by a superposition of four waves chosen so that +the four lines z = Ú, z = ø, =0, and = ö are all nodes. In addition we shall +require that all waves have the same frequency, so that the resulting motion +will represent a mode. From our earlier treatment of light refection we know +that (c#9(e~?*e#+#v9) represents a wave travelling in the direction indicated in +Fig. 49-4. Equation (49.6), that is, k = œ/c, still holds, provided +k? = kệ + kệ. (49.10) +lt is clear from the figure that k„ = kcosØ and k„ = ksin 0. +Now our equation for the displacement, say ở, of the rectangular drumhead +takes on the grand form +¿= le“ [e(—/#ez+ikuv) — c(†ikz~+iRyU) _— c(~?Rz=—iRu) +ettRez—/Ryu)], (49.11a) +Although this looks rather a mess, the sum of these things now is not very hard. +'The exponentials can be combined to give sine functions, so that the displacement +turns out to be +ó = |4sin k„z sin k„][e“1. (49.11b) +In other words, it is a sinusoida]l oscillation, all right, with a pattern that is also +sinusoidal in both the z- and the ø-direction. Our boundary conditions are of +--- Trang 876 --- +course satisfed at z = 0 and =0. We also want ¿ to be zero when ø = ø and +when = 0. Therefore we have to put in two other conditions: &„ø must be an +integral multiple oŸ x, and k„b must be another integral multiple of . 5ince we +have seen that k„ = kcosØ and k„ = ksin Ø, we immediately get equations (49.7) +and (49.8) and from these the ñnal result (49.9). +Now let us take as an example a rectangle whose width is twice the height. IÝ +we take œ = 2b and use qs. (49.4) and (49.9), we can calculate the frequencies +of all of the modes: +3 7€ \ˆ 4m + m +u“= (5) —T—' (49.12) +Table 49-1 lists a few of the simple modes and also shows their shape in a +qualitative way. +'The most important point to be emphasized about this particular case is that +the frequencies are not multiples of each other, nor are they multiples of any +number. 'Phe idea that the natural frequencies are harmonically related is not +generally true. l% is not true for a system of more than one dimension, nor is +1t true for one-dimensional systems which are more complicated than a string +with uniform density and tension. Ä simple example of the latter is a hanging +chain in which the tension is higher at the top than at the bottom. TỶ such a +chaïn is set in harmonic oscillation, there are various modes and frequencies, but +the frequencies are not simple multiples of any number, nor are the mode shapes +sinusoidal. +'The modes of more complicated systems are still more elaborate. For example, +Inside the mouth we have a cavity above the vocal cords, and by moving the +tongue and the lips, and so forth, we make an open-ended pipe or a closed-ended +pipe of diferent diameters and shapes; it is a terribly complicated resonator, but +1t is a resonator nevertheless. Now when one talks with the vocal cords, they are +made to produce some kind of tone. The tone 1s rather complicated and there +are many sounds coming out, but the cavity of the mouth further modifes that +tone because of the various resonant frequencies of the cavity. Eor instance, a +Singer can sing various vowels, a, or o, or oo, and so forth, at the same pitch, but +they sound diferent because the various harmoniecs are in resonance in this cavity +to different degrees. The very great importance of the resonant frequencies of a +cavity in modifying the voice sounds can be demonstrated by a simple experiment. +Since the speed of sound goes as the reciprocal of the square root of the density, +the speed of sound may be varied by using diferent gases. lf one uses helium +--- Trang 877 --- +Table 49-1 +__ Modeshpe mm (w/@g` œ/@p ++. —. + 1 3 3.25 1.80 +¬ 2 1 4.25 2.06 +¬. 2 2 5.00 2.24 +Instead of air, so that the density 1s lower, the speed of sound is mụch higher, +and all the frequencies of a cavity will be raised. Consequently 1ƒ one fills oneˆs +lungs with helium before speaking, the character of his voice will be drastically +altered even though the vocal cords may still be vibrating at the same frequency. +49-4 Coupled pendulums +tPinally we should emphasize that not only do modes exist for complicated +continuous systems, but also for very simple mechanical systems. À good example +1s the system of two coupled pendulums discussed in the preceding chapter. In +that chapter it was shown that the motion could be analyzed as a superposition +of two harmonic motions with diferent frequencies. So even this system can +--- Trang 878 --- +be analyzed in terms of harmonic motions or modes. The string has an infnite +number of modes and the two-dimensional surface also has an infnite number of +modes. In a sense it is a double inñnity, if we know how to count infnities. But +a simple mechanical thing which has only two degrees of freedom, and requires +only two variables to describe it, has only two modes. +Fig. 49-5. Iwo coupled pendulums. +Let us make a mathematical analysis of these two modes for the case where +the pendulums are of equal length. Let the displacement of one be z, and the +displacement of the other be , as shown in Fig. 49-5. Without a spring, the force +on the first mass is proportional to the displacement of that mass, because of +gravity. Thhere would be, if there were no spring, a certain natural frequency œg +for this one alone. 'Phe equation of motion without a spring would be +mg = —TnuA#. (49.13) +The other pendulum would swing in the same way If there were no spring. In +addition to the force of restoration due to gravitation, there is an additional force +pulling the first mass. hat force depends upon the excess distance oŸ ø over +and 1s proportional to that diference, so it is some constant which depends on +the geometry, times (œ — ). The same force in reverse sense acts on the second +mass. The equations of motion that have to be solved are therefore +d2+z dˆụ +m nà = —mua — k(œ — 9), mì nà = —mua — k(w— #). (49.14) +In order to fñnd a motion in which both of the masses move at the same +frequency, we must determine how much each mass moves. In other words, +--- Trang 879 --- +pendulum z and pendulum ø will oscillate at the same frequency, but their +amplitudes must have certain values, 4 and , whose relation is fixed. Leb us +try this solution: +œ= Ac“!, ụ= Bể*“!, (49.15) +Tf these are substituted in Bqs. (49.14) and similar terms are collected, the results +(‹: — ư§ — xÌA = _*p, +(49.16) +2 s.k k +(‹ — Ư§ — )#= —— A. +The equations as written have had the common factor e““f removed and have +been divided by m. +Now we see that we have bwo equations for what looks like two unknowns. +But there really are not #ø unknowns, because the whole size of the motion is +something that we cannot determine from these equations. 'Phe above equations +can determine only the rœfio of A to Ð, but the must both giue the same ratio. +'The necessity for both of these equations to be consistent is a requirement that +the frequency be something very special. +Tn this particular case this can be worked out rather easily. Ifthe two equations +are multiplied together, the result is +(‹: — 8 — m) AB= (ñ) AB. (49.17) +The term 4? can be removed from both sides unless Á and Ö are zero, which +means there is no motion at all. If there is motion, then the other terms must +be equal, giving a quadratic equation to solve. The result ¡is that there are bwo +possible frequencies: +7 mu, — U9 muổ + Bà (49.18) +Furthermore, if these values of frequency are substituted back into Eq. (49.16), +we find that for the first frequency A = Ö, and for the second frequency A = —Ö. +These are the “mode shapes,” as can be readily verified by experiment. +It is clear that in the frst mode, where A = ?Ö, the spring is never stretched, +and both masses oscillate at the frequenecy œo, as though the spring were absent. +In the other solution, where A = —?Ö, the spring contributes a restoring force +--- Trang 880 --- +and raises the Írequency. A more interesting case results if the pendulums have +diferent lengths. The analysis is very similar to that given above, and is left as +an exercise for the reader. +49-5 Linear systems +Now let us sunmarize the ideas discussed above, which are all aspects of what +1s probably the most general and wonderful principle of mathematical physics. +l we have a linear system whose character is independent of the time, then +the motion does not have to have any particular simplicity, and in fact may +be exceedingly complex, but there are very special motions, usually a series +of special motions, in which the whole pattern of motion varies exponentially +with the time. For the vibrating systems that we are talking about now, the +exponential is imaginary, and instead of saying “exponentially” we might prefer +to say “sinusoidally” with time. However, one can be more general and say that +the motions will vary exponentially with the time in very special modes, with very +special shapes. The most general motion of the system can always be represented +as a superposition of motions involving each of the diferent exponentials. +This is worth stating again for the case oŸ sinusoidal motion: a linear system +need not be moving in a purely sinusoidal motion, I1.e., at a defnite single +frequency, but no matter how it does move, this motion can be represented as a +superposition of pure sinusoidal motions. The frequency of each of these motions +1s a characteristic of the system, and the pattern or waveform of each motion is +also a characteristic of the system. “he general motion in any such system can +be characterized by giving the strength and the phase of each of these modes, +and adding them all together. Another way of saying this is that any linear +vibrating system ¡is equivalent to a set of independent harmonic oscillators, with +the natural frequencies corresponding to the modes. +W©e conclude this chapter by remarking on the connection of modes with +quantum mechanics. In quantum mechanics the vibrating object, or the thing +that varies in space, is the amplitude of a probability function that gives the +probability of ñnding an electron, or system of electrons, in a given configuration. +This amplitude function can vary in space and time, and satisfes, in fact, a +linear equation. But in quantum mechanics there is a transformation, in that +what we call frequency of the probability amplitude is equal, in the classical +idea, to energy. Therefore we can translate the principle stated above to this +case by taking the word ƒreqguencw and replacing it with energy. It becomes +--- Trang 881 --- +something like this: a quantum-mechanical system, for example an atom, need +not have a defnite energy, jus as a simple mechanical system does not have to +have a defnite frequency; but no matter how the system behaves, its behavior +can always be represented as a superposition of states of definite energy. The +energy of each state is a characteristic of the atom, and so is the pattern of +amplitude which determines the probability of ñnding particles in diferent places. +The general motion can be described by giving the amplitude of each of these +diferent energy states. This is the origin of energy levels in quantum mechanics. +Since quantum mechanics is represented by waves, in the circumstance in which +the electron does not have enough energy to ultimately escape from the proton, +they are confined aues. Like the confned waves of a string, there are delnite +frequencies for the solution of the wave equation for quantum mechanics. The +quantum-mechanical interpretation is that these are defñnite energies. Therefore +a quantum-mechanical system, because it is represented by waves, can have +defnite states of ñxed energy; examples are the energy levels of various atoms. +--- Trang 882 --- +F/ 70/1/7011) +50-1 Musical tones +Pythagoras is said to have discovered the fact that two similar strings under +the same tension and difering only in length, when sounded together give an +effect that is pleasant to the ear #ƒ the lengths of the strings are in the ratio of +two small integers. lf the lengths are as one is to two, they then correspond to +the octave in music. lỶ the lengths are as two is to three, they correspond to the +interval between Œ and Œ, which is called a ñfth. These intervals are generally +accepted as “pleasant” sounding chords. +Pythagoras was so impressed by this discovery that he made it the basis of a +school—Pythagoreans they were called——which held mystie belief§ in the great +powers of numbers. It was believed that something similar would be found out +about the planets—or “spheres.” We sometimes hear the expression: “the music +of the spheres.” The idea was that there would be some numerical relationships +between the orbits of the planets or between other things in nature. People +usually think that this is just a kind of superstition held by the Greeks. But +1s it so diferent from our own scientifc interest in quantitative relationships? +Pythagoras' discovery was the frst example, outside geometry, of any numerical +relationship in nature. It must have been very surprising to suddenly discover +that there was a ƒfac£ of nature that involved a simple numerical relationship. +Simple measurements of lengths gave a prediction about something which had +no apparent connection to geometry——the production of pleasant sounds. 'This +discovery led to the extension that perhaps a good tool for understanding nature +would be arithmetic and mathematical analysis. he results of modern science +Justify that point of view. +Pythagoras could only have made his discovery by making an experimental +observation. Yet this important aspect does not seem to have impressed him. TỶ +--- Trang 883 --- +it had, physics might have had a much earlier start. (Tt is always easy to look +back at what someone else has done and to decide what he sbould have donel) +We might remark on a third aspect of this very interesting discovery: that +the discovery had to do with t6wo notes that sownd pleasant to the ear. We may +question whether +0 are any better off than Pythagoras in understanding +0 +only certain sounds are pleasant to our ear. 'Phe general theory of aesthetics is +probably no further advanced now than in the time of Pythagoras. In this one +discovery of the Greeks, there are the three aspects: experiment, mathematical +relationships, and aesthetics. Physics has made great progress on only the first +two parts. This chapter will deal with our present-day understanding of the +discovery of Pythagoras. +Among the sounds that we hear, there is one kind that we call nø¿se. Noise +corresponds to a sorE oŸ irregular vibration of the eardrum that is produced by the +irregular vibration of some object in the neighborhood. IỶ we make a diagram to +indicate the pressure oŸ the air on the eardrum (and, therefore, the displacement +of the drum) as a function of time, the graph which corresponds to a noise might +look like that shown in Pig. 50-1(a). (Such a noise might correspond roughly +to the sound of a stamped foot.) The sound of muws¿c has a different character. +Music is characterized by the presence of more-or-less swstaimed ‡ones——or musical +PRESSURE +(a) A NOISE +PRESSURE +JV \ JVÀ JV\ [ TIME +— 7T —l +(b) A MUSICAL TONE +Fig. 50-1. Pressure as a function of time for (a) a noise, and (b) a +musical tone. +--- Trang 884 --- +“nobes.” (Musical instruments may make noises as welll) The tone may last for a +relatively short time, as when a key is pressed on a piano, or it may be sustained +almost indefnitely, as when a fñute player holds a long note. +'What is the special character of a musical note from the point of view of the +pressure in the air? A musical note difers from a noise in that there is a periodicity +in its graph. There is some uneven shape to the variation of the air pressure with +time, and the shape repeats itself over and over again. An example oŸ a pressure- +time function that would correspond to a musical note is shown in Eig. 50-1(b). +Musicians will usually speak of a musical tone in terms of three characteristics: +the loudness, the pitch, and the “quality.” 'Phe “loudness” is found to correspond +to the magnitude of the pressure changes. The “pitch” corresponds to the period +of tìme for one repetition of the basic pressure function. (“Low” notes have +longer periods than “high” notes.) The “quality” of a tone has to do with the +difÑferences we may still be able to hear between two notes of the same loudness +and pitch. An oboe, a violin, or a soprano are still distinguishable even when +they sound notes of the same pitch. The quality has to do with the structure of +the repeating pattern. +Let us consider, for a moment, the sound produced by a vibrating string. lf +we pluck the string, by pulling ít to one side and releasing it, the subsequent +motion will be determined by the motions of the waves we have produced. We +know that these waves will travel in both directions, and will be reflected at the +ends. They will slosh back and forth for a long time. Ño matter how complicated +the wave is, however, it will repeat itself. "The period of repetition is Just the +time 7' required for the wave to travel t6wo full lengths of the string. For that +1s just the time required for any wave, once started, to reflect of each end and +return ©o its starting position, and be proceeding in the original direction. 'Phe +time is the same for waves which start out in either direction. Each point on the +string will, then, return to its starting position after one period, and again one +period later, etc. The sound wave produced must also have the same repetition. +'W©e see why a plucked string produces a musical tone. +50-2 The Fourier series +W©e have discussed in the preceding chapter another way of looking at the +motfion of a vibrating system. We have seen that a string has various natural +modes of oscillation, and that any particular kind of vibration that may be set +up by the starting conditions can be thought of as a combination—in suitable +--- Trang 885 --- +proportions—of several of the natural modes, oscillating together. For a string we +found that the normal modes of oscillation had the frequencies œọ, 2o, 3œ, .... +The most general motion of a plucked string, therefore, is composed of the +sum of a sinusoidal oscillation at the fundamental frequenecy œọ, another at the +second harmonie frequenecy 2œ, another at the third harmonic 3œ, etc. Now the +fundamental mode repeats itself every period 71 = 2z/œo. The second harmonie +mode repeats itself every 72 = 27/2. Ib øÏso repeats itself every Tì = 27, +after #uo of its periods. Similarly, the third harmonic mode repeats itself after a +time 71 which is 3 of its periods. We see again why a plucked string repeats its +whole pattern with a periodicity of 7. It produces a musical tone. +W©e have been talking about the motion of the string. But the sound, which +1s the motion of the air, is produced by the motion of the string, so its vibrations +too must be composed of the same harmonics—though we are no longer thinking +about the normal modes of the air. Also, the relative strength of the harmonics +may be diferent in the air than in the string, particularly if the string is “coupled” +to the air via a sounding board. “The efficiency of the coupling to the air is +diferent for diferent harmonics. +Tf we let ƒ(#) represent the air pressure as a function of time for a musical tone +|such as that in Fig. 50-1(b)], then we expect that ƒ(#) can be written as the sum +of a number of simple harmonic functions of time——like cosœ#—for each of the +various harmonic frequencies. If the period of the vibration is 7, the fundamental +angular frequency will be œ = 2/7, and the harmonics will be 2œ, 3œ, etkc. +'There is one slipht complication. Eor each frequency we may expect that the +starting phases will not necessarily be the same for all equencies. We should, +therefore, use functions like cos (œ£ -Ƒ @). It is, however, simpler to use instead +both the sine and cosine functions for cøch frequency. We recall that +cos (É + ở) = (cos Ócos @‡ — sin ở sin œ£) (50.1) +and since ở is a constant, øww sinusoidal oscillation at the frequency ¿ can be +written as the sum of a term with cosœ# and another term with sin œ#. +We conclude, then, that anmg function ƒ(f) that is periodic with the period 7 +can be written mathematically as +ƒŒ) = ao ++øicos + Ùqsin (œ ++ aa cos 2 + ba sin 2t +--- Trang 886 --- ++ ag cos 3£ + ba sỉn 3# ++-:-- +--- (50.2) +where œ = 27/7 and the a*s and *s are numerical constants which tell us how +much of each component oscillation is present in the oscillation ƒ(£). We have +added the “zero-frequency” term øo so that our formula will be completely general, +although ït is usually zero for a musical tone. Ït represents a shift of the average +value (that is, the “zero” level) of the sound pressure. With i9 our formula can +take care of any case. The equality of Eq. (50.2) is represented schematically +in Eig. 50-2. (The amplitudes, œ„ and bạ, of the harmonic functions must be +suitably chosen. 'Phey are shown schematically and without any particular scale +in the ñgure.) The series (50.2) is called the #ouwrier series for ƒ(t). +: -+—_ ++ —: + ^=—: ++ NÓ + ˆV: ++ etc. + etc. +Fig. 50-2. Any periodic function f(£) is equal to a sum of simple +harmonic functions. +W© have said that am periodic function can be made up in this way. We +should correct that and say that any sound wave, or any function we ordinarily +encounter in physics, can be made up of such a sum. The mathematicians can +Invent functions which cannot be made up of simple harmonic funetions——for +instance, a function that has a “reverse twist” so that it has two values for some +values of ! We need not worry about such functions here. +--- Trang 887 --- +50-3 Quality and consonance +Now we are able to describe what it is that determines the “quality” of a +musical tone. lt is the relative amounts of the various harmonics—the values of +the a's and 0s. A tone with only the first harmonic is a “pure” tone. A tone with +many strong harmonics is a “rich” tone. Á violin produces a đifferent proportion +of harmonies than does an oboe. +W© can “manufacture” various musical tones If we connect several “oscillators” +to a loudspeaker. (An oscillator usually produces a nearÌy pure simple harmonie +function.) We should choose the frequencies of the oscillators to be œ, 2, 3œ, ebc. +Then by adjusting the volume control on each oscillator, we can add in any +amount we wish of each harmonic—thereby producing tones of diferent quality. +An electric organ works in much this way. The “keys” select the frequency of +the fundamental oscillator and the “stops” are switches that control the relative +proportions of the harmonics. By throwing these switches, the organ can be +made to sound like a fute, or an oboe, or a violin. +lt is interesting that to produce such “artificial” tones we need only one +oscillator for each frequency——we do not need separate oscillators for the sine +and cosine components. The ear is not very sensitive to the relative phases Of +the harmonics. lt pays attention mainly to the #ø#@l of the sỉne and cosine parts +of each frequency. Our analysis is more accurate than is necessary to explain +the sưub7ectiue aspect of music. The response of a microphone or other physical +instrument does depend on the phases, however, and our complete analysis may +be needed to treat such cases. +The “quality” of a spoken sound also determines the vowel sounds that we +recognize in speech. 'Phe shape of the mouth determines the frequencies of the +natural modes of vibration of the air in the mouth. Some of these modes are +set into vibration by the sound waves om the vocal chords. In this way, the +amplitudes of some of the harmonics of the sound are increased with respect +to others. When we change the shape of our mouth, harmonics of diferent +frequencies are given preference. 'Phese efects account for the diference between +an “e-e-e” sound and an “a-a-a” sound. +W© all know that a particular vowel sound——say “e-e-e”——still “sounds like” +the same vowel whether we say (or sing) it at a hiph or a low piích. From +the mechanism we describe, we would expect that parf¿cular frequencies are +emphasized when we shape our mouth for an “e-e-e,” and that they do no£ +change as we change the pitch of our voice. So the relation of the important +--- Trang 888 --- +harmonics to the fundamental—that is, the “quality”—changes as we change +pitch. Apparently the mechanism by which we recognize speech is not based on +specifc harmonic relationships. +'What should we say now about Pythagoras' discovery? We understand that +two similar strings with lengths in the ratio of 2 to 3 will have fundamental +frequencies in the ratio 3 to 2. But why should they “sound pleasant” together? +Perhaps we should take our clue from the frequencies of the harmonics. “The +second harmonic of the lower shorter string will have the sœme frequenecy as the +third harmonic of the longer string. (Tt is easy to show—or to believe—that a +plucked string produces strongly the several lowest harmonics.) +Perhaps we should make the following rules. Notes sound consonant when +they have harmonics with the same frequency. Notes sound dissonant If their +upper harmonics have frequencies near to each other but far enough apart that +there are rapid beats between the two. Why beats do not sound pleasant, and +why unison of the upper harmonics does sound pleasant, is something that we +do not know how to define or describe. We cannot say from this knowledge of +what sounds good, what ought, for example, to smell good. In other words, our +understanding of it is not anything more general than the statement that when +they are in unison they sound good. It does not permit us to deduce anything +more than the properties of concordance in music. +Tt is easy to check on the harmonic relationships we have described by some +simple experiments with a piano. Let us label the 3 successive C°s near the middle +of the keyboard by C, C”, and C”, and the G”s Just above by G, G”, and G”. +'Then the fundamentals wïll have relative frequencies as follows: +€ -2 G-3 +C“-4 G - 6 +C“-8 G“-12 +'These harmonic relatlonships can be demonstrated in the following way: Suppose +we press C“ sỈouÏ—so that i9 does not sound but we cause the damper to be +lifted. If we then sound €, it will produce its own fundamental and some second +harmonic. 'The second harmonic will set the strings of C7 into vibration. IÝ we +now release Ở (keeping C” pressed) the damper will stop the vibration of the C +sirings, and we can hear (softly) the note C” as it dies away. In a similar way, the +third harmonic of C can cause a vibration of Gf. Or the sixth of C (now getting +much weaker) can set up a vibration in the fundamental of G”. +--- Trang 889 --- +A somewhat diferent result is obtained if we press G quietly and then sound +C7. The third harmonic of C7 will correspond ©o the fourth harmonie oŸ Œ, so +oml the fourth harmonic of G will be excited. We can hear (if we listen closely) +the sound of G”, which is two octaves above the G we have pressedl It is easy to +think up many more combinations for this game. +We may remark in passing that the major scale can be defned just by the +condition that the three major chords (E-A-©); (C=E-G); and (G-B-D) cách +represent tone sequences with the frequency ratio (4: 5 : 6). These ratios—plus +the fact that an octave (C—C”, B-B/, etc.) has the ratio 1 : 2—determine the +whole scale for the “ideal” case, or for what is called “Just intonation.” Keyboard +instruments like the piano are nøø usually tuned in this manner, but a little +“fudging” is done so that the Írequencies are øpprozimatel correct for all possible +sbarting tones. For this tuning, which is called “tempered,” the octave (still 1 : 2) +is đivided into 12 equal intervals for which the frequency ratio is (2)!⁄12, A fifth +no longer has the frequeney ratio 3/2, but 27/12 = 1.499, which is apparently +close enough for most ears. +W© have stated a rule for consonance in terms of the coineidence of harmonics. +Ts this coincidence perhaps the reason that two notes are consonant? One worker +has claimed that two pure tones—tones carefully manufactured to be free of +harmonies—do not give the sensafions of consonance or dissonance as the relative +Írequencies are placed at or near the expected ratios. (Such experiments are +difficult because ït is difficult to manufacture pure tones, for reasons that we +shall see later.) We cannot still be certain whether the ear is matching harmonics +or doïng arithmetic when we decide that we like a sound. +50-4 The Fourier coefficients +Let us return now to the idea that any note—that 1s, a per?odic sound——can +be represented by a suitable combination of harmonics. We would like to show +how we can find out what amount of each harmonic is required. ϧ is, of course, +casy to compute ƒ(£), using Eq. (50.2), iƒ we are giuen all the coefficients ø and b. +The question now is, if we are given ƒ(#) how can we know what the coefficients +of the various harmonie terms should be? (Tt is easy to make a cake from a recipe; +but can we write down the recipe iŸ we are given a cake?) +Fourier discovered that it was not really very difficult. The term ao is certainly +casy. We have already said that it is just the average value of ƒ(#) over one period +(from # = 0 to ý = 7). We can easily see that this is indeed so. The average value +--- Trang 890 --- +Of a sine or cosine function over one period is zero. Over two, or three, or any +whole number of periods, it is also zero. So the average value of all of the terms +on the right-hand side of Eq. (50.2) is zero, except for øo. (Recall that we must +choose œ = 2/7.) +Now the average of a sum is the sum of the averages. So the average of ƒ(£) +1s just the average of øo. But øo 1s a constønt, so 1ts average is just the same as +its value. Recalling the definition oŸ an average, we have +ao—= + lI ƒ0)di. (50.3) +The other coefficients are only a little more dificult. To fnd them we can +use a trick discovered by Eourier. Suppose we multiply both sides of Eq. (50.2) +by some harmonic function—say by cos 7/#. We have then +ƒ() - cos TuÈ = ao - cos 7u ++øqCOS (@Ý - cCOs 7# + bqsỉn œÉ - cOs TúÈ ++ aa cOsS 2£ - cOs 7È + ba sin 2úJÊ - cOS 7 (0È ++ đy COS 7È - cOS 7È + bự sin TúJÊ - cOS 7 (0È ++--- mm (50.4) +No let us average both sides. "The average of øo cos 7 over the tỉme T is +proportional to the average of a cosine over 7 whole periods. But that is just +zoro. The average of øửnost øÏl of the rest of the terms is aso 2ero. Let us look +at the ơi term. We know, in general, that +cos 4cos = š cos(A + B) + š cos (A — ). (50.5) +'The ai term becomes +301 (cos 8£ + cos 6Ÿ). (50.6) +We thus have two cosine terms, one with 8 full periods in 7' and the other with 6. +The both querage ‡o zcro. The average of the a term is therefore zero. +For the ø¿ term, we would fnd a¿ cos 9ý and øa cos 5, each of which also +averages to zero. For the œo term, we would ñnd cos 16 and cos(—2œ#). But +cos (—2/È) is the same as cos 2#, so both oŸ these have zero averages. ÏIb is clear +--- Trang 891 --- +that øÏl of the ø terms will have a zero average ezcep‡ one. And that one is the +ơ;y term. Eor this one we have +súr(cos 14¿£ + cos 0). (50.7) +'The cosine of zero is one, and its average, of course, is one. 5o we have the result +that the average of all of the ø terms oŸ Eq. (50.4) equals d7. +The ö terms are even easier. When we multiply by any cosine term like cos ¿Ý, +we can show by the same method that øl of the b terms have the average value +We see that Eourier's “trick” has acted like a sieve. When we multiply +by cos 7/ and average, all terms drop out except œ;, and we find that +Average [ƒ() - cos 7/f] = ar/2, (50.8) +đự —= rÍ ƒ() - cos Tot di. (50.9) +We© shall leave it for the reader to show that the coefficient b; can be obtained +by multiplying Eq. (50.2) by sin 7# and averaging both sides. The result is +bạ = rÍ ƒ(Ð - sin 7œt di. (50.10) +Now what is true for 7 we expect is true for any integer. So we can summarize +our proof and result in the following more elegant mathematical form. In and mœ +are integers other than zero, and IŸ œ¡ = 2z/T, then +1. I sin noÝ cos mmu‡ dự = 0. (50.11) +TH. ‡ ‡ dt = +/ COS '\UŸ GOS TU 0 iFn #m. +(50.12) +T 7/2 ifn=mm. +TH. I sin nu sin mứt dÉ — +IV. ƒŒ)=so+ » đựy, COS THUÊ ~E » b„ sìn nư‡. (50.13) +m„=l ni +--- Trang 892 --- +V. qọ= rÍ ƒ(@ dt. (50.14) +địạ — Ti ƒ() - cos not đt. (50.15) +bạ —= rÍ ƒ@) - sin nứt dt. (50.16) +In earlier chapters it was convenlent to use the exponential notation for +representing simple harmonic motion. Instead of cos¿# we used Re c““f, the real +part of the exponential function. We have used cosine and sine functions in this +chapter because it made the derivations perhaps a little clearer. Our fñnal result +of Eq. (50.13) can, however, be written in the compact form +ƒŒ) = Re ânc th (50.17) +where â„ is the complex number œ„ — ?b„ (with bọ = 0). IÝ we wish to use the +same notation throughout, we can write also +PNH ; +ân = rỊ ƒ(Đe~"*f đt (n > 1). (50.18) +W©e now know how to “analyze” a periodic wave into its harmonic components. +The procedure is called #ourier analusis, and the separate terms are called +Fourier components. We have øœø‡ shown, however, that once we find all of the +Fourier components and add them together, we do indeed get back our ƒ(/). The +mathematicians have shown, for a wide class of functions, in fact for all that are +of interest to physicists, that IŸ we can do the integrals we will get back ƒ(). +There is one minor exception. TÝ the function ƒ(#) is discontinuous, i.e., iÝ it jumps +suddenly from one value to another, the Fourier sum will give a value at the +breakpoint halfway between the upper and lower values at the discontinuity. So +1ƒ we have the strange function ƒ(£) =0, 0 < £< tạ, and ƒ( = 1 for fạ<£<7, +the Fourier sum will give the right value everywhere ezcep‡ ø‡ tọ, where it will +have the value 3 instead of 1. It is rather unphysical anyway to insist that a +funection should be zero œp #o ứoọ, but 1 r7gh# ø‡ tọ. So perhaps we should make +the “rule” for physicists that any discontinuous function (which can only be a +simplification of a real physical function) should be defined with halfway values +--- Trang 893 --- +17/2 IÏ £ +1 x““.. +_ ]+1 for0+—+——_—--. +x5) (tạp tp tạ tên) +so we learn that the sum of the squares of the reeiprocals of the odd integers is 2/8. +In a similar way, by fñrst obtaining the Eourier series for the function ƒ(£) = +(t— T/2)2 and using the energy theorem, we can prove that 1+ 1/2%-+1/34+::: +is 1/90, a result we needed in Chapter 45. +50-6 Nonlinear responses +Finally, in the theory of harmonics there is an important phenomenon which +should be remarked upon because of its practical Importance—that of nonlinear +tan”!z, Second, we expand the integrand in a series 1/(1 + #2) =1—#2+z*—z8+--- We +integrate the series term by term (from zero to #) to obtain tan” + = z—z3/3-++5/5—œ” /T-+E--: +Setting ø = 1, we have the stated result, since tan” 1 = z/4. +--- Trang 895 --- +efects. In all the systems that we have been considering so far, we have supposed +that everything was linear, that the responses to Íorces, say the displacements or +the accelerations, were always proportional to the forces. Ôr that the currents in +the cireuits were proportional to the voltages, and so on. We now wish to consider +cases where there is not a strict proportionality. We think, at the moment, of +some device in which the response, which we will call #øeụy at the time ứ, 1s +determined by the input z¡nạ at the time . For example, #ø¡n might be the force +and #ou¿ might be the displacement. Ôr z¡„ might be the current and #ou¿ the +voltage. If the device is linear, we would have +#out (9 — Kzu(), (50.24) +where #C is a constant independent of £ and of #¡n. Suppose, however, that the +device is nearly, but not exactly, linear, so that we can write +#out(Đ = K[zin() + cz2 (Đ], (50.25) +where e is small in comparison with unity. Such linear and nonlinear responses +are shown in the graphs of Fig. 50-4. +Xout Xout +/ Xin / Xin +(a) LINEAR (b) NONLINEAR +Xout = Xin Xout = K(xn + ex2) +Fig. 50-4. Linear and nonlinear responses. +Nonlinear responses have several important practical consequences. We shall +discuss some of them now. First we consider what happens 1Ý we apply a pure +tone at the input. We let #¡n — cosœứ. IÝ we plot #ou¿ as a function of time we get +the solid curve shown in Eig. 50-5. The dashed curve gives, for comparison, the +response of a linear system. We see that the output is no longer a cosine function. +lt is more peaked at the top and flatter at the bottom. We say that the output +1s đistortcd. We know, however, that such a wave is no longer a pure tone, that +--- Trang 896 --- +NONLINEAR +W, : +N~- LINEAR N +Fig. 50-5. The response of a nonlinear device to the input cos¿uf. A +linear response Is shown for comparison. +it will have harmonics. We can fnd what the harmonics are. sing #¡n = cos „Ý +with Eq. (50.25), we have +#out(Ê) = K(cosœ£ + ccos? œ£). (50.26) +Erom the equality cos? Ø = š(1 + cos2Ø), we have +#out(#) = Kco œ‡ + s?gc08 21). (50.27) +The output has not only a component at the fundamental frequeney, that was +present at the input, but also has some of its second harmonic. There has also +appeared at the output a constant term “(e/2), which corresponds to the shift +of the average value, shown ïn Eig. 50-5. The process of produecing a shift of the +average value 1s called rectjfication. +A nonlinear response will rectify and will produce harmonics of the frequencies +at its input. Although the nonlinearity we assumed produced only second +harmonics, nonlinearities of higher order—those which have terms like zøÿ, and #‡., +for example—will produce harmonics higher than the second. +Another efect which results from a nonlinear response is rmodulatlion. TỶ +our input function contains two (or more) pure tones, the output will have +not only their harmonics, but still other frequency components. Let #¡n = +Acos¿1f + cosúaf, where now œ1 and œ2 are øø£ intended to be in a harmonic +relation. In addition to the linear term (which is # times the input) we shall +--- Trang 897 --- +have a component in the output given by +#ouy = e(A cosư£ + Ðcos ằ2£)Ÿ (50.28) += Ké(A?cos? 1£ + B cos? ø¿£ + 2 AB cos 1£ cos 02Ÿ). (50.29) +The first two terms in the parentheses of Eq. (50.29) are just those which gave +the constant terms and second harmonic terms we found above. 'Phe last term is +We can look at this new “cross term” A4 cosu1f cos¿a£ in bwo ways. Pirst, +1f the two frequencies are widely diferent (for example, IŸ œị is much greater +than œ2) we can consider that the cross term represents a cosine oscillation of +varying amplitude. That is, we can think of the factors in this way: +ABcosu£cos 2È = C(É) cos 001, (50.30) +ŒŒ) = AB cosu¿t. (50.31) +W© say that the amplitude oŸ cos¿# is modulated with the frequenecy œ2. +Alternatively, we can write the cross term in another way: +AB cosu‡ cos 2Ÿ = " [cos (œ1 -È @2)# + cos (0 — œ2a)Ÿ]. (50.32) +We would now say that two øeu components have been produced, one at the +gưm frequency (œ1 + œ2), another at the đjƒerence frequenecy (œ1 — œ2). +W©e have two different, but equivalent, ways of looking at the same result. +In the special case that œ << ¿¿, we can relate these two diferent views by +remarking that since (œ + œ2) and (dị — œø2) are near to each other we would +expect to observe beats between them. But these beats have just the efect of +modulating the amplitude of the auerage frequenecy œị by one-half the diference +frequency 2/¿. We see, then, why the two descriptions are equivalent. +In summary, we have found that a nonlinear response produces several effects: +rectification, generation of harmonics, and modulation, or the generation of +components with sum and diference frequencies. +We should notice that all these efects (Eq. 50.29) are proportional not only to +the nonlinearity coefficient c, but also to the produet of two amplitudes——either +A?, Bˆ?,or AB. We expect these efects to be much more important for sfrong +signals than for weak ones. +--- Trang 898 --- +'The efects we have been describing have many practical applications. Eirst, +with regard to sound, it is believed that the ear is nonlinear. 'Phis is believed +to account for the fact that with loud sounds we have the sensation that we +hear harmonics and also sum and diference frequencies even ïf the sound waves +contain only pure tones. +'The components which are used in sound-reproducing equipment——amplifiers, +loudspeakers, etc.—always have some nonlinearity. They produce distortions +in the sound—they generate harmonics, etc.—which were not present in the +original sound. 'Phese new components are heard by the ear and are apparentÌy +objectionable. It is for this reason that “Hi-Fi” equipment is designed to be as +linear as possible. (Why the nonlinearities of the eør are of “objectionable” in +the same way, or how we even know that the nonlinearity is in the ioudspeaker +rather than in the eør is not clearl) +Nonlinearities are quite øecessar, and are, in fact, intentionally made large in +certain parts of radio transmitting and receiving equipment. Ín an AM transmitter +the “voice” signal (with frequencies oŸ some kilocycles per second) is combined +with the “carrier” signal (with a requency of some megacycles per second) in a +nonlinear cireuit called a zmodulator, to produce the modulated oscillation that +is transmitted. In the receiver, the components of the received signal are fed +to a nonlinear cireuit which combines the sum and diference frequencies of the +mmodulated carrier to generate again the voice signal. +'When we discussed the transmission of light, we assumed that the induced +oscillations of charges were proportional to the electric fñield of the light—that the +response was linear. That is indeed a very good approximation. lt is only within +the last few years that light sources have been devised (lasers) which produce an +intensity of light strong enough so that nonlinear efects can be observed. ÏIt is +now possible to generate harmonics of light frequencies. When a strong red light +passes through a piece of glass, a little bit of blue light——second harmonic——comes +--- Trang 899 --- +Wœe+es +51-1 Bow waves +Although we have fñnished our quantitative analyses of waves, this added +chapter on the subJect is intended to give some appreciation, qualitatively, for +various phenomena that are associated with waves, which are too complicated +to analyze in detail here. Since we have been dealing with waves for several +chapters, more properly the subject might be called “some oŸ the more complex +phenomena associated with waves.” +The first topic to be discussed concerns the efects that are produced by +a source of waves which is moving faster than the wave velocity, or the phase +velocity. Let us fñrst consider waves that have a defñnite velocity, like sound and +light. If we have a source of sound which is moving faster than the speed of +sound, then something like this happens: Suppose at a given moment a sound +wave is generated from the source at point # in Eig. 51-1; then, in the next +Z7 +Fig. 51-1. The shock wave front lies on a cone with apex at the +source and half-angle 6 = sin 1 c„/v. +--- Trang 900 --- +mmoment, as the source moves to #2, the wave from z¡ expands by a radius r1 +smaller than the distance that the source moves; and, of course, another wave +starts from z¿. When the sound source has moved still farther, to zs, and a wave +1s starting there, the wave from z#a has now expanded to rạ, and the one from #1 +has expanded to rsz. Of course the thing is done continuously, not in steps, and +therefore, we have a series of wave circles with a common tangent line which +goes through the center of the source. We see that instead of a source generating +spherical waves, as iÿ would if it were standing still, it generates a wavefront +which forms a cone in three dimensions, or a pair of lines in two dimensions. The +angle of the cone is very easy to ñgure out. Ín a given amount of time the source +moves a distance, say #s — #1, proportional to 0, the velocity of the source. Ïn +the meantime the wavefront has moved out a distance rs, proportional to c„, the +speed of the wave. Therefore it is clear that the halfangle of opening has a sỉne +equal to the ratio of the speed of the waves, divided by the speed of the source, +and this sine has a solution only if c„ is less than 0, or the speed of the object is +faster than the speed of the wave: +sin 0 = “, (51.1) +Ineidentally, although we implied that it is necessary to have a source 0Ÿ +sound, it turns out, very interestinply, that once the object is moving faster than +the speed of sound, it will make sound. 'That is, it is not necessary that it have a +certain tone vibrational character. Any object moving through a medium faster +than the speed at which the medium carries waves will generate waves on each +side, automatically, just from the motion itself. 'This is simple in the case of +sound, but i% also occurs in the case of light. At frst one might think nothing can +move faster than the speed of light. However, light in glass has a phase velocity +less than the speed of light in a vacuum, and ït is possible to shoot a charged +particle of very high energy through a bloeck of glass such that the particle velocity +1s close to the speed of light in a vacuum, while the speed of light in the glass may +be only Ỹ the speed of light in the vacuum. A particle moving faster than the +speed of light in the medium will produce a conical wave of light with its apex at +the source, like the wave wake from a boat (which is from the same efect, as a +matter of fact). By measuring the cone angle, we can determine the speed of the +particle. This is used technically to determine the speeds of particles as one of +the methods of determining their energy in high-energy research. 'Phe direction +of the light ¡is all that needs to be measured. +--- Trang 901 --- +=~ : kế 2 sp Ñ +S†_S ——. — +> : x —- +Kế S. S225 SAPICC b2 co co +Fig. 51-2. A shock wave Induced in a gas by a projectile moving faster +than sound. +This light is sometimes called Cherenkov radiation, because it was first ob- +served by Cherenkov. How intense this light should be was analyzed theoretically +by Erank and Tamm. The 1958 Nobel Prize for physics was awarded jointly to +all three for this work. +The corresponding cireumstances in the case of sound are illustrated in +Fig. 51-2, which is a photograph of an object moving through a gas at a speed +greater than the speed of sound. 'The changes In pressure produce a change In +refractive index, and with a suitable optical system the edges of the waves can be +made visible. We see that the object moving faster than the speed of sound does, +indeed, produce a conical wave. But closer inspection reveals that the surface is +actually curved. It is straight asyrmptotically, but it is curved near the apex, and +we have now to discuss how that can be, which brings us to the second topic of +this chapter. +51-2 Shock waves +'Wave speed often depends on the amplitude, and in the case of sound the speed +depends upon the amplitude in the following way. An object moving through the +aïr has to move the air out of the way, so the disturbance produced in thỉs case +--- Trang 902 --- +1s some kind of a pressure step, with the pressure higher behind the wavefront +than in the undisturbed region not yet reached by the wave (running along at the +normal speed, say). But the air that is left behind, after the wavefront passes, has +been compressed adiabatically, and therefore the temperature is increased. Now +the speed of sound increases with the temperature, so the speed in the region +behind the jump is faster than in the air in front. 'Phat means that any other +disturbanece that is made behind this step, say by a continuous pushing of the body, +or any other disturbance, will ride faster than the front, the speed increasing with +higher pressure. Pigure 51-3 illustrates the situation, with some little bumps of +pressure added to the pressure contour to aid visualization. We see that the higher +pressure regions at the rear overtake the front as time goes on, until ultimately +the compressional wave develops a sharp front. lÝ the strength is very high, +“ultimately” means right away; if it as rather weak, it takes a long time; it may be, +in fact, that the sound is spreading and dying out before I% has time to do this. +ÔN VN +ằ ta >fi b tạ > fs a +Distance +Fig. 51-3. Wavefront “snapshots” at successive Instants In time. +The sounds we make in talking are extremely weak relative to the atmospheric +pressure—only 1 part in a million or so. But for pressure changes of the order +of 1 atmosphere, the wave velocity increases by about twenty percent, and the +wavefront sharpens up at a correspondingly high rate. In nature nothing happens +tn[initclu rapidly, presumably, and what we call a “sharp” front has, actually, +a very slight thickness; it is not infnitely steep. The distances over which it is +varying are of the order of one mean free path, in which the theory of the wave +equation begins to fail because we did not consider the structure of the gas. +Now, referring again to Fig. 51-2, we see that the curvature can be understood +1Í we appreciate that the pressures near the apex are higher than they are farther +back, and so the angle Ø is greater. That is, the curve is the result of the fact that +the speed depends upon the strength of the wave. 'Pherefore the wave from an +atomie bomb explosion travels much faster than the speed of sound for a while, +until it gets so far out that it is weakened to such an extent from spreading that +--- Trang 903 --- +the pressure bump is small compared with atmospheric pressure. The speed of +the bump then approaches the speed of sound in the gas into which it is goïng. +(Incidentally, ¡it always turns out that the speed of the shock is higher than the +speed of sound in the gas ahead, but is lower than the speed of sound in the gas +behind. 'That is, mpulses from the back will arrive at the front, but the front +rides into the medium in which it is going faster than the normal speed of signals. +So one cannot tell, acoustically, that the shock is coming until it is too late. The +light from the bomb arrives first, but one cannot tell that the shoeck is coming +until it arrives, because there is no sound signal coming ahead of it.) +'This is a very interesting phenomenon, this piling up of waves, and the main +point on which it depends is that after a wave is present, the speed of the resulting +wave should be higher. Another example of the same phenomenon is the following. +Consider water flowing in a long channel with ñnite width and ñnite depth. If +a piston, or a wall across the channel, is moved along the channel fast enough, +water piles up, like snow before a snow plow. Now suppose the situation 1s as +shown in Fig. 51-4, with a sudden step in water height somewhere in the channel. +Tlt can be demonstrated that long waves in a channel travel faster in deeper water +than they do in shallow water. 'Pherefore any new bumps or irregularities in +energy supplied by the piston run of forward and pile up at the front. Again, +ultimately what we have is just water with a sharp front, theoretically. However, +Figure 51-4 +--- Trang 904 --- +as Eig. 51-4 shows, there are complications. Pictured is a wave coming up a +channel; the piston is at the far right end of the channel. At first it might have +appeared like a well-behaved wave, as one might expect, but farther along the +chamnel, it has become sharper and sharper until the events pictured occurred. +'There is a terrible churning at the surface, as the pieces of water fall down, but +1t 1s essentially a very sharp rise with no disturbance of the water ahead. +Actually water is much more complicated than sound. However, just to +1llustrate a point, we will try to analyze the speed of such a so-called bore, zn +a channel. 'Phe point here is not that this is of any basic importance Íor our +purposes—it is not a great generalization—it is only to illustrate that the laws of +mechanics that we already know are capable of explaining the phenomenon. +| — Ï +Kxv At ~u At +XI X2 X3 X4 +Fig. 51-5. TWo cross sections of a bore in a channel, with (b) at an +interval At later than (a). +Imagine, for a moment, that the water does look something like Fig. 51-5(a), +that water at the higher height ha is moving with a velocity 0, and that the front +is moving with velocity œ into undisturbed water which is at height hị. We would +like to determine the speed at which the front moves. In a time A£ a vertical +plane initially at z¡ moves a distance ø Af to z¿, while the front of the wave has +moved œ Af. +Now we apply the equations of conservation of matter and momentum. First, +the former: Per unit channel width, we see that the amount hạ A£ of matter +--- Trang 905 --- +that has moved past z¡ (shown shaded) is compensated by the other shaded +region, which amounts to (hạ — hị)u At. So, dividing by At, 0hạ = u(hạ — hì). +That does not yet give us enough, because although we have hạ and hị, we do +not know either œ or ø; we are trying to get both of them. +Now the next step is to use conservation of momentum. We have not discussed +the problems oŸ water pressure, or anything in hydrodynamics, but it is clear +anyway that the pressure of water at a given depth is just enough to hold up +the column of water above it. 'Pherefore the pressure of water is equal to ø, the +density of water, times ø, times the depth below the surface. Since the pressure +Increases linearly with depth, the average pressure over the plane at #1, say, +1s 3 pgha, which is also the average force per unit width and per unit height +pushing the plane toward #¿. 5o we multiply by another hạ to get the total force +which is acting on the water pushing from the left. On the other hand, there is +pressure in the water on the right also, exerting an opposite force on the region +in question, which is, by the same kind of analysis, 5 0gh2. Ñow we must balance +the forces against the rate of change of the momentum. Thus we have to ñgure +out how much more momentum there is in situation (b) in Eig. 51-5 than there +was in (a). We see that the additional mass that has acquired the speed 0 is +Just øhaw At — phzu At (per unit width), and multiplying this by 0 gives the +additional momentum to be equated to the impulse #' Af: +(phu At — phu Af)0 = (š3pghã — spghŸ) AI. +T we eliminate ø from this equation by substituting 0hạ = u(hạ — hị), already +found, and simplify, we get finally that u2 = gha¿(hị + ha) /2hạ. +Tí the height difference is very small, so that hị and hạ are nearly equal, this +says that the velocily = v⁄gh. As we will see later, that is only true provided the +wavelength of the wave is longer than the depth of the channel. +W© could also do the analogous thing for sound waves—including the conser- +vation of internal energy, not the conservation of entropy, because the shock is +Irreversible. In fact, If one checks the conservation of energy in the bore problem, +one fñnds that energy is not conserved. lf the height difference is smaill, it 1s +almost perfectly conserved, but as soon as the height diference becomes very +appreciable, there is a net loss of energy. This is manifested as the falling water +and the churning shown in Fig. 51-4. +In shock waves there is a corresponding apparent loss of energy, from the +point of view of adiabatic reactions. The energy in the sound wave, behind the +--- Trang 906 --- +shock, goes into heating of the gas after shock passes, corresponding to churning +of the water in the bore. In working it out, three equations for the sound case +turn out to be necessary for solution, and the temperature behind the shoeck is +not the same as the temperature in front, as we have seen. +Tf we try to make a bore that is upside down (ha < hị), then we ñnd that the +energy Ìoss per second is negative. Since energy is not available from anywhere, +that bore cannot then maintain itself; it is unstable. If we were to start a wave +of that sort, it would fatten out, because the speed dependence on height that +resulted in sharpening in the case we discussed would now have the opposite +efect. +51-3 Waves in solids +The next kind of waves to be discussed are the more complicated waves in +solids. We have already discussed sound waves in gas and ĩn liquid, and there +1s a direct analog to a sound wave in a solid. If a sudden push ¡is applied to +a solid, it is compressed. It resists the compression, and a wave analogous to +sound is started. However there is another kind of wave that is possible in a +solid, and which is not possible in a Ñuid. IỶ a solid is distorted by pushing ït +sideways (called sheøring), then it tries to pull itself back. That is by definition +what distinguishes a solid from a liquid: if we distort a liquid (internally), hold +1 a minute so that i9 calms down, and then let go, it will stay that way, but If +we take a solid and push ït, like shearing a piece of “Jello,” and let it go, it flies +back and starts a sheør wave, travelling in the same way the compressions travel. +In all cases, the shear wave speed 1s less than the speed of longitudinal waves. +'The shear waves are somewhat more analogous, so far as their polarizations are +concerned, to light waves. Sound has no polarization, it is Just a pressure wave. +Light has a characteristic orientation perpendicular to its direction of travel. +In a solid, the waves are of both kinds. Pirst, there is a compression wave, +analogous to sound, that runs at one speed. Tf the solid is not crystalline, then a +shear wave polarized in any direction will propagate at a characteristic speed. +(Of course all solids are crystalline, but iŸ we use a block made up o£ microcrystals +of all orientations, the crystal anisotropies average out.) +Another interesting question concerning sound waves is the following: What +happens if the wavelength in a solid gets shorter, and shorter, and shorter? How +short can it get? It is interesting that it cannot get any shorter than the space +between the atoms, because If there is supposed to be a wave in which one +--- Trang 907 --- +point goes up and the next down, ete., the shortest possible wavelength is clearly +the atom spacing. In terms of the modes of oscillation, we say that there are +longitudinal modes, and transverse modes, long wave modes, short wave modes. +As we consider wavelengths comparable to the spacing between the atoms, then +the speeds are no longer constant; there is a dispersion efect where the velocity +1s not independent of the wave number. But, ultimately, the highest mode of +transverse waves would be that in which every atom ¡s doing the opposite of +neiphboring atoms. +Now from the poïnt of view of atoms, the situation is like the two pendulums +that we were talking about, for which there are two modes, one in which they +both go together, and the other in which they go apart. It is possible to analyze +the solid waves another way, in terms of a system of coupled harmonic oscillators, +like an enormous number of pendulums, with the highest mode such that they +oscillate oppositely, and lower modes with diferent relationships of the timing. +The shortest wavelengths are so short that they are not usually available +technically. However they are of great interest because, in the theory of thermo- +dynamics of a solid, the heat properties of a solid, for example specific heats, can +be analyzed in terms of the properties of the short sound waves. Going to the +extreme of sound waves of ever shorter wavelength, one necessarily comes to the +individual motions of the atoms; the two things are the same ultimately. +A very interesting example of sound waves in a solid, both longitudinal and +transverse, are the waves that are in the solid earth. Who makes the noises we do +not know, but inside the earth, from time to time, there are earthquakes——some +rock slides past some other rock. 'That is like a little noise. So waves like sound +waves start out from such a source very much longer in wavelength than one +usually considers in sound waves, but still they are sound waves, and they travel +around in the earth. “The earth is not homogeneous, however, and the properties, +of pressure, density, compressibility, and so on, change with depth, and therefore +the speed varies with depth. Thhen the waves do not travel in straight lines—there +is a kind of index of refraction and they go in curves. The longitudinal waves +and the transverse waves have diferent speeds, so there are diferent solutions for +the diferent speeds. 'Therefore if we place a seismograph at some location and +watch the way the thing jiggles after there has been an earthquake somewhere +else, then we do not just get an irregular jiggling. We might get a jiggling, and +a quieting down, and then another jiggling—what happens depends upon the +location. lÝ it were close enough, we would first receive longitudinal waves from +the disturbance, and then, a few moments later, transverse waves, because they +--- Trang 908 --- +SOURCE $ +; " =. STATION +PKPI Ñ Z +PKPPKP ` ¬. +LONGITUDINAL (P,K) Ỷ TÁC 6) +xa -_-_—_ thÔ ho +Fig. 51-6. Schematic of the earth, showing paths of longitudinal and +transverse sound waves. +travel more slowly. By measuring the time diference between the two, we can +tell how far away the earthquake is, if we know enough about the speeds and +composition of the interior regions involved. +An example of the behavior pattern of waves in the earth is shown in Fig. 51-6. +The 6wo kinds of waves are represented by different symbols. lÝ there were an +earthquake at the place marked “source,” the transverse waves and longitudinal +waves would arrive at diferent times at the station by the most direct routes, +and there would also be reflections at discontinuities, resulting in other paths +and times. It turns out that there is a core in the earth which does not carry +transverse waves. lf the station is opposite the source, transverse waves still +arrive, but the timing is not right. What happens is that the transverse wave +comes to the core, and whenever the transverse waves come to a surface which +1s oblique, bebween two materials, two new waves are generated, one transverse +and one longitudinal. But inside the core of the earth, a transverse wave is not +propagated (or at least, there is no evidence for it, only for a longitudinal wave); +1t comes out again in both forms and comes to the station. +Tt is from the behavior of these earthquake waves that it has been determined +that transverse waves cannot be propagated within the inner circle. 'This means +that the center of the earth ¡is liquid in the sense that it cannot propagate +transverse waves. The only way we know what is inside the earth is by studying +earthquakes. 5o, by using a large number of observations of many earthquakes +at diferent stations, the details have been worked out——the speed, the curves, +--- Trang 909 --- +etc. are all known. We know what the speeds of various kinds of waves are +at every depth. Knowing that, therefore, it is possible to fgure out what the +normal modes of the earth are, because we know the speed of propagation of +sound waves——in other words, the elastic properties of both kinds of waves at +every depth. Suppose the earth were distorted into an ellipsoid and let go. lt is +Just a matter of superposing waves travelling around in the ellipsoid to determine +the period and shapes in a free mode. We have figured out that if there is a +disturbance, there are a lot of modes, from the lowest, which is ellipsoidal, to +higher modes with more structure. +The Chilean earthquake of May 1960 made a loud enough “noise” that the +signals went around the earth many times, and new seismographs of great delicacy +were made just in time to determine the frequencies of the fundamental modes +of the earth and to compare them with the values that were calculated from the +theory of sound with the known velocities, as measured from the independent +earthquakes. The result of this experiment is illustrated in Eig. 51-7, which is a +plot of the strength of the signal versus the frequency of its oscillation (a Fourier +analus¿s). Note that at certain particular frequencies there is mụuch more being +received than at other frequencies; there are very defnite maxima. These are +the natural frequencies of the earth, because these are the main frequencies at +which the earth can oscillate. In other words, if the entire motion of the earth is +made up of many diferent modes, we would expect to obtain, for each station, +19 H1 /#M— S24 NSSINS +NT TIIBIINEEIIBSIRSEEEESRSS +HN. lIBIIBINI Led +RE E0NIIU Si SIIẾIEIHPTRINMMIIRMRNN +.E'IITMIINBNNIHII0IU0108I005 EIRIRRVEN +gang ( M INN t4 Á 4 | Â Ì Lị \ Ì | | +“TL D007 HH CỤ +NET Lại g1 M0 VI) (NI) +FREQUENCY IN CYCLES PER MINUTE FREQUENCY IN CYcLes PEn À[ MÑUTE +: Fig. 51-7. Power versus frequency as detected at seismographs In +Naña, Peru, and lsabella, California. The coherence is a measure of +the coupling between the stations. [From Benioff, Press and Smith, _J. +Geoph. Research 66, 605 (1961)]. +--- Trang 910 --- +Irregular bumpings which indicate a superposition of many frequencies. lÝ we +analyze this in terms of frequencies, we should be able to fñnd the characteristic +frequencies of the earth. The vertical dark lines in the fñgure are the calculated +frequencies, and we fnd a remarkable agreement, an agreement due to the fact +that the theory oŸ sound is right for the inside of the earth. +400 == +FOURIER ANALYSIS ø s +ISABELLA STRAIN +T = 16000 MỊN h +-- IIIN +T?0Ƒ2Ss;FaSaSsS;Fa +S3 250 œ +lì B5 +ễ |: Ì +Š 15o tr 1 +IMIRIRRIRN +0.0180 0.0182 0.0184 0.0186 0.0188 0.0190 0.0192 +FREQUENCY IN CYCLES PER MINUTE +Fig. 51-8. High-resolution analysis of one of the selsmograph records, +showing spectral doublet. +A very curious point is revealed in Fig. 51-8, which shows a very careful +mmeasurement, with better resolution of the lowest mode, the ellipsoidal mode of +the earth. Note that it is not a single maximum, but a double one, 54.7 minutes +and 53.1 minutes—slightly diferent. The reason for the two diÑferent Írequencies +was not known at the time that it was measured, although it may have been +found ¡in the meantime. There are at least two possible explanations: One would +be that there may be asymmetry in the earth”s distribution, which would result +in two similar modes. Another possibility, which is even more interesting, is this: +TImagine the waves goiïng around the earth in bwo directions from the source. +The speeds will not be equal because of efects of the rotation of the earth In +--- Trang 911 --- +the equations of motion, which have not been taken into account in making the +analysis. Motion in a rotating system is modifed by Coriolis forces, and these +may cause the observed splitting. +Regarding the method by which these quakes have been analyzed, what is +obtained on the seismograph is not a curve of amplitude as a function of frequency, +but displacement as a function of time, always a very irregular tracing. To fnd +the amount of all the diferent sine waves for all diferent frequencies, we know +that the trick is to multiply the data by a sine wave of a given frequency and +integrate, i.e., average it, and in the average all other frequencies disappear. The +fñgures were thus plots of the integrals found when the data were multiplied by +sine waves of diferent cycles per minute, and integrated. +51-4 Surface waves +Now, the next waves of interest, that are easily seen by everyone and which +are usually used as an example of waves in elementary courses, are water waves. +As we shall soon see, they are the worst possible example, because they are in +no respects like sound and light; they have all the complications that waves can +have. Let us start with long water waves in deep water. lf the ocean is considered +infnitely deep and a disturbance is made on the surface, waves are generated. All +kinds of irregular motions occur, but the sinusoidal type motion, with a very small +disturbance, might look like the common smooth ocean waves coming in toward +the shore. Now with such a wave, the water, of course, on the average, is standing +siiH, but the wave moves. What is the motion, is it transverse or longitudinal? It +must be neither; it is not transverse, nor is it longitudinal. Although the water +at a given place is alternately trough or hill, it cannot simply be moving up and +down, by the conservation of water. That is, If it goes down, where is the water +goïing to go? The water is essentially incompressible. The speed oŸ compression +of waves—that is, sound in the water——is much, much higher, and we are not +considering that now. 5ince water is incompressible on this scale, as a hill comes +down the water must move away from the region. What actually happens is that +particles of water near the surface move approximately in circles. When smooth +swells are coming, a person foating in a tire can look at a nearby obJect and +see it going in a circle. So it is a mixture of longitudinal and transverse, to add +to the confusion. At greater depths in the water the motions are smaller circles +until, reasonably far down, there is nothing left of the motion (Eig. 51-9). +--- Trang 912 --- +Wave direction +A water wave _ +Water molecules move in \ +circular orbits when Wave trough +Wave passes by +Fig. 51-9. Deep-water waves are formed from particles moving In +circles. Note the systematic phase shifts from circle to circle. How +would a floating object move? +To ñnd the velocity oŸ such waves is an interesting problem: ¡it must be some +combination of the density of the water, the acceleration of gravity, which is the +restoring force that makes the waves, and possibly of the wavelength and of the +depth. If we take the case where the depth goes to infnity, it will no longer +depend on the depth. Whatever formula we are going to get for the velocity of +the phases of the waves must combine the various factors to make the proper +dimensions, and If we try this in various ways, we fnd only one way to combine +the density, ø, and À in order to make a velocity, namely, ⁄gA, which does not +include the density at all. Actually, this formula for the phase velocity is not +exactly right, but a complete analysis of the dynamics, which we will not go into, +shows that the factors are as we have them, except for 2z: +Đphase = V0À/27z (for gravity waves). +Tt is interesting that the long waves go faster than the short waves. Thus if a +boat makes waves far out, because there is some sports-car driver in a motorboat +travelling by, then after a while the waves come to shore with slow sloshings at +first and then more and more rapid sloshings, because the first waves that come +are long. The waves get shorter and shorter as the time goes on, because the +velocities go as the square root of the wavelength. +One may objJect, “hat is not right, we must look at the group velocity in +order to fñgure it out!” Of course that is true. The formula for the phase velocity +does not tell us what is goïng to arrive first; what tells us ¡is the group velocity. +So we have to work out the group velocity, and it is left as a problem to show +1t to be one-half of the phase velocity, assuming that the velocity goes as the +square root of the wavelength, which is all that is needed. 'Phe group velocity +also goes as the square root of the wavelength. How can the group velocity go +--- Trang 913 --- +half as fast as the phase? If one looks at the bunch of waves that are made by a +boat travelling along, following a particular crest, he fnds that it moves forward +in the group and gradually gets weaker and dies out in the front, and mystically +and mysteriously a weak one in the back works its way forward and gets stronger. +Tn short, the waves are moving through the group while the group is only moving +at half the speed that the waves are moving. +sc cớ __ +S—======_- +` Xe kế “-xessee sec —=—=..-. ` +- À &. ——— NGssiei +“t2 ` NESSteoc--smnnngsc--oa. ca2226 +——. —-.„— +3 XS €90460sy0ndjyopxzz/74f% +` : TS SỐ. cu +vang Ế Đa +Fig. 51-10. The wake of a boat. +Because the group velocities and phase velocities are not equal, then the +waves that are produced by an objJect moving through are no longer simply a +cone, but it is mụch more interesting. We can see that in Fig. 51-10, which +shows the waves produced by an object moving through the water. Note that it +is quite diferent than what we would have for sound, in which the velocity is +independent of wavelength, where we would have wavefronts only along the cone, +travelling outward. Instead of that, we have waves in the back with fronts moving +parallel to the motion of the boat, and then we have little waves on the sides at +other angles. 'Phis entire pattern of waves can, with ingenuity, be analyzed by +--- Trang 914 --- +knowing only this: that the phase velocity is proportional to the square root of +the wavelength. “The trick 1s that the pattern of waves 1s stationary relative to +the (constant-velocity) boat; any other pattern would get lost from the boat. +The water waves that we have been considering so far were long waves in +which the force of restoration is due to gravitation. But when waves get very +short in the water, the main restoring force is capillary attraction, i.e., the energy +of the surface, the surface tension. For surface tension waves, it turns out that +the phase velocity is +Đphase = V/2#T'/Aø (for ripples), +where 7' is the surface tension and ø the density. It is the exac% opposite: the +phase velocity 1s hZøher, the shorter the wavelength, when the wavelength gets +very small. When we have both gravity and capillary action, as we always do, +we get the combination of these bwo together: +Uphase — V Tkịp + g/k, +where k = 27/A is the wave number. So the velocity of the waves of water is really +quite complicated. The phase velocity as a function of the wavelength is shown +in Eig. 51-11; for very short waves it is fast, for very long waves iW is fast, and +there is a minimum speed at which the waves can go. The group velocity can be +^A (cm) +Fig. 51-11. Phase velocity vs. wavelength for water. +--- Trang 915 --- +calculated from the formula: it goes to Ỷ the phase velocity for ripples and D the +phase velocity for gravity waves. To the left of the minimum the group velocity +1s hipher than the phase velocity; to the right, the group velocity is less than the +phase velocity. There are a number of interesting phenomena associated with +these facts. In the first place, since the group velocity 1s increasing so rapidly as +the wavelength goes down, iŸ we make a disturbance there will be a slowest end of +the disturbance goïing at the minimum speed with the corresponding wavelength, +and then in front, going at higher speed, will be a short wave and a very long +wave. It is very hard to see the long ones, but it is easy to see the short ones in +a water tank. +So we see that the ripples often used to illustrate simple waves are quite +interesting and complicated; they do not have a sharp wavefront at all, as is the +case for simple waves like sound and light. The main wave has little ripples which +run out ahead. A sharp disturbance in the water does not produce a sharp wave +because of the dispersion. Pirst come the very fñne waves. Incidentally, if an +object moves through the water at a certain speed, a rather complicated pattern +results, because all the diferent waves are going at different speeds. One can +demonstrate this with a tray of water and see that the fastest ones are the fne +caplllary waves. There are slowest waves, of a certain kind, which go behind. By +inelining the bottom, one sees that where the depth is lower, the speed is lower. +TỶ a wave comes in at an angle to the line of maximum slope, it bends and tends +to follow that line. In this way one can show various things, and we conclude +that waves are more complicated in water than in air. +The speed of long waves in water with circulational motions is slower when +the depth is less, faster in deep water. “Thus as water comes toward a beach +where the depth lessens, the waves go slower. But where the water is deeper, the +wawves are faster, so we get the efects of shock waves. 'Phis time, since the wave +1s not so simple, the shocks are much more contorted, and the wave over-curves +itself, in the familiar way shown in Fig. 51-12. This is what happens when waves +come into the shore, and the real complexities in nature are well revealed in such +a circumstance. No one has yet been able to fñgure out what shape the wave +should take as it breaks. It is easy enough when the waves are small, but when +one gets large and breaks, then it is much more complicated. +An interesting feature about capillary waves can be seen in the disturbances +made by an object moving through the water. From the point of view of the +obJject itself, the water is Ñowing past, and the waves which ultimately sit around +1t are always the waves which have just the right speed to stay still with the +--- Trang 916 --- +_" HP. ....Ềẻẽ” sÍ s@ ¬ " +¬ TS. & kLV( SAU La & +là \ CC “Số 4 +b NNG - |: sả b Š : +» ` 8P ¬ xì “` — +NNGG xa... " = +Fig. 51-12. A water wave. +object in the water. Similarly, around an object in a stream, with the stream +fowing by, the pattern of waves is stationary, and at just the right wavelengths to +go at the same speed as the water going by. But ïf the group velocity is less than +the phase velocity, then the disturbances propagate out backwards in the stream, +because the group velocity is not quite enough to keep up with the stream. Tf +the group velocity is faster than the velocity of the phase, the pattern of waves +will appear in front of the object. If one looks closely at objects in a stream, one +can see that there are little ripples in front and long “slurps” in the back. +Another interesting feature of this sorE can be observed in pouring liquids. If +milk is poured fast enough out of a bottle, for instance, a large number of lines +can be seen crossing both ways in the outgoing stream. They are waves starting +from the disturbance at the edges and running out, mụch like the waves about an +object in a stream. 'Phere are efects rom both sides which produce the crossed +pattern. +We have investigated some of the interesting properties of waves and the +various complications of dependence of phase velocity on wavelength, the speed of +the waves on depth, and so forth, that produce the really complex, and therefore +Interesting, phenomena of nature. +--- Trang 917 --- +Sggrmmaofrgg ít FPhạysícetl E«aers +52-1 Symmetry operations +The subject of this chapter is what we may call sựmwmetru ïn phụsical laus. VWG +have already discussed certain features of symmetry in physical laws in connection +with vector analysis (Chapter I1), the theory of relativity (Chapter 16), and +rotation (Chapter 20). +Why should we be concerned with symmetry? In the fñrst place, symmetry is +fascinating to the human mỉnd, and everyone likes objects or patterns that are in +some way symmetrical. It is an interesting fact that nature often exhibits certain +kinds of symmetry in the objects we fñnd in the world around us. Perhaps the +most symmetrical object imaginable is a sphere, and nature is full of spheres—— +stars, planets, water droplets in clouds. 'Phe crystals found in rocks exhibit many +diferent kinds of symmetry, the study of which tells us some Important things +about the structure of solids. Even the animal and vegetable worlds show some +degree of symmetry, although the symmetry of a fower or oŸ a bee is not as +perfect or as fundamental as is that of a crystal. +But our main concern here is not with the fact that the øb7ecfs of nature +are often symmetrical. Rather, we wish to examine some oŸ the even more +remarkable symmetries of the universe—the symmetries that exist in the basi/c +laus themselues which govern the operation of the physical world. +tirst, what 2s symmetry? How can a physical øu be “symmetrical”? The +problem of delning symmetry is an interesting one and we have already noted +that Weyl gave a good defñnition, the substance of which is that a thing is +symmetrical if there is something we can do to it so that after we have done it, +1t looks the same as it did before. For example, a symmetfrical vase 1s of such +a kind that if we refect or turn ï$, it will look the same as it did before. The +question we wish to consider here is what we can do to physical phenomena, or +to a physical situation in an experiment, and yet leave the result the same. A +list of the known operations under which various physical phenomena remain +invariant is shown in Table 52-1. +--- Trang 918 --- +Table 52-1 +Symmetry Operations +'Translation in space +'Translation in time +Rotation through a fñxed angle +Uniform velocity in a straight +line (Lorentz transformation) +Reversal of time +Reflection of space +Interchange of identical atoms +or identical particles +Quantum-mechanical phase +Matter-antimatter (charge conjugation) +52-2 Symmetry ỉn space and tỉme +'The first thing we might try to do, for example, 1s to trønslate the phenomenon +in space. lf we do an experiment in a certain region, and then build another +apparatus at another place in space (or move the original one over) then, whatever +went on in one apparatus, in a certain order in time, will occur in the same way +1ƒ we have arranged the same condition, with all due attention to the restrictions +that we mentioned before: that all of those features of the environment which +make i% not behave the same way have also been moved over——we talked about +how to defne how much we should include in those cireumstances, and we shall +not go into those details again. +In the same way, we also believe today that đisplacement ?n từne will have +no effect on physical laws. (That is, øs far aøs tue knot todœ——all of these things +are as far as we know today!) That means that if we build a certain apparatus +and start it at a certain time, say on Thursday at 10:00 a.m., and then build +the same apparatus and start it, say, three days later in the same condition, the +two apparatuses will go through the same motions in exactly the same way as +a function of time no matter what the starting time, provided again, of course, +that the relevant features of the environment are also modifed appropriately In +tứmne. That symmetry means, of course, that if one bought General Motors stock +three months ago, the same thỉng would happen to it if he bought it nowl +--- Trang 919 --- +We have to watch out for geographical diferences too, for there are, Of €OUrse, +variations in the characteristics of the earth's surface. So, for example, IŸ we +measure the magnetic field in a certain region and move the apparatus to some +other region, it may not work in precisely the same way because the magnetic +ñeld is diÑerent, but we say that is because the magnetic fñeld is associated with +the earth. We can imagine that if we move the whole earth and the equipment, +it would make no diference in the operation of the apparatus. +Another thing that we discussed in considerable detail was rotation in space: +1Ý we turn an apparatus at an angle it works Just as well, provided we turn +everything else that is relevant along with it. In fact, we discussed the problem of +symmetry under rotation in space in some detail in Chapter 11, and we invented +a mathematical system called øector ønalsis to handle it as neatly as possible. +Ơn a more advanced level we had another symmetry——the symmetry under +uniform velocity in a straight line. That is to say——a rather remarkable efect— +that ïf we have a piece of apparatus working a certain way and then take the +same apparatus and put it in a car, and move the whole car, plus all the relevant +Surroundings, at a uniform velocity in a straight line, then so far as the phenomena, +inside the car are concerned there is no diference: all the laws of physics appear +the same. We even know how to express this more technically, and that is that the +mathematical equations of the physical laws must be unchanged under a Loren‡z +transƒormation. As a matter of fact, it was a study of the relativity problem that +concentrated physicistsˆ attention most sharply on symmetry in physical laws. +Now the above-mentioned symmetries have all been of a geometrical nature, +time and space being more or less the same, but there are other symmetries of a +diferent kind. Eor example, there is a symmetry which describes the fact that +we can replace one atom by another of the same kind; to put ¡it diferently, there +đre atoms of the same kind. lt is possible to find groups of atoms such that iŸ we +change a pair around, it makes no diference—the atoms are identical. Whatever +one atom of oxygen of a certain type will do, another atom of oxygen of that type +will do. One may say, “hat is ridiculous, that is the deffnion of equal typesl” +'That may be merely the defnition, but then we still do not know whether there +gre any “atoms of the same type”; the ƒacứ is that there are many, many atoms of +the same type. Thus it does mean something to say that it makes no dilerence +1ƒ we replace one atom by another of the same type. 'Phe so-called elementary +particles of which the atoms are made are also identical particles in the above +sense—all electrons are the same; all protons are the same; all positive pions are +the same; and so on. +--- Trang 920 --- +After such a long list of things that can be done without changing the +phenomena, one might think we could do practically anything; so let us give +some examples to the contrary, just to see the diference. Suppose that we ask: +“ Are the physical laws symmetrical under a change of scale?” Suppose we bưuild a +certain piece of apparatus, and then buïld another apparatus five times bigger In +every part, will it work exactly the same way? "The answcr is, in this case, ?øol +The wavelength of light emitted, for example, by the atoms inside one box of +sodium atoms and the wavelength of light emitted by a gas of sodium atoms five +times in volume is not five times longer, but is in fact exactly the same as the +other. So the ratio of the wavelength to the size of the emitter will change. +Another example: we see in the newspaper, every once in a while pictures +of a great cathedral made with little matchsticks—a tremendous work of art$ +by some retired fellow who keeps gluing matchsticks together. Ït is mụch more +elaborate and wonderful than any real cathedral. If we imagine that this wooden +cathedral were actually built on the scale of a real cathedral, we see where the +trouble is; it would not last—the whole thing would collapse because of the fact +that scaled-up matchsticks are just not strong enough. “Yes,” one might say, +“but we also know that when there is an inÑuence from the outside, it also must +be changed in proportion!” We are talking about the ability of the object to +withstand gravitation. So what we should do is frst to take the model cathedral +of real matchsticks and the real earth, and then we know ït ¡is stable. hen we +should take the larger cathedral and take a bigger carth. But then it is even +worse, because the gravitation is increased still morel +Today, of course, we understand the fact that phenomena depend on the +scale on the grounds that matter is atomic in nature, and certainly if we built an +apparatus that was so smaill there were only five atoms in it, it would clearly be +something we could not scale up and down arbitrarlly. The scale of an individual +atom is not at all arbitrary—it is quite deñnite. +The fact that the laws of physics are not unchanged under a change of scale +was discovered by Galileo. He realized that the strengths of materials were not +in exactly the right proportion to their sizes, and he illustrated this property +that we were just discussing, about the cathedral oŸ matchsticks, by drawing two +bones, the bone of one dog, in the right proportion for holding up his weight, +and the imaginary bone of a “super dog” that would be, say, ten or a hundred +times bigger—that bone was a big, solid thing with quite diferent proportions. +W©e do not know whether he ever carried the argument quite to the conclusion +that the laws of nature must have a defnite scale, but he was so impressed with +--- Trang 921 --- +this discovery that he considered it to be as important as the discovery of the +laws of motion, because he published them both ïn the same volume, called “Ôn +'Iwo New Sciences.” +Another example in which the laws are not symmetrical, that we know quite +well, is this: a system in rotation at a uniform angular velocity does not give the +same apparent laws as one that is not rotating. If we make an experiment and then +put everything in a space ship and have the space ship spinning ïn empty space, +all alone at a constant angular velocity, the apparatus will not work the same way +because, as we know, things inside the equipment will be thrown to the outside, +and so on, by the centrifugal or Coriolis forces, etc. In fact, we can tell that the +earth is rotating by using a so-called Eoucault pendulum, without looking outside. +Next we mention a very interesting symmetry which is obviously false, 1.e., +reuersiblitụ ïm từmne. The physical laws apparently cannot be reversible in time, +because, as we know, all obvious phenomena are irreversible on a large scale: +“he moving fñnger writes, and having writ, moves on.” So far as we can tell, this +iIrreversibility is due to the very large number of partieles involved, and if we +could see the individual molecules, we would not be able to discern whether the +machinery was working forward or backwards. 'To make it more precise: we build +a small apparatus in which we know what all the atoms are doïing, in which we +can watch them jiggling. Now we build another apparatus like it, but which starts +10s motion in the fñnal condition of the other one, with all the velocitles precisely +reversed. Ï# uill then go through the same mottions, Du‡ exactlU ím reuerse. Putting +1t another way: if we take a motion picture, with sufficient detail, of all the inner +works of a piece of material and shine it on a screen and run it backwards, no +physicist will be able to say, “That is against the laws of physics, that is doïng +something wrong!” TỶ we do not see all the details, of course, the situation will be +perfectly clear. lÝ we see the egg splattering on the sidewalk and the shell cracking +open, and so on, then we will surely say, “ Phat is irreversible, because If we run the +moving picture backwards the egg will all collect together and the shell will go back +together, and that is obviously ridiculousl” But if we look at the individual atoms +themselves, the laws look completely reversible. This is, of course, a much harder +discovery to have made, but apparently ït is true that the fundamental physical +laws, on a microscopic and fundamental level, are completely reversible in timel +52-3 Symmetry and conservation laws +The symmetries of the physical laws are very interesting at this level, but they +turn out, in the end, to be even more interesting and exciting when we come to +--- Trang 922 --- +quantum mechanics. For a reason which we cannot make clear at the level of the +present discussion—a fact that most physicists still fnd somewhat staggering, a +most profound and beautiful thing, is that, in quantum mechanics, ƒor cach oƒ the +rules 0ƒ sụmmetru there is œ corresponding conseruation lau; there 1s a defnite +connection between the laws of conservation and the symmetries of physical laws. +W© can only state this at present, without any attempt at explanation. +'The fact, for example, that the laws are symmetrical for translation in space +when we add the principles of quantum mechanies, turns out to mean that +mormnentum 1s conserued. +'That the laws are symmetrical under translation in time means, in quantum +mnechanics, that energụ ?s conserued. +Invariance under rotation through a fixed angle in space corresponds to the +conseruation oŸ angular mnormentwm. These connections are very interesting and +beautiful things, among the most beautiful and profound things in physics. +Incidentally, there are a number of symmetries which appear in quantum +mmechanics which have no classical analog, which have no method of description +in classical physics. One of these is as follows: Tf is the amplitude for some +process or other, we know that the absolute square of + is the probability that +the process will occur. Now ïif someone else were to make his calculations, not +with this ý, but with a ÿˆ which difers merely by a change in phase (let A be +some constant, and multiply e⁄Ê times the old +), the absolube square oŸ ', +which is the probability of the event, is then equal to the absolute square of 4: +Ụ =ÚcÔŠ; — J1 = |0: (52.1) +Therefore the physical laws are unchanged if the phase of the wave function +is shifted by an arbitrary constant. That is another symmetry. Physical laws +must be of such a nature that a shift in the quantum-mechanical phase ma.kes +no diference. As we have just mentioned, in quantum mechanics there is a +conservation law for every symmetry. The conservation law which is connected +with the quantum-mechanical phase seems to be the conseruation oƒ clectrical +charge. Thịs is altogether a very interesting businessl +52-4 Mirror reflections +Now the next question, which is going to concern us for most of the rest of +this chapter, is the question of symmetry under reffecfion ?n space. The problem +--- Trang 923 --- +1s this: Are the physical laws symmetrical under reflection? We may put it this +way: Suppose we build a piece of equipment, let us say a clock, with lots of +wheels and hands and numbers; it ticks, it works, and it has things wound up +inside. We look at the clock in the mirror. How it iooks in the mirror is not the +question. But let us actually buziđ another clock which is exactly the same as +the frst clock looks in the mirror—every time there is a secrew with a right-hand +thread in one, we use a screw with a left-hand thread in the corresponding +place of the other; where one is marked “2” on the face, we mark a “$” on +the face of the other; each coiled spring is twisted one way in one clock and +the other way in the mirror-image clock; when we are all ñnished, we have bwo +clocks, both physical, which bear to each other the relation of an object and its +mirror image, although they are both actual, material objects, we emphasize. +Now the question is: lf the two clocks are started in the same condition, the +springs wound to corresponding tightnesses, will the two clocks tiek and go +around, forever after, as exact mirror images? (This is a physical question, not +a philosophical question.) Qur intuition about the laws of physics would suggest +that they ould. +W©e would suspect that, at least in the case of these clocks, reflection in space +is one of the symmetries of physical laws, that if we change everything from +“right” to “left” and leave it otherwise the same, we cannot ©ell the diference. Let +us, then, suppose for a moment that this is true. lf it is true, then it would be +impossible to distinguish “right” and “left” by any physical phenomenon, just as +1 1s, for example, impossible to defne a particular absolute velocity by a physical +phenomenon. So it should be impossible, by any physical phenomenon, to defne +absolutely what we mean by “right” as opposed to “left,” because the physical +laws should be symmetrical. +Of course, the world does not haøe to be symmetrical. For example, using +what we may call “geography,” surely “right” can be defñned. For instance, we +siand in New Orleans and look at Chicago, and Florida is to our right (when our +feet are on the groundl). So we can define “right” and “left” by geography. Of +course, the actual situation in any system does not have to have the symmetry +that we are talking about; it is a question of whether the la+s are symmetrical——in +other words, whether it 1s agœ#nst the phụs¿cal laus to have a sphere like the +earth with “left-handed dirt” on i9 and a person like ourselves standing looking +at a city like Chicago from a place like New Orleans, but with everything the +other way around, so Florida is on the other side. It clearly seems not impossible, +not against the physical laws, to have everything changed left for right. +--- Trang 924 --- +Another point is that our definition of “right” should not depend on history. +An easy way to distinguish right from left is to go to a machine shop and pick +up a screw at random. 'Phe odds are i% has a right-hand thread——not necessarily, +but it is mụch more likely to have a right-hand thread than a left-hand one. This +1s a question of history or convention, or the way things happen to be, and 1s +again not a question of fundamental laws. As we can well appreciate, everyone +could have started out making left-handed screwsl +So we must try to fnd some phenomenon in which “right hand” is involved +fundamentally. The next possibility we discuss is the fact that polarized light +rotates its plane of polarization as it goes through, say, sugar water. Às we saw +in Chapter 33, it rotates, let us say, to the right in a certain sugar solution. That +is a way of defning “right-hand,” because we may dissolve some sugar in the +water and then the polarization goes to the right. But sugar has come from living +things, and If we try to make the sugar artificially, then we discover that i% đoes +not rotate the plane of polarization! But if we then take that same sugar which +1s made artificially and which does not rotate the plane of polarization, and put +bacteria in it (they eat some of the sugar) and then filter out the bacteria, we +fñnd that we still have sugar left (almost half as much as we had before), and this +tỉme it does rotate the plane of polarization, but #Öe other uaul Ït seems very +confusing, but is easily explained. +'Take another example: Ône of the substances which is common to all living +creatures and that is fundamental to life is protein. Proteins consist of chains of +amino acids. Figure 52-1 shows a model of an amino acid that comes out of a +protein. 'This amino acid is called alanine, and the molecular arrangement would +look like that in Eig. 52-1(a) 1Í it came out oŸ a protein of a real living thing. +On the other hand, if we try to make alanine from carbon dioxide, ethane, and +ammonia (and we cøn make it, it is not a cormplicated molecule), we discover that +we are making equal amounts of this molecule and the one shown in Fig. 52-1(b)l +The fñrst molecule, the one that comes from the living thing, is called L-alanine. +'The other one, which is the same chemically, in that ït has the same kinds of atoms +and the same connections of the atoms, is a “right-hand” molecule, compared +with the “left-hand” L-alanine, and ït is called D-alan¿ne. 'The interesting thing is +that when we make alanine at home in a laboratory from simple gases, we get an +equal mixture of both kinds. However, the only thing that life uses is L-alanine. +(This is not exactly true. Here and there in living creatures there is a special use +for D-alanine, but it is very rare. All proteins use L-alanine exclusively.) Ñow +1ƒ we make both kinds, and we feed the mixture to some animal which likes to +--- Trang 925 --- +ˆ % - ki +lY. `. +“củ h...:\ VY 7 +„"= .¡ 6 xẻ += + ' _““.__.” _ výT vn: - _= +Fig. 52-1. (a) L-alanine (left), and (b) D-alanine (right). +“eat,” or use up, alanine, it cannot use D-alanine, so it only uses the L-alanine; +that is what happened to our sugar——after the bacteria eat the sugar that works +well for them, only the “wrong” kind is left! (Left-handed sugar tastes sweet, but +not the same as right-handed sugar.) +So 1ÿ looks as though the phenomena, of life permit a distinction bebween +“right” and “left,” or chemistry permits a distinction, because the two molecules +are chemically diferent. But no, i§ does not! So far as physical measurements can +be made, such as of energy, the rates of chemical reactions, and so on, the bwo +kinds work exactly the same way if we make everything else in a mirror image +too. One molecule will rotate light to the right, and the other will rotate It 0o +the left in precisely the same amount, through the same amount of Ñuid. 'Thus, +so far as physics is concerned, these two amino acids are equally satisfactory. So +far as we understand things today, the fundamentals of the Schrödinger equation +have it that the two molecules should behave in exactly corresponding ways, so +that one is to the right as the other is to the left. Nevertheless, in life ¡it ¡s all +one wayl +Tt is presumed that the reason for this is the following. Let us suppose, for +example, that life is somehow at one moment in a certain condition in which +all the proteins in some creatures have left-handed amino acids, and all the +enzymes are lopsided——every substanee in the living creature is lopsided——it is +not symmetrical. 5o when the digestive enzymes try to change the chemicals in +the food from one kind to another, one kind of chemical “ñts” into the enzyme, +but the other kind does not (like Cinderella and the slipper, except that it is a +--- Trang 926 --- +“left foot” that we are testing). So far as we know, in principle, we could buïld +a frog, for example, in which every molecule is reversed, everything is like the +“left-hand” mirror image of a real frog; we have a left-hand frog. 'This left-hand +frog would go on all right for a while, but he would ñnd nothing to eat, because +1ƒ he swallows a Ñy, his enzymes are not built to digest it. 'Phe Ñy has the wrong +“kind” of amino acids (unless we give him a left-hand fy). 5o as far as we know, +the chemical and life processes would continue in the same manner if everything +were reversed. +TỶ life is entirely a physical and chemical phenomenon, then we can understand +that the proteins are all made in the same corkscrew only from the idea that +at the very beginning some living molecules, by accident, got started and a few +won. Somewhere, once, one organic molecule was lopsided in a certain way, +and from this particular thing the “right” happened to evolve in our particular +geography; a particular historical accident was one-sided, and ever since then the +lopsidedness has propagated itself. Once having arrived at the state that it is in +now, of course, it will always continue—all the enzymes digest the right things, +manufacture the right things: when the carbon dioxide and the water vapor, +and so on, go in the plant leaves, the enzymes that make the sugars make them +lopsided because the enzymes are lopsided. IÝ any new kind oÝ virus or living +thing were to originate at a later time, it would survive only if it could “eat” the +kind of living matter already present. Thus it, too, must be of the same kind. +There is no conservation of the number of right-handed molecules. Ônce +started, we could keep increasing the number of right-handed molecules. So the +presumption is, then, that the phenomena in the case of life do not show a lack +of symmetry in physical laws, but do show, on the contrary, the universal nature +and the commonness of ultimate origin of all creatures on earth, in the sense +described above. +52-5 Polar and axial vectors +Now we go further. We observe that in physics there are a lot of other places +where we have “right” and “left” hand rules. As a matter of fact, when we +learned about vector analysis we learned about the right-hand rules we have to +use in order to get the angular momentum, torque, magnetic field, and so on, to +come out right. 'Phe force on a charge moving in a magnetic field, for example, +1s E! —= qu x B. In a given situation, in which we know #'!, 0, and #ØÖ, isnt that +cquation enough to defne right-handedness? As a matter of fact, if we go back +--- Trang 927 --- +and look at where the vectors came from, we know that the “right-hand rule” +was merely a convention; it was a trick. 'Phe original quantities, like the angular +momenta and the angular velocities, and things of this kind, were not really +vectors at alll Thhey are all somehow associated with a certain plane, and it is just +because there are three dimensions in space that we can associate the quantity +with a direction perpendicular to that plane. Of the two possible directions, we +chose the “right-hand” direction. +So 1ƒ the laws of physics are symmetrical, we should fnd that If some demon +were to sneak into all the physics laboratories and replace the word “right” for +“left” in every book in which “right-hand rules” are given, and instead we were to +use all “left-hand rules,” uniformly, then it should make no diference whatever +in the physical laws. +Fig. 52-2. A step In space and Its mirror image. +Let us give an illustration. There are two kinds of vectors. There are “honest” +vectors, for example a step 7 in space. lfin our apparatus there 1s a piece here +and something else there, then in a mirror apparatus there will be the image +piece and the image something else, and if we draw a vector from the “piece7” +to the “something else,” one vector is the mirror image of the other (Fig. 52-2). +The vector arrow changes its head, just as the whole space turns inside out; such +a vector we call a polar 0ector. +But the other kind of vector, which has to do with rotations, is of a different +nature. For example, suppose that in three dimensions something is rotating +as shown in Fig. 52-3. Then If we look at it in a mirror, it will be rotating as +indicated, namely, as the mirror image of the original rotation. Now we have +agreed to represent the mirror rotation by the same rule, it is a “vector” which, +on reflection, does øoø change about as the polar vector does, but is reversed +relative to the polar vectors and to the geometry of the space; such a vector 1s +called an azial 0ector. +Now if the law of relection symmetry is right in physics, then it must be true +that the equations must be so designed that if we change the sign of each axial +--- Trang 928 --- +NÌm lÌ +Fig. 52-3. A rotating wheel and Its mirror image. Note that the +angular velocity “vector” Is not reversed ¡In direction. +vector and each cross-product of vectors, which would be what corresponds to +refection, nothing will happen. For instance, when we write a formula which says +that the angular momentum is Ù = r x p, that equation is all right, because if we +change to a left-hand coordinate system, we change the sign of b, but p and r +do not change; the cross-product sign is changed, since we must change from a +right-hand rule to a left-hand rule. Äs another example, we know that the force +on a charge moving in a magnetic fñeld is #' = gu x Ö, but if we change from a +right- to a left-handed system, since # and ø are known to be polar vectors the +sien change required by the cross-product must be cancelled by a sign change +in Ö, which means that must be an axial vector. In other words, if we make +such a reflection, Ö must go to —. 5o ïf we change our coordinates from right +to left, we must also change the poles oŸ magnets from north to south. +Let us see how that works in an example. Suppose that we have two magnets, +as in Eig. 52-4. One is a magnet with the coils going around a certain way, and +with current in a given direction. The other magnet looks like the refection +of the first magnet in a mirror—the coil will wind the other way, everything +that happens inside the coil is exactly reversed, and the current goes as shown. +Now, from the laws for the production of magnetic fields, which we do not know +yet officially, but which we most likely learned in high school, ít turns out that +TH: J «b +IRqm, J8: +Fig. 52-4. A magnet and Its mirror image. +--- Trang 929 --- +the magnetic field is as shown in the fgure. In one case the pole is a south +magnetic pole, while in the other magnet the current is going the other way and +the magnetic fñeld is reversed—it is a north magnetic pole. So we see that when +we go from right to left we must indeed change om north to southl +Never mind changing north to south; these too are mere conventions. Let us +talk about phenomenaø. Suppose, now, that we have an electron moving through +one field, goïng into the page. 'Then, If we use the formula for the force, x +(remember the charge is minus), we find that the electron will deviate in the +indicated direction according to the physical law. 5o the phenomenon is that we +have a coïl with a current going in a specifed sense and an electron curves in a +certain way—that is the physics—never mind how we label everything. +Now let us do the same experiment with a mirror: we send an electron through +in a corresponding direction and now the force is reversed, if we calculate it from +the same rule, and that is very good because the corresponding rmmofions are then +mirror imagesl +52-6 Which hand is right? +So the fact of the matter is that in studying any phenomenon there are aÌways +two right-hand rules, or an even number of them, and the net result is that the +phenomena always look symmetrical. In short, therefore, we cannot t$ell right +from left if we also are not able to tell north from south. However, i% may seem +that we can tell the north pole of a magnet. The north pole of a compass needle, +for example, is one that points to the north. But of course that is again a local +property that has to do with geography of the earth; that is just like talking +about in which direction is Chicago, so it does not count. If we have seen compass +needles, we may have noticed that the north-seeking pole is a sort of bluish color. +But that is just due to the man who painted the magnet. These are all local, +conventional criteria. +However, if a magnet were to have the property that if we looked at it closely +enough we would see small hairs growing on its north pole but not on its south +pole, if that were the general rule, or if there were ø? unique way to distinguish +the north from the south pole of a magnet, then we could tell which of the two +cases we actually had, and #høt uould be the end oƒ the lau oƒ reflecHlion sựmmetrg. +To illustrate the whole problem still more clearly, imagine that we were talking +to a Martian, or someone very far away, by telephone. We are not allowed to +send him any actual samples to inspect; for instance, if we could send light, we +--- Trang 930 --- +could send him right-hand circularly polarized light and say, “Phat is right-hand +light—just watch the way it is going.” But we cannot øiue him anything, we can +only talk to him. He is far away, or in some strange location, and he cannot see +anything we can see. For instance, we cannot say, “Look at Ủrsa major; now +see how those stars are arranged. What we mean by “right is...” We are only +allowed to telephone him. +Now we want to tell him all about us. Of course, first we start defning +numbers, and say, “Tiek, tick, #uo, tick, tick, tick, ứhree,...,” so that gradually +he can understand a couple of words, and so on. After a while we may become +very familiar with this fellow, and he says, “What do you guys look like?” We start +to describe ourselves, and say, “Well, we are six feet tall” He says, “Wait a minute, +what is sỉx feet?” Is it possible to tell him what six feet is? Certainly! We say, “You +know about the diameter of hydrogen atoms——we are 17,000,000,000 hydrogen +atoms highl” 'That is possible because physical laws are not invariant under +change of scale, and therefore we can define an absolute length. And so we define +the size of the body, and tell him what the general shape is—it has prongs with +fñve bumps sticking out on the ends, and so on, and he follows us along, and we +fnish describing how we look on the outside, presumably without encountering +any particular dificulties. He is even making a model of us as we go along. He +says, “My, you are certainly very handsome fellows; now what is on the inside?” +So we start to describe the various organs on the inside, and we come to the +heart, and we carefully describe the shape of it, and say, “Ñow put the heart on +the left side.” He says, “Duhhh—the left side?” Now our problem is to describe +to him which side the heart goes on without his ever seeing anything that we see, +and without our ever sending any sample to him of what we mean by “right”——no +standard right-handed object. Can we do it? +52-7 Parity is not conservedl +Tt turns out that the laws oŸ gravitation, the laws of electricity and magnetism, +nuclear forces, all satisfy the principle of refection symmetry, so these laws, or +anything derived ữom them, cannot be used. But associated with the many +particles that are found in nature there is a phenomenon called betø đeca, or tUueak +đecø. One of the examples of weak decay, in connection with a particle discovered +in about 1954, posed a strange puzzle. 'There was a certain charged particle +which disintegrated into three r-mesons, as shown schematically in Eig. 52-5. +This particle was called, for a while, a r-meson. Now in Eig. 52-5 we also see +--- Trang 931 --- ++ = _"an—>~>~F +T + + +—= l Si +Fig. 52-5. A schematic diagram of the disintegration of a 7” and +a 8* particle. +another particle which disintegrates into #ưo mesons; one must be neutral, from +the conservation of charge. This particle was called a Ø-meson. 5o on the one +hand we have a particle called a 7, which disintegrates into three -mesons, and +a Ø, which disintegrates into two 7-mesons. Now iÿ was soon discovered that +the 7 and the Ø are almost equal in mass; in fact, within the experimental error, +they are equal. Next, the length of time it took for them to disintegrate into +three 7s and two 7s was found to be almost exactly the same; they live the +same length of time. Next, whenever they were made, they were made in the +same proportions, say, 14 percent 7”s to 86 percent Ø's. +Anyone in his right mỉnd realizes immediately that they must be the same +particle, that we merely produce an object which has two different ways of +disintegrating——not two diferent particles. This object that can disintegrate in +two diferent ways has, therefore, the same lifetime and the same production +ratio (because this is simply the ratio of the odds with which ¡it disintegrates into +these tEwo kinds). +However, it was possible bo prove (and we cannot here explain at all hou), +from the principle of refection symmetry in quantum mechanics, that it was +#mpossible to have these both come from the same particle—the same particle +could not disintegrate in both of these ways. The conservation law corresponding +to the principle of reflection syrmnmetry is something which has no classical analog, +and so this kind of quantum-mechanical conservation was called the conseruøiion +0ƒ parifụ. So, it was a result of the conservation of parity or, more precisely, from +the symmetry of the quantum-mechanical equations of the weak decays under +refection, that the same particle could not go into both, so it must be some kind +of coincidence of masses, lifetimes, and so on. But the more it was studied, the +more remarkable the coincidence, and the suspicion gradually grew that possibly +the deep law of the reflection symmetry of nature may be false. +As a result of this apparent failure, the physicists Lee and Yang suggested +that other experiments be done in related decays to try to test whether the law +--- Trang 932 --- +was correct in other cases. The first such experiment was carried out by Miss Wu +from Columbia, and was done as follows. sing a very strong magnet at a very +low temperature, it turns out that a certain isotope of cobalt, which disintegrates +by emitting an electron, is magnetie, and iŸ the temperature is low enough that +the thermail oscillations do not jiggle the atomic magnets about too much, they +line up in the magnetic field. So the cobalt atoms will all line up in this strong +fñeld. They then disintegrate, emitting an electron, and it was discovered that +when the atoms were lined up in a field whose Ö vector points upward, most of +the electrons were emitted in a downward direction. +TÝ one is not really “hep” to the world, such a remark does not sound like +anything of significance, but If one appreciates the problems and interesting +things in the world, then he sees that it is a most dramatic discovery: When +we put cobalt atoms in an extremely strong magnetic field, more disintegration +electrons go down than up. Therefore if we were to put i in a corresponding +experiment in a “mirror,” in which the cobalt atoms would be lined up ín the +opposite direction, they would spit their electrons p, not doưn; the action 1s +unsumwmctrical. The magnet has groun. hairsf The south pole of a magnet 1s of +such a kind that the electrons in a Øđ-disintegration tend to go away ữom it; that +distinguishes, in a physical way, the north pole from the south pole. +After this, a lot of other experiments were done: the disintegration of the +into / and 1; into an electron and two neutrinos; nowadays, the Ä into proton +and z; disintegration of 37s; and many other disintegrations. In fact, in almost +all cases where it could be expected, all have been found øø to obey reflection +symmetryl Pundamentally, the law of refection symmetry, at this level in physics, +18 IncOrrect. +In short, we can tell a Martian where to put the heart: we say, “Listen, build +yourself a magnet, and put the coils in, and put the current on, and then take +some cobalt and lower the temperature. Arrange the experiment so the electrons +go from the foot to the head, then the direction in which the current goes through +the coils is the direction that goes in on what we call the right and comes out on +the left.” So it is possible to define right and left, now, by doïng an experiment of +this kind. +'There are a lot of other features that were predicted. For example, it turns out +that the spin, the angular momentum, of the cobalt nucleus before disintegration +1s 5 units of #, and after disintegration it is 4 units. The electron carries spin +angular momentum, and there is also a neutrino involved. It is easy to see from +this that the electron must carry its spin angular momentum aligned along its +--- Trang 933 --- +direction of motion, the neutrino likewise. So ¡it looks as though the electron is +spinning to the left, and that was also checked. In fact, it was checked right here +at Caltech by Boehm and Wapstra, that the electrons spin mostly to the left. +(There were some other experiments that gave the opposite answer, but they +wore wrongl) +The next problem, of course, was to fnd the law of the failure of parity +conservation. What is the rule that tells us how strong the failure is going to +be? “The rule is this: it occurs only in these very slow reactions, called weak +decays, and when i% occurs, the rule is that the particles which carry spin, like +the electron, neutrino, and so on, come out with a spin tending to the left. That +1s a lopsided rule; it connects a polar vector velocity and an axial vector angular +mmomentum, and says that the angular momentum is more likely to be opposite +to the velocity than along ït. +Now that is the rule, but today we do not really understand the whys and +wherefores of it. WM/hg is this the right rule, what is the fundamental reason +for it, and how is it connected to anything else? At the moment we have been +so shocked by the fact that this thíng is unsymmetrical that we have not been +able to recover enough to understand what it means with regard to all the other +rules. However, the subject is interesting, modern, and still unsolved, so it seems +appropriate that we discuss some of the questions associated with ït. +52-8 Antimatter +The fñrst thing to do when one oŸ the symmetries is lost is to immediately go +back over the list of known or assumed symmetries and ask whether any of the +others are lost. Now we did not mention one operation on our list, which must +necessarily be questioned, and that is the relation between matter and antimatter. +Dirac predicted that in addition to electrons there must be another particle, called +the positron (discovered at Caltech by Anderson), that is necessarily related +to the electron. All the properties of these two particles obey certain rules of +correspondence: the energies are equal; the masses are equal; the charges are +reversed; but, more important than anything, the bwo of them, when they come +together, can annihilate each other and liberate their entire mass in the form +of energy, say +-rays. The positron is called an ønf2parf¿cle to the electron, +and these are the characteristics of a particle and its antiparticle. It was clear +from Dirac's argument that all the rest of the particles in the world should also +have corresponding antiparticles. For instance, for the proton there should be an +--- Trang 934 --- +antiproton, which is now symbolized by a7. The would have a negative electrical +charge and the same mass as a proton, and so on. The most important feature, +however, is that a proton and an antiproton coming together can annihilate each +other. The reason we emphasize this is that people do not understand it when we +say there is a neutron and also an antineutron, because they say, “A neutron is +neutral, so how cøn it have the opposite charge?” “The rule of the “anti” is not Just +that it has the opposite charge, it has a certain set of properties, the whole lot of +which are opposite. “The antineutron is distinguished from the neutron in this way: +1ƒ we briỉng two neutrons together, they just stay as two neutrons, but if we bring +a neutron and an antineutron together, they annihilate each other with a great +explosion of energy being liberated, with various -mesons, +-rays, and whatnot. +Now 1Í we have antineutrons, antiprotons, and antielectrons, we can make +antiatoms, in principle. “They have not been made yet, but it is possible In +principle. For instance, a hydrogen atom has a proton in the center with an +electron goïng around outside. Now imagine that somewhere we can make an +antiproton with a positron going around, would it go around? Woll, first of all, +the antiproton is electrically negative and the antielectron is electrically positive, +so they attract each other in a corresponding manner—the masses are all the +same; everything is the same. It is one of the principles of the symmetry of +physics, the equations seem to show, that if a clock, say, were made of matter on +one hand, and then we made the same clock of antimatter, it would run in this +way. (Of course, iŸ we put the clocks together, they would annihilate each other, +but that is diferent.) +An immediate question then arises. We can build, out of matter, two clocks, +one which is “left-hand” and one which is “right-hand.” EFor example, we could +bưild a clock which is not built in a simple way, but has cobalt and magnets and +electron detectors which detect the presence of đ-decay electrons and count them. +Bach time one is counted, the second hand moves over. hen the mirror clock, +receiving fewer electrons, will not run at the same rate. 5o evidentÏy we can +make ©wo clocks such that the left-hand clock does not agree with the right-hand +one. Let us make, out of matter, a clock which we call the standard or right-hand +clock. Now let us make, also out of matter, a clock which we call the left-hand +clock. We have just discovered that, in general, these two will no run the same +way; prior to that famous physical discovery, it was thought that they would. +Now it was also supposed that matter and antimatter were equivalent. That is, 1Ý +we made an antimatter clock, right-hand, the same shape, then it would run the +same as the right-hand matter clock, and if we made the same clock to the left it +--- Trang 935 --- +would run the same. In other words, in the beginning it was believed that all +ƒour of these clocks were the same; now of course we know that the right-hand +and left-hand matter are not the same. Presumably, therefore, the right-handed +antimatter and the left-handed antimatter are not the same. +So the obvious question is, which goes with which, If either? In other words, +does the right-handed matter behave the same way as the right-handed antimatter? +Or does the right-handed matter behave the same as the left-handed antimatter? +Ø-decay experiments, using positron decay instead of electron decay, indicate +that this is the interconnection: matter to the “right” works the same way as +antimatter to the “left” +Therefore, at long last, it is really true that right and left symmetry is still +maintainedl TỶ we made a left-hand clock, but made it out of the other kind of +matter, antimatter instead of matter, it would run in the same way. So what has +happened is that instead of having two independent rules in our list of symmetries, +two of these rules go together to make a new rule, which says that matter to the +right is symmetrical with antimatter to the left. +So iƒ our Martian is made of antimatter and we give him instructions to make +this “right” handed model like us, it will, of course, come out the other way +around. What would happen when, after much conversation back and forth, we +cach have taught the other to make space ships and we meet halfway in empty +space? We have instructed each other on our traditions, and so forth, and the +two of us come rushing out to shake hands. Well, ¡if he puts out his left hand, +watch outl +52-9 Broken symmetries +The next question is, what can we make out of laws which are nearlu symmet- +rical? The marvelous thing about ït all is that for such a wide range of important, +strong phenomena——nuclear forces, electrical phenomena, and even weak ones +like gravitation——over a tremendous range of physics, all the laws for these seem +to be symmetrical. Ôn the other hand, this little extra piece says, “No, the laws +are not symmetricall” How is it that nature can be almost symmetrical, but not +perfectly symmetrical? What shall we make of this? First, do we have any other +examples? The answer is, we do, in fact, have a few other examples. Eor instance, +the nuclear part of the force between proton and proton, between neutron and +neutron, and between neutron and proton, is all exactly the same——there is a +symmetry for nuclear forces, a new one, that we can interchange neutron and +proton——=but it evidently is not a general symmetry, for the electrical repulsion +--- Trang 936 --- +bebween ©wo protons at a distance does not exist for neutrons. So ït 1s not +generally true that we can aÌø/s replace a proton with a neutron, but only to a +good approximation. Why øoođ? Because the nuclear forces are much stronger +than the electrical forces. 5o this is an “almost” symmetry also. So we do have +examples in other things. +W©e have, in our minds, a tendency to accept symmetry as some kind of +perfection. In fact it is like the old idea of the Greeks that circles were perfect, +and it was rather horrible to believe that the planetary orbits were not cireles, +but only nearly circles. he diference between being a circle and beiïng nearly +a circle is not a small diference, it is a fiundamental change so far as the mind +is concerned. 'There is a sign of perfection and symmetry in a circle that is not +there the moment the circle is slightly of—that ¡is the end of it—it is no longer +symmetrical. Then the question is why it is only meari a circle—that is a much +more difficult question. The actual motion of the planets, in general, should +be ellipses, but during the ages, because of tidal forces, and so on, they have +been made almost symmetrical. Now the question is whether we have a similar +problem here. 'Phe problem from the point of view of the cireles is If they were +perfect circles there would be nothing to explain, that is clearly simple. But since +they are only nearly circles, there is a lot to explain, and the result turned out +to be a big dynamical problem, and now our problem is to explain why they are +nearly symmetrical by looking at tida]l forces and so on. +So our problem is to explain where symmetry comes from. Why is nature +so nearly symmetrical? No one has any idea why. The only thing we might +suggest is something like this: There is a gate in Japan, a gate in Neiko, which is +sometimes called by the Japanese the most beautiful gate in all Japan; it was +built in a time when there was great influence from Chỉinese art. 'Phis gate is +very elaborate, with lots of gables and beautiful carving and lots oŸ columns and +dragon heads and princes carved into the pillars, and so on. But when one looks +closely he sees that in the elaborate and complex design along one of the pillars, +one of the small design elements is carved upside down; otherwise the thing 1s +completely symmetrical. If one asks why this is, the story 1s that it was carved +upside down so that the gods will not be jealous of the perfection of man. So +they purposely put an error in there, so that the gods would not be jealous and +get angry with human beings. +We might like to turn the idea around and think that the true explanation +of the near symmetry of nature is this: that God made the laws only nearly +symmetrical so that we should not be jealous of His perfectionl +--- Trang 937 --- +Nnclox +A Air troupgh, 10-7 +Aberration, 27-12 , 34-18 AIgebra, 22-1 +Chromatie ~, 27-13 Four-vector ~, 17-12 ff +Spherical ~, 27-13, 36-6 Greek ~, 8-4 +of an electron microscope, II-29-10 Matrix ~, III-5-24, III-11-5, HII-20-28 +Absolute zero, 1-8, 2-10, 44-19, 44-22 Tensor ~>, III-8-6 +Absorption, 31-14 f Vector ~, 11-10 f, II-2-3, I-2-13, +of light, III-9-23 TI-2-21 f, I-3-1, II-3-21 f, II-27-6, +of photons, III-4-13 TI-27-8, III-5-25, III-8-2 f, IIT-8-6 +Absorption coefficient, II-32-13 AIlgebraic operator, III-20-4 +Acceleration, 8-13 ff Alnico V, II-36-23, II-37-20 +Angular ~, 18-5 Alternating-current circuits, II-22-1 +Componentfs of ~, 9-4 ff Alternating-current generator, II-17-11 +of gravity, 9-6 Amber, II-1-20, II-37-27 +Accelerator guide fields, II-29-10 Ammeter, II-16-2 +Acceptor, IHI-14-10 Ammonia maser, III-9-1 +Acetylcholine, 3-4 Ammonia molecule, TII-8-17 f +Activation energy, 3-6, 42-12 f States of an ~, III-9-1 ff +Active circuit element, II-22-9 Ampère's law, II-13-6 f +Actomyosin, 3-4 Ampèrian current, II-36-4 +Adenine, 3-9 Amplitude modulation, 48-6 ff +Adiabatic compression, 39-8 Amplitude of oscillation, 21-6 +Adiabatic demagnetization, II-35-18 f, Amplitudes, II-8-1 f +TI-35-18 f Interfering ~, III-5-16 +Adiabatic expansion, 44-10 Probability ~, 37-16, III-1-16, III-3-1 f, +Adjoint, III-11-39 TH-16-1 +Hermitian ~, III-20-5 5pace dependence of ~, III-13-7, +Affective future, 17-7 f IH-16-1 +Affective past, 17-7 Time dependence of ~, II-7-1 +TNDEX-1 +--- Trang 938 --- +'Transformation of ~, III-6-1 and parity conservation, III-18-5 +Analog computer, 25-15 Attenuation, 31-15 +Angle Avogadro's number, 41-18, II-8-9 +Brewster°s ~, 33-10 Axial vector, 20-6, 52-10 f, 52-17 +of incidence, 26-6, II-33-1 +of precession, II-34-7, I[I-34-7 B +of reflection, 26-6, II-33-1 Bar (unit), 47-7 +Angstrom (unit), 1-4 Barkhausen effect, II-37-19 +Angular acceleration, 18-5 Baryons, III-11-23 +Angular frequency, 21-5, 29-4, 29-6, 49-4 Base states, III-5-13 , II-12-1 +Angular momentum, 7-13, 18-8 f, 20-1, of the world, III-8-8 ff +TII-18-1 f, II-20-22 Battery, II-22-13 +Composition of ~, III-18-25 ff Benzene molecule, III-10-17 f, III-15-11 +Conservation of ~, 4-13 Bernoulli's theorem, II-40-10 +of a rigid body, 20-14 Bessel function, II-23-11, II-23-14, +of circularly polarized light, 33-18 1I-23-19, II-24-7 +Orbital ~, HI-19-2 Betatron, II-17-8 f, I-29-15 +Angular velocity, 18-4 f Binocular vision, 36-6, 36-8 f +Anomalous dispersion, 31-14 Biology and physiecs, 3-3 +Anomalous refraction, 33-15 Biot-Savart law, II-14-18 f, II-21-13 +Antiferromagnetic material, II-37-23 Birefringence, 33-4 f, 33-16 +Antimatter, 52-17 f, III-11-27 Birefringent material, 33-16 , II-33-6, +Antiparticle, 2-12, III-11-23 1I-39-14 +Antiproton, THI-11-23 Blackbody radiation, 41-5 ff +Argon, III-19-30 f Blackbody spectrum, III-4-15 +Associated Legendre functions, III-19-16 Bohr magneton, II-34-19, II-35-18, II-37-2, +Astronomy and physics, 3-10 f TI-12-19, III-34-19, III-35-18 +Atom, 1-3 Bohr radius, 38-12, III-2-12, III-19-5, +Metastable ~, 42-17 TIH-19-9 +Rutherford-Bohr model, II-5-4 Boltzmamn energy, lII-36-24 +Stability of ~s, II-5-4 f Boltzmamn factor, III-14-8 +'Thomson model, II-5-4 Boltzmamn”s constant, 41-18, II-7-14, +Atomie clock, 5-10, HI-9-22 TH-14-7 +Atomie currents, II-13-9 f, II-32-6 f, Boltzmamn”s law, 40-4 f +1I-36-4 Boltzmamn theory, III-21-13 +Atomic hypothesis, 1-3 f Boron, LIII-19-30 +Atomie orbiws, II-1-15 Bose particles, II-4-1 , III-15-10 f +Atomic particles, 2-12 ff Boundary layer, II-41-15 +Atomie polarizability, II-32-3 Boundary-value problems, II-7-2 +Atomic processes, I-8 Boyle's law, 40-16 +TNDEX-2 +--- Trang 939 --- +“Boys” camera, II-9-21 Centrifugal force, 7-9, 12-18, 16-3, 19-13 f, +Bragg-Nye crystal model, II-30-22 f 20-14, 43-7, 52-5, II-34-12, II-41-18, +Breaking-drop theory, II-9-18 f TI-19-20, III-19-25, III-34-12 +Bremsstrahlung, 34-12 f Centripetal force, 19-15 f +Brewster°s angle, 33-10 Charge +Brownian motion, 1-16, 6-8, 41-1 Œ, 46-2f, — Conservation of ~, +14, I-13-2 f +46-9 TImage ~, II-6-17 +BĐrush discharge, II-9-20 Line oÊ~, IL-5-6 f +Bulk modulus, II-38-6 Motion oŸ ~, H-29-1 ff +on electron, 12-12 +lọ) Point ~, II-1-3 +Caleulus Polarization ~s, II-10-6 +Differential ~, I-2-1 f Sheet of ~, I-ð-7 ff +Integral ~, IL3-1 f Sphere of ~, II-5-10 f +s y2 Charged conductor, II-6-14 £, II-8-4 ff +of variations, II-19-6 . +Cantilever beam, II-38-19 Chàng noparalion in a thunđer cloud, +Côpachance toa 3g Chemical bonds, II-30-5 f +. ; Chemical energy, 4-3 +Capacitor, 23-8, I-225 ữ Chemical kinetics, 42-11 +at hich frequencies, II-23-4 Chemical reaction. 1-12 f +Parallel-plate ~, 14-16 f, I-6-22 f, Chemistry and physics, 3Iữ +H-8-5 Cherenkov radiation, 51-3 +Capacity, II-6-23 Chlorophyll molecule, III-15-20 +of a condenser, [I-8-4 Chromatic aberration, 27-13 +Capillary action, 51-16 Chromaticity, 35-11 f +Carnot cycle, 44-8 ff, 45-4, 45-7 Circuit elements, II-23-1 f +Carriers Active ~, II-22-9 +Negative ~, III-14-3 Passive ~‹. II-22-9 +Positive ~, II-14-3 Cireuis +Carrier signal, 48-6 Alternating-current ~, II-22-1 +Catalyst, 42-13 Equivalent ~, II-22-22 f +Cavendish's experiment, 7-15 Circular motion, 21-6 f +Cavity resonators, II-23-1 f Circular polarization, 33-3 +Cells Circulation, II-1-8, II-3-14 ff +Cone ~, 35-2 f, 35-9, 35-14, 35-17 f, Classical electron radius, 32-6, II-28-5 +36-2 f, 36-8 Classical limit, III-7-16 ff +Rod ~, 35-2 f, 35-9, 35-17 f, 36-8, Clausius-Clapeyron equation, 45-10 ff +36-10 f, II-13-16 Clausius-Mossotti equation, II-11-13 f, +Center of mass, 18-1 , 19-1 1I-32-11 +TNDEX-3 +--- Trang 940 --- +Cleavage plane, II-30-3 of angular momentum, 4-13, 18-11 f, +Clebsch-Gordan coefficients, III-18-29, 20-8 +TI-18-34 of baryon number, III-11-23 +Coaxial line, I-24-2 of charge, 4-14, LII-13-2 ff +Coeficient of energy, 3-3, 4-1 f, II-27-1 f, I-42-24, +Absorption ~, II-32-13 IH-7-9 +Clebsch-Gordan ~‹s, II-18-29, II-18-34 of linear momentum, 4-13, 10-1 +Drag ~, II-41-11 of strangeness, III-11-21 +Einstein ~s, III-9-25 Conservative force, 14-5 ff +of coupling, II-17-25 Constant +of friction, 12-6 Boltzmamn”s ~, 41-18, II-7-14, HII-14-7 +of ViscOsity, TI-41-2 Dielectric ~; II-10-1 +Collision, 16-10 Gravitational ~, 7-17 +Elastie ~, 10-13 f Planck?s ~, 4-13, 5-19, 17-14, 37-18, +Collision cross section, 43-5 f 1L15-16, I-19-18 f, L-28-17, +Colloidal particles, II-7-13 HIE1-18, HH-20-24, TH-21-2 +Color vision, 35-1 f, 36-1 ff . ....A©. +. l onstrained motion, 14- +¬.-. ". ữ Contraction hypothesis, 15-8 f +. Ẻ Coriolis force, 19-14 , 20-8, 51-13, 52-5, +Complex impedance, 23-12 IL-34-12. IH-34-12 +Complex numbers, 22-11 Cornea. 35-1 36-5 f 36-18 +and harmonic motion, 23-1 Cornu's spiral 30-16 +Complex variable, II-7-3 ff Cosmic rays, 2-9, I-9-4 +. (Insect) eye, 36-12 ff Cosmic synchrotron radiation, 34-10 +OImpr©s5Ion Couette fow, II-41-17 ff +Adiabatic ~, 39-8 Coulomb's law, 28-1, 28-3, II-1-4 f, T-1-11, +Ă otnerminl ~, 44-10 1-4-3 f, I-4-8, IL-4-12, I-4-19, +ondensor IL-5-11 # +Energy of a ~, LL-8-4 ff Coupling, coefficient of, II-17-25 +Parallel-plate ~, I-6-22 ff, II-8-5 Covalent bonds, II-30-5 +Conduction band, LIH-14-2 Cross product, II-2-14, II-31-14 f +Conductivity, II-32-16 ross section, 5-15 +lomic ~, 43-9 Collision ~, 43-5 f +'Thermal ~, II-2-16, II-12-3, II-12-6 Nuclear ~, 5-15 +of a gas, 43-16 f Scattering ~, 32-12 +Conductor, II-1-3 'Thomson scattering ~, 32-13 +Cone cells, 35-2 f, 35-9, 35-14, 35-17 f, Crystal, II-30-1 +36-2 f, 36-8 Geometry of ~s, II-30-1 ff +Conservation lonic ~, II-8-8 +TNDEX-4 +--- Trang 941 --- +Molecular ~, II-30-5 D +Crystal difraction, 38-8 f, II-2-8 f Dˆ'Alembertian operator, II-25-13 +Crystal lattice, II-30-7 Damped oscillation, 24-4 ff +Cubic ~, II-30-17 Debye length, II-7-15 +Hexagonal ~, II-30-16 Definite energy, states of, III-13-5 f +Imperfections in a ~, II-13-16 Degrees of freedom, 25-3, 39-19, 40-1 +Monoclinic ~, I-30-16 Demagnetization, adiabatic, II-35-18 f, +Orthorhombie ~, II-30-17 . 11-35-18 Í +Propagation in a ~, III-13-1 Density, 1-6 +Current ~, II-13-2 +Tetragonal ~, II-30-17 +mm" Energy ~, II-27-3 +Tricinic ~, 1-30-15 Probability ~„ 6-13, 6-15, TI-16-9 +Trigonal ~, II-30-16 Derivative, 8-9 f +Cubic lattice, II-30-17 Partial ~, 14-15 +Curie point, II-37-7, II-37-20, II-37-26 Diamagnetism, II-34-1 , II-34-9 f, +Curie”s law, [I-11-9 II-34-1 f, IIL-34-9 +Curie temperature, II-36-29, II-36-31, Diamond lattice, II-14-1 +1I-37-2, I-37-6, II-37-23 Dielectric, II-10-1 f, TI-11-1 +Curie-Weiss law, II-11-20 Dielectric constant, II-10-1 ff +Curl operator, II-2-15, II-3-1 Differential calculus, 8-7, II-2-1 +Current Diffraction, 30-1 ff +Ampèrian ~, II-36-4 by a screen, 31-17 +Atomiec ~s, II-13-9 f, I-32-6 f,II-36-4f X-ray ~, 30-14, 38-9, II-8-9, I-30-3, +Eddy ~, II-16-11 IH-2-9 +Eleetric ~, I-13-2 Diffraction grating, 30-6 +in the atmosphere, II-9-4 ff Resolving power of a ~, 30-10 f +Induced ~s, II-16-10 Difusion, 43-1 f +Current density, [I-13-2 Molecular ~, 43-11 f +Curtate cycloid, 34-5, 34-8 ¬- eutrons, H-12-12 f +Curvature . . Electric ~, II-6-2 +in three-dimensional space, II-42-11 Magnetie ~, I-14-13 ff +Intrinsic ~, II-42-11 Molecular ~, II-11-1 +Mean ~, H-42-14 Oscillating ~, I-21-§ +Negative ~, H-42-11 Dipole moment, 12-9, II-6-5 +Positive ~, II-42-11 Magnetic ~, II-14-15 +Curved space, II-42-1 f Dipole potential, II-6-8 +Cutoff frequency, TII-22-30 Dipole radiator, 28-7 f, 29-6 +Cyclotron, II-29-10, II-29-15 Dirac equation, 20-11 +Cytosine, 3-9 Dislocations, II-30-19 +TNDEX-ð +--- Trang 942 --- +and crystal growth, II-30-20 f Doppler ~, 17-14, 23-18, 34-13 , 38-11, +Screw ~, II-30-20 f 1I-42-21, II-2-11, LII-12-15 +Slip ~, II-30-20 Hall ~, HI-14-12 +Dispersion, 31-10 ff lKerr ~, 33-8 +Anomalous ~, 31-14 Meissner ~, III-21-14 f, II-21-22 +Normal ~, 31-14 Mössbauer ~, II-42-24 +Dispersion equation, 31-10 Purkinje ~, 35-4 +Distance, 5-1 ff Effective mass, III-13-12 +Distance measurement tEfficiency of an ideal engine, 44-13 f +by the color-brightness relationship of Eigenstates, III-11-38 +stars, 5-12 Eigenvalues, III-11-38 +by triangulation, 5-10 BEinstein coefficients, III-4-15, III-9-25 +Distribution Einstein-Podolsky-Rosen paradox, +Normal (Gaussian) ~, 6-15, III-16-12, TH-18-16 +TII-16-14 Einstein?s equation of motion, II-42-30 +Probability ~, 6-13 ff Binstein”s feld equation, II-42-29 +Divergence tlastica, curves of the, II-38-25 +of four-vectors, II-25-11 Elastic collision, 10-13 f +Divergence operator, II-2-14, II-3-1 Elastic constants, II-39-9, II-39-19 +DNA, 3-8 Elastic energy, 4-3, 4-11 f +Domain, II-37-11 Elasticity, II-38-1 +Donor site, III-14-9 Elasticity tensor, II-39-6 +Doppler efect, 17-14, 23-18, 34-13 f, Elastic materials, II-39-1 +38-11, IT-42-21, II-2-11,IIT-12-15 Electret, II-11-16 +Dot product, II-2-9 Electrical energy, 4-3, 4-12, 10-15, +of four-vectors, II-25-6 1I-15-5 +Double stars, 7-10 Electrical forces, 2-5 ff, II-1-1 , II-13-1 +Drag coefficient, LI-41-11 in relativistic notation, II-25-1 +“Dry” water, II-40-1 Jilectric charge density, II-2-15, III-21-10 +Dyes, III-10-21 f Electric current, II-13-2 +Dynamical (ø-) momentum, III-21-8 in the atmosphere, II-9-4 +Dynamics, 9-1 Jlectric current density, LII-2-15 +Development of ~, 7-4 Electric dipole, II-6-2 ff +of rotation, 18-ð f Electric dipole matrix element, III-9-25 +Relativistic ~, 15-15 ff Electric ñeld, 2-6, 12-11 fŒ, II-1-4 fŒ, +1-6-1 Œ, II-7-1 ++ Relativity of , LI-13-13 +Eddy current, II-16-11 Electric ñux, II-1-8 +bfect Electric generator, II-16-1 Œ, II-22-9 +Barkhausen ~, II-37-19 Electric motor, II-16-1 +TNDEX-6 +--- Trang 943 --- +Electric potential, II-4-6 ff Boltzmamn ~, II-36-24 +Electric susceptibility, LI-10-7 Chemical ~, 4-3 +Electrodynamiecs, II-1-5 Conservation of ~, 3-3, 4-1 f, II-27-1 f, +Jlectromagnetic energy, 29-3 f 1I-42-24, III-7-9 +Electromagnetic field, 2-3, 2-7, 10-15 f Elastic ~, 4-3, 4-11 f +Electromagnetic mass, II-28-1 Electrical ~, 4-3, 4-12, 10-15, II-15-5 ff +Electromagnetic radiation, 26-1, 28-1 ff Electromagnetic ~, 29-3 f +Electromagnetic waves, 2-7, II-21-1 Electrostatic ~, II-8-1 +Electromagnetism, II-1-1 ff in nuclei, II-8-12 +Laws of ~, II-1-9 of a point charge, II-8-22 f +Electromotive force (ME), II-16-ð of charges, II-8-1 +Electron, 2-6, 37-2, 37-7 f, II-1-1, of ionic crystals, II-8-8 ff +TII-1-6 Eield ~, I-27-1 +Charge on ~, 12-12 Gravitational ~, 4-3 +Classical ~ radius, 32-6, II-28-5 Heat ~, 4-3, 4-11 f, 10-15 +Electron cloud, 6-20 in the electrostatic field, II-8-18 +Electron configuration, III-19-29 Kinetic ~, 1-13, 2-10, 4-3, 4-10 £f +Electronic polarization, II-11-2 f and temperature, 39-10 ff +Electron microscope, II-29-9 f Magnetic ~, II-17-22 ff +Electron-ray tube, 12-15 Mass ~, 4-3, 4-12 +Electron volt (unit), 34-7 Mechanical ~, II-15-5 f +Jlectrostatic energy, lII-8-1 Nuclear ~, 4-3 +in nuelei, II-8-12 of a condensor, II-8-4 ff +of a point charge, II-8-22 f Potential ~, 4-7, 13-1 , 14-1 f, +of charges, II-8-1 f TH-7-9 +of ionic crystals, II-8-8 Radiant ~, 4-3, 4-12, 7-20, 10-15 +Electrostatic equations Relativistie ~, 16-1 +with dielectries, II-10-10 ff Rotational kinetie ~, 19-12 +Electrostatic field, II-5-1 f, II-7-1 Rydberg ~, III-10-6, HII-19-5 +tEnergy in the ~, II-8-18 Wall ~, IIL-37-11 +of a grid, II-7-17 tEnergy density, II-27-3 +Jlectrostatic lens, II-29-5 f tEnergy diagram, III-14-2 +Electrostatic potential, equations of the, Energy Rux, II-27-3 +1-6-1 f Energy level diagram, III-14-6 +BElectrostatics, LII-4-1 f, II-5-1 ff Energy levels, 38-13 f, III-2-13 ff, +Eillipse, 7-2 TH-12-12 +Emission of photons, III-4-13 of a harmonie oscillator, 40-17 f +tEmissivity, LI-6-28 tEnergy theorem, 50-13 +Energy, 4-1 f, II-22-24 ff Enthalpy, 45-9 +Activation ~, 3-6, 42-12 f Entropy, 44-19 f, 46-9 +TINDEX-7 +--- Trang 944 --- +bquation bquilibrium, 1-12 +Clausius-Mossotti ~, II-11-13 f, Equipotential surfaces, II-4-20 +1I-32-11 Equivalent circuits, I-22-22 +Difusion ~ Jthylene molecule, III-15-13 +Heat ~, Euclidean geometry, l-1, 12-4, 12-19, 17-4 +Neutron ~, II-12-13 kEuler force, II-38-23 +Dirac ~, 20-11 Evaporation, 1-10, 1-12 +Dispersion ~, 31-10 of a liquid, 40-5 f, 42-1 ff +Einstein”s fñeld ~, II-42-29 Excess radius, II-42-9 ff, II-42-13 f, +Einstein”s ~ of motion, II-42-30 II-42-29 +Laplace ~, lI-7-2 Exchange force, II-37-3 +Maxwells ~s, 46-12, 47-12, I-2-1, Jxcited state, II-8-14, III-13-15 +1I-2-15, II-4-1 £, II-6-1, II-7-11, Exciton, LIII-13-16 +1I-8-20, II-10-11, II-13-7, II-13-13, Exclusion principle, IH-4-23 +1I-13-22, II-15-24 f, II-18-1 ff, Expansion +TI-22-1 £, II-22-13 £f, II-22-16, Adiabatie ~, 44-10 +1I-22-23, II-23-6, II-23-13 f, Isothermail ~, 44-10 +1I-23-20 f, I-24-9, II-25-12, Exponential atmosphere, 40-1 +TI-25-15, II-25-19, II-26-3, II-26-20, Eye +TI-26-23, II-27-5, II-27-8, II-27-13, Compound (insect) ~, 36-12 f +1I-27-17, II-28-1, II-32-7, II-33-2, Human ~, 35-1 +1I-33-5 , II-33-12 £, II-33-16, +1I-34-14, II-36-2, II-36-5, II-36-11, E +TII-36-26, II-38-4, II-39-14, I-42-29, Farad (unit), 25-14, II-6-24 +TI-34-14 Faradayˆs law of induction, II-17-3, +for four-vectors, II-25-17 TI-17-6, II-18-1, II-18-15, II-18-18 +General solution of ~, II-21-6 tFermat”s principle, 26-5, 26-7, 26-9, 26-11, +in a dielectric, II-32-4 26-13 , 26-17 f +Modifications of ~, II-28-10 Eermi (unit), 5-18 +Solutions of ~ in free space, II-201 f_ Eermi particles, III-4-1 f, III-15-11 +5olutions oŸ ~ with currents and Ferrites, II-37-25 f +charges, II-21-1 f terroelectricity, II-11-17 +Solving ~, II-18-17 terromagnetic insulators, II-37-25 +Poisson ~, II-6-2 terromagnetic materials, II-37-19 ff +Saha ~, 42-9 Ferromagnetism, II-36-1 f, II-37-1 ff +Schrödinger ~, II-15-21, II-41-20, Field, 14-12 f +TII-16-6, IH-16-18 Blectrie ~, 2-6, 12-11 f, IT-1-4 f, +for the hydrogen atom, III-19-1 1-6-1 Œ, II-7-1 +in a classical context, III-21-1 Electromagnetic ~, 2-3, 2-7, 10-15 f +Wave ~, 47-1 , II-18-17 Electrostatic ~, II-5-1 , II-7-1 +TNDEX-8 +--- Trang 945 --- +of a grid, II-7-17 of a lens, 27-7 ff +tFlux of a vector ~, II-3-4 of a spherical surface, 27-2 ff +Gravitational ~, 12-13 , 13-13 ff Focus, 26-11, 27-4 +in a cavity, II-5-17 tForce +Magnetic ~, 12-15 f, II-1-4 f, II-13-1 f, Centrifugal ~, 7-9, 12-18, 16-3, 19-13 f, +II-14-1 20-14, 43-7, 52-5, II-34-12, II-41-18, +of steady currents, II-13-6 ff TH-19-20, III-19-25, III-34-12 +Magnetizing ~, II-36-15 Centripetal ~, 19-15 f +of a charged conductor, II-6-14 f Components of ~, 9-4 ff +of a conductor, II-5-16 f Conservative ~, 14-5 +Relativity of electric ~, II-13-13 Coriolis ~, 19-14 , 20-8, 51-13, 52-5, +Relativity of magnetic ~, II-13-13 1I-34-12, III-34-12 +Scalar ~, II-2-3 ff Electrical ~s, 2-5 , II-1-1 Œ, II-138-1 +Superposition of ~s, 12-15 in relativistic notation, II-25-1 +'Two-dimensional ~s, II-7-3 ff Electromotive ~ (EMPF), II-16-5 +Vector ~, II-1-8 f, II-2-3 f kEuler ~, II-38-23 +Eield-emission microscope, II-6-27 ff Exchange ~, II-37-3 +Jield energy, II-27-1 Gravitational ~, 2-4 +of a point charge, II-28-1 f Lorentz ~, II-18-1, II-15-25 +Eield index, II-29-13 Magnetic ~, 12-15 f, II-1-4, II-13-1 +Eield-ion microscope, II-6-27 on a current, II-13-5 f +Eield lines, II-4-20 Molecular ~s, 12-9 ff +Eield momentum, II-27-1 ff Moment of ~, 18-8 +of a moving charge, II-28-3 f Nonconservative ~, 14-10 f +Field strength, II-1-6 Nuelear ~s, 12-20 f, II-1-2 f, II-8-12 f, +Eilter, H-22-30 1I-28-18, II-28-20 ff, III-10-10 +Flow Pseudo ~, 12-17 +Pluid ~, I-12-16 Fortune teller, 17-8 +Heat ~, LII-2-16 f, I-12-2 f Foucault pendulum, 16-3 +Irrotational ~, II-12-16 f, I-40-9 ff Fourier analysis, 25-7, 50-3 ff, 50-8 ff +Steady ~, II-40-10 ff Fourier theorem, II-7-17 +Viscous ~, II-41-6 Fourier transforms, 25-⁄ +Fluid fow, II-12-16 ff Four-potential, II-25-15 +Flux, II-4-12 Four-vector algebra, 17-12 +Electric ~, II-1-8 Four-vectors, 15-14 f, 17-8 f, I-25-1 f +Energy ~, II-27-3 Fovea, 35-2 f, 35-5, 35-18 +of a vector field, II-3-4 Frequency +Flux quantization, III-21-16 Angular ~, 21-5, 29-4, 29-6, 49-4 +Flux rule, II-17-1 Larmor ~, II-34-12, HII-34-12 +Focal length of oscillation, 2-7 +TINDEX-9 +--- Trang 946 --- +Plasma ~, II-7-12, I-32-18 Guanine, 3-9 +tresnels refection formulas, 33-15 Gyroscope, 20-9 +Eriction, 10-7, 12-4 +Coefficient of ~, 12-6 H +Origin of ~, 12-9 Haidinger's brush, 36-14 +Hail efect, III-14-12 +ŒG Hamiltonian, III-8-16 +Galilean relativity, 10-5, 10-11 Hamiltonian matrix, III-8-1 ff +Galilean transformation, 12-18, 15-4 Hamilton's first principal function, +Gallium, III-19-32 f TI-19-16 +Galvanometer, II-1-17, II-16-2 Harmonic motion, 21-6 f, 23-1 ff +Gamma rays, 2-8 Harmonic oscillator, 10-1, 21-1 ff +Garnets, II-37-25 f tEnergy levels of a ~, 40-17 f +Gauss (unit), 34-7, II-36-12 Forced ~, 21-9 , 23-4 +Gaussian distribution, 6-15, LIII-16-12, Harmonics, 50-1 +TII-16-14 Heat, 1-5, 13-5 +Gauss' law, II-4-18 f 5pecifc ~, 40-13 f, II-37-7 +Applications of ~, II-5-1 and the failure of classical physics, +for fñeld lines, II-4-21 40-16 +Gaussˆ theorem, II-3-8 f, III-21-7 at constant volume, 45-3 +Generator Heat conduction, II-3-10 +Alternating-current ~, II-17-11 Heat difusion equation, II-3-10 ff +Electric ~, II-16-1 , I-22-9 f Heat energy, 4-3, 4-11 f, 10-15 +Van de Graaff ~, II-5-19, II-8-14 Heat engines, 44-1 ff +Geology and physics, 3-12 f Heat fow, II-2-16 f, II-12-2 +Geometrical opties, 26-2, 27-1 Helium, 1-8, 3-11 f, 49-9 f, III-19-27 +Gradient operator, II-2-8 f, II-3-1 Liquid ~, LI-4-22 f +Gravitation, 2-4, 7-1 , 12-2, I-42-1 Helmholtz”s theorem, II-40-22 f +Theory of ~, II-42-28 Henry (unit), 25-13 +Gravitational acceleration, 9-6 Hermitian adjoint, III-20-5 +Gravitational constant, 7-17 Hexagonal lattice, II-30-16 +Gravitational energy, 4-3 f High-voltage breakdown, II-6-25 f +Gravitational fñeld, 12-13 , 13-13 ff Hooke”s law, 12-11, II-10-12, II-30-29, +Gravitational force, 2-4 1I-31-22, II-38-1 , II-38-6, II-39-6, +Gravity, 13-5 , II-42-17 f 1I-39-18 +Acceleration oŸ ~, 9-6 Human eye, 35-1 f +Greeks' difficulties with speed, 8-4 f Hydrodynamics, II-40-5 +Green's function, 25-8 Hydrogen, III-19-26 f +Ground state, II-8-14, III-7-3 Hyperfine splitting in ~, LIII-12-1 +Group velocity, 48-11 f Hydrogen atom, IH-19-1 +TNDEX-10 +--- Trang 947 --- +Hydrogen molecular ion, III-10-1 Interference, 28-10 , 29-1 +Hydrogen molecule, III-10-13 and diffraction, 30-1 +Hydrogen wave functions, [I-19-21 Two-slit ~, LII-3-8 +Hydrostatic pressure, II-40-1 Interfering amplitudes, III-5-16 ff +Hydrostatics, II-40-1 Interfering waves, 37-6, LII-1-6 +Hyperfne splitting in hydrogen, III-12-1 fÐ Interferometer, 15-8 +Hysteresis curve, II-37-10 lon, 1-11 +Hysteresis loop, II-36-16 lonic bonds, II-30-5 +lonic conductivity, 43-9 +T1 lonic crystal, II-8-8 +Ideal gas law, 39-16 lonic polarizability, LI-11-17 +Identical particles, II-3-16 f, III-4-1 lonization energy, 42-8 +IHumination, II-12-20 of hydrogen, 38-12, III-2-12 +Image charge, II-6-17 lonosphere, II-7-9, II-7-12, II-9-6, II-32-22 +Impedance, 25-15 f, II-22-1 Irreversibility, 46-9 +Complex ~, 23-12 Irrotational ñow, II-12-16 f, II-40-9 +of a vacuum, 32-3 Isotherm, II-2-5 +Impure semiconductors, II-14-8 ff Isothermal atmosphere, 40-3 +Incidence, angle of, 26-6, II-33-1 lsothermal compression, 44-10 +Inclined plane, 4-7 Isothermal expansion, 44-10 +Independent particle approximation, Isothermail surfaces, II-2-5 +TH-15-1 lsotopes, 3-7, 3-12, 39-17 +Eield ~, II-29-13 J +of refraction, 31-1 f, II-32-1 Johnson noïse, 41-4, 41-14 +Induced currents, II-16-10 Josephson junction, [I-21-25 +Inductance, 28-10, II-16-7 Œ, II-17-16 f, Joule (unit), 13-5 +1I-22-3 f Joule heating, 24-3 +Mutual ~, II-17-16 f, II-22-36 f +Self£-~, II-16-8, II-17-20 K +Induction, laws of, II-17-1 ff Kármán vortex street, II-41-13 +Inductor, 25-13 Kepler°s laws, 7-2 f, 7-5, 7-7, 9-1, 18-11 +Inertia, 2-4, 7-20 Kerr cell, 33-8 +Moment of ~, 18-12 f, 19-1 lerr efect, 33-8 +Principle of ~, 9-l Kilocalorie (unit), II-8-9 +Infrared radiation, 2-8, 23-14, 26-1 Kinematic (mo-) momentum, TII-21-8 +Insulator, II-1-3, II-10-1 Kinetic energy, 1-13, 2-10, 4-3, 4-10 f +Integral, 8-11 and temperature, 39-10 ff +Line ~, II-3-1 Rotational ~, 19-12 +Integral calculus, II-3-1 JKinetic theory +TNDEX-11 +--- Trang 948 --- +Applications of ~, 42-1 ff Newton”s ~s, 2-9, 7-10, 7-12, 9-1 f, +Of gases, 39-1 f 10-1 £, 10-5, 11-3 f, 11-7 f, 12-1 f, +irchhoff's laws, 25-16, II-22-14 f, 12-4, 12-18, 12-20, 13-1, 14-10, +II-22-27 15-1 , 15-5, 15-16, 16-4, 16-13 f, +Kronecker delta, II-31-10 18-1, 19-4, 20-1, 28-5, 39-1, 39-3, +Krypton, III-19-32 f 39-17, 41-2, 46-1, 46-9, 47-4 f, +1-7-9, II-19-2, II-42-1, I-42-28 +L in vector notation, 11-13 ff +Lagrangian, II-19-15 of refection, 26-3 +Lamé elastic constants, II-39-9 Ohms ~, 23-9, 25-12, 43-11, II-19-26, +Lamb-Retherford measurement, II-5-14 TH-14-12 +Landé g-factor, II-34-6, LIII-34-6 Rayleigh?s ~, 41-10 +Laplace equation, II-7-2 Snells ~, 26-5, 26-7, 26-14, 31-4, I-33-1 +Laplacian operator, II-2-20 Laws +Larmor frequency, II-34-12, III-34-12 of electromagnetism, II-1-9 +Larmor°'s theorem, II-34-11 , III-34-11 f of induction, II-17-1 ff +Laser, 5-4, 32-9, 42-17 f, 50-17, IH-9-21 Least action, principle of, II-19-1 ff +Law Least time, principle of, 26-1 ff +Ampère's ~, II-13-6 Legendre functions, associated, III-19-16 +Applications of Gauss' ~, II-5-1 Legendre polynomials, I[II-18-23, III-19-16 +Biot-Savart ~, II-14-18 f, II-21-13 Lens +Boltzmanmn”s ~, 40-4 f Electrostatic ~, II-29-5 ff +Boyle's ~, 40-16 Magnetic ~, II-29-7 f +Coulombs ~, 28-1, 28-3, II-1-4 f, Quadrupole ~, II-7-6, II-29-15 f +TI-1-11, II-4-3 f, II-4-8, II-4-12, Lens formula, 27-11 +1I-4-19, II-5-11 Lenz's rule, II-16-9 f, II-34-2, II-34-2 +Curie's ~, [I-11-9 Liếnard-Wiechert potentials, II-21-16 ff +Curie-Weiss ~, II-11-20 Light, 2-7, I-21-1 f +Faraday's ~ of induction, II-17-3, Absorption of ~, III-9-23 +TI-17-6, II-18-1, II-18-15, II-18-18 Momentum of ~, 34-20 ff +Gauss° ~, II-4-18 f Polarized ~, 32-15 +for field lines, II-4-21 Refection of ~, II-33-1 +Hooke's ~, 12-11, II-10-12, II-30-29, Refraction of ~, II-33-1 +1I-31-22, II-38-1 f, II-38-6, II-39-6, Scattering of ~, 32-1 +1I-39-18 5peed of ~, 15-1, II-18-16 f +Ideal gas ~, 39-16 Light cone, 17-6 +Kepler°s ~s, 7-2 Í, 7-5, 7-7, 9-1, 18-11 Lightning, II-9-21 +lirchhoffs ~s, 25-16, H-22-14 f, Light pressure, 34-20 +1I-22-27 Light waves, 48-1 +Lenz's ~, II-16-9 f, II-34-2, II-34-2 Linear momentum +TNDEX-12 +--- Trang 949 --- +Conservation of ~, 4-13, 10-1 ff Dia>, LII-34-1 Ế, II-34-9 f, III-34-1 f, +Linear systems, 25-1 IIH-34-9 +Linear transformation, 11-11 Ferro~>, II-36-1 , II-37-1 +Line integral, II-3-1 Para~>, II-34-1 f, II-35-1 Œ, II-34-1 ff, +Line of charge, II-ð-6 f IIH-35-1 +Liquid helium, HHI-4-22 f Magnetization currents, II-36-1 ff +Lithium, III-19-27 f Magnetizing field, II-36-15 +Lodestone, II-1-20, II-37-27 Magnetostatics, II-4-2, II-13-1 +Logarithms, 22-3 Magnetostriction, II-37-12, I-37-21 +Lorentz contraction, 15-13 Magnification, 27-10 f +Lorentz force, II-13-1, II-15-25 Magnons, III-15-6 +Lorentz formula, II-21-21 Maser, 42-17 +Lorentz group, II-25-5 Ammonia ~, III-9-1 +Lorentz transformation, 15-4 f, 17-1, Mass, 9-2, 15-1 +34-15, 52-3, II-25-1 Center of ~, 18-1 f, 19-1 +of fields, II-26-1 ff Effective ~, L[II-13-12 +Lorenz condition, II-25-15 Electromagnetic ~, II-28-1 ff +Lorenz gauge, II-18-20, II-25-15 Relativistic ~, 16-9 +Mass energy, 4-3, 4-12 +M Mass-energy equivalence, 15-17 f +Mach number, II-41-11 Mathematics and physics, 3-1 +Magenta, I[I-10-21 Matrix, III-5-9 +Magnetic dipole, II-14-13 Matrix algebra, III-5-24, III-11-5, +Magnetic dipole moment, II-14-15 II-20-28 +Magnetic energy, II-17-22 ff Maxwells demon, 46-8 f +Magnetic field, 12-15 f, I-1-4 , IIL-13-1 Í, Maxwells equations, 15-3 f, 25-5, 25-8, +TI-14-1 46-12, 47-12, II-2-1, II-2-15, +of steady currents, II-13-6 f 1-4-1 f, II-6-1, II-7-11, I-8-20, +Relativity of ~, II-13-13 TI-10-11, II-18-7, II-13-13, II-13-22, +Magnetic force, 12-15 , II-1-4, II-13-1 II-15-24 f, II-18-1 f, II-22-1 f, +on a current, [I-13-ð5 f 1I-22-13 £, I-22-16, II-22-23, +Magnetic induction, 12-17 1I-23-6, II-23-13 f, II-23-20 f, +Magnetic lens, II-29-7 ff 1I-24-9, II-25-12, II-25-15, II-25-19, +Magnetic materials, II-37-1 ff 1I-26-3, II-26-20, II-26-23, II-27-5, +Magnetic moments, II-34-4 f, III-11-8, 1I-27-8, II-27-13, II-27-17, II-28-1, +IH-34-4 f 1I-32-7, II-33-2, II-33-5, II-33-7 f, +Magnetic resonanece, II-35-1 f, III-35-1 1I-33-12 £, II-33-16, II-34-14, +Nuelear ~, II-35-19 f, II-35-19 1I-36-2, II-36-5, II-36-11, II-36-26, +Magnetic susceptibility, II-35-14, III-35-14 1I-38-4, II-39-14, II-42-29, III-10-12, +Magnetism, 2-7, II-34-1 f, III-34-1 ff TH-21-11, IIT-21-24, II-34-14 +TNDEX-13 +--- Trang 950 --- +for four-vectors, II-25-17 Angular ~, 18-8 , 20-1, IIT-18-1 f, +General solution of ~, II-21-6 IH-20-22 f +in a dielectrie, II-32-4 Composition of ~, LII-18-25 ff +Modifications of ~, II-28-10 Conservation of ~, 4-13, 18-11 , 20-8 +Solutions of ~ in free space, II-20-1 o£ a rigid body, 20-14 +Solutions of ~ with currents and Conservation of angular ~, +charges, II-21-1 Conservation of linear ~, 4-13, 10-1 +Solving ~, II-18-17 ff Dynamical (p-) ~, II-21-8 +Mean free path, 43-4 ff Eield ~, IE27-1 +Mean square distance, 6-9, 41-15 in quantum mechanics, 10-16 f +Mechanical energy, II-15-5 f Kinematic (mø-) ~, IH-21-8 +Meissner efect, IH-21-14 f, II-21-22 of light, 34-20 f +Metastable atom, 42-17 Relativistie ~, 10-14 , 16-1 f +Meter (unit), 5-18 Momentum operator, III-20-4, II-20-15 +MeV (unit), 2-14 Momentum spectrometer, II-29-2 +Michelson-Morley experiment, 15-ð fF, Momentum spectrum, ]I-29-4 +15-13 Monatomic gas, 39-7 ff, 39-11, 39-17 f, +Mi 40-13 f +icroscope ¬ . +Electron ~„ II-29-9 f Monoclinic lattice, II-30-16 +. - Motion, ð-1 f, 8-1 +Fiold-emission ~›, [6-27 Brownian ~, 1-16, 6-8, 41-1 , 46-2 f +Eield-ion ~, II-6-27 46-9 k ; ; ; ; +Minkowski space, II-31-23 Circular ~, 21-6 # +Modes, 49-1 Constrained ~, 14-4 f +Normal ~, 48-17 f Harmonic ~, 21-6 f, 23-1 +Mole (unit), 39-17 of charge, II-29-1 +Molecular crystal, II-30-5 Orbital ~, II-34-5, HI-34-5 +Molecular difusion, 43-11 Parabolie ~„ 8-17 +Molecular dipole, II-11-1 Perpetual ~, 46-3 +Molecular forces, 12-9 ff Planetary ~, 7-1 , 9-11 , 13-9 +Molecular motion, 41-1 Motor, electric, II-16-1 +Molecule, 1-4 Moving charge, ñeld momentum of, +Nonpolar ~, II-11-1 IL28-3 f +Polar ~, II-11-1, II-11-5 Muscle +Mössbauer efect, II-42-24 Smooth ~, 14-3 +Moment Striated (skeletal) ~, 14-3 +Dipole ~, 12-9, II-6-5 Music, 50-2 +of force, 18-8 Mutual capacitance, II-22-38 +of inertia, 18-12 f, 19-1 ff Mutual inductance, II-17-16 f, II-22-36 f +Momentum, 9-1 , 38-3 , III-2-3 ff mmu-momentum, III-21-8 +TINDEX-14 +--- Trang 951 --- +N Nutation, 20-12 f +Nabla operator (V), II-2-12 +Negative carriers, LIII-14-3 O +Neon, III-19-30 Oersted (unit), II-36-12 +Nernst heat theorem, 44-22 Ohm (unit), 25-12 +Neutral K-meson, III-11-21 Ohm”s law, 23-9, 25-12, 43-11, II-19-26, +Neutral pion, III-10-11 TH-14-12 +Neutron difusion equation, II-12-13 One-dimensional lattice, III-13-1 +Neutrons, 2-6 Obperator, III-8-7, II-20-1 +Difusion of ~, II-12-12 Algebraic ~, III-20-4 +NÑewton (unit), 11-10 Curl ~, II-2-15, II-3-1 +Newton - meter (unit), 13-5 D'Alembertian ~, II-25-13 +Newton's laws, 2-9, 7-10, 7-12, 9-1 f, Divergence ~, II-2-14, II-3-1 +10-1 , 10-5, 11-3 f, 11-7 f, 12-1 f, Gradient ~, II-2-8 f, II-3-1 +12-4, 12-18, 12-20, 13-1, 14-10, Laplacian ~, HI-2-20 +15-1 , 15-5, 15-16, 16-4, 16-13 f, Momentum ~, III-20-4, III-20-15 +18-1, 19-4, 20-1, 28-5, 39-1, 39-3, Nabla ~ (V), II-2-12 +39-17, 41-2, 46-1, 46-9, 47-4 f, Vector ~, II-2-12 +1-7-9, II-19-2, II-42-1, II-42-28 Optic axis, 33-5 +in vector notation, 11-13 Optic nerve, 35-3 +Nodes, 49-3 Optics, 26-1 +Noise, 50-2 Geometrical ~, 26-2, 27-1 ff +Nonconservative force, 14-10 Orbital angular momentum, III-19-2 +Nonpolar molecule, II-11-1 Orbital motion, II-34-5, III-34-5 +Normal dispersion, 31-14 Orientation polarization, II-11-5 ff +Normal distribution, 6-15, I[II-16-12, Oriented magnetic moment, II-35-7, +TII-16-14 TIH-35-7 +Normal modes, 48-17 f Orthorhombic lattice, II-30-17 +n-type semiconductor, III-14-10 Oscillating dipole, II-21-8 ff +Nuelear cross section, 5-15 Oscillation +Nuelear energy, 4-3 Amplitude of ~, 21-6 +Nuelear forces, 12-20 f, II-1-2 f, II-8-12 f, Damped ~, 24-4 +1I-28-18, II-28-20 f, III-10-10 ff tFrequency of ~, 2-7 +Nuelear g-factor, II-34-6, IH-34-6 Periodie ~, 9-7 +Nuelear interactions, II-8-14 Period of ~, 21-4 +Nuelear magnetic resonance, II-35-19 ff, Phase of ~, 21-6 +IH-35-19 Plasma ~s, II-7-9 ff +Nueleon, III-11-5 Oscillator, 5-4 +Nueleus, 2-6, 2-9 f, 2-12 Forced harmonic ~, 21-9 , 23-4 +Numerical analysis, 9-11 Harmonic ~, 10-1, 21-1 ff +TNDEX-15 +--- Trang 952 --- +P Photosynthesis, 3-4 +Pappus, theorem of, 19-6 f Physics +Parabolic antenna, 30-12 f Astronomy and ~, 3-10 f +Parabolic motion, 8-17 before 1920, 2-4 +Parallel-axis theorem, 19-9 Biology and ~, 3-3 +Parallel-plate capacitor, 14-16 f, II-6-22 ff, Chemistry and ~, 3-1 ff +T1I-8-ð Geology and ~, 3-12 f +Paramagnetism, II-34-1 f, II-35-1 f, Mathematics and ~, 3-1 +TI-34-1 Œ, II-35-1 Psychology and ~, 3-13 f +Paraxial rays, 27-3 Relationship to other sciences, 3-1 fŸ +Partial derivative, 14-15 Piezoelectricity, II-11-16, II-31-23 +Particles Planek's constant, 4-13, 5-19, 17-14, 37-18, +Bose ~, II-4-1 Ế, III-15-10 f TI-15-16, II-19-18 f, I-28-17, +tFermi ~, IHII-4-1 , II-15-11 TH-1-18, IH-20-24, IH-21-2 +Identical ~, III-3-16 fŒ, IH-4-1 Plane lattice, II-30-12 +Spin-one ~, [II-5-1 ff Planetary motion, 7-1 f, 9-11 f, 13-9 +Spin one-half ~, III-6-1 , II-12-1 f Plane waves, II-20-1 +Precession of ~, [I-7-18 Plasma, II-7-9 +Pascal's triangle, 6-7 Plasma frequency, II-7-12, II-32-18 ff +Passive circuit element, II-22-9 Plasma oscillations, II-7-9 ff +Pauli exclusion principle, II-36-31 ø-momentum, III-21-8 +Pauli spin exchange operator, LIII-12-12, Poincaré stress, II-28-7 f +TI-15-3 Point charge, II-1-3 +Pauli spin matrices, III-11-1 f Electrostatic energy of a ~, II-8-22 f +Pendulum, 5-3 Jield energy of a ~, II-28-1 f +Coupled ~s, 49-10 Poisson equation, II-6-2 +Pendulum clock, 5-3 Poisson”s ratio, II-38-3, II-38-6, II-38-21 +Periodic table, 2-14, 3-2, III-19-25 Polarization, 33-1 ff +Period of oscillation, 21-4 Circular ~, 33-3 +Permalloy, II-37-22 Electronice ~, II-11-2 +Permeability, II-36-18 of matter, II-32-1 ff +Relative ~, II-36-18 of scattered light, 33-4 +Perpetual motion, 46-3 Orientation ~, II-11-5 +Phase of oscillation, 21-6 Polarization charges, II-10-6 ff +Phase shift, 21-6 Polarization vector, II-10-4 ff +Phase velocity, 48-10, 48-12 Polarized light, 32-15 +Photon, 2-11, 17-14, 26-2, 37-13, IH-1-12 Polar molecule, II-11-1, II-11-5 +Absorption of ~s, III-4-13 f Polar vector, 20-6, 52-10 +Emission of ~s, HII-4-13 Positive carriers, LIII-14-3 +Polarization states of the ~, III-11-15 _ Potassium, III-19-31 f +TNDEX-16 +--- Trang 953 --- +Potential Proton spin, II-8-12 +tour-~>, II-25-15 Pseudo force, 12-17 +Quadrupole ~, II-6-14 Psychology and physics, 3-13 +Vector ~, II-14-1 f, II-15-1 f ø-type semiconductor, III-14-10 +of known currents, LI-14-5 ff Purkinje efect, 35-4 +Potential energy, 4-7, 13-1 , 14-1 E, Pyroelectricity, II-11-16 +II-7-9 +Potential gradient of the atmosphere, Q +I-9-1 Quadrupole lens, II-7-6, II-29-15 f +Quadrupole potential, II-6-14 +Power, 13-4 +h Quantized magnetic states, II-35-1 f, +Poynting vector, II-27-9 +Precession t am cl suod ics, 2-12 f, 2-17 +Angle of ~, II-34-7, III-34-7 Quan "`. tong ynanuW, 424 4°b +of atomic magnets, II-34-7 ff, III-34-7 QUỢ uc +Pressure. 1-6 and point charges, II-28-16 +2v. Quantum mechanical resonance, III-10-6 +Hydrostatic ~, II-40-1 . +Lisht ~. 34-20 Quantum mechanics, 2-3, 2-9 f, 6-17 f, +Š l 37-1 , 38-1 , HI-1-1 Ế, II-2-1 f, +of a gas, 39-3 +. TH-3-1 +Radiation ~, 34-20 . +¬ and vector potential, II-15-14 f, +Principal quantum number, III-19-22 +Principle II-2I-2 +of equivalence, II-42-17 f l +of inertia, 9-1 1T +of least action, II-19-1 Rabi molecular-beam method, II-35-7 f, +of superposition, II-1-5, H-4-4 IH-35-7 +of virtual work, 4-10 Radiant energy, 4-3, 4-12, 7-20, 10-15 +Uncertainty ~, 2-9 f, 6-17 , 7-21, Radiation +37-14 £, 37-18 , 38-5, 38-11 f, Blackbody ~, 41-5 +38-15, III-1-14, II-1-17 Œ, IIH-2-5, Bremsstrahlung, 34-12 f +TH-2-10 Ế, IIH-2-15 Cherenkov ~, 51-3 +and stability of atoms, II-1-2, II-5-5 Cosmic rays, 2-9, II-9-4 +Probability, 6-1 Cosmic synchrotron ~, 34-10 +Probability amplitudes, 37-16, III-1-16, Electromagnetic ~, 26-1, 28-1 ff +TII-3-1 , IIH-16-1 Gamma rays, 2-8 +Probability density, 6-13, 6-15, III-16-9 Infrared ~, 2-8, 23-14, 26-1 +Probability distribution, 6-13 , III-16-9 Light, 2-7 +Propagation, in a crystal lattice, I[I-13-1 Relativistic efects in ~, 34-1 +Propagation factor, II-22-31 Synchrotron ~, 34-6 +Protons, 2-6 UlItraviolet ~, 2-8, 26-1 +TNDEX-17 +--- Trang 954 --- +X-rays, 2-8, 26-1, 31-11, 34-8, 48-10, Theory of ~, 7-20 f, 17-1 +48-12 Resistance, 23-9 +Radiation damping, 32-1 Resistor, 23-9, 41-4 f, 41-14, I-22-7 f +Radiation pressure, 34-20 Resolving power, 27-14 f +Radiation resistance, 32-1 of a diÑraction grating, 30-10 f +Radioactive clock, 5-6 ff Resonance, 23-1 f +Radioactive isotopes, 3-7, 5-8, 52-16 Electrical ~, 23-8 ff +Radius in nature, 23-12 +Bohr ~, 38-12, HII-2-12, III-19-5, Quantum mechanical ~, III-10-6 +TII-19-9 Resonant cavity, II-23-11 +Classical electron ~, 32-6, II-28-5 Resonant circuits, II-23-22 f +Jxcess ~, II-42-9 f, II-42-13 , I-42-29 Resonant mode, II-23-21 +Random walk, 6-8 f, 41-14 Resonator, cavity, LII-23-1 ff +Ratchet and paw] machine, 46-1 ff Retarded time, 28-4 +Rayleigh”s criterion, 30-11 Retina, 35-1 f +Rayleigh”s law, 41-10 Reynolds number, II-41-8 +Rayleigh waves, II-38-16 Rigid body, 18-1, 20-1 +Reactance, II-22-25 f Angular momentum of a ~, 20-14 +Reciprocity principle, 26-9, 30-12 Rotation of a ~, 18-4 +Rectification, 50-15 Ritz combination principle, 38-14, III-2-14 +Rectifer, II-22-34 Rod cells, 35-2 f, 35-9, 35-17 f, 36-8, +Refected waves, II-33-14 ff 36-10 , III-13-16 +Reflection, 26-3 Root-mean-square (RMS) distance, 6-10 +Angle of ~, 26-6, II-33-1 Rotation +of light, II-33-1 ff in space, 20-1 +Total internal ~, II-33-22 in two dimensions, 18-1 +Refraction, 26-3 f of a rigid body, 18-4 +Anomalous ~, 33-15 f of axes, 11-4 +Index of ~, 31-1 , II-32-1 Plane ~, 18-1 +of light, II-33-1 ff Rotation matrix, III-6-6 +Relative permeability, II-36-18 Rutherford-Bohr atomic model, II-5-4 +Relativistic dynamics, 15-15 f Rydberg (unit), 38-12, III-2-12 +Relativistic energy, 16-1 Rydberg energy, III-10-6, III-19-5 +Relativistic mass, 16-9 +Relativistic momentum, 10-14 , 16-1 5 +Relativity Saha equation, 42-9 +Galilean ~, 10-5, 10-11 Scalar, 11-8 +of electric field, II-13-13 ff Scalar field, II-2-3 +of magnetic field, II-13-13 ff Scalar product, 11-15 +Special theory of ~, 15-1 ff of four-vectors, LI-25-5 ff +TNDEX-18 +--- Trang 955 --- +Scattering of light, 32-1 f Special theory of relativity, 15-1 +Schrödinger equation, II-15-21, II-41-20, 5pecific heat, 40-13 , I-37-7 +TII-16-6, II-16-18 , HI-20-28 and the failure of classical physics, +for the hydrogen atom, III-19-1 40-16 +in a classical context, LII-21-1 at constant volume, 45-3 +Scientific method, 2-2 5peed, 8-4 ff +Screw dislocations, II-30-20 f and velocity, 9-3 f +Screw jack, 4-8 Greeks' dificulties with ~, 8-4 f +Second (unit), 5-10 of light, 15-1, II-18-16 f +Seismograph, 51-9 of sound, 47-12 f +Self-inductance, II-16-8, II-17-20 ff Sphere of charge, II-5-10 f +Semiconductor junction, III-14-15 Spherical aberration, 27-13, 36-6 +Rectification at a ~, LII-14-19 ff of an electron microscope, II-29-10 +Semiconductors, III-14-1 Spherical harmonies, III-19-13 +Impure ~, IIH-14-8 f Spherically symmetric solutions, III-19-4 +n-type ~, LIH-14-10 Spherical waves, II-20-20 f, II-21-4 +ø-type ~, II-14-10 Spinel (MgAlzO¿), II-37-24 +Shear modulus, II-38-10 pin one-half particles, III-6-1 ff, +Shear waves, 5Í-8, II-38-11 ff IH-12-1 +Sheet of charge, II-5-7 Precession of ~, III-7-18 +Side bands, 48-7 ff Spin-one particles, III-ð-1 ff +Sigma electron, III-12-5 Spin orbit, II-8-13 +Sigma matrices, LII-11-3 Đpin-orbit interaction, II-15-25 +Sigma proton, [II-12-6 pin waves, III-15-1 +Sigma vector, III-11-7 5pontaneous emission, 42-15 +Simultaneity, 15-13 f 5pontaneous magnetization, II-36-24 +Sinusoidal waves, 29-4 ff Standard deviation, 6-15 +Skin depth, II-32-18 States +Slip dislocations, II-30-20 Eigen>, LII-11-38 +Smooth muscle, 14-3 Excited ~, II-8-14, III-13-15 +Snell's law, 26-5, 26-7, 26-14, 31-4, II-33-1 Ground ~, IH-7-3 +Sodium, III-19-30 f of defñnite energy, III-13-5 +Solenoid, II-13-11 Stationary ~, III-7-1 f, IH-11-38 +Solid-state physics, II-8-11 'Time-dependent ~, III-13-10 ff +Sound, 2-5, 47-1 f, 50-4 State vector, III-8-1 +5peed of ~, 47-12 f Resolution of ~s, III-8-4 +Space, 2-4, 8-4 Stationary states, [II-7-1 , III-11-38 +Curved ~, II-42-1 Statistical ñuctuations, 6-4 +5pace-time, 2-9, 17-1 f, II-26-22 Statistical mechanics, 3-2, 40-1 ff +Geometry of ~, 17-1 Steady fow, II-40-10 +TNDEX-19 +--- Trang 956 --- +Steap leader, II-9-21 T +Stefan-Boltzmamn constant, 45-14 'Taylor expansion, II-6-14 +Stern-Gerlach apparatus, [II-5-1 Temperature, 39-10 ff +Stern-Gerlach experiment, II-35-4 f, Tension +IH-35-4 Surface ~, II-12-8 +Stokes' theorem, II-3-17 Tensor, I-26-15, I-31-1 f +Strain, I-38-3 of elasticity, 1I-39-6 +Volume ~, II-38-6 of Imertia, 1L31-11 hà +Strain tensor, II-31-22, II-39-1 of polarizability, 1r31-1 +Strain ~, II-31-22, II-39-1 +Strangeness, III-11-21 +. Stress ~, II-31-15 ff +Conservation of ~, III-11-21 - +Transformation of ~ components, +“Strangeness” number, 2-14 I-31-4 +“Strange” particles, II-8-14 Tensor algebra, III-8-6 +Streamlines, II-40-10 Tensor 8eld. II-31-21 +Stress, [I-38-3 Tetragonal lattice, II-30-17 +Poincaré ~, II-28-7 f "Theorem +Volume ~, II-38-6 Bernoullis ~, II-40-10 ff +Stress tensor, II-31-15 Fourier ~, II-7-17 +Striated (skeletal) muscle, 14-3 Gauss' ~, II-3-8 ff, II-21-7 +Superconductivity, III-21-1 Helmholtz's ~, II-40-22 f +Supermalloy, II-36-18 Larmor?s ~, II-34-11 , IIH-34-11 +Superposition, II-13-22 f 0okes' ~, [I-3-17 f +of fñelds, 12-15 'Theory of gravitation, II-42-28 +Principle of ~, 25-3 f, 28-3, 47-11, "Thermal conductivity, 1I-2-16, 1I-12-3, +1-1-5, I-4-4 I-12-6 +Surface ` a .. Tên # +Equipotential ~s, H-4-20 f 1herma SgU1AĐTHM, 45 +Isothermal ^s. II-2-5 'Thermal ionization, 42-8 ff +Lo 'Thermodynamics, 39-3, 45-1 f, II-37-7 ff +Surface tension, II-12-8 Laws of ^.. 441 +susceptibility Thomson atomic model, II-5-4 +Electric ~, I-10-7 'Thomson scattering cross section, 32-13 +Magnetic ~, II-35-14, III-35-14 Three-body problem, 10-1 +Symmetry, l-8, II-] f 'Three-dimensional lattice, II-13-12 f +in physical laws, 52-1 f Three-dimensional waves, II-20-13 +Synchrotron, 2-8, 15-16, II-17-9, II-29-10, Three-phase power, II-16-16 +1I-29-15 f, I-29-20 'Thunderstorms, II-9-9 ff +Synchrotron radiation, 34-6 f, II-17-9 'Thymine, 3-9 +Cosmic ~, 34-10 Tides, 7-8 +TNDEX-20 +--- Trang 957 --- +Time, 2-4, 5-1 f, 8-1 Uncertainty principle, 2-9 f, 6-17 , 7-21, +Retarded ~, 28-4 37-14 f, 37-18 , 38-5, 38-11 f, +Standard of ~, 5-9 f 38-15, LII-1-14, III-1-17 Œ, IH-2-5, +'Transformation of ~, 15-9 f TH-2-10 f, IH-2-15 +'Time-dependent states, III-13-10 ff and sbability of atoms, IL-I-2, IL-ð-ð +Torque, 18-6, 20-1 Unit cell, 38-9, LI-2-9 +Torsion, II-38-11 Unit matrix, LIII-11-4 +'Total internal refection, II-33-22 ff Unit vector, 11-18, HI-2-6 +Transformation Unworldliness, II-25-18 +Fourler ~, 25-7 +Galilean ~, 12-18, 15-4 M +Linear ~, 11-11 Van de Graaff generator, II-5-19, II-8-14 +Lorentz ~, 15-4 f, 17-1, 34-15, 52-3, Vector, 11-1 f +I-25-1 Axial ~, 20-6, 52-10 +of fields, I-26-1 f Componentfs of a ~, 11-9 +of tỉme. 15-9 Four-~s, 15-14 f, 17-8 f, I-25-1 +of veloeity 16-5 Polar ~, 20-6, 52-10 +Ẻ Polarization ~, II-10-4 +'Transformer, II-16-7 Poynting, II-27-9 +'Transforming amplitudes, III-6-1 State ~¿ II- g1 +'Transient response, 21-10 Resolution of ^s. IIL-8-4 +Transients, 24-1 ff Unit ~, 11-18, IL2-6 +Electrical ~, 247 Vector algebra, 11-10 Œ, II-2-3, II-2-13, +Transistor, HI-14-21 f I-2-31 f, I-3-1, I-3-21 f, I-27-6, +Translation of axes, 11-2 IL27-8, II-5-25, III-8-2 f, III-§-6 +'Transmission line, II-24-1 Four-~, 17-12 +Transmitted waves, II-33-14 Vector analysis, 11-8 +Travelling fñeld, I-18-9 Vector field, II-1-8 f, II-2-3 ff +Triclinic lattice, II-30-15 Flux of a ~, II-3-4 +Trigonal lattice, II-30-16 Vector integrals, II-3-1 +'Triphenyl cyclopropenyl molecule, Vector operator, II-2-12 +TIH-15-233 Vector potential, II-14-1 , II-15-1 +'Twenty-one centimeter line, III-12-15 and quantum mechanics, II-15-14 f, +Twin paradox, 16-4 Ÿ TII-21-2f +'Two-dimensional fields, II-7-3 of known currents, II-14-5 +'Two-slit interference, III-3-8 ff Vector produet, 20-6 +'Iwo-state systems, III-10-1 f, IIIL-11-1 Velocity, §-6 +Angular ~, 18-4 f +U Components of ~, 9-4 ff +UIltraviolet radiation, 2-8, 26-1 Group ~, 46-11 f +TINDEX-21 +--- Trang 958 --- +Phase ~, 48-10, 48-12 Wave packet, III-13-11 +5peed and ~, 9-3 f Waves, 51-1 f +Transformation of ~, 16-5 Electromagnetic ~, 2-7, II-21-1 +Velocity potential, II-12-17 Light ~, 48-1 +Virtual image, 27-6 Plane ~, II-20-1 +Virtual work, principle of, 4-10 Refected ~, II-33-14 ff +Viscosity, II-41-1 Shear ~, 51-8, II-38-11 +Coeficient of ~, II-41-2 Sinusoidal ~, 29-4 +Viscous fow, II-41-6 Spherical ~, I-20-20 f, II-21-4 ff +Vision, 36-1 Ế, III-13-16 pin ~, LII-15-1 +Binocular ~, 36-6, 36-8 f Three-dimensional ~, II-20-13 f +Color ~, 35-1 , 36-1 Transmitted ~, II-33-14 +Physiochemistry of ~, 35-15 f “Wet” water, II-41-1 +Neurology of ~, 36-19 Work, 13-1 , 14-1 +Visual cortex, 36-6, 36-8 +Visual purple, 35-15, 35-17 X +Voltmeter, II-16-2 X-ray difraction, 30-14, 38-9, II-8-9, +Volume strain, II-38-6 1I-30-3, III-2-9 +Volume stress, II-38-6 X-rays, 2-8, 26-1, 31-11, 34-8, 48-10, 48-12 +Vortex lines, II-40-21 +Vorticity, II-40-9 4 +Young's modulus, II-38-3, II-38-11 +wW Yukawa “photon”, II-28-23 +'Wall energy, II-37-11 Yukawa potential, II-28-22, III-10-11 +Watt (unit), 13-5 +Wave cquation, 47-1 f, II-18-17 VẢ +Wavefront, 33-16 f, 47-6, 51-2, 51-4 Zeeman effect, III-12-19 +Wave function, LII-16-7 Zeeman splitting, IIH-12-15 +Meaning of the ~, III-21-10 f Zero, absolute, 1-8, 2-10 +Waveguides, II-24-1 ff Zero curl, II-3-20 f, II-4-2 +'Wavelength, 26-1, 29-5 Zero divergence, II-3-20 f, I-4-2 +Wave nodes, III-7-17 Zero mass, 2-17 +Wave number, 29-5 Zinc, III-19-31 f +TINDEX-22 +--- Trang 959 --- +NNưmao In‹ưÏlo+x +A Brown, Robert (1773-1858), 41-1 +Adams, John C. (1819-92), 7-10 +Aharonov, Yakir (1932-), I-15-21 C +Ampère, André-Marie (1775-1836), Carnot, N. L. Sadi (1796-1832), 4-3, +TI-13-7, I-18-17, I-20-17 44-4 ff, 45-6, 45-12 +Anderson, Carl D. (1905-91), 52-17 Cavendish, Henry (1731-1810), 7-16 +Aristotle (384-322 BC), 5-1 Cherenkov, Pavel A. (1908-90), 5l-3 +Avogadro, L. R. Amedeo Ơ. (1776-1856), Clapeyron, Benoft Paul Emile +30-3 (1799-1864), 44-4 +Copernicus, Nicolaus (1473-1543), 7-1 +B Coulomb, Charles-Augustin de +Becquerel, Antoine Henri (1852-1908), (1736-1806), IE5-14 +28-5 D +Bell, Alexander G. (1847-1922), II-16-6 Dedekind, J. W. Richard (1831-1916) +Bessel, Friedrich W. (1784-1846), I-23-11 29-5 Í +Boehm, Felix H. (1924), 52-17 Dicke, Robert H. (1916-97), 7-20 +Bohm, David (1917-92), IE7-13, I-15-21 pac, Pau] A, M, (1902 81), 52-17, +Bohr, Niels (1885-1962), 42-14, 1I-5-4, TI-2-2, TI-28-12 f, TII-28-17, TIT-3-1, +HE16-22, HH-19-8 HI-3-3, HI-8-3 f, III-8-6, TH-12-11 f, +Boltzmamn, Ludwig (1844-1906), 41-2 TII-16-15, II-16-22 +Bopp, Friedrich A. (1909-87), II-28-13 f, +II-28-16 f E +Born, Max (1882-1970), 37-2, 38-16, Binstein, Albert (1879-1955), 2-9, 4-13, +1I-28-12, II-28-17, III-1-1, IH-2-16, 6-18, 7-20 f, 12-15, 12-19 f, 15-1 f, +II-3-1, II-21-10 15-5, 15-15, 15-17, 16-1, 16-8, +Bragg, William Lawrence (1890-1971), 16-15, 41-1, 41-15, 42-14 f, 43-15, +1I-30-22 IL-13-13, II-25-19, I-26-23, +Brewster, David (1781-1868), 33-9 IT-27-18, II-28-7, II-42-1, II-42-11, +Briggs, Henry (1561-1630), 22-10 IL-42-14, II-42-17 £, II-42-21, +NAME INDEX-I +--- Trang 960 --- +TI-42-24, II-42-28 f, III-4-15, Helmholtz, Hermanmn von (1821-94), +TH-18-16 35-15, II-40-21, II-40-23 +Eötvös, Roland von (1848-1919), 7-20 Hess, Victor F. (1883-1964), II-9-4 +Euclid (c. 300 B©), 2-4, 5-10, 12-4, Huygens, Christiaan (1629-95), 15-3, 26-3, +II-42-8 f 33-16 +Earaday, Michael (1791-1867), II-10-1, Infeld, Leopold (1898-1968), II-28-12, +1I-10-4, II-16-3 f, II-16-7, II-16-17, 1I-28-17 +II-16-21, II-17-2 f, I-18-17, +IT-20-17 Bj +Fermat, Pierre de (1601-65), 26-5 f, 26-15 jeans, James H. (1877-1946), 40-17, +EFermi, Enrico (1901-54), 5-18 41-11, 41-13, II-2-12 +Feynman, Richard P. (1918-88), II-21-9, Jensen, .J. Hans D. (1907-73), III-15-25 +TI-28-13, II-28-17 Josephson, Brian D. (1940-), III-21-25 +Eourier, J. B. Joseph (1768-1830), 50-8 +Frank, Ilya M. (1908-90), 51-3 K +Franklin, Benjamin (1706-90), II-5-13 Kepler, Johannes (1571-1630), 7-2 +Galileo Galilei (1564-1642), 5-1, 5-3, 7-4, Lamb, Willis E. (1913-2008), II-5-14 +9-1, 10-7 f, 52-4 Laplace, Pierre-Simon de (1749-1827), +Gauss, J. Carl F. (1777-1855), II-3-10, 47-12 +II-16-3, II-36-12 Lawton, Willard E. (1899-1946), II-5-14 f +Geiger, Johann W. (1882-1945), II-5-4 Leibniz, Gottfried Willhelm (1646-1716), +Gell-Mamn, Murray (1929-), 2-14, 8-7 +TH-11-21 f, III-11-27 , III-11-33 Le Verrier, Urbain (1811-77), 7-10 +Gerlach, Walther (1889-1979), II-35-4, Liénard, Alfred-Marie (1869-1958), +II-35-6, II-35-4, III-35-6 IL-21-21 +Goeppert-Mayer, Maria (1906-72), Lorentz, Hendrik Antoon (1853-1928), +III-15-25 15-4, 15-8, II-21-21, II-21-24, +IT-25-19, II-28-6, II-28-12, II-28-20 +Hamilton, William Rowan (1805-65), M +TI-8-16 MacCullagh, James (1809-47), II-1-18 +Heaviside, Oliver (1850-1925), II-21-9 Marsden, Ernest (1889-1970), II-5-4 +Heisenberg, Werner K. (1901-76), 37-2, Maxwell, James Clerk (1831-79), 6-1, +37-14, 37-18 f, 38-16, I-19-19, 6-16, 28-1, 28-4, 40-16, 41-13, 46-8, +TH-1-1, HI-1-14, HI-1-17 , TI-1-16 f, I-1-20, II-5-14 f, II-17-3, +TH-2-16, II-16-14, II-20-27 f IT-18-1, II-18-3 f, II-18-6, II-18-8, +NAME INDEX-2 +--- Trang 961 --- +TI-18-15, II-18-17, II-18-21, Poynting, John Henry (1852-1914), +II-20-17, I-21-8, II-28-5, I-32-5 IL-27-5, II-28-5 +Mayer, Julius R. von (1814-78), 3-3 Priestley, Joseph (1733-1804), II-5-14 +Mendeleev, Dmitri I. (1834-1907), 2-14 Ptolemy, Claudius (c. 2nd cent.), 26-4 f +Michelson, Albert A. (1852-1931), 15-5, Pythagoras (c. 6th cent. BC), 50-1 f +Miller, William C. (1910-81), 35-4 ¬. +Minkowski, Hermamn (1864 1909), 17-14 Rabi, _. (1898-1988), H-35-7, +Motor (08 0Ì, nay he am E Anh +15-8 l Ẻ Ẻ Retherford, Robert C. (1912-81), II-5-14 +Roemer, Ole (1644-1710), 7-9 +N Rushton, William A. H. (1901-80), 35-17 f +Nernst, Walter H. (1864-1941), 4-21 Rutherford, Ernest (1871-1937), I-5-4 +Newton, Isaac (1643-1727), 7-4 f, 7-17, s +7-20, 8-7, 9-1 †, 9-6, 10-2 f, 10-16 Ï, S$ehrödinger, Erwin (1887-1961), 35-10, +11-2, 12-2, 12-14, 14-10, 15-1 f, 37-2, 38-16, II-19-19, TI-1-1, +16-2, 16-10, 18-11, 37-2, 47-12, II-2-16, III-3-1, III-16-6, +1I-4-20, II-19-14, II-42-1, IH-1-1 II-16-20 , II-20-27 f, III-21-10 +Nishijima, Kazuhiko (1926-2009), 2-14, Shannon, Claude E. (1916-2001), 44-4 +IH-11-21 f Smoluchowski, Marian (1872-1917), 41-15 +Nye, John F. (1923), II-30-22 Snellus), Willebrord (1580-1626), 26-5 +Stern, Otto (1888-1969), II-35-4, II-35-6, +O IH-35-4, III-35-6 +Oersted, Hans C. (1777-1851), II-18-17, Stevin(us), Simon (1548/49-1620), 4-8 +1I-36-12 +P Tamm, Igor Y. (1895-1971), 51-3 +Pais, Abraham (Bram) (1918-2000), Thomson, Joseph John (1856-1940), II-5-4 +TI-11-21, IH-11-27 f, II-11-33 Tycho Brahe (1546-1601), 7-2 +Pasteur, Louis (1822-95), 3-16 V +Panl, Wolfgang E. (1900 58), H45, Vi Teonardo da (1453-1519), 36-4 +HI-11-3 von NÑeumanmn, John (1903-57), II-12-17, +Pines, David (1924-), II-7-13 II-40-6 +Planck, Max (1858-1947), 40-19, 41-11 f, +42-13 f, 42-16, II-4-22 W +Plimpton, Samuel J. (1883-1948), Wapstra, Aaldert Hendrik (1922-2006), +1I-5-14 f 52-17 +Poincaré, J. Henri (1854-1912), 15-5, 15-9, Weber, Wilhelm E. (1804-91), II-16-3 +16-1, I-28-7 Weyl, Hermann (1885-1955), 11-1, 52-1 +NAME INDEX-3 +--- Trang 962 --- +Wheeler, John A. (1911-2008), II-28-13, Yukawa, Hideki (1907-81), 2-13, II-28-21, +1I-28-17 TH-10-10 +Wiechert, Emil Johann (1861-1928), Yustova, Elizaveta N. (1910-2008), 35-15, +TI-21-21 35-18 +Wilson, Charles 'T. R. (1869-1959), +1I-9-19 f +Young, Thomas (1773-1829), 35-13 Zeno of Elea (c. 5th cent. BC), 8-5 +NAME INDEX-4 +--- Trang 963 --- +X}isế of Sgyrrebols +| | absolute value, 6-9 +(2) binomial coefficlent, mœ over &, 6-7 +dể complex conjugate of a, 23-1 : +L1? D'Alembertian operator, L] = nh — V2, II-25-13 +( ) expectation value, 6-9 +2 . ¬. 9? 8? +V Laplacian operator, V4“ = m2 + ðy + 2z TI-2-20 +X4 nabla operator, W = (9/9z,9/Ø0,9/9»), 14-15 +|1). |2) a specific choice of base vectors for a two-state system, III-9-1 +|7, |1) a specific choice of base vectors for a two-state system, III-9-3 +(ø| state @ written as a bra vector, [II-8-ä3 +(ƒ|s) amplitude for a system prepared in the starting state | s) to be +found in the ñnal state | ƒ), TII-3-3 +|ø) state @ written as a ket vector, III-8-3 += approximately, 6-16 +~ of the order, 2-17 +œ proportional to, 5-2 +œ angular acceleration, 18-5 +^ heat capacity ratio (adiabatic index or specifc heat ratio), 39-8 +€ọ dielectric constant or permittivity oŸ vacuum, cọ = 8.854187817x +10—†12 EF/m, 12-12 +E Boltzmann's constant, ø = 1.3806504 x 10~23 J/K, TIT-14-7 +K relative permittivity, II-10-8 +E thermal conductivity, 43-16 +À wavelength, 17-14 +À reduced wavelength, À = À/2z, II-15-16 +LIST OF SYMBOLS-I +--- Trang 964 --- +ụu coefficient of friction, 12-6 +ụu magnetic moment, II-14-15 +Uu magnetic moment vector, II-14-15 +ụ shear modulus, II-38-10 +U frequency, 17-14 +p density, 47-6 +p electric charge density, II-2-15 +ơ cross section, 5-lð +ơ Pauli spin matrices vector, III-11-7 +Ơy, Ơu, Ơ; Pauli spin matrices, III-11-3 +Ø Poisson”s ratio, II-38-3 +ơ Stefan-Boltzmamn constant, ơ = 5.6704 x 108 W/m2K$, 45-14 +T torque, 18-7 +T torque vector, 20-7 +Ọ electrostatic potential, II-4-9 +®ọ basic ñux unit, [H-21-21 +X electric susceptibility, II-10-7 +œ angular velocity, 18-4 +lu angular velocity vector, 20-7 +MWỷ vorticity, I-40-9 +b7 acceleration vector, 19-3 +đạ, đụ, Œy cartesian components of the acceleration vector, 8-16 +G magnitude or component of the acceleration vector, 8-13 +A area, 5-l7 +A„u = (ø, A) four-potential, II-25-15 +A vector potential, II-14-2 +A„, Ây, Az cartesion components of the vector potential, II-14-2 +bB magnetic ñeld vector (magnetic induction), 12-17 +Đ„, Bụ, B„ cartesian components of the magnetic field vector, 12-17 +C speed of light, c = 2.99792458 x 108 m/s, 4-13 +lổi capacitance, 23-9 +lổi Clebsch-Gordan coefficients, III-18-34 +Œvy specifc heat at constant volume, 45-3 +d distance, 12-10 +D electric displacement vector, II-10-11 +LIST OF SYMBOLS-2 +--- Trang 965 --- +Cự unit vector in the direction 7ø, 28-2 +t electric fñeld vector, 12-13 +đy, Jụ, F; cartesian components of the electric field vector, 12-17 +b energy, 4-13 +đJ2gap energy øap, lII-14-7 +cự transverse electric field vector, III-14-14 +Lợi electric fñield vector, III-9-8 +ễ electromotive force, II-17-2 +ễ energy, 33-19 +ƒ focal length, 27-4 +đu electromagnetic tensor, II-26-12 ++ force vector, 11-9 +Từ, Fụ, F; cartesian components of the force vector, 9-5 +F magnitude or component oŸ the force vector, 7-l +g acceleration of gravity, 9-6 +G gravitational constant, 7-1 +h heat fow vector, II-2-6 +h Planck”s constant, h = 6.62606896 x 10” Js, 17-14 +h reduced Planck constant, = h/2z, 2-9 +H magnetizing fñeld vector, II-32-7 +? iImaginary unit, 22-11 +% unit vector in the direction zø, 11-18 +T electric current, 25-9 +T Intensity, 30-2 +T mmoment of inertia, 18-12 +1; tensor of inertia, II-31-13 +J Intensity, L[II-9-23 +3 electric current density vector, II-2-15 +Jz› ?ụ› 7z cartesian components of the electric current density vector, ITI- +3 unit vector in the direction , 11-18 +/ÿƑ angular momentum vector oŸ electron orbit, II-34-4 +Jo(z) Bessel function of the frst kind, II-28-11 +k Boltzmanmn*s constant, k = 1.3806504 x 10~?3 J/K, 39-16 +LIST OF SYMBOLS-3 +--- Trang 966 --- +kụ = (œ,k) four-wave vector, 34-18 +k unit vector in the direction z, 1-18 +k wave vector, 34-l7 +kự„, Kụ, k; cartesian components of the wave vector, 34-17 +k magnitude or component of the wave vector, wave number, 29-6 +K bulk modulus, II-38-6 +h angular momentum vector, 20-7 +Iý magnitude or component of the angular momentum vector, 18-8 +L self-inductanee, 23-10 +5 Lagrangian, II-19-15 +® self-inductanee, II-17-20 +|1) left-hand circularly polarized photon state, III-11-19 +m mass, 4-13 +Tneq efective electron mass in a crystal lattice, II-13-12 +mọ rest mass, 10-15 +M magnetfization vector, II-35-14 +MM mmutual inductanee, II-22-36 +9t mmutual induectanee, II-17-18 +3 bending momert, II-38-19 +n Index of refraction, 26-7 +n the øth Roman numecral, so that n takes on the values f, !, +.--; 1N, HI-I1-37 +T: unit normal vector, II-2-6 +Nụ number of electrons per unit volume, THI-14-7 +Áp number of holes per unit volume, THI-14-7 +p dipole moment vector, II-6-ð +p magnitude or component oŸ the dipole moment vector, II-6-ð +Đụ = (E,p) four-momentum, 17-12 +p mmomentum vector, 15-16 +Đa: ĐDụ: Dz cartesian components of the momentum vector, 10-15 +p magnitude or component of momentum vector, 2-9 +p pressure, II-40-3 +Tšpin exch Pauli spin exchange operator, TII-12-12 +P polarization vector, II-10-5 +P magnitude or component of the polarization vector, II-10-7 +LIST OF SYMBOLS-4 +--- Trang 967 --- +P power, 24-2 +Pp pressure, 39-4 +P(k,m) Bernoulli or binomial probability, 6-8 +P(1) probability of observing event 4, 6-2 +q electric charge, 12-11 +Q heat, 44-5 +T radius (position) vector, 11-9 +r radius or distance, 5-15 +R resistance, 23-9 +Mà Reynold”s number, II-41-10 +|? right-hand circularly polarized photon state, III-11-19 +8 distance, 8-2 +S action, II-19-6 +S entropy, 44-19 +S Poynting vector, II-27-3 +S “strangeness” number, 2-14 +ĐT stress tensor, II-31-17 +t time, 5-2 +T absolute temperature, 39-16 +T half-life, 5-6 +T kinetic energy, 13-l +tu velocity, 15-2 +U internal energy, 39-7 +U(:, 1) operator designating the operation waiting from tỉme q until £a, +THI-8-12 +U potential energy, 13-1 +U unworldliness, II-25-18 +Đ velocity vector, I1-12 +Uạ, Đụ, Uy cartesian components of the velocity vector, 8-15 +Đ magnitude or component of velocity vector, 8-7 +V velocity, 4-11 +V voltage, 23-9 +V volume, 39-4 +y voltage, II-17-21 +LIST OF SYMBOLS-ð +--- Trang 968 --- +W weight, 4-7 +W work, 14-2 +% cartesian coordinate, Í-l1 +%„ = (t,) four-position, 34-18 +Ụ cartesian coordinate, Í-l1 +Vì m(6, ð) spherical harmonics, III-19-13 +Y Young ”s modulus, II-38-3 +# cartesian coordinate, Í-l1 +Z complex impedance, 23-12 +LIST OF SYMBOLS-6 + + + + + + +--- Trang 1 --- +l'eyn?nan +LECTURESON PHYSICS +Feynman - Leighton - Sands +--- Trang 2 --- +l)e H: ¿711 +LECTURES ON +NEW MILLENNIUM EDITION +FEYNMANsLEIGHTONsSANDS +BASIC BOOKS VOLUME II +--- Trang 3 --- +Copyright © 1964, 2006, 2010 by California Institute of “Technology, +Michael A. Gottlieb, and Rudolf Pfeifer +Published by Basic Books, +A Member of the Perseus Books Group +AII rights reserved. Printed in the Ủnited States of America. +No part of this book may be reproduced in any manner whatsoever without written permission +except in the case of brief quotations embodied mm critical articles and reviews. +For Informatlon, address Basic Books, 250 West 57th Street, 15th Floor, New York, NY 10107. +Books published by Basic Books are available at special discounts for bulk purchases +in the United States by corporations, institutions, and other organizations. +Tor more Informatlon, please contact the Speclal Markets Department at the +Perseus Books Group, 2300 Chestnut Street, Suite 200, Philadelphia, PA 19103, +or call (800) 810-4145, ext. 5000, or e-mail speclal.markets@)perseusbooks.com. +A CTP catalog record for the hardcover edition of +this book 1s available from the Library of. Congress. +LCCN: 2010938208 +J-book ISBN: 978-0-465-07998-8 +--- Trang 4 --- +Abouét Hichear-‹cl[ Foggrtrrterrt +Born in 1918 in New York City, Richard P. Eeynman received his Ph.D. +from Princeton in 1942. Despite his youth, he played an important part in the +Manhattan Project at Los Alamos during World War II. Subsequently, he taught +at Cornell and at the California Institute of Technology. In 1965 he received the +Nobel Prize in Physics, along with Sin-ltiro Tomonaga and Julian Schwinger, for +his work in quantum electrodynamics. +Dr. Feynman won his Nobel Prize for successfully resolving problems with the +theory of quantum electrodynamics. He also created a mathematical theory that +accounts for the phenomenon of superfluidity in liquid helium. Thereafter, with +Murray Gell-Mamn, he did fundamental work in the area of weak interactions such +as beba decay. In later years Feynman played a key role in the development of +quark theory by putting forward his parton model of high energy proton collision +DrOC©SSGS. +Beyond these achievements, Dr. Eeynman introduced basic new computa- +tional techniques and notations into physics—above all, the ubiquitous Feynman +diagrams that, perhaps more than any other formalism in recent scientific history, +have changed the way in which basic physical processes are conceptualized and +calculated. +teynman was a remarkably efective educator. Of all his numerous awards, +he was especially proud of the Oersted Medal for Teaching, which he won in +1972. The Feunman Lectures on Phụsics, originally published in 1963, were +described by a reviewer in Scientiic American as “tough, but nourishing and full +of flavor. After 25 years it is /he guide for teachers and for the best of beginning +students.” In order to increase the understanding of physics among the lay public, +Dr. Feynman wrote 7e Character oƑ Phụsical Lao and QED: The Strange +Theor oƒ Light and Matter. He also authored a number of advanced publications +that have become classic references and textbooks for researchers and students. +Richard Feynman was a constructive public man. His work on the Challenger +commission is well known, especially his famous demonstration of the susceptibility +of the O-rings to cold, an elegant experiment which required nothing more than +a glass of Ice water and a C-clamp. Less well known were Dr. Eeynman's eforts +on the California State Curriculum Committee in the 1960s, where he protested +the mediocrity of textbooks. +A recital of Richard Feynman's myriad scientific and educational accomplish- +ments cannot adequately capture the essence of the man. Âs any reader of +even his most technical publications knows, Feynman's lively and multi-sided +personality shines through all his work. Besides being a physicist, he was at +varlous times a repairer of radios, a picker of locks, an artist, a dancer, a bongo +player, and even a decipherer of Mayan Hieroglyphics. Perpetually curious about +his world, he was an exemplary empiricist. +Richard Feynman died on Eebruary 15, 1988, in Los Angeles. +--- Trang 5 --- +MProftco ếo (lo 'Voar IWilloraraitrrie EZcÏfffG@re +Nearly ffty years have passed since Richard Eeynman taught the introductory +physics course at Caltech that gave rise to these three volumes, 7e Fewrwnan +Lectures on Phụsics. In those fñfty years our understanding of the physical +world has changed greatly, but The Feynman Lectures on Phụsics has endured. +teynman's lectures are as powerful today as when frst published, thanks to +Feynmans unique physics insights and pedagogy. 'They have been studied +worldwide by novices and mature physicists alike; they have been translated +into at least a dozen languages with more than 1.5 millions copies printed in the +tnglish language alone. Perhaps no other set of physics books has had such wide +Impact, for so long. +This Neu MiiHennzwm Edition ushers in a new era for The Feunman Lectures +ơn Phụsics (FLP): the twenty-flrst century era of electronic publishing. ÖP +has been converted to eF'LÖP, with the text and equations expressed in the IÃTERX +electronic typesetting language, and all fñgures redone using modern drawing +SOftware. +The consequences for the przn# version of this edition are no startling; it +looks almost the same as the original red books that physics students have known +and loved for decades. 'The main differences are an expanded and improved index, +the correction of 885 errata found by readers over the fve years since the first +primting of the previous edition, and the ease of correcting errata that future +readers may fnd. To this I shall return below. +'The eBook Wersion of this edition, and the Enhanced Electronic Version are +electronic innovations. By contrast with most eBook versions of 20th century tech- +mical books, whose equations, fñgures and sometimes even text become pixellated +when one tries 0o enlarge them, the IÃIEX manuscript of the Weu MiiHenniun +bdition makes it possible to create eBooks of the highest quality, in which all +features on the page (except photographs) can be enlarged without bound and +retain their precise shapes and sharpness. And the nhanced Electronic Version, +with Its audio and blackboard photos from Feynmanở”s original lectures, and is +links to other resources, is an innovation that would have given Feynman great +pleasure. +IMormaeeortos oŸ Fopgrartederrts Loe£mrros +These three volumes are a selcontained pedagogical treatise. They are also a +historical record of Feynmanˆs 1961-64 undergraduate physics lectures, a course +required of all Caltech freshmen and sophomores regardless of their majors. +Readers may wonder, as l have, how Feynman'”s lectures impacted the students +who attended them. Feynman, in his Preface to these volumes, ofered a somewhat +negative view. “[ don't think I did very well by the students,” he wrote. Matthew +Sands, in his memoir in Feywman's Tips on Phụsics expressed a far more positive +view. Out of curiosity, in spring 2005 I emailed or talked to a quasi-random set +o£ 17 students (out oŸ about 150) rom Feynman”s 1961-63 class—some who had +great dificulty with the class, and some who mastered it with ease; majors in +biology, chemistry, engineering, geology, mathematics and astronomy, as well as +in physics. +The intervening years might have glazed their memories with a euphoric tim, +but about 80 percent recall Feynman's lectures as highlights of their college years. +--- Trang 6 --- +“lt was like going to church” “The lectures were “a transformational experience, ” +“the experience of a lifetime, probably the most Important thing I got from +Caltech” “l was a biology major but Feynman's lectures stand out as a high +point in my undergraduate experience... though I must admit T couldn't do +the homework at the time and I hardly turned any of it in.” “Í was among the +least promising of students in this course, and Ï never missed a lecture.... Ï +remember and can still feel Feynman's joy of discovery.... His lectures had an +.... emotional impact that was probably lost in the printed Lectures.” +By contrast, several of the students have negative memories due largely to Ewo +issues: (1) “You couldn't learn to work the homework problems by attending the +lectures. Feynman was too slick——he knew tricks and what approximations could +be made, and had intuition based on experience and genius that a beginning +student does not possess.” Feynman and colleagues, aware of this faw in the +course, addressed it in part with materials that have been incorporated into +tecuwmans Tips on Phụsïcs: three problem-solving lectures by Feynman, and +a set Of exercises and answers assembled by Robert B. Leighton and Rochus +Vogt. (1) “The insecurity of not knowing what was likely to be discussed in +the next lecture, the lack of a text book or reference with any connection to +the lecture material, and consequent inability for us to read ahead, were very +frustrating.... I found the lectures exciting and understandable in the hall, but +they were Sanskrit outside [when I tried to reconstruct the details]” 'This problem, +OŸ course, was solved by these three volumnes, the printed version of The FeWnwman +Lectures on Phụsics. Thhey became the textbook from which Caltech students +studied for many years thereafter, and they live on today as one of Feynman's +greatest legacies. +A HHistorg, oŸ FErrddÉ( +The Feunman Lectures on Phụsics was produced very quickly by Eeynman +and his co-authors, Robert B. Leighton and Matthew Sands, working from +and expanding on tape recordings and blackboard photos of Eeynman”s course +lectures# (both of which are incorporated into the Enhaneced Electromic Version +of this Weu Miillenmum Edition). Given the high speed at which Feynman, +Leighton and Sands worked, it was inevitable that many errors crept into the fñrst +edition. Feynman accumulated long lists of claimed errata over the subsequent +years—errata found by students and faculty at Caltech and by readers around +the world. In the 1960°s and early 70s, Eeynman made time in his intense life +to check most but not all of the claimed errata for Volumes I and II, and insert +corrections into subsequent printings. But Eeynman”s sense of duty never rose +high enough above the excitement of discovering new things to make him deal +with the errata in Volume III.† After his untimely death in 1988, lists of errata +for all three volumes were deposited in the Caltech Archives, and there they lay +forgotten. +In 2002 Ralph Leighton (son of the late Robert Leighton and compatriot of +Feynman) informed me of the old errata and a new long list compiled by Ralph's +friend Michael Gottlieb. Leighton proposed that Caltech produce a new edition +of The Feunman. Lectures with all errata corrected, and publish it alongside a new +volume of auxiliary materlal, Feynwmans Tips on Phụsics, which he and Gottlieb +W©T© DI€pAring. +teynman was my hero and a close personal friend. When I saw the lists of +errata and the content of the proposed new volume, Ï quickly agreed to oversee +this project on behalf of Caltech (Feynman's long-time academic home, to which +* Eor descriptions of the genesis of Feynman”s lectures and of these volumes, see Feynman's +Preface and the Forewords to each of the three volumes, and also Matt Sands' Memoir in +teụnman*s Tips on Phụs¿cs, and the Special Preface to the Commemoratiue Edilion of FPLP, +written in 1989 by David Goodstein and Gerry Neugebauer, which also appears in the 2005 +Definstiue Edition. +† In 1975, he started checking errata for Volume III but got distracted by other things and +never fñnished the task, so no corrections were made. +--- Trang 7 --- +he, Leighton and Sands had entrusted all rights and responsibilities for The +Feunman Lectures). After a year and a ha]f of meticulous work by Gottlieb, and +careful scrutiny by Dr. Michael Hartl (an outstanding Caltech postdoc who vetted +all errata plus the new volume), the 2005 Defimiliue EdiHon oƒƑ The Feyrmaen +Lectures on Phụsics was born, with about 200 errata corrected and accompanied +by Feunmans Tips on Phụsics by Feynman, Gottlieb and Leighton. +1 thought that edition was goïng to be “Defnitive” What I dịd not antic- +Ipate was the enthusiastic response of readers around the world to an appeal +trom Gottlieb to identify further errata, and submit them via a website that +Gottlieb created and continues to maintain, 7e Feunman Lectures Website, +www.feynmanlectures.info. In the fve years sỉnce then, 965 new errata have +been submitted and survived the meticulous scrutiny of Gottlieb, Hartl, and Nate +Bode (an outstanding Caltech physics graduate student, who succeeded Hartl +as Caltech”s vetter of errata). Of these, 965 vetted errata, 80 were corrected in +the fourth printing of the 2efinilioe Ediion (August 2006) and the remaining +885 are correcbed in the first printing of this Weu Mllenniưm Edition (332 in +volume I, 263 in volume II, and 200 in volume IIT). For details of the errata, see +www .feynmanlectures.in£o. +Clearly, making The Fewwman Lectures on Phụsics error-free has become a +world-wide community enterprise. Ôn behalf of Caltech I thank the 50 readers +who have contributed since 2005 and the many more who may contribute over the +coming years. 'he names of all contributors are posted at www. feynmanlectures. +info/flp_errata.htm1. +Almost all the errata have been of three types: (ï) typographical errors +in prose; (ii) typographical and mathematical errors in equations, tables and +fgures—sign errors, incorrect numbers (e.g., a 5 that should be a 4), and missing +subscripts, summation signs, parentheses and terms in equations; (ii) incorrecE +cross references to chapters, tables and fgures. Thhese kinds of errors, though +not terribly serilous to a mature physicist, can be frustrating and confusing to +Feynman”s primary audience: students. +lt is remarkable that among the 1165 errata corrected under my auspices, +only several do Ï regard as true errors in physics. An example is Volume TT, +page 5-9, which now says “... no static distribution of charges inside a closed +grounded conductor can produce any |electric] ñelds outside” (the word grounded +was omited in previous editions). Thỉs error was pointed out to Feynman by a +number of readers, including Beulah Elizabeth Cox, a student at The College of +William and Mary, who had relied on Feynmanˆs erroneous passage in an exam. +To Ms. Cox, Feynman wrote in 1975,* “Your instrucbor was right not to give +you any points, Íor your answer was wrong, as he demonstrated using Gauss's +law. You should, in science, believe logic and arguments, carefully drawn, and +not authorities. You also read the book correctly and understood it. I made a +mistake, so the book is wrong. I probably was thinking oŸ a grounded conducting +sphere, or else of the fact that moving the charges around in diferent places +inside does not afect things on the outside. I am not sure how I did it, but +goofed. And you goofed, too, for believing me” +MNHoar thís 'Voar IWĩillorartrirrre EăÏfffGore Ấ (qiao Éo lo +Between November 2005 and .July 2006, 340 errata were submitted to 7 he +teunman Lectures Website www. feynman1ectures.info. Remarkably, the bulk +of these came from one person: Dr. Rudolf Pfeifer, then a physics postdoctoral +fellow at the University of Vienna, Austria. The publisher, Addison Wesley, fixed +80 errata, but balked at fñxing more because of cost: the books were being printed +by a photo-offset process, working from photographic images of the pages from +the 1960s. Correcting an error involved re-typesetting the entire page, and to +©enSure no new errors crept in, the page was re-typeset twice by two diferent +* Pages 288-289 of Perƒfectllu Reasonable Deuiations j[rom the Beaten Track, The Letters oŸ +Richard P. Fenman, ed. Michelle Feynman (Basic Books, New York, 2005). +--- Trang 8 --- +people, then compared and proofread by several other people—a very costly +process indeed, when hundreds of errata are involved. +Gottlieb, Pfeifer and Ralph Leighton were very unhappy about this, so they +formulated a plan aimed at facilitating the repair of all errata, and also aimed +at produecing eBook and enhanced electronic versions of The Feynwman Ùbectures +on Phụsics. They proposed their plan to me, as Caltechˆs representative, in +2007. I was enthusiastic but cautious. After seeing further details, including a +one-chapter demonstration of the Enhanced Electronic Version, Ï recommended +that Caltech cooperate with Gottlieb, Pfeifer and Leighton in the execution of +their plan. The plan was approved by three successive chairs of Caltech?s Division +of Physics, Mathematics and Astronomy—— Tom Tombrello, Andrew Lange, and +Tom Soifer—and the complex legal and contractual details were worked out by +Caltech?s Intellectual Property Counsel, Adam Cochran. With the publication of +this Neu Miilennium Edition, the plan has been executed successfully, despite +its complexity. 5pecifically: +Pfeifer and Gottlieb have converted into LÃTEX all three volumes of 'LP +(and also more than 1000 exercises from the Feynman course for incorporation +into Peywmans Tips on Phụsics). The PLP figures were redrawn in modern +electronic form in India, under guidance of the "'LP German translator, Henning +Heinze, for use in the German edition. Gottlieb and Pfeifer traded non-exclusive +use of their IATEX equations in the German edition (published by Oldenbourg) +for non-exclusive use of Heinze”s Ññgures in this Weu Milennium English edition. +Pfeifer and Gottlieb have meticulously checked all the IÃTEX text and equations +and all the redrawn fñgures, and made corrections as needed. Nate Bode and +1, on behalf of Caltech, have done spot checks of text, equations, and figures; +and remarkably, we have found no errors. Pfeifer and Gottlieb are unbelievably +meticulous and accurate. Gottlieb and Pfeifer arranged for John Sullivan at the +Huntington Library to digitize the photos of Feynmans 1962-64 blackboards, +and for George Blood Audio to digitize the lecture tapes—with financial support +and encouragement from Caltech Professor Carver Mead, logistical support from +Caltech Archivist Shelley Erwin, and legal support from Cochran. +The legal issues were serious: In the 1960s, Caltech licensed to Addison Wesley +rights to publish the print edition, and in the 1990s, rights to distribute the audio +of Feynman's lectures and a variant of an electronic edition. In the 2000s, through +a sequence of acquisitions of those licenses, the print rights were transferred to +the Pearson publishing group, while rights to the audio and the electronic version +were transferred to the Perseus publishing group. Cochran, with the aid of Ike +Williams, an attorney who specializes in publishing, succeeded in uniting all of +these rights with Perseus (Basic Books), making possible this Neu Millennium +bdiữtion. +AcEreo:r-loclqgrrorsÉs +Ơn behalf of Caltech, I thank the many people who have made this Neu +MMiilennium PEdition possible. Specifically, T thank the key people mentioned +above: Ralph Leighton, Michael Gottlieb, Tom Tombrello, Michael Hartl, Rudolf +Pfeifer, Henning Heinze, Adam Cochran, Carver Mead, Nate Bode, Shelley Erwin, +Andrew Lange, Tom Soifer, Ike Williams, and the 50 people who submitted errata +(Isted at www.feynmanlectures.info). And I also thank Michelle Feynman +(daughter of Richard Feynman) for her continuing support and advice, Alan Rice +for behind-the-scenes assistance and advice at Caltech, Stephan Puchegger and +Calvin Jackson for assistance and advice to Pfeifer about conversion of #'LP to +IATEX, Michael Figl, Manfred Smolik, and Andreas Stangl for discussions about +corrections of errata; and the Staff of Perseus/Basic Books, and (for previous +editions) the staf of Addison Wesley. +lip S. Thorne +'The Feynman Professor of 'heoretical Physics, Emeritus +California Institute of Technology Ociober 2010 +--- Trang 9 --- +MAINLY ELECTROMAGNETISM AND MATTER +RICHARD P. FEYNMAN +Richard Chace Tolman Professor 0ƒ Theoretical Physics +Califormia Insfitufe oƒ Technoloey +ROBERT B. LEIGHTON +Professor 0ƒ Physics +Califormia Insfitufe oƒ Technoloey +MATTHEW SANDS +Professor 0ƒ Physics +Califormia Insfitufe oƒ Technoloey +--- Trang 10 --- +Copyright © 1964 +CALIFORNIA INSTITUTE OEF TECHNOLOGY +Primted in the United States oƒ Ámerica +ALL RIGHTS RESERVED. THIS BOOK, OR PARTS THEREOEF +MAY NOT BE REPRODUCED IN ANY FORM WITHOUT +WRITTEN PERMISSION OF THE COPYRIGHT HOLDER. +Library oƒ Congress Catalog Card No. 63-20717 +Sixth priming, February 1977 +TS5BN 0-201-02117-X-P +0-201-02011-4-R +BBCCDDEEFFGG-MU-898 +--- Trang 11 --- +( lÌ l d +' lý Ỉ Á, +-®„Ì : +MO... +Mrogyrtraterre s Profqe© +These are the lectures in physics that I gave last year and the year before +to the freshman and sophomore classes at Caltech. The lectures are, of course, +not verbatim——they have been edited, sometimes extensively and sometimes less +so. The lectures form only part of the complete course. The whole group of 180 +students gathered in a big lecture room twice a week to hear these lectures and +then they broke up into small groups of 15 to 20 students in recitation sections +under the guidance of a teaching assistant. In addition, there was a laboratory +Session once a week. +The special problem we tried to get at with these lectures was to maintain +the interest of the very enthusiastic and rather smart students coming out of +the high schools and into Caltech. They have heard a lot about how interesting +and exciting physics is—the theory of relativity, quantum mechanies, and other +modern ideas. By the end of two years oŸ our previous course, many would be +very discouraged because there were really very few grand, new, modern ideas +presented to them. 'Phey were made to study inclined planes, electrostatics, and +so forth, and after two years it was quite stultifying. The problem was whether +or not we could make a course which would save the more advanced and excited +student by maintaining his enthusiasm. +'The lectures here are not in any way meant to be a survey course, but are very +serious. ÏI thought to address them to the most intelligent in the class and to make +sure, if possible, that even the most intelligent student was unable to completely +encompass everything that was in the lectures—by putting in suggestions of +applications of the ideas and concepts in various directions outside the main line +of attack. Eor this reason, thouph, I tried very hard to make all the statements +as accurate as possible, to point out in every case where the equations and ideas +fitted into the body of physics, and how—when they learned more—things would +be modifed. I also felt that for such students ït is important to indicate what +1b is that they should—Tf they are suficiently clever——be able to understand by +deduction from what has been said before, and what is being put in as something +new. When new ideas came in, [ would try either to deduce them if they were +deducible, or to explain that it œøs a new idea which hadn't any basis in terms of +things they had already learned and which was not supposed to be provable——=but +was just added ïn. +At the start of these lectures, Ï assumed that the students knew something +when they came out oŸ high school—such things as geometrical optics, simple +chemistry ideas, and so on. I also didn”t see that there was any reason to +make the lectures in a defñnite order, in the sense that I would not be allowed +--- Trang 12 --- +to mention something until Ï was ready to discuss i% in detail. There was a +great deal oŸ mention of things to come, without complete discussions. 'These +more complete discussions would come later when the preparation became more +advanced. Examples are the discussions oŸ inductance, and of energy levels, which +are at fñrst brought in in a very qualitative way and are later developed more +completely. +At the same time that Ï was aiming at the more active student, I also wanted +to take care of the fellow for whom the extra fireworks and side applications are +merely disquieting and who cannot be expected to learn most of the material +in the lecture at all. For such students I wanted there to be at least a central +core or backbone of material which he could get. Even ïf he didn't understand +everything ín a lecture, I hoped he wouldn't get nervous. I didn”t expect him to +understand everything, but only the central and most direct features. It takes, +of course, a certain intelligence on his part to see which are the central theorems +and central ideas, and which are the more advanced side issues and applications +which he may understand only in later years. +In giving these lectures there was one serious difficulty: in the way the course +was given, there wasn”t any feedback from the students to the lecturer 6o indicate +how well the lectures were goïing over. This is indeed a very serious difficulty, and +T don't know how good the lectures really are. The whole thing was essentially +an experiment. And ïf I đid it again I wouldn” do ¡it the same way——I hope Ï +đon?† have to do it again! I think, though, that things worked out——so far as the +physics is concerned——quite satisfactorily in the first year. +In the second year Ï was not so satisled. In the first part of the course, dealing +with electricity and magnetism, I couldn't think of any really unique or diferent +way of doing it —of any way that would be particularly more exciting than the +usual way of presenting it. 5o I don't think TI did very much ïn the lectures on +electricity and magnetism. At the end of the second year I had originally intended +to go on, after the electricity and magnetism, by giving some more lectures on +the properties of materials, but mainly to take up things like fundamental modes, +solutions of the difusion equation, vibrating systems, orthogonal functions, ... +developing the first stages of what are usually called “the mathematical methods +of physics.” In retrospect, I think that if Ï were doing i% again I would go back +to that original idea. But since it was not planned that I would be giving these +lectures again, it was suggested that it might be a good idea to try to give an +introduction to the quantum mechanics—what you will ñnd in Volume TH. +lt is perfectly clear that students who will major in physics can wait until +theïr third year for quantum mechanies. On the other hand, the argument was +made that many of the students in our course study physics as a background for +their primary interest in other fields. And the usual way of dealing with quantum +mnechanics makes that subJect almost unavailable for the great majority of students +because they have to take so long to learn it. Yet, in its real applications—— +especially ím its more complex applications, such as in electrical engineering +and chemistry—the full machinery of the diferential equation approach is not +actually used. So I tried to describe the prineiples of quantum mechanics in +a way which wouldn”t require that one first know the mathematics of partial +diferential equations. Even for a physicist I think that is an interesting thing +to try to do—to present quantum mechanics in this reverse fashion——for several +reasons which may be apparent in the lectures themselves. However, I think that +the experiment in the quantum mechanics part was not completely successful——in +large part because I really did not have enough time at the end (TI should, for +Instance, have had three or four more lectures in order to deal more completely +with such matters as energy bands and the spatial dependence of amplitudes). +Also, I had never presented the subject this way before, so the lack of feedbaeck +was particularly serious. Ï now believe the quantum mechaniecs should be given +at a later time. Maybe Ƒ]l have a chance to do it again someday. Then Ƒ]I do it +right. +The reason there are no lectures on how to solve problems is because there +were recitation sections. Although I did put ¡in three lectures in the frst year on +--- Trang 13 --- +how to solve problems, they are not included here. Also there was a lecture on +inertial guidance which certainly belongs after the lecture on rotating systems, +but which was, unfortunately, omitted. 'Phe fñifth and sixth lectures are actually +due to Matthew Sands, as Ï was out of town. +'The question, of course, is how well this experiment has succeeded. My own +point of view—which, however, does not seem to be shared by most of the people +who worked with the students——is pessimistic. I donˆt think I did very well by +the students. When I look at the way the majority of the students handled the +problems on the examinations, I think that the system is a failure. Of course, +my friends point out to me that there were one or ©wo dozen students who——very +surprisingly——understood almost everything in all of the lectures, and who were +quite active in working with the material and worrying about the many points +in an excited and interested way. Thhese people have now, I believe, a first-rate +background in physics—and they are, after all, the ones Ï was trying to get at. +But then, “The power of instruction is seldom of mụuch efficacy except in those +happy dispositions where it is almost superfuous” (Gibbon) +StI, I didn't want to leave any student completely behind, as perhaps T did. +T think one way we could help the students more would be by putting more hard +work into developing a set of problems which would elucidate some of the ideas +in the lectures. Problems give a good opportunity to fñll out the material of the +lectures and make more realistic, more complete, and more settled in the mind +the ideas that have been exposed. +1 think, however, that there isn't any solution to this problem of education +other than to realize that the best teaching can be done only when there is a +direct individual relationship between a student and a good teacher—a situation +in which the student discusses the ideas, thinks about the things, and talks about +the things. It's impossible to learn very much by simply sitting in a lecture, or +even by simply doing problems that are assigned. But in our modern tỉimes we +have so many students to teach that we have to try to ñnd some substitute for +the ideal. Perhaps my lectures can make some contribution. Perhaps in some +small place where there are individual teachers and students, they may get some +inspiration or some ideas from the lectures. Perhaps they will have fun thinking +them through—or goïng on to develop some of the ideas further. +RICHARD P. FEEYNMAN +Jưnec, 1968 +--- Trang 14 --- +Morosrcor-‹[ +For some forty years Richard P. Feynman focussed his curiosity on the +mysterious workings of the physical world, and bent his intellect to searching out +the order in its chaos. Now, he has given two years of his ability and his energy +to his Lectures on Physics for beginning students. For them he has distilled the +essence of his knowledge, and has created in terms they can hope to grasp a +picture of the physicist's universe. 'Io his lectures he has brought the brilliance +and clarity of his thought, the originality and vitality of his approach, and the +contagious enthusiasm of his delivery. It was a joy to behold. +The first yearˆs lectures formed the basis for the fñrst volume of this set of +books. We have tried in this the second volume to make some kind of a record of +a part of the second yearˆs lectures—which were given to the sophomore cÌass +during the 1962-1963 academic year. The rest of the second yearˆs lectures will +make up Volume TII. +Of the second year of lectures, the fñrst two-thirds were devoted to a fairly +complete treatment of the physics of electricity and magnetism. Ïts presentation +was intended to serve a dual purpose. We hoped, first, to give the students a +complete view of one of the great chapters of physics—from the early gropings +of Franklin, through the great synthesis of Maxwell, on to the Lorentz electron +theory of material properties, and ending with the still unsolved dilemmas of the +electromagnetic selfenergy. And we hoped, second, by introducing at the outset +the calculus of vector fields, to give a solid introduection to the mathematics of +ñeld theories. 'To emphasize the general utility of the mathematical methods, +related subjects rom other parts of physics were sometimes analyzed together +with their electric counterparts. W©e continually tried to drive home the generality +of the mathematics. (“The same equations have the same solutions.”) And we +emphasized this point by the kinds of exercises and examinations we gave with +the cOUrse. +Following the electromagnetism there are two chapters each on elasticity and +ñuid fiow. In the fñrst chapter of each pair, the elementary and practical aspects +are treated. The second chapter on each subject attempts to give an overview of +the whole complex range of phenomena which the subjJect can lead to. 'These +four chapters can well be omitted without serious loss, since they are not at all a +necessary preparation for Volume TIT. +The last quarter, approximately, of the second year was dedicated to an +Introduction to quantum mechanics. 'This material has been put into the third +volume. +In this record of the Feynman Lectures we wished to do more than provide a +transcription of what was said. We hoped to make the written version as clear an +exposition as possible of the ideas on which the original lectures were based. Eor +some of the lectures this could be done by making only minor adjustments of the +wording in the original transcript. Eor others of the lectures a major reworking +and rearrangement of the material was required. Sometimes we felt we should +add some new material to improve the clarity or balance of the presentation. +Throughout the process we beneftted from the continual help and advice of +Professor Feynman. +--- Trang 15 --- +'The translation of over 1,000,000 spoken words into a coherent text on a tight +schedule is a formidable task, particularly when 1t is accompanied by the other +onerous burdens which come with the introduction of a new course—preparing for +recitation sections, and meeting students, designing exercises and examinations, +and grading them, and so on. Many hands—and heads—were involved. Ïn some +Instances we have, I believe, been able to render a faithful image—or a tenderly +retouched portrait—of the original Feynman. In other instances we have fallen +far short of this ideal. Our successes are owed to all those who helped. “Phe +failures, we regret. +As explained in detail in the Eoreword to Volume I, these lectures were +but one aspecE oŸ a program ¡initiated and supervised by the Physics Course +Revision Committee (R. B. Leighton, Chairman, H. V. Neher, and M. Sands) +at the California Institute of Technology, and supported fñnancially by the Ford +Foundation. In addition, the following people helped with one aspect or another of +the preparation of textual material for this second volume: 'F. K. Caughey, M. L. +R. W. Kavanagh, R. B. Leighton, J. Mathews, M. S. Plesset, F. L. Warren, W. +'Whaling, C. H. Wilts, and B. Zimmerman. Others contributed indirectly through +their work on the course: J. Blue, G. E. Chapline, M. J. Clauser, R. Dolen, H. H. +HH, and A. M. Title. Professor Gerry Neugebauer contributed in all aspects +of our task with a diligence and devotion far beyond the dictates of duty. The +story of physics you fñnd here would, however, not have been, except for the +extraordinary ability and industry of Richard P. Feynman. +MATTHEW SANDS +March, 1964 +--- Trang 16 --- +(toref©reés +CHAPTER l. ELECTROMAGNETISM CHAPTER 6. “THE ELECTRIC PIELD IN VARIOUS +1-6 Electromagnetism in science and technology .. . 1-10 6-5 The dipole approximation for an arbitrary distribu- +"2 aAa. . . Ta +2-6 The diferential equation ofheatfow ....... 2-8 +CHAPTER 3. VECTOR INTEGRAL CALCULUS 7-1 Methods for ñnding the electrostaticfeld ...... 7-1 +7-2_ 'Two-dimensional fields; functions of the complex +3-6 The circulation around a square; Stokes' theorem 3-9 CHAPTER 8. ELECTROSTATIC ENERGY +3-7 Curl-free and divergence-freefields......... 3-10 +CHAPTER 4. ELECTROSTATICS 8-2_ The energy of a condenser. Forces on charged con- +du€fOTS... . . . Q23 +... 8-2 +¬...- ôSTTaNTaa II II NIHaăa:g. 8-3 The electrostatic energy ofan ioniccrystal .... 8-4 +4-8 Eield lines; equipotential surfaces.......... 411 9-1 “The electric potential gradient of the atmosphere 9-1 +9-2 Electric currentsin theatmosphere ........ 9-2 +CHAPTER 5. ÁPPLICATION OE QAUSS' LAW 9-3 Origin of the atmospheric currents ......... 9-4 +--- Trang 17 --- +CHAPTER l1. ÍNSIDE DIELECTRICS CHAPTER 17. “HE LAWS OE INDUCTION +11-3 Polar molecules; orientation polarization ..... 11-3 17-3 Particle acceleration by an induced electric field; +11-ð The dielectric constant of liquids; the Clausius- 1-4 Á paradox . ¬ +CHAPTER 12. ELECTROSTATIC ÂNALOGS CHAPTER 18. “HE MAXWELL EQUATIONS +12-5 Irrotational fuid fow; the fow past asphere ... 12-8 WAV€ €QUAÙOH v2 kh kẽ 8-9 +12-6 Illumination; the uniform lighting of a plane ..... 12-10 +19-1 A special lecture—almost verbatim ........ 19-1 +18-2 Electric current; the conservation ofcharge .... 13-1 FREE SPACE +13-3 The magnetic Íorce on acurrent .......... 13-2 +13-5 The magnetic feld of a straight wire and of a 20-2 Three-dimensional waves . ki 42323 + + „+ 20-8 +13-7 The transformation of currents and charges..... 13-11 +13-8 Superposition; the right-hand rule ......... 13-11 CHAPTER 2l. SOLUTIONS OF MAXWELLS EQUATIONS WITH +CURRENTS AND CHARGES +21-2 Spherical waves from a point source .......... 21-2 +14-5 The 8eld of a small loop; the magnetic dipole .. 14-7 21-6 'The potentials for a charge moving with constant +CHAPTER 22. AC CIRCUITS +15-1 The forces on a current loop; energy ofa dipole . 15-1 22-3 Networks of ideal elements; Kirchhofsrules .... 22-7 +15-5 The vector potential and quantum mechanies... 15-8 —————————.. +CHAPTER 23. CAVITY RESONATORS +CHAPTER l6. INDUCED CURRENTS +--- Trang 18 --- +CHAPTER 24. WAVEGUIDES CHAPTER 30. 'HE ÏNTERNAL GEOMETRY OF CRYSTALS +CHAPTER 25. ELECTRODYNAMICS IN RELATIVISTIC CHAPTER 31. 'TENSORS +NOTATION +25-5 The four-potential of a moving charge ........ 25-9 31-6 The tensor of SWĐSS vu về kh hen E8 +25-6 'The invariance of the equations of electrodynamics 25-10 3I-7 Tensors of higher rank ...... ¬ . 3I-H +31-8 The four-tensor of electromagnetic momentum .. 31-12 +CHAPTER 26. LORENTZ TRANSFORMATIONS OF THE EIELDS CHAPTER 32. REFRACTIVE [NDEX OF DENSE MATERIALS +26-2 The fields of a point charge with a constant velocity 26-2 32-2 Maxwells equations in a dielectric ......... 32-3 +CHAPTER 27. EIELD EBNERGY AND EIELD MOMENTUM 32-7 Low-frequency and high-frequency approximations; +the skin depth and the plasma frequency ..... 32-11 +27-2 Energy €onservaflon and electromagnetism .... . 27-2 CHAPTER 33. REFLECTION FROM SURFACES +27-3 Energy density and energy fÑow in the electromag- +CHAPTER 28. ELECTROMAGNETIC MASS +28-1 The field energy ofa pointcharge ......... 28-1 CHAPTER 34. THE MAGNETISM OF MATTER +34-6 Classical physics gives neither diamagnetism nor +CHAPTER 29. 'HE MOTION OF CHARGES IN ELECTRIC AND 34-7 Angular momentum in quantum mechanics.... 34-8 +29-1 Motion in a uniform electric or magnetic feld .. 29-1 +--- Trang 19 --- +CHAPTER 36. EERROMAGNETISM CHAPTER 40. “HE ELOW OEF DRY WATER +HAPTER 41. HE ELOW OF WET WATER +37-5 Extraordinary magnetic materials ......... 37-11 CHAPTER 42. CURVED SPAOE +42-1 Curved spaces with twodimensions ........ 42-1 +CHAPTER 38. ELASTICITY 42-2 Curvature in three-dimensionalspace ....... 42-5 +CHAPTER 39. ELASTIC MATERIALS INDEX +--- Trang 20 --- +Mlocfrorttrjrt©f£rSsrtt +1-1 Electrical forces +Consider a force like gravitation which varies predominantly inversely as the 1-1 Electrical forces +square of the distance, but which is about a b7ữon-bitlion-bdllion-billion tìmes 1-2 Electric and magnetic felds +stronger. And with another diference. Thhere are two kinds of “matter,” which we 1-3 Characteristics of vector ñelds +can call positive and negative. Like kinds repel and unlike kinds attract——unlike : +. : . 1-4 The laws of electromagnetism +gravity where there is only attraction. What would happen? +A bunch of positives would repel with an enormous force and spread out in all 1š What are the fields? +directions. A bunch of negatives would do the same. But an evenly mixed bunch l6 Electromagnetism in sclence and +OŸ positives and negatives would do something completely diferent. The opposite technology +pieces would be pulled together by the enormous attractions. The net result +would be that the terrifc forces would balance themselves out almost perfectly, +by forming tight, ñne mixtures of the positive and the negative, and between two +separate bunches of such mixtures there would be practically no attraction or +repulsion at all. +There is such a force: the electrical force. And all matter is a mixture of Reuieu: Chapter 12, Vol. Lj Character- +positive protons and negative electrons which are attracting and repelling with ñstics 0ƒ Force +this great force. 5o perfect 1s the balance, however, that when you stand near +someone else you don” feel any force at all. If there were even a little bit of +unbalance you would know it. If you were standing at arm's length om someone +and each of you had øne percen‡ more electrons than protons, the repelling force +would be incredible. How great? Enough to lift the Empire State Building? Nol +To lift Mount Everest? Nol “The repulsion would be enough to lift a “weight” +cqual to that of the entire earthl +'With such enormous forces so perfectly balanced in this intimate mixture, it +1s not hard to understand that matter, trying to keep its positive and negative +charges in the fñnest balance, can have a great stifness and strength. The Empire +State Building, for example, swings less than one inch in the wind because the +electrical forces hold every electron and proton more or less in its proper place. +On the other hand, ¡if we look at matter on a scale small enough that we see only +a few atoms, any small piece will not, usually, have an equal number of positive +and negative charges, and so there will be strong residual electrical forces. Even +when there are equal numbers of both charges in two neighboring small pieces, +there may still be large net electrical forces because the forces between individual +charges vary inversely as the square of the distance. AÁ net force can arise iÝ a +negative charge of one piece is closer to the positive than to the negative charges +of the other piece. 'Phe attractive forces can then be larger than the repulsive +ones and there can be a net attraction between two small pieces with no excess +charges. The force that holds the atoms together, and the chemical forces that +hold molecules together, are really electrical forces acting in regions where the +balance of charge is not perfect, or where the distances are very small. +You know, of course, that atoms are made with positive protons in the nucleus +and with electrons outside. You may ask: “If this electrical force is so terrifc, +why don” the protons and electrons just get on top oŸ each other? If they want +to be in an intimate mixture, why isn't it still more intimate?” 'Phe answer has to +do with the quantum efects. If we try to confine our electrons in a region that is +very close to the protons, then according to the uncertainty principle they must +have some mean square momentum which is larger the more we try to conflne +them. It is this motion, required by the laws of quantum mechaniecs, that keeps +the electrical attraction from bringing the charges any closer together. +--- Trang 21 --- +'There is another question: “What holds the nueleus together”? In a nucleus +there are several protons, all of which are positive. Why dont they push them- +selves apart? It turns out that in nuclei there are, in addition to electrical forces, +nonelectrical forces, called nuclear forces, which are greater than the electrical +forces and which are able to hold the protons together in spite of the electrical +repulsion. The nuclear forces, however, have a short range—their force falls of +much more rapidly than 1/r2. And this has an important consequence. lf a +nucleus has too many protons in it, it gets too big, and ¡ít will not stay together. +An example is uranium, with 92 protons. The nuclear forces act mainly between +cach proton (or neutron) and is nearest neighbor, while the electrical forces act +over larger distances, giving a repulsion between each proton and all of the others +in the nucleus. “The more protons in a nucleus, the stronger is the electrical +repulsion, until, as in the case of uranium, the balance is so delicate that the +nucleus 1s almost ready to y apart from the repulsive electrical force. lÝ such a +nucleus is just “tapped” lightly (as can be done by sending in a sÌow neutron), it +breaks into two pieces, each with positive charge, and these pieces fly apart by +electrical repulsion. “The energy which is liberated is the energy of the atomic Lower case Greek letters +bomb. This energy is usually called “nuelear” energy, but it is really “electrical” and commonly used capitals +energy released when electrical forces have overcome the attractive nuclear Íorces. +W©e may ask, finally, what holds a negatively charged electron together (since ¬ alpha +it has no nuclear forces). lf an electron is all made of one kind of substance, each 8 beta +part should repel the other parts. Why, then, doesn't it fly apart? But does the + ÏT gamma +electron have “parts”? Perhaps we should say that the electron is Just a point ổ SA delta +and that electrical forces only act between đjferent point charges, so that the e epsilon +electron does not act upon itself. Perhaps. All we can say is that the question of ẹ zeta +what holds the electron together has produced many diffculties in the attempts 1 cta +to form a complete theory of electromagnetism. The question has never been ƯANG theta +answered. We will entertain ourselves by discussing this subjJec some more in t lota +later chapters. “ kappa +As we have seen, we should expect that it is a combination of electrical forces À A_ lambda +and quantum-mechanieal efects that will determine the detailed structure of H „mũ +materials in bulk, and, therefore, their properties. Some materials are hard, some HỘ HH +are soft. Some are electrical “eonductors”——because their electrons are free to § = xi (ksi) +move about; others are “insulators”——because their electrons are held tightly to ” 0mIcron +individual atoms. We shall consider later how some of these properties come ml pi +about, but that is a very complicated subject, so we will begin by looking at the Ø rho +electrical forces only in simple situations. We begin by treating only the laws of Z » sigma +electricity——including magnetism, which is really a part of the same subject. T tau. +We have said that the electrical force, like a gravitational foree, decreases u T1 upsion +inversely as the square of the distance between charges. This relationship is $9 phi . +called Coulomb”s law. But it is not precisely true when charges are moving—the X chỉ (khi) +electrical forces depend also on the motions of the charges in a complicated U W psi +way. One part of the force bebween moving charges we call the maønetic force. œ $} omega +lt is really one aspectE of an electrical efect. 'Phat is why we call the subject +“electromagnetism.” +There is an important general principle that makes it possible to treat elec- +tromagnetic forces in a relatively simple way. We find, from experiment, that the +force that acts on a particular charge—no matter how many other charges there +are or how they are moving——depends only on the position of that particular +charge, on the velocity of the charge, and on the amount of charge. We can write +the force #' on a charge g moving with a velocity as +t=q(E+ox Đ). (1.1) +We call E the elecfric field and B the magnetic field at the location of the charge. +The important thing is that the electrical forces from all the other charges in the +universe can be summarized by giving just these two vectors. Theïr values will +depend on +0here the charge is, and may change with £ữne. Furthermore, iŸ we +replace that charge with another charge, the force on the new charge will be just +--- Trang 22 --- +in proportion to the amount of charge so long as all the rest of the charges in +the world do not change their positions or motions. (In real situations, oŸ course, +cach charge produces forces on all other charges in the neighborhood and may +cause these other charges to move, and so in some cases the fields cøn change if +we replace our particular charge by another.) +W©e know from Vol. I how to ñnd the motion of a particle if we know the force +on it. Equation (1.1) can be combined with the equation of motion to give +z1 mi =EF=q(E+ox Đ). (1.2) +So 1ƒ E and ® are given, we can fnd the motions. Ñow we need to know how +the 7s and Ö's are produced. +One of the most important simplifying principles about the way the fields are +produced ïs this: Suppose a number of charges moving in some manner would +produce a field #, and another set of charges would produce #2. If both sets of +charges are in place at the same time (keeping the same locations and motions +they had when considered separately), then the field produced is just the sum ++ = Eị+ E›. (1.3) +'This fact is called £he principle oƒ superposition of fñelds. Tt holds also for magnetic +ñelds. +This principle means that if we know the law for the electric and magnetic +fñelds produced by a singie charge moving in an arbitrary way, then all the laws of +electrodynamics are complete. If we want to know the force on charge Á we need +only calculate the # and #Ö produced by each of the charges , Œ, D, etc., and +then add the #”s and s from all the charges to ñnd the fñelds, and from them +the forces acting on charge A. If it had only turned out that the fñeld produced +by a single charge was simple, this would be the neatest way to describe the laws +of electrodynamics. We have already given a description of this law (Chapter 28, +Vol. T) and it is, unfortunately, rather complicated. +lt turns out that the form in which the laws of electrodynamics are simplest +are not what you might expect. It is no simplest to give a formula for the force +that one charge produces on another. It is true that when charges are standing +still the Coulomb force law is simple, but when charges are moving about the +relatlons are complicated by delays in time and by the efects of acceleration, +among others. As a result, we do not wish to present electrodynamics only +through the force laws between charges; we find it more convenient to consider +another point oŸ view——a point of view in which the laws of electrodynamics +appear to be the most easily manageable. +1-2 Electric and magnetic ñelds +First, we must extend, somewhat, our ideas of the electric and magnetic +vectors, # and Ö. We have defñned them in terms of the forces that are felt by a +charge. We wish now to speak of electric and magnetic fñelds ø# ø pøoïn# even when +there is no charge present. We are saying, in efect, that since there are Íorces +“acting on” the charge, there is still “something” there when the charge is removed. +T a charge located at the point (z,,2z) at the time £ feels the force #" given +by E4q. (1.1) we associate the vectors and Ö with (he poin£ in space (#, U, 2). +We may think of E(z, , z,£) and B(z,,z,£) as giving the forces that œould be +experienced at the time £ by a charge located at (z,, 2), tu“th the cơndition that +placing the charge there đid no‡ đisturb the positions or motions of all the other +charges responsible for the felds. +Following this idea, we associate with euerw point (z, , 2) in space Ewo vecbors +E and , which may be changing with time. The electric and magnetic fñelds are, +then, viewed as 0ecfor ƒuncfions oŸ ø, ụ, z, and . Since a vector is specified by +1ts components, each of the fields (+, ø, z,£) and B(z, 9, z, É) represents three +mathematical functions oŸ ø, , z, and . +--- Trang 23 --- +It is precisely because # (or ) can be specifed at every point in space that it «X +is called a “feld.” A “field” is any physical quantity which takes on diferent values +at diferent points in space. Temperature, for example, is a fñield——in this case a ..—> 6 +scalar field, which we write as 7z, 0,2). The temperature could also vary in tỉme, =‹ «* +and we would say the temperature field is time-dependent, and write 7z, , z, ‡). +Another example is the “velocity field” of a flowing liquid. We write (+, 9, z, È) —> >~ +for the velocity of the liquid at each poïnt in space at the time £. It is a vector field. c. _—~ = +Returning to the electromagnetic felds—although they are produced by +charges according to complicated formulas, they have the following important — +characteristic: the relationships between the values of the fields at one po#n‡ and ^^ +the values at a nearbu poïn‡ are very simple. With only a few such relationships Fig. 1-1. A vector field may be repre- +in the form of diferential equations we can describe the fields completely. lt is in sented by drawing a set of arrows whose +terms of such equations that the laws of electrodynamies are most simply written. magnitudes and directions indicate the val- +'There have been various inventions to help the mind visualize the behavior of ues of the vector field at the points from +fields. The most correct is also the most abstract: we simply consider the fields which the arrows are drawn. +as mathematical functions of position and time. We can also attempt to get a +mental picture of the field by drawing vectors at many points in space, each of +which gives the fñeld strength and direction at that point. Such a representation +is shown in Fig. I-I. We can go further, however, and draw lines which are +everywhere tangent to the vectors—which, so to speak, follow the arrows and +keep track of the direction of the feld. When we do this we lose track of the ⁄ +lengths of the vectors, but we can keep track of the strength of the fñeld by +drawing the lines far apart when the fñeld is weak and close together when it is +strong. We adopt the convention that the nưmber oƒ lines per wnit area at right +angles to the lines is proportional to the field strength. 'This is, oŸ course, only an ——~— S„—T- +approximation, and it will require, in general, that new lines sometimes start up ——— +in order to keep the number up to the strength of the feld. The feld of Fig. 1-1 ¬-—ẰẴẴẰẴẰ—— +is represented by feld lines in EFig. 1-2. +1-3 Characteristics of vector ñelds “ÔN +There are two mathematically Important properties of a vector feld which +we will use in our description of the laws of electricity from the fñeld poïnt of Elg. 1-2. A vector field can be represented +view. Suppose we imagine a closed surface of some kind and ask whether we by drawing lines which are tangent to the di- +are losing “something” from the inside; that is, does the field have a quality of rectlon of the tield vector at cach poInt, and +“outflow”? EFor instance, for a velocity field we might ask whether the velocity is k drawing the density of Ines proportional +ì o the magnitude of the field vector. +always outward on the surface or, more generally, whether more Ñuid fows out +(per unit time) than comes in. We call the net amount of fluid going out through +the surface per unit time the “fux of velocity” through the surface. The flow +through an element of a surface is just equal to the component of the velocity +perpendicular to the surface times the area of the surface. For an arbitrary closed +surface, the net owktuard [lo ——or ƒfu——is the average outward normal component +of the velocity, times the area of the surface: Ị \ +Flux = (average normal component) - (surface area). (1.4) ựwem +In the case of an electric field, we can mathematically defne something h ⁄⁄ +analogous to an outfow, and we again call it the Ñux, but of course it is not the /2¡ +fow of any substance, because the electric feld is not the velocity of anything. lt / ` Component perpendicular +turns out, however, that the mathematical quantity which is the average normal to the surface +component of the field still has a useful sipgnificance. We speak, then, of the Surface +electric fiuz——also delned by †q. (1.4). Finally, it is also useful to speak of the +ñux not only through a completely closed surface, but through any bounded +surface. As before, the ñux through such a surface is defned as the average / ⁄ +normal component of a vector times the area of the surface. These ideas are +1llustrated in Flg. I-3. Fig. 1-3. The flux of a vector field +'There is a second property of a vector fñeld that has to do with a line, rather through a surface is defined as the aver- +than a surface. Suppose again that we think of a velocity field that describes the age value of the normal component of the +fow of a liquid. We might ask this interesting question: Is the liquid circulating? vector times the area of the surface. +--- Trang 24 --- +By that we mean: ls there a net rotational motion around some loop? Suppose (a) +that we instantaneously freeze the liquid everywhere except inside of a tube +which is of uniform bore, and which goes in a loop that closes back on itself as +in Eig. I-4. Outside of the tube the liquid stops moving, but inside the tube 1$ +may keep on moving because of the momentum in the trapped liquid——that 1s, +1ƒ there is more momentum heading one way around the tube than the other. +W© define a quantity called the czrculation as the resulting speed of the liquid in +the tube times its cireumference. We can again extend our ideas and defne the +“eirculation” for any vector field (even when there isn't anything moving). For +any vector field the cứculatlion around an tmagined closed curue is deñned as +the average tangential component of the vector (in a consistent sense) multiplied (P) - — +by the circumference of the loop (Fig. 1-5): _— =N. +Circulation = (average tangential component) - (distance around). (1.5) “ươm N ì ` +—”, T~~~—-_. ` \ I +You will see that this defnition does indeed give a number which is proportional `. ` \ ) Ị +to the circulation velocity in the quickly frozen tube described above. Tube __ ` Z H / ị +With just these two ideas—fux and circulation—we can describe all the laws m.———<= ⁄ ' +of electricity and magnetism at once. You may not understand the significance : "Xã x ˆv +of the laws right away, but they will give you some idea of the way the physics of TỦ vu xã +electromagnetism will be ultimately described. ¬ .. +1-4 The laws of electromagnetism ¬ TY nà ¬- +The first law of electromagnetism describes the fux of the electric field: ãn Z Z cư ¬.Mm. vẽ +The fux of E through any closed surface —= the net charge insidc, (1.6) " ` côn hệ ¬__ 2 Sun, +€0 _. ` “ã. — , +where eo is a convenient constant. (The constant co is usually read as “epsilon- TRỪ. kar.crz/ẽs +zero” or “epsilon-naught”.) TỶ there are no charges inside the surface, even though ¬-— +there are charges nearby outside the surface, the aueraøe normal component of Fig. 1-4. (a) The velocity field in a liquid. +is zero, so there is no net fux through the surface. To show the power of this Imagine a tube of uniform cross section that +type of statement, we can show that Eq. (1.6) is the same as Coulomb”s law, follows an arbitrary closed curve as In (b). lf +provided only that we also add the idea that the field from a single charge is the liquid were suddenly frozen everywhere +spherically symmetric. Eor a point charge, we draw a sphere around the charge. except inside the tube, the liquid in the tube +Then the average normal component is just the value of the magnitude of E at would circulate as shown in (c). +any point, since the field must be directed radially and have the same strength +for all points on the sphere. Our rule now says that the field at the surface of the +sphere, times the area of the sphere—that is, the outgoing fux——is proportional +to the charge inside. IÝ we were to make the radius of the sphere bigger, the area +would increase as the square oŸ the radius. The average normal component of +the electric field times that area must still be equal to the same charge inside, +and so the field must decrease as the square of the distance—we get an “inverse +square” field. +Tí we have an arbitrary stationary curve in space and measure the circulation +of the electric field around the curve, we will fnd that it is not, in general, zero +(although it is for the Coulomb field). Rather, for electricity there is a second +law that states: for any surface Š (not closed) whose edge is the curve Œ, ++ direction “ +: . đ "+ +Circulation of E around Œ = —atlux of through 59). (1.7) ⁄ - +W© can complete the laws of the electromagnetic field by writing Ewo corre- +sponding equations for the magnetic fñeld #Ö: h +Flux of through any closed surface = 0. (1.8) Arbitary TS lồi +Closed Curve À„ =—== +For a surface Š bounded by the curve Œ, — — +d Fig. 1-5. The circulation of a vector field +œ (circulation of Ö around €) = q¡ị ñux of È through 5) Is the average tangential component of the +fiux of electric current through Ø vector (in a consistent sense) times the cir- ++“————__——. (1.49) cumference of the loop. +--- Trang 25 --- +(et maane) ⁄4` + TERMINAL +(on wire) +— TERMINAL | +SỈ] BAR MAGNET +Fig. 1-6. A bar magnet gives a field +at a wire. When there is a current along +the wire, the wire moves because of the +force F = qv x B. +The constant e2 that appears in Eq. (1.9) is the square of the velocity of light. +lt appears because magnetism is in reality a relativistic efect of electricity. The +constant eo has been stuck in to make the units of electric current come out in a +convenient way. +Equations (1.6) through (1.9), together with Bq. (1.1), are all the laws of +electrodynamicsẺ. As you remember, the laws of NÑewton were very simple to +write down, but they had a lot of complicated consequences and it took us a long +time to learn about them all. 'Phese laws are not nearly as simple to write down, +which means that the consequences are going to be more elaborate and it will +take us quite a lot of time to fgure them all out. +W© can illustrate some of the laws of electrodynamics by a series of small ex- +periments which show qualitatively the interrelationships of electric and magnetic +fñelds. You have experienced the fñrst term of Eq. (1.1) when combing your haiïr, +so we wont show that one. 'The second part oŸ Bq. (1.1) can be demonstrated +by passing a current through a wire which hangs above a bar magnet, as shown +in Eig. I-6. "The wire will move when a current is turned on because of the +force È' = gu x B. When a current exists, the charges inside the wire are moving, +so they have a velocity , and the magnetic fñeld from the magnet exerts a Íforce +on them, which results in pushing the wire sideways. +'When the wire is pushed to the left, we would expect that the magnet must +feel a push to the right. (Otherwise we could put the whole thing on a wagon +and have a propulsion system that didn't conserve momentuml) Although the +force is too small to make movement of the bar magnet visible, a more sensitively +supported magnet, like a compass needle, wïll show the movement. +How does the wire push on the magnet? "The current in the wire produces a +magnetic field of its own that exerts forces on the magnet. According to the last +Lines of B & TO +from wire ⁄ + TERMINAL +— TERMINAL Ƒ (on magnet) +°LÌBAR MAGNET +Fig. 1-7. The magnetic field of the wire +exerts a force on the magnet. +* We need only to add a remark about some conventions for the s¿øw of the circulation. +--- Trang 26 --- +Fig. 1-8. Two wires, carrying current, +“HH, exert forces on each other. +term in Eq. (1.9), a current must have a circulation of B——in this case, the lines +of are loops around the wire, as shown in Eig. I-7. This B-field is responsible +for the force on the magnet. +Equation (1.9) tells us that for a fixed current through the wire the circulation +öoŸ is the same for an curve that surrounds the wire. For curves—say circles— +that are farther away from the wire, the cireumference is larger, so the tangential +component of must decrease. You can see that we would, in fact, expect to +decrease linearly with the distance from a long straight wire. +Now, we have said that a current through a wire produces a magnetic field, +and that when there is a magnetic ñeld present there is a Íorce on a wire carrying +a current. Then we should also expect that if we make a magnetic fñeld with a +current in one wire, it should exert a force on another wire which also carries +a current. 'Phis can be shown by using two hanging wires as shown in Fig. 1-8. +'When the currents are in the same direction, the two wires attract, but when +the currents are opposite, they repel. +In short, electrical currents, as well as magnets, make magnetic fields. But +wait, what is a magnet, anyway? If magnetic fñelds are produced by moving +charges, is it not possible that the magnetic ñeld tom a piece of iron is really the +result of currents? It appears to be so. We can replace the bar magnet of our +experiment with a coil of wire, as shown in Fig. 1-9. When a current is passed +throupgh the coil—as well as through the straight wire above 1t —we observe a +motion of the wire exactly as before, when we had a magnet instead of a coil. In +other words, the current in the coil imitates a magnet. Ït appears, then, that a +plece of iron acts as thouph it contains a perpetual circulating current. We can, +in fact, understand magnets in terms of permanent currents in the atoms of the +iron. The force on the magnet in EFig. 1-7 is due to the second term in Eq. (1.1). +B. TO +đrem coi) 4 ||T TERMINAL +(on wire) +— TERMINAL +ẻ COIL OF WIRE +nướn Fig. 1-9. The bar magnet of Fig. 1-6 can +be replaced by a coil carrying an electrical +current. A similar force acts on the wire. +--- Trang 27 --- +Where do the currents come from? One possibility would be from the motion +of the electrons in atomic orbits. Actually, that is not the case for iron, although +1È is for some materials. In addition to moving around in an atom, an electron also +spins about on I1ts own axis—something like the spin of the earth—and ït is the +current from this spin that gives the magnetic field in iron. (WS say “something +like the spin of the earth” because the question is so deep in quantum mechanics +that the classical ideas do not really describe things too well.) In most substances, +some electrons spin one way and some spin the other, so the magnetism cancels +out, but in iron—for a mysterious reason which we will discuss later=many of +the electrons are spinning with theïr axes lined up, and that is the source of the +1nagnetism. +Since the fields of magnets are from currents, we do not have to add any +extra term to Eqs. (1.8) or (1.9) 6o take care of magnets. We just take aiÏ +currents, including the circulating currents of the spinning electrons, and then +the law is right. You should also notice that Bq. (1.8) says that there are no +magnetic “charges” analogous to the electrical charges appearing on the right +side of Eq. (1.6). None has been found. +BẠ ⁄ BÀ +Current ⁄ - Current +Fig. 1-10. The circulation of B around è—————_—_—---- [————` +the curve C ¡is given either by the current ⁄⁄ ⁄ +passing through the surface S, or by the sÑ ⁄⁄⁄4 - +rate of change of the flux of E through the Z⁄ +surface Sa. Curve C +Surface Sị Surface Sa +The first term on the right-hand side of Eq. (1.9) was discovered theoretically +by Maxwell and is of great importance. It says that changing clecfric fields +produce magnetic efects. In fact, without this term the equation would not +make sense, because without it there could be no currents in circuits that are +not complete loops. But such currents do exist, as we can see in the following +example. Imagine a capacitor made of two flat plates. It is being charged by +a current that fows toward one plate and away from the other, as shown in +Fig. I-10. We draw a curve Œ around one of the wires and fiÏl it in with a surface +which crosses the wire, as shown by the surface 5 in the fñgure. According to +Ea. (1.9), the cireulation of Ö around Œ (times c2) is given by the current in the +wire (divided by co). But what if we fll in the curve with a đjƒerenf surface 652, +which is shaped like a bowl and passes between the plates of the capacitor, staying +always away from the wire? 'There is certainly no current through this surface. +But, surely, Jjust changing the location of an imaginary surface is not going to +change a real magnetic fieldl 'The circulation of must be what i9 was before. +The first term on the right-hand side of Eq. (1.9) does, indeed, combine with the +second term to give the same result for the bwo surfaces 5 and %2. EFor 5a the +circulation of Ö is given in terms of the rate of change of the ñux of # between +the plates of the capacitor. And it works out that the changing # ¡is related to +the current in just the way required for Bq. (1.9) to be correct. Maxwell saw +that it was needed, and he was the first to write the complete equation. +'With the setup shown in Fig. I-6 we can demonstrate another of the laws of +electromagnetism. We disconnect the ends of the hanging wire from the battery +and connect them to a galvanometer which tells us when there is a current through +the wire. When we øush the wire sideways through the magnetic field of the +magnet, we observe a current. 5uch an efect is again just another consequence of +Eq. (1.1)—the electrons in the wire feel the force #' = gu x . The electrons have +a sidewise velocity because they move with the wire. This ø with a vertical +from the magnet results in a force on the electrons directed ølong the wire, which +starts the electrons moving toward the galvanometer. +--- Trang 28 --- +Suppose, however, that we leave the wire alone and move the magnet. We +guess from relativity that it should make no diference, and indeed, we observe +a similar current in the galvanometer. How does the magnetic fñeld produce +forces on charges at rest? According to Eq. (1.1) there must be an electric +field. A moving magnet must make an electric fñeld. How that happens is said +quantitatively by Eq. (1.7). This equation describes many phenomena of great +practical interest, such as those that occur in electric generators and transformers. +'The most remarkable eonsequence of our equations is that the combination of +Eq. (1.7) and Eaq. (1.9) contains the explanation of the radiation of electromagnetic +efects over large distances. The reason is roughly something like this: suppose +that somewhere we have a magnetic field which is increasing because, say, a +current is turned on suddenly in a wire. Then by Eq. (1.7) there must be a +circulation of an electric feld. As the electric fñeld builds up to produce its +circulation, then according to Eq. (1.9) a magnetic circulation will be generated. +But the building up of £52s magnetic fñeld will produce a new circulation of the +electric ñeld, and so on. In this way fields work their way through space without +the need of charges or currents except at their source. That is the way we see +cach otherl It is all in the equations of the electromagnetic fields. +1-5 What are the fields? +W©e now make a few remarks on our way of looking at this subject. You may +be saying: “All this business of ñuxes and circulations is pretty abstract. There +are electric fields at every point in space; then there are these “laws.. But what is +acEuallu happening? Why can't you explain it, for instance, by whatever it ¡s that +goes between the charges.” Well, ¡it debends on your prejudices. Many physicists +used to say that direct action with nothing in bebween was inconceivable. (How +could they ñnd an idea inconceivable when it had already been conceived?) They +would say: “Look, the only forces we know are the direct action of one piece oŸ +matter on another. It is impossible that there can be a force with nothing to +transmit it” But what really happens when we study the “direct action” of one +plece of matter right against another? We discover that it is not one piece right +against the other; they are slightly separated, and there are electrical forces acting +on a tỉny scale. Thus we find that we are goïng to explain so-called direct-contact +action in terms of the picture for electrical forces. It is certainly not sensible to +try to insist that an electrical force has to look like the old, familiar, muscular +push or pull, when ¡it will turn out that the muscular pushes and pulls are going +to be interpreted as electrical forcesl The only sensible question is what is the +most conuenien‡ way to look at electrical efects. Some people prefer to represent +them as the interaction at a distance of charges, and to use a complicated law. +Others love the fñeld lines. They draw feld lines all the time, and feel that writing +Esand B35 is too abstract. The feld lines, however, are only a crude way of +describing a field, and it is very diffcult to give the correct, quantitative laws +directly in terms of field lines. Also, the ideas of the field lines do not contain +the deepest principle of electrodynamics, which is the superposition principle. +ven though we know how the fñeld lines look for one set of charges and what +the fñeld lines look like for another set of charges, we don” get any idea about +what the field line patterns will look like when both sets are present together. +trom the mathematical standpoint, on the other hand, superposition 1s easy——we +simply add the two vectors. The field lines have some advantage in giving a vivid +picture, but they also have some disadvantages. The direct interaction way of +thinking has great advantages when thinking of electrical charges at rest, but +has great disadvantages when dealing with charges in rapid motion. +'The best way ¡is to use the abstract field idea. “That ït is abstract is unfortunate, +but necessary. 'Phe attempts to try to represent the electric field as the motion +of some kind of gear wheels, or in terms of lines, or of stresses In some kind of +material have used up more efort of physicists than it would have taken simply +to get the right answers about electrodynamics. It is interesting that the correct +cquations for the behavior of light were worked out by MacCullagh in 1839. +--- Trang 29 --- +But people said to him: “Yes, but there is no real material whose mechanical +properties could possibly satisfy those equations, and since light is an oscillation +that must vibrate In sormethzng, we cannot believe this abstract equation business.” +T people had been more open-minded, they might have believed in the right +cequations for the behavior of light a lot earlier than they did. +In the case of the magnetic fñeld we can make the following point: Suppose +that you fñnally succeeded in making up a picture of the magnetic feld in terms +of some kind of lines or of gear wheels running through space. 'Phen you try to +explain what happens to two charges moving in space, bot©h at the same speed +and parallel to each other. Because they are moving, they will behave like two +currents and will have a magnetic feld associated with them (Iike the currents in +the wires of Eig. 1-8). An observer who was riding along with the two charges, +however, would see both charges as stationary, and would say that there is øo +magnetic fñield. The “gear wheels” or “lines” disappear when you ride along with +the objectl All we have done is to invent a øœeœ problem. How can the gear +wheels disappear?l The people who draw field lines are in a similar dificulty. +Not only is it not possible to say whether the field lines move or do not move +with charges—they may disappear completely in certain coordinate frames. +'What we are saying, then, is that magnetism is really a relativistic efect. In +the case of the two charges we just considered, travelling parallel to each other, we +would expect to have to make relativistic corrections to their motion, with terms +of order 02/c?. These corrections must correspond to the magnetie force. But +what about the force between the two wires in our experiment (Eig. I-8). There +the magnetic force is the :0hole force. It didn”t look like a “relativistic correction.” +Also, if we estimate the velocities of the electrons in the wire (you can do this +yourself), we fnd that their average speed along the wire is about 0.01 centimeter +per second. 8o ø2/c? is about 10~25. Surely a negligible “correction” But nol +Although the magnetic force is, in this case, 107? of the “normal” electrical +force between the moving electrons, remember that the “normal” electrical forces +have disappeared because of the almost perfect balancing out——because the wires +have the same number of protons as electrons. 'Phe balance is much more precise +than one part in 1027, and the small relativistic term which we call the magnetie +force is the only term left. It becomes the dominant term. +Tt is the near-perfect cancellation of electrical efects which allowed relativity +cfects (that is, magnetism) to be studied and the correct equations—to or- +der 02/c2—to be discovered, even though physiecists didn't knou that's what was +happening. And that is why, when relativity was discovered, the electromagnetic +laws didn't need to be changed. 'They——unlike mechanics—were already correct +to a preecision oŸ 02 /cŸ. +1-6 Electromagnetism ỉn science and technology +Let us end this chapter by pointing out that among the many phenomena, +studied by the Greeks there were two very strange ones: that if you rubbed a +piece of amber you could lift up little pieces of papyrus, and that there was a +strange rock from the island of Magnesia which attracted iron. It is amazing to +think that these were the only phenomena known to the Greeks in which the +efects of electricity or magnetism were apparent. The reason that these were +the only phenomena that appeared is due primarily to the fantastic precision +of the balancing of charges that we mentioned earlier. Study by scientists who +came after the Greeks uncovered one new phenomenon after another that were +really some aspect of these amber and/or lodestone efects. Ñow we realize that +the phenomena of chemical interaction and, ultimately, of life itself are to be +understood in terms of electromagnetism. +At the same time that an understanding of the subject of electromagnetism was +being developed, technical possibilities that defed the imagination of the people +that came before were appearing: it became possible to signal by telegraph over +long distances, and to talk to another person miles away without any connections +between, and to run huge power systems——a great water wheel, connected by +--- Trang 30 --- +filaments over hundreds of miles to another engine that turns in response to the +master wheel—many thousands of branching fñlaments—ten thousand engines in +ten thousand places running the machines of industries and homes—all turning +because of the knowledge of the laws of electromagnetism. +Today we are applying even more subtle efects. 'he electrical forces, enormous +as they are, can also be very tiny, and we can control them and use them in very +many ways. So delicate are our instruments that we can tell what a man is doing +by the way he affects the electrons in a thin metal rod hundreds of miles away. +All we need to do is to use the rod as an antenna for a television receiverl +trom a long view of the history of mankind—seen from, say, ten thousand +years rom now——there can be little doubt that the most signiicant event of the +19th century will be judged as Maxwell's discovery of the laws of electrodynamics. +The American Ơivil War will pale into provincial insignificance in comparison +with this Important scientifc event of the same decade. +--- Trang 31 --- +Mfforortfi(l ẤÍcrrlrrs oŸ Voeceor' Frol‹ls +2-1 Understandiỉng physiỉcs +The physicist needs a facility in looking at problems from several points of 2-1 Understanding physics +view. The exact analysis of real physical problems is usually quite complicated, 2-2_ Scalar and vector fields—T +and any particular physical situation may be too complicated to analyze directly and h +by solving the diferential equation. But one can still get a very good idea of 2-3 Derivatives of 8elds— the +the behavior of a system if one has some feel for the character of the solution gradient +in diferent circumstances. Ideas such as the field lines, capacitance, resistance, +and inductance are, for such purposes, very useful. 5o we will spend much of our 24 The 0perator +time analyzing them. In this way we will get a feel as to what should happen in 2-5 Operations with V +diferent electromagnetic situations. On the other hand, none of the heuristic 2-6 The diferential equation of heat +models, such as field lines, ¡is really adequate and accurate for all situations. flow +There is only one precise way of presenting the laws, and that is by means of 2-7 Second derivatives ofvector fields +diferential equations. 'PThey have the advantage of being fundamental and, so 2-8_ PitRlls +far as we know, precise. lf you have learned the diferential equations you can +always go back to them. There is nothing to unlearn. +lt will take you some time to understand what should happen in diferent +circumstances. You will have to solve the equations. Each time you solve +the equations, you will learn something about the character of the solutions. +To keep these solutions in mỉnd, it will be useful also to study their meaning +in terms of field lines and of other concepts. This is the way you will really +“understand” the equations. That is the diference between mathematics and Reuieu: Chapter 11, Vol. 1, Weefors +physics. Mathematicians, or people who have very mathematical minds, are often +led astray when “studying” physics because they lose sight of the physics. They +say: “Look, these diferential equations—the Maxwell equations—are all there is +to electrodynamies; it is admitted by the physicists that there is nothing which is +not contained in the equations. The equations are complicated, but after all they +are only mathematical equations and ïf Ï understand them mathematically inside +out, I will understand the physics inside out.” Only it doesn't work that way. +Mathematicians who study physics with that point of view—and there have been +many of them——usually make little contribution to physics and, in fact, little to +mathematics. 'Phey fail because the actual physical situations in the real world +are so complicated that it is necessary to have a much broader understanding of +the equations. +'What it means really to understand an equation—that is, in more than a +strictly mathematical sense—was described by Dirac. He said: “I understand +what an equation means ïf [ have a way of fguring out the characteristics 0Ý is +solution without actually solving it.” So if we have a way of knowing what should +happen in given circumstances without actually solving the equations, then +we “understand” the equations, as applied to these cireumstances. Á physical +understanding is a completely unmathematical, imprecise, and inexact thing, but +absolutely necessary for a physicist. +Ordinarily, a course like this is given by developing gradually the physical +ideas—by starting with simple situations and goïing on to more and more compli- +cated situations. 'Phis requires that you continuously forget things you previousÌy +learned—things that are true in certain situations, but which are not true in +general. Eor example, the “law” that the electrical foree depends on the square +of the distance is not øÈøs true. We prefer the opposite approach. We prefer +to take first the cornplete laws, and then to step back and apply them to simple +--- Trang 32 --- +situations, developing the physical ideas as we go along. And that is what we +are going to do. +Our approach is completely opposite to the historical approach in which one +develops the subject in terms of the experiments by which the information was +obtained. But the subject of physics has been developed over the past 200 years +by some very ingenious people, and as we have only a limited time to acquire +our knowledge, we cannot possibly cover everything they did. Unfortunately +one of the things that we shall have a tendeney to lose in these lectures is the +historical, experimental development. lt is hoped that in the laboratory some of +this lack can be corrected. You can also fll in what we must leave out by reading +the Encyclopedia Britannica, which has excellent historical articles on electricity +and on other parts of physics. You will also ñnd historical information in many +textbooks on electricity and magnetism. +2-2 Scalar and vector fields—T' and h +W© begin now with the abstract, mathematical view of the theory of electricity +and magnetism. The ultimate idea is to explain the meaning of the laws given in +Chapter 1. But to do this we must first explain a new and peculiar notation that +we want to use. 5o let us forget electromagnetism for the moment and discuss +the mathematics of vector fñelds. lt is of very great importance, not only for +electromagnetism, but for all kinds of physical cireumstances. Just as ordinary ¬ +điferential and integral calculus is so important to all branches of physics, so Ghượn) ~~~« ¬ +.~d. +also is the diferential calculus of vectors. We turn to that subJect. +Listed below are a few facts from the algebra of vectors. It is assumed that $ +you already know them. xa ph Xa +—- _ _ +A - B=sealar= A„D„ + AyB, + A,D, (2.1) E ~ 8 na P, +AxB=vector (2.2) 01322 c¬“†> +(AxB),= A,P,— AybB, E +(Ax B);= A,B, — A,Ðy ~“~` +(AxB),= A,B,— A,bB, Wa ho BỊ ae isaah +A:(Axb)=0 (2.4) HI JktUmÁA +A-(BxÄC)=(Axb).C (2.5) +Ax(Bx€) = B(A -C) - C(A - B) (2.6) Ð PP @R 5 7T Ю +Also we will want to use the two following equalities from the calculus: V W xX # Z +lôi lôi 8 , +Af(x,w.z)S= 2Ÿ Az+ 9 Ay+ SỀ Az, (2.7) ÂU on aad +3z Øụ 3z +6 & (+ ad e +0Ƒ Ø°ƒ In: +—“—— =——: (2.8) , +ÔxØu OụÔz In: " kEÐ 2 xm w% +The first equation (2.7) is, of course, true only in the limit that Az, Aw, and Az +go toward zero. 2 ? † xð 4 # « +'The simplest possible physical feld is a scalar field. By a field, you remember, “.Ar x z +we mean a quantity which depends upon position in space. By a scalar fteld we k +merely mean a field which is characterized at each point by a single number——a &a~ ¿ ~ +scalar. Of course the number may change in time, but we need not worry about Ẳ ử ` +that for the moment. We will talk about what the fñeld looks like at a given +instant. As an example of a scalar feld, consider a solid block of material which +has been heated at some places and cooled at others, so that the temperature of +the body varies from point to point in a complicated way. Then the temperature +will be a function of z, , and z, the position in space measured in a rectangular +coordinate system. 'Temperature is a scalar ñeld. +--- Trang 33 --- +⁄ T=40° +.~“” T— 30 _EFÍg. 21. Temperature m= an example of a scalar +T(x,y,z) field. With each point (x, y, z) in space there is asso- +⁄ | T~20° ciated a number T(x,y,z). All points on the surface +Cold marked T7 = 20” (shown as a curve at z = 0) are at the +⁄ \ h same temperature. The arrows are samples of the heat +cold T= 10 flow vector h. +One way of thinking about scalar fields is to imagine “contours” which are +imaginary surfaces drawn through all points for which the fñeld has the same +value, just as contour lines on a map connect points with the same height. Eor +a temperature field the contours are called “isothermal surfaces” or isotherms. +Jigure 2-1 illustrates a temperature fñeld and shows the dependence oŸ 7' on # +and when z = 0. Several isotherms are drawn. +There are also vector fields. The idea is very simple. Á vector is given for each +point in space. The vector varies from point to point. Ás an example, consider a +rotating body. The velocity of the material of the body at any point is a vector +which is a function of position (Eig. 2-2). As a second example, consider the fow “EẾTATION +of heat in a block of material. If the temperature in the block 1s high at one +place and low at another, there will be a ow of heat from the hotter places %O +the colder. 'Phe heat will be owing in different directions in diferent parts of : : : +the block. The heat flow is a directional quantity which we call h. Its magnitude 1g: = The velocity Of the atoms In +. ` „ a rotating object is an example of a vector +is a measure of how much heat is fiowing. Examples of the heat Ñow vector are field. +also shown in Fig. 2-1. +heat flow Fig. 2-3. Heat flow Is a vector field. The vector h +points along the direction of the flow. lts magnitude is +the energy transported per unit time across a surface +x element oriented perpendicular to the flow, divided by +the area of the surface element. +Let's make a more precise deflnition of h: The magnitude of the vector heat +fow at a point is the amount of thermal energy that passes, per unit time and +per unit area, through an infinitesimal surface element at right angles to the +direction of fow. The vector points in the direction of fow (see Eig. 2-3). In +symbols: If A.J is the thermal energy that passes per unit time through the +surface element Az, then +h= =- C/, (2.9) +where ez is a wnit 0ector in the direction of fow. +The vector can be defned in another way-—in terms of its components. VWe +ask how much heat ows through a small surface at ønmy angle with respect to the +--- Trang 34 --- +fow. In Fig. 2-4 we show a small surface Aaa inclined with respect to Aø+, which n +is perpendicular to the fow. 'PThe ii 0ector ?w is normal to the surface Aas. The +angle Ø bebween ?ø and h is the same as the angle between the surfaces (since h ‹ +is normal to Aa). Ñow what is the heat flow per un¿t area through Aaa? The _⁄“ N n +fow through Aasa is the same as through Aax; only the areas are diferent. In ".« ` +fact, Aøi = Aøa cos0. The heat fow through Aaa is c⁄ +" = " cos Ø = Ìh - Tt. (2.10) Aai +We interpret this equation: the heat fow (per unit time and per unit area) through Aa +am surface element whose unit normal is , is given by h -r. EBqually, we could : . +say: the component of the heat fow perpendicular to the surface element Aas Flg. 2-4. The heat flow through A4 is +1sh-rw. Wo can, IŸ we wish, consider that these statements đefine h. We will be the same as through Aai. +applying the same ideas to other vector fñelds. +2-3 Derivatives of ñelds—the gradient +When fñelds vary in time, we can describe the variation by giving their +derivatives with respect to ý. We want to describe the variations with position +in a similar way, because we are interested in the relationship between, say, the +temperature in one place and the temperature at a nearby place. How shall +we take the derivative of the temperature with respect to position? Do we +diferentiate the temperature with respect to +? Ôr with respect to , or z7? +Useful physical laws do not depend upon the orientation of the coordinate +system. Thhey should, therefore, be written in a form in which either both sides +are scalars or both sides are vectors. What is the derivative of a scalar field, +say ØT /Ø+? Is it a scalar, or a vector, or what? It is neither a scalar nor a +vecbor, as you can easily appreciate, because iŸ we took a different z-axis, Ø?'/Øz +would certainly be diferent. But notice: We have three possible derivatives: +ØT/0z, ØT/Øụ, and ØT/Øz. Since there are three kinds of derivatives and we +know that it takes three numbers to form a vector, perhaps these three derivatives +are the components of a vector: +(an) ** a vector. (2.11) +Of course it is not generally true that an three numbers form a vector. Ït is +true only iÍ, when we rotate the coordinate system, the components of the vector +transform among themselves in the correct way. So i% is necessary to analyze +how these derivatives are changed by a rotation of the coordinate system. We +shall show that (2.11) is indeed a vector. The derivatives do transform in the +correct way when the coordinate system is rotated. +W© can see this in several ways. One way is to ask a question whose answer +is independent of the coordinate system, and try to express the answer in an +“invariant” form. Eor instance, if 9S = A- , and if A and Ö are vectors, we +know—because we proved it in Chapter l1 of Vol. I—that Š is a scalar. We +knmou that Š 1s a scalar without investigating whether it changes with changes in +coordinate systems. It cøn7, because is a dot produect of two vectors. Similarly, +1f we have three numbers , Ø;, and s and we find out that for euerg vector A +A„Bi+ AyÐ: + A4;Ö› = 6S, (2.12) +where Š is the same for any coordinate system, then it rz%sứ be that the three +numbers ị, ›, ạ are the components B„, Ö,, B; of some vector . +Now let's think of the temperature field. Suppose we take two points ị +and , separated by the small interval AR. The temperature at ¡ is 71 and +at ¿ is T2, and the diference AT = Tạ — 7ì. The temperatures at these real, +physical points certainly do not depend on what axis we choose for measuring +the coordinates. In particular, AT is a number independent of the coordinate +system. Ïl% is a scalar. +--- Trang 35 --- +Tf we choose some convenient set of axes, we could write 71 = 7z, , z) and y +1a = T(z + Az, + Au,z+ A2), where Az, Aw, and Az are the components of +the vector ARR (Eig. 2-5). Remembering Eq. (2.7), we can write "¬.. +"+ AZ.“¡ +ØT ØT ØT `. JAy (— Ta TP, !Ay +AT'= —Az+—— Ayu+ —— Az. (2.13) Si Ị _””” +Ồz ỡy Ôz ọ ¡ PyZZ~“—---—+ +ZAz xả à k⁄ ⁄ +The left side of Bq. (2.13) is a scalar. The right side is the sum of three products +with Az, Aw, and Az, which are the components of a vector. It follows that the - +three numbers Az.^S=E ¬ +ØT ðT ðT 3 TY +Ô6z` Øụ` Ôz W +are also the z-, -, and z-components of a vector. We write this new vector Fig. 2-5. The vector AR, whose compo- +with the symbol V7' The symbol V (called “del?) is an upside-down A, and nents are Âx, Ấy, and Az. +1s supposed to remind us of diferentiation. People read Win various ways: +“del-7,” or “gradient of 7,” or “grad 7;”* +ØT' ØT ðï +rad7'=V7'= | —.——.— |. 2.14 +5 ts Qụ ` Ôz ) G1) +Using this notation, we can rewrite Eq. (2.13) in the more compact form +AT =VT. AR. (2.15) +In words, this equation says that the diference in temperature between ÿwo +nearby points is the dot produect of the gradient of 7" and the vector displacement +between the points. "The form of Eq. (2.15) also illustrates clearly our proof above +that W7' ¡is indeed a vector. +Perhaps you are still not convinced? Let”s prove it in a diferent way. (AI- +thouph if you look carefully, you may be able to see that itˆs really the same proof +in a longer-winded form!) We shall show that the components of V7 transform +in just the same way that components of l do. If they do, V7? is a vector +according to our original definition of a vector in Chapter 11 of Vol. I. We take a +new coordinate system 4, , z”, and in this new system we calculate Ø?'/9z, y +ØT/0V, and ØT/8z”. To make things a little simpler, we let z = Z2”, so that we ⁄ (a) +can forget about the z-coordinate. (You can check out the more general case for +yourself.) _—_x nh +We© take an z//-system rotated an angle Ø with respect to the #-system, as .x - +in Eig. 2-6(a). Eor a point (z,z) the coordinates in the prime system are y +B4 xỉ +z'= #øcosØ +sin6, (2.16) 9 +=—zsinØ + cos 6. (2.17) x +Ór, solving for øz and , ` +# = # cos 8 — 1 sin Ø, (2.18) ⁄ (b) +ụ = #'sin0 + cos0. (2.19) _“ ga \A⁄ +Db.®—=~— 'Ax ~ ®P, +TÝ any païr of numbers transforms with these equations in the same way that +and ø do, they are the components oŸ a vector. “, +Now let”s look at the diference in temperature between the Ewo nearby points +Đị and ạ, chosen as in Eig. 2-6(b). TÝ we calculate with the z- and /-coordinates, +we would write * +AT= 07 Az (2.20) Fig. 2-6. (a) Transformation to a rotated +Øz coordinate system. (b) Special case of an +—since A# is zero. Interval AR parallel to the x-axIs. +* In our notation, the expression (ø, b,c) represents a vector with components a, Ò, and c. lŸ +you like to use the unit vectors 2, 7, and k&, you may write +ðT ðØT Ø7T' +T=i—=+j—+k—-. +y ; ðz + ðy + Ôz +--- Trang 36 --- +'What would a computation in the prime system give? We would have written +ØT ØT +AT'= <— Az'+—— AW. 2.21 +9z z+ Øự ỹ ( ) +Looking at Fig. 2-6(b), we see that +Az=_ Azcos0 (2.22) +Ad' = —Azsin 0, (2.23) +since A# is negative when Az is positive. Substituting these in Eq. (2.21), we +fnd that đm đm +A7'= —— AzcosØT— —— AzsinØ (2.24) +3z” Ø9 +ØT ØT += lấn cos ØÐ — ny s9) Am. (2.25) +Comparing Eq. (2.25) with (2.20), we see that +Ø1 ØT ØT`. +Đ = nay COS Ú — pự SA. (2.26) +This equation says that Ø7 /Øz is obtained rom Ø7 /9z and Ø7 /Øw", just as # +is obtained from z and in Ed. (2.18). So Ø7'/Øz is the z-component of a +vector. The same kind of arguments would show that Ø7/Øy and ؆?'/Ôz are g- +and z-components. So W7' is defnitely a vector. It is a vector fñeld derived from +the scalar field 7'. +2-4 The operator V +Now we can do something that is extremely amusing and ingenious—and +characteristic of the things that make mathematics beautiful. The argument +that grad T, or V7), is a vector did not depend upon +0hø‡ scalar field we were +diferentiating. All the arguments would go the same ïf T7' were replaced by an +scalar ficld. Since the transformation equations are the same no matter what +we differentiate, we could just as well omit the 7' and replace Eq. (2.26) by the +operator equation +lô) lôi 9. +Ôz — a C050 — aụ S0. (2.27) +We leave the operators, as Jeans said, “hungry for something to diferentiate.” +Since the diferential operators themselves transform as the components of a +vector should, we can call them components of a 0ecfor operœtor. We can write +8 8 ôÔ +V_=|l--.-.= 2.28 +(a2) (2.38) +which means, OŸ cOurse, +V„==—, Vy==-, Vy=-—-. 2.29 +Øz 1Ø Øz (229) +We have abstracted the gradient away from the 7—that is the wonderful idea. +You must always remember, of course, that V is an operator. Alone, it means +nothing. If W by itself means nothing, what does it mean if we multiply # by a +scalar—say 7—to get the product 7V? (One can always multiply a vector by a +scalar.) It still does not mean anything. Its z-component is +TT 2.30 +c: (2.30) +which is not a number, but is still some kind of operator. However, according to +the algebra of vectors we would still call 7V a vector. +--- Trang 37 --- +Now let's multiply V by a scalar on the other side, so that we have the +product (V7). In ordinary algebra +TA = AI, (2.31) +but we have to remember that operator algebra is a little diferent from ordinary +vector algebra. With operators we must always keep the sequenece right, so that +the operations make proper sense. You will have no difficulty if you just remember +that the operator V obeys the same convention as the derivative notation. What is +to be diferentiated must be placed on the right of the V. “The order is important. +JKeeping in mind this problem of order, we understand that 7T V is an operator, +but the product V7 is no longer a hungry operator; the operator is completely +satisied. It ¡is indeed a physical vector having a meaning. lt represents the +spatial rate of change of 7'. The z-component of VT' is how fast 7' changes in +the z-direction. What is the direction of the vector WfT? We know that the rate +of change of 7' in any direction is the component of V7 in that direction (see +Eq. 2.15). It follows that the direction of W? is that in which it has the largest +possible component——in other words, the direction in which 7' changes the fastest. +The gradient of 7' has the direction of the steepest uphill slope (mn 7). +2-5 _ Operations with V +Can we do any other algebra with the vector operator W? Let us try combining +19 with a vector. We can combine two vectors by making a dot product. We +could make the products +(a vector) - V, Or W:- (a vector). +The first one doesn't mean anything yet, because it is still an operator. What +19 might ultimately mean would depend on what it is made to operate on. 'Phe +second produect is some scalar field. (A - Ð is always a scalar.) +Let's try the dot product of with a vector feld we know, say h. We write +out the components: +V:-h= V„h„ + Vyh„ + V„yhz„ (2.32) +Qhy„ _ 0h Ôh„ +V:-h=——+— ~+—.. 2.33 +Øz làn Øz (2.33) +'The sum ïs invariant under a coordinate transformation. If we were to choose a +diferent system (indicated by primes), we would have* +Qh„ Oh Qh„ +W.h= “+ ở +_—_` 2.34 +9z + Øy + 8z! ` (239) +which is the sazme number as would be gotten from Eq. (2.33), even though it +looks diferent. 'That 1s, +V':h=V-h (2.35) +for every point in space. So V -h, is a scalar field, which must represent some +physical quantity. You should realize that the combination of derivatives in V: h +is rather special. There are all sorts of other combinations like Øh„/Øz, which +are neither scalars nor components of vectOrs. +The scalar quantity - (a vector) is extrermely useful in physics. It has been +given the name the đ?uergence. Eor example, +W:h =divh = “divergence of h” (2.36) +As we diịd for V7', we can ascribe a physical significance to WV-h. We shall, +however, postpone that until later. +* We think of h as a phụs¿cøl quantity that depends on position in space, and not strictly +as a mathematical function of three variables. When h is “diferentiated” with respect to +z, , and z, or with respect to +, ', and z, the mathematical expression for h must first be +expressed as a function of the appropriate variables. +--- Trang 38 --- +First, we wish to see what else we can cook up with the vector operator V. +What about a cross product? We must expect that +Vxh=a vector. (2.37) +Tt is a vector whose components we can write by the usual rule for cross products +(see lq. 2.2): +Similarly, +(Vxh)„ = Vụh; — V„hụ = 2y — % (2.39) +0h 9h„ +The combination V x Ö ¡is called “the curÍ of h.” 'The reason for the name +and the physical meaning of the combination will be discussed later. +Summarizing, we have three kinds of combinations with V: +V?' =grad?7'—a vector, +V:h =divh —ascalar, +Vxh=curlh =a vector. +sing these combinations, we can write about the spatial variations of fields in +a convenient way——in a way that is general, in that it doesn't depend on any +particular set of axes. +As an example of the use of our vector diferential operator V, we write a set Z +of vector equations which contain the same laws of electromagnetism that we é +gave in words in Chapter I1. They are called Maxwells equations. ọ ⁄ n +Mazuells Equations x +p Flow +(1) V.E=— ( +(2) VxE=--—_ +ðt (2.41) = +(3) V{.b-=0 ⁄ Area A +(4) cÔỒVxEB 0E + j Ầ +€ =——+— +Øt €0 F—s—¬ (a) +where ø (rho), the “electric charge density,” is the amount oŸ charge per unit +volume, and 7, the “electric current density,” is the rate at which charge ows +through a unit area per second. 'Phese four equations contain the complete +classical theory of the electromagnetic field. You see what an elegantly simple +form we can get with our new notationl bé +2-6 The diferential equation of heat flow +Let us give another example of a law of physics written in vector notation. The Area Aề +law is not a precise one, but for many metals and a number of other substances that +conduct heat it is quite accurate. You know that if you take a slab of material and +heat one face to temperature 7; and cool the other to a diferent temperature 7T] ISOTHERMALS +the heat will fow through the material from 7; to 71 [Fig. 2-7(a)|. The heat fow +is proportional to the area A of the faces, and to the temperature diference. Ït +is also inversely proportional to đ, the distance between the plates. (For a given T1 AT T +temperature diference, the thinner the slab the greater the heat fow.) Letting J ¬ r +be the thermal energy that passes per unit time through the slab, we write @œ) +J = R(T; — TÌ) 4 (2.42) Fig. 2-7. (a) Heat flow through a slab. +d (b) An infinitesimal slab parallel to an +The constant oŸ proportionality (kappa) is called the £hermal conductiuitg. isothermal surface in a large block. +--- Trang 39 --- +'What will happen in a more complicated case? Say in an odd-shaped block of +material in which the temperature varies in peculiar ways? Suppose we look at a +tỉny piece of the block and imagine a slab like that of Fig. 2-7(a) on a miniature +scale. We orient the faces parallel to the isothermal surfaces, as in EFig. 2-7(b), +so that Bq. (2.42) is correct for the small slab. +Tf the area of the small slab is AA, the heat flow per unit tỉme is +A/V=gAT —— 2.43 +RAT SE, (3.43) +where As is the thickness of the slab. Now AJ/AA we have defined earlier as +the magnitude of h, whose direction is the heat fow. The heat fow will be om +Tị + AT toward 71 and so it will be perpendicular to the isotherms, as drawn +in Eig. 2-7(b). Also, A7/As is just the rate of change of 7' with position. And +since the position change is perpendicular to the isotherms, our A7 '/As is the +maximum rate of change. It is, therefore, Just the magnitude of V7". NÑow since +the direction of W7” is opposite to that of h, we can write (2.43) as a vector +equation: +h =—kœVT. (2.44) +(The minus sign is necessary because heat fows “downhill? in temperature.) +Ebquation (2.44) is the differential equation of heat conduction in bulk materials. +You see that It is a proper vector equation. Each side is a vector I1 œ is just a +number. It is the generalization to arbitrary cases oŸ the special relation (2.42) for +rectangular slabs. Later we should learn to write all sorts of elementary physics +relations like (2.42) in the more sophisticated vector notation. This notation is +useful not only because 1% makes the equations /ook simpler. It also shows most +clearly the ph#s¿cal con‡en£ of the equations without reference to any arbitrarily +chosen coordinate system. +2-7 Second derivatives of vector ñelds +So far we have had only frst derivatives. Why not second derivatives? We +could have several combinations: +(a) V:(V7) +(b) Vx(V7) +(c) V(V:h) (2.45) +(dd) V:(Vxh) +(e) Wx(Vxh) +You can check that these are all the possible combinations. +Let”s look first at the second one, (b). It has the same form as +A x(A7) =(Ax A)T =0, +since Á x A is always zero. So we should have +curl(grad 7) = W x (V7) =0. (2.46) +W©e can see how this equation comes about if we go through once with the +componenfs: +(V x (V7)]¿ = V„(V7)„ — Vu(VT)z +8 (Øðï 8 (8T +==— | — l_- —=|—l]. (2.47) +Øz \Øy Øy\ Øz +which is zero (by Eq. 2.8). It goes the same for the other components. So +Wx(V7) =0, for any temperature distribution——in fact, for ø scalar function. +Now let us take another example. Let us see whether we can fnd another +zero. 'Phe dot produect of a vector with a cross product which contains that vector +18 Z©TO: +A:(Axb)=0, (2.48) +--- Trang 40 --- +because A x is perpendicular to A, and so has no components in the direction A. +The same combination appears in (d) of (2.45), so we have +V:(Vxh) =div(curlh) = 0. (2.49) +Again, it is easy to show that it is zero by carrying through the operations with +components. +Now we are going to state two mathematical theorems that we will not prove. +They are very interesting and useful theorems for physicists to know. +In a physical problem we frequently fñnd that the curl of some quantity——say +of the vector feld Á——is zero. Now we have seen (Eq. 2.46) that the curl of a +gradient is zero, which is easy to remember because of the way the vectors work. +lt could certainly be, then, that A is the gradient of some quantity, because then +1ts curl would necessarily be zero. The interesting theorem is that if the curl A +is zero, then A is øøays the gradient of something—there is some scalar field +(psi) such that 4 ¡is equal to grad ý. In other words, we have the +'THEOREM: +Tf VxA=0 +there is a Dh +such that 4= Vụ. (2.50) +There is a similar theorem ïf the divergence of Á is zero. We have seen in +Eq. (2.49) that the divergence of a curl oŸ something is always zero. lỶ you come +across a vector ñeld D for which div D is zero, then you can conclude that 1 is +the curl of some vector ñeld Œ. +'THEOREM: +there is a C +such that 2= V x C. (2.51) +In looking at the possible combinations of two operators, we have found +that two of them always give zero. Now we look at the ones that are o‡ zero. +Take the combination Ð - (W7), which was first on our list. It is not, in general, +zoro. We write out the components: +Vĩ'=¿V„T'+7Vụ„T + kV,T. +V:(V7) = V„(V;„7) + Vyụ(Vyạ7) + V¿(V;7) +0T 0*T7 6@T +==ò]>+.—.atđ..- 2.52 +0x2 0g ` 0z) (2.52) +which would, in general, come out to be some number. lt is a scalar field. +You see that we do not need to keep the parentheses, but can write, without +any chance of confusion, +V:(V7)=WVWV-V7=(V-V)7 = VẺT. (2.53) +We look at V2 as a new operator. It is a scalar operator. Because it appears +often in physics, it has been given a special name—the Laplacian. +lầu 82 82 +Laplacian = VÏ= -—; + +: 2.54 +aplacian 2x2 + Øy? + oz5 (2.54) +Since the Laplacian is a scalar operator, we may operate with it on a vector——by +which we mean the same operation on each component in rectangular coordinates: +V*h = (Vˆh„, V?h„,V”h,). +--- Trang 41 --- +Let's look at one more possibility: V x (VW x h), which was (e) in the +list (2.45). NÑow the curl of the curÌ can be written diferently if we use the vector +cquality (2.6): +Ax(BxC)= Bb(A-C)- C(A:Đ). (2.55) +In order to use this formula, we should replace A and Ö by the operator V and +put C = h. If we do that, we get +#Wfx(Vxh)=V(V:-h) - h(V-V)...??? +Wait a minutel Something is wrong. The first two terms are vectors all right (the +operators are satisfed), but the last term doesn”t come out to anything. Its stil +an operator. The trouble is that we haven”t been careful enough about keeping +the order of our terms straight. IÝ you look again at Eq. (2.55), however, you see +that we could equally well have written i as +Ax(BxC)= Bb(A-C)-(A: B)C. (2.56) +The order of terms looks better. Now let's make our substitution in (2.56). We +#Wfx(Vxh)=V(V:h) - (V - V)h. (2.57) +This form looks all right. It is, in fact, correct, as you can verify by computing +the components. “The last term is the Laplacian, so we can equally well write +Vx(Vxh)=V(V:h) - V°h. (2.58) +W©e have had something to say about all of the combinations in our list of +double W”s, except for (c), VW(W -h). It is a possible vector fñield, but there is +nothing special to say about it. It's just some vector field which may occasionally +come up. +It will be convenient to have a table of our conclusions: +(a) W:(V7) =V”T =a scalar ñeld +(b) Vx(V7)=0 +(c) VW(VW:h)=a vector fñeld +(2.59) +(dd) V:(Vxh)=0 +(e) Vx(Vxh)=V(V:h)- V°h +()Q (V:V)h = Vˆh = a vector fñeld +You may notice that we haven't tried to invent a new vector operator (VW x Vì). +Do you see why? +2-8 Pitfalls +We© have been applying our knowledge of ordinary vector algebra to the algebra, +of the operator V. We have to be careful, though, because 1È is possible to go +astray. There are ©wo pitfalls which we will mention, although they will not +come up in this course. What would you say about the following expression, that +involves the two scalar functions and ø (phì): +(Vú) x (Vó)? +You might want to say: it must be zero because it's Just like +(Aaø) x (A)), +which is zero because the cross product of bwo egual vectors Á x A is always zero. +But in our example the two operators V are not equall “The first one operates +on one function, ý; the other operates on a different function, ¿. So although +we represent them by the same symbol V, they must be considered as diferent +--- Trang 42 --- +operators. Clearly, the direction of Wớ depends on the function ÿ, so it is not +likely to be parallel to Vọ: +(Vú) x(Vøỏ) #0 (gencrally). +Fortunately, we wonˆt have to use such expressions. (What we have said doesn't +change the fact that VW x Vụ = 0 for any scalar field, because here both W?s +operate on the same function.) +Pitfall number ©wo (which, again, we need not get into in our course) is the +following: “The rules that we have outlined here are simple and nice when we +use rectangular coordinates. Eor example, if we have V?h and we want the +#-component, 1t 1s +(VỶh)„ = lấn + nại + 32) h„ = V°hụ. (2.60) +'The same expression would no work if we were to ask for the rađ¿al component +of Vˆ2h. The radial component of V2] is not equal to V2h„. The reason is that +when we are dealing with the algebra of vectors, the directions of the vectors are +all quite defnite. But when we are dealing with vector fields, their directions +are difÑferent at diferent places. If we try to describe a vector field in, say, polar +coordinates, what we call the “radial” direction varies from point to point. So +we can get into a lot of trouble when we start to diferentiate the componenfs. +For example, even for a constan£ vector feld, the radial component changes from +point to poïnt. +lt is usually safest and simplest just to stick 6o rectangular coordinates +and avoid trouble, but there is one exception worth mentioning: Since the +Laplacian V2, is a scalar, we can write it in any coordinate system we want to +(for example, in polar coordinates). Đut since ït is a diferential operator, we +should use it only on vecbors whose components are in a fxed direction——that +means rectangular coordinates. So we shall express all of our vector fñelds in +terms of their z-, -, and z-components when we write our vector diferential +equatlons out in components. +--- Trang 43 --- +Woc£or' Ireéoqggr‹ä[ Ế «ÍcrrÏrrs +3-1 Vector integrals; the line integral of V+Ù +W© found in Chapter 2 that there were various ways of taking derivatives of 3-1 Vector integrals; the line integral +fields. Some gave vector fields; some gave scalar fields. Although we developed of Vụ +many diferent formulas, everything in Chapter 2 could be summarized in one 3-2_ The flux of a vector fñeld +rule: the operators Ø/Øz, Ø/Ø, and Ø/Øz are the three components oŸ a vecbor 3-3 The fñux from a cube; Gauss' +operator V. We would now like to get some understanding of the significance theorem +of the derivatives of fields. We will then have a better feeling for what a vector 3-4 Heat conduction; the đi8usion +fñeld equation means. R +We have already discussed the meaning of the gradient operation (W on a cquailon . +scalar). NÑow we turn to the meanings of the divergence and curl operations. The d-š The circulation ofa vector field +interpretation of these quantities is best done in terms of certain vector integrals 3-6 The circulation around a square; +and equations relating such integrals. Thhese equations cannot, unfortunately, be 5tokes° theorem +obtained from vector algebra by some easy substitution, so you will just have to 3-7 Curl-free and divergence-free +learn them as something new. OÝ these integral formulas, one is practically trivial, fields +but the other two are not. We will derive them and explain their mmplications. 3-8. Summary +The equations we shall study are really mathematical theorems. 'Phey will be +useful not only for interpreting the meaning and the content of the divergence and +the curl, but also in working out general physical theories. 'Phese mathematical +theorems are, for the theory of fields, what the theorem of the conservation of +energy is to the mechanies of particles. General theorems like these are important +for a deeper understanding of physics. You will fnd, though, that they are not +very useful for solving problems——except in the simplest cases. It ¡is delightful, +however, that in the beginning of our subject there will be many simple problems +which can be solved with the three integral formulas we are going to treat. We +will see, however, as the problems get harder, that we can no longer use these +simple methods. +We take up frst an integral formula involving the gradient. "The relation Vụ +contains a very simple idea: Since the gradient represents the rate of change of a (2) +ñeld quantity, if we integrate that rate of change, we should get the total change. +Suppose we have the scalar field (+, ,z). At any ©wo points (1) and (2), the Curve T +function ¿ will have the values (1) and (2), respectively. [We use a convenient +notation, in which (2) represents the poïnt (sa, a2, z2) and (2) means the same ds +thing as (4a, 0a, Z2).| TT (gamma) is any curve joining (1) and (2), as in Eig. 3-1, +the following relation is true: Œ) +'THEOREM 1. Fig. 3-1. The terms used in Edq. (3.1). +@) The vector V+ÿ is evaluated at the line ele- +6) 00) = [- (V6) cdẽ, G-1)— mensds +dong T +The integral is a ¿ne in‡egral, from (1) to (2) along the curve T, of the dot product +of Vú——a vector—with đs—another vector which is an infnitesimal line element VƯ`C — (vụ), +of the curve T` (directed away from (1) and toward (2)). ) (2) +Pirst, we should review what we mean by a line integral. Consider a scalar l§ +function ƒ(z, ,z), and the curve T joining two points (1) and (2). We mark of ⁄Z CurveT +the curve at a number of points and join these points by straight-line segments, +as shown in Eig. 3-2. Each segment has the length Az;, where ¿ is an index that As As +runs 1, 2, 3,... By the line integral @ As, c +(2) “Na; +k Jds Fig. 3-2. The line integral is the limit of +along 3a Sum. +--- Trang 44 --- +we mean the limit of the sum +» . đAs¿, +where ƒ; is the value of the function at the ;th segment. 'Phe limiting value is +what the sum approaches as we add more and more segments (in a sensible way, +so that the largest Az; —> 0). +The integral in our theorem, Eq. (3.1), means the same thing, although it +looks a little diferent. Instead of ƒ, we have another scalar—the component +of Vụ in the direction of A4. TỶ we write (Wø)¿ for thịs tangential component, +1t 1s clear that +(V)¿ As =(Vụ)- A3. (3.2) +The integral in Eq. (3.1) means the sum oŸ such terms. +Now lets see why 4q. (3.1) is true. In Chapter 2, we showed that the +component of Vú along a small displacement AF was the rate of change of +in the direction of Ai. Consider the line segment As from (1) to point ø in +Fig. 3-2. According to our definition, +Aúi = (4) — Ú() = (VỤ): - Asi. (3.3) +Also, we have +0(b) — 0(a) = (VỤ); - A$a, (3.4) +where, of course, (Wø)¡ means the gradient evaluated at the segment Asi, +and (Vø)a, the gradient evaluated at Asa. T we add Eqs. (3.3) and (3.4), we get +00) — 0(1) = (Vú): - Asi + (VỤ)¿ - A5. (3.5) +You can see that if we keep adding such terms, we get the result +00) = 00) = À ).(V0)¡ - Ai. (3.6) +The left-hand side doesn'6 depend on how we choose our intervals—if (1) and (2) +are kept always the same——so we can take the limit of the right-hand side. We +have therefore proved Eq. (3.1). +You can see from our proof that just as the equality doesn't depend on how +the points ø Ù, c,..., are chosen, similarly it doesn't depend on what we choose +for the curve Ƒ' to join (1) and (2). Our theorem is correct for an curve from (1) +to (2). +One remark on notation: You will see that there is no confusion 1Ý we write, +for convenience, +(Vụ) - ds = Vụ - d3. (3.7) +With this notation, our theorem is +'THEOREM I1. (3) +(2) - (1) = J ) Vụ - ds. (3.8) +2ny X — ” h +1) to (2 +Closed s6 +Surface S +3-2 The ñÑux ofa vector field ⁄ ⁄7 Zˆ n +Before we consider our next integral theorem——a theorem about the divergence Volume V ⁄ | | : +——we would like to study a certain idea which has an easily understood physical _. +significance in the case of heat ñow. We have defned the vector h, which represents / // — +the heat that fows through a unit area in a unit time. Suppose that inside a —— +block of material we have some closed surface Š which encloses the volume W — +(Fig. 3-3). We would like to find out how much heat is flowing out of this 0olwme. +We can, of course, ñnd it by calculating the total heat Ñow out of the surface S. Fig. 3-3. The closed surface S defines +We write đø for the area of an element of the surface. The symbol stands for the volume V. The unit vector n is the +a two-dimensional diferential. Tf, for instance, the area happened to be in the outward normal to the surface element đa, +zu-plane we would have and ñh ¡s the heat-flow vector at the surface +da = dz dụ. element. +--- Trang 45 --- +Later we shall have integrals over volume and for these i is convenient to consider +a diferential volume that is a little cube. So when we write đV we mean +dV = d+z dụ dz. +Some people like to write đ^ø instead of da to remind themselves that it is +kind of a second-order quantity. They would also write đỶV instead of dV. We +will use the simpler notation, and assume that you can remember that an area +has two dimensions and a volume has three. +The heat fow out through the surface element dø is the area times the +component of h perpendicular to da. We have already defñned ?ø as a unit vector +pointing outward at right angles to the surface (Eig. 3-3). The component of h +that we wanf 1s +hạ =h -mn. (3.9) +'The heat fow out through da is then +h - n da. (3.10) +To get the total heat ñow through any surface we sum the contributions from all +the elements of the surface. In other words, we integrate (3.10) over the whole +surface: +Total heat fow outward through S9 = J h - n da. (3.11) +W© are also going to call this surface integral “the Ñux of h through the +surface.” Originally the word fux meant ow, so that the surface integral jus$ +means the fow of h through the surface. We may think: h is the “current density” +of heat fñow and the surface integral of it is the total heat current directed out of +the surface; that is, the thermal energy per unit time (joules per second). +We would like to generalize this idea to the case where the vector does not +represent the flow of anything; for instance, it might be the electric ñeld. We can +certainly still integrate the normal component of the electric ñeld over an area iÍ +we wish. Althouph it is not the fow of anything, we still call it the “fux” We say +Flux of through the surface 9 = J E-n da. (3.12) +We generalize the word “ñÑux” to mean the “surface Integral of the normal +component” of a vector. We will also use the same defnition even when the +surface considered is not a closed one, as it is here. +Returning to the special case of heat fow, let us take a situation in which +heqt ¡s conserued. Eor example, imagine some material in which after an initial +heating no further heat energy is generated or absorbed. 'Then, if there is a net +heat ñow out of a closed surface, the heat content of the volume inside must +decrease. So, In circumstances in which heat would be conserved, we say that +... (3.13) +where @ is the heat inside the surface. 'Phe heat fux out of Š is equal to minus +the rate of change with respect to time of the total heat Q inside of S. This +Interpretation is possible because we are speaking of heat fow and also because +we supposed that the heat was conserved. We could not, of course, speak of the +total heat inside the volume if heat were being generated there. +Now we shall poïint out an interesting fact about the ñux of any vector. You +may think of the heat ñow vector if you wish, but what we say will be true for +any vector ñeld Œ. Imagine that we have a closed surface Š that encloses the +volume W. We now separate the volume into two parts by some kind of a “cut,” +as in Fig. 3-4. Now we have two closed surfaces and volumes. The volume VỊ is +enclosed in the surface 51, which is made up of part of the original surface %„ and +of the surface of the cut, S„;. The volume V2 is enelosed by 5+, which is made up +--- Trang 46 --- +Sap h V +Fig. 3-4. A volume V/ contained inside the surface S c +is divided into two pieces by a “cut” at the surface Sap. h _=* +We now have the volume Ví enclosed in the surface ! ạ +SỊ = Sa+ Sa; and the volume \⁄2 enclosed in the surface 4 +Sa = Sb + Sàp. “ +of the rest of the original surface %p and closed of by the cut S„y. Now consider +the following question: Suppose we calculate the fux out through surface 5 and +add to it the ñux through surface 5+. Does the sum equal the fux through the +whole surface that we started with? The answer is yes. Phe ñux through the +part of the surfaces S„; common to both J5 and 52 just exactly cancels out. Eor +the ux of the vector out of VU we can write +Flux through 5 = J C-nda + C -mị da, (3.14) +S% Sạp +and for the Ñux out of V2, +Flux through S2 = J C-nda + C - na da. (3.15) +Sp Sab +Note that in the second integral we have written ?øœ for the outward normal +for S„p when it belongs to 51, and m»z when ¡it belongs to 5, as shown ïn Fig. 3-4. +Clearly, rị — —m¿, so that +J C mì da = = | C - nạ da. (3.16) +S&b S«b +T we now add Eqs. (3.14) and (3.15), we see that the sum of the Ñuxes through +S5 and S52 is just the sum of two integrals which, taken together, give the ñux +through the original surface Š = ®%„ + S%b. +W© see that the Ñux through the complete outer surface Š can be considered (x,y+Ay,z) 4 +as the sum of the Ñuxes from the two pieces into which the volume was broken. 5 +W© can similarly subdivide again—say by cutting VỊ into bwo pieces. You see 3` +that the same arguments apply. So for an way of dividing the original volume, c +1ÿ must be generally true that the Ñux through the outer surface, which is the .x Ả +original integral, is equal to a sum of the ñuxes out of all the little interior pieces. n TẤT _—= l +(XVZ) —_ +3-3 The Ñux from a cube; Gauss° theorem 6 & Ax (x+ Ax.y.2) +⁄ ⁄ Z +We now take the special case of a small cube# and fñnd an interesting formula ú +for the Ññux out of it. Consider a cube whose edges are lined up with the axes Giyz + Az) 3 +as in Eig. 3-5. Let us suppose that the coordinates of the corner nearest the Fig. 3-5. Computation of the flux of C +origin are #, , z. Leb Az be the length of the cube ïn the z-direction, A» be out of a small cube. +the length in the g-direction, and Az be the length in the z-direction. We wish +to ñnd the ñux of a vector fñeld Œ through the surface of the cube. We shall do +this by making a sum of the fuxes through each of the six faces. First, consider +the face marked 1 in the fgure. The ñux ou£uørd on this face is the negative of +the z-component of Œ, integrated over the area of the face. 'Phis ñux is +— J > dụ dz. +* The following development applies equally well to any rectangular parallelepiped. +--- Trang 47 --- +Since we are considering a smail cube, we can approximate this integral by the +value of „ at the center of the face—which we call the point (1)—multiplied by +the area of the face, A Az: +Flux out oŸ 1 = —Œz(1) Au Az. +Similarly, for the ñux out of face 2, we write +Flux out of 2 = €z„(2) Au Az. +Now Œ„(1) and „(2) are, in general, slightly diferent. If Az is small enough, +W©€ Can WTIt ôC +Œ„(2) =Œ,(1)+ <“ Az. +There are, of course, more terms, but they will involve (Az)2 and higher powers, +and so will be negligible if we consider only the limit of small Az. So the fux +through face 2 1s +Flux out of 2 = |C„z(1) + _ Azl AuAz. +Summing the fuxes for faces 1 and 2, we get +Flux out of 1 and 2 = =_" Az Au Az. +The derivative should really be evaluated at the center of face 1; that is, at +[~,+(A/2).z + (Az/2)|J. But in the limit of an infinitesimal cube, we make a +negligible error iÝ we evaluate it at the corner (z, , 2). +Applying the same reasoning to each of the other pairs of faces, we have +Flux out of 3 and 4= n Az AuAz +Flux out oŸ ð and 6 = Đc Az Au Az. +'The total ñux through all the faces is the sum of these terms. We fnd that +93C 9C, 9C +ŒC-nda= [| - “+-- “+. “|AzAyA +J T. da (5t tư tin )A»AyAs +and the sum of the derivatives is just V -C. Also, Az Ay Az = AV, the volume +of the cube. So we can say that for an ứn[imitesimal cube +J Œ -nda =(V-C) AV. (3.17) +surface +W© have shown that the outward fux from the surface of an infñnitesimal cube is +cqual to the divergence of the vector multiplied by the volume of the cube. We +now see the “meaning” of the divergence of a vector. 'Phe divergence of a vector +at the point is the ñux—the outgoing “flow” of C—per uuit 0olumne, in the +neighborhood of ?. +W© have connected the divergence of C to the Ñux of out ofeach infnitesimal +volume. For any fñnite volume we can use the fact we proved above—that the +total ñux from a volume is the sum of the fuxes out of each part. We can, that +1s, Integrate the divergence over the entire volume. 'Phis gives us the theorem +that the integral of the normal component oŸ any vector over any closed surface +can also be written as the integral of the divergence of the vector over the volume +enclosed by the surface. This theorem is named after Gauss. +GAUSS” THEOREM. +C-naa= | V-CaY, (3.18) +where Š is any closed surface and V is the volume inside 1t. +--- Trang 48 --- +3-4 Heat conduction; the difusion equation +Let”s consider an example of the use of this theorem, just to get familiar with +19. Suppose we take again the case of heat fow in, say, a metal. Suppose we +have a simple situation in which all the heat has been previously put in and the +body is just cooling of. There are no sources of heat, so that heat is conserved. +Then how mụch heat is there inside some chosen volume at any time? lt must +be decreasing by just the amount that fows out of the surface of the volume. Tf +our volume is a little cube, we would write, following Eaq. (3.17), +Heat out = =...` (3.19) +But this must equal the rate of loss of the heat inside the cube. If g is the heat +per unit volume, the heat in the cube is g AV, and the rate of Ïoss is +ôi (g@AV)= Đi AV. (3.20) +Comparing (3.19) and (3.20), we see that +bên V:h. (3.21) +Take careful note of the form of this equation; the form appears often in +physics. Ït expresses a conservation law—here the conservation of heat. We have +expressed the same physical fact in another way in Eq. (3.13). Here we have the +diƒerential form oŸ a conservation equation, while Eq. (3.13) is the Znfegral form. +W© have obtained Ea. (3.21) by applying Eq. (3.13) to an infinitesimal cube. +W©e can also go the other way. Eor a big volume W bounded by Š, Gauss' law +says that +hinda= | V-hat (3.22) +Using (3.21), the integral on the right-hand side is found to be jusb —đ@Q/đ, and +again we have Eq. (3.13). +Now let”s consider a different case. Imagine that we have a block of material h +and that inside it there is a very tỉny hole in which some chemical reaction is _ 1 +taking place and generating heat. Or we could imagine that there are some wires 7 v* +¬. . . . . . ~—l R +running into a tiny resistor that is being heated by an electric current. We shall J +suppose that the heat is generated practically at a point, and let W represent Source `T ~Ñ +the energy liberated per second at that point. We shall suppose that in the rest of heat +of the volume heat is conserved, and that the heat generation has been going on Block of metal +for a long time——so that now the temperature is no longer changing anywhere. +The problem is: What does the heat vector h look like at various places in the +metal? How much heat fow is there at each point? Fig. 3-6. In the region near a point source +W© know that if we integrate the normal component of h over a closed surface of heat, the heat flow is radially outward. +that encloses the source, we will always get W. AlI the heat that is being +generated at the point source must fow out through the surface, since we have +supposed that the fÑow is steady. We have the difficult problem of ñnding a vector +fñeld which, when integrated over any surface, always gives W/. We can, however, +fñnd the fñeld rather easily by taking a somewhat special surface. We take a +sphere of radius †, centered at the source, and assume that the heat fow is radial +(Fig. 3-6). Our intuition tells us that should be radial if the block of material +is large and we don't get too close to the edges, and it should also have the same +magnitude at all points on the sphere. You see that we are adding a certain +amount oŸ guesswork——usually called “physical intuition”—to our mathematics +in order to fñnd the answer. +When h is radial and spherically symmetric, the integral of the normal +component of h over the area is very simple, because the normal component +--- Trang 49 --- +1s just the magnitude of h and is constant. The area over which we integrate +is 4rR2. We have then that +J h-n da = h- 4rR2 (3.23) +(where h is the magnitude of h). Thịis integral should equal W, the rate at which +heat is produced at the source. WWe get +h— 4x2 €r, (3.24) +where, as usual, e„ represents a unit vector in the radial direction. Our result +says that h is proportional to W and varies inversely as the square of the distance +from the source. +The result we have just obtained applies to the heat ñow In the vicinity of +a point source of heat. Let's now try to fnd the equations that hold in the +most general kind of heat ñow, keeping only the condition that heat is conserved. +We will be dealing only with what happens at places outside of any sources or +absorbers of heat. +The diferential equation for the conduction of heat was derived in Chapter 2. +According to Eq. (2.44), +h =—kœVT. (3.25) +(Remember that this relationship is an approximate one, but fairly good for +some materials like metals.) It is applicable, oŸ course, only in regions of the +material where there is no generation or absorption of heat. We derived above +another relation, Eq. (3.21), that holds when heat is conserved. TỶ we combine +that equation with (3.25), we get +——_=V-h=_—V:(kV?T), +ôi V7) +2 =œV:V7T=wV°T, (3.26) +1Í œ is a constant. You remember that g is the amount of heat in a unit volume +and W: V = V2 is the Laplacian operator +82 92 lầu +V?=_—+~+d—-. +8z2 + Øy? + 8z? +Tf we now make one more assumption we can obtain a very interesting equation. +W© assume that the temperature of the material is proportional to the heat content +per unit volume——that is, that the material has a defnite specifc heat. When +this assumption is valid (as it ofben is), we can write +Aq=cœ,AT' +—-=Œy——: 3.27 +ðt ` “" ôi 3⁄27) +'The rate of change of heat is proportional to the rate of change of temperature. +'The constant of proportionality c„ is, here, the specifc heat per unit 0olưme oŸ +the material. Using Eq. (3.27) with (3.26), we get +lÚ ĐI, +—=—=_— VỀ†. 3.28 +Ô† — œ (3.28) +W©e find that the #ữne rate of change of T——at every point—Is proportional to +the Laplacian of 7, which is the second derivative of its spatial dependence. We +have a diÑferential equation—=in zø, , z, and £—for the temperature 7. +--- Trang 50 --- +The diferential equation (3.28) is called the heat djƒƑfusion cquation. TW is +often written as đt +Tin DV⁄“T, (3.29) +where Ï is called the đjƒƒus¿on constant, and is here equal to &/đ. +'The difusion equation appears in many physical problems——in the difusion of +gases, in the difusion of neutrons, and in others. We have already discussed the +physics of some of these phenomena in Chapter 43 of Vol. I. NÑow you have the +complete equation that describes diÑusion in the most general possible situation. +At some later time we will take up ways of solving the difusion equation to ñnd +how the temperature varles in particular cases. We turn back now to consider +other theorems about vector fñelds. +3-5 The circulation of a vector field +We wish now to look at the curl in somewhat the same way we looked at +the divergence. We obtained Gaussˆ theorem by considering the integral over C +a surface, although it was not obvious at the beginning that we were going to Loop E ¿ +be dealing with the divergence. How did we know that we were supposed to `“ +integrate over a surface in order to get the divergence? lý was not at all clear +that this would be the result. And so with an apparent equal lack of justification, + +we shall calculate something else about a vector and show that it is related to +the curl. This time we calculate what is called the circulation of a vector field. +lf Œ is any vector feld, we take its component along a curved line and take the +Integral of this component all the way around a complete loop. “The integral ; +1s called the circulation of the vector field around the loop. We have already 7 +considered a line integral of Vụ earlier in this chapter. Now we do the same +kind of thing for øny vector field ŒC. Fig. 3-7. The circulation of C around +Let T' be any closed loop in space—imaginary, of course. An example is given the curve [' is the line integral of C;, the +in Eig. 3-7. The line integral of the tangential component of C around the loop tangential component of C. +1s wrltten as +‡ ty ds = 1 C - ds. (3.30) +You should note that the integral is taken all the way around, not from one poïnt +to another as we did before. 'The little circle on the integral sign is to remind +us that the integral is to be taken all the way around. This integral is called +the circulation of the vector fñeld around the curve I`. The name came originally +from considering the circulation of a liquid. But the name——like Ñux—has been +extended to apply to any field even when there is no material “circulating.” +Playing the same kind of game we did with the fux, we can show that the q) +circulation around a loop is the sum of the circulations around two partial loops. Tp +Suppose we break up our curve of Fig. 3-7 into two loops, by joining two points là +(1) and (2) on the original curve by some line that cuts across as shown in Fig. 3-8. +'There are now two loops, Dị and Ùạ. E is made up of Ứ„, which is that part of +the original curve to the left of (1) and (2), plus Ƒạ», the “short cut.” Ứs is made +up of the rest of the original curve plus the short cut. (2) +'The circulation around Ủ is the sum of an integral along ¿ and along Lạ;. +Similarly, the circulation around L's is the sum of two parts, one along Ủy and the Fig. 3-8. The circulation around the +other along Ùạ;. The integral along Dạy will have, for the curve 2, the opposite whole loop is the sum of the circulations +sign from what it has for L', because the direction of travel is opposite—we must around the two loops: F = Fạ + Fạp and +take both our line integrals with the same “sense” of rotation. [a =lIb;~+ Tạp. +Following the same kind of argument we used before, you can see that the +sum of the two circulations will give just the line integral around the original +curve Ï`, The parts due to „; cancel. The circulation around the one part plus +the circulation around the second part equals the circulation about the outer +line. We can continue the process of cutting the original loop into any number +of smaller loops. When we add the circulations of the smaller loops, there 1s +always a cancellation of the parts on their adjacent portions, so that the sum is +equivalent to the circulation around the original single loop. +--- Trang 51 --- +Now let us suppose that the original loop is the boundary of some surface. c-—¬> Loop F +There are, of course, an infinite number of surfaces which all have the original Ấ-L TL 1^>—. +loops as the boundary. Our results will not, however, depend on which surface 3200nnnmmm= +we choose. Eirst, we break our original loop into a number of small loops that 5055 r†} }°}?!9| |Ì +all lie on the surface we have chosen, as in Eig. 3-9. NÑo matter what the shape W1] †T®EšJP] +of the surface, if we choose our small loops small enough, we can assume that P1? 9e) +cach of the small loops will enclose an area which is essentially fat. Also, we can =7 7 7 => +choose our small loops so that each is very nearly a square. NÑow we can calculate +the circulation around the big loop L` by ñnding the circulations around all of Fig. 3-9. Some surface bounded by the +the little squares and then taking their sum. loop [ is chosen. The surface Is divided +Into a number of small areas, each approxiI- +3-6 The circulation around a square; Stokes° theorem mately 3 SQUaF6. The circulation 2round Ï +is the sum of the circulations around the +How shall we fnd the circulation for each little square? Ône question is, how little loops. +1s the square oriented in space? We could easily make the calculation If it had a +special orilentation. Eor example, iŸ it were in one of the coordinate planes. Since +we have not assumed anything as yet about the orientation of the coordinate +axes, we can just as well choose the axes so that the one littÌe square we are y +concentrating on at the moment lies in the z-plane, as in Eig. 3-10. lÝ our result +1s expressed In vector notation, we can say that it will be the same no matter +what the particular orientation of the plane. 3 +W©e want now to find the circulation of the field Œ around our little square. lt +will be easy to do the line integral if we make the square small enough that the | Œ}-->c +vector Œ doesnˆt change mụch along any one side of the square. (The assumption Ay 5, +is better the smaller the square, so we are really talking about infnitesimal +squares.) Starting at the point (+, )—the lower left corner oŸ the ñgure—we go ? +around in the direction indicated by the arrows. Along the first side—marked (1)—— h c +the tangential component is C„(1) and the distance is Az. The first part of the c : +integral is C„(1) Az. Along the second leg, we get Œ„(2) Ay. Along the third, we Gx.x) ì +get —Œ„(3) Az, and along the fourth, —Œ,(4) A+». The minus signs are required | Ax +because we want the tangential ecomponent in the direction of travel. 'The whole +line integral is then - ++c - dø = Œ„(1) Az + Œy(2) Au — C„(3) Az — Œy(4) Aw. (3.31) Fig. 3-10. Computing the circulation of C +around a small square. +Now let?s look at the fñrst and third pieces. Together they are +[C„(1) — €z(3)] Az. (3.32) +You might think that to our approximation the diference is zero. That is true +to the first approximation. We can be more accurate, however, and take into +account the rate of change of C„. If we do, we may write +Œ„(3) = €„(1) + 2% Aw. (3.33) +TÝ we included the next approximation, it would involve terms in (A2), but since +we will ultimately think of the limit as A —> 0, such terms can be neglected. +Putting (3.33) together with (3.32), we fnd that +[Œ„(1) — €z(3)] Az = —ag. Az Aw. (3.34) +The derivative can, to our approximation, be evaluated at (z, 0). +Similarly, for the other two terms in the circulation, we may write +Œy(2) Au— Œy(4) Au = 2 Az Au. (3.35) +'The circulation around our square is then +K- — ®) Az Au, (3.36) +--- Trang 52 --- +which is interesting, because the bwo terms in the parentheses are just the z- +component of the curl. Also, we note that Az Aø# is the area of our square. So +we can write our circulation (3.36) as +(Vx€Œ); Aa. +But the z-component really means the component normal to the surface element. +We can, therefore, write the cireulation around a diferential square in an invariant +vector form: +{C -ds= (V xi), Aa = (V x C) cứu, (3.37) +Our result is: the circulation of any vector around an infñnitesimal square +1s the component of the curl of C normail to the surface, times the area. of the +SquAre. < +The circulation around any loop ` can now be easily related to the curl of Loop T +the vector fñield. We fill in the loop with any convenient surface Š, as in Fig. 3-11, +and add the circulations around a set of inñnitesimal squares in this surface. The Surface +sum can be written as an integral. Our result is a very useful theorem called +Stokes'` theorem (after Mr. Stokes). +STOKES” 'HEOREM. +‡ C-ds = Jv x Ơ), da, (3.38) Ạ +T S 7 +where Š is any surface bounded by T'. nu j c +W© must now speak about a convention of siegns. In Fig. 3-10 the z-axis would Fig. 3-11. The circulation of C around F +point #øouørd you in a “usual?”—that is, “right-handed”——system of axes. When is the surface integral of the normal compo- +we took our line integral with a “positive” sense of rotation, we found that the nent of Ý x€. +circulation was equal to the z-component of Ÿ x Œ. Tf we had gone around the +other way, we would have gotten the opposite sign. Now how shall we know, +in general, what direction to choose for the positive direction of the “normal” +component of Ÿ x C? "The “positive” normal must always be related to the sense +of rotation, as in Fig. 3-10. It is indicated for the general case in Fig. 3-11. +One way of remembering the relationship is by the “right-hand rule.” IÝ you +make the ñngers of your zøh# hand go around the curve L`, with the fñngertips +pointed in the direction of the positive sense of đs, then your thumb points in +the direction of the øos?fz»e normal to the surface Z5. +3-7 Curl-free and divergence-free fields +'W©e would like, now, to consider some consequences of our new theorems. Take (2) +frst the case of a vector whose curÌ is eueruhere zero. hen 5tokes' theorem +says that the circulation around any loop is zero. Now if we choose ÿwo points +(1) and (2) on a closed curve (Fig. 3-12), it follows that the line integral of the +tangential component from (1) to (2) is independent of which of the two possible +paths is taken. We can conclude that the integral from (1) to (2) can depend C +only on the location of these points—that is to say, 1t is some function of position ._. +only. The same logic was used in Chapter 14 of Vol. Ï, where we proved that q) +1f the integral around a closed loop of some quantity is always zero, then that . . . . +. . . sua Fig. 3-12. lÝWW xC is zero, the circulation +integral can be represented as the diference of a function of the position of the : . +. . . . around the closed curve [is zero. The line +two ends. 'This fact allowed us to invent the idea oEa potential. We proved, integral from (1) to (2) along a must be +furthermore, that the vector feld was the gradient of this potential function (see the same as the line integral along b. +Eq. 14.13 of Vol. ]). +Tt follows that any vector ñeld whose cur] is zero is equal to the gradient of +some scalar function. That is, i x =0, everywhere, there is some (psi) +for which C = WVj——a useful idea. We can, if we wish, describe this special kind +of vector feld by means of a scalar field. +Let's show something else. Suppose we have ønww scalar ñeld ó (phi). IHf we +take Its gradient, Vớ, the integral of this vector around any closed loop must be +--- Trang 53 --- +zero. Its line integral from poïnt (1) to point (2) ¡is [@(2) — ø(1)]|. HT (1) and (2) +are the same points, our Theorem 1, Eq. (3.8), tells us that the line integral is +ZGTO: +‡ Vọộ - ds =0. +Using Stokesˆ theorem, we can conclude that +Jv x (Wỏ))„ da =0 +over ø/ surface. But if the integral is zero over ø? surface, the integrand must +be zero. So +Wx(Vø)=0, always. +W© proved the same result in 5ection 2-7 by vector algebra. +Let's look now at a special case in which we fill in a smaøil loop l` with a +large surface ®, as indicated im Fig. 3-13. We would like, in fact, to see what +happens when the loop shrinks down to a point, so that the surface boundary +disappears—the surface becomes closed. Now If the vector is everywhere fñnite, () BI, - +the line integral around I` must go to zero as we shrink the loop—the integral is Lodp L +roughly proportional to the cireumference of [', which goes to zero. According to ~ +Stokes” theorem, the surface integral of (W x )„ must also vanish. Somehow, Surface S vxe +as we close the surface we add in contributions that cancel out what was there . . "¬ +Fig. 3-13. Going to the limit of a closed +before. 5o we have a new theorem: . , +surface, we find that the surface Iintegral +f(VWx(C)n +† ¡sh. +J (Ý x Ở);„ da = 0. (3.30) — °HỮYXC); must vank +any closed +surface +Now this is interesting, because we already have a theorem about the surface +Integral of a vector field. Such a surface integral is equal to the volume integral +of the divergence oŸ the vector, according to Gauss” theorem (Eq. 3.18). Gauss' +theorem, applied to x Œ, says +J (V x Ở)„ da = J V-(Vx Ơ)dV. (3.40) +closed volumne +surface inside +So we conclude that the second integral must also be zero: +J W:(Vx€C)dV =0, (3.41) +volume +and this is true for any vector field whatever. Since Eq. (3.41) is true for ng +0olurne, it must be true that at e0erw po¿n‡ In space the integrand is zero. We +W:(VxŒC)=0, always. +But this is the same result we got from vector algebra in Section 2-7. Now we +begin to see how everything fits together. +3-8 Summary +Let us summarize what we have found about the vector calculus. These are +really the salient points of Chapters 2 and 3: +1. The operators Ø/Øz, Ø/Øụ, and Ø/Øz can be considered as the three com- +ponents of a vector operator V, and the formulas which result from vector +algebra by treating this operator as a vector are cOrrect: +8 Ø8 Ô +V=|[—.-.—]- +9z Ôu' Øz +--- Trang 54 --- +2. The diference of the values of a scalar field at Ewo points is equal to the line +Integral of the tangential component of the gradient of that scalar along +any curve at all bebween the first and second points: +(2) — (1) = J Vụ - da. (3.42) +3. The surface integral of the normal component of an arbitrary vector over a +closed surface is equal to the integral of the divergence of the vector over +the volume interior to the surface: +J C - nda = J V-CdV. (3.43) +closed volume +surface inside +4. The line integral of the tangential component of an arbitrary vector around +a closed loop is equal to the surface integral of the normal component of +the curl of that vector over any surface which is bounded by the loop: +J C -ds= J (VđxCŒC):-nda. (3.44) +boundary surface +--- Trang 55 --- +Mlocfrostqaffe©s +4-1 Statics +We begin now our detailed study of the theory of electromagnetism. All of 4-41 Statics +electromagnetism is contained in the Maxwell equations. 4-2_ Coulomb3s law; superposition +Maszuells equalions: 4-3 blectric potential +ÿ.E—#, (41) +4 E=-Vớ +«0 4-5 The fux of E +VxE= _- (4.2) 4-6 Gauss' law; the divergence of # +ý : 4-7 Eield ofa sphere of charge +cẦVxB-= _ + = (4.3) 4-8. Field lines; equipotential surfaces +V.B-=0. (4.4) +The situations that are described by these equations can be very complicated. +W©e will consider first relatively simple situations, and learn how to handle them +before we take up more complicated ones. 'Phe easiest circumstance to treat 1s +one in which nothing depends on the time——called the s#af#c case. All charges Reuicu: Chapters 13 and 14, Vol. T, +are permanently ñxed in space, or ¡if they do move, they move as a steady fow Work and Potential Energụ +in a circuit (so ø and 7 are constant in tỉme). In these circumstances, all of the +terms in the Maxwell equations which are time derivatives of the field are zero. +In this case, the Maxwell equations become: +blectrostatics: +W.E=Ú, (4.5) „— 10 +€0 €oC“ — mm +VxE-=0. (4.6) _? ox109 +- 47€o +Magnetostalics: : [eo] — coulomb2/newton-meter2 +VxB=-”=, (4.7) +V.B-=0. (4.8) +You will notice an interesting thing about this set of four equations. It can +be separated into two pairs. The electric fñeld # appears only in the first two, +and the magnetic ñeld Ö appears only in the second two. “The two fields are +not interconnected. This means that clecfricit and rnagnelism are đístinct +phenomena so long œs charges and curren‡s are static. 'Phe interdependence of E +and #Ö does not appear until there are changes in charges or currents, as when a +condensor is charged, or a magnet moved. Only when there are sufficiently rapid +changes, so that the time derivatives in Maxwell's equations become significant, +will E and #Ö depend on each other. +Now ïf you look at the equations of statics you will see that the study of +the ©wo subjects we call electrostatics and magnetostatics is ideal from the +point of view of learning about the mathematical properties of vector fields. +tlectrostatics is a neat example oŸ a vector fñeld with zero curL and a giuen +điuergence. Magnetostatics is a neat example of a fñield with zero điuergence +and a giuen curi. “he more conventional—and you may be thinking, more +satisfactory——wawy of presenting the theory of electromagnetism is to start fñrst +with electrostatics and thus to learn about the divergence. Magnetostatics and +the curl are taken up later. Einally, electricity and magnetism are put together. +--- Trang 56 --- +We have chosen to start with the complete theory of vector calculus. Now we +shall apply it to the special case of electrostatics, the field of # given by the frst +pair of equations. +We will begin with the simplest situations—ones in which the positions of all +charges are specifed. If we had only to study electrostatics at this level (as we +shall do in the next two chapters), life would be very simple——in fact, almost trivial. +tverything can be obtained from Coulomb's law and some integration, as you +will see. In many real electrostatic problems, however, we do not. knou, initially, +where the charges are. We know only that they have distributed themselves in +ways that depend on the properties of matter. The positions that the charges take +up depend on the # field, which in turn depends on the positions of the charges. +Then things can get quite complicated. lf, for instance, a charged body is brought +near a conductor or insulator, the electrons and protons in the conductor or +insulator will move around. "The charge density ø in Eq. (4.5) may have one +part that we know about, from the charge that we brought up; but there will +be other parts om charges that have moved around in the conductor. And all +of the charges must be taken into account. Ône can get into some rather subtle +and interesting problems. So although this chapter is to be on electrostatics, 1% +will not cover the more beautiful and subtle parts of the subject. It will treat +only the situation where we can assume that the positions of all the charges are +known. Naturally, you should be able to do that case before you try to handle +the other ones. +4-2 Coulomb?s law; superposition +It would be logical to use Bqs. (4.5) and (4.6) as our starting points. It will +be easier, however, if we start somewhere else and come back to these equations. +The results will be equivalent. We will start with a law that we have talked +about before, called Coulombs law, which says that between two charges at rest +there is a force directly proportional to the produect of the charges and inversely +proportional to the square of the distance between. The force is along the straight +line from one charge to the other. +Coulomb ˆs lau: 1 +EFìi=—— “P2ej=—E. (4.9) +47€0 Tía +F'`\ is the force on charge gị, €a is the unit vector in the direction £o gi from qa, +and ra is the distance between g¡ and 4s. The force F2 on qg› is equal and +opposite to F'. +The constant of proportionality, for historical reasons, is written as 1/47eo. +In the system of units which we use—the mks system——it is defned as exactÌy +10—T times the speed of light squared. Now since the speed of light is approximately +3 x 10Ÿ meters per second, the constant is approximately 9 x 10, and the unit +turns out to be newton-meter2 per coulomb or volt-meter per coulomb. +" = 1072 (by defnition) += 9.0 x 10 (by experiment). (4.10) +Unit: newton-meter2/coulomb, +or volt-meter/coulomb. +'When there are more than two charges present—the only really interesting +times—we must supplement Coulombs law with one other fact of nature: the +force on any charge is the vector sum of the Coulomb forces from each of the +other charges. 'This fact ¡is called “the principle of superposition.” 'Phat”s all +there is to electrostatics. IÝ we combine the Coulomb law and the principle of +superposition, there is nothing else. Equations (4.5) and (4.6)——the electrostatic +equations——say no more and no less. +'When applying Coulombs law, it is convenient to introduce the idea of an +electric fñeld. We say that the fñeld (1) is the force per un#t charge on gi (due +--- Trang 57 --- +to all other charges). Dividing Eq. (4.9) by gi, we have, for one other charge +besides q, +EU)=—— ““es. (4.11) +47€o T1s +Also, we consider that (1) describes something about the point (1) even i ø +were not there—assuming that all other charges keep their same positions. We +say: (1) is the electric feld a# the point (1). +The electric fñeld # is a vector, so by Eq. (4.11) we really mean three equa- +tỉons—one for each component. Writing out explicitly the z-component, Eq. (4.11) +Œ2 %1 — #2 +„(1,1 Z1) = — — =>. 4.12 +(#1, 1; Z1) 4meo [(# — #2)2 + (\ — a)2 + (z4 — z2)2]3/2 ( ) +and similarly for the other components. +TÍ there are many charges present, the field # at any point (1) is a sum of +the contributions from each of the other charges. Each term of the sum will look +like (4.11) or (4.12). Letting g; be the magnitude of the jth charge, and 7+; the +displacement from g; to the point (1), we write +:(1) = —— -3x-€l;. 4.13 +ú) 2 47co TỶ, #17 ) +Which means, of course, +1 q;(#1 — #¿}) +E„a,.z)= ` =—————bnPDS— 7? (414 +——".—=.ốằ. ..e. 5a +and so on. +Often it is convenient to ignore the fact that charges come in packages like +electrons and protons, and think of them as being spread out in a continuous +smear——or in a “distribution,” as it is called. "This is O.K. so long as we are +not interested in what is happening on too small a scale. We describe a charge +distribution by the “charge density,” ø(z, , z). IÝ the amount of charge in a small +volume AVW2 located at the point (2) is Aøa, then ø is delned by +Aq = p(2)A. (4.15) +To use Coulombs law with such a description, we replace the sums of Eqs. (1); Éxi, vì, Z1) +(4.13) or (4.14) by integrals over all volumes containing charges. Then we have Họ +Ø(x. Y. Z) +1 2 dV; +E()=—— J 20)e d1, (4.16) +47€o Tía +space ~—— +Some people prefer to write +ca 12 (2); (xa, va, Z2) +12 — ) +T12 +where 71a is the vector displacement fo (1) fom (2), as shown in Fig. 4-1. The Fig. 4-1. The electric field E at point (1), +integral for is then written as from a charge distribution, ¡is obtained from +an integral over the distribution. Point (1) +EQ)= 1 J p0); Là Š (417) could also be inside the distribution. +47€o Tịa +When we want to calculate something with these integrals, we usually have +to write them out in explicit detail. For the z-component of either Eq. (4.16) +or (4.17), we would have +(#1 — #2)0(%a, 9a, Z2) d+a dụa dza +E„(41,91, Z1) = ————— =-—¬-—_=na-: 4.18 +(#1, 1, Z1) ị 4mco[(#+ — #a)2 + (0 — 9s)2 + (z\ — za)2]3/2 ( ) +--- Trang 58 --- +W© are not goïng to use this formula mụch. We write it here only to emphasize +the fact that we have completely solved all the electrostatic problems in which +we know the locations of all of the charges. Given the charges, what are the +fields? Ansuer: Do this integral. So there is nothing to the subject; i is just a +case of doïng complicated integrals over three dimensions—strictly a job for a +computing machinel +With our integrals we can fnd the ñelds produced by a sheet of charge, from a +line of charge, from a spherical shell of charge, or from any specifed distribution. +lt is important to realize, as we go on to draw field lines, to talk about potentials, +or to calculate divergences, that we already have the answer here. It is merely a +matter of it being sometimes easier to do an integral by some clever guesswork +than by actually carrying ¡it out. The guesswork requires learning all kinds of +strange things. In practice, it might be easier to forget trying to be clever and +always to do the integral directly instead of beïng so smart. We are, however, +goïing to try to be smart about it. We shall go on to discuss some other features +of the electric ñeld. +4-3 Electric potential +First we take up the idea of electric potential, which is related to the work +done in carrying a charge from one poïnt to another. There is some distribution F +of charge, which produces an electric feld. We ask about how much work I1 b +would take to carry a small charge from one place to another. "The work done ; +agœ”nst the electrical forces in carrying a charge along some path is the œegaf2ue one path +of the component of the electrical force in the direction of the motion, integrated another +along the path. lIf we carry a charge from point ø to point b, path +W= -ƒ l-ds, Fig. 4-2. The work done in carrying a +š charge from a to b ¡is the negative of the +where #' is the electrical force ơn the charge at each point, and đs is the điferential integral of F - ds along the path taken. +vector displacement along the path. (See Fig. 4-2.) +lt is more interesting for our purposes to consider the work that would be +done in carrying øne wn#‡ of charge. Then the force on the charge is numerically +the same as the electric field. Calling the work done against electrical forces in +this case W/(unit), we write +W(unit) = -ƒ đ - ds. (4.19) +Now, in general, what we get with this kind of an integral depends on the path +we take. But if the integral of (4.19) depended on the path from ø to Ù, we could +get work out of the field by carrying the charge to ö along one path and then +back to ø on the other. We would go to ö along the path for which Wƒ is smaller +and ðøck along the other, getting øu more work than we put ?n. +There is nothing impossible, in principle, about getting energy out of a ñeld. +W© shall, in fact, encounter fields where ït is possible. It could be that as you +move a charge you produce forces on the other part of the “machinery.” If the +“machinery” moved against the force it would lose energy, thereby keeping the +total energy in the world constant. For elecfrostatics, however, there is no such +“machinery.” We know what the forces back on the sources of the fñeld are. They +are the Coulomb forces on the charges responsible for the ñeld. If the other charges +are fxed in position——as we assume in elec‡rostatics only——these back forces can +do no work on them. “Phere is no way to get energy from them——provided, of +course, that the prineiple of energy conservation works for electrostatic situations. +We believe that it will work, but letˆs Just show that it must follow from Coulomb”s +law of force. +W©e consider fñrst what happens in the fñeld due to a single charge g. Let +point ø be at the distance r„ from g, and point 0 at r,. NÑow we carry a diferent +charge, which we will call the “test” charge, and whose magnitude we choose to +--- Trang 59 --- +be one unit, from ø to Ö. Let”s start with the easiest possible path to calculate. +W©e carry our test charge first along the arc of a circle, then along a radius, as +shown in part (a) of Fig. 4-3. Ñow on that particular path it is chỉld?s play to +fnd the work done (otherwise we wouldnt have picked it). EFirst, there is no +work done at all on the path from ø to a/. The feld is radial (rom Coulomb)s +law), so it is at right angles to the direction of motion. Next, on the path from +œ' to b, the field is in the direction of motion and varies as 1/rz?. Thus the work +done on the test charge in carrying it from ø to b would be P2 +b b (a) +-[ Bs=-< [ Set (Ta): (4.20) +P 4mreo Jạ¿ r2 47€0 \Ta Tp +Now let”s take another easy path. Eor instance, the one shown in part (b}) Ầ +of Eig. 4-3. It goes for awhile along an are of a circle, then radially for awhile, +then along an arc again, then radially, and so on. Every time we go along the ệ + +circular parts, we do no work. Every time we go along the radial parts, we must +just integrate 1/r?. Along the first radial stretch, we integrate from r„ to r4, b +then along the next radial stretch from 7x to r„, and so on. 'Phe sum of all +these integrals is the same as a single integral directly from r„ to r;. W© get the +same answer for this path that we did for the first path we tried. It is clear that +we would get the same answer for øn% path which is made up of an arbitrary (b) +number of the same kinds of pieces. 2" +What about smooth paths? Would we get the same answer? We discussed : +this point previously in Chapter 13 of Vol. I. Applying the same arguments used , š +there, we can conclude that work done in carrying a unit charge from œ to Ö is 3 +independent of the path. Fig. 4-3. In carrying a test charge from +a to b the same work ¡is done along either +M. b path. +=— J E - d3. +Since the work done depends only on the endpoints, it can be represented as +the difÑference between two numbers. We can see this in the following way. Let's +choose a reference point g and agree to evaluate our integral by using a path +that always goes t0aw øƒ poïnt Pụ. Let ð(a) stand for the work done against +the field in going from Pụ to point ø, and let ó(b) be the work done in going +rơm Pụ to point b (Eig. 4-4). The work in going foø Pb from ø (on the way to Ù) +is the negative of j(ø), so we have that +— J - ds = ó(Ù) — oð(a). (4.21) W(a —> b)= Ó(b)— Ó(3) 2p +Since only the diference in the function ó at two points is ever involved, we W(P; —¬ b) = ó(b) +do not really have to specify the location of Pụ. Once we have chosen some +reference point, however, a number ø is determined for ønự point in space; ở 1s . +then a scalar ficld. It is a funection of ø, , z. We call this scalar function the W(f — a) = j(2) P +clectrostatic potential at any point. +Fig. 4-4. The work done In going along +Elecrostatic potenHial: F any path from a to b ¡is the negative of the +work from some point fụ to a plus the work +9Œ?) =— IR 1ịds. (4.22) from to b. +For convenience, we will often take the reference point at infnity. Then, for a +single charge at the origin, the potential ó is given for any point (z, , z)——using +Eq. (4.20): +Ó(#, 1,z) = 1e ¬ (4.23) +7€0 T' +The electric ñeld from several charges can be written as the sum of the electric +field from the first, from the second, from the third, etc. When we integrate +the sum to find the potential we get a sum of integrals. Each of the integrals +--- Trang 60 --- +1s the negative of the potential from one of the charges. We conclude that +the potential ó from a lot of charges is the sum of the potentials from all the +individual charges. 'Phere is a superposition principle also for potentials. Using +the same kind of arguments by which we found the electric fñeld from a group of +charges and for a distribution of charges, we can get the complete formulas for +the potential ¿ at a poin we call (1): +1)= ——_-= 4.24 +90) =À đun rụ) (429 +2A0 =¡c | (425) +s. 47co T12 l l +Remember that the potential ¿ has a physical signifcanee: it is the potential +energy which a unit charge would have If brought to the specified poïnt in space +from some reference poiïnt. +4-4 E— —Vọ +'Who cares about ý? Eorces on charges are given by #, the electric fñeld. The +poïnt is that # can be obtained easily from j@—Ït is as easy, in fact, as taking a +derivative. Consider bwo points, one at # and one at (+ Az), but both at the +same and z, and ask how much work is done in carrying a unit charge from +one point to the other. The path is along the horizontal line from # to # + Az. +'The work done is the diference in the potential at the two points: +AW = j(œ + Az,U, z) — Ó(, Ù; z) — ðz Az., +But the work done against the field for the same path is +AI == [E-ds= —E, An, +We see that 0ó +„ =——. 4.26 +5x (4.26) +Similarly, #„ = —Ø0/Ø, E; = —Øó/Ôz, or, summarizing with the notation of +vector analysis, +—=-—Vọ. (4.27) +This equation is the diferential form of Eq. (4.22). Any problem with specified +charges can be solved by computing the potential from (4.24) or (4.25) and +using (4.27) to get the field. Equation (4.27) also agrees with what we found +from vector calculus: that for any scalar field ¿ +J Wộ- ds = ó(Ù) — ð(a). (4.28) +According to Eq. (4.25) the scalar potential ó is given by a three-dimensional +integral similar to the one we had for #. Is there any advantage to computing ¿ +rather than #? Yes. 'There is only one integral for ó, while there are three +integrals for E—because it is a vector. Furthermore, l/r is usually a little easier +to integrate than #/zỞ. It turns out in many practical cases that it is easier to +calculate ó and then take the gradient to fñnd the electric fñeld, than it is to +evaluate the three integrals for #. It is merely a practical matter. +There is also a deeper physical signifcance to the potential ó. We have +shown that / of Coulombˆs law is obtained from ##j = — grad ở, when ở is given +by (4.22). But if E is cqual to the gradient of a scalar field, then we know from +the vector calculus that the curl of E# must vanish: +VxE-=0. (4.29) +--- Trang 61 --- +But that is just our second fundamental equation of electrostatics, Eq. (4.6). We +have shown that Coulomb'°s law gives an # field that satisfes that condition. So +far, everything is all right. +W© had really proved that V x was zero before we defñned the potential. +We had shown that the work done around a closed path is zero. That is, that +‡ E;-ds—=0 +for ønụ path. We saw in Chapter 3 that for any such ñeld V x # must be zero +everywhere. The electric field in electrostatics is an example of a curl-free field. +You can practice your vector calculus by proving that VW x # is zero in a +diferent way——by computing the components of V x # for the field of a poïnt +charge, as given by Eq. (4.11). IÝ you get zero, the superposition principle says +you would get zero for the field of any charge distribution. +W© should point out an important fact. EFor any rad¿aøl force the work done +1s independent of the path, and there exists a potential. lf you think about +it, the entire areument we made above to show that the work integral was +independent of the path depended only on the fact that the force from a single +charge was radial and spherically symmetric. It did not depend on the fact that +the dependenece on distance was as 1/r2—there could have been any r dependence. +'The existence of a potential, and the fact that the curl of # is zero, comes really +only from the s/mmetrw and đirection of the electrostatic forces. Because of this, +Eq. (4.28)—or (4.29)—can contain only part of the laws of electricity. +4-5 The fux of F +W© will now derive a fñeld equation that depends specifcally and directly on +the fact that the force law is inverse square. That the fñeld varies inversely as +the square of the distance seems, for some people, to be “only natural,” because +“that ”s the way things spread out.” Take a light source with light streaming out: +the amount of light that passes through a surface cut out by a cone with its +apex at the source is the same no matter at what radius the surface ¡is placed. lt +must be so if there is to be conservation of light energy. The amount of light per +unit area—the intensity——must vary inversely as the area cut by the cone, 1.e., +inversely as the square of the distance from the source. Certainly the electric ñeld +should vary inversely as the square of the distance for the same reasonl But there +is no such thing as the “same reason” here. Nobody can say that the electric ñeld +measures the ow of something like light which must be conserved. Jƒ we had +a “model” of the electric field in which the electric field vector represented the +direction and speed—say the current—of some kind of little “bullets” which were +fying out, and if our model required that these bullets were conserved, that none +could ever disappear once it was shot out of a charge, then we might say that +we can “see” that the inverse square law is necessary. Ôn the other hand, there +would necessarily be some mathematical way to express this physical idea. If the +electric ñeld œere like conserved bullets going out, then iÿ would vary inversely +as the square of the distance and we would be able to describe that behavior by +an equation—which is purely mathematical. Now there is no harm in thinking +this way, so long as we do not say that the electric fñeld 2s rmade out of bullets, +but realize that we are using a model to help us fñnd the right mathematics. +Suppose, indeed, that we imagine for a moment that the electric fñeld did +represent the fow of something that was conserved——everywhere, that is, except +at charges. (It has to start somewherel) We imagine that whatever it is fiows out +OŸ a charge into the space around. Tf E were the vector oŸ such a flow (as h is for +heat fow), it would have a 1/r2 dependence near a point source. NÑow we wish +to use ©his model to fnd out how to state the inverse square law in a deeper or +more abstract way, rather than sỉmply saying “inverse square.” (You may wonder +why we should want to avoid the direct statement of such a simple law, and want +instead to imply the same thing sneakily in a diferent way. Patiencel It will +turn out to be useful.) +--- Trang 62 --- +_. BC +Closed Surface S Eo _ 2 E +_Z x Tu c +@⁄ “ˆ Fig. 4-5. The flux of E out of the sur- @2 Fig. 4-6. The flux of E out of the sur- +Point Charge face S is zero. Point Charge face S is zero. +W© ask: What is the “fow” of E out of an arbitrary closed surface in the +neighborhood of a poïnt charge? First let's take an easy surface—the one shown +in Eig. 4-5. If the # feld ¡s like a Ñow, the net ow out of this box should be zero. +That is what we get if by the “flow” om this surface we mean the surface integral +of the normal component of ——that is, the Ñux of . Ôn the radial faces, the +normal component is zero. Ôn the spherical faces, the normal component Ï2„ is +Just the magnitude oŸ #—minus for the smaller face and plus for the larger face. +The magnitude of E decreases as 1/z2, but the surface area is proportional to r2, +so the product is independent ofz. The ñux of E into face a is Just cancelled by the ⁄⁄ Surface Š +ñux out of face 0. The total ñow out oŸ S is zero, which is to say that for this surface => +J E„ da =0. (4.30) V 7 +Next we show that the two end surfaces may be tilted with respect to the radial I8 +line without changing the integral (4.30). Although it is true in general, for our 2” +purposes it is only necessary to show that this is true when the end surfaces are @Œ +small, so that they subtend a small angle from the source—in fact, an infnitesimal +angle. In Fig. 4-6 we show a surface Š whose “sides” are radial, but whose “ends” : +. . . . Fig. 4-7. Any volume can be thought +are tilted. The end surfaces are not small in the fñgure, but you are to imagine of as completely made up of infinitesimal +the situation for very small end surfaces. Then the fñeld will be suficiently truncated cones. The flux of E from one +uniform over the surface that we can use just its value at the center. When we end of each conical segment is equal and +tilt the surface by an angle Ø, the area is increased by the factor 1/cosØ. But mạ, opposite to the flux from the other end. The +the component of # normal to the surface, is decreased by the factor cosØ. The total flux from the surface S is therefore +product 2 Aa is unchanged. The Hux out of the whole surface Š is still zero. zero. +Now it is easy to see that the ñux out of a volume enclosed by ømyw surface S +must be zero. Any volume can be thought of as made up of pieces, like that in +Fig. 4-6. The surface will be subdivided completely into pairs of end surfaces, +and since the fuxes in and out of these end surfaces cancel by pairs, the ©otal +ñux out of the surface will be zero. The idea is ilHustrated in Fig. 4-7. We have +the completely general result that the total ñux of E out oŸ am surface 5 in the +fñeld of a point charge 1s zero. +But noticel Our proof works only if the surface S9 does no‡ surrownd the +charge. What would happen if the point charge were ¿ns7de the surface? We +could stilH divide our surface into pairs of areas that are matched by radial lines =. +through the charge, as shown in Fig. 4-8. The Ñuxes through the two surfaces +are still equal—by the same arguments as before—only now they have the sœme % +sien. The ñux out of a surface that surrounds a charge is no£ zero. hen what +1s 1t? We can find out by a little trick. Suppose we “remove” the charge from ⁄ +the “inside” by surrounding the charge by a little surface S5” totally inside the +original surface Š, as shown in Fig. 4-9. Now the volume enclosed befeen the +two surfaces 9 and 5Š“ has no charge in it. The total Hux out of this volume E, ạ +(including that through Š”) is zero, by the arguments we have given above. The +arguments tell us, in fact, that the ñux zn£o the volume through 5” is the same Fig. 4-8. lf a charge is inside a surface, +as the fux outward through 5S. the flux out is not zero. +--- Trang 63 --- +W© can choose any shape we wish for S7, so let°s make it a sphere centered +on the charge, as in Fig. 4-10. Then we can easily calculate the Ñux through it. +Tf the radius of the little sphere is z, the value of # everywhere on its surface 1s Surface +_1 đ | [ ( ( +47co r3' Point Charge ( +Í-À. 4 +and is directed always normal to the surface. We find the total ñux through ®” if Surface \ Š \ \ \ +we multiply this normal component of by the surface area: Sở ¬v +Flux through the surface Š” = n #) (4mr7) = + (4.31) +47g r2 cọ” +a number independent of the radius of the spherel We know then that the Ñux +outward through ®Š is also g/cọ—a value independent of the shape of 8 so long Elg. 49. The flux through S is the same +as the charge g is inside. as the flux through S'. +We can write our conclusions as follows: +0; — goutside S +J Tín da — 4, q inside Š (4.32) +any surface Š €0 +Let”s return to our “bullet” analogy and see iŸit makes sense. Our theorem says +that the net flow of bullets through a surface is zero if the surface does not enclose +the gun that shoots the bullets. If the gun is enclosed in a surface, whatever size E +and shape it is, the number of bullets passing through is the same—it is given by +the rate at which bullets are generated at the gun. lt all seems quite reasonable for +conserved bullets. But does the model tell us anything more than we get simply by @s +writing Eq. (4.32)? No one has succeeded in making these “bullets” do anything +else but produce this one law. After that, they produce nothing but errors. That bò +is why today we prefer to represent the electromagnetic fñeld purely abstractly. +46 Gauss' law; the divergence oŸ É Fig. 4-10. The flux through a spherical +Our nice result, Eq. (4.32), was proved for a single point charge. NÑow suppose surface containing a point charge q is g/eo. +that there are two charges, a charge g¡ at one point and a charge ga at another. +The problem looks more difficult. "The electric fñeld whose normal component we +Integrate for the fux is the feld due to both charges. That is, if E represents +the electric fñeld that would have been produced by g¡ alone, and #2; represents +the electric field produced by q;¿ alone, the total electric field is # = E + Ea. +The ñux through any closed surface Š is +lứa + an) da = J 1„ da + J b22„ da. (4.33) +S S S +The fux with both charges present ¡is the ñux due to a single charge plus the Ñux +due to the other charge. If both charges are outside 5, the fux through Š is zero. +TÍ gi is inside 3Š but qs is outside, then the first integral gives g¡ /co and the second +integral gives zero. If the surface encloses both charges, each will give its contribu- +tion and we have that the fux is (gi + g2)/co. The general rule is clearly that the +total Ñux out of a closed surface is equal to the total charge #nside, divided by eo. +Our result is an important general law of the electrostatic fñeld, called Gauss' +Gaussˆ lau: ¬ +J Eạ da = sum of charges mmgide (4.34) +any closed +surface ý +J E-n da = ân (4.35) +any closed ` +surface S +--- Trang 64 --- +in: = » q;. (4.36) +inside S +Tf we describe the location of charges in terms of a charge density ø, we can +consider that each infñnitesimal volumne đV contains a “point” charge ødV. The +sum over all charges is then the integral +¬.... (137) +volume +inside S +trom our derivation you see that Gauss' law follows from the fact that the +exponent in Coulomb's law is exactly two. A 1/zỞ field, or any 1/r* fñeld with +m z# 2, would not give Gauss' law. So Gauss' law is just an expression, in +a diferent form, of the Coulomb law of forces between two charges. In fact, +working back from Gauss' law, you can derive Coulombs law. The two are quite +equivalent so long as we keep in mind the rule that the forces bebween charges +are radial. +W©e would now like to write Gauss' law in terms of derivatives. To do this, we +apply Gauss' law to an infinitesimal cubical surface. We showed in Chapter 3 +that the Ñux of E out of such a cube is V - times the volume đW of the cube. +The charge inside of đV, by the defñnition of ø, is equal to øđV, so Gauss” law +V-Eav =0, +ÿ.E=f. (4.38) +'The diferential form of Gauss' law is the first of our fundamental field equations +Of electrostatics, Eq. (4.5). We have now shown that the two equations of +electrostatics, Ðqs. (4.5) and (4.6), are equivalent to Coulomb?s law of force. We +will now consider one exarmnple of the use of Gauss' law. (We will come later to +many more exarmnples.) +4-7 Field of a sphere of charge +One of the difficult problems we had when we studied the theory of gravi- z ` E +tational attractions was to prove that the force produced by a solid sphere of / ` “é +matter was the same at the surface of the sphere as it would be ïf all the matter ⁄ \ +were concentrated at the center. For many years Newton didn't make public | 4 277 +his theory of gravitation, because he couldn”t be sure this theorem was true. Charge ⁄⁄⁄% R BS +W©e proved the theorem in Chapter 13 of Vol. I by doing the integral for the Distribution \_. Surface S +potential and then ñnding the gravitational force by using the gradient. NÑow we Ø NV Z⁄ +can prove the theorem in a most simple fashion. OÔnly this time we will prove — +the corresponding theorem for a uniform sphere of electrical charge. (Since the Fig. 4-11. Using Gauss' law to find the +laws of electrostatics are the same as those of gravitation, the same proof could field of a uniform sphere of charge. +be done for the gravitational field.) +W© ask: What is the electric feld at a point P anywhere outside the surface +of a sphere flled with a uniform distribution of charge? 5ince there is no “special” +direction, we can assume that # is everywhere directed away from the center of +the sphere. We consider an imaginary surface that is spherical and concentric +with the sphere of charge, and that passes through the point P (Fig. 4-11). Eor +this surface, the ñux outward 1s +. = E-4nT. +Gauss' law tells us that this ñux is equal to the total charge @Q of the sphere +(over co): +E-AnR? = kà +--- Trang 65 --- +_ — ¬— ¬~ +⁄ =1+— ` +⁄/ >ự » ` +/ ⁄ ¬" ` ` Lines of E +l Z#†ExéN \ +| l LÍ < +| C1 2 N/Í | +` ~- ⁄ ở = Constant +\ N — | —_ ⁄ L +" —_ _— ~Z +N ⁄ ⁄ +¬ ~ _ ~ +Fig. 4-12. Field lines and equipotential surfaces for a positive point charge. +tk= mm. (4.39) +which is the same formula we would have for a point charge Q. We have proved +Newton's problem more easily than by doïng the integral. It is, of course, a false +kind of easiness—it has taken you some time to be able to understand Gauss' +law, so you may think that no time has really been saved. But after you have +used the theorem more and more, it begins to pay. Ï% is a question of eficieney. +4-8 Eield lines; equipotential surfaces +W©e would like now to give a geometrical description of the electrostatic ñeld. +The two laws of electrostatics, one that the ñux is proportional to the charge +inside and the other that the electric fñeld is the gradient of a potential, can also +be represented geometrically. We illustrate this with two examples. +First, we take the feld of a point charge. We draw lines in the direction of +the fñeld——lines which are always tangent to the feld, as in Eig. 4-12. These are +called field iines. 'Phe lines show everywhere the direction of the electric vector. +But we also wish to represent the magnitude of the vector. We can make the +rule that the strength of the electric ñeld will be represented by the “density” of +the lines. By the density of the lines we mean the number of lines per unit area +through a surface perpendicular to the lines. With these two rules we can have +a picture of the electric fñeld. For a point charge, the density of the lines must +decrease as 1/72. But the area of a spherical surface perpendicular to the lines +at any radius ? mereases as r2, so if we always keep the same @wmber of lines +for aÏỦ distances from the charge, the đenszt will remain in proportion to the +magnitude of the field. We can guarantee that there are the same number of +lines at every distance iŸ we insist that the lines be confnuous—that once a line +1s sbarted from the charge, it never stops. In terms of the field lines, Gauss'ˆ law +says that lines should start only at plus charges and stop at minus charges. The +number which ieøe a charge q must be equal to g/eo. +--- Trang 66 --- +~T g > ~ _Jm ` +/ X. » ` +/ ⁄ X \ +) Tai. ắ +Ị X\ƒ; Ñ J +` 2 RBmBRRm. Z2: +| XI mmass l | +\ C 2 / +` ` Z ⁄ +¬ ¬ —— 7 >> —_ ” < +Fig. 4-13. Field lines and equipotentials for two equal and opposite point charges. +Now, we can fnd a similar geometrical picture for the potential ó. 'Phe easiest +way to represent the potential is to draw surfaces on which ở is a constant. WWe +call them egu2potential surfaces—surfaces of equal potential. Now what is the +geometrical relationship of the equipotential surfaces to the field lines? The +electric field is the gradient of the potential. 'Phe gradient is in the direction +of the most rapid change of the potential, and is therefore perpendicular to an +equipotential surface. If # were noøf perpendicular to the surface, it would have +a component 7w the surface. 'Phe potential would be changing in the surface, but +then it wouldnˆt be an equipotential. 'The equipotential surfaces must then be +everywhere at right angles to the electric field lines. +For a point charge all by itself, the equipotential surfaces are spheres centered A Note about Units +at the charge. We have shown in Fig. 4-12 the intersection of these spheres with +a plane through the charge. Quanti6 mái +As a second example, we consider the feld near two equal charges, a positive # newton +one and a negative one. To get the feld is easy. The feld is the superposition of @ coulomb +the felds from each of the two charges. So, we can take two pictures like Fig.4-12 È Tneter +and superimpose them——impossiblel Then we would have field lines crossing each wW 3 joule 3 +other, and that”s not possible, because # can't have #uo directions at the same p~ Q/1 2¬ coulomb/ motet 3 +. . . : . . : 1/eo ~ FL2/Qˆ— newton-meter“/coulomb +point. The disadvantage of the field-line picture is now evident. By geometrical E~F/Q newton/coulomb +arguments ït is Impossible to analyze in a very simple way where the new lines ~ W/Q joule/coulomb = volt +go. From the £wo independent pictures, we can” get the combined picture. The „. j/L volt/meter +principle of superposition, a simple and deep principle about electric fields, does 1/cạ ~. Z12/Q — volt:meter/coulomb +not have, in the field-line picture, an easy representation. +The field-line picture has Its uses, however, so we might still like to draw +the picture for a pair of equal (and opposite) charges. If we calculate the ñelds +from E4q. (4.13) and the potentials from (4.24), we can draw the fñeld lines and +equipotentials. Figure 4-13 shows the result. But we first had to solve the +problem mathematicallyl +--- Trang 67 --- +Appliceafiorm ©@Ÿ Ấxerrssˆ Lee +5-1 Electrostatics is Gauss' law plus... +'There are t©wo laws of electrostatics: that the fux of the electric ñeld from a 5-1 Electrostatics is Gauss` law +volume is proportional to the charge inside—=auss' law, and that the circulation of plus ... +the electric field is zero——# is a gradient. EFrom these two laws, all the predictions 5-2 _ Equilibrium in an electrostatic +of electrostatics follow. But to say these things mathematically is one thing; to use ñeld +them easily, and with a certain amount oŸ ingenuity, is another. In this chapter 5-3 Equilibrium with conductors +we will work through a number of calculations which can be made with Gauss' " +law directly. We will prove theorems and describe some efects, particularly in 5-4 Stabilty of Aloms +conduetors, that can be understood very easily from Causs' law. Causs lawby 55 The ñeld ofa line charge +itself cannot give the solution of any problem because the other law must beobeyed — 5-6 Á sheet of charge; two sheets +too. So when we use Gauss' law for the solution of particular problems, we will 5-7 A sphere of charge; a spherical +have to add something to it. We will have to presuppose, for instance, some idea shell +of how the field looks——based, for example, on arguments of symmetry. Ôr we may 5-8 Is the feld of a point charge +have to introduece specifically the idea that the field is the gradient of a potential. exactly 1/r?? +5-9 The fields ofa conductor +5-2 Equilibrium in an electrostatic ñeld 5-10 The feld in a cavity of a +Consider first the following question: When can a point charge be in stable conductor +mmechanical equilibrium in the electric field of other charges? As an example, +imagine three negative charges at the corners of an equilateral triangle in a +horizontal plane. Would a positive charge placed at the center of the triangle +remain there? (It will be simpler if we ignore gravity for the moment, although +including it would not change the results.) The force on the positive charge is +zero, but is the equilibrium stable? Would the charge return to the equilibrium +position if displaced slightly? “The answer is no. +There are øø points of stable equilibrium in ønø% electrostatic field——except +right on top of another charge. Ũsing Gauss' law, it is easy to see why. Eirst, for +a charge to be in equilibrium at any particular point Fụ, the fñeld must be zero. +Second, if the equilibrium is to be a stable one, we require that if we move the +charge away from ụ in ømy direction, there should be a restoring force directed +opposite to the displacement. The electric field at øÏi nearby points must be +pointing inward—toward the point Pụ. But that is in violation of Gauss' law If +there is no charge at ụ, as we can easily see. +Consider a tiny imaginary surface that encloses ụ, as in Eig. 5-1. lƒ the _Ằ +electric ñeld everywhere in the vicinity is pointed toward Tù, the surface integral rÁ ` +of the normal eomponent is certainly not zero. For the case shown in the ñgure, lo Pạ 1 +the ñux through the surface must be a negative number. But Gauss' law says < x 2naginary +that the ñux of electric fñeld through any surface is proportional to the total ¬ J _x⁄ surrounding fb +charge inside. lf there is no charge at ụ, the fñeld we have imagined violates ¬ +Gauss' law. It is impossible to balance a positive charge in empty space—at Fig. 5-1. lf Fạ were a position of stable +a poïnt where there is not some negative charge. AÁ positive charge cøn be in equllibrium for a positive charge, the electric +equilibrium if it is in the middle of a distributed negative charge. Of course, — field eerywhere in the neighborhood would +the negative charge distribution would have to be held in place by other than point toward . +electrical forcesl +Our result has been obtained for a point charge. Does the same coneclusion +hold for a complicated arrangement of charges held together in fxed relative +positions—with rods, for example? We consider the question for two equal charges +fñxed on a rod. Is ít possible that this combination can be in equilibrium in some +electrostatic ñeld? 'The answer is again no. The #ø/al force on the rod cannot be +restoring for displacements in every direction. +--- Trang 68 --- +Call ' the total force on the rod in any position—F' is then a vector ñeld. +Following the argument used above, we conclude that at a position of stable +equilibrium, the divergence of ' must be a negative number. But the total force +on the rod is the first charge times the fñield at its position, plus the second charge +times the field at its position: +†#' =giEì +qsE:. (5.1) +'The divergence of F' is given by +V:F=4i(V:E¡)+q›(V - E›). +Tí each of the two charges g¡ and g¿ is in free space, both V - Eạ and VW- Hs are +zoero, and W - È! is zero—not negative, as would be required for equilibrium. You +can see that an extension of the argument shows that no rigid combination of any +number of charges can have a position of stable equilibrium in an electrostatic +field in free space. +— CC —x> @ x~- — +Fig. 5-2. A charge can be in equilibrium `^⁄ +If there are mechanical constraints. | H +Now we have not shown that equilibrium i¡s forbidden if there are pivofS Or +other mechanical constraints. As an example, consider a hollow tube in which a +charge can move back and forth freely, but not sideways. Now it is very easy to +devise an electric feld that points inward at both ends of the tube 1f it ¡s allowed +that the ñeld may point laterally outward near the center of the tube. We simply +place positive charges at each end of the tube, as in Eig. 5-2. There can now be +an equilibrium point even though the divergence of is zero. The charge, of +course, would not be in stable equilibrium for sideways motion were it not for +“nonelectrical” forces from the tube walls. +5-3 Equilibrium with conductors +'There is no stable spot in the fñield of a system of fñxed charges. What about +a system of charged conductors? Can a system of charged conductors produce a +fñeld that will have a stable equilibrium poiïnt for a point charge? (We mean at +a point other than on a conductor, oŸ course.) You know that conductors have +the property that charges can move freely around in them. Perhaps when the +point charge is displaced slightly, the other charges on the conductors will move +in a way that will give a restoring force to the point charge? 'Phe answer is still +no—although the proof we have just given doesn”t show it. The proof for this +case is more difficult, and we will only indicate how i% goes. +flirst, we note that when charges redistribute themselves on the conductors, +they can only do so ïiŸ their motion decreases their total potential energy. (Some +energy is losb to heat as they move in the conductor.) Ñow we have already +shown that if the charges producing a fñeld are s/aiionar, there is, near any +zero point ụ in the fñeld, some direction for which moving a point charge away +from ụ will decrease the energy of the system (since the force is a+0øy from ). +Any readjustment of the charges on the conductors can only lower the potential +energy still more, so (by the principle of virtual work) their motion will only +#ncrease the force in that particular direction away from ụ, and not reverse it. +Our conclusions do not mean that it is not possible to balance a charge by +electrical forces. It is possible iŸ one is willing to control the locations or the sizes +of the supporting charges with suitable devices. You know that a rod standing +on its point in a gravitational feld is unstable, but this does not prove that 1 +cannot be balanced on the end of a ñnger. 5imilarly, a charge can be held in +one spot by electric fñields if they are 0arable. But not with a passive—that is, a +stafic——system. +--- Trang 69 --- +5-4 Stability of atoms +Tí charges cannot be held stably in position, it is surely not proper to imagine Tin +matter to be made up of static poin# charges (electrons and protons) governed FHrR--R------ UNIFORM SPHERE +only by the laws of electrostatics. Such a static confguration is impossible; it tr. Kĩ AnQ T +would collapsel Bnnannnnrsxnnnnninn +lt was once suggested that the positive charge oŸ an atom could be distributed 'FHh—#=-- NEGATIVE CHARGE +uniformly in a sphere, and the negative charges, the electrons, could be at rest LTERT—T-——=-- concenwrnAreD +inside the positive charge, as shown in Eig. 5-3. This was the first atomic model, ESsss5nssnninnsm AT THE CENTER +proposed by Thomson. But Rutherford concluded from the experiment of Geiger Tnnnnnnnnrnn +and Marsden that the positive charges were very much concentrated, in what he +called the nucleus. Thomson'”s static model had to be abandoned. Rutherford and Fig. 5-3. The Thomson model of an atom. +Bohr then suggested that the equilibrium might be dynamic, with the electrons +revolving in orbits, as shown in Fig. 5-4. The electrons would be kept from falling +in toward the nucleus by their orbital motion. We already know at least one +difficulty with this picture. With such motion, the electrons would be accelerating +(because of the circular motion) and would, therefore, be radiating energy. They +would lose the kinetic energy required to stay in orbit, and would spiral in toward +the nucleus. Again unstablel +'The stability of the atoms is now explained in terms of quantum mechanics. (Ở) POSITIVE NUCLEUS +'The electrostatic forces pull the electron as close to the nucleus as possible, but © AT THE CENTER +the electron is compelled to stay spread out in space over a distance given by C53), +the uncertainty principle. lÝ it were confined in too small a space, it would have = +yĐ p pace, © +a great uncertainty in momentum. But that means that it would have a high ⁄ _—NEGATIVE +expected energy——which it would use to escape from the electrical attraction. œ PLANETARV OEBITS +The net result is an electrical equilibrium not too diferent from the idea of +Thomson—only it is the øegøf#ioe charge that is spread out (because the mass of +the electron is so mụuch smaller than the mass of the probon). Fig. 5-4. The Rutherford-Bohr model of +an atom. +5-5 The fñeld of a line charge +Gauss' law can be used to solve a number of electrostatic feld problems +involving a special symmetry——usually spherical, cylindrical, or planar symmetry. +In the remainder of this chapter we will apply €Gauss' law to a few such problems. +The ease with which these problems can be solved may give the misleading +Impression that the method is very powerful, and that one should be able to go +on to many other problems. It is unfortunately not so. One soon exhausts the +list oŸ problems that can be solved easily with Gauss' law. In later chapters we +will develop more powerful methods for investigating electrostatic fields. +As our first example, we consider a system with cylindrical symmetry. Suppose +that we have a very long, uniformly charged rod. By this we mean that electric +charges are distributed uniformly along an indefinitely long straight line, with the +charge À per unit length. We wish to know the electric feld. 'Phe problem can, +. . . . E +of course, be solved by integrating the contribution to the fñield from every part ¬ +of the line. We are goïng to do i% without integrating, by using Gauss' law and +some guesswork. First, we surmise that the electric fñeld will be directed radially +outward from the line. Any axial component from charges on one side would be +accompanied by an equal axial component from charges on the other side. “The re- 7 +sult could only be a radial fñeld. It also seems reasonable that the fñeld should have Ề +the same magnitude at all points equidistant from the line. This is obvious. (It GAUSSIAN |" +may not be easy to prove, but iE is true iŸ space is symmetric—as we believe i is.) SURPACE CHARGE +W© can use Gauss” law in the following way. We consider an #naginarw surface +in the shape of a cylinder coaxial with the line, as shown in Fig. 5-5. According Fig. 5-5. A cylindrical gaussian surface +to Gauss' law, the total fux of # from this surface is equal to the charge inside coaxial with a line charge. +divided by co. Since the field is assumed to be normal to the surface, the normal +component is the magnitude of the field. Let”s call it #. Also, let the radius of +the cylinder be z, and its length be taken as one unit, for convenience. The Ñux +through the cylindrical surface is equal to # times the area of the surface, which +is 27r. The Ñux through the two end faces is zero because the electric field is +--- Trang 70 --- +tangential to them. "The total charge inside our surface is just À, because the +length of the line inside is one unit. Gauss' law then gives +1-2mr = À/eo, +E=_——. (5.2)  +27cogT \ +The electric fñeld of a line charge depends inversely on the ƒirs power of the zz +distance rom the line. SHEET +5-6 A sheet of charge; two sheets \ +As another example, we will caleulate the fñeld from a uniform plane sheet ` bài +of charge. Suppose that the sheet is infinite in extent and that the charge per x ứ ` +unit area is ơ. We are going to take another guess. Considerations ofsymmetry #‡_ ì I`. ` E¡ +lead us to believe that the field direction is everywhere normal to the plane, and NỀN L~ +tƒ tuc haue no field from an other charges ïn the tuorid, the fñelds must be the NÀNG ⁄⁄ §\ +same (in magnitude) on each side. Thỉis time we choose for our Gaussian surface | bé NZ SN GAUSSIAN +a recbtangular box that cuts through the sheet, as shown in Eig. 5-6. The two SURFACE +faces parallel to the sheet will have equal areas, say A. The field is normal to +these two faces, and parallel to the other four. The total ñux is # times the area +of the first face, plus # times the area of the opposite face—with no contribution +from the other four faces. The total charge enclosed in the box is ơA. Pquating +the ñux to the charge inside, we have +BA+ BA= SẮ, ` +from which ơ Fig. 5-6. The electric field near a uni- +E=.——, (5.3) formly charged sheet can be found by apply- +2eg - , l . +ing Gauss' law to an imaginary box. +a simple but important result. +You may remember that the same result was obtained in an earlier chapter +by an integration over the entire surface. Gauss' law gives us the answer, in this +instance, much more quickly (although it is not as generally applicable as the +earlier method). Ị Ị +'W©e emphasize that this result applies omiu to the field due to the charges on +the sheet. If there are other charges in the neighborhood, the total ñeld near the + R +sheet would be the sum of (5.3) and the field of the other charges. Gauss' law + _ +would then tell us only that +€0 + — +where and s are the fñelds directed outward on each side of the sheet. +'The problem of two parallel sheets with equal and opposite charge densities, + R ++ơ and —ơ, is equally simple if we assume again that the outside world is + — +quite symmetric. Either by superposing two solutions for a single sheet or by +constructing a gaussian box that includes both sheets, it is easily seen that the ñeld (b) 1 -E/ [ +1s zoro ou#side of the two sheets (Fig. 5-7a). By considering a box that includes + +only one surface or the other, as in (b) or (c) of the figure, it can be seen that the +fñeld between the sheets must be twice what it is for a single sheet. The result is 7 s +#2(between the sheets) = ø/eg, (5.5) : ++Ì |+E/2 — +#2 (outside) =0. (5.6) (©) II +5-7 A sphere of charge; a spherical shell +W©e have already (in Chapter 4) used Gauss' law to fnd the fñeld outside a +uniformly charged spherical region. The same method can also give us the fñeld Fig. 5-7. The field between two charged +at points 7nside the sphere. Eor example, the computation can be used to obtain sheets is Ø/eo. +a good approximation to the field inside an atomic nucleus. In spite of the fact +--- Trang 71 --- +that the protons in a nucleus repel each other, they are, because of the strong +nuclear forces, spread nearly uniformly throughout the body of the nucleus. +uppose that we have a sphere of radius # ñlled uniformly with charge. Let ø +be the charge per unit volume. Âgain using arguments of symmetry, we assume ⁄⁄ +the feld to be radial and equal in magnitude at all points at the same distance 277 UNIEORM +from the center. To ñnd the fñeld at the distance z from the center, we take a ⁄Z CHARGE +spherical gaussian surface of radius z (r < ?Ÿ), as shown in Fig. 5-8. The fux out ⁄⁄ 7 DENSITY +of this surface is <1 +Amr?E. ⁄⁄7 | +The charge inside our gaussian surface is the volume inside times /ø, or | +31 Ð- | +Using Gauss' law, it follows that the magnitude of the field is given by «ữ < = +E=fˆ (r<ñ). (5.7) | +You can see that this formula gives the proper result for z = #. The electric field f +1s proportional to the radius and is directed radially outward. Fig. 5-8. Gauss' law can be used to find +The arguments we have just given for a uniformly charged sphere can be the field inside a uniformly charged sphere. +applied also to a thin spherical shell of charge. Assuming that the field is +everywhere radial and is spherically symmetric, one gets immediately from Gauss' +law that the feld outside the shell is like that of a point charge, while the field +everywhere inside the shell is zero. (A gaussian surface inside the shell will +contain no charge.) +5-8 Is the feld of a point charge exactly 1/72? +T we look in a little more detail at how the field inside the shell gets to be +Zoro, we can see more clearly why it is that Gauss' law is true only because +the Coulomb force depends exactly on the square of the distance. Consider any +point ? inside a uniform spherical shell of charge. Imagine a small cone whose +apex Is at P and which extends to the surface of the sphere, where it cuts out +a small surface area Aa, as in Eig. 5-9. An exactly symmetric cone diverging +from the opposite side of ? would cut out the surface area Aaa. TÝ the distances +from to these two elements of area are 7z and ra, the areas are in the ratio +Aas rổ +(You can show this by geometry for any point ? inside the sphere.) +Tf the surface of the sphere is uniformly charged, the charge Aø on each of n +the elements of area is proportional to the area, so +Aø: _ Aas P +Am s. Aai ' +Coulomb°s law then says that the magnitudes of the fñelds produced at by f› +these Ewo surface elements are in the ratio Aaa +s2 _ Aq/ r3 — 1 +k 1 Aq 1 / r‡ ' +The fields cancel exactly. Since all parts of the surface can be paired of in the Fig. 5-9. The field is zero at any point P +same way, the total field at ? is zero. But you can see that it would not be so if inside a spherical shell of charge. +the exponent oŸ r in Coulomb”s law were not exactly bwo. +'The validity of Gaussˆ law depends upon the inverse square law of Coulomb. +Tí the force law were not exactly the inverse square, it would not be true that the +field inside a uniformly charged sphere would be exactly zero. Eor instance, If +the force varied more rapidly, like, say, the inverse cube of r, that portion of the +surface which is nearer to an interior point would produce a fñeld which is larger +--- Trang 72 --- +than that which is farther away, resulting in a radial inward fñeld for a positive +surface charge. These conclusions suggest an elegant way of ñnding out whether +the inverse square law 1s precisely correct. We need only determine whether or +not the ñeld inside of a uniformly charged spherical shell is precisely zero. +lt is lucky that such a method exists. It is usually dificult to measure a +physical quantity to high precision—a one percent result may not be too difficult, +but how would one go about measuring, say, Coulomb”s law to an accuracy of +one part in a billion? It is almost certainly not possible with the best available +techniques to measure the ƒorce between two charged objects with such an +accuracy. But by determining only that the electric ñelds inside a charged sphere +are smaller than some value we can make a highly accurate measurement of +the correctness of Gauss” law, and hence of the inverse square dependence of +Coulomb'?s law. What one does, in efect, is cornpare the force law to an ideal +Inverse square. Such comparisons of things that are equal, or nearly so, are +usually the bases of the most precise physical measurements. +How shall we observe the feld inside a charged sphere? Ône way is to try +to charge an object by touching it to the inside of a spherical conductor. You +know that if we touch a small metal ball to a charged object© and then touch +1È to an electrometer the meter will become charged and the pointer will move +from zero (EFig. 5-10a). The ball picks up charge because there are electric felds +outside the charged sphere that cause charges to run onto (or of) the little ball. (a) +TỶ you do the same experiment by touching the little ball to the #wszde of the CHARGED „ „ — +charged sphere, you ñnd that no charge is carried to the electrometer. With such SPHERE _ ÿ ý : ˆ " +an experiment you can easily show that the fñeld inside is, at most, a few percent + + +of the feld outside, and that Gauss' law 1s at least approximately correct. + + +lt appears that Benjamin Eranklin was the frst to notice that the field inside NỀN +INSULATOR ELECTROMETER +a conducting shell is zero. The result seemed strange to him. When he reported +his observation to Priestley, the latter suggested that it might be connected +with an inverse square law, since it was known that a spherical shell of matter +produced no gravitational ñeld inside. But Coulomb didnˆt measure the inverse @œ) +square dependence untfil 18 years later, and Gauss' law came even later still. Hà _> +Gauss” law has been checked carefully by putting an electrometer inside a +% _ +large sphere and observing whether any deflections occur when the sphere is + + +charged to a high voltage. A null result is always obtained. Knowing the geometry ` / +of the apparatus and the sensitivity of the meter, it is possible to compute the 7N +minimum field that would be observed. From this number ït is possible to place +an upper limit on the deviation of the exponent from two. lf we write that the +electrostatic force depends on r—?†*, we can place an upper bound on €. By this Fig. 5-10. The electric field is zero inside +method Maxwell determined that e was less than 1/10,000. The experiment was a closed conducting shell. +repeated and improved upon in 1936 by Plimpton and Lawton. They found that +Coulombs exponent difers from %wo by less than one part in a billion. +Now that brings up an interesting question: How accurate do we know this +Coulomb law to be in various circumstances? The experiments we just described +measure the dependence of the field on distance for distances of some tens of +centimeters. But what about the distances inside an atom——in the hydrogen +atom, for instance, where we believe the electron is attracted to the nucleus +by the same inverse square law? It is true that quantum mechanics must be +used for the mechanical part of the behavior of the electron, but the force is the +usual electrostatic one. In the formulation of the problem, the potential energy +of an electron must be known as a function of distance from the nucleus, and +Coulomb”s law gives a potential which varies inversely with the first power of the +distance. How accurately is the exponent known for such small distances? Às +a result of very careful measurements in 1947 by Lamb and Retherford on the +relative positions of the energy levels of hydrogen, we know that the exponent is +correcE again to one part in a billion on the atomic scale—that is, at distances of +the order of one angstrom (10~Š centimeter). +'The accuracy ofthe Lamb-Retherford measurement was possible again because +of a physical “accident.” Two of the states of a hydrogen atom are expected to +have almost identical energies on if the potential varies exactly as l/r. A +--- Trang 73 --- +mmeasurement was made of the very slight đjƒference in energies by ñnding the +frequency œ of the photons that are emitted or absorbed in the transition from +one state to the other, using for the energy diference A = hư. Computations +showed that A# would have been noticeably diferent from what was observed if +the exponent in the force law 1/z2 difered from 2 by as much as one part in a +bilion. +ls the same exponent correct at still shorter distances? EFYrom measurements +in nuclear physics it is found that there are electrostatic forces at typical nuclear +đistanees—at about 10~13 centimeter—and that they still vary approximately +as the inverse square. We shall look at some of the evidenee in a later chapter. +Coulomb'?s law is, we know, still valid, at least to some extent, at distances of +the order of 10~†3 centimeter. +How about 10~14 centimeter? 'This range can be investigated by bombarding +protons with very energetic electrons and observing how they are scattered. +Results to date seem to indicate that the law fails at these distances. 'Phe +electrical force seems to be about 10 times too weak at distances less than +101“ centimeter. Now there are two possible explanations. One is that the +Coulomb law does not work at such small distances; the other is that our objects, +the electrons and protons, are not point charges. Perhaps either the electron or +proton, or both, is some kind of a smear. Most physieists prefer to think that the +charge of the proton is smeared. We know that protons interact strongly with +mesons. 'Phis implies that a proton will, from time to time, exisÈ as a neutron +with a T meson around it. Such a configuration would act—on the average—like +a little sphere oŸ positive charge. We know that the fñeld from a sphere oŸ charge +does not vary as 1/z2 all the way into the center. It is quite likely that the proton +charge is smeared, but the theory of pions is still quite inecomplete, so it may also +be that Coulomb's law fails at very small distances. The question ïs still open. +One more point: 'The inverse square law is valid at distances like one meter +and also at 1019 m; but is the coefficient 1/4zco the same? The answer is y@s; +at least to an accuracy of 15 parts in a million. +W© go back now to an important matter that we slighted when we spoke of +the experimental verification of Gauss' law. You may have wondered how the +experiment of Maxwell or of Plimpton and Lawton could give such an accuracy +unless the spherical conductor they used was a perfect sphere. Ân accuracy +of one part in a billion is really something to achieve, and you might well ask +whether they could make a sphere which was that precise. 'There are certain to +be slight irregularities in any real sphere and ïf there are irregularities, wïll they +not produce fields inside? We wish to show now that it is not necessary to have +a perfect sphere. Ït is possible, in fact, to show that there is no field inside a +closed conducting shell of an shape. In other words, the experiments depended +on 1/72, but had nothing to do with the surface being a sphere (except that with +a sphere it is easier to calculate what the fields +0ouid be if Coulomb had been +wrong), so we take up that subjJecb now. To show thỉs, it is necessary to know +some of the properties of electrical conductors. +5-9 The fñelds of a conductor +An electrical conduector is a solid that contains many “free” electrons. The +electrons can move around freely ?w the material, but cannot leave the surface. +In a metal there are so many free electrons that any electric fñeld will set large +numbers of them into motion. Either the current of electrons so set up must +be continually kept moving by external sources of energy, or the motion of the +electrons will cease as they discharge the sources producing the initial fñeld. In +“electrostatic” situations, we do not consider continuous sources of current (they +will be considered later when we study magnetostatics), so the electrons move +only until they have arranged themselves to produece zero electric field everywhere +inside the conduector. (This usually happens in a small fraction oŸ a second.) Tf +there were any field left, this fñield would urge still more electrons to move; the +only electrostatic solution is that the fñeld is everywhere zero inside. +--- Trang 74 --- +Now consider the ?nferior of a charged conducting object. (By “interior” we +mean in the mefaÏ itself.) Since the metal is a conductor, the interior field must +be zero, and so the gradient of the potential ở is zero. 'Phat means that ¿ does +not vary from point to point. Every conduector is an equipotential region, and its +surface is an equipotential surface. Since in a conducting material the electric +ñeld is everywhere zero, the divergence of #/ is zero, and by Gauss' law the charge +density in the #mferior oŸ the conductor must be zero. +Tf there can be no charges in a conductor, how can it ever be charged? What + +do we mean when we say a conductor is “charged”? Where are the charges? The + ` +answer is that they reside at the surface of the conductor, where there are strong +forces to keep them from leaving—they are not completely “free.” When we study CONDUCTOR \ +solid-state physics, we shall ñnd that the excess charge of any conductor is on the + E¡ =0 +average within one or two atomiec layers of the surface. For our present purposes, + +1E is accurate enough to say that if any charge is put on, or 7n, a conductor it all GAUSSIAN +accumulates on the surface; there is no charge in the interior of a conduetor. + /:⁄ SURFACE +W© note also that the electric field 7usử ou#side the surface of a conductor 5 E,= 7 +must be normal to the surface. There can be no tangential component. If there + c0 +were a tangential component, the electrons would move øiong the surface; there + +are no forces preventing that. Saying it another way: we know that the electric SUREACE CHARGE +fñeld lines must always go at right angles to an equipotential surface. _¬<⁄“” DENSITY ø +W© can also, using Gauss' law, relate the field strength just outside a conductor +to the local density of the charge at the surface. For a gaussian surface, we take a Fig. 5-11. The electric field just outside +small cylindrical box half inside and half outside the surface, like the one shown the surface of a conductor Is proportional +in Eig. 5-11. There is a contribution to the total Ñux of E only from the side of to the local surface density of charge. +the box outside the conductor. The field just outside the surface of a conductor +is then +Ou‡stde a conductor: ơ +tE=_—, (5.8) +where øơ is the /ocøl surface charge density. +Why does a sheet of charge on a conductor produce a diferent field than +just a sheet of charge? In other words, why is (5.8) twice as large as (5.3)? The +reason, of course, 1s that we have øøf said for the conductor that there are no +“other” charges around. 'There must, in fact, be some to make # = 0 in the +conductor. “The charges in the immediate neighborhood of a point on the +surface do, in facb, give a field loeai = Ølocal/2eo both inside and outside the +surface. But all the rest of the charges on the conductor “conspire” to produce +an additional fñeld at the poïnt equal in magnitude to #locai. The total ñeld +inside goes to zero and the field outside to 22locai = đ/€o. +5-10 The field ỉin a cavity of a conductor +W© return now to the problem of the hollow container——a conduetor with a : +cavity. There is no field in the rme‡al, but what about in the ca? We shall * = +show that if the cavity is emp# then there are no fields in it, no rnaiter that the . z2 +shøpe of the conductor or the cavity——say for the one in Fig. 5-12. Consider a + tớ) +gaussian surface, like S in Eig. 5-12, that encloses the cavity but stays everywhere E=2 #2 +in the conducting material. Everywhere on Š the field is zero, so there is no ñux ?- h +through Š and the £o£øl charge inside Š is zero. Eor a spherical shell, one could + › _ __Ƒý +then argue from symmetry that there could be øoø charge inside. But, in general, : k, « : }r +we can only say that there are equal amounts of positive and negative charge on `: 2 4 +the inner surface of the conductor. 'Phere could be a positive surface charge on + Z2 ⁄⁄ /⁄⁄” +one part and a negative one somewhere else, as indicated in Fig. 5-12. Such a S ⁄% >. N4 +thing cannot be ruled out by Gauss' law. _ +'What really happens, of course, is that any equal and opposite charges on the + Suốce Z⁄Z : B +inner surface would slide around to meet each other, cancelling out completely. VWe ` + +can show that they must cancel completely by using the law that the circulation ' +Of E is always zero (electrostatics). Suppose there were charges on sorne parts Fig. 5-12. What is the field in an empty +of the inner surface. We know that there would have to be an equal number of cavity of a conductor, for any shape? +--- Trang 75 --- +opposite charges somewhere else. Now any lines of # would have to start on the +positive charges and end on the negative charges (since we are considering only +the case that there are no free charges in the cavity). Now imagine a loop T that +crosses the cavity along a line of force from some positive charge to some negative +charge, and returns to its starting point via the conductor (as in Fig. 5-12). The +integral along such a line of force from the positive to the negative charges would +not be zero. The integral through the metal is zero, since # = 0. So we would +‡ +E- ds # 0??? +But the line integral of # around any closed loop in an electrostatic field is always +zero. 5o there can be no fñelds inside the empty cavity, nor any charges on the +inside surface. +You should notice carefully one Important qualification we have made. We +have always said “inside an emøpứU” cavity. If some charges are øÏaced at some +ñxed locations in the cavity—as on an insulator or on a small conductor insulated +from the main one—then there cøn be fñields in the cavity. But then that is not +an “empty” cavity. +W© have shown that Iƒ a cavity is cormpletely enclosed by a conduector, no statie +distribution of charges ou#side can ever produce any fields inside. 'Phis explains +the principle of “shielding” electrical equipment by placing it in a metal can. The +same arguments can be used to show that no static distribution of charges ?mside +a closed grounded conductor can produce any fñelds ow#s¿de. Shielding works +both waysl In electrostatics—but not in varying fields—the fields on the bwo +sides of a closed grounded conducting shell are completely independent. +Now you see why 1% was possible to check Coulombs law to such a great +precision. 'Phe shape of the hollow shell used doesnˆt matter. It doesn”t need to be +spherical; it could be squarel If Gauss' law is exact, the feld inside is always zero. +Now you also understand why it is safe to sit inside the high-voltage terminal +of a million-volt Van de Graaff generator, without worrying about getting a +shock——because of Gauss' law. +--- Trang 76 --- +Theo Elocfric Fioldl trẻ V(r@rrs ẤTr'cttrrtSÉcrft(©S +6-1 Equations of the electrostatic potential +This chapter will describe the behavior of the electric field in a number of 6-1 Equations of the electrostatic +diferent circumstances. It will provide some experience with the way the electric potential +field behaves, and will describe some of the mathematical methods which are 6-2 The electric dipole +—_ là BAN t thất the ghol thematieal oroblem E the solnti 6-3 Remarks on vector equations +e begin by pointing out that the whole mathematical problem is the solution 6-4 The dipole potentialas a gradient +of two equations, the Maxwell equations for electrostatics: : . . +6-5 The dipole approximation for an +ÿ.E=f, (6.1) arbitrary distribution +«0 6-6 The fields of charged conductors +VxE=0. (6.2) 6-7 The method of images +In fact, the two can be combined into a single equation. Erom the second equation, 6-8 A poïnt charge near a conducting +we know at once that we can describe the ñeld as the gradient of a scalar (see plane +Section 3-7): 6-9 A point charge near a conducting +E=-Vó. (6.3) sphere +W©e may, if we wish, completely describe any particular electric field in terms 6-10 Condensers; parallel plates +of its potential ở. We obtain the diferential equation that @ must obey by 6-11 High-voltage breakdown +substituting Eq. (6.3) into (6.1), to get 6-12 The field-emission microscope +V.Vo=_—/, (6.4) +The divergence of the gradient of ó is the same as VỶ operating on ý: +9$ 0°3¿ Ø2 +Ý.V¿ø=V?¿=—_—+-—_—+— 6.5 +Ó ớ 02 Ì 0p + z2 (6.5) +so we write bq. (6.4) as +W?¿=_—£. (6.6) +€0 Tcuieu: Chapter 23, Vol. I, Tiesonance +The operator VZ is called the Laplaeian, and E4q. (6.6) is called the Poisson +equation. The entire subject of electrostatics, from a mathematical point of view, +is merely a study of the solutions of the single equation (6.6). Once ø is obtained +by solving Eq. (6.6) we can fnd immediately from Eq. (6.3). +We take up fñrst the special class of problems in which ø is given as a funection +of z, , z. In that case the problem 1s almost trivial, for we already know the +solution o£ Eq. (6.6) for the general case. We have shown that 1Ý ø is known at +every point, the potential at point (1) is +ø(2) dW› +1)= | ——— 6.7 +2a) = | (67) +where ø(2) is the charge density, đW2 is the volume element at point (2), and ra +is the distance bebween points (1) and (2). The solution of the đjfferential +cquation (6.6) is reduced to an ?mtegrœiion over space. The solution (6.7) should +be especially noted, because there are many situations in physics that lead to +cquations like +V(something) = (something else), +and Ea. (6.7) is a prototype oÊ the solution for any of these problems. +The solution of electrostatic fñeld problems is thus completely straightforward +when the positions of all the charges are known. Let”s see how it works in a few +examples. +--- Trang 77 --- +6-2 The electric dipole +First, take two point charges, +g and —g, separated by the distance d. Let W +the z-axis go through the charges, and pick the origin halfway between, as shown +in Eig. 6-1. Then, using (4.24), the potential from the two charges is given by +P(x, y,Z) +@(%, 1J, Z) S +_-_1_ _——— + ———l: (6.8) +4o | [z — (d/2)]2++2 +2 v[z+(d/2)]2 + +2 + g2 +W© are not goïing to write out the formula for the electric fñeld, but we can always +calculate it once we have the potential. So we have solved the problem of bwo : +charges. ——-+- y +There is an important special case in which the bwo charges are very cÌose _ +together——which is to say that we are interested ¡in the fñelds only at distances 2 +from the charges large in comparison with their separation. We call such a close +pair of charges a đ¿pole. Dipoles are very common. +A “dipole” antenna can often be approximated by two charges separated by a : ¬ +small distance—if we dont ask about the field too close to the anbenna. (W© are ảnd Ca the ma. charges +4 +usually interested in antennas with mmouzng charges; then the equations of statics +do not really apply, but for some purposes they are an adequate approximation.) +More important perhaps, are atomie dipoles. lf there is an electric fñeld in any +material, the electrons and protons feel opposite forces and are displaced relative +to each other. In a conductor, you remermber, some of the electrons move to +the surfaces, so that the field inside becomes zero. In an insulator the electrons +cannot move very far; they are pulled back by the attraction of the nucleus. They +do, however, shift a little bít. So although an atom, or molecule, remains neutral +in an external electric field, there is a very tiny separation of its positive and +negative charges and i% becomes a microscopic dipole. If we are interested in the +fields of these atomie dipoles in the neighborhood of ordinary-sized objects, we +are normally dealing with distances large compared with the separations of the +pairs of charges. +In some molecules the charges are somewhat separated even in the absence +of external fields, because of the form of the molecule. In a water molecule, for +example, there is a net negative charge on the oxygen atom and a net positive = +charge on each of the two hydrogen atoms, which are not placed symmetrically +but as in Eig. 6-2. Although the charge of the whole molecule is zero, there is a +charge distribution with a little more negative charge on one side and a little more +positive charge on the other. This arrangement is certainly not as simple as Ewo +point charges, but when seen from far away the system acts like a dipole. Âs we ⁄) Cs) +shall see a little later, the fñeld at large distances is not sensitive to the fine details. + + +Let”s look, then, at the field of two opposite charges with a small separation d. +Tf đ becomes zero, the two charges are on top of each other, the two potentials Flg. 6-2. The water molecule HaO. The +cancel, and there is no fñeld. But ïf they are not exactly on top oŸ each other, we nydrogen atoms have slightly less than ther +Ỉ . - ` share of the electron cloud; the oxygen, +can get a good approximation to the potential by expanding the terms of (6.8) in slightly more. +a power series in the small quantity ở (using the binomial expansion). Keeping +terms only to fñrst order in ở, we can write +(:-š) z? — zd. +Tt 1s convenlent to write +#2 + 2 + z2 — r, +(:- 3 tiở +iể S rổ si =vÊ[L= 5} +1 - 1 1 ( : „) 1⁄2 +vĩz- (4/2)P++z?+2 ` vr5-=(zd/rÐ)] r rẻ +--- Trang 78 --- +Using the binomial expansion again for [1 — (zd/r?)] 1⁄2——and throwing away +terms with the square or higher powers of d—we get +1 1+ 1 zd +r 2r27/ +Similarly, +1 ˆ -Í 1 1 3) +V[z+(d/2)]2+z2 +2” 2727 +The diference of these two terms gives for the potential +=—— -+dd. 6.9 +6(s..*) = TT x34 (6.9) +'The potential, and hence the field, which is its derivative, is proportional to qd, +the product of the charge and the separation. “Phis product is defned as the +đipolÌe mnormment oŸ the two charges, for which we will use the symbol p (do =oø# +confuse with momentuml): +p= qd. (6.10) +Equation (6.9) can also be written as +1 pcosØ +=———— 6.11 +Ó(%, 1, 2) mm (6.11) +since z/# = cosØ, where Ø is the angle between the axis of the dipole and +the radius vector to the poïnt (+, ,z)—see Fig. 6-1. The potental oŸ a dipole p +decreases as 1/72 for a given direction from the axis (whereas for a point charge +it goes as 1/z). The electric ñeld # of the dipole will then decrease as 1/3. +We can put our formula into a vector form If we defne ø as a vector whose +magnitude is p and whose direction is along the axis of the dipole, pointing from ọ ƒ +—q toward +g. Then +pcos0 = p:€,, (6.12) +where e„ is the unit radial vector (Fig. 6-3). We can also represent the poiïnt P]⁄ +(z,,z) by r. Then +Dipole potential: Fig. 6-3. Vector notation for a dipole. +l1 p-e, l1 p-r +— _# T“—__ Ý — 6.13 +2ữ) 4mcg_ TỶ 4mcg_ rở ( ) +'This formula is valid for a dipole with any orientation and position ïŸ represenfs +the vector from the dipole to the point of interest. +Tí we want the electric fñeld of the dipole we can get i% by taking the gradient +of ó. For example, the z-component of the feld is —Øj/Øz. For a dipole oriented +along the z-axis we can use (6.9): +0p Ø0(/zÀ_ p (1 32 +6z — 4meoeÔz\r3j — 4mee\r3 — rõ j) +p 3cos?0— 1 +E¿==—————. 6.14 +47g r ( ) +The z- and #-components are +p 3zz p 3ZỤ +Œ„= ——_——~, Eụ ==———.. +41g T5 # 47meg TỔ +'These two can be combined to give one component directed perpendicular to the +z-axis, which we will call the transverse component #¡: +Eìị = E2 + E = _T— V2 +2 +# 47mcg rŠ +3 cos Øsin 9 +2 (6.15) +47��o rỏ +--- Trang 79 --- +The transverse component F/¡ is in the #z-plane and points directly away from +the azs of the dipole. 'Phe total feld, of course, is +EZ= vEF2+ E}. +The dipole fñeld varies inversely as the cube of the distance from the dipole. +On the axis, at Ø = 0, it is Ewice as strong as at Ø = 90°. At both of these special +angles the electric fñeld has only a z-component, but of opposite sign at the tEwo +places (Fig. 6-4). +6-3 Remarks on vector equations +This is a good place to make a general remark about vector analysis. The +fundamental proofs can be expressed by elegant equations in a general form, but +in making various calculations and analyses it is always a good idea to choose +the axes in some convenient way. Notice that when we were ñnding the potential +of a dipole we chose the z-axis along the direction of the dipole, rather than at +some arbitrary angle. This made the work much easier. But then we wrote the p S Ei +cquations in vector form so that they would no longer depend on any particular 2% E +coordinate system. After that, we are allowed to choose any coordinate system CC ) +we wish, knowing that the relation is, in general, true. It clearly doesnˆt make +any sense to bother with an arbitrary coordinate system at some complicated +angle when you can choose a neat system for the particular problem——provided +that the result can fñnally be expressed as a vector equation. So by all means take +advantage of the fact that vector equations are independent of any coordinate +system. +On the other hand, if you are trying to calculate the divergence of a vector, Fig. 6-4. The electric field of a dipole. +instead of just looking at V - E and wondering what it is, don't forget that it +can always be spread out as +ØE„ + ðRv + 0E, +Øz ỡy Õz +l you can then work out the z-, -, and z-components of the electric field +and diferentiate them, you will have the divergence. There often seems to be +a feeling that there is something inelegant—some kind of defeat involved——in +writing ou§ the components; that somehow there ought always to be a way to +do everything with the vector operators. Thhere is often no advantage to it. The +first time we encounter a particular kind of problem, ït usually helps to write out +the components to be sure we understand what is goïng on. 'Phere is nothing +Inelegant about putting numbers into equations, and nothing inelegant about +substituting the derivatives for the fancy symbols. In fact, there is often a certain +cleverness in doing just that. Of course when you publish a paper in a professional +journal it will look better—and be more easily understood——if you can write +everything in vector form. Besides, it saves print. +6-4 The dipole potential as a gradient +W©e would like to point out a rather amusing thing about the dipole formula, +Eq. (6.13). The potential can also be written as +=———p-V|-]}. 6.16 +¿=-—p:Y( ) (6.16) +Tf you calculate the gradient of 1/z, you get +xv — =—-_~—=—-.; +( r ) rồ r2 +and Eq. (6.16) is the same as Eq. (6.13). +How did we think of that? We just remembered that ez/r2 appeared in the +formula for the feld of a point charge, and that the fñeld was the gradient of a +potential which has a 1/r dependence. +--- Trang 80 --- +There is a pñ#s¿cal reason for beïng able to write the dipole potential in the +form of Eq. (6.16). Suppose we have a point charge g at the origin. The potential +at the point P at (z,9, 2) 1s +Óo ==. +(Let”s leave off the 1/4zeo while we make these arguments; we can stick it in at +the end.) NÑow if we move the charge +q up a distance Az, the potential at ÐP +will change a little, by, say, Aø+. How mụuch is Aø +? Woll, it is just the amount +that the potential œø0ould change if we were to leœue the charge at the origin and z +move douward by the same distance Az (Fig. 6-5). That is, +ð /7p.ÂZ +Aó, =—0A¿, ⁄⁄ 2P +where by Az we mean the same as đ/2. 5o, using óo = g/zr, we have that the ⁄⁄ +potential from the positive charge 1s ⁄ +g_ Ø(4\d AzL Z +==—x-| “lc- 6.17 +#+ rÖz Œ) 2 (617) Ø ⁄ +Applying the same reasoning for the potential from the negative charge, we +can write ô : x +_=— + >~-|— |-. 6.18 +% T + =[ T )› ( ) +The total potential is the sum of (6.17) and (6.18): Flg. 6-5. The potential at P from 3 poInt +charge at Az above the origin is the same +Ô (q as the potential at P“ (Az below P) from +¿=ðó+_ +ó = —g; Œ) đ (6.19) the same charge at the origin. +LÂY: +=—.~_|_ Jqd +Øz () 1 +For other orlentations of the dipole, we could represent the displacement of +the positive charge by the vector Ar,. We should then write the equation above +Eq. (6.17) as +Ad+ = —Vớa - Ar-, +where A7 is then to be replaced by đ/2. Completing the derivation as before, +Eq. (6.19) would then become +=—V| - | -qd. +Thịs is the same as Eq. (6.16), if we replace gđ = p, and put back the 1/47eo. +Looking at i9 another way, we see that the dipole potential, Eq. (6.13), can be +Interpreted as +=~p: VẰa, (6.20) +where ®oọ = 1/4reor is the potential of a n#t point charge. +Although we can always ñnd the potential of a known charge distribution +by an integration, it is sometimes possible to save time by getting the answer +with a clever trick. Eor example, one can often make use of the superposition +principle. IÝ we are given a charge distribution that can be made up of the sum +of two distributions for which the potentials are already known, it is easy to nd +the desired potential by just adding the two known ones. One example of this is +our đderivation of (6.20), another is the following. +Suppose we have a spherical surface with a distribution of surface charge that +varies as the cosine of the polar angle. “The integration for this distribution is fairly +messy. But, surprisingly, such a distribution can be analyzed by superposition. +For imagine a sphere with a uniform øolznae density of positive charge, and +another sphere with an equal uniform volume density of negative charge, originally +superposed to make a neutral—that is, uncharged——sphere. If the positive sphere +--- Trang 81 --- +Fig. 6-6. Two uniformly charged spheres, +superposed with a slight displacement, are +equivalent to a nonuniform distribution of +surface charge. ————————— ————— ^- +(a) + (b) = (c) +is then displaced slightly with respect to the negative sphere, the body of the +uncharged sphere would remain neutral, but a little positive charge will appear on +one side, and some negative charge will appear on the opposite side, as illustrated +in Eig. 6-6. If the relative displacement of the two spheres is small, the net charge +is equivalent to a surface charge (on a spherical surface), and the surface charge +density will be proportional to the cosine of the polar angle. +Now if we want the potential from this distribution, we do not need to do an +Integral. We know that the potential from each of the spheres of charge Is——Íor +points outside the sphere—the same as from a point charge. The two displaced +spheres are like two point charges; the potential is just that of a dipole. +In this way you can show that a charge distribution on a sphere of radius œ +with a surface charge density +Ø = ØoCOS8 +produces a feld outside the sphere which is Just that of a dipole whose moment is +c— 4rơgdaŠ +p= 3g.” +lt can also be shown that inside the sphere the field is constant, with the value +tb=_—. +Tí Ø is the angle from the positive z-axis, the electric field inside the sphere is in +the negøiue z-direction. The example we have just considered is not as artifcial +as 1È may appear; we will encounter it again in the theory of dielectrics. +6-5 The dipole approximation for an arbitrary distribution +The dipole field appears in another circumstance both interesting and im- +portant. Suppose that we have an object that has a complicated distribution +of charge—like the water molecule (Fig. 6-2) —and we are interested only in +the fields far away. We will show that it is possible to fnd a relatively simple +expression for the fields which is appropriate for distances large compared with +the size of the obJect. +W©e can think of our object as an assembly of point charges q;¿ In a certain +limited region, as shown in Eig. 6-7. (W©e can, later, replace g; by @đV iIŸ we +wish.) Let each charge g¿ be located at the displacement đ; from an origin chosen +Fig. 6-7. Computation of the potential +at a point Ð at a large distance from a set <á| « +of charges. +--- Trang 82 --- +somewhere in the middle of the group of charges. What is the potential at the +point ?, located at , where i is much larger than the maximum dđ;? The +potential from the whole collection is given by +=—— — 6.21 +“=1 » ¬ (6.21) +where ?¿ is the distance from ? to the charge q; (the length of the vector Jề— đ;). +Now ïf the distance from the charges to , the point of observation, is enormous, +cach of the r;'s can be approximated by #. Each term becomes g;/Ï, and we +can take 1/] out as a factor in front of the summation. This gives us the simple +result L1 Q +=——= ¡ = —— 6.22 +ứ 4meo Tỉ . 47eg Tỉ ) +where is just the total charge of the whole object. Thus we ñnd that for points +far enough from any lump of charge, the lump looks like a point charge. The +result is not too surprising. +But what if there are equal numbers oŸ positive and negative charges? Then +the total charge @ of the object is zero. This is not an unusual case; in fact, as +we know, objects are usually neutral. The water molecule is neutral, but the +charges are not all at one poïnt, so if we are close enough we should be able to see +some efects of the separate charges. We need a better approximation than (6.22) +for the potential rom an arbitrary distribution of charge in a neutral object. +Equation (6.21) is still precise, but we can no longer just set r; = l?. We need +a more accurate expression for r¿. lf the point ? is at a large distance, r¿ will +differ from ?#‡ to an excellent approximation by the projection of đ on đ, as can +be seen from Eig. 6-7. (You should imagine that P is really farther away than +is shown in the fgure.) In other words, if ep is the unit vecbor in the direction +of ñ, then our next approximation to r¿ 1s +r; R— d,-en. (6.23) +What we really want is 1/r;, which, since d¿ < Ï, can be written to our +approximation as +1 1 đ, ':“h +—#ư—=|l+——_—_]. 6.24 +Substituting this in (6.21), we get that the potential is +1 Q đ; '.Ch +=——| + ¡na T+-'' ]- 6.25 +“=(§ Tiếp (6.25) +The three dots indicate the terms of higher order in d;/? that we have neglected. +These, as well as the ones we have already obtained, are successive terms in a +Taylor expansion of 1/r¿ about 1/?? in powers of dạ /P. +The frst term in (6.25) is what we got before; it drops out iŸ the object is +neutral. The second term depends on 1/R”, just as for a dipole. In fact, if we +p=}À ;údi (6.26) +as a property of the charge distribution, the second term oŸ the potential (6.25) +Ì P:€n +=——_—- 6.27 +“=1 ng (6.27) +precisclU œ dipole potential. The quantity p is called the dipole moment of the +distribution. It is a generalization of our earlier deñnition, and reduces to it for +the special case of two point charges. +Our result is that, far enough away from ønw mess of charges that is as a +whole neutral, the potential is a dipole potential. It decreases as 1/2 and varies +--- Trang 83 --- +as cos Ø8——and its strength depends on the dipole moment of the distribution of +charge. It is for these reasons that dipole felds are important, since the simple +case of a pair of point charges is quite rare. +The water molecule, for example, has a rather strong dipole moment. "The +electric fields that result from this moment are responsible for some of the +important properties of water. EFor many molecules, for example CÕs, the dipole +mmoment vanishes because of the symnmetry of the molecule. Eor them we should +expand still more accurately, obtaining another term in the potential which +decreases as 1/RỞ, and which is called a quadrupole potential. We will discuss +such cases later. +6-6 The fñelds of charged conductors +We have now finished with the examples we wish to cover of situations in ⁄ B +which the charge distribution is known from the start. It has been a problem A +without serious complications, involving at most some integrations. We turn now _— Z= ¬ +to an entirely new kind of problem, the determination of the fields near charged ⁄ › +conductors. _ = +Suppose that we have a situation in which a total charge Œ is placed on an é XU X ` +arbitrary conductor. Now we will not be able to say exactly where the charges KT Ị +are. They will spread out in some way on the surface. How can we know how ` V2 s +the charges have distributed themselves on the surface? 'Phey must distribute +themselves so that the potential of the surface is constant. If the surface were ¬ _⁄ +not an equipotential, there would be an electric ñeld inside the conductor, and TT” 6 +the charges would keep moving until it became zero. The general problem of this +kind can be solved in the following way. We guess at a distribution of charge and +calculate the potential. If the potential turns out to be constant everywhere on Fig. 6-8. The field lines and equipoten- +the surface, the problem is ñnished. If the surface is not an equipotential, we have tials for two point charges. +guessed the wrong distribution of charges, and should guess again—hopefully +with an improved guessl 'This can go on forever, unless we are judicious about +the successive guesses. +The question of how to guess at the distribution is mathematically dificult. +Nature, of course, has time to do it; the charges push and pull until they all +balance themselves. When we try to solve the problem, however, it takes us so +long to make each trial that that method is very tedious. With an arbitrary +group of conduectors and charges the problem can be very complicated, and in +general it cannot be solved without rather elaborate numerical methods. Such +numerical computations, these days, are set up on a computing machine that +will do the work for us, once we have told it how to proceed. +On the other hand, there are a lot of little practical cases where it would be +nice to be able to ñnd the answer by some more direct method——without having í +to write a program for a computer. Fortunately, there are a number of cases +where the answer can be obtained by squeezing it out oŸ Nature by some trick or Xứ +other. The first trick we will describe involves making use of solutions we have <7 cốt +already obtained for situations in which charges have specified locations. X7 +CONDUCTOR '/ +6-7 The method of images +We have solved, for example, the fñeld of two point charges. Pigure 6-8 shows +some of the field lines and equipotential surfaces we obtained by the computations : : +in Chapter 4. Now consider the equipotential surface marked A4. Suppose we F1g. 69. The field outside 3 Conductor +were to shape a thin sheet of metal so that it Just fits this surface. If we place it shaped like the equipotential Á of Fig. 6-8. +right at the surface and adjust its potential to the proper value, no one would +ever know it was there, because nothing would be changed. +But noticel We have really solved a new problem. We have a situation in +which the surface of a curved conductor with a given potential is placed near a +point charge. lf the metal sheet we placed at the equipotential surface eventually +closes on itself (or, in practice, 1Ý it goes far enough) we have the kind of situation +considered in Section 5-10, in which our space is divided into bwo regions, one +--- Trang 84 --- +inside and one outside a closed conducting shell. We found there that the fields +in the two regions are quite independent of each other. So we would have the +same fields outside our curved conductor no matter what is inside. W©e can even +fll up the whole inside with conducting material. We have found, therefore, the +fñelds for the arrangement of Fig. 6-9. In the space outside the conductor the +ñeld is just like that of two poïint charges, as in Fig. 6-8. Inside the conduector, +1b is zero. Also—as it must be—the electric field just outside the conduector is +normal to the surface. +Thus we can compute the fñelds in Fig. 6-9 by computing the ñeld due to g +and to an imaginary point charge —q at a suitable point. The point charge we +“imagine” existing behind the conducting surface is called an #mage charge. +In books you can fnd long lists oŸ solutions for hyperbolic-shaped conductors +and other complicated looking things, and you wonder how anyone ever solved +these terrible shapes. They were solved backwardsl Someone solved a simple +problem with given charges. He then saw that some equipotential surface showed +up in a new shape, and he wrote a paper in which he pointed out that the field +outside that particular shape can be described in a certain way. +6-8 A point charge near a conducting plane +As the simplest application of the use of this method, let°s make use of the +plane equipotential surface of Eig. 6-8. With it, we can solve the problem of a +charge in Íront of a conducting sheet. We just cross out the left-hand half of the +picture. "The field lines for our solution are shown in Eig. 6-10. Notice that the +plane, since it was halfway between the two charges, has zero potential. We have +solved the problem of a positive charge next to a grounded conducting sheet. +W© have now solved for the total ñeld, but what about the real charges that +are responsible for it? 'Phere are, in addition to our positive point charge, some +induced negative charges on the conducting sheet that have been attracted by +the positive charge (from large distances away). NÑow suppose that for some +\ ' / — +` \ / ` +N \ ; Z SN +\ SN _ +À À Ị ⁄ ` +` \ ⁄ tù +` \ |CONDUCTING ` +`Y \ | PLATE z K +NV N l ⁄ ni : +TU ` XI “ Z SN +" ` \ À +xỐ ` \Í / ⁄ +~e ¬ \V[/⁄“ +>> XẺN ⁄_ ⁄Z R +~ Z—~ h \ +——————-ÌMAGE CHARGE— <=--k» TL +~Z“ Z //I\XSS ` +_” ⁄ˆ^ | N ~ +~ ⁄ / / \ \ ` =M +sứ ⁄ \ ` +~ ⁄ ⁄ [\ ` +⁄ ⁄ / TA +⁄ 71A N ` +⁄ / Ị \ › SN +⁄ / \ tàn +⁄ / ` ` +⁄ / ` ` +⁄ / \ ` ` +⁄ l \ ¬ +/ Ị \ = +Fig. 6-10. The field of a charge near a plane conducting surface, found by the method +of images. +--- Trang 85 --- +technical reason——or out of curiosity—you would like to know how the negative +charges are distributed on the surface. You can find the surface charge density +by using the result we worked out in Section 5-9 with Gauss' law. The normal +component of the electric ñeld just outside a conductor is equal to the density of +surface charge ơ divided by co. We can obtain the density of charge at any point +on the surface by working backwards from the normal component of the electric +fñeld at the surface. We know that, because we know the fñeld everywhere. +Consider a point on the surface at the distance ø from the poïnt directly +beneath the positive charge (Eig. 6-10). The electric fñeld at this point is normal +to the surface and is directed into it. The component normal to the surface of +the feld from the poszfzue point charge is +đJm¡ — _~ (451 p2)3/5: (6.28) +To this we must add the electric fñeld produced by the negative Image charge. +That just doubles the normal component (and cancels all others), so the charge +density ø at any point on the surface is +ơ(ø) = cof(0) = “wœ (6.29) +An interesting check on our work is to integrate ø over the whole surface. We +fnd that the total induced charge 1s —g, as it should be. +One further question: Is there a force on the point charge? Yes, because there +is an attraction from the induced negative surface charge on the plate. Now that +we know what the surface charges are (from Eq. 6.29), we could compute the +force on our positive point charge by an integral. But we also know that the force +acting on the positive charge is exactly the same as it t0ould be with the negative +Image charge instead of the plate, because the fields in the neighborhood are +the same in both cases. The point charge feels a force toward the plate whose +magnitude is +.¬: 6.30 +“1n. Ga)” (6.30) +We have found the force much more easily than by integrating over all the negative +charges. +6-9 A point charge near a conducting sphere +'What other surfaces besides a plane have a simple solution? 'Phe next most ` +simple shape is a sphere. Let's ñnd the fields around a grounded metal sphere PÀ hn +which has a point charge g near it, as shown in Eig. 6-11. NÑow we must look for / Ñ +a simple physical situation which gives a sphere for an equipotential surface. If N q +we look around at problems people have already solved, we fnd that someone ` g=-2q +has noticed that the feld of two weqgual point charges has an equipotential that b +isa sphere. Ahal If we choose the location of an image charge—and pick the +right amount of charge—maybe we can make the equipotential surface ft our = +sphere. Indeed, it can be done with the following prescription. Fig. 6-11. The point charge g induces +Assume that you want the equipotential surface to be a sphere of radius œ charges on a grounded conducting sphere +with its center at the distance Ò from the charge g. Put an image charge of whose fields are those of an image charge q +strength g' = —q(ø/b) on the line from the charge to the center of the sphere, placed at the point shown. +and at a distance a2/b from the center. The sphere will be at zero potential. +'The mathematical reason stems from the fact that a sphere is the locus of all +points for which the distances from two points are in a constant ratio. Referring +to Eig. 6-11, the potential at P from q and đƒ is proportional to +TỊ T2 +'The potential wïll thus be zero at all points for which +LỚN. ra +—=—— OF _“=_—_—. +T2 TỊ T1 g +--- Trang 86 --- +TÝ we place g' at the distance a2/b from the center, the ratio z2/r¡ has the constant +value ø/b. Then if +the sphere is an equipotential. Its potential is, in fact, zero. +'What happens if we are interested in a sphere that is not at zero potential? +That would be so only ïf its total charge happens accidentally to be g“. Of course +1Í it is grounded, the charges induced on iÿ would have to be just that. But what +1Ý it is insulated, and we have put no charge on it? Or if we know that the total +charge @Q has been put on it? Or just that it has a given potential øø# equal +to zero? All these questions are easily answered. We can always add a point +charge g” at the center of the sphere. “The sphere still remains an equipotential +by superposition; only the magnitude of the potential will be changed. +lí we have, for example, a conducting sphere which is initially uncharged and +insulated from everything else, and we bring near to it the positive point charge q, +the total charge of the sphere will remain zero. The solution is found by using +an image charge gøˆ as before, but, in addition, adding a charge g” at the center +of the sphere, choosing +qg =-qg = pứ. (6.32) +The fñelds everywhere outside the sphere are given by the superposition of the +fields of ạ, g, and q”. The problem is solved. +W©e can see now that there will be a force of attraction between the sphere +and the point charge g. lt is not zero even though there is no charge on the +neutral sphere. Where does the attraction come from? When you bring a positive +charge up to a conducting sphere, the positive charge attracts negative charges to +the side closer to itself and leaves positive charges on the surface of the far side. +The attraction by the negative charges exceeds the repulsion from the positive +charges; there is a net attraction. We can fnd out how large the attraction is +by computing the force on g in the field produced by @' and gˆ”. The total force +is the sum of the attractive force between g and a charge qg' = —(a/b)q, at the +distance b— (a2/b), and the repulsive force bebween g and a charge g” = +(a/b)q +at the distance Ù. +Those who were entertained in childhood by the baking powder box which +has on its label a picture of a baking powder box which has on its label a piebure +of a baking powder box which has... may be interested in the following problem. +Two equal spheres, one with a total charge of + and the other with a total +charge of —Œ, are placed at some distance from each other. What is the force +between them? The problem can be solved with an infñnite number oÝ images. +One first approximates each sphere by a charge at its center. Thhese charges will +have image charges in the other sphere. The image charges will have images, etc., +ebc., ebc. The solution is like the picture on the box of baking powder——and 1$ +converges pretty fast. ++ơ Asa =A +6-10 Condensers; parallel plates xxx" mm ' +We take up now another kind of a problem involving conductors. Consider two ZTZZZZZ.ZZ.ZZZ.ZZZZZZZZZZZa +large metal plates which are parallel to each other and separated by a distance . +small compared with their width. Let”s suppose that equal and opposite charges Fig. 6-12. A parallel-plate condenser. +have been put on the plates. The charges on each plate will be attracted by the +charges on the other plate, and the charges will spread out uniformly on the inner +surfaces of the plates. The plates will have surface charge densities +øơ and —ơ, +respectively, as In Eig. 6-12. From Chapter 5 we know that the field between the +plates is ơ/eo, and that the fñeld outside the plates is zero. The plates will have +diferent potentials ói and øs. For convenience we will call the diference V; it is +often called the “voltage”: +Óị — 0a = V. +(You will nd that sometimes people use WV for the pobential, but we have chosen +to use ở.) +--- Trang 87 --- +The potential diference W is the work per unit charge required to carry a +small charge from one plate to the other, so that +V= Ed= d6 (6.33) +where -FŒ is the total charge on each plate, A4 is the area of the plates, and đ is +the separation. +We fñnd that the voltage is proportional to the charge. Such a proportionality +between W and @ is found for any two conductors in space ïIf there is a plus +charge on one and an equal minus charge on the other. The potential diference +between them—that is, the voltage——will be proportional to the charge. (We are +assuming that there are no other charges around.) +'Why this proportionality? Just the superposition principle. Suppose we know +the solution for one set of charges, and then we superimpose two such solutions. +'The charges are doubled, the fñelds are doubled, and the work done in carrying a +unit charge from one point to the other is also doubled. Thherefore the potential +diference between any bwo poinfs is proportional to the charges. In particular, +the potential diference between the two conductors is proportional to the charges +on them. Someone originally wrote the equation of proportionality the other way. +'That is, they wrote +Q=CY, +where Œ is a constant. This coefficient of proportionality is called the capacitg, +and such a system of two conduectors is called a condenser.X For our parallel-plate +condenser +C= TT (parallel plates). (6.34) +This formula is not exact, because the fñield is not really uniform everywhere +between the plates, as we assumed. “The field does not just suddenly quit at the +edges, but really is more as shown in Fig. 6-13. The total charge is not øơ Á, as we +have assumed——there is a little correction for the effects at the edges. To find out +what the correction is, we will have to calculate the field more exactly and find XNNNNNNNNNNNNNNNNNANNNNN +out just what does happen at the edges. 'Phat ¡is a complicated mathematical +problem which can, however, be solved by techniques which we will not describe +now. 'Phe result of such calculations is that the charge density rises somewhat +near the edges of the plates. This means that the capacity of the plates is a little +higher than we computed. +W© have talked about the capacity for two conductors only. Sometimes people 5S S3ŠŠŠšŠŠš +talk about the capacity ofa single object. 'They say, for instance, that the capacity +oŸ a sphere of radius œ is 4reoa. What they imagine is that the other terminal is +another sphere of infinite radius—that when there is a charge -+Q on the sphere, +the opposite charge, —C, is on an infinite sphere. Ône can also speak of capacities +when there are three or more conductors, a discussion we shall, however, defer. . ca. +Suppose that we wish to have a condenser with a very large capacity. We of n Tel nake field near the edge +could get a large capacity by taking a very big area and a very small separation. l +W©e could put waxed paper between sheets of aluminum foil and roll i9 up. (Tf +we seal it in plastic, we have a typical radio-type condenser.) What good is it? +lt is good for storing charge. lÝ we try to store charge on a ball, for example, +1ts potential rises rapidly as we charge it up. It may even get so high that the +charge begins to escape into the air by way of sparks. But iŸ we put the same +charge on a condenser whose capacity is very large, the voltage developed across +the condenser will be small. +In many applications in electronie circuits, it is useful to have something +which can absorb or deliver large quantities of charge without changing is +potential much. AÁ condenser (or “capacitor”) does just that. There are also +many applications in electronic instruments and in computers where a condenser +* Some people think the words “capacitance” and “capacitor” should be used, instead of +“capacity” and “condensor.” We have decided to use the older terminology, because it is still +more commonly heard in the physics laboratory——even if not in textbooksl +--- Trang 88 --- +1s used to get a specified change in voltage in response to a particular change +in charge. We have seen a similar application in Chapter 23, Vol. Ï, where we +described the properties of resonant circuis. +Erom the delnition of Ở, we see that its unit is one coulomb/volt. This unit 1 +is also called a farøœd. Looking at Eq. (6.34), we see that one can express the +units oŸ eo as farad/meter, which is the unit most commonly used. T'ypical sizes +of condensers run from one micro-microfarad (1 picofarad) to millifarads. Small +condensers of a few picofarads are used in high-frequency tuned circuits, and +capacities up to hundreds or thousands of microfarads are found in power-supply +filters. A pair of plates one square centimeter in area with a one millimeter +separation have a capacity of roughly one micro-microfarad. +6-11 High-voltage breakdown +W©e would like now to discuss qualitatively some of the characteristics of the +felds around conductors. lÝ we charge a conductor that is not a sphere, but one +that has on it a point or a very sharp end, as, for example, the object sketched +in Fig. 6-14, the ñeld around the poïnt is much higher than the fñeld in the other TT†rrrrE---L_ +regions. 'Phe reason is, qualitatively, that charges try to spread out as much as ¬ Ñ +possible on the surface of a conduector, and the tip of a sharp point is as far away ---_---+-- Ƒ TTr1/⁄ +as it is possible to be from most of the surface. Some of the charges on the plate h +get pushed all the way to the tip. A relatively srnall amount of charge on the tỉp bà, +can sfill provide a large surface đensit; a high charge density means a high field CONDUCTOR ` ⁄ ự +Just outside. / NZ +One way to see that the fñeld is highest at those places on a conductor where LẦX mã +the radius of curvature is smallest is to consider the combination oŸ a big sphere «< Xí +and a little sphere connected by a wire, as shown in PFig. 6-15. It is a somewhat ứ ⁄ +idealized version of the conductor of Eig. 6-14. The wire will have little iniuence ú +on the fields outside; it is there to keep the spheres at the same potential. Now, ‹⁄ +which ball has the biggest field at its surface? If the ball on the left has the +radius ø and carries a charge Q, its potential is about Fig. 6-14. The electric field near a sharp +1 Q point on a conductor ¡s very high. +ở =——.. +4mcg œ +(Of course the presence of one ball changes the charge distribution on the other, +so that the charges are not really spherically symmetric on either. But if we are +interested only in an estimate of the fields, we can use the potential of a spherical +charge.) IÝ the smaller ball, whose radius is Ò, carries the charge g, its potential +is about 1g +j2 = 41meo b +But ởi = đa, sO WIRE ⁄ +On the other hand, the feld at the surface (see Bq. 5.8) is proportional to the +surface charge density, which is like the total charge over the radius squared. We +get that Fig. @-15. The field of a pointed object +Đụ — Q/a2 — b (6.35) can be approximated by that of two spheres +AN q/b2 mm. ' at the same potential. +'Therefore the field is higher at the surface of the small sphere. The fields are in +the inverse proportion of the radi1. +'This result is technically very important, because air will break down if the +electric field is too great. What happens is that a loose charge (electron, or ion) +somewhere in the air is accelerated by the fñeld, and ïf the field is very great, +the charge can pick up enough speed before it hits another atom to be able to +knock an electron of that atom. As a result, more and more ions are produeced. +'Their motion constitutes a discharge, or spark. lf you want to charge an object +to a hiph potential and not have it discharge itself by sparks in the air, you must +be sure that the surface is smooth, so that there is no place where the field is +abnormally large. +--- Trang 89 --- +6-12 The field-emission mỉicroscope +There is an interesting application of the extremely high electric field which —== TT NG +surrounds any sharp protuberance on a charged conductor. The field-em“ssion ZZ“ à +mứúcroscope depends for is operation on the hiph felds produced at a sharp Z +metal point.* It is built in the following way. A very fñne needle, with a tỉp ƒ Ầ +whose diameter is about 1000 angstroms, is placed at the center of an evacuated />S \ +glass sphere (Fig. 6-16). The inner surface of the sphere is coated with a thin A-== =— | +conduecting layer of Ñuorescent material, and a very high potential diferenece is CS J j +applied bebween the fuorescent coating and the needle. \ CC +Let”s first consider what happens when the needle is negative with respect to j 7 GROUND +the Ñuorescent coating. The field lines are highly concentrated at the sharp point. +The electric fñeld can be as high as 40 million volts per centimeter. In such intense GLẦSS BULB +fields, electrons are pulled out of the surface of the needle and accelerated across +the potential diference between the needle and the Ñuorescent layer. When they TO j +arrive there they cause light to be emitted, just as in a television picture tube. Mi j +'The electrons which arrive at a given point on the Ñuorescent surface are, tO +an excellent approximation, those which leave the other end of the radial fñeld l +line, because the electrons will travel along the field line passing from the point$ J+ HIGH VOLTAGE +to the surface. Thus we see on the surface some kind oŸ an image of the tỉp of Eid. 6-16. Field-emission microscope +the needle. More precisely, we see a picture of the ermm2ss?uity of the surface of the 3 l pc: +needle—that is the ease with which electrons can leave the surface of the metal +tip. lf the resolution were high enough, one could hope to resolve the positions +of the individual atoms on the tip of the needle. With electrons, this resolution +1s not possible for the following reasons. First, there is quantum-mechanical +difraction of the electron waves which blurs the image. Second, due to the +internal motions of the electrons in the metal they have a small sideways initial +velocity when they leave the needle, and this random transverse component of +the velocity causes some smearing of the image. The combination oŸ these two +effects limits the resolution to 25 Ä or so. +Tí, however, we reverse the polarity and introduce a small amount of helium gas s KH cường, +into the bulb, much higher resolutions are possible. When a helium atom collides - lv Lô SÁU QÊI #hêo +with the tip of the needle, the intense field there strips an electron of the helium SP Hiện) XS XE CN : +atom, leaving it positively charged. The helium ion is then accelerated outward Về Vệ h Làn ti TC Vui ⁄ +along a feld line to the fuorescent sereen. Since the helium ion is so muchheavicr Kãt sở T Sa s22 VG sc +than an electron, the quantum-mechanical wavelengths are much smaller. If the 2 SỆ Án LH NĂU L3 T//7 h2 Êo }2 +temperature is not too high, the efect of the thermal veloeities is also smaller `. 2. ẽe-. Sà +than in the electron case. With less smearing of the image a much sharperpictuc Sa... 2 8i +of the point is obtained. It has been possible to obtain magnifcations up to ng ng ưa, SN như, = +2,000,000 times with the positive ion field-emission microscope—a magnification nhớ = s0 c CS Êng X = KH kb C +ten tỉmes better than is obtained with the best electron microscope. K ~". ẻ.... +Pigure 6-17 is an example of the results which were obtained with a field-ion sW =5... 29 +microscope, using a tungsten needle. The center of a tungsten atom ionizes a 29s 1 2: VI Ji aia (0n ko š +helium atom at a slightly different rate than the spaces between the tungsten su cả g lá TƯANGWS. 2 tật % +atoms. The pattern of spots on the fuorescent screen shows the arrangement of ¬:À Ty vi lề cuIẾP,— (21007 52 ¬.. +the indiuidual atorms on the tungsten tip. The reason the spots appear in rings HN: TC óc Là nh KEuệ; SẾC LÍ : +can be understood by visualizing a large box of balls packed in a rectangular tÒy 2 z : ĐH ta ẾN Mộc Quy +array, representing the atoms in the metal. If you cut an approximately spherical HiN Lư ú7 7¬ 2: SIẾ GIRADfb +section out of this box, you will see the Tỉng pattern characteristic of the atomie Eig. 6-17. Image produced by a field- +structure. 'The field-ion microscope provided human beings with the means of emission microscope. [Courtesy of Erwin W. +seeing atoms for the frst time. 'This is a remarkable achievement, considering Miiler, Research Prof. of Physics, Pennsyl- +the simplicity of the instrument. vania State University.] +* See E. W. Miller: “'Phe field-ion microscope,” Aduønces ?m Electronmics ơnd Electrow +Phạs¿cs, 13, 83-179 (1960). Academic Press, New York. +--- Trang 90 --- +Theo Elocfric Fioldl trẻ V(r@rrs ẤTr'cttrrtSÉcrft(©S +(€ortfirerio«Ïl) +7-1 Methods for ñnding the electrostatic feld +This chapter is a continuation of our consideration of the characteristics of 7-1 Methods for ñnding the +electric fñelds in various particular situations. We shall frst describe some of the electrostatic ñeld +more elaborate methods for solving problems with conduectors. Ït is not expected 7-2_ Two-dimensional 8elds; functions +that these more advanced methods can be mastered at this time. Yet it may be of the complex variable +of interest to have some idea about the kinds of problems that can be solved, 7-3 Plasma oscillations +using techniques that may be learned in more advanced courses. hen we take : : : +up two examples in which the charge distribution 1s neither fxed nor is carried 7-4 Colloidal paricles m an +by a conductor, but instead is determined by some other law of physỉcs. electrolyte . . +As we found in Chapter 6, the problem of the electrostatic field is fundamen- 7-5 _ The electrostatic field ofa grid +tally simple when the distribution of charges is specified; it requires only the +evaluation of an integral. When there are conductors present, however, compli- +cations arise because the charge distribution on the conduectors is not initially +known; the charge must distribute itself on the surface of the conduector in such +a way that the conductor is an equipotential. The solution of such problems is +neither direct nor simple. +W© have looked at an indirect method of solving such problems, in which we +fnd the equipotentials for some specified charge distribution and replace one of +them by a conducting surface. In this way we can build up a catalog of special +solutions for conductors in the shapes of spheres, planes, etc. The use of Images, +described in Chapter 6, is an example of an indirect method. We shall describe +another in this chapter. +Tf the problem to be solved does not belong to the class of problems for which +we can construct solutions by the indirect method, we are forced to solve the +problem by a more direct method. “The mathematical problem of the direc +method is the solution of Laplace°s equation, +V”¿ =0, (7.1) +subject to the condition that ở is a suitable constant on certain boundaries—the +surfaces of the conduectors. Problems which involve the solution of a diferential +ñeld equation subject to certain bowndar conditions are called boundaru-oalue +problems. They have been the object oŸ considerable mathematical study. In the +case of conductors having complicated shapes, there are no general analytical +methods. ven such a simple problem as that of a charged cylindrical metal can +closed at both ends——a beer can——presents formidable mathematical dificulties. +lt can be solved only approximately, using numerical methods. The on general +xmnethods of solution are numerical. +There are a few problems for which Eq. (7.1) can be solved directly. Eor +example, the problem of a charged conductor having the shape of an ellipsoid +of revolution can be solved exactly in terms of known special functions. 'Phe +solution for a thin disc can be obtained by letting the ellipsoid become infnitely +oblate. In a similar manner, the solution for a needle can be obtained by letting +the ellipsoid become infinitely prolate. However, it must be stressed that the +only direct methods of general applicability are the numerical techniques. +Boundary-value problems can also be solved by measurements of a physical +analog. Laplace°s equation arises in many diferent physical situations: in steady- +state heat fow, ín irrotational ñưid Ñow, in current fow in an extended medium, +--- Trang 91 --- +and ín the deflection of an elastic membrane. ÏI§ is frequently possible to set up +a physical model which is analogous to an electrical problem which we wish to +solve. By the measurement of a suitable analogous quantity on the model, the +solution to the problem of interest can be determined. An example of the analog +technique is the use of the electrolytic tank for the solution of two-dimensional +problems in electrostatics. Thịis works because the diferential equation for the +potential in a uniform conducting medium is the same as i is for a vacuum. +There are many physical situations in which the variations of the physical +fñelds in one direction are zero, or can be neglected in comparison with the +variations in the other two directions. Such problems are called two-dimensional; +the ñeld depends on two coordinates only. Eor example, if we place a long charged +wire along the z-axis, then for points not too far rom the wire the electric feld +depends on z and , but not on z; the problem is two-dimensional. Since in a +two-dimensional problem Øj/Øz = 0, the equation for ó in free space is +22 + Đó =0. (7.2) +9z2 Ôy2 +Because the ©wo-dimensional equation is comparatively simple, there is a wide +range of conditions under which it can be solved analytically. There is, in fact, +a very powerful indirect mathematical technique which depends on a theorem +from the mathematics of functions of a complex variable, and which we will now +describe. +7-2 Two-dimensional ñelds; functions of the complex variable +'The complex variable ¿ is defned as +ậ=# +09. +(Do not confuse 4 with the z-coordinate, which we ignore in the following discussion +because we assume there is no z-dependence of the fields.) Every point in # and +then corresponds to a complex number ¿. We can use 3 as a single (complex) +variable, and with it write the usual kinds of mathematical functions #4). Eor +example, +FQ) =3Ỷ. +Ƒ§) = 1/3. +#@) =aln4, +and so forth. +Given any particular (4) we can substitute 4 = #-+~?, and we have a function +of z and —with real and imaginary parts. For example, +3? = (+ iu)° = +? — uˆ + 2izg. (7.3) +Any function 4) can be writben as a sum of a pure real part and a pure +Imaginary part, each part a function of z and ø: +F#) = U(,.) +iV(#, 9). (7.4) +where (+, ) and V{(z, g) are real functions. Thus from any complex function (4) +two new functions (z,) and WV(z,) can be derived. For example, F4) = sŸ +gives us the two functions +U(#, 1U) = #Ÿ — (7.5) +V{(z, 9) = 2+. (7.6) +Now we come to a miraculous mathematical theorem which is so delightful +that we sha]ll leave a proof of i9 for one of your courses in mathematics. (We +--- Trang 92 --- +should not reveal all the mysteries of mathematies, or that subject matter would +become too dull.) It is this. For any “ordinary function” (mathematicians will +defñne it better) the functions Ứ and V øutomaticaliu satisfy the relations +9U ØV +T=<, (7.7) +Ø@V ØU +=—=——-- (7.8) +Tt follows immediately that cach of the functions U and V satisfy Laplace's +equation: +2U Ø?U +—=s +—=s=0. (7.9) +8z2 ôÔy2 +93V Ø3V +=s+.a¬z—=0, (7.10) +3z2 — ÐØụy2 +These equations are clearly true for the functions of (7.5) and (7.6). +Thus, starting with any ordinary function, we can arrive at ©6wo functions +U(z,) and V(z,g), which are both solutions of Laplace°s equation in two +dimensions. Each function represents a possible electrostatic potential. We can +pick a1 funection (4) and it should represent søzne electric ñeld problem——in +fact, #o problems, because Ữ and V each represent solutions. We can write +down as many solutions as we wish—by just making up functions—then we just +have to ñnd the problem that goes with each solution. It may sound backwards, +but is a possible approach. +/ X \ +E7 1Ð AV) +Mộ lã ` +<Š / ` ồ +“*3x >^‹/ ! si \X >⁄ +X X *⁄2 ⁄ — \ à X⁄< +P ZS3z \ ) ` ^ ` +XS XS? ự 1N 2V ca +~ _¬Z ⁄ =1 B=1 X >> ~ +“Ì \ À~ A=0 A=0 "¿_ | Ị" +“la la là Ìđ AIL 2| 3Ÿ 4| - +_JƑ 7¬ -jÐm=i B=-1L_ TY +> = ^\ B=0 Z _ - +¬ ` /"⁄⁄2 A=0 A=%C 2N 2 “- +` \ A=—1/ ⁄ ~ +` = 3 -3 ~ - +SN ⁄.N#⁄ +ˆ X4 M =s +» TP ẴN \ / ⁄Z < +3S XxX \ La! ⁄ `X< +Ñ ` / 2 +Fig. 7-1. Two sets of orthogonal curves which can represent equipo- +tentials in a two-dimensional electrostatic field. +As an example, lets see what physics the function Ƒ{4) = ¿2 gives us. From +it we get the two potential functions of (7.5) and (7.6). To see what problem the +function belongs to, we solve for the equipotential surfaces by setting Ứ = A, +a constant: +xz?—2= A. +This is the equation of a rectangular hyperbola. For various values of A, we +get the hyperbolas shown in Eig. 7-I. When A = 0, we get the special case of +diagonal straight lines through the origin. +--- Trang 93 --- +⁄ €ONDUCTOR + +etc. lÌ ÍÌ etc. +_¬ z7. ZZ7Z7Z7ZZZZZZZ::-› +Fig. 7-2. The field near the point € is V.. — +the same as that In Fig. 7-1. +Such a set of equipotentials corresponds to several possible physical situations. +Flirst, it represents the fñne details of the field near the point halfway between two +equal point charges. Second, it represents the feld at an inside right-angle corner +of a conductor. lf we have two electrodes shaped like those in Fig. 7-2, which are +held at diferent potentials, the field near the corner marked Œ will look just like +the fñeld above the origin in Fig. 7-I. The solid lines are the equipotentials, and +the broken lines at right angles correspond to lines of E. Whereas at points or +protuberaneces the electric fñeld tends to be hiph, ít tends to be ioœ in dents or +hollows. +'The solution we have found also corresponds to that for a hyperbola-shaped +electrode near a right-angle corner, or for two hyperbolas at suitable potentials. +You will notice that the fñeld of Eig. 7-I has an interesting property. The zø- +component of the electric fñeld, „, is given by +Ty — `. = —21. +'The electric field is proportional to the distance from the axis. This fact is used to +make devices (called quadrupole lenses) that are useful for focusing particle bearms +(see Section 29-7). The desired feld is usually obtained by using four hyperbola $@=+V +shaped electrodes, as shown in Eig. 7-3. Eor the electric fñeld lines in Eig. 7-3, +we have simply copied from Eig. 7-1 the set of broken-line curves that represent +V = constant. We have a bonusl "The curves for V = constant are orthogonal to +the ones for = constant because of the equations (7.7) and (7.8). Whenever +we choose a function #4), we get from and V both the equipotentials and @=-V @=—V +fñeld lines. And you will remermber that we have solved either of two problems, +depending on which set of curves we call the equipotentials. +As a second example, consider the function +#) = v3. (7.11) ⁄ CONDUCTOR +. @=+V +TÍ we write +ậ—=#+iu= 0c", Fig. 7-3. The field in a quadrupole lens. +tan 8 = /+, +Fq) — p1⁄2e9/2 +—= p2 (eo Ÿ + jsin +=p (e 2 + ;sin 2): +--- Trang 94 --- +B=4 ' A=4 = +/ ⁄ _“ +⁄ ⁄) ~“ +? B=3⁄ | A=3 ~ +/ ⁄ : x“ +/ »> +\ \ » T~ +: \ ` T——-_ +N SN ¬ Fig. 7-4. Curves of constant U(x, y) +` | ¬ and V(x, y) from Edq. (7.12). +\ | ` T^ ` +N ` ` ~ +from which +2 291/2 1/2 2 241/2 _— „11⁄2 +4“ + +# .|(œ“ + H5 +#4) = (@ +) “+z -+ÿ (+) -# . (7.12) +The curves for (z,) = A and V{(z,9) = Ö, using U and V from Eq. (7.12), +are plotted in Eig. 7-4. Again, there are many possible situations that could be +described by these fields. One of the most interesting is the field near the edge +of a thin plate. If the line = 0—to the right of the -axis—represents a thin +charged plate, the field lines near i% are given by the curves for various values +of A. The physical situation is shown in Eig. 7-5. +Further examples are +F) = 377, (7.13) +which yields the fñeld ou#side a rectangular corner +#§) =In, (7.14) +which yields the fñeld for a line charge, and +F) = 1/ạ, (7.15) +which gives the field for the two-dimensional analog of an electric dipole, 1.e., Ewo ___ +parallel line charges with opposite polarities, very close together. GROUNDED +W©e will not pursue this subJect further in this course, but should emphasize +that although the complex variable technique is often powerful, it is limited to E +two-dimensional problems; and also, it is an indirect method. +7-3 Plasma oscillations +. . . . . . . . Fig. 7-5. The electric field near the edge +W© consider now some physical situations in which the fñeld is determined of a thin grounded plate. +neither by ñxed charges nor by charges on conducting surfaces, but by a com- +bination of two physical phenomena. In other words, the feld will be governed +simultaneously by two sets of equations: (1) the equations from electrostatics +relating electric felds to charge distribution, and (2) an equation from another +part of physics that determines the positions or motions of the charges in the +presence of the field. +The frst example that we will discuss is a dynamic one in which the motion +of the charges is governed by Newton”s laws. A simple example of such a +--- Trang 95 --- +situation occurs in a plasma, which is an ionized gas consisting of ions and free +electrons distributed over a region in space. 'Phe Ionosphere—an upper layer of +the atmosphere—is an example of such a plasma. "The ultraviolet rays from the +sun knock electrons of the molecules of the air, creating free electrons and ions. +In such a plasma. the positive ions are very mụch heavier than the electrons, so +we may neglect the ionic motion, in comparison to that of the electrons. +Let mọ be the density of electrons in the undisturbed, equilibrium state. +Assuming the molecules are singly ionized, this must also be the density of +positive lons, since the plasma is electrically neutral (when undisturbed). Ñow +we suppose that the electrons are somehow moved from equilibrium and ask what +happens. If the density of the electrons in one region is increased, they will repel +cach other and tend to return to their equilibrium positions. As the electrons +move toward their original positions they pick up kinetic energy, and instead of +coming to rest in their equilibriun confguration, they overshoot the mark. They +will oscillate back and forth. "The situation is similar to what occurs in sound +waves, in which the restoring force is the gas pressure. In a plasma, the restoring +force is the electrical force on the electrons. 5. P +To simplify the discussion, we will worry only about a situation in which +the motions are all in one dimension, say ø. Let us suppose that the electrons +originally at z are, at the instant ¿, displaced from theïr equilibrium positions +by a small amount s(+,£). Since the electrons have been displaced, their density TT Ax ——¬ +will, in general, be changed. 'Phe change in density is easily calculated. Referring , , +to Fig. 7-6, the electrons initially contained bebween the two planes ø and b have 5 ⁄⁄ +moved and are now contained between the planes ø“ and #. "The number of Hs sẲÁs +electrons that were between ø and Ù is proportional to mgAz; the sarme number | ⁄⁄⁄ +are now contained in the space whose width is Az-+ As. The density has changed Ị +to “==——x#s—————Ax+As————| +ngẪz Ttọ +n— Az+As — 1+(As/Az)' (7.16) Fig. 7-6. Motion in a plasma wave. The +electrons at the plane a move to z, and +Tf the change in density is smaill, we can write [using the binomial expansion those at b move to Đ. +for (1+) 1] +?t — Tìo ( Tp) (7.17) +We assume that the positive ions do not move appreciably (because of the mụuch +larger inertia), so their density remains mạ. Each electron carries the charge —qe, +so the average charge density at any poïnt is given by +Ø0 = ~(n— nìo)qe: +0 = nöqe ~— (7.18) +(where we have written the diferential form for As/Az). +'The charge density is related to the electric fñeld by Maxwell's equations, in +particular, +ÿ.E=f. (7.19) +Tf the problem is indeed one-dimensional (and if there are no other fields but the +one due to the displacements of the electrons), the electric ñeld # has a single +component #⁄„. Equation (7.19), togebher with (7.18), gives +ĐR„ _ noqe Ös: (7.20) +3z cọ. ỞZ +Integrating Eq. (7.20) gives +Đ„S= sự R, (7.21) +Since „ = 0 when s = 0, the integration constant #C is zero. +--- Trang 96 --- +'The force on an electron in the displaced position 1s +F„ẹ —————, (7.22) +a restoring force proportional to the displacement s of the electron. 'Phis leads +to a harmonie oscillation of the electrons. 'Phe equation of motion of a displaced +electron is 2 : +đ“s Troq2 +my =———. 7.23 +: d2 €0 ( ) +We fnd that s will vary harmonically. Its time variation will be as cos(p#, +or—using the exponential notation of Vol. [—as +c«t, (7.24) +The frequency of oscillation œ; is determined from (7.23): +2 nụq2 +6 = ——; (7.25) +and ïs called the plasmaø ƒrequencg. It 1s a characteristic number of the plasma. +When dealing with electron charges many people prefer to express their +answers in terms of a quantity e2 defned by +c? = —“— = 2.3068 x 10” ”Š newton-meterẺ. (7.26) +Using this convention, Eq. (7.25) becomes +=———, 7.27 +¬=-. (727) +which is the form you will fnd in most books. +'Thus we have found that a disturbance of a plasma. will set up free oscillations +of the electrons about their equilibrium positions at the natural frequenecy ứ;, +which is proportional to the square root of the density of the electrons. "The +plasma electrons behave like a resonant system, such as those we described in +Chapter 23 of Vol. I. +'This natural resonance of a plasma has some interesting efects. For example, +1ƒ one tries to propagate a radiowave through the ionosphere, one finds that it can +penetrate only if its frequeney is higher than the plasma frequency. Otherwise the +signal is refected back. We must use hiph frequencies if we wish to communicate +with a satellite in space. Ôn the other hand, 1Ý we wish to communicate with a +radio station beyond the horizon, we must use frequencies lower than the plasma +frequency, so that the signal will be refected back to the earth. +Another interesting example of plasma oscillations occurs in metals. In a +metal we have a contained plasma of positive Ions, and free electrons. The +density ?+o is very high, so œp is also. But ¡t should still be possible to observe +the electron oscillations. Now, according to quantum mechanies, a harmonic +oscillator with a natural frequency œ„ has energy levels which are separated +by the energy increment ñœ„. lf, then, one shoots electrons through, say, an +aluminum foïil, and makes very careful measurements of the electron energies on +the other side, one might expect to fnd that the electrons sometimes lose the +energy ñœ„ to the plasma oscillations. This does indeed happen. lt was first +observed experimentally in 1936 that electrons with energies oŸ a few hundred +to a few thousand electron volts lost energy in jumps when scattering from or +going through a thin metal foil. The efect was not understood until 1953 when +Bohm and Pines# showed that the observations could be explained in terms of +quantum excitations of the plasma oscillations in the metal. +* For some recent work and a bibliography see C. .J. Powell and J. B. Swann, Phs. Reu. +115, 869 (1959). +T7 +--- Trang 97 --- +7-4 Colloidal particles in an electrolyte +W© turn to another phenomenon in which the locations of charges are governed +by a potential that arises in part from the same charges. The resulting efects +inHuence in an important way the behavior of colloids. A colloid consists of a +suspension in water of small charged particles which, though microscopic, from +an atomie poïnt of view are still very large. If the colloidal particles were not +charged, they would tend to coagulate into large lumps; but because of their +charge, they repel each other and remain in suspension. +Now If there is also some salt dissolved in the water, it will be dissociated +into positive and negative ions. (Such a solution of ions is called an electrolyte.) +The negative lons are attracted to the colloid particles (assuming their charge +is positive) and the positive ions are repelled. We will determine how the lons +which surround such a colloidal particle are distributed in space. +'To keep the ideas simple, we will again solve only a one-dimensional case. lIf we +think of a colloidal particle as a sphere having a very large radius—on an atomic +scalel—we can then treat a small part oŸ its surface as a plane. (Whenever one is +trying to understand a new phenomenon ït is a good idea to take a somewhat +oversimplifed model; then, having understood the problem with that model, one +is better able to proceed to tackle the more exact calculation.) +W© suppose that the distribution of lons generates a charge density ø(+), and +an electrical potential ó, related by the electrostatie law V2@ = —/ø/co or, for +fields that vary in only one dimension, by +c? =_#, (7.28) +NÑow supposing there were such a potential Ø(z), how would the ions distribute +themselves in it? 'Phis we can determine by the principles of statistical mechanics. +Our problem then is to determine ở so that the resulting charge density from +sbatistical mechanics aÍso satisfies (7.28). +According to statistical mechanics (see Chapter 40, Vol. I), particles in thermal +equilibrium in a force field are distributed in such a way that the density ø of +particles at the position z is given by +n(#) = nọc U@)/ET. (7.29) +where (+) is the potential energy, & is Boltzmann's constant, and ?' is the +absolute temperature. +We assume that the ions carry one electronic charge, positive or negative. At +the distance zø from the surface of a colloidal particle, a positive ion will have +potential energy qe2(#), so that +U(#) = qe0(2). +The density of positive ions, œ, is then +nà (3) = nạc— %90)/ET, +Similarly, the density of negative ions is +n_() = nục†%9)/kT, +'The total charge density is +0 = qe†!+ — qe†T!'—; +p0 = qeng(c 19/87 — ¿†úc9/T), (7.30) +Combining this with Eq. (7.28), we fnd that the potential ¿ must satisfy +đ?¿ đe?T0 +—==— — (c %9/*T — c†de9/K?), 7.31 +dự? ¬ ( ) (7.31) +--- Trang 98 --- +Thịỉs equation is readily solved in general [multiply both sides by 2(dj/dz), and +integrate with respect to z], but to keep the problem as simple as possible, we +will consider here only the limiting case in which the potentials are smaill or the +temperature 7' is high. The case where ở is small corresponds to a dilute solution. +For these cases the exponent is small, and we can approximate ++ac6/KT — 1 + CÓ, 7.32 +e kT (7.32) +Equation (7.31) then gives +d?¿ 2nod +—s> =T1l— . 7.33 +1Ð — TQ 90) (7.33) +Notice that this time the sign on the right is positive. 'Phe solutions for ô are +not oscillatory, but exponential. +The general solution of Eq. (7.33) is += Ae */P + BeT*/ÐP, (7.34) +with k7 +D?=.—.. 7.35 +2nod2 (35) +The constants 4 and must be determined from the conditions of the problem. +In our case, must be zero; otherwise the potential would go to infinity for +large ø. 5o we have that += Ac"*/?, (7.36) +in which A is the potential at z = 0, the surface of the colloidal particle. +Fig. 7-7. The variation of the potential +near the surface of a colloidal particle. ƒ2 Is +the Debye length. +0 D 2D 3D x +The potential decreases by a factor 1/e each time the distance increases by Ï, +as shown in the graph of Fig. 7-7. The number J is called the Debwe length, and +1s a measure of the thickness of the ion sheath that surrounds a large charged +particle in an electrolyte. Equation (7.35) says that the sheath gets thinner with +increasing concentration oŸ the ions (nọ) or with decreasing temperature. +The constant 4 in Eq. (7.36) is easily obtained if we know the surface charge +density ơ on the colloid particle. We know that +E„ = E„(0) = ˆ. (7.37) +But # is also the gradient of ở: +t„(0)=— —| =+~, 7.38 +(0)=- | =+5 (7.38) +from which we get +A=“”., (7.39) +--- Trang 99 --- +Using this result in (7.36), we find (by taking z = 0) that the potential of the +colloidal particle 1s +ø(0) = “7. (7.40) +You will notice that this potential is the same as the potential difference across a +condenser with a plate spacing D and a surface charge density ø. +W©e have said that the colloidal particles are kept apart by their electrical +repulsion. But now we see that the field a little way from the surface of a particle +is reduced by the ion sheath that collects around it. If the sheaths get thin +enouph, the particles have a good chance of knocking against each other. They +will then stick, and the colloid will coagulate and precipitate out of the liquid. +trom our analysis, we understand why adding enough salt to a colloid should +cause it to precipitate out. The process is called “salting out a colloid.” +Another interesting example is the efect that a salt solution has on protein +molecules. A protein molecule is a long, complicated, and flexible chain of amino +acids. "The molecule has various charges on it, and it sometimes happens that +there is a net charge, say negative, which is distributed along the chain. Because +of mutual repulsion of the negative charges, the protein chain is kept stretched +out. Also, if there are other similar chain molecules present in the solution, +they will be kept apart by the same repulsive efects. We can, therefore, have +a suspension of chain molecules in a liquid. But if we add salt to the liquid +we change the properties of the suspension. As salt is added to the solution, +decreasing the Debye distance, the chain molecules can approach one another, +and can also coil up. If enough salt ¡is added to the solution, the chain molecules +will precipitate out of the solution. 'Phere are many chemical efects of this kind +that can be understood in terms of electrical forces. +7-5 The electrostatic ñeld of a grid +As our last example, we would like to describe another interesting property of +electric ñelds. It is one which is made use ofin the design of electrical instruments, +in the construction of vacuum tubes, and for other purposes. 'Phis is the character +of the electric fñeld near a grid of charged wires. To make the problem as simple +as possible, let us consider an array of parallel wires lying in a plane, the wires +beïng infinitely long and with a uniform spacing between them. +Tf we look at the feld a large distance above the plane of the wires, we see +a constant electric feld, Just as though the charge were uniformly spread over +a plane. Ás we approach the grid of wires, the field begins to deviate from the +uniform field we found at large distances from the grid. We would like to estimate +how close to the grid we have to be in order to see appreciable variations in +the potential. Figure 7-8 shows a rough sketch of the equipotentials at various +distances from the grid. The closer we get to the grid, the larger the variations. +As we travel parallel to the grid, we observe that the field fuctuates in a periodic +Immanner. +Fig. 7-8. Equipotential surfaces above a +uniform grid of charged wires. +--- Trang 100 --- +NÑow we have seen (Chapter 50, Vol. I) that any periodic quantity can be +expressed as a sum of sine waves (EFourier”s theorem). Let”s see if we can fñnd a +suitable harmonie function that satisfes our fñeld equations. +T the wires lie in the zz-plane and run parallel to the -axis, then we might +try terms like +ð(,z) = Fa(2)cos ““Ẽ, (7.41) +where ø is the spacing of the wires and ø is the harmonic number. (We have +assumed long wires, so there should be no variation with .) A complete solution +would be made up of a sum of such terms for ?ø+ = 1, 2, 3,.... +T this is to be a valid potential, it must satisfy Lbaplace's equation in the +region above the wires (where there are no charges). That is, +9z2 — Ôz2 +Trying this equation on the ở in (7.41), we fnd that +4n2n2 2mnz d21, 27n4 +— _—.. €os —— + "2z C08 —— = 0, (7.42) +or that F„(2) must satisfy +d?F,„ 4mn2 +So we must have +Hạ = Aae 7/9, (7.44) +—— (7.45) +Zn = R * +W© have found that ïf there is a Fourier component of the field of harmonie n, fhø‡ +component will decrease exponentially with a characteristic distance zo = œ/27mm. +Eor the first harmonic (œ = 1), the amplitude falls by the factor e”2” (a large +decrease) each tỉme we increase z by one grid spacing ø. The other harmonics fall +of even more rapidly as we move away from the grid. We see that if we are only +a few times the distance ø away from the grid, the fñeld is very nearly uniform, +1.e., the oscillating terms are smaill. Thhere would, of course, aÌways remain the +“zero harmonic” fñeld +óo = — EoZ +to give the uniform field at large z. For a complete solution, we would combine +this term with a sum of terms like (7.41) with F„ rom (7.44). The coeficients Á„ +would be adjusted so that the total sum would, when differentiated, give an +electric ñeld that would ft the charge density À of the grid wires. +The method we have just developed can be used to explain why electrostatic +shielding by means oŸ a screen is often just as good as with a solid metal sheet. +Except within a distance from the screen a few times the spacing of the screen +wires, the fields inside a closed screen are zero. We see why copper screen—— +lighter and cheaper than copper sheet——is often used to shield sensitive electrical +equipment from external disturbing fields. +--- Trang 101 --- +MliocfrosteaffC Frnor'4t/ +8-1 The electrostatic energy of charges. Á uniform sphere +In the study of mechanies, one of the most interesting and useful discoveries 8-1 The electrostatic energy of +was the law of the conservation of energy. The expressions for the kinetic and charges. Á uniform sphere +potential energies of a mechanical system helped us to discover connections §-2_ The energy ofa condenser. Forces +between the states of a system at two diferent times without having to look into on charged conductors +the details of what was occurring in between. We wish now to consider the energy 8-3 The electrostatic energy of an +of electrostatic systems. In electricity also the principle of the conservation of ionic crystal +energy will be useful for discovering a number of interesting things. : : : +. ¬Ằ có. ; 8-4 Electrostatic energy in nuclei +'The law of the energy of interaction in electrostatics is very simple; we have, Ộ l +in fact, already discussed it. Suppose we have two charges g¡ and g¿ separated by 8õ Energy in the electrostatic feld +the distance ra. 'Phere is some energy in the system, because a certain amount 8-6 The energy ofa point charge +of work was required to bring the charges together. We have already calculated +the work done in bringing two charges together from a large distance. lt is +_—, (8.1) +4m €0T12 +W© also know, from the prineciple of superposition, that if we have many charges Reuicu: Chapter 4, Vol. Ï, Conserua- +present, the total force on any charge is the sum of the forces from the others. lt tion 0ƒ Energu +follows, therefore, that the tota]l energy of a system of a number of charges is the Chapters 13 and 14, Vol. I, +sum of terms due to the mutual interaction of each pair of charges. lỶ g; and g; Work and Potential Energụ +are any two of the charges and r7¿; is the distance between them (Fig. 8-1), the +energy of that particular païr is +Tay (8.2) +The total electrostatic energy is the sum of the energies of all possible pairs oŸ ° ° +charges: S So +q;q; O +U= » Trong” (8.3) dø ° o +all pairs ¬ +Tƒ we have a distribution of charge specifed by a charge density ø, the sum of ° So S N Hy 9 © +Eq. (8.3) is, of course, to be replaced by an integral. N +We shall concern ourselves with two aspects of this energy. (One is the S O ¬ +application of the concept of energy to electrostatic problems; the other is the 'Ow +cualuation of the energy in different ways. 5ometimes iE is easier to compute the o o ° +work done for some special case than to evaluate the sum in Eq. (8.3), or the +corresponding integral. As an example, let us calculate the energy required to Flg. 8-1. The electrostatic energy of a +assemble a sphere of charge with a uniform charge density. The energy isjust — SvStem of particles is the sum of the elec- +the work done in gathering the charges together from infinity. trostatlc energy of cach palr. +lImagine that we assemble the sphere by building up a succession of thin +spherical layers of infinitesimal thickness. At each stage of the process, we gather +a small amount of charge and put it in a thin layer om z to r + dr. W© continue +the process until we arrive at the fñnal radius ø (Fig. S-2). IÝ Q„ ¡is the charge of +the sphere when it has been built up to the radius z, the work done in bringing +a charge đ@) to it is +dỤ = h9, (8.4 +47cogTr +--- Trang 102 --- +Tf the density of charge in the sphere is ø, the charge QQz 1s +and the charge đ@) is +dQ = p- 4mr? dr. +Equation (S.4) becomes +4mp2r` dr ⁄⁄⁄Z ⁄⁄) R_ dQ +đỮ = —————. 8.ð < P +: ° (01 +The total energy required to assemble the sphere is the integral of đỮ from + = 0 2 +tO?= đa, OT >4 +4mnp2a5 +U=-———. 8.6 +lỗco ' ) : - +Ór ïf we wish to express the result in terms of the total charge Q of the sphere, Flg. 8-2. The energy of a uniform sphere +of charge can be computed by imagining +3 Q2 that it is assembled from successive spherical +U=-_——. 8.7 +5 4mcod ( ) shells. +'The energy is proportional to the square of the total charge and inverselÌy propor- +tional to the radius. We can also interpret Eq. (8.7) as saying that the average +of (1/7¿z) for all pairs of points in the sphere is 6/5a. +8-2 The energy of a condenser. Forces on charged conductors +W© consider now the energy required to charge a condenser. If the charge Q +has been taken from one of the conductors of a condenser and placed on the +other, the potential diference bebween them is +V=~, 8.8 +: (3.8) +where is the capacity of the condenser. How much work is done in charging +the condenser? Proceeding as for the sphere, we imagine that the condenser +has been charged by transferring charge from one plate to the other in small +Increments đ@). The work required to transfer the charge đ) is +dU = Vdq. +Taking V rom Ead. (8.8), we write +dŨ = ——. +Ór integrating from zero charge to the fñnal charge Q, we have +U=-—. 8.09 +5Ø (8.9) +This energy can also be written as +U = 3CV”. (8.10) +Recalling that the capacity oŸ a conducting sphere (relative to inũnity) is +šsphere — 47cod, +we can immediately get rom E4q. (8.9) the energy of a charged sphere, +U=-_——. 8.11 +2 4mcoa ( ) +'This, of course, is also the energy of a thin sphericøk shell of total charge Q and +is just 5/6 of the energy of a wniƒormi charged sphere, Bq. (8.7). +--- Trang 103 --- +W©e now consider applications of the idea of electrostatic energy. Consider +the following questions: What is the force between the plates of a condenser? +Or what is the torque about some axis of a charged conductor in the presence +of another with opposite charge? Such questions are easily answered by using +our result Eq. (8.9) for electrostatic energy of a condenser, together with the +principle of virtual work (Chapters 4, 13, and 14 of Vol. I). +Let's use this method for determining the force between the plates of a +parallel-plate condenser. lÝ we imagine that the spacing of the plates is increased +by the small amount Az, then the mechanical work done from the outside in +moving the plates would be +AW =FAz, (8.12) +where #' is the force between the plates. This work must be equal to the change +in the electrostatic energy of the condenser. +By Eq. (8.9), the energy of the condenser was originally +U= 2S, +The change in energy (ïf we do not let the charge change) is +AU=+@?A(3 (8.13) +—3 C7 : +Equating (8.12) and (S.13), we have +PAz=—A|—]. 8.14 +>> (814) +'This can also be written as +Tt'Az =—_>: AC. 8.15 +Ẩ 2Œ^2 ( ) +'The force, of course, results from the attraction of the charges on the plates, but +we see that we do not have to worry in detail about how they are distributed; +everything we need is taken care ofin the capacity Ở. +lt is easy to see how the idea is extended to conduectors of any shape, and for +other components of the force. In Eq. (8.14), we replace #' by the component we +are looking for, and we replace Az by a small displacement in the corresponding +direction. Ôr if we have an electrode with a pivot and we want to know the +torque 7, we write the virtual work as +AW =r A0, +where A0 is a small angular displacement. Of course, A(1/C) must be the change đị +in 1/C which corresponds to A0. We could, in this way, fnd the torque on the +movable plates in a variable condenser oŸ the type shown in Eig. 8-3. +Returning to the special case of a parallel-plate condenser, we can use the +formula we derived in Chapter 6 for the capacity: +—==—; 8.16 +lôi coA ( ) +Fig. 8-3. What is the torque on a variable +where 4 is the area of cach pÌate. IÝ we increase the separation by Az, capacitor? +A(+ì- Azs +Erom Eaq. (8.14) we get that the force between the plates is +tˆ'=_-—.. 8.17 +2cgA ( ) +--- Trang 104 --- +Let”s look at Eq. (S.17) a little more closely and see if we can tell how the +force arises. If for the charge on one plate we write +Eq. (8.17) can be rewritten as +?=-Q—. +2 Q €0 +©r, since the electric field between the plates is +To — =—, +”ˆ.=jQE¡. (8.18) +One would immediately guess that the force acting on one plate is the charge Q +on the plate times the field acting on the charge. But we have a surprising factor +of one-half. The reason is that #o is not the field ø¿ the charges. If we imagine ⁄ +that the charge at the surface of the plate occuples a thin layer, as indicated in +Fig. S-4, the fñeld will vary from zero at the inner boundary of the layer to Eoin — CONDUCIING LAYER OF +. : PLATE SURFACE +the space outside of the plate. The average field acting on the surface charges CHARGE ơ +is Eo/2. That is why the factor one-half is in Eq. (8.18). +You should notice that in computing the virtual work we have assumed that +the charge on the condenser was constant—that it was not electrically connected E +to other objects, and so the total charge could not change. -~ +uppose we had imagined that the condenser was held at a constant potential +diference as we made the virtual displacement. 'Phen we should have taken +|E| Eo +U = §CV” +and in place of Eq. (S.15) we would have had +FAz= $§V2ACŒ, +which gives a force equal in magnitude to the one in EBq. (8.15) (because V = Q/C), Flg. 8-4. The field at the surface of a +. " : conductor varies from zero to Eạ = đ/eo, +but with the opposite sien! Surely the force bebween the condenser plates doesn”t +¬ › . - . as one passes through the layer of surface +reverse in sign as we disconnect it from its charging source. Also, we know charge. +that bwo plates with opposite electrical charges must attract. "The principle of +virtual work has been incorrectly applied in the second case—we have not taken +into account the virtual work done on the charging source. “that is, to keep +the potential constant at V as the capacity changes, a charge W AC must be +supplied by a source of charge. But this charge is supplied at a potential V, so +the work done by the electrical system which keeps the potential constant is +V?AC. The mechanical work 'Az pius this electrical work V2 AC together +make up the change in the total energy sV2 AC of the condenser. "Therefore +F.Azis —3V? AC, as before. +8-3 The electrostatic energy of an ionic crystal +W©e now consider an application of the concept of electrostatic energy in +atomic physics. We cannot easily measure the forces between atoms, but we are +often interested in the energy diferences between one atomiec arrangement and +another, as, for example, the energy of a chemical change. 5ïnce atomic forces are +basically electrical, chemical energles are in large part Just electrostatic energies. +Let?s consider, for example, the electrostatic energy of an ionie lattice. An +ionic crystal like NaC] consists of positive and negative ions which can be thought +OŸ as rigid spheres. They attract electrically until they begin to touch; then there is +a repulsive force which goes up very rapidly if we try to push them closer together. +For our first approximation, therefore, we imagine a set of rigid spheres that +represent the atoms in a salt crystal. "The structure of the lattice has been +determined by x-ray difÑfraction. It is a cubie lattice—like a three-dimensional +--- Trang 105 --- +checkerboard. Figure 8-5 shows a cross-sectional view. The spacing of the ions is +2.81 Ä (— 2.81 x 10” em). +TỶ our picture of this system is correct, we should be able to check it by +asking the following question: How much energy will it take to puÌl all these +lons apart——that is, to separate the crystal completely into ions? "Phis energy +should be equal to the heat of vaporization of NaC1 plus the energy required to ">*S<<ˆ^~Z< <7 +dissociate the molecules into Ilons. “This total energy to separate NaGC] to Ilons X:XX:XX:XX +1s determined experimentally to be 7.92 electron volts per molecule. sing the K> <> <> <> <> <> <) +COnversion +1 eV = 1.602 x 10” joule, Ộ Ộ Ộ Ộ \ \ \ +and Avogadro's number for the number of molecules in a mole, \X:X-X:X:X:X-X +¬ j8ö666066. +the energy of dissociation can also be given as xà» < <> <> <> <> <\ +W = 7.64 x 10” joules/mole. ¬-. +Physical chemists prefer for an energy unit the kilocalorie, which is 4190 joules; Eiq. 8-5. Cross section of a salt cr/stal +so that 1 eV per molecule is 23 kilocalories per mole. AÁ chemist would then say on 3 atonic scale. The checkerboard sr- +that the dissociation energy of NaC] is rangement of Na and Cl ions is the same in +" the two cross sections perpendicular to the +W = 183 kcal/mole. one shown. (See Vol. l, Fig. 1-7.) +Can we obtain this chemical energy theoretically by computing how much +work it would take to puÌl apart the crystal? According to our theory, this work is +the sum of the potential energies of all the pairs of ions. he easiest way to fñgure +out this sum is to pick out a particular ion and compute its potential energy with +cach of the other ions. That will give us #w2ce the energy per ion, because the +energy belongs to the øø¿rs of charges. lÝ we want the energy to be associated +with one particular ion, we should take half the sum. But we really want the +energy øer rnolecule, which contains two ions, so that the sum we compute will +give directly the energy per molecule. +The energy of an ion with one of its nearest neighbors is e2/a, where e2 = +qŠ/4meo and a is the center-to-center spacing between ions. (We are considering +monovalent ions.) Thịis energy is 5.12 eV, which we already see is goỉng to give +us a result of the correct order of magnitude. But it is still a long way om the +infñnite sum of terms we need. +Let's begin by summing all the terms from the ions along a straight line. +Considering that the ion marked Nain Eig. 8-5 is our special ion, we shall consider +first those lons on a horizontal line with it. There are two nearest CÌ ions with +negative charges, each at the distance ø. hen there are two positive ions at the +distance 2a, etc. Calling the energy of this sum 1, we write +€ 2.2 2 2 +Ứu==—-|-“+<“—-“+“=... +tTn ( 113.3 1T ) +2c? 11 1 +=——|Ìl—==+;z—_—+---]. 8.19 +g ( 2 + 3 4 ) ) +The series converges slowly, so it is dificult to evaluate numerically, but ït is +known to be equal to ln2. So +U¡ =—“—In2= —1.386 —. (8.20) +Now consider the next adjacent line of ions above. The nearest is negative +and at the distance ø. Then there are two positives at the distance 2a. The +next pair are at the distance v5 a, the next at 10a, and so on. So for the whole +line we get the series +€ 1 2 2 2 +—[ + =-_-=+-=~-- ]: 8.21 +a ( 1 v2 v5 v10 ) (8.21) +--- Trang 106 --- +There are ƒowr such lines: above, below, in front, and in back. 'Then there are +the four lines which are the nearest lines on diagonals, and on and on. +Tf you work patiently through for all the lines, and then take the sum, you +ñnd that the grand total is : +U 1.747 —, +which is Jjust somewhat more than what we obtained in (8.20) for the first line. +Using e?/a = 5.12 eV, we get +U =—8.94 œV. +Our answer is about 10% above the experimentally observed energy. It shows +that our idea that the whole lattice is held together by electrical Coulomb forces +is fundamentally correct. This is the first time that we have obtained a specifc +property of a macroscopic substance from a knowledge of atomic physics. We +will do much more later. The subject that tries to understand the behavior of +bulk matter in terms of the laws of atomic behavior is called sol2d-state phụsic3. +Now what about the error in our calculation? Why is it not exactly right? +Tt is because oŸ the repulsion between the ions at close distances. They are not +perfectly rigid spheres, so when they are close together they are partly squashed. +They are not very soft, so they squash only a little bit. Some energy, however, +1s used in deforming them, and when the ions are pulled apart this energy is +released. 'Phe actual energy needed to pull the ions apart is a little less than the +energy that we calculated; the repulsion helps in overcoming the electrostatic +attraction. +ls there any way we can make an allowance for this contribution? We could +1Ý we knew the law of the repulsive force. We are not ready to analyze the +details of this repulsive mechanism, but we can get some idea of its characteristics +from some large-scale measurements. From a measurement of the cornpressibifitụ +of the whole crystal, it is possible to obtain a quantitative idea of the law of +repulsion between the ions and therefore of its contribution to the energy. In this +way iÈ has been found that this contribution must be 1/9.4 of the contribution +from the electrostatic attraction and, of course, of opposite sign. If we subtract +this contribution from the pure electrostatic energy, we obtain 7.99 eV for the +dissociation energy per molecule. It is much closer to the observed result of +7.92 eV, but still not in perfect agreement. There is one more thing we haven”§ +taken into account: we have made no allowance for the kinetic energy of the +crystal vibrations. lf a correction is made for this efect, very good agreement +with the experimental number is obtained. “The ideas are then correct; the major +contribution to the energy of a crystal like NaC] is electrostatic. +8-4 Electrostatic energy in nuclei +W© will now take up another example of electrostatic energy In atomie physics, +the electrical energy of atomic nuelei. Before we do this we will have to discuss +some properties of the main forces (called nuclear forces) that hold the protons +and neutrons together in a nucleus. In the early days of the discovery of nuclei— +and of the neutrons and protons that make them up——it was hoped that the law +of the strong, nonelectrical part of the force bebween, say, a proton and another +proton would have some simple law, like the inverse square law of electricity. Eor +once one had determined this law of force, and the corresponding ones between +a proton and a neutron, and a neutron and a neutron, it would be possible to +describe theoretically the complete behavior of these particles in nuclei. 'Pherefore +a big program was started for the study of the scattering of protons, in the hope +of ñnding the law of force between them; but after thirty years of efort, nothing +simple has emerged. A considerable knowledge of the force between proton and +proton has been accumulated, but we find that the force is as complicated as it +can possibly be. +'What we mean by “as complicated as it can be” is that the force depends on +as many things as it possibly can. +--- Trang 107 --- +First, the force is not a simple function of the distance between the two +probtons. Ất large distances there is an attraction, but at closer distances there is +a repulsion. “The distance dependence is a complicated function, still imperfectly +known. +Second, the force depends on the orientation of the protons' spin. The protons +have a spin, and any two interacting protons may be spinning with theïir angular a b +mmomenta in the same direction or in opposite directions. And the force is different Ộ © Ộ +when the spins are parallel rom what it is when they are antiparallel, as in (a) S +and (b) of Fig. 8-6. The diference is quite large; it is not a smaill efect. +Third, the force is considerably diferent when the separation of the bwo +protons is in the direction øarailel to their spins, as in (c) and (d) of Fig. 8-6, c d +than it is when the separation is in a direction perpendicular to the spins, as In Ộ Ộ +(a) and (b). +Fourth, the force depends, as it does In magnetism, on the velocity of the Ộ +protons, only much more strongly than in magnetism. And this velocity-dependent +force is not a relativistic efect; i% is strong even at speeds much less than the +speed of light. Eurthermore, this part of the force depends on other things besides +the magnitude of the velocity. Eor instance, when a proton is moving near another m- r -~—-~ +proton, the force is diferent when the orbital motion has the same direction of ⁄ Ộ ` ⁄ Ộ › +rotation as the spin, as in (e) of Fig. §-6, than when it has the opposite direction +Of rotation, as in (f). This is called the “spin orbit” part of the force. +The force between a proton and a neutron and between a neutron and a : +. l l Fig. 8-6. The force between two protons +neutron are also equally complicated. To this day we do not know the machinery depends on every possible parameter. +behind these forces—that is to say, any simple way of understanding them. +'There is, however, one important way in which the nucleon forces are sữnpler +than they could be. That is that the nœøweclear force between ©wo neutrons is the +same as the force between a proton and a neutron, which is the same as the +force bebtween two protonsl TỸ, in any nuclear situation, we replace a proton +by a neutron (or vice versa), the wwclear ứnkeractions are not changed. "The 10.61 +“fundamental reason” for this equality is not known, but it is an example of an +Important principle that can be extended also to the interaction laws of other sửa cS=== +strongly interacting particles—such as the r-mesons and the “strange” particles. +Thịs fact is nicely ïllustrated by the locations of the energy levels in similar g43— CS +nuclei. Consider a nucleus like B!! (boron-eleven), which is composed of fve +protons and six neutrons. In the nucleus the eleven particles interact with one +another in a most complicated dance. Now, there is one configuration of all the so — | +possible interactions which has the lowest possible energy; this is the normal +state of the nucleus, and is called the ground state. TÝ the nucleus is disturbed +(for example, by being struck by a high-energy probon or other particle) it can be 5.03 +put into any number of other configurations, called ezcited states, each of which +will have a characteristic energy that is higher than that of the ground state. In +nuclear physics research, such as is carried on with Van de Graaff generator (for +example, in Caltechs Kellogg and Sloan Laboratories), the energies and other +properties of these excited states are determined by experiment. The energles of 214 +the ffteen lowest known excited states of B!1 are shown in a one-dimensional = +graph on the left half of Fig. 8-7. The lowest horizontal line represents the ground +state. The frst excited state has an energy 2.14 MeV higher than the ground +state, the next an energy 4.46 MeV higher than the ground state, and so on. +The study of nuclear physics attempts to find an explanation for this rather B!' 1982 c1 +complicated pattern of energies; there is as yet, however, no complete general : +theory of such nuclear energy levels. Fig. 8-7. The energy levels of B!! and +TỶ we replace one of the neutrons in B†! with a proton, we have the nucleus of C† (energies in MeV). The ground state of +an isotope of carbon, C!!, 'The energies of the lowest sixteen excited states o£ C11 C'' is 1.982 MeV higher than that of B. +have also been measured; they are shown in the right half of Fig. 8-7. (The broken +lines indicate levels for which the experimental information is questionable.) +Looking at Eig. 8-7, we see a striking similarity between the pattern of the +energy levels in the two nuclei. The fñrst excited states are about 2 MeV above +the ground states. There is a large gap of about 2.3 MeV to the second excited +state, then a small jump of only 0.5 MeV to the third level. Again, bebtween +--- Trang 108 --- +the fourth and fifth levels, a big Jump; but between the fifth and sixth a tiny +separation of the order of 0.1 MeV. And so on. After about the tenth level, the +correspondence seems to become lost, but can still be seen if the levels are labeled +with their other defñning characteristics—for instance, their angular momentum +and what they do to lose their extra energy. +The striking similarity of the pattern of the energy levels of B!! and ClÍ is +surely not just a coincidence. ÏIt must reveal some physical law. It shows, in fact, +that even in the complicated situation in a nucleus, replacing a neutron by a +proton makes very little change. This can mean only that the neutron-neutron +and proton-proton forces must be nearly identical. ÔOnly then would we expect +the nuclear confgurations with fñve protons and six neutrons to be the same as +with six protons and five neutrons. +Notice that the properties of these two nuclei tell us nothing about the +neutron-proton force; there are the same number of neutron-proton combinations +in both nuelei. But if we compare two other nuclei, such as C†“, which has six +protons and eight neutrons, with NÑ!“, which has seven of each, we ñnd a similar +correspondence of energy levels. So we can conclude that the p-p, n-n, and p-n +forces are identical ín all their complexities. 'There is an unexpected principle in +the laws of nuclear forces. Even though the force bebween each pair of nuclear +particles is very complicated, the force between the three possible diferent pairs +is the same. +But there are some small diferences. The levels do not correspond exactly; +also, the ground state of C1! has an absolute energy (its mass) which is higher +than the ground state of B!! by 1.982 MeV. All the other levels are also higher +in absolute energy by this same amount. So the forces are not exactly equal. But +we know very well that the cormpie‡e forces are not exactly equal; there is an +clectrical force between two protons because each has a positive charge, while +between bwo neutrons there is no such electrical force. Can we perhaps explain +the điferences between B!! and C!! by the fact that the electrical interaction +of the protons is diferent in the two cases? Perhaps even the remaining minor +difÑferences in the levels are caused by electrical efects? Since the nuclear forces +are so mụch stronger than the electrical force, electrical efects would have only +a small perturbing efect on the energies of the levels. +In order to check this idea, or rather to fnd out what the consequences of this +idea are, we first consider the diference in the ground-state energies of the bwo +nuclei. To take a very simple model, we suppose that the nuclei are spheres of +radius 7 (to be determined), containing Z protons. If we consider that a nucleus +1s like a sphere with uniform charge density, we would expect the electrostatic +energy (from Eq. 8.7) to be +3(Z q)” +U= 5 Aner” (8.22) +where q is the elementary charge of the proton. Since Z is fve for B!! and six +for C!!, their electrostatic energies would be different. +With such a small number oŸ protons, however, Eq. (8.22) is not quite correct. +TỶ we compute the electrical energy between all pairs of protons, considered as +points which we assume to be nearly uniformly distributed throughout the sphere, +we find that in Eq. (S.22) the quantity Z2 should be replaced by Z(Z - 1), so +the energy is : +=3 7Œ ~ 14 _ 3 ZỨ - lộc (8.23) +5 _ 47cor 5 r +Tf we knew the nuclear radius z, we could use (8.23) to find the electrostatic +energy diference between B!! and C!!, But let's do the opposite; let”s instead +use the observed energy difference to compute the radius, assuming that the +energy diference is all electrostatie in origin. +That is, however, not quite right. The energy diference of 1.982 MeV bebween +the ground states of B!! and C!H includes the rest energies— that is, the en- +ergy mc2—of all the particles. In goïng from B†† to C1!, we replace a neutron by +a probon and an electron, which have less mass. 5o part of the energy difference +--- Trang 109 --- +1s the diference in the rest energies of a neutron and that of a proton plus an +electron, which ¡is 0.784 MeV. 'The diference, to be accounted for by electrostatic +energy, is thus more than 1.982 MeV; it is +1.982 MeV + 0.784 MeV = 2.766 MeV. +Using this energy in Eq. (8.23), for the radius of either B!! or C!! we find +r= 3.12 x 103 em. (8.24) +Does this number have any meaning? To see whether i% does, we should +compare it with some other determination of the radius of these nuclei. For +example, we can make another measurement of the radius of a nucleus by seeing +how It scatters fast particles. From such measurements it has been found, in fact, +that the đensity oŸ matter in all nuclei is nearly the same, i.e., their volumes are +proportional to the number of particles they contain. If we let A be the number +oŸ protons and neutrons in a nucleus (a number very nearly proportional to its +mass), it is found that its radius is given by +r— ALŠrg, (8.25) +ro = 1.2 x 1013 em. (8.26) +Erom these measurements we find that the radius of a B!! (or a C†) nueleus +is expected to be +r = (11)1⁄3(1.2 x 1013) em = 2.7 x 10”13 em, +Comparing this result with (8.24), we see that our assumptions that the energy +điferenece between B†!†! and C! is electrostatic is fairly good; the discrepaney is +only about 15% (not bad for our frst nuclear computation!). +The reason for the discrepancy is probably the following. According to the +current understanding of nuclei, an even number of nuclear particles——in the case +of B!!, ñve neutrons together with five protons—makes a kind of core; when one +more particle is added to this core, it revolves around on the outside to make a +new spherical nucleus, rather than being absorbed. If this is so, we should have +taken a different electrostatic energy for the additional proton. We should have +taken the excess energy of C!! over B!! to be just +which is the energy needed to add one more proton to the outside of the core. +Thịis number is just 5/6 of what Bq. (8.23) predicts, so the new prediction for the +radius is 5/6 of (8.24), which is in much closer agreement with what is directly +mneasured. +W© can draw two conclusions from this agreement. One is that the electrical +laws appear to be working at dimensions as small as 10”! em. The other is that +we have verified the remarkable coincidence that the nonelectrical part of the +forces between proton and proton, neutron and neutron, and proton and neutron +are all equal. +8-5 Energy in the electrostatic ñeld +WS now consider other methods of calculating electrostatic energy. They +can all be derived from the basic relation Eq. (8.3), the sum, over all pairs of +charges, of the mutual energies of each charge-pair. First we wish to write an +expression for the energy of a charge distribution. Äs usual, we consider that +cach volume element đV contains the element of charge odV. Then Eq. (8.3) +should be written h (Da) +U=r_ ——————dVidV:. 8.27 +2 J 4m €0T12 , ? ( ) +space 8-9 +--- Trang 110 --- +Notice the factor 3 which is introduced because in the double integral over +đVì and đV2 we have counted all pairs of charge elements twice. (There is no +convenient way of writing an integral that keeps track of the pairs so that each +pair is counted only once.) Next we notice that the integral over đW2 in (8.27) is +Jjust the potential at (1). That is, +Ti mì = a0), +4m €0T12 +so that (8.27) can be written as +=5 | ø(ó0)46. +Ór, since the point (2) no longer appears, we can simply write +U= si] sóaV (8.28) +This equation can be interpreted as follows. "The potential energy of the +charge øđV is the product of this charge and the potential at the same point. +The total energy is therefore the integral over @øđV. But there is again the +factor 3. Tt is still required because we are counting energies twice. 'Phe mutual +energies of two charges is the charge oŸ one times the potential at it due to the +other. z, it can be taken as the second charge times the potential at it from +the frst. Thus for two point charges we would write +U =iø(1) =ứi —— +7i€0T'12 +U = q92) = q4 1rephia' +Notice that we could also write +Ư = ÿ|ai9(1) + œ(2)]. (8.29) +The integral in (S.28) corresponds to the sum of both terms in the brackets +of (8.29). That is why we need the factor s. +An interesting question is: Where is the electrostatic energy located? One +might also ask: Who cares? What is the meaning of such a question? Tf there is +a païr of interacting charges, the combination has a certain energy. Do we need +to say that the energy is located at one of the charges or the other, or at both, +or in between? 'Phese questions may not make sense because we really know only +that the total energy is conserved. 'Phe idea that the energy is located someuhere +1S IOẲ TIec©sSary. +Yet suppose that it đzd make sense to say, in general, that energy is located +a% a certain place, as it does for heat energy. We might then ez¿end our principle +of the conservation of energy with the idea that if the energy in a given volume +changes, we should be able to account for the change by the fow oÝ energy into +or out of that volume. You realize that our early statement of the prineiple of +the conservation of energy is still perfectly all right If some energy disappears at +one place and appears somewhere else far away without anything passing (that +is, withoub any special phenomena occurring) in the space between. We are, +therefore, now discussing an extension of the idea of the conservation of energy. +W©e might call it a principle of the Íocal conservation of energy. Such a principle +would say that the energy in any given volume changes only by the amount that +fows into or out of the volume. lt is indeed possible that energy 1s conserved +locally in such a way. Tf it is, we would have a much more detailed law than the +simple statement of the conservation of total energy. I% does turn out that in +nature energu 1s conserued locallu. We can fnd formulas for where the energy is +located and how ït travels from place to place. +--- Trang 111 --- +There is also a phs?cœÏ reason why it is imperative that we be able to say +where energy is located. According to the theory of gravitation, all mass is a +source of gravitational attraction. We also know, by #2 = mc2, that mass and +energy are equivalent. All energy is, therefore, a source of gravitational foree. +Tƒ we could not locate the energy, we could not locate all the mass. We would +not be able to say where the sources of the gravitational field are located. 'Phe +theory of gravitation would be incomplete. +TÍ we restrict ourselves to electrostatics there is really no way to tell where the +energy is located. 'Phe complete Maxwell equations of electrodynamics give us +much more information (although even then the answer is, strictly speaking, not +unique.) We will therefore discuss this question in detail again in a later chapter. +We will give you now only the result for the particular case of electrostatics. 'T he +energy is located in space, where the electric field is. 'Phis seems reasonable +because we know that when charges are accelerated they radiate electric fields. +We would like to say that when light or radiowaves travel from one point to +another, they carry their energy with them. But there are no charges in the +waves. So we would like to locate the energy where the electromagnetic field 1s +and not at the charges from which it came. We thus describe the energy, not in +terms of the charges, but in terms of the felds they produce. We can, in fact, +show that bq. (8.28) is mưmnericallu equal to +U= $9 [E- bar (8.30) +W© can then interpret this formula as saying that when an electric field is present, +there is located in space an energy whose đensify (energy per unit volume) is +€ọ cọE2 +=—E.E=-—_—. 8.31 +-= 2 (831) ~ +'This idea is illustrated in Fig. 8-8. +To show that Ea. (8.30) is consistent with our laws of electrostatics, we begin +by introduecing into Eq. (8.28) the relation between ø and ở that we obtained in +Chapter 6: E +0—=—€o VWˆ2¿. +We get dV +¬= CỮ, +U= -ÿ Jøy ằ@dV. (8.32) +'Writing out the components of the integrand, we see that 2 +04022 Ø?J +$V°¿= sl= + ð„2 + 9z2 Fig. 8-8. Each volume element dV = +2 2 2 dx dy đz In an electric field contains the +- (¿224 _(#\.26#2A-(/25)-2623- (ề energy (eo/2)E? dV. +Øz (` Øz Øz Øw\~ Øyụ Øy Øz\ ` Øz Øz +=V:(óVó) — (Võ) - (Với). (8.33) +Our energy integral is then +U= + J(Vø)-(Vó)dV — | V-:(@Vó) áV. +We© can use Gauss' theorem to change the second integral into a surface integral: +Jv -(@Wð) dV = J (ó Vó) -n da, (8.34) +vol. surface +We evaluate the surface integral in the case that the surface goes to infinity +(so the volume integrals become integrals over all space), supposing that all the +charges are located within some fñnite distance. The simple way to proceed is to +--- Trang 112 --- +take a spherical surface of enormous radius # whose center is at the origin of +coordinates. We know that when we are very far away from all charges, ở varies +as 1/R and Vọ as 1/R2. (Both will decrease even faster with #† if there the +net charge in the distribution is zero.) Since the surface area of the large sphere +increases as ƒ?2, we see that the surface integral falls of as (1/R)(1/R2)R2 = (1/R) +as the radius of the sphere increases. So if we include all space in our integration +(J — oo), the surface integral goes to zero and we have that +U=S J (Vó) -(Vð)dV =9 [ B: Bát, (8.35) +all all +space space +We© see that it is possible for us to represent the energy of any charge distribution +as being the integral over an energy density located in the fñeld. +8-6 The energy of a point charge +Our new relation, Đq. (8.35), says that even a single point charge g will have +some electrostatic energy. In this case, the electric field is given by +b=_ T.., +47cor2 +So the energy density at the distance z from the charge is +€0 E2 s— q? +2 32m2cgr1' +We can take for an element of volume a spherical shell of thickness dz and +area 4mr?. The total energy is +_x 2 2 1 r=oo +U= [ gn#=-gcÿ (8.36) +8zegr2 87€o T|„—o +Now the limit at z = co gives no difficulty. But for a point charge we are +supposed to integrate down to r —= 0, which gives an infinite integral. Equa- +tion (S.35) says that there is an infnite amount oŸ energy in the feld of a point +charge, although we began with the idea that there was energy only be#ueen +point charges. In our original energy formula for a collection of point charges +(Eq. 8.3), we did not include any interaction energy of a charge with itself. What +has happened is that when we went over to a continuous distribution of charge +in Eq. (8.27), we counted the energy of interaction of every ?nfinitesimal charge +with all other infinitesimal charges. The same account is included in Eq. (8.35), +so when we apply it to a fimite point charge, we are including the energy it would +take to assemble that charge from infinitesimal parts. You will notice, in fact, +that we would also get the result in Eq. (S.36) if we used our expression (8.11) +for the energy of a charged sphere and let the radius tend toward zero. +W© must conclude that the idea of locating the energy in the fñeld is inconsistent +with the assumption of the existence of point charges. One way out o£the difficulty +would be to say that elementary charges, such as an electron, are not points but +are really small distributions of charge. Alternatively, we could say that there +1s something wrong in our theory of electricity at very small distaneces, or with +the idea of the local conservation of energy. There are dificulties with either +point of view. Thhese difficulties have never been overcome; they exist to this +day. Sometime later, when we have discussed some additional ideas, such as the +qmomentum in an electromagnetic field, we will give a more complete account of +these fundamental dificulties in our understanding oŸ nature. +--- Trang 113 --- +Mglocfricrftg, íra ho đmteospphor©e +9-1 The electric potential gradient of the atmosphere +Ơn an ordinary day over flat desert country, or over the sea, as One ØOes 9-1 The electric potential gradient of +upward from the surface of the ground the electrie potential increases by about the atmosphere +100 volts per meter. Thus there is a vertical electric ñeld # of 100 volts/m in 9-2_ Flectric currents in the +the air. The sign of the field corresponds to a negative charge on the earth”s atmosphere +surface. This means that outdoors the potential at the height of your nose is 9-3 Origin of the atmospheric +200 volts higher than the potential at your feetl You might ask: “Why don”t we currents +Just stick a pair of electrodes out in the air one meter apart and use the 100 volts +.— da » . . . 9-4 Thunderstorms +to power our electric lights?” Or you might wonder: “If there is reallgy a potential . +diference of 200 volts bebween my nose and my feet, why is it I don” get a shock 9-5 The mechanism of charge +when I go out into the street?” separatiom +We will answer the second question frst. Your body is a relatively good 9-6 Lightning +conductor. lÝ you are in contact with the ground, you and the ground will tend to +make one equipotential surface. Ordinarily, the equipotentials are parallel to the +surface, as shown in Fig. 9-1(a), but when you are there, the equipotentials are +đistorted, and the field looks somewhat as shown in Eig. 9-1(b). 5o you still have +very nearly zero potential diference between your head and your feet. There are +charges that come from the earth to your head, changing the fñeld. Some of them Refcrencc: Chalmers, Jj. Alan, Atmo- +may be discharged by ions collected from the air, but the current oŸ these is very spheric Plectricitụ, Perga- +small because air is a poor conductor. mon Press, London (1957). +330X 7 ¬ +30V _ _ _ _ TT — —_—_—_—— _.ẮẶẶẶẰ—Ằ— +„200 < _ — —_ > ¬ +200V CC 7C 7C C7 777777 7 ky c¬0V +N c— c— +|:- 100 V/m —_ cào _ _ ~ —— +=5 TT inlhliIaaaaaaaaa. — — +ZZZZZZ up ZZZZZz ⁄ GROUND ⁄ +(a) (b) +Fig. 9-1. (a) The potential distribution above the earth. (b) The potential +distribution near a man in an open flat place. +How can we measure such a fñield if the fñield is changed by putting something +there? 'Phere are several ways. One way is to place an insulated conductor at +some distance above the ground and leave it there until it is at the same potential +as the air. lÝ we leave i% long enough, the very small conductivity in the air will +let the charges leak off (or onto) the conductor until it comes to the potential at +its level. hen we can bring i% back to the ground, and measure the shift of its +potential as we do so. A faster way is to let the conductor be a bucket of water +with a small leak. As the water drops out, it carries away any excess charges +and the bucket will approach the same potbential as the air. (The charges, as you +know, reside on the surface, and as the drops come of “pieces of surface” break +of.) We can measure the potential of the bucket with an electrometer. +--- Trang 114 --- +There is another way to directly measure the potential građien#. Since there +is an electric field, there is a surface charge on the earth (ø = eo). If we place +a at metal plate at the earthˆs surface and ground it, negative charges appear | E +on i (Eig. 9-2a). TỶ this plate is now covered by another grounded conducting | | +cover ?Ö, the charges will appear on the cover, and there will be no charges on the +củ CONNECTION +original plate A. IỶ we measure the charge that ows om plate A to the ground TO GROUND \_ _ _ _— _ „ METAL PLATEA +(by, say, a galvanometer in the grounding wire) as we cover it, we can find the _.Ằ.ÃỶ.ằÃn-.. `. về xứ nG. +surface charge density that was there, and therefore also fnd the electric fñeld. (a) +Having suggested how we can measure the electric field in the atmosphere, we +now continue our description of it. Measurements show, first of all, that the field +continues to exist, but gets weaker, as one goes up to high altitudes. By about | | E | +50 kilometers, the feld is very small, so most of the potential change (the integral +Of #2) is at lower altitudes. The total potential diference from the surface of the „EOVER PLATE B +earth to the top of the atmosphere is about 400,000 volts. +V GROUND +9-2 Electric currents ïn the atmosphere +Fig. 9-2. (a) A grounded metal plate will +Another thing that can be measured, in addition to the potential gradient, is have the same surface charge as the earth. +the current in the atmosphere. 'Phe current density is small—about 10 micromi- (b) If the plate is covered with a grounded +croamperes crosses each square meter parallel to the earth. The air is evidently conductor it will have no surface charge. +not a perfect insulator, and because of this conductivity, a small current——caused +by the electric fñeld we have just been describing——passes from the sky down to +the earth. +Why does the atmosphere have conductivity? Here and there among the air +molecules there is an ion——a molecule of oxygen, say, which has acquired an extra +electron, or perhaps lost one. 'These ions do not stay as single molecules; because +of their electric fñeld they usually accumulate a few other molecules around them. +Each ion then becomes a little lump which, along with other lumps, drifts in the +ñeld—moving slowly upward or downward—making the observed current. Where +do the 7øns come from? It was first guessed that the ions were produced by the Ì ® ======dl. xe +radioactivity of the earth. (Ib was known that the radiation from radioactive _|+ đề — _AIR +materials would make air conducting by ionizing the air molecules.) Particles like ——Vv IONS¿‡— 7 —— +Ø-rays coming out of the atomic nuclei are moving so fast that they tear elecbrons - ` PS +from the atoms, leaving ions behind. 'This would imply, of course, that iŸ we were BÀ. +to go to higher altitudes, we should ñnd less ionization, because the radioactivity ELECTROMETER +1s all in the dirt on the ground——in the traces of radium, tranium, potassium, ch. Eig. 9-3. Measuring the conductivity of +To test this theory, some physicists carried an experiment up in balloons air due to the motion of ions. +to measure the ionization of the air (Hess, in 1912) and discovered that the +opposite was true—the ionization per unit volume #wcreased with altitudel (The +apparatus was like that of Fig. 9-3. 'The two plates were charged periodically to +the potential W. Due to the conductivity of the air, the plates slowly discharged; +the rate of discharge was measured with the electrometer.) This was a most +mysterious result—the most dramatic ñnding in the entire history of atmospherie +electricity. It was so dramatie, in fact, that it required a branching of of an +entirely new subject——cosmic rays. Atmospheric electricity itself remained less +dramatic. lonization was evidently being produced by something from outside +the earth; the investigation of this source led to the discovery of the cosmic +rays. W©e will not discuss the subject of cosmic rays now, except to say that they +maintain the supply ofions. Although the ions are being swept away all the time, +new ones are being created by the cosmic-ray particles coming from the outside. +To be precise, we must say that besides the ions made of molecules, there are +also other kinds of ions. Tỉny pieces of dirt, like extremely ñne bits of dust, Ñoat +in the air and become charged. They are sometimes called “nuclei” Eor example, +when a wave breaks in the sea, little bits of spray are thrown into the air. When +one of these drops evaporates, it leaves an infnitesimal crystal of NaC] foating +in the air. 'These tiny crystals can then pick up charges and become ions; they +are called “large ions.” +The small ions—those formed by cosmic rays—are the most mobile. Because +they are so small, they move rapidly through the air—with a speed of about +--- Trang 115 --- +1 cm/sec in a feld of 100 volts/meter, or 1 volt/cm. The much bigger and heavier +ions move much more slowly. It turns out that ifƒ there are many “nuclei,” they will +pick up the charges from the small ions. 'Then, since the “large ions” move so sÌlowÌy +in a fñeld, the total conduectivity is reduced. The conductivity of air, therefore, is +quite variable, since it is very sensitive to the amount of “dirt” there is in it. There +1s mụch more of such dirt over land——where the winds can blow up dust or where +man throws all kinds of pollution into the air—than there is over water. lt is not +surprising that from day to day, from moment to moment, from place to place, +the conductivity near the earth”s surface varies enormously. The voltage gradient +observed at any particular place on the earth”s surface also varies greatly because +roughly the same current Ñows down from high altitudes in diferent places, and +the varying conductivity near the earth results in a varying voltage gradient. +The conductivity of the air due to the drifting of ions also increases rapidly +with altitude—for two reasons. First of all, the ionization from cosmic rays +increases with altitude. Secondly, as the density of air goes down, the mean free +path of the ions increases, so that they can travel farther in the electric fñeld before +they have a collision——resulting in a rapid increase of conductivity as one goes up. +Although the electric current-density in the air is only a few micromicroam- +peres per square meter, there are very many square meters on the earth's surface. +The total electric current reaching the earth's surface at any time is very nearly +constant at 1800 amperes. This current, of course, is “positive”——it carrles plus +charges to the earth. So we have a voltage supply of 400,000 volts with a current +of 1800 amperes—a power of 700 megawattsl +With such a large current coming down, the negative charge on the earth CONDUCTIVITY +should soon be discharged. In fact, it should take only about half an hour to 50,000 m=~ ~C~==~=~~=~~~#= =ể~— +discharge the entire earth. But the atmospheric electric ñeld has already lasted | CURRENT +more than a hal£hour since is discovery. How is it maintained? What maintains 400,000 [Em +the voltage? And between what and the earth? There are many questions. VOLTS +The earth is negative, and the potential in the air is positive. If you go high +enouph, the conductivity is so great that horizontally there is no more chance SEA Ma... +for voltage varlations. "The air, for the scale of times that we are talking about, EARTH'S SURFACE +becomes effectively a conductor. Thịs OCCUTS aW a height in the neighborhood Fig. 9-4. Typical electrical conditions in +of 50 kilometers. 'Phis is not as high as what is called the “ionosphere,” in a clear atmosphere. +which there are very large numbers of ions produced by photoelectricity from the +sun. Nevertheless, for our discussions of atmospheric electricity, the air becomes +suficiently conductive at about 50 kilometers that we can imagine that there is +practically a perfect conducting surface at this height, from which the currents +come down. Our picture of the situation is shown in Pig. 9-4. The problem is: +How is the positive charge maintained there? How is it pumped back? Because E(V/m) +1 it comes down to the earth, it has to be pumped back somehow. 'That was one +of the greatest puzzles of atmospheric electricity for quite a while. 120 +tBach piece of information we can get should give a clue or, at least, tell +you something about it. Here is an interesting phenomenon: lÝ we measure „9 +the current (which is more stable than the potential gradient) over the sea, for 100 +instance, or in careful conditions, and average very carefully so that we get rid of +the irregularities, we discover that there is still a daily variation. 'Phe average of no +many measurements over the oceans has a variation with time roughly as shown +in Eig. 9-5. The current varies by about +15 percent, and ït is largest at 7:00 P.M. “———+——+>—az—a— +in London. 'Phe strange part of the thing is that no matter œhere you measure HOURS GMT +the current——in the Atlantic Ocean, the Pacific Ocean, or the Arctic Ocean—it is Fig. 9-5. The average daily variation of +at 1ts peak value when the clocks in London say 7:00 P.M.! All over the world the the atmospheric potential gradient on a clear +current is at its maximum at 7:00 P.M. London tỉme and it is at a minimum at day over the oceans; referred to Greenwich +4:00 A.M. London time. In other words, it depends upon the absolute time on the time. +earth, no upon the local time at the place of observation. In one respect this is not +mysterious; it checks with our idea that there is a very high conductivity laterally +at the top, because that makes it impossible for the voltage diference from the +ground to the top to vary locally. Any potential variations should be worldwide, +as indeed they are. What we now know, therefore, is that the voltage at the “top” +surface is dropping and rising by lỗ percent with the absolute time on the earth. +--- Trang 116 --- +9-3 Origin of the atmospheric currents +We must next talk about the source of the large negative currents which +must be fowing from the “top” $o the surface of the earth to keep charging ¡§ +up negatively. Where are the batteries that do this? The “battery” is shown in +Fig. . lE is the thunderstorm and ïts lightning. It turns out that the bolts of +lightning do not “discharge” the potential we have been talking about (as you +might at fñrst guess). Lightning storms carry +egaiiue charges to the earth. When +a lightning bolt strikes, nine times out of ten it brings down negative charges to +the earth in large amounts. lt is the thunderstorms throughout the world that +are charging the earth with an average of 1800 amperes, which is then being +discharged through regions of fair weather. +There are about 40,000 thunderstorms per day all over the earth, and we +can think of them as batteries pumping the electricity to the upper layer and +maintaining the voltage diference. Then take into account the geography of the +earth—there are thunderstorms in the afternoon in Brazil, tropical thunderstorms +in Africa, and so forth. People have made estimates of how much lightning is +striking world-wide at any time, and perhaps needless to say, their estimates +more or less agree with the voltage diference measurements: the total amount +of thunderstorm activity is highest on the whole earth at about 7:00 P.M. in +London. However, the thunderstorm estimates are very difficult to make and were +made only afer it was known that the variation should have occurred. 'These +things are very difficult because we don't have enough observations on the seas +and over all parts of the world to know the number of thunderstorms accurately. +But those people who think they “do it right” obtain the result that there are +about 100 lightning flashes per second world-wide with a peak in the activity at +7:00 P.M. Greenwich Mean 'Time. +_ÂW(:: on. / ¬¬ +P / ¬ +Ñ... .. : +Fig. 9-6. The mechanism that generates atmospheric electric field. [Photo by William L. Widmayer.] +--- Trang 117 --- +In order to understand how these batteries work, we will look at a thunderstorm +in detail. What is going on inside a thunderstorm? We will describe this insofar +as it is known. Ás we get into this marvelous phenomenon of real nature——instead +of the idealized spheres of perfect conductors inside of other spheres that we can +solve so neatly—we discover that we dont know very much. Yet it is really quite +exciting. Anyone who has been in a thunderstorm has enjoyed it, or has been +frightened, or at least has had some emotion. And in those places in nature where |25:990 =16C +we get an emotion, we find that there is generally a corresponding complexity -/2/./1 1. ` +and mystery about it. It is not goiïng to be possible to describe exactly how a 20.000 -.... +thunderstorm works, because we do not yet know very much. But we will try to ¬ . ¬ +disevibe a idle bậ abont hạt bappers +15,000 Á ... = - — . +In the first place, an ordinary thundersborm is made up of a number of “cells” ##—‡ ===—— - ——— +8< +fairly close together, but almost independent of each other. So i is best to +analyze one cell at a time. By a “cell” we mean a region with a limit area in the 5,000 ¬.. .... +17C +horizontal direction in which all of the basic processes occur. sually there are ¬¬————— +several cells side by side, and in each one about the same thing ¡is happening, +although perhaps with a diferent timing. Figure 9-7 indicates in an idealized - Ệ ZTCCONONCOCONCONOCONCONCOSCOSCESGCESCEC +fashion what such a cell looks like in the early stage of the thunderstorm. lt turns +out that in a certain place in the air, under certain conditions which we shall Ír¿zevaxoseaeÔ 12 g2. , sney +describe, there is a general rising of the air, with higher and higher velocities +near the top. As the warm, moist air at the bottom rises, it cools and the water Fig. 9-7. A thunderstorm cell in the early +vapor in it condenses. In the fgure the little stars indicate snow and the dots stages of development. [From U.S. Depart- +indicate rain, but because the updraft currents are great enough and the drops ment of Commerce Weather Bureau Report, +are small enough, the snow and rain do not come down at thịs stage. 'This is the dJune 1949.] +beginning stage, and not the real thunderstorm yet——in the sense that we don”t +have anything happening at the ground. At the same time that the warm air +rises, there is an entrainment of air from the sides—an important point which +was neglected for many years. Thus it is not just the air from below which is +rising, but also a certain amount of other air from the sides. +'Why does the air rise like this? As you know, when you go up in altitude +the air is colder. The ground is heated by the sun, and the re-radiation of heat +to the sky comes from water vapor high in the atmosphere; so at high altitudes +the air is cold——very cold——whereas lower down it is warm. You may say, “Phen À +1t)s very simple. Warm air is lighter than cold; therefore the combination is & ` +mechanically unstable and the warm aïr rises.” Of course, if the temperature Là ` +is diferent at difÑferent heights, the air 7s unstable ¿hermodwnamnscallu. Left to m SN R +1tself inñnitely long, the air would all come to the same temperature. But it is : ¬ S +not left to itself; the sun is always shining (during the day). So the problem is E ` d +indeed not one of thermodynamic equilibrium, but of mechancøl equilibrium. NI +3uppose we plot—as in Fig. 9-8—the temperature of the air against height above ĐỒNG 2 +the ground. In ordinary circumstances we would get a decrease along a curve ` +like the one labeled (a); as the height goes up, the temperature goes down. How ALTITUDE +can the atmosphere be stable? Why doesnt the hot air below simply rise up +into the cold air? The answer is this: if the air were to go up, its pressure would Fig. 9-8. Atmospheric temperature. +go down, and iŸ we consider a particular parcel oŸ air going up, it would be (a) Static atmosphere; (b) adiabatic cooling +expanding adiabatically. (There would be no heat coming in or out becausein — 9f®ry ai; (C) adiabatic cooling of wet air, +the large dimensions considered here, there isnˆt time for much heat fow.) Thus (d) wet alr with some mixing of ambient air. +the parcel of air would cool as it rises. Such an adiabatic process would give a +temperature-height relationship like curve (b) in Fig. 9-§. Any air which rose +from below would be coider than the environment it goes into. Thus there is no +reason for the hot air below to rise; if it were to rise, iÿ would cool to a lower +temperature than the air already there, would be heavier than the air there, and +would just want to come down again. Ôn a good, bright day with very little +humidity there is a certain rate at which the temperature in the atmosphere falls, +and this rate is, in general, lower than the “maximum stable gradient,” which is +represented by curve (b). The air is in stable mechanical equilibrium. +--- Trang 118 --- +On the other hand, if we think of a parcel of air that contains a lot of water |reer +vapor being carried up Into the aïr, its adiabatic cooling curve will be diferent. +As it expands and cools, the water vapor in i% will condense, and the condensing T77 Ta ` +waf6er will liberate heat. Moist air, therefore, does not cool nearly as much as dry Tá. ¬—— +air does. So 1Ý air that is wetter than the average starts to rise, its temperature Ca ca = NV, R +will follow a curve like (e) in Eig. 9-§. It will cool off somewhat, but will still TT TY NNg , Hh tt. “ ⁄”» +be warmer than the surrounding air at the same level. If we have a region of : NNN APEC a» +warm moist air and something starts it rising, it will always ñnd itself lighter and +warmer than the air around it and will continue to rise until it gets to enormous — —=. “+ — T— ` = ———— +heights. 'Phis is the machinery that makes the air in the thunderstorm cell rise. +For many years the thunderstorm cell was explained simply in this manner. |2 TT HH —== GIải c——=c +But then measurements showed that the temperature of the cloud at diferent c\L! 1 1.1. „ở% ¬-= - +heights was not nearly as high as indicated by curve (c). The reason is that as lisooo — “j2 0c +the moist air “bubble” goes up, it entrains air from the environment and is cooled đài | | Zà. an _ï +of by it. The temperature-versus-height curve looks more like curve (d), which | „„ \ . : / | ] TU +is much closer to the original curve (a) than to curve (©). Ẳ su mượn nhe... +After the convection just described gets under way, the cross section of | - „ * j 1/7 //“' +a thunderstorm cell looks like Fig. 9-9. We have what is called a “mature” [” Mi F .. +thunderstorm. “There is a very rapid updraft which, in this stage, goes up to .. ~m0070U VN >>» +about 10,000 to 15,000 meters—sometimes even much higher. The thunderheads, Š#%=====<<< ï0MÌNHAMVVNQ GEEE- - +with their condensation, climb way up out of the general cloud bank, carried by +an updraft that is usually about 60 miles an hour. As the water vapor is carricd |[PZZVdg Set 56x -IcCsiak +up and condenses, it forms tiny drops which are rapidly cooled tO temperatures Eig. 9-9. A mature thunderstorm cell +below zero degrees. 'They should freeze, but do not freeze Immediately—they +"+ . l [From U.S. Department of Commerce +are “supercooled.” Water and other liquids will usually cool well below their Weather Bureau Report, June 1949.] +freezing points before crystallizing If there are no “nuclei” present to start the +crystallization process. Ônly if there is some small piece of material present, like +a tỉny crystal of NaC], will the water drop freeze into a little piece of ice. 'Phen +the equilibrium is such that the water drops evaporate and the ice crystals øgrow. +'Thus at a certain point there is a rapid disappearance of the water and a rapid +buildup ofice. Also, there may be direct collisions between the water drops and +the ice—collisions In which the supercooled water becomes attached to the Ice +crystals, which causes it to suddenly crystallize. 5o at a certain point in the +cloud expansion there is a rapid accumulation of large ice particles. +'When the ice particles are heavy enough, they begin to fall through the rising +air—they get too heavy to be supported any longer in the updraft. As they come +down, they draw a little air with them and start a downdraft. And surprisingly +enough, it is easy to see that once the downdraft is started, it will maintain itself. +The air now drives itself downl +Notice that the curve (đ) in Eig. 9-8 for the actual distribution of temperature +in the cloud is not as steep as curve (c), which applies to web air. So iŸ we +have wet air falling, its temperature will drop with the sÌope oŸ curve (c) and +will go belou the temperature of the environment if it gets down far enough, as +indicated by curve (e) in the fñgure. The moment it does that, it is denser than +the environment and continues to fall rapidly. You say, ““Phat is perpetual motion. +first, you argue that the air should rise, and when you have i% up there, you +argue equally well that the air should fall” But ït isn't perpetual motion. When +the situation is unstable and the warm air should rise, then clearly something +has to replace the warm air. It is equally true that cold air coming down would +energetically replace the warm air, but you realize that what is coming down 1s +no‡ the original air. 'Phe early arguments, that had a particular cloud without +entrainment going up and then coming down, had some kind of a puz⁄zle. They +needed the rain to maintain the downdraft—an argument which is hard to believe. +As soon as you realize that there is a lot of original air mixed in with the rising +air, the thermodynamic argument shows that there can be a descent of the cold +air which was originally at some great height. 'Phis explains the picture of the +active thunderstorm sketched in Eig. 9-9. +As the air comes down, rain begins to come out of the bottom of the thun- +derstorm. In addition, the relatively cold air spreads out when it arrives at the +--- Trang 119 --- +earth's surface. So just before the rain comes there is a certain little cold wind +that gives us a fÍorewarning of the coming storm. In the storm itself there are +rapid and irregular gusts of air, there is an enormous turbulence in the cloud, +and so on. But basically we have an updraft, then a downdraft——in general, a +very complicated process. +The moment at which precipitation starts is the same moment that the large +downdraft begins and is the same moment, in fact, when the electrical phenomena +arise. Before we describe lightning, however, we can fñnish the story by looking +at what happens to the thunderstorm cell after about one-half an hour to an +hour. The cell looks as shown in Fig. 9-10. 'Phe updraft stops because there is +no longer enough warm air to maintain it. 'Phe downward precipitation continues +far a while, the last little bits of water come out, and things get quieter and +quieter——although there are small ice crystals left way up in the air. Because +the winds at very great altitude are in diferent directions, the top of the cloud +usually spreads into an anvil shape. The cell ecomes to the end of its life. +Z“ - - - — — Ế +40,000 _- T- _ _ _ -- -- _ C +L _ DRAFTS IN THIŠ REGION _ _ +35,000, LESS THAN 10 EEET PER SECOND -38C +15,000 L —————-_——-—_- 0C 0C +mê : +10,000 —————————— — +8C +Horizontal Scale_ Ô T/” Ïmj y Ran man: ¬— _ +18130 + Snow +_# 3 + ĐHÌU (vớt 3g + d +Draft Vector Scale A1 f/sec ~lee Crystals .=⁄⁄=⁄⁄=⁄⁄=⁄Z=⁄⁄=⁄⁄=⁄Z⁄=⁄⁄=⁄⁄=⁄ˆ] +Fig. 9-10. The late phase of a thunderstorm cell. Fig. 9-11. The distribution of electrical charges in a mature +[From U.S. Department of Commerce Weather Bu- thunderstorm cell. [From U.S. Department of Commerce +reau Report, June 1949.] Weather Bureau Report, June 1949.] +9-5 The mechanism of charge separation +We want now to discuss the most important aspect for our purposes—the +development of the electrical charges. lxperiments of various kinds——including +fying airplanes through thunderstorms (the pilots who do this are brave menl)— +tell us that the charge distribution in a thunderstorm cell is something like that +shown in Eig. 9-11. The top of the thunderstorm has a positive charge, and the +bottom a negative one—except for a small local region of positive charge in the +bottom of the cloud, which has caused everybody a lo of worry. No one seems to +know why it is there, how important it is—whether it is a secondary efect of the +positive rain coming down, or whether it is an essential part of the machinery. +Things would be much simpler ïf it weren't there. Anyway, the predominantly +negative charge at the bottom and the positive charge at the top have the correct +sign for the battery needed to drive the earth negative. “he positive charges are 6 +or 7 kilometers up in the air, where the temperature is about —20”°ƠC, whereas the +negative charges are 3 or 4 kilometers high, where the temperature is between +Zero and —10°Ỡ. +--- Trang 120 --- +The charge at the bottom of the cloud is large enough to produce potential +diferences of 20, or 30, or even 100 million volts between the cloud and the +earth—much bigger than the 0.4 million volts from the “sky” to the ground in a +clear atmosphere. 'These large voltages break down the aïr and create gianÈ are +discharges. When the breakdown occurs the negative charges at the bottom of +the thunderstorm are carried down to the earth in the lightning strokes. +Now we will describe in some detail the character of the lightning. Eirst +of all, there are large voltage diferences around, so that the air breaks down. +There are lightning strokes between one piece oŸ a cloud and another piece of +a cloud, or between one cloud and another cloud, or between a cloud and the +carth. In cach of the independent discharge fashes—the kind of lightning strokes +you see there are approximately 20 or 30 coulombs of charge brought down. +One question is: How long does i% take for the cloud to regenerate the 20 or +30 coulombs which are taken away by the lightning bolt? 'Phis can be seen by +measuring, far from a cloud, the electric fñield produced by the cloud's dipole +mmoment. In such measurements you see a sudden decrease in the fñeld when the +lightning strikes, and then an exponential return to the previous value with a +time constant which is slightly diferent for diferent cases but which is in the +neighborhood of ð seconds. It takes a thunderstorm only 5 seconds after each +lightning stroke to build its charge up again. hat doesnˆt necessarily mean +that another stroke is goiïng to occur in exactly 5 seconds every time, because, +of course, the geometry is changed, and so on. The strokes occur more or less x⁄ZZ*š3N\. +Irregularly, but the important point is that it takes about 5 seconds to recreate ZỐ ¬ +the original condition. 'PThus there are approximately 4 amperes of current in ` +the generating machine of the thunderstorm. This means that any model made +to explain how this storm generates its electricity must be one with plenty of +Julce—it must be a big, rapidly operating deviee. +Before we go further we shall consider something which is almost certainly +completely irrelevant, but nevertheless interesting, because it does show the efect x2; +of an electric fñeld on water drops. We say that it may be irrelevant because it 2 +relates to an experiment one can do in the laboratory with a stream of water ' +to show the rather strong efects of the electric fñeld on drops of water. In a +thunderstorm there is no stream of water; there is a cloud of condensing ice and +drops of water. So the question of the mechanisms at work in a thunderstorm TO WATER +1s probably not at all related to what you can see in the simple experiment we SUPPLY +will describe. If you take a small nozzle connected to a water faucet and direct it Fig. 9-12. A jet of water with an electric +upward at a steep angle, as in Fig. 9-12, the water will come out in a ñne stream field near the nozzle. +that eventually breaks up into a spray of fne drops. lf you now put an electric +fñeld across the stream at the nozzle (by bringing up a charged rod, for example), +the form of the stream will change. With a weak electric fñeld you will ñnd that +the stream breaks up into a smaller number of large-sized drops. But if you apply +a stronger field, the stream breaks up into many, many fne drops—smaller than +before.* With a weak electric field there is a tendency to inhibit the breakup of +the stream into drops. With a stronger fñeld, however, there is an increase in the +tendency to separate into drops. +The explanation of these efects is probably the following. If we have the +stream of water coming out of the nozzle and we put a small electric fñield across +it one side of the water gets slightly positive and the other side gets slightly +negative. Then, when the stream breaks, the drops on one side may be positive, +and those on the other side may be negative. 'They will attract each other and +will have a tendency to stick together more than they would have before—the +stream doesn”t break up as much. Ôn the other hang, ïf the field is stronger, the +charge in each one of the drops gets much larger, and there is a tendency for +the charge #sejƒ to help break up the drops through theïr own repulsion. Each +drop will break into many smaller ones, each carrying a charge, so that they are +all repelled, and spread out so rapidly. So as we increase the field, the stream +* A handy way to observe the sizes of the drops is to let the stream fall on a large thin +metal plate. The larger drops make a louder noise. +--- Trang 121 --- +becomes more fñnely separated. The only point we wish to make is that in certain +circumstaneces electric fñelds can have considerable infuence on the drops. The +exact machinery by which something happens in a thunderstorm is not at all +known, and is not at all necessarily related to what we have Jjust described. We +have included ït just so that you will appreciate the complexities that could come +into play. In fact, nobody has a theory applicable to clouds based on that idea. +'W©e would like to describe two theories which have been invented to account for +the separation of the charges in a thunderstorm. All the theories involve the idea +that there should be some charge on the precipitation particles and a difÑferent +charge in the air. Then by the movement of the precipitation particles—the +water or the ice—through the air there is a sebaration of electric charge. The +only question is: How does the charging of the drops begin? One of the older +theories is called the “breaking-drop” theory. Somebody discovered that If you +have a drop of water that breaks into two pieces in a windstream, there is positive +charge on the water and negative charge in the air. This breaking-drop theory +has several disadvantages, among which the most serious is that the sign is wrong. +Second, in the large number of temperate-zone thunderstorms which do exhibit +lightning, the precipitation efects at hiph altitudes are in ice, no in water. DROP +Hrom what we have just said, we note that iŸ we could imagine some way +for the charge to be diferent at the top and bottom of a drop and If we could E +also see some reason why drops in a high-speed airstream would break up into +unequal pieces—a large one in the front and a smaller one in the back because of +the motion through the air or something—we would have a theory. (Diferent +from any known theory!) Then the small drops would not fall through the air œ@` +as fast as the big ones, because of the air resistance, and we would get a charge @) +separation. You see, it is possible to concoct all kinds of possibilities. v +One of the more ingenious theories, which is more satisfactory in many respects LARGE IÖNS +than the breaking-drop theory, is due to Ơ. T. R. Wilson. We will describe it, +as Wilson did, with reference to water drops, although the same phenomenon Fig. 9-13. C.T. R. Wilson's theory of +would also work with ice. Suppose we have a water drop that is falling in the charge separation in a thundercloud. +electric ñeld of about 100 volts per meter toward the negatively charged earth. +The drop will have an induced dipole moment—with the bottom of the drop +positive and the top oŸ the drop negative, as drawn in Fig. 9-13. Now there are +in the air the “nuclei” that we mentioned earlier—the large slow-moving ions. +(The fast ions do not have an important efect here.) Suppose that as a drop +comes down, it approaches a large ion. If the ion is positive, it is repelled by the +positive bottom of the drop and is pushed away. So it does not become attached +to the drop. If the ion were to approach from the top, however, it might attach +to the negative, top side. But since the drop is falling through the air, there +is an air drift relative to it, going upwards, which carries the ions away iŸ their +motion through the air is slow enough. Thus the positive ions cannot attach at +the top either. This would apply, you see, only to the large, slow-moving 1ons. +The positive ions of this type will not attach themselves either to the front or +the back of a falling drop. Ôn the other hand, as the large, slow, negøtoe lons +are approached by a drop, they will be attracted and will be caught. The drop +will acquire negative charge—the sign of the charge having been determined by +the original potential diference on the entire earth—and we get the right sign. +Negative charge will be brought down to the bottom part of the cloud by the +drops, and the positively charged ions which are left behind will be blown to the +top oŸ the cloud by the various updraft currents. The theory looks pretty good, +and it at least gives the right sign. Also it doesn”t depend on having liquid drops. +We will see, when we learn about polarization in a dielectric, that pieces 0 ice +will do the same thing. They also will develop positive and negative charges on +their extremities when they are in an electric ñeld. +There are, however, some problems even with this theory. First of all, the +total charge involved in a thunderstorm is very hiph. After a short time, the +supply of large ions would get used up. So Wilson and others have had to propose +that there are additional sources of the large ions. Once the charge separation +starts, very large electric felds are developed, and in these large fñelds there may +--- Trang 122 --- +be places where the air will become ionized. If there is a highly charged point, +or any small object like a drop, it may concentrate the fñeld enough to make a +“brush discharge.” When there is a strong enough electric fñeld—let us say iW is +positive—electrons will fall into the field and wi]l pick up a lot of speed bebween +collisions. Their speed will be such that in hitting another atom they will tear +electrons of at that atom, leaving positive charges behind. 'Phese new electrons +also pick up speed and collide with more electrons. So a kind of chain reaction or +avalanche occurs, and there is a rapid accumulation of ions. The positive charges +are left near their original positions, so the net effect is to distribute the positive +charge on the point into a region around the point. 'Phen, of course, there is +no longer a strong fñeld, and the process stops. 'This is the character of a brush +discharge. It is possible that the fñields may become strong enough in the cloud +to produee a little bit of brush discharge; there may also be other mechanisms, +once the thing is started, to produce a large amount of ionization. But nobody +knows exactly how it works. So the fundamental origin of lightning is really not +thoroughly understood. We know it comes from the thunderstorms. (And we +know, of course, that thunder comes from the lightning—from the thermal energy +released by the bolt.) +At least we can understand, in part, the origin of atmospheric electricity. Due â sơn NÊb +to the air currents, ions, and water drops on ice particles in a thunderstorm, : hề +positive and negative charges are separated. The positive charges are carried Ệ Ầ +upward to the top of the cloud (see Eig. 9-11), and the negative charges are Ặ : ` +dumped into the ground ïn lightning strokes. The positive charges leave the top ¡: £# z x \ 8 +of the cloud, enter the high-altitude layers oŸ more highly conducting aïir, and ⁄/ \ +spread throughout the earth. In regions of clear weather, the positive charges in j ⁄ `. ` +this layer are slowly conducted to the earth by the ions in the air—ions formed À. : +by cosmic rays, by the sea, and by man”s activities. The atmosphere is a busy [Doys camera | ` ' +electrical machinel bờ: VNI 790701 ì +9-6 Lightning \ Lê =- Ñ \ +The frst evidence of what happens in a lightning stroke was obtained in `: . ị “#7 +photographs taken with a camera held by hand and moved back and forth with § SN ñ ẵ +the shutter open——while pointed toward a place where lightning was expected. 3. ›D L/ +The first photographs obtained this way showed clearly that lightning strokes are Ki E «4 +usually multiple discharges along the same path. Later, the “Boys” camera, which : +has #uo lenses mounted 1802 apart on a rapidly rotating disc, was developed. Fig. 9-14. Photograph of a lightning flash +The image made by each lens moves across the fñlm——the picture is spread out in taken with a "Boys” camera. [From Schon- +time. If, for instance, the stroke repeats, there will be two Images side by side. land, Malan, and Collens, Proc. Roy. Soc. +By comparing the images of the two lenses, it is possible to work out the details London, Vol. 152 (1935). +of the time sequence of the flashes. Figure 9-14 shows a photograph taken with +a “Boys” camera. +We will now describe the lightning. Again, we donˆt understand exactly how +1t works. We will give a qualitative description of what it looks like, but we won't +go into any details of :0hự it does what i9 appears to do. We will describe only the +ordinary case of the cloud with a negative bottom over fat country. Its potential +is much more negative than the earth underneath, so negative electrons will be +accelerated toward the earth. What happens is the following. It all starts with a +thing called a “step leader,” which is not as bright as the stroke of lightning. Ôn +the photographs one can see a little bright spot at the beginning that starts from +the cloud and moves downward very rapidly—at a sixth of the speed of lightl +lt goes only about 50 meters and stops. It pauses for about 50 microseconds, +and then takes another step. It pauses again and then goes another step, and +So on. Ït moves in a series of steps toward the ground, along a path like that +shown in Eig. 9-15. In the leader there are negative charges from the cloud; the +whole columm is full of negative charge. Also, the air is becoming ionized by the +rapidly moving charges that produce the leader, so the air becomes a conductor +along the path traced out. The moment the leader touches the ground, we have +a conducting “wire” that runs all the way up to the cloud and is full of negative +--- Trang 123 --- +charge. Now, at last, the negative charge of the cloud can simply escape and +run out. The electrons at the bottom of the leader are the fñrs% ones to realize ⁄ +this; they dump out, leaving positive charge behind that attracts more negative ⁄ +charge from higher up in the leader, which In its turn pours out, ebc. So finally — / = +all the negative charge in a part® of the cloud runs out along the column in a — CLOUD ⁄ ” +rapid and energetic way. So the lightning stroke you see runs 10ards from the = ⁄ _ +ground, as indicated in Eig. 9-16. In fact, this main stroke—by far the brightest ự +part——is called the return stroke. It is what produces the very bright light, and “ +the heat, which by causing a rapid expansion of the air makes the thunder clap. — +The current in a lightning stroke is about 10,000 amperes at its peak, and it \ +carries down about 20 coulombs. — +But we are still not finished. After a time of, perhaps, a few hundredths of — +a second, when the return stroke has disappeared, another leader comes down. = — +But this time there are no pauses. Ït ¡is called a “dart leader” this time, and it r- +goes all the way down—from top to bottom in one swoop. Ït goes full steam on +exactly the old track, because there is enough debris there to make it the easiest ` +route. 'Phe new leader is again full of negative charge. The moment it touches +the ground——zingl—there is a return stroke going straight up along the path. So 77VYV xế 277V 2^Z¿ +you see the lightning strike again, and again, and again. Sometimes it strikes +only once or bwice, sometimes five or ten times—once as many as 42 times on Fig. 9-15. The formation of the “step +the same track was seen——but always in rapid succession. leader.” +Sometimes things get even more complicated. For instance, after one oÝ is +pauses the leader may develop a branch by sending out #o steps——both toward +the ground but in somewhat diferent directions, as shown in Eig. 9-15. What +happens then depends on whether one branch reaches the ground defnitely +before the other. If that does happen, the bright return stroke (of negative charge _ ) ⁄ — +dumping into the ground) works its way up along the branch that touches the — 7 7 ⁄Z ” +ground, and when it reaches and passes the branching point on its way up to =———= << ⁄ Z +the cloud, a bright stroke appears to go doun the other branch. Why? Because +negative charge is dumping out and that is what lights up the bolt. Thịs charge `" +begins to move at the top of the secondary branch, emptying successive, longer _ +pleces of the branch, so the bright lightning bolt appears to work its way down +that branch, at the same time as it works up toward the cloud. Ilf, however, ˆ +one of these extra leader branches happens to have reached the ground almost “e +simultaneously with the original leader, it can sometimes happen that the đart _ +leader of the second stroke will take the second branch. 'Phen you will see the +first main flash ïn one place and the second flash in another place. Ït is a variant +of the original idea. +Also, our description is oversimplified for the region very near the ground. +'When the step leader gets to within a hundred meters or so from the ground, 1⁄7” 1⁄7 „ +there is evidence that a discharge rises from the ground to meet it. Presumably, l : : +the field gets big enough for a brush-type discharge to occur. Tf, for instance, Fig. 9-16. The return lightning stroke +there is a sharp object, like a building with a point at the top, then as the leader rụns back up the path made by the leader. +comes down nearby the fñelds are so large that a discharge starts from the sharp +point and reaches up to the leader. The lightning tends to strike such a poïnt. +lt has apparently been known for a long time that high objects are struck by +lightning. There is a quotation of Artabanus, the advisor to Xerxes, giving his +master advice on a contemplated attack on the Greeks—during Xerxes` campaign +to bring the entire known world under the control of the Persians. Artabanus +said, “See how God with his lightning always smites the bigger animals and will +not sufer them to wax insolent, while these of a lesser bulk chafe him not. How +likewise his bolts fall ever on the highest houses and tallest trees.” And then he +explains the reason: “So, plainly, doth he love to bring down everything that +exalts itself” +Do you think—now that you know a true account oŸ lightning striking tall +trees—that you have a greater wisdom in advising kings on military matters than +did Artabanus 2400 years ago? Do not exalt yourself. You could only do it less +poetically. +--- Trang 124 --- +I0 +M)rolocfrrcs +10-1 The dielectric constant +Here we begin to discuss another of the peculiar properties of matter under 10-1 “The dielectric constant +the infuence of the electric field. In an earlier chapter we considered the behavior 10-2 The polarization vector P +Of conductors, in which the charges move freely in response to an electric field 10-3 Polarization charges +to such points that there is no field left inside a conductor. NÑow we will discuss R : : +. . . " . . 10-4 The electrostatic equations with +ứnsulators, materials which do not conduct electricity. One might at first believe dielectrics +that there should be no efect whatsoever. However, using a simple electroscope . ¬¬ . +and a parallel-plate capacitor, Earaday discovered that this was not so. His 10-ã Eields and forces with dielectrics +experiments showed that the capacitance of such a capacitor 1s 7ncreøsed when +an insulator is put between the plates. If the insulator completely fills the space +between the plates, the capacitance is increased by a factor which depends +only on the nature of the insulating material. Insulating materials are also called +điclectrics; the factor œ 1s then a property of the dielectrie, and is called the +điclectric constant. The dielectric constant of a vacuum is, Of course, unity. +Our problem now is to explain why there is any electrical efect if the insu- +lators are indeed insulators and do not conduet electricity. We begin with the +experimental fact that the capacitance is increased and try to reason out what +might be going on. Consider a parallel-plate capacitor with some charges on +the surfaces of the conductors, let us say negative charge on the top plate and +positive charge on the bottom plate. Suppose that the spacing between the plates +is d and the area of each plate is A. As we have proved earlier, the capacitance is +C= x (10.1) +and the charge and voltage on the capacitor are related by +Q=(CY. (10.2) +Now the experimental fact is that if we put a piece of insulating material like +lucite or glass bebween the plates, we fnd that the capacitance is larger. hat +means, of course, that the voltage is lower for the same charge. But the voltage +diference is the integral of the electric field across the capacitor; so we mus$ +conclude that inside the capacitor, the electric field is reduced even though the +charges on the plates remain unchanged. +mec CONDUCTOR +( ⁄ ⁄ ⁄ x ⁄ x „ ⁄ # {ˆ1⁄:zˆzZ=zˆz⁄=1- —~ | +NKTRHNE +BH SRRRN „ " +MNWNNNEMNMMNWNMNERNRRNMNRWN Fig. 10-1. A parallel-plate capacitor with +[LL⁄2⁄*'/*22⁄2 2 +LN tl: 2 + Ÿ Ý # V #Ý 7 7 | a dielectric. The lines of E are shown. +"mœ CONDUCTOR +Now how can that be? We have a law due to Gauss that tells us that the +ñux of the electric fñeld is directly related to the enclosed charge. Consider the +gaussian surface Š shown by broken lines in Fig. 10-1. 5ince the electric field is +reduced with the dielectric present, we conclude that the net charge inside the +--- Trang 125 --- +surface must be lower than it would be without the material. There is only one +possible coneclusion, and that is that there must be positive charges on the surface +of the dielectric. Since the fñeld is reduced but is not zero, we would expect this +positive charge to be smaller than the negative charge on the conductor. 5o the +phenomena can be explained if we could understand in some way that when a +dielectric material is placed in an electric ñeld there is positive charge induced +on one surface and negative charge induced on the other. +CONDUCTOR +f££77771770771118 +⁄⁄2 4 4 4 4 L b +Fig. 10-2. lf we put a conducting plate CONDUCTOR b d +In the gap of a parallel-plate condenser, the ⁄2 +induced charges reduce the field in the con- FLEÉEE!L] 1}! 1Ì | |} Í 2 +ductor t . +mem ““**Z** 777 za +W©e would expect that to happen for a conductor. Eor example, suppose that +we had a capacitor with a plate spacing ở, and we put between the plates a +neutral conductor whose thickness is Ò, as in Fig. 10-2. 'Phe electric ñeld induces +a positive charge on the upper surface and a negative charge on the lower surface, +so there is no field inside the conductor. 'Phe field in the rest of the space is the +same as it was without the conductor, because it is the surface density of charge +divided by eo; but the distance over which we have to integrate to get the voltage +(the potential điference) is reduced. The voltage is +V= “(a-%). +The resulting equation for the capacitance is like Eq. (10.1), with (đ— b) substi- +tuted for đ: Ạ +C=———. (10.3) +đ[1 — (b/4)] +The capacitanee is increased by a factor which depends upon (b/đ), the proportion +of the volume which is occupied by the conductor. +'This gives us an obvious model for what happens with dielectrics—that inside +the material there are many little sheets of conducting material. The trouble +with such a model ¡is that it has a specific axis, the normal to the sheets, whereas +most dielectrics have no such axis. However, this difficulty can be eliminated if we +? >2 zzy ty ;nyzyY +assume that all insulating materials contain small conducting spheres separated 22ao2aooo +from each other by insulation, as shown in EFig. 10-3. The phenomenon of the 20o2ooo +dielectric constant is explained by the efect of the charges which would be induced 2222222» +on each sphere. 'Phis is one of the earliest physical models of dielectrics used to +explain the phenomenon that Faraday observed. More specifcally, it was assumed Fig. 10-3. A model of a dielectric: small +that each of the atoms of a material was a perfect conduector, but insulated from conducting spheres embedded in an idealized +the others. The dielectric constant s would depend on the proportion of space insulator. +which was occupied by the conducting spheres. 'Phis is not, however, the model +that is used today. +10-2 The polarization vector ?? +Tf we follow the above analysis further, we discover that the idea of regions of +perfect conductivity and insulation is not essential. Each of the small spheres +acts like a dipole, the moment of which is induced by the external fñeld. The only +thing that is essential to the understanding of dielectrics is that there are many +little dipoles induced in the material. Whether the dipoles are induced because +there are tiny conducting spheres or for any other reason is irrelevant. +--- Trang 126 --- +'Why should a fñeld induce a dipole moment in an atom ïf the atom is not +a conducting sphere? 'This subject will be discussed in much greater detail in +the next chapter, which will be about the inner workings of dielectric materials. +However, we give here one example to illustrate a possible mechanism. Ân atom +has a positive charge on the nucleus, which is surrounded by negative electrons. +In an electric field, the nucleus wiïll be attracted in one direction and the electrons +in the other. The orbits or wave patterns of the electrons (or whatever picture +is used in quantum mechanics) will be distorbed to some extent, as shown in +Fig. 10-4; the center of gravity of the negative charge will be displaced and will +no longer coincide with the positive charge of the nucleus. We have already +discussed such distributions of charge. lf we look from a distance, such a neutral +configuration is equivalent, to a first approximation, to a little dipole. ELÉCTRON DISTRIBUTION +Tt seems reasonable that if the fñield is not too enormous, the amount of induced +dipole moment will be proportional to the ñeld. “That is, a small ñeld will displace +the charges a little bit and a larger fñeld will displace them further—and in +proportion to the feld——unless the displacement gets too large. For the remainder _— ~ +of this chapter, it will be supposed that the dipole moment is exactly proportional ¬.. +to the ñeld. ...... +We will now assume that in each atom there are charges g separated by a E ____9+#_ +distance ổ, so that gổ is the dipole moment per atom. (We use ổ because we are ¬ +already using đ for the plate separation.) IÝ there are Ý atoms per unit volume, ¬ +there will be a đipole mmomen‡ per un#t 0ofume equal to Ngõ. 'Thịis đipole moment TC +per unit volume will be represented by a vector, . Needless to say, it is in the =_—_— +direction of the individual dipole moments, i.e., in the direction of the charge +separation ổ: Fig. 10-4. An atom ¡in an electric field +P= Ngõ. (10.4) has Its distribution of electrons displaced +with respect to the nucleus. +In general, will vary from place to place in the dielectric. However, at any +point in the material, ? is proportional to the electric fñeld . 'Phe constant of +proportionality, which depends on the ease with which the electron are displaced, +will depend on the kinds of atoms in the material. +'What actually determines how this constant of proportionality behaves, how +accurately it is constant for very large fñelds, and what is going on inside diferent +materials, we will discuss at a later time. For the present, we will simply suppose +that there exists a mechanism by which a dipole moment is induced which is +proportional to the electric field. +10-3 Polarization charges +Now let us see what this model gives for the theory of a condenser with a +dielectric. Eirst consider a sheet of material in which there is a certain dipole +moment per unit volume. WIll there be on the average any charge density +produced by this? Not if P is uniform. Tƒ the positive and negative charges being +displaced relative to each other have the same average density, the fact that they +are displaced does not produce any net charge inside the volume. Ôn the other +hand, if P were larger at one place and smaller at another, that would mean +that more charge would be moved into some region than away from it; we would +then expect to get a volume density of charge. Eor the parallel-plate condenser, +we suppose that ? ¡is uniform, so we need to look only at what happens at the +surfaces. Át one surface the negative charges, the electrons, have efectively +moved out a distance ở; at the other surface they have movcd in, leaving some +positive charge efectively out a distance ổ. As shown in Fig. 10-5, we will have a +surface density of charge, which will be called the surface polar¿zation chargc. +“=....e.. cớ tnrđẽốốn nr ha ốnn +Fig. 10-5. A dielectric slab in a uniform +ỗ field. The positive charges displaced the +h— — — — — — — — — -—L_ distance ổ with respect to the negatives. +——. "rẽ... +--- Trang 127 --- +This charge can be calculated as follows. If A is the area of the plate, the +number of electrons that appear at the surface is the product of 4 and ®, the +number per unit volume, and the displacement ổ, which we assume here is +perpendicular to the surface. The total charge is obtained by multiplying by the +electronic charge ge. To get the surface density of the polarization charge induced +on the surface, we divide by A. The magnitude of the surface charge density is +Øpol — N qe ỗ. +But this is just equal to the magnitude ? of the polarization vector , Eq. (10.4): +Øpoi = P. (10.5) +The surface density of charge is equal to the polarization inside the material. 'The +surface charge is, of course, positive on one surface and negative on the other. +Now let us assume that our slab ¡is the dielectric of a parallel-plate capacitor. +The piafes of the capacitor also have a surface charge, which we will call Øtree, +because they can move “freely” anywhere on the conductor. 'Thịis is, of course, the +charge that we put on when we charged the capacitor. It should be emphasized +that Øpoi exists only because of Øgee. lÝ Øpyee is removed by discharging the +capacitor, then øpoi will disappear, not by goỉng out on the discharging wire, but +by moving back into the material—by the relaxation of the polarization inside +the material. +W©e can now apply Gauss' law to the gaussian surface Š in EFig. 10-1. The +electric fñeld # in the dielectric is equal to the £o#øÏ surface charge density divided +by eo. lt is clear that Øpo¡ and Øwee have opposite signs, sO +bám. (10.6) +Note that the field #o between the metal plate and the surface of the dielectric +1s higher than the field #; it corresponds to ơrzee alone. But here we are concerned +with the fñeld inside the dielectric which, 1ƒ the dielectric nearly fills the gap, 1s +the field over nearly the whole volume. sing Eq. (10.5), we can write +D6 (10.7) +This equation doesn't tell us what the electric field is unless we know what ? +1s. Here, however, we are assuming that depends on E_—in fact, that it is +proportional to #. This proportionality is usually written as +ÐP =x‹gE. (10.8) +The constant x (Greek “khi”) is called the electric susceptibiitg of the dielectric. +Then Edq. (10.7) becomes +Ơfree 1 +E=———., 10.9 +sọ (1+X}) 089) +which gives us the factor 1/(1 + x) by which the feld is reduced. +'The voltage between the plates is the integral of the electric feld. Since the +fñeld is uniform, the integral is just the product of # and the plate separation đ. +W©e have that +V— Eả— _ hen +co(1 + x}) +The total charge on the capacitor is øyyee4, so that the capacitance defned +by (10.2) becomes +eoA(I+x) “eạA +Œ=——————=_—_. 10.10 +We have explained the observed facts. When a parallel-plate capacitor 1s +fled with a dielectric, the capacitance is increased by the factor +KE=lI+X, (10.11) +--- Trang 128 --- +which is a property of the material. Qur explanation, of course, is not complete +until we have explained—as we will do later—=how the atomie polarization comes +about. +Let's now consider something a little bi more complicated——the situation +in which the polarization ? is not everywhere the same. As mentioned earlier, +1f the polarization is not constant, we would expect in general to ñnd a charge +density in the volume, because more charge might come into one side of a small +volume element than leaves it on the other. How can we find out how much ` +charge is gained or lost from a small volume? N N +First let°s compute how much charge moves across any imaginary surface Na ` +when the material is polarized. The amount of charge that øoes across a surface ` SN À »XN VN SN +1s just times the surface area If the polarization 1s n=ormal to the surface. Of > “ad +; ; ; ; - - ựcos0 +course, iƒ the polarization is tangen#ial to the surface, no charge moves across it. +Following the same arguments we have already used, it is easy to see that NI ` +the charge moved across any surface element is proportional to the cormnponent ` +of P perpendicular to the surface. Compare Fig. 10-6 with Eig. 10-5. We see Fig. 10-6. The charge moved across an +that Eq. (10.5) should, in the general case, be written element of an imaginary surface in a dielec- +tric is proportional to the component of P +Øpoi = P-1r. (10.12) normal to the surface. +lÍ we are thinking of an imagined surface element /nside the dielectric, +Eq. (10.12) gives the charge moved across the surface but doesnˆt result in ẢNNNN ` +a net surface charge, because there are equal and opposite contributions from 'DIELECTRIC SN +the dielectric on the two sides of the surface. ` y +The displacements of the charges can, however, result in a øolurne charge ‹ ` +density. "The total charge displaced øuý of any volume V by the polarization AQ +1s the integral of the outward normal component of P over the surface Š that NI +bounds the volume (see Fig. 10-7). An cqual excess charge of the opposite sign Volume V +is left behind. Denoting the net charge inside V by AQ);ø¡ we write ` uc +A P.nd 10.13 ` Ầ N +=— -=da. : +Qua =— (10.13) +We can attribute AQp¿¡ to a volume distribution of charge with the density Øpai, ` +and so Fig. 10-7. A nonuniform polarization P +AQjbpai = J Øpol đV. (10.14) can result in a net charge in the body of a +V dielectric. +Combining the two equations yields +J Øpoi ỞV = -Í P.nda. (10.15) +We© have a kind of Gauss' theorem that relates the charge density from polarized +materials to the polarizatilon vector . We can see that it agrees with the +result we got for the surface polarization charge or the dielectric in a parallel- +plate capacitor. sing Eq. (10.15) with the gaussian surface of Fig. 10-1, the +surface integral gives P.AA, and the charge inside is øpoi AÁ, so we get again +that Øpol — P. +Just as we dịd for Gauss' law of electrostatics, we can convert Eq. (10.15) to +a diferential form——using Gauss' mathematical theorem: +| P.na= | V- PdaV. +We get +Øpoai =—VW -P. (10.16) +TÍ there is a nonuniform polarization, its divergence gives the net density of charge +appearing in the material. We emphasize that this is a perfectly real charge +density; we call it “polarization charge” only to remind ourselves how i% got there. +--- Trang 129 --- +10-4 The electrostatic equations with dielectrics +Now let's combine the above result with our theory of electrostatics. The +fundamental equation is +ÿ.E=f. (10.17) +The ø here is the density of aÌl electric charges. 5ince it is no easy to keep track +of the polarization charges, it is convenient to separate ø into bwo parts. Again +we call øpo¡ the charges due to nonuniform polarizations, and call ø¿« all the +rest. Usually øyyee is the charge we put on conductors, or at known places in +space. Equation (10.17) then becomes +ÿ.E- free + pol — free — V.P +Ã. (=: ) - Em, (10.18) +Of course, the equation for the curl of # is unchanged: +VxE-=0. (10.19) +Taking P from Ea. (10.8), we get the simpler equation +ÿ:[1+x)E| = V:(äE) = #ÈS, (10.20) +These are the equations of electrostatics when there are dielectrics. hey don't, +of course, say anything new, but they are in a form which is more convenient for +computation in cases where ørzee 1s known and the polarization ? is proportional +Notice that we have not taken the dielectrie “constant,” &, out of the divergence. +That is because i9 may not be the same everywhere. lÝit has everywhere the same +value, it can be factored out and the equations are just those of electrostatics with +the charge density øyee divided by œ. In the form we have given, the equations +apply to the general case where different dielectrics may be in diferent places in +the fñeld. 'hen the equations may be quite difficult to solve. +'There is a matter of some historical importance which should be mentioned +here. In the early days of electricity, the atomic mechanism of polarization was +not known and the existence oŸ øpoi was not appreciated. The charge /Øwec WaS +considered to be the entire charge density. In order to write Maxwell's equations +in a simple form, a new vector 2 was defñned to be equal to a linear combination +of E and ?: +D-=‹cọoE+P. (10.21) +As a result, Eqs. (10.18) and (10.19) were written in an apparently very simple +form: +V-D=pwee, VxE=0. (10.22) +Can one solve these? Only if a third equation is given for the relationship between +D and E. When Ed. (10.8) holds, this relationship is +D= cạ(1+ x)E = keo. (10.23) +'This equation was usually written +D=cEÈ, (10.24) +where e is still another constant for describing the dielectric property of materials. +It is called the “permittivity” (Ñow you see why we have co in our equations, it +is the “permittivity of empty space”) Evidently, +c€= keo = (1+ xso. (10.25) +--- Trang 130 --- +Today we look upon these matters from another point of view, namely, that +we have simpler equations in a vacuum, and if we exhibit in every case all the +charges, whatever their origin, the equations are always correct. If we separate +some of the charges away for convenience, or because we do not want to discuss +what is goïing on in detail, then we can, IŸ we wish, write our equations in any +other form that may be convenient. +One more point should be emphasized. An equation like 2 = cE is an +attempt to describe a property of matter. But matter is extremely complicated, +and such an equation is in fact not correct. Eor instance, if E gets too large, +then ñÐ ¡is no longer proportional to #. For some substances, the proportionality +breaks down even with relatively small ñelds. Also, the “constant” oŸ propor- +tionality may depend on how fast changes with time. Therefore this kind of +cquation is a kind of approximation, like Hooke's law. It cannot be a deep and +fundamental equation. Ôn the other hand, our fundamental equations for #, +(10.17) and (10.19), represent our deepest and most complete understanding of +electrostatics. +10-5 Eields and forces with dielectrics +We© will now prove some rather general theorems for electrostatics in situations +where dielectrics are present. We have seen that the capacitance of a parallel-plate +capacitor is increased by a defñnite factor if it is fñlled with a dielectric. We can +show that this is true for a capacitor of an shape, provided the entire reglon in +the neighborhood of the ©wo conduectors is filled with a uniform linear dielectric. +Without the dielectric, the equations to be solved are +—— and Vx Eọ =0. +With the dielectric present, the first of these equations is modified; we have +instead the equations +Ý:(E) = =n and — WYxE=0. (10.26) +Now since we are taking œ to be everywhere the same, the last two equations +can be written as +Ý:(E) = "¬ and Wx(wE)=0. (10.27) +W©e therefore have the same equations for &# as for 2o, so they have the +solution œ = E¿g. In other words, the field is everywhere smaller, by the +factor 1/, than in the case without the dielectric. Since the voltage difference is +a line integral of the fñeld, the voltage is reduced by this same factor. 5ince the +charge on the electrodes of the capacitor has been taken the same in both cases, +Eaq. (10.2) tells us that the capacitance, in the case of an everywhere uniform +dielectric, is increased by the factor &. +Let us now ask what the ƒforce would be between two charged conductors +in a dielectric. We consider a liquid dielectric that is homogeneous everywhere. +We have seen earlier that one way to obtain the force is to differentiate the +energy with respect to the appropriate distance. Tf the conductors have equal and +opposite charges, the energy = Q2/2Œ, where Œ is their capacitance. Using +the principle of virtual work, any component is given by a diferentiation; for +example, : +F;=-c=~Ý z(Ð): (10.28) +Øz 2 Øz\C +Since the dielectric increases the capacity by a factor , all forces will be reduced +by this same factor. +One point should be emphasized. What we have said is true only ïif the +dielectric is a liquid. Any motion of conductors that are embedded in a solid +--- Trang 131 --- +dielectric changes the mechanical stress conditions of the dielectric and alters its +electrical properties, as well as causing some mechanical energy change in the +dielectric. Moving the conductors in a liquid does not change the liquid. “The +liquid moves to a new place but its electrical characteristics are not changed. +Many older books on electricity start with the “fundamental” law that the +force between bwo charges is +. (10.29) +4eokr2 +a point of view which is thoroughly unsatisfactory. For one thiíng, it is not true +in general; it is true only for a world filled with a liquid. Secondly, ¡ depends +on the fact that œ is a constant, which is only approximately true for most real +materials. It is much better to start with Coulomb”s law for charges in a 0acwwm, +which is always right (for stationary charges). +'What does happen in a solid? 'Phis is a very difficult problem which has not +been solved, because ït is, in a sense, indeterminate. lf you put charges inside +a dielectric solid, there are many kinds oŸ pressures and strains. You cannot +deal with virtual work without including also the mechanical energy required to +compress the solid, and it is a difficult matter, generally speaking, to make a +unique distinction between the electrical forces and the mechanical forces due +to the solid material itself. Fortunately, no one ever really needs to know the +answer to the question proposed. He may sometimes want to know how much +strain there is going to be in a solid, and that can be worked out. But it is much +more complicated than the simple result we got for liquids. +A surprisingly complicated problem in the theory of dielectrics is the following: +'Why does a charged obJect pick up little pieces of dielectrie? If you comb your +haïr on a dry day, the comb readily picks up small scraps of paper. If you thought +casually about it, you probably assumed the comb had one charge on it and the —— +paper had the opposite charge on it. But the paper is initially electrically neutral. +lt hasn't any net charge, but it is attracted anyway. lt is true that sometimes the +paper will come up to the comb and then fy away, repelled immediately after E +1t touches the comb. “The reason is, of course, that when the paper touches the +comb, it picks up some negative charges and then the like charges repel. But b +that doesnt answer the original question. Why did the paper come toward the +comb in the first place? DIELECTRIC +The answer has to do with the polarization of a dielectric when ït is placed OBJECT +in an electric feld. 'There are polarization charges of both signs, which are +attracted and repelled by the comb. 'Phere is a net attraction, however, because N +the fñeld nearer the comb is stronger than the fñeld farther away——the comb 1s +not an infnite sheet. Its charge is localized. A neutral piece of paper will not be Fig. 10-8. A dielectric object in a nonuni- +attracted to either plate inside the parallel plates of a capacitor. The variation form field feels a force toward regions of +of the field is an essential part of the attraction mechanism. higher field strength. +As illustrated in Eig. 10-8, a dielectric is always drawn from a region of weak +field toward a region of stronger field. In fact, one can prove that for small +objects the force is proportional to the gradient of the sguøre of the electric +fñeld. Why does it depend on the square of the fñeld? Because the induced +polarization charges are proportional to the fields, and for given charges the +forces are proportional to the field. However, as we have just indicated, there will +be a ne‡ force only if the square of the feld is changing from point to point. So +the force is proportional to the gradient of the square of the field. 'The constant +of proportionality involves, among other things, the dielectric constant of the +obJect, and it also depends upon the size and shape of the obJect. +There is a related problem in which the force on a dielectric can be worked +out quite accurately. If we have a parallel-plate capacitor with a dielectric +slab only partially inserted, as shown in Fig. 10-9, there will be a force driving +the sheetin. A detailed examination of the force is quite complicated; it is +related to nonuniformities in the field near the edges of the dielectric and the +plates. However, iŸ we do not look at the details, but merely use the principle of +conservation of energy, we can easily calculate the force. We can find the force +--- Trang 132 --- +CONDUCTOR +j...Vư6ưU6HE HE. L.eLreB5B +r Nó. den +SN Fig. 10-9. The force on a dielectric sheet +VˆưT ý ⁄jÍ 3Ÿ ⁄/ 3⁄/ } } } }y } 1T} In a parallel-plate capacitor can be com- +W puted by applying the principle of energy +* Conservation. +from the formula we derived earlier. Equation (10.28) is equivalent to +ØU V2 9Œ +F„=———=+— —-. 10.30 +` Øz 2 Øz ' ) +W© neecd only fnd out how the capacitance varies with the position of the dielectric +Let's suppose that the total length of the plates is b, that the width of the +plates is W/, that the plate separation and dielectric thickness are đ, and that +the distance to which the dielectric has been inserted 1s z. The capacitance 1s +the ratio of the total free charge on the plates to the voltage between the plates. +We© have seen above that for a given voltage V the surface charge density of Íree +charge is coV/d. So the total charge on the plates is +keo V coV +Q=— —zW+- —(L-z)W, +from which we get the capacitance: +C= “TT (K# + =3). (10.31) +Using (10.30), we have +V2 coW +Now this equation is not particularly useful for anything unless you happen to +need to know the force in such circumstances. We only wished to show that +the theory of energy can often be used to avoid enormous complications in +determining the forces on dielectric materials—as there would be in the present +Our discussion of the theory of dielectrics has dealt only with electrical +phenomena, accepting the fact that the material has a polarization which is +proportional to the electric field. Why there is such a proportionality is perhaps +Of greater interest to physics. Once we understand the origin of the dielectric +constants from an atomic point of view, we can use electrical measuremenfs of +the dielectric constants in varying circumstances to obtain detailed information +about atomie or molecular structure. This aspect will be treated in part in the +next chapter. +--- Trang 133 --- +}rrsrclo Hfolocfrrcs +11-1 Molecular dipoles +In this chapter we are going to discuss why it is that materials are dielectric. 11-1 Molecular dipoles +W© said in the last chapter that we could understand the properties of electrical 11-2 Electronic polarization +systems with dielectrics once we appreciated that when an electric field is applied 11-3 Polar molecules; orientation +to a dielectric it induces a dipole moment in the atoms. Specifically, if the polarization +electric teld + induces an average dipole moment per unit volume ?, then “, 11-4 Electric 8elds in cavities ofa +the dielectric constant, is given by R : +dielectric +c—1= P- (11.1) 11-5 The dielectric constant of liquids; +cọ l the Clausius-Mossotti equation +W©e have already discussed how this equation is applied; now we have to 1I-6 5old dielectrics : +discuss the mechanism by which polarization arises when there is an electric ñeld 11-ĩ Eerroelectricity; BaTiOs +inside a material. We begin with the simplest possible example—the polarization +of gases. But even gases already have complications: there are bwo types. The +mmolecules of some gases, like oxygen, which has a symmetric païr of atoms in each +molecule, have no inherent dipole moment. But the molecules of others, like water +vapor (which has a nonsyrmmmetric arrangement of hydrogen and oxygen atoms) +carry a permanent electric dipole moment. Âs we pointed out in Chapter 6, there +1s in the water vapor molecule an average plus charge on the hydrogen atoms +and a negative charge on the oxygen. 5ince the center of gravity of the negative Reuieu: Chapter 31, Vol. L, The Ôrigin +charge and the center of gravity of the positive charge do not coinecide, the total 0ƒ the Relractiue Indez +charge distribution of the molecule has a dipole moment. Such a molecule is Chapter 40, Vol. Lj The Prin- +called a polar molecule. In oxygen, because of the symmetry of the molecule, the ciples oƒ Statistical Mechanics +centers of gravity of the positive and negative charges are the same, so 1È 1S a +nonpolar molecule. Tt does, however, become a dipole when placed in an electric +ñeld. The forms of the two types of molecules are sketched in Fig. 11-1. +11-2 Electronic polarization - - +We will first điscuss the polarization of non polar molecules. We can start with — R ' R +the simplest case oŸ a monatomic gas (for instance, heliun). When an atom of _ —ẢNN — +such a gas is in an electric field, the electrons are pulled one way by the field while _ N- — +the nucleus is pulled the other way, as shown in Eig. 10-4. Although the atoms _ _ CENTER OE +are very stif with respect to the electrical forces we can apply experimentally, — + AND — CHARGE +there is a slight net displacement of the centers of charge, and a dipole moment (a) +is induced. Eor small fields, the amount of displacement, and so also the dipole +mmoment, is proportional to the electric fñeld. 'Phe displacement of the electron +distribution which produces this kind ofinduced dipole moment is called electronic +polarization. +We have already discussed the inÑuence of an electric fñeld on an atom in ⁄% +Chapter 3Í of Vol. I, when we were dealing with the theory of the index of +refraction. If you think about it for a moment, you will see that what we must Thôn ARGE: +do now is exactly the same as we did then. But now we need worry only about C*3 +fields that do not vary with time, while the index of refraction depended on CENTER OF +. : + CHARGE +time-varying fñelds. () +In Chapter 31 of Vol. Ï we supposed that when an atom ¡is placed in an +oscillating electric fñeld the center of charge of the electrons obeys the equation Fig. 11-1. (a) An oxygen molecule +with zero dipole moment. (b) The wa- +m d?œ + mu2a+ = q.E (1 2) ter molecule has a permanent dipole mo- +dị2 0 de l ment po. +--- Trang 134 --- +The frst term is the electron mass times its acceleration and the second is a +restoring force, while the right-hand side is the force from the outside electric +fñeld. Iƒ the electric field varies with the frequenecy œ, Eq. (11.2) has the solution +=——s—mx 11.3 +. m(08 — 02) ` 13) +which has a resonance at œ = œạọ. When we previously found this solution, we +interpreted it as saying that œ was the Írequency at which light (in the optical +region or in the ultraviolet, depending on the atom) was absorbed. Eor our +Dpurposes, however, we are interested only in the ease of constant fields, i.e., +for œ = Ö, so we can disregard the acceleration term in (11.2), and we fñnd that +the displacement 1s +r= ST, (11.4) +trom this we see that the dipole moment p oŸ a single atom is += =—=:. 11.5 +p=dq.x mu ( ) +In this theory the dipole moment ø is indeed proportional to the electric field. +People usually write +Ð= ằœcuE. (11.6) +(Again the eo is put in for historical reasons.) The constant œ is called the +polarizability of the atom, and has the dimensions FỞ. Tt is a measure of how +casy i is to induce a moment in an atom with an electric ñeld. Comparing (11.5) +and (11.6), our simple theory says that +2 4me2 +an (11.7) +€ọ7nu§ Tnu§ +Tf there are / atoms in a unit volume, the polarization the dipole moment +per unit volume——is given by +P_—=Np= NeacgE. (11.8) +Putting (11.1) and (11.8) together, we get +—=l=-_—=ÄÑN 11.9 +E ẶP ữa (11.9) +or, using (11.7), +R—l= (11.10) +Erom E4q. (11.10) we would predict that the dielectric constant of different +gases should depend on the density of the gas and on the frequency œ OŸ its +optical absorption. +Our formula is, of course, only a very rough approximation, because in +Eq. (11.2) we have taken a model which ignores the complications oŸ quantun +mechanics. For example, we have assumed that an atom has only one resonant +frequency, when it really has many. To calculate properly the polarizability œ of +atoms we must use the complete quantum-mechanical theory, but the classical +ideas above give us a reasonable estimate. +Let”s see if we can get the right order oŸ magnitude for the dielectric constant +of some substance. Suppose we try hydrogen. We have once estimated (Chap- +ter 38, Vol. I) that the energy needed to ionize the hydrogen atom should be +approximately +Ex_~-.. 11.11 +5p (11.11) +--- Trang 135 --- +For an estimate of the natural frequency œọ, we can set this energy equal to ñ¿u— +the energy of an atomie oscillator whose natural frequency is œọ. WWe get +1 me° +œọ ~ ———. +TÍ we now use this value of œạ in Eq. (11.7), we ñnd for the electronic polarizability ++ 16r|——| . 11.12 +œ 7 lục | ( ) +The quantity (ñ2/me?) is the radius of the ground-state orbit of a Bohr atom (see +Chapter 38, Vol. I) and equals 0.528 angstroms. In a gas at standard pressure and +temperature (1 atmosphere, 0°C) there are 2.69 x 1012 atoms/cmở, so Eq. (11.9) +Ø1V©S US +œ = 1 + (2.69 x 10!2)16z(0.528 x 103)” = 1.00020. (11.13) +'The dielectric constant for hydrogen gas is measured to be +Eexp —= 1.00026. +W© see that our theory is about right. We should not expect any better, because +the measurements were, of course, made with normal hydrogen gas, which +has diatomic molecules, not single atoms. We should not be surprised 1ƒ the +polarization of the atoms in a molecule is not quite the same as that of the +separate atoms. "The molecular efect, however, is not really that large. An exact +quantum-mechanical calculation of œ for hydrogen atoms gives a result about +12% higher than (11.12) (the 16z is changed to 187), and therefore predicts a +dielectric constant somewhat closer to the observed one. In any case, iÈ 1s clear +that our model of a dielectric is fairly good. +Another check on our theory is to try Eq. (11.12) on atoms which have a +higher frequenecy of excitation. Eor instance, it takes about 24.6 electron volts to ` \ F4 +pull the electron of helium, compared with the 18.6 electron volts required to S b +lonize hydrogen. We would, therefore, expect that the absorption frequency œg \ \ / ® +for helium would be about twice as big as for hydrogen and that œ would be á ` +one-quarter as large. We expect that $ ¬e. ộ ¬o- +Ebeliun 2 1.000050. Lá \ ế SN +Experimentally, m +Eheliun — 1.000068, (a) +So you see that our rough estimates are coming out on the right track. So we +have understood the dielectric constant oŸ nonpolar gas, but only qualitatively, ứ co ậ +because we have not yet used a correct atomic theory of the motions of the atomie Lị L4 +electrons. kì ộ t. \ ` +11-3 Polar molecules; orientation polarization D) s4 ộ k +Next we will consider a molecule which carries a permanent dipole moment øo—— ø ộ mổ ø L4 +such as a water molecule. With no electric field, the individual dipoles point +in random directions, so the net moment per unit volume is zero. But when (b) +an electric field is applied, two things happen: First, there is an extra dipole +moment induced because of the forces on the electrons; this part gives just the Fig. 11-2. (a) In a gas of polar molecules, +same kind of electronic polarizability we found for a nonpolar molecule. For very the individual moments are Oriented at ran- +accurate work, this efect should, of course, be included, but we will neglect it dom; the SN Ti ma Mã. ve +for the moment. (It can always be added in at the end.) Second, the electric TNG lề ⁄609. (b) When there San 6ece +l ¬ . . field, there is some average alignment of +field tends to line up the individual dipoles to produce a net moment per unit the molecules +volume. Tf all the dipoles in a gas were to line up, there would be a very large : +polarization, but that does not happen. Ät ordinary temperatures and electric +fñelds the collisions of the molecules in their thermal motion keep them from lining +--- Trang 136 --- +up very much. But there is some net alignment, and so some polarization (see +Eig. 11-2). The polarization that does occur can be computed by the methods of +statistical mechanics we described in Chapter 40 of Vol. T. +To use this method we need to know the energy of a dipole in an electric +fñeld. Consider a dipole of moment øạ in an electric ñeld, as shown in Fig. 11-3. +The energy of the positive charge is gớ(1), and the energy of the negative charge +is —gØ(2). Thus the energy of the dipole is +U = gó(1) — qó(2) = qd- Vỏ, +U =_—pog- È = —poE cos0, (11.14) (1) : +where Ø is the angle between øạ and . As we would expect, the energy is lower › +when the dipoles are lined up with the feld. +W© now fnd out how much lining up occurs by using the methods oŸ statistical =g"@) +mechanics. We found in Chapter 40 of Vol. I that in a state of thermal equilibrium, +the relative number of molecules with the potential energy is proportional to Fig. 11-3. The energy of a dipole pọ in +the field E Is —po - E. +c~U/RT. (11.15) & +where (z,g,z) is the potential energy as a function oŸ position. The same +arguments would say that using Eq. (11.14) for the potential energy as a function +of angle, the number of molecules at Ø per wn2t sold œngle 1s proportional +to e—U/T, +Letting m0) be the number of molecules per unit solid angle at Ø, we have +n(0) = nạc}Pocos0/ET, (11.16) +For normal temperatures and fields, the exponent is small, so we can approximate +by expanding the exponential: +o1 cos 8 +0) —= 1+ ———— |. 11.17 +T".... (1117) +W© can find nọ 1Ý we integrate (11.17) over all angles; the result should be +Just , the total number of molecules per unit volume. 'Phe average value of cos Ø +over all angles is zero, so the integral is just nọ times the total solid angle 4m. +We get +=—. 11.18 +nọ = (1118) +We see from (11.17) that there will be more molecules oriented along the fñeld +(cos Ø = 1) than against the field (cosØ = —1). So in any small volume containing +many molecules there will be a net dipole moment per unit volume—that is, a +polarization . To calculate , we want the vector sum of all the molecular +mmoments in a unit volume. Since we know that the result is going to be in the +direction oŸ E, we will just sum the components in that direction (the components +at right angles to EZ will sum to zero): +P= » Ðo COS Ổ;. +vỏitme +We can evaluate the sum by integrating over the angular distribution. 'Phe +solid angle at Ø is 2z sỉn Ø đ0, so +P= J n0()po cos 0 27 sin Ø d0. (11.19) +Substituting for ø(Ø) from (11.17), we have +P= ÿJ ( + TT cos0 pm cosd[eosổ) +--- Trang 137 --- +which is easily integrated to give +P= SkT ” (11.20) +'The polarization is proportional to the feld #, so there will be normal dielectric +behavior. Also, as we expect, the polarization depends inversely on the temper- +ature, because at higher temperatures there is more disalignment by collisions. +Thịs 1/7! dependence is called Curies law. The permanent moment øo appears +squared for the following reason: In a given electric fñeld, the aligning force +depends upon øạ, and the mean moment that is produced by the lining up is +again proportional to pọ. The average induced moment is proportional to p§. c1 H +We should now try to see how well Eq. (11.20) agrees with experiment. Lets 0.004 ⁄ +look at the case of steam. Since we don't know what ?øọ is, we cannot compute / +directly, but Eq. (11.20) does predict that œ — 1 should vary inversely as the z +temperature, and this we should check. # +Erom (11.20) we get ⁄ +Pp Ngệ 0.003 / +&k—l]=-—=_--.., (11.21) ⁄ +cọ 3cokT ⁄ +SO & — l should vary in direcE proportion to the density , and inversely as ⁄ +the absolute temperature. The dielectric constant has been measured at several ooo2 ⁄ứ +diferent pressures and temperatures, chosen such that the number of molecules ⁄ +in a unit volume remained ñxed.* [Notice that if the measurements had all been /⁄ +taken at constant pressure, the number of molecules per unit volume would “ +decrease linearly with inereasing temperature and s — 1 would vary as T7? oooi ⁄ +instead of as 7'~!,] In Eig. 11-4 we plot the experimental observations for ø — 1 ⁄ +as a function of 1/7. The dependence predicted by (11.21) is followed quite well. H +'There is another characteristic of the dielectric constant of polar molecules—— ứ +1ts variation with the frequency of the applied field. Due to the moment of inertia 0 +. . : 0 0.001 0.002 0.003 +of the molecules, it takes a certain amount of time for the heavy molecules to turn sụn +toward the direction of the field. So if we apply frequencies in the high microwave 7K ) +region or above, the polar contribution to the dielectric constant begins to fall Fig. 11-4. Experimental measurements +away because the molecules cannot follow. In contrast to this, the electronic of the dielectric constant of water vapor at +polarizability still remains the same up to optical frequencies, because of the Various temperatures. +smaller inertia in the electrons. +11-4 Electric fields in cavities of a dielectric ⁄4 Sổ ⁄ +W©e now turn to an interesting but complicated question——the problem of the L4 1⁄2 h ⁄__⁄⁄_⁄/_ +dielectric constant in dense materials. Suppose that we take liquid helium or liquid (Ly 2 É 6 22 TZ- Z2 +argon or some other nonpolar material. We still expect electronie polarization. 1 #1 ⁄ v4 “ “Z7? +But in a dense material, can be large, so the fñeld on an individual atom will '⁄4 ; 41 +be iniuenced by the polarization of the atoms in its close neighborhood. 'Phe c⁄ +question is, what electric field acts on the individual atom? ⁄ +Imagine that the liquid is put between the plates of a condenser. lf the (a) (c) +plates are charged they will produce an electric field in the liquid. But there are +also charges in the individual atoms, and the total ñeld # is the sum of both 4 ⁄⁄ ⁄ +of these efects. This true electric fñeld varies very, very rapidly from point to T ⁄5 +point in the liquid. It is very hiph inside the atoms——particularly right next to h =~—=<==<⁄“- +the nucleus—and relatively small between the atoms. The potential diference ⁄ F. +between the plates is the line integral of this total fñeld. IÝ we ignore all the ⁄ +ñne-grained variations, we can think of an øerøge electric field #, which is 4 ⁄ +⁄2 ZZ⁄ +just V/d. (This is the ñeld we were using in the last chapter.) We should think T ⁄ +of this ñeld as the average over a space containing many atoms. (b) (4) +Now you might think that an “average” atom in an “average” location would . ¬ . +feel this average field. But it is not that simple, as we can show by considerin . FIg. ¬ The fieldin a slot củt In a +5 ÐĐc, y 5 +. . . . . . . . dielectric depends on the shape and orienta- +what happens IÝ we imagine diferent-shaped holes in a dielectric. Eor instance, tion of the slot. +suppose that we cut a slot in a polarized dielectric, with the slot oriented parallel +* Sãnger, Steiger, and Gächter, Heluetica Phụsi¿ca Acta 5, 200 (1932). +--- Trang 138 --- +to the fñeld, as shown in part (a) of Fig. 11-5. Since we know that V x E=0, +the line integral oŸ # around the curve, [', which goes as shown in (b) of the +fñgure, should be zero. The fñeld inside the slot must give a contribution which +Just cancels the part from the field outside. 'Therefore the field #g actually found +in the center of a long thin slot is equal to #2, the average electric ñeld found in +the dielectric. +Now consider another slot whose large sides are perpendicular to #, as shown +in part (c) of Fig. 11-5. In this case, the ñeld 2o in the slot is not the same as +because polarization charges appear on the surfaces. lf we apply Gauss' law to a +surface Š drawn as in (d) of the figure, we find that the field Jọ ?n the sÌot is +given by +tọo= E+—, (11.22) +where # is again the electric field in the dielectric. (The gaussian surface +contains the surface polarization charge Øpoi = P.) We mentioned in Chapter 10 +that co + P ïs often called J2, so eo o = Do is equal to D in the dielectric. +Barlier in the history of physics, when it was supposed to be very Important +to deflne every quantity by direct experiment, people were delighted to discover +that they could defñne what they meant by # and D in a dielectric without +having to crawl around between the atoms. The average field # is numerically +cqual to the field #o that would be measured ïn a slot cut parallel to the field. +And the field DĐ could be measured by fñnding ọ in a slot cu normal to the +field. But nobody ever measures them that way anyway, so It was Just one of +those philosophical things. +27277 (7 Z⁄ 2> Fig. 11-6. The field at any point A in a +2/7 1/27 — J6 27 + ( ì dielectric can be considered as the sum of +Í 21/27/27 jP V77 lZ “4 the field in a spherical hole plus the field due +(7⁄77 M2212 ⁄⁄Z to a spherical plug. +124/22/4 1⁄⁄2⁄Z +For most liquids which are not too complicated in structure, we could expect +that an atom finds itself, on the average, surrounded by the other atoms in what +would be a good approximation to a spherical hole. And so we should ask: “What +would be the fñeld in a spherical hole?” We can fñnd out by noticing that iŸ we +imagine carving out a spherical hole in a uniformly polarized material, we are just +removing a sphere of polarized material. (We must imagine that the polarization +is “frozen in” before we cut out the hole.) By superposition, however, the felds +inside the dielectric, before the sphere was removed, is the sum of the fields from +all charges outside the spherical volume plus the fñelds from the charges within DIPOLE EIELD +the polarized sphere. That is, if we call E the fñeld in the uniform dielectric, we OUTSIDE +can wWrIte À+l>/ +E = Ehole + Epiug, (11.23) | ĐỀ +where Fuoie is the field in the hole and ji¿g is the field inside a sphere which đ | lUU) +is uniformly polarized (see Fig. 11-6). The fields due to a uniformly polarized tIH +sphere are shown in Eig. 11-7. The electric fñeld inside the sphere is uniform, and >_+€ +1ts value is +Fblug — _— 11.24) +plug — 3eg: (1. +Using (11.23), we get +Phole = # + _ (11.25) +0 k AP +The field in a spherical cavity is greater than the average field by the amount P/3so. +(The spherical hole gives a feld 1/3 of the way between a slot parallel to the ñeld Fig. 11-7. The electric field of a uniformly +and a slot perpendicular to the feld.) polarized sphere. +--- Trang 139 --- +11-5 The dielectric constant of liquids; the Clausius-Mossotti equation +In a liquid we expect that the field which will polarize an individual atom +is more like FsJ¿ than just E. T we use the hej¿ oŸ (11.25) for the polarizing +ñeld in Eq. (11.6), then Bq. (11.8) becomes +PE=Nœeg[ + — ], (11.26) +P=——-a(uử. 11.27 +1_ (Na/3) ° 0127) +Remembering that & — 1 is Just P/cạ#, we have +—=l=————_ 11.28 +. 1—(Na/3)` 01.28) +which gives us the dielectric constant of a liquid in terms of œ, the atomic +polarizability. This ¡is called the Clauszus-Mlossotti equation. +Whenever œ is very small, as it is for a gas (because the density is small), +then the term Wø/3 can be neglected compared with 1, and we get our old result, +Eq. (11.9), that +g— 1= No. (11.29) +Let”°s compare Eq. (11.28) with some experimental results. It is frst necessary +to look at gases for which, using the measurement of z, we can fnd œ from +Eq. (11.29). Eor instance, for carbon disulfide at zero degrees centigrade the +dielectric constant is 1.0029, so /ơ ¡s 0.0029. Now the density of the gas is easily +worked out and the density of the liquid can be found in handbooks. At 20°€, the +density of lquid C5a is 381 times higher than the density of the gas at 0°Ơ. This +means that / is 381 times higher in the liquid than it is in the gas so, that—If +we make the approximation that the basic atomic polarizability of the carbon +disulũde doesnˆt change when it is condensed into a liquid——œ ïn the liquid +is equal to 381 times 0.0029, or 1.11. Notice that the Nœ/3 term amounts to +almost 0.4, so it is quite sipnificant. With these numbers we predict a dielectric +constant of 2.76, which agrees reasonably well with the observed value of 2.64. +In Table 11-1 we give some experimental data on various materials (taken from +the Handbook oƒ Chemistru and Phụsics), together with the dielectric constants +calculated rom EBq. (11.28) in the way just described. The agreement between +observation and theory is even better for argon and oxygen than for C5a2——and not +so good for carbon tetrachloride. On the whole, the results show that Eq. (11.28) +works very well. +Table 11-1 +Computation of the dielectric constants of liquids +from the dielectric constant of the gas. +C52 1.0029 0.0029 0.00339 | 1.293 381 | 1.11 2.76 2.64 +O2 1.000523 | 0.000523 | 0.00143 | 1.19 832 | 0.435 1.509 1.507 +COl¿ 1.0030 0.0030 0.00489 | 1.59 325 | 0.977 2.45 2.24 +Ar 1.000545 | 0.000545 | 0.00178 | 1.44 810 | 0.441 1.517 1.54 +† Ratio = density of liquid/density of gas. +Our derivation of Eq. (11.28) is valid only for electronie polarization in liquids. +Tt is not right for a polar molecule like HạO. If we go through the same calculations +for water, we get 13.2 for Nœ, which means that the dielectric constant for the +liquid is megøaizue, while the observed value of œ ¡is 80. “The problem has to do +--- Trang 140 --- +with the correct treatment of the permanent dipoles, and Ônsager has pointed +out the ripht way to go. We do not have the time to treat the case now, but +1f you are interested it is discussed in Kittels book, Introduction to Solid State +Phụsics. +11-6 Solid dielectrics +Now we turn to the solids. 'Phe frst interesting fact about solids is that there +can be a permanent polarization built in—which exists even without applying an +electric fñeld. An example occurs with a material like wax, which contains long +molecules having a permanent dipole moment. lf you melt some wax and put a +strong electric fñeld on it when ït is a liquid, so that the dipole moments get partÌy +lined up, they will stay that way when the liquid freezes. The solid material will +have a permanent polarization which remains when the field is removed. Such a +solid is called an electrcet. +An electret has permanent polarization charges on its surface. It is the h +electrical analog of a magnet. I§ is not as useful, though, because free charges ¬ ¬ +from the air are attracted to its surfaces, eventually cancelling the polarization C3 3 3 C3 3 +charges. he electret is “discharged” and there are no visible external fields. ©@|@®@@|l@@|@œe@l@@ +A permanent internal polarization ? is also found occurring naturally in — | C C C ® C " +some crystalline substances. In such crystals, each unit cell of the lattice has an '©|©|Ol©l© +identical permanent dipole moment, as drawn in Eig. 11-8. All the dipoles point __ 19 ©J@@|@@j@@|OO@|_ +in the same direction, even with no applied electric fñeld. Many complicated C3 C3 C3 C3 C3 +crystals have, in fact, such a polarization; we do not normally notice it because eœ@leeleeleelee +the external fñelds are discharged, just as for the electrets. _~ TT +Tf these internal dipole moments oŸ a crystal are changed, however, external C3 3 3 C3 3 +fields appear because there is not time for stray charges to gather and cancel @©@@|@®@@|l@@lœ©e@œlœ@ +the polarization charges. lf the dielectric is in a condenser, free charges will be x pc +induced on the electrodes. For example, the moments can change when a dielectric : : : : : : +is heated, because of thermal expansion. “The efect is called pụroelectricitg. Fig. 11-8. A complex crystal lattice can +Similarly, if we change the stresses in a crystal—for instance, iŸ we bend it— have a permanent intrinsic polarization P. +again the moment may change a little bit, and a small electrical efect, called +piezoclectricit, can be detected. +For crystals that do not have a permanent moment, one can work out a +theory of the dielectric constant that involves the electronic polarizability of the +atoms. ÏIt goes mụch the same as for liquids. Some crystals also have rotatable +dipoles inside, and the rotation of these dipoles will also contribute to œ. Ín ionic +crystals such as NaC] there is also ?onic polarizabily. The crystal consists of a +checkerboard of positive and negative ions, and ín an electric field the positive @ +ions are pulled one way and the negatives the other; there is a net relative motion +of the plus and minus charges, and so a volume polarization. We could estimate © +the magnitude of the ionic polarizability from our knowledge of the stifness of ệ ` +salt crystals, but we will not go into that subject here. © +11-7 Ferroelectricity; BaTiOas Ty +. . . . @— Ms +We want to describe now one special class of crystals which have, just by . VQ +accident almost, a built-in permanent moment. 'Phe situation is so marginal Ỏ s% Ị +that if we increase the temperature a little bit they lose the permanent moment 4 +completely. Ôn the other hand, ¡f they are nearly cubic crystals, so that their © +mmoments can be turned in diferent directions, we can detect a large change in +the moment when an applied electric field is changed. All the moments fẨlip over O AM +and we get a large efect. 5ubstances which have this kind of permanent moment +are called ƒerroelectric, after the corresponding ferromagnetic efects which were eTi“2 OBa2 @©o° +first discovered in ïron. +We would like to explain how ferroelectricity works by describing a partic- Fig. 11-9. The unit cell of BaTiOs. The +ular example of a ferroelectric material. 'There are several ways in which the atoms really fill up most of the space; for +ferroelectric property can originate; but we will take up only one mysterious clarity, only the positions of their centers +case—that of barium titanate, Ba'1Os. This material has a crystal lattice whose are shown. +--- Trang 141 --- +basic cell is sketched in Eig. 11-9. It turns out that above a certain temperature, +specifically 118°Ơ, barium titanate is an ordinary dielectric with an enormous +dielectric constant. Below this temperature, however, it suddenly takes on a +permanent moment. +In working out the polarization of solid material, we must first fnd what are +the local fields in each unit cell. We must include the fields from the polarization +1tself, Just as we did for the case of a liquid. But a crystal is not a homogeneous +liquid, so we cannot use for the local fñeld what we would get in a spherical +hole. IÝ you work it out for a crystal, you ñnd that the factor 1/3 in Eq. (11.24) +becomes slightly diferent, but not far from 1/3. (For a simple cubic crystal, it +is Just 1/3.) We will, therefore, assume for our preliminary discussion that the +factor is 1/3 for BaTiOa. +Now when we wrote Eq. (11.28) you may have wondered what would happen +1ƒ Nơ became greater than ä. It appears as though would become negative. But +that surely cannot be right. Let's see what should happen If we were gradually +to increase œ in a particular crystal. As œ gets larger, the polarization gets +bigger, making a bigger local fñeld. But a bigger local ñeld will polarize each +atom more, raising the local ñelds still more. If the “give” of the atoms is enough, +the process keeps going; there is a kind of feedback that causes the polarization +to increase without limit—assuming that the polarization of each atom increases +in proportion to the fñeld. The “runaway” condition occurs when WMœ = 3. The +polarization does not become infinite, of course, because the proportionality +between the induced moment and the electric field breaks down at hiph fields, so +that our formulas are no longer correct. What happens is that the lattice gets +“locked in” with a high, self-generated, internal polarization. +In the case of Ba TiOs, there is, in addition to an electronic polarization, aÌso +a rather large ionic polarization, presumed to be due to titanium ions which can +move a little within the cubic lattice. 'Phe lattice resists large motions, so after +the titanium has gone a little way, iE jams up and stops. But the crystal cell is +then left with a permanent dipole moment. +In most crystals, this is really the situation for all temperatures that can be +reached. “The very interesting thing about barium titanate is that there is such +a delicate condition that if Nœ is decreased Just a little bit it comes unstuck. +Since decreases with increasing temperature—because of thermal expansion—— +we can vary j)œ by varying the temperature. Below the critical temperature +1% 1s Just barely stuck, so it is easy——by applying an external fñeld——to shift the +polarization and have it lock in a diferent direction. +Let's see IÝ we can analyze what happens in more detail. We call 74 the +critical temperature at which Vơø is exactly 3. As the temperature increases, ý +goes down a. little bit because of the expansion of the lattice. Since the expansion +is small, we can say that near the critical temperature +Nœ=3— 8(T —TT.), (11.30) +where Ø is a small constant, of the same order of magnitude as the thermal +expansion coeffieient, or about 10— to 10~8 per degree C. Now if we substitute +this relation into Eq. (11.28), we get that +g—1= 3— 8Œ — T,.) +8 ứ x 1.)/ 3. +Since we have assumed that đ(7' — T() is small compared with one, we can +approximate this formula by +&— Ì Irmxnï (11.31) +This relation is right, of course, only for 7' > 7¿. We see that just above +the critical temperature œ is enormous. Because œ is so close to 3, there +1s a tremendous magnification efect, and the dielectric constant can easily be +--- Trang 142 --- +as high as 50,000 to 100,000. It is also very sensitive to temperature. For +Increases in temperature, the dielectric constant goes down inversely as the +temperature, but, unlike the case of a dipolar gas, for which œ& — l goes inversely +as the øbsolute temperature, for ferroelectrics it varles inversely as the difference +between the absolute temperature and the critical temperature (this law is called +the Curie-Weiss law). +'When we lower the temperature to the critical temperature, what happens? +TÍ we imagine a lattice of unit cells like that in Fig. 11-9, we see that it is possible +to pick out chains of ions along vertical lines. One of them consists of alternating +oxygen and titanium ions. There are other lines made up of either barium or +oxygen ions, but the spacing along these lines is greater. We make a simple +model to imitate thìs situation by imagining, as shown in Fig. I1I-10(a), a series +of chaiïns of ions. Along what we call the main chain, the separation of the ions Ƒ—— 22 —— +1s, which is høiƒ the lattice constant; the lateral distance between identical +chaïns is 2a. There are less-dense chains in bebtween which we will ignore for the r$ ` ‡ +moment. 'To make the analysis a little easier, we will also suppose that all the a +ions on the main chain are identical. (It is not a serious simplification because + +all the important efects will still appear. 'This is one of the tricks of theoretical ‡ ộ +physics. One does a diferent problem because 1 is easier to figure out the first +time—then when one understands how the thing works, it is time to put in all +the complications.) $ ` ộ +Now let”s try to fnd out what would happen with our model. We suppose +that the dipole moment of each atom is øp and we wish to calculate the fñeld at +one of the atoms of the chain. We must find the sum of the fields from all the $ ộ +other atoms. We will fñrst calculate the fñeld from the dipoles in only one vertical +chain; we will talk about the other chains later. 'The field at the distance z from +a đipole in a direction along its axis is given by $ ° ộ +g—_L #9. (11.32) (a) +47g rŠ +At any given atom, the dipoles at equal distances above and below it give fields +in the same direction, so for the whole chain we get ‡ \ ‡ +2 2.2 2 : +Easn = TẾT đc (Đ+ tp + + } C Tê, (11.33) +[t is not too hard to show that if our model were like a completely cubic crystal— ‡ ‡ +that is, ¡f the next identical lines were only the distance ø away——the number 0.383 +would be changed to 1/3. In other words, if the next lines were at the distance ø ‡ ‡ ‡ +they would contribute only —0.050 unit to our sum. However, the next main chain +we are considering is at the distance 2ø and, as you remember from Chapter 7, +the fñield from a periodic structure dies of exponentially with distance. 'Pherefore ‡ ‡ +these lines contribute much less than —0.050 and we can just ignore all the other +chains. +lt 1s necessary now to fnd out what polarizability œ is needed to make the ‡ { ‡ +runaway process work. Suppose that the induced moment ø of each atom of the +chain is proportional to the ñeld on it, as in Eq. (11.6). We get the polarizing +ñeld on the atom from #2iaiạ using Bq. (11.32). So we have the two equations (b) +p= œgEbain Fig. 11-10. Models of a ferroelectric: +(a) corresponds to an antiferroelectric, and +and (b) to a normal ferroelectric. +0.383 p +J/chain — 8 +'There are two solutions: #4na¡„ and ø both zero, or +Z— 0.388) +with nai; and ø both finite. Thus iŸ œ is as large as a3/0.383, a permanent +polarization sustained by its own field will set in. 'This critical equality must be +--- Trang 143 --- +reached for barium titanate at jus the temperature 7¿. (Notice that IŸ œ were +larger than the critical value for small felds, it would decrease at larger fields +and at equilibrium the same equality we have found would hold.) +Eor BaTiOs, the spacing ø is 2 x 10” em, so we must expect that œ = +21.8 x 10? em. We can compare this with the known polarizabilities of the +individual atoms. For oxygen, œ = 30.2 x 10~? em; we're on the right trackl +But for titanium, œ = 2.4x 102 cmỞ; rather small. To use our model we should +probably take the average. (We could work out the chain again for alternating +atoms, but the result would be about the same.) So œ(average) = 16.3x 10~? em”, +which is not high enough to give a permanent polarization. +But wait a momentl We have so far only added up the electronic polarizabilities. +'There is also some ionic polarization due to the motion of the titanium ion. All +we need is an ionie polarizability of 9.2 x 10~2† em. (A more precise computation +using alternating atoms shows that actually 11.9 x 1072 emở is needed.) To +understand the properties of Ba'iOs, we have to assume that such an Ionic +polarizability exisbs. +'Why the titanium ion in barium titanate should have that much ionie polariz- +ability is not known. Eurthermore, why, at a lower temperature, it polarizes along +the cube diagonal and the face diagonal equally well is not clear. IỶ we fgure +out the actual size of the spheres in Eig. 11-9, and ask whether the titanium is a +little bit loose in the box formed by is neighboring oxygen atoms—which is what +you would hope, so that it could be easily shifted——you fnd quite the contrary. +Tt ñts very tightly. Phe barzwm atoms are slightly loose, but if you let them be +the ones that move, it doesn't work out. So you see that the subJect is really not +one-hundred percent clear; there are still mysteries we would like to understand. +Returning to our simple model of Fig. 11-10(a), we see that the feld from one +chain would tend to polarize the neighboring chain in the opposie direction, which +means that although each chain would be locked, there would be no net permanent +moment per unit volumel (Although there would be no external electric efects, +there are still certain thermodynamic effects one could observe.) Such systems +exist, and are called antiferroelectric. 5o what we have explained is really an +antiferroelectric. Barium titanate, however, is really like the arrangement in +Eig. 11-10(b). The oxygen-titanium chaïins are all polarized in the same direction +because there are intermediate chains of atoms in between. Although the atoms +in these chains are not very polarizable, or very dense, they will be somewhat +polarized, in the direction antiparallel to the oxygen-titanium chains. 'Phe small +fields produced at the next oxygen-titanium chain will get it started parallel to +the first. So BaiOs is really ferroelectric, and ï§ is because of the atoms in +between. You may be wondering: “But what about the direct efect between the +two O-'Li chains?” Remember, though, the direct efect dies of exponentially +with the separation; the efect of the chain of sfrong dipoles at 2a can be less +than the efect of a chain of weak ones at the distance ø. +This completes our rather detailed report on our present understanding of +the dielectric constants of gases, of liquids, and of solids. +--- Trang 144 --- +MglocfrosteaffC reerÏoggs +12-1 The same equations have the same solutions +'The total amount of information which has been acquired about the physical 12-1 The same equations have the +world since the beginning of scientific progress is enormous, and it seems almost same solutions +Impossible that any one person could know a reasonable fraction of it. But it is 12-2 The fow of heat; a point source +actually quite possible for a physicist to retain a broad knowledge of the physical near an infũnite plane boundary +world rather than to become a specialist in SOIH€ TIATTOW ôT€A. The T€ôSOnS for 12-3 The stretched membrane +this are threefold: First, there are great principles which apply to all the diferent 12-4 The đifusion of neutrons; a +kinds of phenomena—such as the principles of the conservation of energy and R l : +. ¬ : uniform spherical source ỉn a +of angular momentum. A thorough understanding of such principles gives an homogeneous medium +understanding of a great deal all at once. Second, there is the fact that many . . +complicated phenomena, such as the behavior of solids under compression, really 12-ã Irrotational Huid fow; the fow +basically depend on electrical and quantum-mechanical forces, so that if one past a sphere +understands the fundamental laws of electricity and quantum mechanies, there is 12-6 IHumination; the uniform +at least some possibility of understanding many of the phenomena that occur lighting of a plane +in complex situations. EFinally, there is a most remarkable coincidence: The 12-7 The “underlying unity” of nature +cquations ƒor nang difjerent phụsicalL situations hœue cractlụ the same appearancc. +OŸÝ course, the symbols may be diferent——one letter is substituted for another—— +but the mathematical form of the equations is the same. This means that having +studied one subject, we immediately have a great deal of direct and precise +knowledge about the solutions of the equations of another. +W© are now finished with the subject of electrostatics, and will soon go on to +study magnetism and electrodynamies. But before doing so, we would like to +show that while learning electrostatics we have simultaneously learned about a +large number of other subjects. We will ñnd that the equations of electrostatics +appear in several other places in physics. By a direct translation of the solutions +(of course the same mathematical equations must have the same solutions) it is +possible to solve problems in other fields with the same ease—or with the same +difculty—as in electrostatics. +'The equations of electrostatics, we know, are +(6E) = “9, (12.1) +VxE-=0. (12.2) +(We take the equations of electrostatics with dielectrics so as to have the most +general situation.) The same physics can be expressed in another mathematical +form: +t=-Vọ, (12.3) +W-:(xVớj) =—_——. (12.4) +Now the poïnt is that there are many physics problems whose mathematical equa- +tions have the same form. 'There is a potential (2) whose gradient multiplied by a +scalar function (&) has a divergence equal to another scalar function (—/wee/€o)- +'Whatever we know about electrostatics can immediately be carried over into +that other subject, and 0c 0ersø. (It works both ways, oŸ course—if the other +subJect has some particular characteristics that are known, then we can apply +that knowledge to the corresponding electrostatic problem.) We want to consider +a series of examples from different subJects that produce equations of this form. +--- Trang 145 --- +12-2 The flow of heat; a point source near an infinite plane boundary +W© have discussed one example earlier (Section 3-4)——the fow of heat. Imagine +a block of material, which need not be homogeneous but may consist of diferent +materials at diferent places, in which the temperature varies from point to point. +As a consequence of these temperature variations there is a ow of heat, which +can be represented by the vector h. It represents the amount of heat energy +which ñows per unit time through a unit area perpendicular to the fow. The +divergence of h, represents the rate per unit volume at which heat is leaving a +T©eg1ON: +{:h = rate of heat out per unit volume. +(We could, oŸ course, write the equation in integral form—just as we did in +electrostatics with Gauss' law—which would say that the Ñux through a surface +is equal to the rate of change of heat energy inside the material. We will not +bother to translate the equations back and forth between the diferential and the +integral forms, because it goes exactly the same as in electrostatics.) +The rate at which heat is generated or absorbed at various places depends, +Of course, on the problem. Suppose, for example, that there is a source of heat +inside the material (perhaps a radioactive source, or a resistor heated by an +electrical current). Let us call s the heat energy produced per unit volume per +second by this source. There may also be losses (or gains) of thermal energy to +other internal energies in the volume. lÝ is the internal energy per unit volume, +—đu/dt will also be a “source” of heat energy. We have, then, +V-h=s~ (12.5) +W© are not going to discuss just now the complete equation in which things +change with time, because we are making an analogy to electrostatics, where +nothing depends on the time. We will consider only s(eady heaf-fiou problems, +in which constant sources have produced an equilibrium state. In these cases, +V:.h=s. (12.6) +Tt is, of course, necessary to have another equation, which describes how the +heat Ñows at various places. In many materials the heat current is approximately +proportional to the rate of change of the temperature with position: the larger +the temperature diference, the more the heat current. As we have seen, the +0uec‡or heat current is proportional to the temperature gradient. The constant of +proportionality , a property of the material, ¡is called the ¿hermal conducliuitg. +h =—KVI. (12.7) +TÍ the properties of the material vary from place to place, then # = (+, 0, 2), +a function of position. [Equation (12.7) is not as fundamental as (12.5), which +expresses the conservation of heat energy, since the former depends upon a special +property of the substance.]| IÝ now we substibute Eq. (12.7) into Bq. (12.6) we +W:(KVT) = —s, (12.8) +which has exactly the same form as (12.4). sSteadu heaft-flou problems œnd +electrostatic problems are the samne. The heat flow vector h corresponds to F, +and the temperature 7 corresponds to ó. We have already noticed that a poïnt +heat source produces a termmperature feld which varies as 1/? and a heat fow +which varies as 1/rz?. This is nothing more than a translation of the statements +from electrostatics that a point charge generates a potential which varies as l/z +and an electric feld which varies as 1/r2. W© can, in general, solve static heat +problems as easily as we can solve electrostatic problems. +Consider a simple example. Suppose that we have a cylinder of radius ø at +the temperature 71, maintained by the generation of heat in the cylinder. (It +could be, for example, a wire carrying a current, or a pipe with steam condensing +--- Trang 146 --- +inside.) The cylinder is covered with a concentric sheath of insulating material +which has a conductivity #. Say the outside radius oŸ the insulation is b and the +outside is kept at temperature 72 (Eig. 12-la). We want 0o ñnd out at what rate ZZ7%>>, +heat will be lost by the wire, or steampipe, or whatever it is in the center. Let 4 ` +the total amount of heat lost from a length Ù of the pipe be called G—which is < ` +what we are trying to ñnd. N2 +How can we solve this problem? We have the diferential equations, but J Lé2? ìN: +since these are the same as those of electrostatics, we have really already solved Ó ©522 / +the mathematical problem. 'Phe analogous problem is that of a conductor of ` SS +radius ø at the potential ó¡, separated from another conductor of radius ö at the sà X⁄ T +potential ó2, with a concentric layer of dielectric material in bebween, as drawn <>>ZZ +in Eig. 12-1(b). NÑow since the heat ow b corresponds to the electric field E, +the quantity Œ that we want to fnd corresponds to the fux of the electric ñeld (a) +from a unit length (in other words, to the electric charge per unit length over eo). +W©e have solved the electrostatic problem by using €Gauss' law. We follow the ZZ<2>> +same procedure for our heat-ow problem. Tội ` +trom the symmetry of the situation, we know that h depends only on the < \Š +distance from the center. So we enclose the pipe in a gaussian cylinder of length Ẳ h'ã ⁄⁄23À S0 +and radius r. Erom Gauss' law, we know that the heat fow h multiplied by the ý J9 ` +area 2mrL of the surface must be equal to the total amount of heat generated Ó s22 ⁄ +inside, which is what we are calling Œ: ` `2 +2arLh=G_ on chẽ CC, (12.9) 2ZZ +'The heat fow is proportional to the temperature gradient: 0) +Fig. 12-1. (a) Heat flow ¡in a cylindrical +h=—EVT, geometry. (b) The corresponding electrical +. . . . problem. +or, in this case, the radial component of h is +h=—K an” +This, together with (12.9), gives +dr — 2nKlr. 12.10) +Integrating from ? = ø to r = Ù, we get +ho (12.11) +Solving for GŒ, we fnd +G= 2nKL(- - 1) (12.12) +In(b/a) +This result corresponds exactly to the result for the charge on a cylindrical +condenser: +Q= 2coL(@1 — óa) +4 In(0/a) l +"The problems are the same, and they have the same solutions. From our knowledge +of electrostatics, we also know how much heat is lost by an insulated pipe. +Let's consider another example of heat Ñow. Suppose we wish to know the +heat fow in the neighborhood of a point source of heat located a little way +beneath the surface of the earth, or near the surface oŸ a large metal block. The +localized heat source might be an atomie bomb that was set of underground, +leaving an intense source of heat, or it might correspond to a small radioactive +source inside a block of Iron—there are numerous possibilities. +We will treat the idealized problem of a point heat source of strength G +at the distance ø beneath the surface of an infinite block of uniform material +whose thermal conductivity is #. And we will neglect the thermal conductivity +--- Trang 147 --- +of the air outside the material. We want to determine the distribution of the +temperature on the surface of the block. How hot is it right above the source +and at various places on the surface of the block? +How shall we solve it? It is like an electrostatic problem with two materials +with diferent dielectric coefficients on opposibe sides of a plane boundary. Ahal +Perhaps it is the analog of a point charge near the boundary between a dielectric +and a conductor, or something similar. Let?s see what the situation is near the +surface. The physical condition is that the normal component of h on the surface +1s zero, since we have assumed there is no heat fow out of the block. We should +ask: In what electrostatic problem do we have the condition that the normal +component of the electric fñeld (which is the analog of h) is zero at a surface? +There is nonel +'That is one of the things that we have to watch out for. For physical reasons, +there may be certain restrictions in the kinds of mathematical conditions which +arise in any one subject. So iŸ we have analyzed the diferential equation only for +certain limited cases, we may have missed some kinds of solutions that can occur +in other physical situations. For example, there is no material with a dielectric +constant of zero, whereas a vacuum does have zero thermal conductivity. So +there is no electrostatic analogy for a perfect heat insulator. VWe can, however, ¬ ` l ⁄ Z +still use the same rmethods. We can try to #nagine what would happen i1f the ` —À “ +dielectric constant øere zero. (Of course, the dielectric constant is never zero in " ¬¬ N | Z "4 = +any real situation. But we might have a case in which there is a material with a ThS 4< “_~“” K=0 +very hígh dielecbriec constant, so that we could neglect the dielectric constant oŸ " .. .. +the air outside.) ' —.. ! NG K +How shall we ñnd an electrie fñeld that has „ø component perpendicular to the V FZZTZ 77 [Z1 +surface? That is, one which is always #angent at the surface? You will notice that a .n.6),oun +our problem is opposite to the one ofa point charge near a plane conductor. There mm án +we wanted the feld to be perpendicular to the surface, because the conductor ƯA Nợ, H +was all at the same potential. In the electrical problem, we invented a solution by x.Kx +imagining a point charge behind the conducting plate. We can use the same idea X+X> +again. We try to pick an “image source” that will automatically make the normal T = Constant “ñ +component of the field zero at the surface. The solution is shown in Eig. 12-2. T +An image source of the same sign and the same strength placed at the distance œ +above the surface will cause the feld to be always horizontal at the surface. The TENPERATURE +normal components of the two sources cancel out. +Thus our heat ow problem ¡s solved. The temperature everywhere is the 0 3 22p +same, by direct analogy, as the potential due to two equal point chargesl The Fig. 12-2. The heat flow and isothermals +temperature 7 at the distance z from a single point source G in an infnite near a point heat source at the distance a +medium is G below the surface of a good thermal con- +T= nh (12.13) ductor. +(This, of course, is just the analog of ở = g/4eog?.) The temperature for a poïnt +source, together with its Image source, is +1= “1... (12.14) +4mlfr, 4mlra +This formula gives us the temperature everywhere in the block. Several isothermal +surfaces are shown in Eig. 12-2. Also shown are lines of h, which can be obtained +from h = —EVT. +W© originally asked for the temperature distribution on the surface. Eor a +point on the surface at the distance ø from the axis, rỊ = ra = 4⁄02 + a2, so +T(surface) = 1E Mr>x-i (12.15) +This function is also shown in the fñgure. The temperature is, naturally, higher +right above the source than it is farther away. This is the kind of problem that +geophysicists often need to solve. We now see that it is the same kind of thing +we have already been solving for electricity. +--- Trang 148 --- +12-3 The stretched membrane +Now let us consider a completely diferent physical situation which, nev- ~. +In the static case—where Ø/Ø = 0—we have Eq. (12.4) all over again! We +can use our knowledge of electrostatics to solve problems about the difusion of ⁄ | l ` +neutrons. So let”s solve a problem. (You may wonder: IW do a problem iÝ we +have already done all the problems in electrostatics? We can do it ƒføsfer this l h +time because we høøe done the electrostatic problemsl) $ Ị +3uppose we have a block of material in which neutrons are being generated—— +say by uranium fssion——uniformly throughout a spherical region of radius ø +(Fig. 12-7). We would like to know: What is the density of neutrons everywhere? +How uniform is the density of neutrons in the region where they are being Ị +generated? What is the ratio of the neutron density at the center to the neutron 0 a T +density at the surface of the source region? Finding the answers is easy. The +source density ,%o replaces the charge density ø, so our problem is the same as &) +the problem of a sphere of uniform charge density. Pinding ẢÑ is just like ñnding Fig. 12-7. (a) Neutrons are produced uni- +the potential Ọ. We have already worked out the fields inside and outside of a, formly throughout a sphere of radius a In +uniformly charged sphere; we can integrate them to get the potential. Outside, a large graphite block and diffuse outward. +the potential is Q/4zcor, with the total charge Q given by 4za3ø/3. So The neutron density Ñ is found as a function +of r, the distance from the center of the +0a3 source. (b) The analogous electrostatic sit- +Óoutside 3eạr` 2.23) uation: a uniform sphere of charge, where /M +corresponds to ở and J corresponds to E. +For points inside, the field is due only 6o the charge Q(z) inside the sphere of +radius ?, Q(r) = 4mxr3o/3, so +tE= 3aọ: (12.24) +The fñeld increases linearly with r. Integrating to get ó, we have +Ôinsde = —£ — +a constant. +--- Trang 151 --- +At the radius ø, Ø¡w¡ae must be the same as Óoutside, sO the constant must +be øa2/2co. (We are assuming that ó is zero at large distances from the source, +which will correspond to W being zero for the neutrons.) Therefore, +Ởinside = mm S — s): (12.25) +W© know immediately the neutron density in our other problem. “The answer +TNoutside — Tnạ (12.26) +ÄNinsiae — sp 5 — 5) (12.27) +ÑN is shown as a function oŸ r in Eig. 12-ĩ. +Now what is the ratio of density at the center to that at the edge? At the +center (? = 0), it is proportional to 3ø2/2. At the edge (r = ø) it is proportional +to 242/2, so the ratio of densities is 3/2. A uniform source doesn't produce a +uniform density of neutrons. You see, our knowledge of electrostatics gïves us a +good start on the physics oŸ nuclear reactOrS. +There are many physical circumstances in which difusion plays a big part. +The motion of ions through a liquid, or of electrons through a semiconduector, +obeys the same equation. We fñnd again and again the same equations. +12-5 Irrotational ñuid fow; the flow past a sphere +Let's now consider an example which is not really a very good one, because the +cequations we will use will not really represent the subject with complete generality +but only in an artificial idealized situation. We take up the problem of ueter +ffou. In the case of the stretched sheet, our equations were an approximation +which was correct only for small defleclons. For our consideration of water ñow, +we will not make that kind of an approximation; we must make restrictions that +do not apply at all to real water. We treat only the case of the steady fow of an +tncompressible, nonuiscous, circulation-free liquid. 'Then we represent the flow by +giving the velocity (r) as a function oŸ position r. TẾ the motion is steady (the +only case for which there is an electrostatic analog) is independent of time. Tf +p 1s the density of the fuid, then ø is the amount of mass which passes per unit +time through a unit area. By the conservation of matter, the divergence oŸ Ø0 +will be, in general, the time rate of change of the mass of the material per unit +volume. We will assume that there are no processes for the continuous creation or +destruction of matter. The conservation of matter then requires that V - ø = 0. +(It should, in general, be equal to —Øø/Ø#, but since our fuid is incompressible, +ø cannot change.) Since ø is everywhere the same, we can factor it out, and our +cquation is simply +V-‹u=0. +Goodl WS have electrostatics again (with no charges); it's just like V - = 0. +Not sol Electrostatics is nof simply V - =0. It is a pưa¿r of equations. Ône +equation does not tell us enough; we need still an additional equation. To +match electrostatics, we should have also that the curÏ of is zero. But that +1s not generally true for real liquids. Most liquids will ordinarily develop some +circulation. 5o we are restricted to the situation in which there is no circulation +of the fuid. Such flow is often called rrotational. Anyway, iŸ we make all our +assumptions, we can magine a case of fuid fow that is analogous to electrostatics. +So we take +V.u=0 (12.28) +Vxø=(0. (12.29) +--- Trang 152 --- +We want to emphasize that the number of cireumstances in which liquid +fow follows these equations is far rom the great majority, but there are a Íew. +They must be cases in which we can neglect surface tension, compressibility, +and viscosity, and in which we can assume that the fÑow ïs irrotational. Thhese +assumptions are valid so rarely for real water that the mathematician John +von Neumamn said that people who analyze Eqs. (12.28) and (12.29) are studying +“dry water”! (We take up the problem o£ ñuid fow in more detail in Chapters 40 +and 41.) +Because V x ø = 0, the velocity of “dry water” can be written as the gradient +of some potential: +0 =— VỤ. (12.30) +'What is the physical meaning of ? 'There isn't any very useful meaning. The +velocity can be written as the gradient of a potential simply because the fñow is +irrotational. And by analogy with electrostatics, is called the 0elocitụ potential, +but it is not related to a potential energy in the way that ó is. Since the divergence +Of ® is zero, we have +:(Vú) = V?ụ =0. (12.31) +The velocity potential obeys the same diferential equation as the electrostatic +potential in free space (ø = 0). +Let”s pick a problem ïn irrotational fow and see whether we can solve it by the +methods we have learned. Consider the problem of a spherical ball falling through +a liquid. lf it is goïng too slowly, the viscous forces, which we are disregarding, +will be important. IÝit is goïng too fast, little whirlpools (turbulence) will appear p \v +in its wake and there will be some circulation of the water. But ïf the ball is +going neither too fast nor too sÌow, it is more or less true that the water fow will x +ft our assumptions, and we can describe the motion of the water by our simple +equations. +Tt is convenient to describe what happens In a frame of reference fzed in the +sphcre. In this Íframe we are asking the question: How does water fow past a +sphere at rest when the fow at large distances is uniform? 'Phat is, when, far from +the sphere, the fow is everywhere the same. 'Phe fow near the sphere will be as +shown by the streamlines drawn in Fig. 12-8. These lines, always parallel to ®, +correspond to lines of electric field. We want to get a quantitative description for +the velocity field, i.e., an expression for the velocity at any point ?. +W©e can find the velocity from the gradient of ý, so we first work out the Fig. 12-8. The velocity field of irrota- +potential. We want a potential that satisfies Eq. (12.31) everywhere, and which tional fluid flow past a sphere. +also satisfles two restrictions: (1) there is no fow in the spherical region inside the +surface of the baill, and (2) the ow is constant at large distances. To satisfy (1), +the component of 0 normal ©o the surface of the sphere must be zero. 'Phat +means that Øj/Ôr is zero at r = a. To satisfy (2), we must have Øj/Øz = 0ạọ at +all points where r >> ø. Strictly speaking, there is no electrostatic case which +corresponds exactly to our problem. It really corresponds to putting a sphere +of dielectric constant zero in a uniform electric field. If we had worked out the +solution to the problem of a sphere of a dielectric constant & in a uniform field, +then by putting = 0 we would immediately have the solution to this problem. +We have not actually worked out this particular electrostatie problem in detail, +but let's do it now. (WSe could work directly on the Huid problem with ø and ở, +but we will use and ở because we are so used to them.) +The problem is: Find a solution of V?ó = 0 such that # = —Wó is a constant, +say ọ, for large r, and such that the radial component of # is equal to zero +atr=a. That 1s, +5 =0. (12.32) +Our problem involves a new kind of boundary condition, not one for which ó +is a constant on a surface, but for which Øj/Ôï is a constant. That is a little +diferent. It is not easy to get the answer immediately. First of all, without +the sphere, @ would be —oz. Then would be in the z-direction and have +--- Trang 153 --- +the constant magnitude lo, everywhere. Now we have analyzed the case of a +dielectric sphere which has a uniform polarization inside ï%, and we found that +the field inside such a polarized sphere is a uniform field, and that outside 1t +1s the same as the field of a point dipole located at the center. So let”s guess +that the solution we wanf is a superposition of a uniform field plus the fñield of a +dipole. The potential of a dipole (Chapter 6) is pz/4xeor3. Thus we assume that +=—È ——n: 12.33 +? 02+ 4mreor3 ) +Since the dipole field falls of as 1/rỞ, at large distances we have just the ñeld Hạ. +Our guess will automatically satisfy condition (2) above. But what do we take +for the dipole strength ø? 'To fñnd out, we may use the other condition on ở, +Eq. (12.32). We must differentiate ¿ with respect to z, but oŸ course we must do +So at a constant angle Ø, so it is more convenient If we first express ø in terms of +r and Ø, rather than of z and r. Since z = rcosØ, we get += — Eurcos 0 P SỐ (12.34) += — + ——~. . +ụ 47cogr2 +'The radial component of E is +"`... ..ẻˆ (12.35) +Ør 27cor3 +'This must be zero at ?z = ø for all Ø. Thịs will be true ïf +Ð= —2mcoa3Eù. (12.36) +Note carefully that ¡if both terms in Eq. (12.35) had not had the same Ø- +dependence, it would not have been possible to choose ø so that (12.35) turned +out to be zero at z = ø for all angles. "The fact that it works out means that +we have guessed wisely in writing Bq. (12.33). Of course, when we made the +guess we were looking ahead; we knew that we would need another term that +(a) satisũed V2ø = 0 (any real feld would do that), (b) dependent on cosØ, and +(c) fell to zero at large r. The dipole field is the only one that does all three. +Using (12.36), our potential is +=-—Eù cos 0n 32}: (12.37) +'The solution of the ñuid ñow problem can be written simply as +=— 6 =— ]- 12.38 +Ụ 0o COS ( + 2z) ( ) +Tt is straightforward to ñnd ø from this potential. We will not pursue the matter +further. +12-6 Humination; the uniform lighting of a plane +In this section we turn to a completely diferent physical problem——we want +to illustrate the great variety of possibilities. Thịis time we will do something that +leads to the same kind oŸ zntegral that we found in electrostatics. (If we have a +mathematical problem which gives us a certain integral, then we know something +about the properties of that integral If it is the same integral that we had to +do for another problem.) We take our example from illumination engineering. +Suppose there is a light source at the distance ø above a plane surface. What +1s the illumination of the surface? 'That is, what is the radiant energy per unit§ +tỉme arriving at a unit area of the surface? (See Fig. 12-9.) We suppose that the +Source is spherically symmetrie, so that light is radiated equally in all directions. +Then the amount of radiant energy which passes through a unit area œ r¿ghf +gngÏes to a light fow varies inversely as the square of the distance. It is evident +--- Trang 154 --- +ZZEE s +ĐằạTT—- 2S. mày S980 Fig. 12-9. The illimination f„ of a surface +đãđáxm ¡s the radiant energy per unit time arriving +h at a unit area of the surface. +that the intensity of the light in the direction normal to the ñow is given by the +same kind of formula as for the electric fñeld from a point source. If the light rays +meet the surface at an angle Ø to the normal, then ?„, the energy arriving øer +tun#t œrea of the surface, is only cos Ø as great, because the same energy goes onto +an area larger by 1/cosØ. TỶ we call the strength of our light source ,5, then lạ, +the ïllumination of a surface, is +l„ = ` ©y - Tt, (12.39) +where e„ is the unit vector from the source and ?ø is the unit normal to the +surface. The ïllumination ?„ corresponds to the normal component of the electric +ñeld from a point charge of strength 4zeoS. Knowing that, we see that for any +distribution of light sources, we can ñnd the answer by solving the corresponding +electrostatic problem. We calculate the vertical component of electric field on +the plane due to a distribution oŸ charge in the same way as for that of the light +Sources. +Consider the following example. We wish for some special experimental +situation to arrange that the top surface of a table will have a very uniform +ilumination. We have available long tubular fuorescent lights which radiate +uniformly along their lengths. We can illuminate the table by placing the +ñuorescent tubes in a regular array on the ceiling, which is at the height z above +the table. What ¡is the widest spacing b from tube to tube that we should use +1Ý we want the surface illumination to be uniform to, say, within one part in +a thousand? Ansuer: (1) Find the electric field from a grid of wires with the +spacing b, each charged uniformly; (2) compute the vertical component of the +electric feld; (3) ñnd out what b must be so that the ripples of the field are not +more than one part in a thousand. +In Chapter 7 we saw that the electric ñeld of a grid of charged wires could be +represented as a sum of terms, each one of which gave a sinusoidal variation of +the field with a period of b/n, where ø is an integer. The amplitude of any one +of these terms is given by Eq. (7.44): +JA= Aner2mmnz/b, +We need consider only ?ø+ = 1, so long as we only want the field at points not +too close to the grid. Eor a complete solution, we would still need to determine +the coefficients A„, which we have not yet done (although it is a straightforward +calculation). Since we necd only 4, we can estimate that its magnitude is +roughly the same as that of the average field. "The exponential factor would +then give us directly the relafzue amplitude of the varlations. lÝ we want this +factor to be 103, we fñnd that b must be 0.91z. If we make the spacing of the +* Since we are talking about ¿ncoherent sources whose #nfensities always add linearly, the +analogous electric charges will always have the same sign. Also, our analogy applies only to the +light energy arriving at the top of an opaque surface, so we must include in our integral only +the sources which shine on the surface (and, naturally, not sources located below the surfacel). +--- Trang 155 --- +fuorescent tubes 3/4 of the distance to the ceiling, the exponential factor is +then 1/4000, and we have a safety factor of 4, so we are fairly sure that we will +have the illumination constant to one part in a thousand. (An exact calculation +shows that Á¡ is really twice the average field, so that b 0.83z.) It is somewhat +surprising that for such a uniform illumination the allowed separation of the +tubes comes out so large. +12-7 The “underlying unity” of nature +In this chapter, we wished to show that in learning electrostatics you have +learned at the same time how to handle many subJects in physics, and that by +keeping this in mind, ¡it is possible to learn almost all of physics in a limited +number of years. +However, a question surely suggests itself at the end of such a discussion: +Whụ are the cquations from different phenomena so sửữnidar? WS might say: +“E is the underlying unity of nature.” But what does that mean? What could +such a statement mean? It could mean simply that the equations are similar for +diferent phenomena; but then, of course, we have given no explanation. “The +“underlying unity” might mean that everything is made out of the same stuft, +and therefore obeys the same equations. 'Phat sounds like a good explanation, +but let us think. “The electrostatic potential, the difusion of neutrons, heat +fow—are we really dealing with the same stuf? Can we really imagine that +the electrostatic potential 1s phụs¿caliu identical to the temperature, or 6o the +density of particles? Certainly ó is not ezactl the same as the thermal energy of +particles. "The displacement of a membrane is certainly øø‡ like a temperature. +'Why, then, ¡is there “an underlying unity”? +A closer look at the physics of the various subjects shows, in fact, that the +cquations are not really identical. The equation we found for neutron difusion is +only an approximation that is good when the distance over which we are looking +1s large compared with the mean free path. If we look more closely, we would see +the individual neutrons running around. Certainly the motion of an individual +neutron is a completely diferent thing from the smooth variation we get om +solving the diferential equation. 'Phe diferential equation is an approximation, +because we assume that the neutrons are smoothly distributed in space. +1s it possible that 02s is the clue? 'Phat the thing which is common to all +the phenomena is the spøce, the framework into which the physics is put? As +long as things are reasonably smooth in space, then the important things that +will be involved will be the rates of change of quantities with position in space. +That is why we always get an equation with a gradient. 'Phe derivatives rmusf +appear in the form of a gradient or a divergence; because the laws of physics are +tndependent oƒ direction, they must be expressible in vector form. 'The equations of +electrostatics are the simplest vector equations that one can get which involve only +the spatial derivatives of quantities. Any other sữnpÏe problem——or simplification +of a complicated problem——must look like electrostatics. What is common to +all our problems is that they involve spøce and that we have Zmn#tated what 1s +actually a complicated phenomenon by a simple diferential equation. +'That leads us to another interesting question. Is the same statement perhaps +also true for the elecfrosta#ic equations? Are they also correct only as a smoothed- +out imitation of a really much more complicated microscopic world? Could ¡it +be that the real world consists of little X-ons which can be seen only at 0er +tiny distances? And that in our measurements we are always observing on such +a large scale that we can” see these little X-ons, and that is why we get the +difÑferential equations? +Our currently most complete theory of electrodynamics does indeed have +1ts difficulties at very short distances. So it is possible, in principle, that these +equations are smoothed-out versions of something. 'PThey appear to be correct +at distances down to about 10714 em, but then they begin to look wrong. It +is possible that there is some as yet undiscovered underlying “machinery,” and +that the details of an underlying complexity are hidden in the smooth-looking +--- Trang 156 --- +equations——as is so in the “smooth” difusion of neutrons. But no one has yet +formulated a successful theory that works that way. +Strangely enough, it turns out (for reasons that we do not at all understand) +that the combination of relativity and quantum mechanics as we know them +seems to ƒorbzd the invention of an equation that is fundamentally diferent +from Eq. (12.4), and which does not at the same time lead to some kind of +contradiction. Not simply a disagreement with experiment, but an ?„ernal +contradiction. Ás, for example, the prediction that the sum of the probabilities +of all possible occurrences is not equal to unity, or that energies may sometimes +come out as complex numbers, or some other such idiocy. No one has yet made +up a theory of electricity for which V2ø = —ø/eo is understood as a smoothed-out +approximation to a mechanism underneath, and which does not lead ultimately +to some kind of an absurdity. But, i§ must be added, it is also true that the +assumption that W2ø@ = —//eo is valid for all distances, no matter how small, +leads to absurdities oŸ its own (the electrical energy of an electron is inlnite)— +absurdities from which no one yet knows an escape. +--- Trang 157 --- +I3 +JMqgJnao£osfetff©s +13-1 The magnetic ñeld +The force on an electric charge depends not only on where it is, but also 13-1 The magnetic field +on how fast it is moving. Every point in space 1s characterized by two vector 13-2 Electric current; the conservation +quantities which determine the force on any charge. First, there is the electric of charge +ƒorce, which gives a force component independent of the motion of the charge. We 13-3 The magnetic force on a current +describe it by the electric ñeld, #. Second, there is an additional force component, R +called the magnetic ƒorce, which depends on the velocity of the charge. This 14-4 The magnetic ñeld of sieady +. . . . ¬= currents; Ampère”s law +magnetic force has a strange directional character: At any particular point in . . +space, both the đirection oŸ the force and its magnitude depend on the direction 1ả-5 The magnetic field of a siraight +of motion of the particle: at every instant the force is always at right angles wire and of a solenoid; atomic +to the velocity vector; also, at any particular point, the force is always at right Currents +angles to a fiwed đireclion in spaee (see Fig. 13-1); and fñnally, the magnitude of — 13-6 The relativity of magnetic and +the force is proportional to the cormponent oŸ the velocity at right angles to this electric fields +unique direction. It is possible to describe all of this behavior by defñning the 13-7 The transformation oŸ currents +magnetic field vector Ö, which specifies both the unique direction in space and and charges +the constant of proportionality with the velocity, and to write the magnetic force 13-8 Superposition; the right-hand +as gu < Ö. The total electromagnetic force on a charge can, then, be written as rule +t=q(E+ox Đ). (13.1) +'This 1s called the Joren‡z ƒorce. +The magnetic force is easily demonstrated by bringing a bar magnet close to +a cathode-ray tube. The deflection of the electron beam shows that the presence +of the magnet results in forces on the electrons transverse to their direction of +motion, as we described in Chapter 12 of Vol. I. +The unit of magnetic feld #Ö is evidently one newton-second per coulomb- Reuieu: Chapter 15, Vol. L, The Special +meter. The same unit is also one volt-seceond per meterZ. It is also called one Theoru oƒ Relatiutụ +tueber per squdre 1ne€ter. +13-2 Electric current; the conservation of charge +W© consider first how we can understand the magnetic forces on wires carrying B +electric currents. In order to do this, we deñne what is meant by the current +density. Electric currents are electrons or other charges in motion with a net drift +or fow. We can represent the charge fow by a vector which gives the amount +of charge passing per unit area and per unit time through a surface element at 90° ¿ +right angles to the flow (just as we did for the case of heat fow). We call this the q - +current densit and represent it by the vector 7. It is directed along the motion 90) +of the charges. If we take a small area A5 at a given place in the material, the +amount of charge fÑowing across that area in a unit tỉme is F +J:Tn®A®, (13.2) Fig. 13-1. The velocity-dependent com- +ponent of the force on a moving charge is at +where ?ø is the unit vector normal to A55. right angles to v and to the direction of B. +The current density is related to the average flow velocity of the charges. Sup- ltis also proportional to the component of v +pose that we have a distribution of charges whose average motion is a drift with the at right angles to , that is, to v sin ổ. +velocity 0. As this distribution passes over a surface element A5, the charge Aq +passing through the surface element in a tỉme Aứ is equal to the charge con- +tained in a parallelepiped whose base is AŠ and whose height is A£, as shown in +Eig. 13-2. The volume of the parallelepiped is the projection of AŠ at right angles +--- Trang 158 --- +to times ø A#, which when multiplied by the charge density ø will give Aq. Thus +Aq= pu-nwA®S At. s—- +The charge per unit time is then øo - AS%, from which we get ⁄ ⁄⁄2 ¬ +j — po. (13.3) ⁄ N +Tƒ the charge distribution consists of individual charges, say electrons, each S45 b +with the charge gø and moving with the mean velocity , then the current density is \Ýy v +J — Nụ. (13.4) ⁄4 ⁄ vAt +where # is the number of charges per unit volume. 42 “ +The total charge passing per unit time through any surface Š is called the TT +clectric curren, T. It is equal to the integral of the normal component of the Ñow Fig. 13-2. lf a charge distribution of den- +through all of the elements of the surface: sity ø moves with the velocity v, the charge +per unit time through AS is pv - nAS. +1= | -rò dS (13.5) +(see Fig. 13-3). +The current ƒ out of a closed surface Š represents the rate at which charge +leaves the volume V enclosed by Š. One of the basic laws of physics is that j , +clectric charge is indestructible; it 1s never lost or created. Electric charges can j _ J +move from place to place but never appear from nowhere. We say that charge ¡s 2S +conserued. TỶ there is a net current out of a closed surface, the amount of charge dS +inside must decrease by the corresponding amount (Fig. 13-4). We can, therefore, +write the law of the conservation of charge as +SURFACE S +J J:rd5 = —a(inside): (13.6) Fig. 13-3. The current / through the +any closed surface S is ƒ j- ndS. +surface +'The charge inside can be written as a volume integral of the charge density: +Qinsiae — J pdỰ. (13.7) N \ Ị n# +Insile Š xế +1nSIdG +T we apply (13.6) to a small volune AV, we know that the left-hand integral TÌN Ñ \ \ Z +isV-7 AV. The charge inside is ø AV, so the conservation of charge can aÌso +be written as 8 —=— —Y +W.j=-SP (13.8) +Øi ~— CLOSED +(Gauss` mathematics once againl). 7 Ỉ \ SURLACE +13-3 The magnetic force on a current Fig. 13-4. The integral of j - n over a +Now we are ready to fnd the force on a current-carrying wire in a magnetic Xa Suácg S no Q im Mà rate of change +fñeld. 'Phe current consists of charged particles moving with the velocity along 0TR giải CHAF06 ý HSI66. +the wire. Each charge feels a transverse force +tt —=quxB +(Fig. 13-5a). IÝ there are such charges per unit volume, the number in a small +volune AV of the wire is ý AV. 'The total magnetic foree A#' on the volune AW +1s the sum of the forces on the individual charges, that is, +AF.=(NAV)(qo x B). +But Nựo 1s Just 7, so +AF=j7xbBbBAV (13.9) +(Fig. 13-5b). The force per unit volume is j x Ö. +--- Trang 159 --- +Tf the current is uniform across a wire whose cross-sectional area is 4, we may +take as the volume element a cylinder with the base area A and the length AF. B +'Then _—_—— AL ——. +AF—=7x ĐAAL. (13.10) ị lR +NÑow we can call 7A the vector curren$ Ï in the wire. (Its magnitude is the — h _— _= Ị +electric current in the wire, and its direction is along the wire.) Then h II. Ni _—~ +lẾ ma .< +AF=TIx BAI. (13.11) / z / +'The force per unit length on a wire is Ï x B. (a) +This equation gives the important result that the magnetic force on a wire, +due to the movement of charges in it, depends only on the total current, and not +on the amount of charge carried by each particle—or even its sign! The magnetic +Íforce on a wire near a magnet is easily shown by observing its deflection when a +current is turned on, as was described in Chapter 1 (see EFig. 1-6). I AL ị ¬ +13-4 The magnetic fñeld of steady currents; Ampère?s law ` — ==—=m_ / +We have seen that there is a force on a wire in the presence of a magnetic field, h h Z lẻ h . +produced, say, by a magnet. From the principle that action equals reaction we — : +might expect that there should be a force on the source of the magnetic field, I.e., / / +on the magnet, when there is a current through the wire.* There are indeed such AF (b) +forces, as is seen by the defection of a compass needle near a current-carrying +wire. Now we know that magnets feel forces from other magnets, so that means Fig. 13-5. The magnetic force on a +that when there is a current in a wire, the wire itself generates a magnetic field. current-carrying wire is the sum of the forces +Moving charges, then, produce a magnetic feld. We would like now to try to on the individual moving charges. +discover the laws that determine how such magnetic fñelds are created. 'Phe +question is: Given a current, what magnetic feld does it make? "The answer to +this question was determined experimentally by three critical experiments and a +brilliant theoretical argument given by Ampère. We will pass over this interesting +historical development and simply say that a large number of experiments have +demonstrated the validity of Maxwells equations. We take them as our starting +point. If we drop the terms involving time derivatives in these equations we get +the equations oŸ mmagnetostatics: +V.:B=0 (13.12) +and : +VxB=”*. (13.13) +These equations are valid only if all electric charge densities are constant and +all currents are steady, so that the electric and magnetic ñelds are not changing +with time—all of the fñelds are “static.” +We may remark that it is rather dangerous to think that there is such a +thing as a static magnetic situation, because there must be currents in order to +get a magnetic ñeld at all and currents can come only from moving charges. +“Magnetostatics” is, therefore, an approximation. It refers to a special kind +of dynamic situation with large mwưmbers of charges in motion, which we can +approximate by a s/eadu flow of charge. Only then can we speak oŸ a current +density 7 which does not change with time. The subject should more accurately +be called the study of steady currents. Assuming that all fñelds are steady, we drop +all terms in ØE/Øt and 9B/Ôt from the complete Maxwell equations, Eqs. (2.41), +and obtain the two equations (13.12) and (13.13) above. Also notice that since +the divergence of the curl of any vector is necessarily zero, Eq. (13.13) requires +that V-7 =0. Thịs is true, by Eq. (13.8), only If Øø/Ø£ is zero. But that must +be so 1Ý ¡is not changing with time, so our assumptions are consistent. +'The requirement that V - 7 = 0 means that we may only have charges which +fow in paths that close back on themselves. They may, for instance, fow in wires +*We will see later, however, that such assumptions are øoø generally correct for electromag- +netic forcesl +--- Trang 160 --- +that form complete loops——called circuits. The circuits may, of course, contain +generators or batteries that keep the charges fowing. But they may not include +condensers which are charging or discharging. (WSe will, of course, extend the +theory later to include dynamic fñelds, but we want frst to take the simpler case +Of sbeady currents.) +Now let us look at Eqs. (13.12) and (13.13) to see what they mean. The +frst one says that the divergence of Ö is zero. Comparing i% to the analogous +cquation in electrostatics, which says that V - E = —ø/co, we can conclude that +there is no magnetic analog oŸ an electric charge. There are no mmagnetic charges +from which lines of Ö can emerge. lf we think in terms of “lines” of the vector +field #, they can never start and they never stop. Then where do they come +trom? Magnetic fields “appear” zn the presence oƒ currents; they have a curÏ +proportional to the current density. Wherever there are currents, there are lines +of magnetic field making loops around the currents. 5ince lines oŸ Ö do not begin +or end, they will often close back on themselves, making closed loops. But there +can also be complicated situations in which the lines are not simple closed loops. +But whatever they do, they never diverge from points. No magnetic charges have +ever been discovered, so V- =0. 'This much is true not only for magnetostatics, +1b 1s aluaws true—even for dynamic fñelds. +The connection between the Ö field and currents is contained in Eq. (13.13). B +Here we have a new kind of situation which is quite diferent from electrostatics, LOOPT +where we had V x E =0. That equation meant that the line integral of E ⁄ +around any closed path 1s zero: +‡ +E;- ds =0. ì J) +loop . / +W© got that result from Stokesˆ theorem, which says that the integral around n +any closed path of an vector field is equal to the surface integral of the normal ÿxB +component of the curl of the vector (taken over any surface which has the closed : ¬ +loop as its periphery). Applying the same theorem to the magnetic field vector _L1g. 13-6. The line integral of the tangen- +and using the symbols shown in Eig. 13-6, we get tai component of is equal to the surface +; Integral of the normal component of V x B. +{B-ds— [(Vx:B)cnds (13.14) +Taking the curl of from Edq. (13.13), we have +{B-ds= | jingS, (13.15) +P €CoŒ“ J8 +The integral over 9, according to (13.5), is the total current 7 through the +surface Š. 5ince for steady currents the current through Š is independent of +the shape of Š, so long as it is bounded by the curve L`, one usually speaks of +“the current through the loop I}7 We have, then, a general law: the circulation +of around any closed curve is equal to the current 7ƒ through the loop, divided +by cọc2: +{n - d8 = ˆthrongh T, (13.16) +Thịs law—called Ampère”s Ìaœ——plays the same role in magnetostatics that Gauss” +law played in electrostatics. Ampère's law alone does not determine Ö from +currents; we must, in general, also use V - =0. But, as we will see in the +next section, it can be used to find the field in special cireumstances which have +certain simple symmetries. +13-5 The magnetic ñeld of a straight wire and of a solenoid; atomic currents +We can illustrate the use of Ampère”s law by finding the magnetic fñeld near +a wire. We ask: What ¡is the feld outside a long straight wire with a cylindrical +cross section? We will assume something which may not be at all evident, but +--- Trang 161 --- +which is nevertheless true: that the field lines of Ö go around the wire in closed +circles. TÝ we make this assumption, then Ampère's law, Eq. (13.16), tells us +how strong the field is. From the symmetry of the problem, #Ö has the same +magnitude at all points on a circle concentric with the wire (see Eig. 13-7). We +can then do the line integral of Ö - ds quite easily; it is Just the magnitude of +tỉimes the circumferenee. IÝ r is the radius of the circle, then +{B-4s=B‹ềnr XS +The total current through the loop is merely the current ƒ in the wire, sO +B:-2nr = - +B= 47egc2 TÔ 31) ì +The strength of the magnetic fñeld drops of inversely as r, the distance from ¬ : +the axis of the wire. We can, if we wish, write Eq. (13.17) in vector form. F1g. 18-7. The magnetic field outside of +Remembering that #Ö is at right angles both to T and to r, we have a long wire carrying the current ƒ. +gp__Ì. 21X6. (13.18) +47coc2 r +We have separated out the factor 1/4zcoc2, because it appears often. It is worth +remembering that it is exactly 10— (in the mks system), since an equation +like (13.17) is used to đdefine the unit of current, the armpere. Ất one meter from +a current of one ampere the magnetie ñeld is 2 x 10~” webers per square meter. +Since a current produces a magnetic fñeld, it will exert a force on a nearby +wire which is also carrying a current. In Chapter Í we described a simple +demonstration of the forces between two current-carrying wires. lf the wires are +parallel, each is at ripght angles to the #Ö field of the other; the wires should then +be pushed either toward or away from each other. When currents are in the same +direction, the wires attract; when the currents are moving in opposite directions, +the wires repel. +m...... +s : +lai Z2: +HHNWWWESIAaininii II út Ộ +2H11 HH... 6 J +111111111111152110000)0)): S ⁄Z +\wslslsslslsslslslslstsslstslSlstslSlstslSlsG)sis S_⁄Z⁄ Fig. 13-8. The magnetic field of a long +LINES solenoid. +Let°s take another example that can be analyzed by Ampère's law if we add +some knowledge about the fñeld. Suppose we have a long coil of wire wound in a +tipght spiral, as shown by the cross sections In Fig. 13-S. Such a coil is called a +solenoid. We observe experimentally that when a solenoid is very long compared +with its diameter, the field outside is very small compared with the feld inside. +Ủsing just that fact, together with Ampère's law, we can find the size of the field +Inside. +Since the field sføws inside (and has zero divergence), its lines must go along +parallel to the axis, as shown In Fig. 13-8. 'That being the case, we can use +Ampère's law with the rectangular “curve” I' shown in the figure. 'This loop øgoes +the distance Ù, inside the solenoid, where the field is, say, ọ, then goes at right +angles to the field, and returns along the outside, where the field is negligible. +--- Trang 162 --- +The line integral of Ö for this curve is just ØọL, and it must be 1/coc2 times the +total current through L, which is NT ïf there are Ñ turns of the solenoid in the +length Ù. We have +bBobÙ= xẻ, +Ór, letting øœ be the number of turns per wøw#‡ length of the solenoid (that is, +n= N/L), we get +Po= HỆ, (13.19) +What happens to the lines of when they get to the end of the solenoid? +Presumably, they spread out in some way and return to enter the solenoid at the +other end, as sketched in Eig. 13-9. Such a field is just what is observed outside +of a bar magnet. But what is a magnet anyway? Our equations say that B +comes from the presence of currents. Yet we know that ordinary bars of iron ——>—— +(no batteries or generators) also produce magnetic fields. You might expect that +there should be some other terms on the right-hand side of (18.12) or (13.13) to +represent “the density of magnetie iron” or some such quantity. But there is no +such term. Our theory says that the magnetic efects of iron come Írom some +internal currents which are already taken care of by the 7 term. +Matter is very complex when looked at from a fundamental poïnt oÝ view——as +we saw when we tried to understand dielectrics. In order not to interrupt our +present discussion, we will wait until later to deal in detail with the interior Fig. 13-9. The magnetic field outside of +mmechanisms of magnetic materials like iron: You will have to accept, for the a solenoid. +mmoment, that all magnetism is produced from currents, and that in a permanent +magnet there are permanent internal currents. In the case of iron, these currents +come from electrons spinning around their own axes. Every electron has such +a spin, which corresponds to a tiny circulating current. Of course, one electron +doesn”t produce mụch magnetic field, but in an ordinary plece of matter there are +billions and billions of electrons. Normally these spin and point every which way, +so that there is no net efect. The miracle is that in a very few substances, like +Iron, a large fraction of the electrons spin with their axes in the same direction—— +for iron, two electrons of each atom take part in this cooperative motion. In a bar +magnet there are large numbers of electrons all spinning in the same direction +and, as we will see, their total efect is equivalent to a current circulating on the +surface of the bar. (This ¡is quite analogous to what we found for dielectrics—that +a uniformly polarized dielectric is equivalent to a distribution of charges on its +surface.) It is, therefore, no accident that a bar magnet is equivalent to a solenoid. +13-6 The relativity of magnetic and electric ñelds +'When we said that the magnetic force on a charge was proportional to its +velocity, you may have wondered: “What velocity? With respect to which +reference frame?” It is, in fact, clear from the defnition of Ö given at the +beginning of this chapter that what this vector is will depend on what we choose +as a reference frame for our specification of the velocity of charges. But we have +said nothing about which is the proper frame for specifying the magnetic field. +Tlt turns out that amw inertial frame will do. We will also see that magnetism +and electricity are not independent things—that they should always be taken +together as øne complete electromagnetic feld. Although in the static case +Maxwell's equations separate into two distinct pairs, one pair for electricity and +one pair for magnetism, with no apparent connection between the two fields, +nevertheless, in nature itself there is a very intimate relationship between them +that arises rom the prineciple of relativity. Historically, the principle of relativity +was discovered after Maxwell's equations. It was, in fact, the study of electricity +and magnetism which led ultimately to Einstein's discovery of his principle of +relativity. But let's see what our knowledge of relativity would tell us about +magnetic forces if we assume that the relativity principle is applicable—as 1E +is—to electromagnetism. +--- Trang 163 --- +4(=)—= q +: S : ° +; vị =0 V_=v ⁄? ; vị =—V v.=0 2 +(a) “ Z (b) “ ớ +Fig. 13-10. The Interaction of a current-carrying wire and a particle with the +charge q as seen in two frames. In frame S (part a), the wire is at rest; in frame Sĩ +(part b), the charge is at rest. +Suppose we think about what happens when a negative charge moves with +velocity øo parallel to a current-carrying wire, as in Fig. 13-10. We will try to +understand what goes on in two reference Írames: one fñxed with respect to the +wire, as in part (a) of the ñgure, and one fixed with respect to the particle, as in +part (b). We will call the first frame Š and the second ,S”. +In the S-frame, there is clearly a magnetic force on the particle. 'Phe force +is directed toward the wire, so if the charge were moving freely we would see it +curve in toward the wire. But in the S”-frame there can be no magnetic force +on the particle, because its velocity is zero. Does it, therefore, stay where it is? +'Would we see diferent things happening in the two systems? 'The principle of +relativity would say that in 5“ we should also see the particle move closer to the +wire. We must try to understand why that would happen. +W© return to our atomic description of a wire carrying a current. In a normal +conductor, like copper, the electric currents come from the motion of some of the +negative electrons——called the conduction electrons—while the positive nuclear +charges and the remainder of the electrons stay fñxed in the body of the material. +We let the charge density of the conduction electrons be ø_ and their velocity +in 5 be ø. The density of the charges at rest in Š is ø+, which must be equal to +the negative oŸ ø_, since we are considering an uncharged wire. There is thus no +electric fñeld outside the wire, and the force on the moving particle 1s Just +tF= qUo %X B. +Using the result we found in Eq. (13.18) for the magnetic feld at the distance z +from the axis of a wire, we conclude that the force on the particle is directed +toward the wire and has the magnitude +47coc2 r +Using Eqs. (13.3) and (13.5), the current 7 can be written as ø_øA, where A is +the area of a cross section of the wire. Then +p——L_.20-Ât, (13.20) +4meoc2 r +W© could continue to treat the general case of arbitrary velocities for ø and 0o, +but it will be Just as good to look at the special case in which the velocity 0g +of the particle is the same as the velocity 0 of the conduction electrons. 5o we +write 0o = 0, and Eq. (13.20) becomes +q_p-A7 +EP= 3o (13.21) +NÑow we turn our attention to what happens in S”, in which the particle is at +rest and the wire is running past (toward the left in the figure) with the speed ø. +The positive charges moving with the wire will make some magnetic fñeld Bí at +the particle. But the particle is now at resf, so there is no rmaønetic force on ïtÏ +Tí there is any force on the particle, it must come from an electric ñeld. It must +--- Trang 164 --- +be that the moving wire has produced an electric field. But it can do that only if +1t appears charged——it must be that a neutral wire with a current appears to be +charged when set in motion. +We must look into this. We must try to compute the charge density in the +wire in 5%“ from what we know about it in Š. One might, at first, think they +are the same; but we know that lengths are changed between 9 and ®S” (see +Chapter lỗ, Vol. I), so volumes will change also. Since the charge đensiiies +depend on the volume occupied by charges, the densities wïll change, too. +Before we can decide about the charge đensifies in S”, we must know what +happens to the electric chørgøe of a bunch of electrons when the charges are +moving. We know that the apparent mass of a particle changes by 1/4/⁄1 — 02/2. +Does its charge do something similar? Nol Charges are always the same, moving +or not. Otherwise we would not always observe that the total charge is conserved. +Suppose that we take a block of material, say a conductor, which is initially +uncharged. Now we heat it up. Because the electrons have a different mass +than the protons, the velocities of the electrons and of the protons will change +by diÑerent amounts. If the charge of a particle depended on the speed of the +particle carrying it, in the heated block the charge of the electrons and protons +would no longer balance. A block would become charged when heated. Às we +have seen earlier, a very small fractional change in the charge of all the electrons +in a block would give rise to enormous electric fñelds. No such efect has ever +been observed. +Also, we can point out that the mean speed of the electrons in matter depends +on its chemical composition. If the charge on an electron changed with speed, +the net charge in a piece of material would be changed in a chemical reaction. +Again, a straightforward calculation shows that even a very small dependence of +charge on speed would give enormous fields from the simplest chemical reactions. +No such efect is observed, and we conclude that the electric charge of a single +particle is Independent of its state of motion. +So the charge g on a particle is an invariant scalar quantity, independent of +the frame of reference. 'Phat means that in any frame the charge density of a +distribution of electrons 1s Just proportional to the number of electrons per unit +volume. We need only worry about the fact that the volume cøn change because +of the relativistic contraction of distances. +W©e now apply these ideas to our moving wire. lf we take a length họ of the +wire, in which there is a charge density øo OŸ sứationar charges, it will contain the +total charge @Q = øogo Áo. If the same charges are observed in a diferent frame +to be moving with velocity ø, they will all be found in a piece of the material +with the shorter length +L= LowWl-— 02/c2, (13.22) +but with the same area Áo (since dimensions transverse to the motion are +unchanged). See Eig. 13-11. +Tí we call ø the density of charges in the tame in which they are moving, the +total charge Q will be ob4o. Thịs must also be equal to øgoÁo, because charge +is the same in any system, so that øÙ = øg họ or, from (13.22), +p Ưm.r (13.23) +@ [TỰ _" Đ TY Y7 HT +Q v=0 Area Ao ° — Area Ao +Fig. 13-11. lf a distribution of charged particles at rest has the charge density øo, +the same charges will have the density ø = Øo/+⁄1 — v2/c? when seen from a frame +with the relative velocity v. +--- Trang 165 --- +The charge densit of a moving đistr(bution of charges varies in the same way as +the relativistic mass of a particle. +W© now use this general result for the positive charge density øØ+ of our wire. +These charges are at rest in frame Š. In 5Š”, however, where the wire moves with +the speed 0, the positive charge density becomes +=————. 18.24 ++ /1— ø2/c2 ( ) +The negafioe charges are at rest in S7. So they have their “rest density” øo in +this rame. In Eq. (13.23) øo = ø—, because they have the density ø_ when the +tu#re 1s at rest, i.e., in frame Š, where the speed of the negative charges is 0. For +the conduction electrons, we then have that +0-= —=_——, (13.25) +v1— 12/c2 +ø_—=p V1- 02/e2. (13.26) +Now we can see why there are electric fñelds in S—because in this rame the +wire has the net charge density øˆ given by +p=Ø,+p. +Using (18.24) and (13.26), we have +0+ 1 n2D/a2 +v1—12ƒ/c +Since the stationary wire is neutral, ø_ = —ø+, and we have +“=p_.—————. 13.27 +mm... (13.27) +Our moving wire is positively charged and will produce an electric field E7 at the +external stationary particle. We have already solved the electrostatie problem of +a uniformly charged cylinder. The electric ñeld at the distance r from the axis of +the cylinder 1s +'A A 2/2 +g_ 0A __ prAdjc - (13.28) +27cor 2mcogrv/1— 02/c2 +The force on the negatively charged particle is toward the wire. We have, at +least, a force in the same direction from the two points of view; the electric force +in 5” has the same direction as the magnetic force in đ. +The magnitude of the force in 5” is +A 2/2 +E.— 1 PL^ _ tjc (13.29) +27T v1— 02/c2 +Comparing this result for F” with our result for in Eq. (13.21), we see that +the magnitudes of the forces are almost identical from the two points of view. In +†'=———, (13.30) +v1— 02/2 +so for the small velocities we have been considering, the two forces are equal. +W© can say that for low velocities, at least, we understand that magnetism and +electricity are just “two ways of looking at the same thing.” +But things are even better than that. If we take into account the fact that +ƒorces aÌlso transform when we go om one system to the other, we fnd that the +two ways of looking at what happens do indeed give the same øñ/s¿cal result for +any velocity. +--- Trang 166 --- +One way of seeing this is to ask a question like: What transverse momentum +will the particle have after the force has acted for a little while? We know from +Chapter 16 of Vol. I that the transverse momentum of a particle should be the +same in both the Š- and S”-rames. Calling the transverse coordinate , we want +to compare Aø„ and AĐ. Using the relativistically correct equation of motion, +F' = dp/dt, we expect that after the tìme Af our particle will have a transverse +momentum Aø, in the S-system given by +Ap = P.At. (13.31) +In the Š”-system, the transverse momentum will be +AD — Ƒ AU. (13.32) é +We must, of course, compare Â?ø„ and ADy for corresponding time intervals Af +and A7. We have seen in Chapter 15 of Vol. I that the time intervals referred ⁄ +to a noư��ng particle appear to be longer than those in the rest system of the B Z⁄ +particle. Since our particle is initially at rest in S”, we expect, for small A, that 2 +Ar=-_-E—. (13.33) +V1= 0/2 +and everything comes out O.K. Erom (13.31) and (13.32), +Am F“AU € +ADpw — FAt' +which is Just = I if we combine (13.30) and (13.33). +We have found that we get the same physical result whether we analyze the ⁄ +motion of a particle moving along a wire in a coordinate system at rest with g Z +respect to the wire, or in a system at rest with respect to the particle. In the fñrst Z +instance, the force was purely “magnetiec,” in the second, it was purely “electric.” : : +The two points of view are illustrated in Eig. 13-12 (although there is still a __ Flg. 13-12. In frame 5 the charge density +magnetic field in the second frame, it produces no forces on the stationary IS Zero and the current density Sử: There Is +. only a magnetic field. In S”, there is a charge +particle). - - density øˆ and a different current density ƒ,. +T we had chosen still another coordinate system, we would have found a The magnetic field B” is different and there +diferent mixture of E and Ö felds. Electric and magnetic Íorces are part of is an electric field E”. +one physical phenomenon—the electromagnetic interactions of particles. 'Phe +separation of this interaction into electric and magnetic parts depends very much +on the reference frame chosen for the description. But a complete electromagnetic +description is invariant; electricity and magnetism taken together are consistent +with Einsteinˆs relativity. +Since electric and magnetic fields appear in diferent mixtures if we change +our frame of reference, we must be careful about how we look at the fields E +and #Ö. Eor instance, If we think of “lines” of # or Ö, we must not attach too +much reality to them. The lines may disappear if we try to observe them Írom a +diferent coordinate system. For example, in system S” there are electric field +lines, which we do nø ñnd “moving past us with velocity in system Š” In +system ®$ there are no electric field lines at alll Therefore it makes no sense to +say something like: When I move a magnet, it takes its ñeld with it, so the lines +of are also moved. 'Phere is no way to make sense, in general, out of the idea +of “the speed of a moving feld line.” The fñields are our way of describing what +goes on at a point in space. In particular, # and # tell us about the forces that +will act on a moving particle. "he question “What is the force on a charge from +a mnou#ng magnetic fñeld?” doesn't mean anything precise. 'Phe Íorce is given by +the values of # and Ö at the charge, and the formula (13.1) is not to be altered +1f the sơurce of E or is moving (it is the values of # and that will be altered +by the motion). Qur mathematical description deals only with the fñelds as a +function of z, , z, and £ uith respect to some ?nertial [rame. +We© will later be speaking oŸ “a øøe of electric and magnetic ñelds travelling +through space,” as, for instance, a light wave. But that is like speaking oŸ a œ0e +--- Trang 167 --- +travelling on a string. We don'ˆt then mean that some part of the sfring is moving +in the direction of the wave, we mean that the đisplacemen# of the string appears +first at one place and later at another. Similarly, in an electromagnetic wave, +the 0øue travels; but the magnitude of the fñelds chønøe. So in the future when +we——or someone else—speaks of a “moving” field, you should think of it as just a +handy, short way of describing a changing field in some circumstances. +13-7 The transformation of currents and charges +You may have worried about the simpliication we made above when we +took the same velocity 0 for the particle and for the conduction electrons in the +wire. We could go back and carry through the analysis again for two diferent +velocities, but it is easier to simply notice that charge and current density are +the components of a four-vector (see Chapter 17, Vol. ]). +W© have seen that iŸ øo is the density of the charges in their rest frame, then +in a frame in which they have the velocity œ, the density is +;= —P—— +V1-— 02/2 +In that frame their current density is +j3 =ø0=————. 13.34 +—Ầ... (3.39) +Now we know that the energy and momentum ø of a particle moving with +velocity are given by +mọc? Tnọ“® +U=——-.. P= ——:: +v1_— 032/c2 v1— 032/c2 +where ?mọ 1s 10s rest mass. We also know that and ø form a relativistic four- +vector. Since ø and 7 depend on the velocity ø exactly as do Ữ and ø, we can +conclude that ø and 7 are aÍso the components of a relativistic four-vector. 'This +property is the key to a general analysis of the fñeld of a wire moving with any +velocity, which we would need If we want to do the problem again with the +velocity ọ of the particle diferent from the velocity of the conduction electrons. +lÍ we wish to transform ø and 7 to a coordinate system moving with a +velocity œ in the z-direction, we know that they transform just like £ and (#, , z), +so that we have (see Chapter 15, Vol. l) +„Ằ— +— ut j2 = J„ — tp +1— u2/c2' 7 1— u2/c2 +Ụ =U, ñụ =u: +z=z ) Ÿ: =%z:› +t— uz/c — Uj„/c? +#— cử tu. ø= З Aổy/CỔ (13.35) +v1— u2/c2 v1_— u2/c2 +With these equations we can relate charges and currents in one frame to those +in another. Taking the charges and currents in either Írame, we can solve the +electromagnetic problem in that frame by using our Maxwell equations. 'Phe +result we obtain ƒor the motlions oƒ particles will be the same no matter which +frame we choose. We will return at a later time to the relativistic transformations +of the electromagnetic fields. +13-8 Superposition; the right-hand rule +We will conclude this chapter by making bwo further points regarding the +subject of magnetostatics. Pirst, our basic equations for the magnetic fñeld, +V{V.Bb-=‹(0, VxB=J/c«, +--- Trang 168 --- +are linear in #Ö and 7. That means that the principle of superposition also applies +to magnetic fields. The fñeld produced by two diferent steady currents is the +sum of the individual fields from each current acting alone. Qur second remark +concerns the righ(-hand rules which we have encountered (such as the right-hand +rule for the magnetic field produced by a current). We have also observed that +the magnetization of an iron magnet is to be understood from the spin of the +electrons in the material. The direction of the magnetic ñeld of a spinning electron +1s related to its spin axis by the same right-hand rule. Because #Ö is determined +by a “handed” rule—involving either a cross product or a curl—ït is called an +azial vector. (Vectors whose direction in space does not depend on a reference to +a ripht or left hand are called polar vectors. Displacement, velocity, force, and #, +for example, are polar vectors.) +Phụsicallụ obseruable quantities in electromagnetism are no, however, right- +(or left-) handed. Electromagnetic interactions are syrnmetrical under reflection +(see Chapter 52, Vol. I). Whenever magnetic forces between ÿwo sets oÝ currenbs are +computed, the result is invariant with respect to a change in the hand convention. +Our equations lead, independently of the right-hand convention, to the end result +that parallel currents attract, or that currents in opposite directions repel. (ltry +working out the force using “left-hand rules.”) An attraction or repulsion is a +polar vector. 'Phis happens because in describing any complete interaction, we +use the right-hand rule twice—once to fnd #Ö from currents, again to fnd the +force this Ö produces on a second current. sing the right-hand rule twice is +the same as using the left-hand rule twice. lIf we were to change our conventions +to a left-hand system all our #Ö fields would be reversed, but all forees—or, +what is perhaps more relevant, the observed accelerations of objects—would be +unchanged. +Although physicists have recently found to their surprise that øÏl the laws of +nature are not always invariant for mirror refections, the laws of electromagnetism +do have such a basic symmetry. +--- Trang 169 --- +Tĩìo W(gyrnofic Fioldl ra Verforrs SfferffO@rts +14-1 The vector potential +In this chapter we continue our discussion of magnetic fields associated with 14-1 The vector potential +steady currents—the subject of magnetostatics. The magnetic field is related to 14-2 The vector potential of known +electric currents by our basic equations currents +ÿ:B=0, (14.1) 14-3 A straight wire +- 14-4 A long solenoid +cWxB-— 7, (14.2) 14-5 The ñeld of a small loop; the +€0 magnetic dipole +We want now to solve these equations mathematically in a general way, that is, 14-6 The vector potential oŸ a circuit +without requiring any special symmetry or intuitive guessing. In electrostatics, 14-7 The law of Biot and Savart +we found that there was a straightforward procedure for ñnding the ñeld when +the positions of all electric charges are known: One simply works out the scalar +potential ở by taking an integral over the charges—as in Eq. (4.25). Then if one +wants the electric field, it is obtained from the derivatives of ó. We will now show +that there is a corresponding procedure for fnding the magnetic field # If we +know the current density 7 of all moving charges. +In electrostatics we saw that (because the curÌ of E was always zero) it was +possible to represent # as the gradient of a scalar field ó. Now the curl of +1s no£ always zero, so it is not possible, in general, to represent it as a gradient. +However, the diuergence of is always zero, and this means that we can aÌways +represent Ö as the curi of another vector field. Eor, as we saw In Section 2-8, +the divergence of a curl is always zero. Thus we can always relate Ö to a field +we will call A by +B=VxA. (14.3) +Ór, by writing out the components, +3A 3A +B„,=(VxA)„= —ˆ-ˆ” +“ ( }z ðy Ôz , +ØA, ÔA, +B„=(Vx A),=——_——— 14.4 +y���(VxA)y= TP - Tết, (1449) +3A 3A +B,=(VxA),,=--“—--.~. +z=Ị ): 3z Øụ +Writing = VW x A guarantees that Eq. (14.1) is satisfied, since, necessarily, +W:B=VY:(VxA4)=(0. +The fñeld A is called the uecfor potential. +You will remember that the scalar potential was not completely specified by +1ts defnition. If we have found ø for some problem, we can always find another +potential j“ that is equally good by adding a constant: +ớ =ó+C. +The new potential @ˆ gives the same electric fields, since the gradient W is zero; +ở and ¿ represent the same physics. +Similarly, we can have diferent vector potentials A which give the same +magnetic fñelds. Again, because #Ö is obtained from .A by diferentiation, adding a +constant to Á doesn't change anything physical. But there is even more latitude +--- Trang 170 --- +for A. W©e can add to 4Á any field which is the gradient of some scalar field, +without changing the physics. We can show this as follows. Suppose we have +an A that gives correctly the magnetic ñeld # for some real situation, and ask +in what cireumstanees some other new vector potential A“ will give the sœme +ñeld ïf substituted into (14.3). Then 4 and A/ must have the same curl: +B=YVxA=VxÁA. +'Therefore +VxA-VxA=Vx(A-4A)=0. +But ïf the curl of a vector is zero it must be the gradient of some scalar field, +say Ú, so A“°T— A = Vụ. That means that if A is a satisfactory vector potential +for a problem then, for any , at all, +A=A+Vụ (14.5) +will be an equally satisfactory vector potential, leading to the same field Ö. +]t is usually convenient to take some of the “latitude” out of A by arbitrarily +placing some other condition on it (in much the same way that we found it +convenient—often—to choose to make the potential ở zero at large distances). +We can, for instance, restrict 4 by choosing arbitrarily what the divergence of Á +must be. We can always do that without afecting #Ö. 'This is because although +A' and A have the same curl, and give the same Ö, they do not need to have +the same divergenee. In fact, Wf: A'—=W: A+ V2, and by a suitable choice +of we can make - A“ anything we wish. +What should we choose for V- 4? "The choice should be made to get the +greatest mathematical convenience and will depend on the problem we are doing. +FOr magnetostatics, we will make the simple cholce +V.A=0. (14.6) +(Later, when we take up electrodynamics, we will change our choice.) Qur complete +definition# of A is then, for the moment, W x AÁ = B and V: A=0. +To get some experience with the vector potential, let's look first at what it is +for a uniform magnetic ñeld ọ. Taking our z-axis in the direction of Bọ, we +must have ĐA ĐA +D„ =ễ= -==ẽ= 0, +ØA„ ÔA, +B„= ——~_—-—~=(0, 14.7 +ụ Õz Øz 47) +3A 3A +B;¿ = —>”— —— = hạ. +š Øz Øy ụ +By inspection, we see that one øoss¿ble solution of these equations is +Ay =zbh, Ay=0, A; =0. +Or we could equally well take +Ax = _—bq, Ay=0, A; =0. +Still another solution is a linear combination of the bwo: +Az= —šÐ\, Ay= 3#Ðụ, A; =0. (14.8) +lt is clear that for any particular field #Ö, the vector potential A is not unique; +there are many possibilities. +* Our definition still does not uniquely determine A. For a n2gue specification we would +also have to say something about how the fñeld A behaves on some boundary, or at large +distances. Ït is sometimes convenient, for example, to choose a field which goes to zero at large +đistances. +--- Trang 171 --- +The third solution, Eq. (14.8), has some interesting properties. Since the +#-component is proportional to —z and the ¿-component is proportional to +z, +A must be at right angles to the vector from the z-axis, which we will call z“ (the 3 +“prime” is to remind us that it is mo‡ the vector displacement from the origin). +Also, the magnitude of A is proportional to 4⁄#2 + 2 and, hence, to ??. So Ñ TT" +can be simply written (for our uniform feld) as +A= 3Bo xr, (14.9) +The vector potential A has the magnitude Øọr7/2 and rotates about the z-axis as +shown in Eig. 14-1. If, for example, the #Ö feld is the axial fñeld inside a solenoid, Z1» +then the vector potential circulates in the same sense as do the currents of the — ` x +solenoid. „v +The vector potential for a uniform feld can be obtained in another way. The +cireulation of 4 on any closed loop I` can be related to the surface integral +of WV x A by Stokes' theorem, Eq. (3.38): ) +đA-ds= J (V x A) - n da. (14.10) +r inside +But the integral on the right is equal to the fux of through the loop, so in n, 2 hoc hon co ngDondh to 3 vector +potential A that rotates about the z-axIs, +1a - d8 = J B-n da. (14.11) with the magnitude A = Br//2 (r' ¡s the +L inside T displacement from the z-axis). +So the circulation of Á around øø% loop is equal to the fux of through the +loop. lÝ we take a circular loop, of radius 7” in a plane perpendicular 0o a uniform +fñeld #Ö, the fux is Just +lf we choose our origin on an axis of symmetry, so that we can take Á as +circumferential and a function only oŸ z”, the circulation will be +1A - ds = 2mr'A = mr2B. +W© get, as before, +In the example we have just given, we have calculated the vector potential from +the magnetic field, which is opposite to what one normally does. In complicated +problems it is usually easier to solve for the vector potential, and then determine +the magnetic ñeld from it. We will now show how this can be done. +14-2 The vector potential of known currents +Since is determined by currents, so also is A. We want now to fnd Á in +terms of the currents. We start with our basic equation (14.2): +cẦVxB= 5 +which means, of course, that +cVx(VxA)= 7. (14.12) +This equation is for magnetostatics what the equation +V.-V¿=_—— (14.13) +was for electrostatics. +--- Trang 172 --- +Our equation (14.12) for the vector potential looks even more like that for ở +1ƒ we rewrite V x (W x 4) using the vector identity Eq. (2.58): +Vx(VxA)=V(V:A)- VỶA. (14.14) +Since we have chosen to make V - A = 0 (and now you see why), q. (14.12) +becomes - +V?A=--—. 14.15 +cọạc2 ( ) +'This vector equation means, of course, three equations: +VˆA,=_—-“, VA,=-—-, V2A,=-—-S, (14.16) +coc2 cọc2 cọc? : ï +And each of these equations is rmafhematicaliu tdentical to +W?¿=_—=. (14.17) 2 +All we have learned about solving for potentials when ø is known can be used for +solving for each component of Á when 7 is knownl +We have seen in Chapter 4 that a general solution for the electrostatic +cquation (14.17) is +Fig. 14-2. The vector potential A at +1 0(2) dV2 - l - - +ø(1) = mm..." point 1 ¡is given by an integral over the cur- +€0 12 rent elements / dV at all points 2. +So we know immediately that a general solution for Áz is +| J„(2) dV› +Az(1)=—p | ——— 14.18 +zä) 4mcgc2 J GEN ( ) +and similarly for 4„ and 4;. (Figure 14-2 will remind you oŸ our conventions for +r1a and đW¿.) We can combine the three solutions in the vector form +1 (2) dV: +A()=——D J 720015. (14.19) +47coc2 T12 +(You can verify if you wish, by direct diferentiation of components, that this +integral for A satisies V - A = 0 so long as V - j =0, which, as we saw, mmust +happen for steady currents.) +W©e have, then, a general method for ñnding the magnetic field of steady +currents. 'Phe principle is: the ø-component of vector potential arising from a +current density 7 is the same as the electric potential ó that would be produced +by a charge density ø equal to 7„/c?—=and similarly for the g- and z-components. +(This principle works only with components in fxed directions. "The “radial” +component of Á does not come in the same way from the “radial” component +of 7, for example.) So from the vector current density 7, we can find A using +Ea. (14.19)—that is, we find each component of A by solving three imaginary +electrostatic problems for the charge distributions ø¡ = 7z/c?, øa = jv/€Ÿ, +and øs = 7;z/c2. Then we get Ö by taking various derivatives of A to obtain Wx A. +lt?s a little more complicated than electrostatics, but the same idea. We will now +iHustrate the theory by solving for the vector potential in a few special cases. +14-3 A straight wire +For our frst example, we will again ñnd the ñeld of a straight wire—which we +solved in the last chapter by using Eq. (14.2) and some arguments of symmmetry. +We take a long straight wire of radius ø, carrying the steady current ƒ. Unlike +the charge on a conductor in the electrostatic case, a steady current in a wire +is uniformly distributed throughout the cross section of the wire. If we choose +our coordinates as shown in Fig. 14-3, the current density vector 7 has only a +z-component. lts magnitude 1s +Jz—=—s (14.20) +inside the wire, and zero outside. +--- Trang 173 --- +Đince 7„ and 7„ are both zero, we have immediately +A;y=0, uy =0. z +To get Á; we can use our solution for the electrostatic potential @ of a wire with a Z ⁄2 +uniform charge density ø = j;z/c2. For points outside an infinite charged cylinder, | 77 +the electrostatic potential is 7 J +ớ =_— 2~ca À Ìn TỶ 27 v +7€0 77m = +where ? = w⁄z2 + 2 and À is the charge per unit length, xa2ø. So 4; must be X " 7/ a +A,= _ghj» Inz 27 +: 2zcoc2 ⁄) +for points outside a long wire carrying a uniform current. Since 27; = Ï, we . ¬ . +. Fig. 14-3. A long cylindrical wire along +can also write ¬. ¬" +T the Z-axis with a uniform current density ƒ. +4z=—-———lnr (14.21) +2zcoc2 +NÑow we can fnd from (14.4). There are only two of the six derivatives +that are not zero. We get +] Ø8 T +:R -ˆ'...... ` (14.22) +2zcoc2 Øụ 2zcoc2 r2 +3" T bu +B,= „ —s--Ìnr= „sa, 14.23 +k 27coc2 9z 2zcoc2 r2 ( ) +B,= 0. +W© get the same result as before: #Ö circles around the wire, and has the magnitude Ự +Ị Ị Ị +B=——“. (14.24) " +4mcogc2 r/ ` y +⁄ <=Ƒ ` 3 +14-4 A long solenoid S ⁄ ` +Next, we consider again the infnitely long solenoid with a circumferential Ị ẤN 2À ` +current on the surface oŸ ø„Ï per unit length. (W© imagine there are turns of \ | x +wire per unit length, carrying the current ï, and we neglect the slight pitch of ` h +the winding.) ¬ kZ +Just as we have defñned a “surface charge density” ơ, we defñne here a “surface TƑF +current density” .JJ equal to the current per unit length on the surface of the +solenoid (which is, oŸ course, just the average 7 times the thickness of the thin +winding). The magnitude of .Ƒ is, here, øĩ. This surface current (see Fig. 14-4) +has the componenfs: Fig. 14-4. A long solenoid with a surface +current density J. +J„ = —Jsinó, Jụ = Jcosó, Jy =0. +Ñow we must fnd A for such a current distribution. +first, we wish to nd 4z for points outside the solenoid. "The result is the +same as the electrostatie potential outside a cylinder with a surface charge density +Ø =ơgsinó, +with øo = —J/c?. We have not solved such a charge distribution, but we have done +something similar. This charge distribution is equivalent to Ewo soljd cylinders of +charge, one positive and one negative, with a slight relative displacement of their +axes in the z-direction. 'Phe potential of such a pair of cylinders is proportional +to the derivative with respect to # of the potential of a single uniformly charged +--- Trang 174 --- +cylinder. We could work out the constant of proportionality, but let”s not worry +about it for the moment. +The potential of a cylinder of charge is proportional to lnz”; the potential of +the pair is then +Ølnzr/ ụỤ +lo 0u — r2Ì +So we know that +A„=-—KE „2 (14.25) +where #Ý is some constant. Following the same argument, we would ñnd +Au=E „- (14.26) +Although we said before that there was no rnøønefic field outside a solenoid, +we find now that there 2s an A-field which circulates around the z-axis, as in +Fig. 14-4. The question is: Is its curÌ 2ero? +Clearly, „ and „ are zero, and +lô) % lô) Ụ +5;=——|E— ]- - |_-K-¬ +°z ar X72) = 2y Rịm) +1 2z? 1 2y +So the magnetic field outside a very long solenoid is indeed zero, even though +the vector potential is not. +We can check our result against something else we know: 'The circulation of +the vector potential around the solenoid should be equal to the ñux of Ö inside +the coil (Bq. 14.11). The circulation is A-2Zr7 or, since A = #/r', the circulation +is 21. Notice that it is ndependent of z”. That is just as it should beifthere : „„ +1s no outside, because the Hux is just the magnitude oŸ J ?nside the solenoid ! +tỉmes a2. It is the same for all circles of radius r7“ > a. W©e have found in the ¬ +last chapter that the feld inside is øÏ/coc”, so we can determine the constant ý: |,““ TNG +2rK = ra? ¬ NHÀ +cọc2 l +OF bạ ¬ +Kj=— nIaŠ hờ 3o W +2coc2 ' h +So the vector potential outs¿de has the magnitude +mĩa? 1 +A=.s— 14.27 +and is always perpendicular to the vector ?. +We have been thinking of a solenoidal coil oŸ wire, but we would produce „ +the same felds if we rotated a long cylinder with an electrostatic charge on the +surface. If we have a thin cylindrical shell of radius øa with a surface charge ơ, +rotating the cylinder makes a surface current .J —= ơu, where 0 = đư 1s the velocity +of the surface charge. There will then be a magnetic field = øaœ/coc? inside Fig. 14-5. A rotating charged cylinder +the cylinder. produces a magnetic field inside. A short +Now we can raise an interesting question. Suppose we put a short piece of 0n Síng /oUNg nh le cylinder has +wire W perpendicular to the axis of the cylinder, extending from the axis out to ChAr065 neucee on 1s 6ne5. +the surface, and fastened to the cylinder so that it rotates with it, asin Eig. 14-5. +This wire is moving in a magnetic field, so the ø x #Ö forces will cause the ends +of the wire to be charged (they will charge up until the E-feld from the charges +Just balances the x # force). IÝ the cylinder has a positive charge, the end of +the wire at the axis will have a negative charge. By measuring the charge on the +end of the wire, we could measure the speed of rotation of the system. We would +have an “angular-velocity meter”! +--- Trang 175 --- +But are you wondering: “What if Ï put myself in the frame oŸ reference of the +rotating cylinder? "Then there is just a charged cylinder at rest, and I know that +the electrostatic equations say there will be mœø electric fields inside, so there will +be no force pushing charges to the center. 5o something must be wrong.” But +there is nothing wrong. There is no “relativity of rotation.”” A rotating system is +no‡ an inertial frame, and the laws of physics are diferent. We must be sure to use +equations of electromagnetism only with respect to inertial coordinate systems. +lt would be nice iŸ we could measure the absolute rotation of the earth with +such a charged cylinder, but unfortunately the efect is much too small to observe +even with the most delicate instruments now available. +14-5 The field of a small loop; the magnetic dipole +Let”s use the vector-potential method to fñnd the magnetic field of a small +loop of current. Ás usual, by “small” we mean simply that we are interested in +the fñelds only at distances large compared with the size of the loop. It will turn +out that any smaill loop is a “magnetic dipole.” 'Phat is, it produces a ?magnefic +field like the electric field from an electric dipole. +T7 — /CTTTIELTTTTTTTD +b Øy! - b/ - +V_ Tà ⁄ , a ⁄ +rr++rrrrrrrrrw +++ư+ưtư+ưtưt ca +"1... j +Fig. 14-6. A rectangular loop of wire with the current ƒ. Fig. 14-7. The distribution of /„ in the +What is the magnetic field at P? (3> a and R% b.) current loop of Fig. 14-6. +W© take first a rectangular loop, and choose our coordinates as shown in +Eig. 14-6. There are no currents in the z-direction, so 4; is zero. "There +are currents in the z-direction on the bwo sides of length ø. In each leg, the +current density (and current) is uniform. So the solution for 4z is just like the +electrostatic potential trom two charged rods (see Eig. 14-7). Since the rods have +opposite charges, their electric potential at large distances would be just the +dipole potential (Section 6-5). At the point P ¡in EFig. 14-6, the potential would +=———. 14.28 +—.¬ (14.28) +where ø is the dipole moment of the charge distribution. The dipole moment, in +this case, is the total charge on one rod times the separation between them: +Ð= Ànb. (14.29) +The dipole moment points in the negative ¿-direction, so the cosine of the angle +between ## and ø is —1/ (where is the coordinate of P). So we have +¿= 1 Àab +_— 4rmeo R2 R +We get A„ simply by replacing À by I/c: +Az=———. 14.30 +. 4mcoc2 l3 ( ) +By the same reasoning, +lab ø +Au=——sza: 14.31 +# Ameoc2 R3 ( ) +--- Trang 176 --- +Again, A„ is proportional to ø and 4; is proportional to —, so the vector +potential (at large distances) goes in circles around the z-axis, circulating in the +same sense as Ï in the loop, as shown in Eig. 14-8. +The strength of A is proportional to 7œb, which is the current times the area +of the loop. This product is called the rmagnetic đipole mmormnemt (or, often, just +“magnetic moment”) of the loop. We represent it by : z +tụ = TaÙ. (14.32) +The vector potential of a small plane loop of an shape (circle, triangle, etc.) is A +also given by Eqs. (14.30) and (14.31) provided we replace Ïab by +tứ = Ï- (area of the loop). (14.33) +We leave the proof of this to you. _ +We can put our equation in vector form If we defne the direction of the +vector /# to be the normail to the plane of the loop, with a positive sense given +by the right-hand rule (Eig. 14-8). Then we can write , +1 xi 1 x TỰ ' +A=— “5=. (14.34) +4mcoc^ T 40c T Fig. 14-8. The vector potential of a small +W© have still to ñnd . Using (14.33) and (14.34), together with (14.4), we current loop at the origin (In the xy-plane); +t a magnetic dipole field. +8 % 3z +P„=—=——-——asxx—''. 14.35 +Ôz 4mcoc2 R3 Tịnh ( ) +(where by --- we mean /u/4Zcoc?), +8 ụỤ 3z +B„,= —|—-...-_Ì=... +JØz ( mm) R°` +lôi % lô Ụ +Đ,=-——|--'—l| _- = | _—-''' 14.36 +0r[ PB)“ MỆT —: 8) 0429 +c— 1 3z? +=—''Ím—J: +The components of the -field behave exactly like those of the -feld for +a đipole oriented along the z-axis. (See Eqs. (6.14) and (6.15); also Eig. 6-4.) +That's why we call the loop a magnetic dipole. The word “dipole” is slightly +misleading when applied to a magnetic ñeld because there are øø magnetic “poles” +that correspond to electric charges. The magnetie “dipole fñeld” is not produced +by two “charges,” but by an elementary current loop. +]t is curious, though, that starting with completely diferent laws, W- E = ø/co +and W x Ö = 7/eạc?, we can end up with the same kind of a feld. Why should +that be? It is because the dipole fñelds appear only when we are far away Írom +all charges or currents. So through most of the relevant space the equations Íor +t2 and B are identical: both have zero divergence and zero curl. So they give +the same solutions. However, the sources whose configuration we summarize by +the dipole moments are physically quite diferent——in one case, it's a circulating +current; in the other, a pair of charges, one above and one below the plane of the +loop for the corresponding fñeld. +14-6 The vector potential of a circuit +W© are often interested in the magnetic ñelds produced by circuits of wire in +which the diameter of the wire is very small compared with the dimensions of +the whole system. In such cases, we can simplify the equations for the magnetic +ñeld. For a thin wire we can write our volume element as +dV = Sds +--- Trang 177 --- +where Š is the cross-sectional area of the wire and đs is the element of distance +along the wire. In fact, since the vector đs is in the same direction as 7, as +shown in Fig. 14-9 (and we can assume that 7 is constant across any given €ross ⁄ ¬ +section), we can write a vector equation: N +jdV =8 da. (14.37) ⁄2 +But 76 ïs Jjust what we call the current Ï in a wire, so our integral for the vector +potential (14.19) becomes +A( | Tảs; Fig. 14-9. For a fine wire j dV ¡is the +1)=——; |“ (14.38) 3- J +47coc2 T13 same as Í ds. +(see Fig. 14-10). (W© assume that Ï is the same throughout the circuit. Tf +there are several branches with diferent currents, we should, of course, use the +appropriate 7 for each branch.) +Again, we can fñnd the felds from (14.38) either by integrating directly or by +solving the corresponding electrostatie problems. +14-7 The law of Biot and Savart nạ 1 +In studying electrostatics we found that the electric feld of a known charge +distribution could be obtained directly with an integral, Eq. (4.16): O 2 +1 0(2)©1a dV› +7T€0 ría +As we have seen, i% is usually more work to evaluate this integral—there are ó x_ Hỗ magneic na of NHÀ +really three integrals, one for each component—than to do the integral for the crcuik 050aIne6 HO. 4D 1016914) afoUnG tne +potential and take its gradient. l +There is a similar integral which relates the magnetic ñeld to the currents. +We already have an integral for A, Eq. (14.19); we can get an integral for by +taking the curl of both sides: +I 72) du +bB(1)=VWxA(1I)=V —— | ————|. 14.39 +0=VxAd)=vx| CS [#5 (14.39) +Now we must be careful: The curl operator means taking the derivatives of A(1), +that is, it operates only on the coordinates (#1, 1,21). We can move the Ÿx op- +erator inside the integral sign if we remember that it operates only on variables +with the subscript 1, which of course, appear only in +mịa = [(Œ1 — #2)” + (Mì — 92) + (ì — z2)°|!2, (14.40) +'W©e have, for the z-component of Ö, +B,= 8A; — OðAy +Øụi — Øzi +1 lô 1 Ø 1 +=——n lz—=—| —]Ì—2%=—| — | |dV: 14.41 +4mcoc2 I1 Øy (-) “ Øzi (S)) Ễ ) +1 . UL— 2. ZL— Z2 +=——— ——— — —ÿju—x~-|đU. +4mcoc? J° BC ⁄ lC +The quantity in brackets is just the negative of the #-component oŸ +JX71a — 3 X ©1a +rỶ; ri +Corresponding results will be found for the other components, so we have +1 J(2)x +B()= “— “—==. (14.42) +4meoc2 Tía +--- Trang 178 --- +The integral gives Ö directly in terms of the known currents. 'Phe geometry +involved is the same as that shown in Fig. 14-2. +Tí the currents exist only in circuits of small wires we can, as in the last section, +immediately do the integral across the wire, replacing 7 đV by Ids, where đs is +an element of length of the wire. Then, using the symbols in Fig. 14-10, +B() = “_- —== (14.43) +4mxegc2 r% +(The minus sign appears because we have reversed the order of the cross product.) +'This equation for #Ö is called the P/of-Saoart la+, after its discoverers. Ït gives +a formula for obtaining directly the magnetic fñeld produced by wires carrying +Currents. +You may wonder: “What ¡is the advantage of the vector potential if we can +find Ö directly with a vector integral? After all, A also involves three integrals!” +Because of the cross product, the integrals for Ö are usually more complicated, +as is evident from Eq. (14.41). Also, since the integrals for A are like those of +electrostatics, we may already know them. Einally, we will see that in more +advanced theoretical matters (in relativity, in advanced formulations of the laws of +mechanics, like the prineiple of least action to be discussed later, and in quantum +mechanics) the vector potential plays an important role. +--- Trang 179 --- +The Voc£or' FPo£onmfi(fl +15-1 The forces on a current loop; energy of a dipole +In the last chapter we studied the magnetic field produced by a small rectan- 15-1 The forces on a current loop; +gular current loop. We found that ït is a dipole fñield, with the dipole moment energy of a dipole +given by 15-2 Mechanical and electrical +u=1A, (15.1) energies +where T is the current and 4 is the area of the loop. The direction of the moment 1ã-3 The energy of steady currents +is normal to the plane of the loop, so we can also write 1ã-4 versus Á +15-5 The vector potential and +u = TẦn, quantum mechanics +15-6 What is true for statics is false ft +where #ø is the unit normal to the area A. 5-0 d ..¬ 08180168 13 0a086 (0E +A current loop—or magnetie dipole—not only produces magnetic felds, but M +will also experience forces when placed in the magnetic feld of other currenfs. +We will look frst at the forces on a rectangular loop in a uniform magnetic feld. +Let the z-axis be along the direction of the feld, and the plane of the loop be +placed through the ¿-axis, making the angle Ø with the z#-plane as in Fig. 15-1. +'Then the magnetic moment of the loop——which is normal to its plane—will make +the angle Ø with the magnetic feld. +Since the currents are opposite on opposite sides of the loop, the forces are +also opposite, so there is no net force on the loop (when the field is uniform). +Because of forces on the two sides marked 1 and 2 in the fñgure, however, there +1s a torque which tends to rotate the loop about the -axis. The magnitude of +these forces 1 and 5 is +Fì = Fạ = TBÌ. +'Their moment arm is +øasin 6, +so the torque 1s +T = Tab Bsin0, +or, since Tab is the magnetic moment of the loop, Là +— h BA ọ +T = uBsin0. F Ủ⁄ ñ +'The torque can be written in vector notation: s4 *x +T=ux Ö. (15.2) 3. si * +` Ặ; ÂU: +Although we have only shown that the torque is given by Eq. (15.2) in one rather & N4 ò +special case, the result is right for a small loop of any shape, as we will see. The a b +same kind of relationship holds for the torque of an electric dipole in an electric v +ñeld: +T—=px E. +Fig. 15-1. A rectangular loop carrying the +We now ask about the mechanical energy of our current loop. Since there current / sits in a uniform field B (in the +1s a torque, the energy evidently depends on the orientation. The principle of z-direction). The torque on the loop is 7 = +virtual work says that the torque is the rate of change of energy with angle, so „ở < B, where the magnetic moment ú = +W© Can WTIt© lab. +dŨ = r d0. +--- Trang 180 --- +Sctting 7 = BsinØ, and integrating, we can write for the energy +U = —bùBcos0 + a constant. (15.3) +(The sign is negative because the torque tries to line up the moment with the +fñeld; the energy is lowest when # and Ö are parallel.) +For reasons which we will discuss later, this energy is n=øf the total energy +of a current loop. (We have, for one thing, not taken into account the energy +required to maintain the current in the loop.) We will, therefore, call this energy +mecn; to remind us that it is only part of the energy. Also, since we are leaving +out some of the energy anyway, we can set the constant of integration equal %o +zero in Pq. (15.3). So we rewrite the equation: +haech = —U- B. (15.4) +Again, this corresponds to the result for an electric dipole: +U=-p: E. (15.5) +Now the electrostatic energy Ù in Eq. (15.5) is the true energy, but Ùjmech +in (15.4) is not the real energy. It cøn, however, be used in computing forces, +by the principle of virtual work, supposing that the current in the loop——or at +least —is kept constant. +W©e can show for our rectangular loop that Umecn also corresponds to the +mmechanical work done in bringing the loop into the fñeld. 'Phe total force on the +loop is zero only in a uniform field; in a nonuniform field there øre net forces on +a current loop. In putting the loop into a region with a field, we must have gone +through places where the feld was not uniform, and so work was done. 'To make +the calculation simple, we shall imagine that the loop is brought into the fñeld +with its moment pointing along the field. (It can be rotated to its ñnal position +after it is in place.) +Imagine that we want to move the loop in the z-direction——toward a region +of stronger feld——and that the loop is oriented as shown in Fig. 15-2. We start +somewhere where the field is zero and integrate the force times the distance as +we bring the loop into the feld. +Fị 2 Œ l1) 2 ⁄ F¿ +Fig. 15-2. A loop is carried along the x- ,. mẽ... +direction through the field B, at right angles ⁄ 3 ⁄ x +to x. XI Xa +First, let°s compute the work done on each side separately and then take the +sum (rather than adding the forces before integrating). "The forces on sides 3 +and 4 are at right angles to the direction of motion, so no work is done on them. +The force on side 2 is Jb(z) in the z-direction, and to get the work done against +the magnetic forces we must integrate this from some + where the field is zero, +Say at # = —o©, fO #a, 1s present position: +W› = -J ‡2 da = Ti 5Œ) da. (15.6) +—œ© —œ© +Similarly, the work done against the forces on side 1 is +#1 T1 +tị =— ƒ ị de = 1b | B(z) da. (15.7) +—œ© —œ© +--- Trang 181 --- +To find each integral, we need to know how Ö(z) depends on z. But notice that +side 1 follows along right behind side 2, so that is integral includes most of the +work done on side 2. In fact, the sum oÊ (15.6) and (15.7) is just +W= Ki B(z) da. (15.8) +But if we are in a region where is nearly the same on both sides 1 and 2, we +can write the integral as +J Đ(#) d+ = (za — zì)B = aB, +where is the field at the center of the loop. “The total mechanical energy we +have put in is +ech = W = —Tlab B = —uB. (15.9) +The result agrees with the energy we took for Eq. (15.4). +W©e would, of course, have gotten the same result if we had added the forces +on the loop before integrating to fnd the work. If we let ¡ be the field at side 1 +and 5; be the fñeld at side 2, then the total force in the z-direction 1s +tạ = Tb(Ha — Bì). +Tf the loop is “small,” that is, If Ba and ị are not too diferent, we can write +9B 8B +Ba = Bị+— A+ = Bị + — a. +So the Íorce is +tụy = Tlab ——. (15.10) +'The total work done on the loop by ezternal forces is +-J E, dự = —lab | TC dự = —IabB, +—œe 3z +which is again just —/B. Only now we see why it is that the ƒforce on a small +current loop is proportional to the derivative of the magnetic field, as we would +expect from +Ty AÁ�� = —Amech = —A(—g- ). (15.11) +Our result, then, is that even though nech — —#: may not include all +the energy of a system——it is a fake kind oŸ energy—it can still be used with the +principle of virtual work to fnd the forces on steady current loops. +15-2 Mechanical and electrical energies +W©e want now to show why the energy mẹcn discussed in the previous section +is not the correct energy associated with steady currents—that it does not keep +track of the total energy in the world. We have, indeed, emphasized that it can +be used like the energy, for computing forces from the principle of virtual work, +prouided that the current in the loop (and all øo#her currents) do not change. +Let°s see why all this works. +TImagine that the loop in EFig. 15-2 is moving in the -+z-direction and take the +z-axis In the direction of Ö. The conduction electrons in side 2 wilÌ experience +a force along the wire, in the -direction. But because of their ow—as an +electric current—there is a component of their motion in the same direction as +the force. Each electron is, therefore, having work done on it at the rate Fy, +where ơ„, is the component of the electron velocity along the wire. We will call +this work done on the electrons elecfr¿ical work. Now it turns out that 1ƒ the +loop is moving in a nớ#orm fñeld, the total electrical work is zero, since positive +work is done on some parts of the loop and an equal amount of negative work is +--- Trang 182 --- +done on other parts. But this is not true if the circuit is moving in a nonuniform +fñeld—then there ⁄lJ be a net amount of work done on the electrons. ÍIn general, +this work would tend to change the flow of the electrons, but if the current is +being held constant, energy must be absorbed or delivered by the battery or +other source that is keeping the current steady. 'This energy was not included +when we computed 7»eeù in Ed. (15.9), because our computations included only +the mechanical forces on the body of the wire. +You may be thinking: But the force on the electrons depends on how ƒasf +the wire is moved; perhaps If the wire is moved slowly enough this electrical +energy can be neglected. It is true that the raf#e at which the electrical energy is +delivered is proportional to the speed of the wire, but the #o£al energy delivered +1s proportional also to the #ữne that this rate goes on. So the tobal electrical +energy is proportional to the velocity times the time, which is just the distance +moved. For a given distance moved in a fñeld the same amount of electrical work +is done. +Let's consider a segment of wire of unit length carrying the current Ï and +moving in a direction perpendicular to itself and to a magnetic fñeld Ö with the +speed øwie. Because of the current the electrons will have a drift velocity 0arify +along the wire. 'Phe component of the magnetic force on each electron in the +direction of the drift 1s qe0wire. So the rate at which electrical work is being +done is 0aritt —= (đeUwire)0arie. TỶ there are conduction electrons in the unit +length of the wire, the total rate at which electrical work is being done 1s +_. = NgeUwirePUartt. +But Nge«0ariy = l, the current in the wire, so +_. =Ĩ Ðwire * +Now since the current ¡is held constant, the forces on the conduction electrons +do not cause them to accelerate; the electrical energy is not going into the +electrons but into the source that is keeping the current constant. +But notice that the force on the re is FB, so Iuyiye 1s also the rate of +mechanical tuork done on the wire, đUmeen “(dt = TBoyue. We conclude that the +mechanical work done on the wire is just equal to the electrical work done on +the current source, so the energy of the loop 2s ø constzn# +'This is not a coincidenee, but a consequence of the law we already know. 'Phe +total force on each charge in the wire is +t=q(E+ox Đ). +'The rate at which work is done is +0+ Et'=q[u-: E+o-(o x Đ). (15.12) +Tí there are no electric felds we have only the second term, which is always 2ero. +W© shall see later that changïng magnetic fñelds produce electric fields, so our +reasoning applies only to moving wires in steady magnetic fields. +How is it then that the principle of virtual work gives the right answer? +Because we s7! have not taken into account the #o#al energy of the world. We +have not included the energy of the currents that are produc¿ng the magnetic +fñeld we start out with. +Suppose we imagine a complete system such as that drawn in Fig. 15-3(a), +in which we are moving our loop with the current 1¡ into the magnetic ñeld ị +produeced by the current Ï¿ in a coil. Now the current 1¡ in the loop will also be +producing some magnetic ñeld ; at the coil. If the loop is moving, the fñeld Ba +will be changing. As we shall see in the next chapter, a changing magnetic field +generates an #-feld; and this #-fñeld will do work on the charges in the coil. +'This energy must also be included ïn our balance sheet of the total energy. +--- Trang 183 --- +Bị B: +`... x.. +†B› †B› +h Loop h +(a) (@œ) +Fig. 15-3. Finding the energy of a small loop in a magnetic field. +W© could wait until the next chapter to ñnd out about this new energy term, +but we can also see what it will be if we use the principle of relativity in the +following way. When we are moving the loop toward the stationary coil we know +that its electrical energy is just equal and opposite to the mechanical work done. So +Dhuech + slect (loop) =0. +Suppose now we look at what is happening from a diferent point of view, in +which the loop is at rest, and the coïil is moved toward ït. The coil is then moving +into the fñeld produced by the loop. The same arguments would give that +Dueen + 2lee¿(coil) = 0. +The mechanical energy is the same in the two cases because it comes from the +force between the bwo circuits. +'The sum of the two equations gives +2mech + slect (loop) + Uslect (coil) =0. +'The total energy of the whole system is, of course, the sum of the two electrical +energies plus the mechanical energy taken only onwece. So we have +Uotai — slect (loop) + Uclect (coil) + Duech — —uech- (15.13) +'The total energy of the world is really the m=egaf2ue of Umecn. LÝ we want the +true energy of a magnetic dipole, for example, we should write +otai — +U - B. +Tt is only iŸ we make the condition that all currents are constant that we can +use only a part of the energy, [7mee, (which is always the negative of the true +energy), to find the mechanical forces. In a more general problem, we must be +careful to include all energies. +We have seen an analogous situation in electrostatics. We showed that the +energy of a capacitor is equal to Q°/2Œ. When we use the principle of virtual +work to ñnd the force between the plates of the capacitor, the change in energy +is equal to Q2/2 tỉmes the change in 1/C. That is, +Q? 1 Q? AC +AU=—Al|—=]=-—-—.. 15.14 +2 lổi 2_ €2 ( ) +Now suppose that we were to calculate the work done in moving two conductors +subject to the diferent condition that the voltage between them ¡s held constant. +'Then we can get the right answers for force from the principle of virtual work iŸ we +do something artifcial. Since Q = CV, the real energy is sCV?. But if we deñne +an artificial energy equal to —;ŒC V2, then the prineiple of virtual work can be used +to get forces by setting the change in the artificial energy equal to the mechanical +work, provided that we insist that the voltage W be held constant. Then +CV? V2 +--- Trang 184 --- +which is the same as Eq. (15.14). We get the correct result even though we are +neglecting the work done by the electrical system to keep the voltage constant. +Again, this electrical energy is just bwice as big as the mechanical energy and +of the opposite sign. +Thus iƒ we calculate artifcially, disregarding the fact that the source of the +potential has to do work to maintain the voltages constant, we get the right +answer. Ï% is exactly analogous to the situation in magnetostatics. +15-3 The energy of steady currents +W©e can now use our knowledge that Utesai = —mech to ñnd the true energy +of steady currents in magnetic ñelds. We can begin with the true energy of a +small current loop. Calling ox¿aị Just Ứ, we write +U=u:-Ö. (15.16) +Although we calculated this energy for a plane rectangular loop, the same result +holds for a small plane loop of any shape. +W© can fñnd the energy of a circuit of any shape by imagining that ¡it is made Ũ +up of small current loops. Say we have a wire in the shape of the loop ' of <<) —>— +Fig. 15-4. We fñll in this curve with the surface Š, and on the surface mark out a “ta re Loop F +large number of small loops, each of which can be considered plane. If we let the tì +current ƒ circulate around eøch of the little loops, the net result will be the same tr rrr.h¬AÀ +as a current around Ï, since the currents will cancel on all lines internal to I. ST. TT +Physically, the system of little currents is indistinguishable from the original S772 1L}7 +circuit. The energy must also be the same, and so is just the sum of the energies _*=S—— 77 +of the little loops. Ị Surface S +T the area of each little loop is Aø, its ©nergy is TAaB„, where „ ¡is the Fig. 15-4. The energy of a large loop in +cormponent normal to Aa. The total energy is a magnetic field can be considered as the +sum of energies of smaller loops. +U=À 1B, Aa. +Goïng to the limit of infnitesimal loops, the sum becomes an integral, and +U =1 | Bị da =1 [ Bnda (15.17) +where ?øw is the unit normal to da. +Tf we set Ở = V x A, we can connect the surface integral to a line integral, +using Stokesˆ theorem, +T(V xA) nai =1 Ệ Ads (15.18) +where đs is the line element along `. 5o we have the energy for a circuit of any +shape: +U=I ‡ A-ds, (15.19) +circuit +In this expression A refers, of course, to the vector potential due to those currents +(other than the 7 in the wire) which produce the field Ö at the wire. +Now any distribution of steady currents can be imagined to be made up of +filaments that run parallel to the lines of current fow. For each pair of such +circuits, the energy is given by (15.19), where the integral is taken around one +circuit, using the vector potential A from the other circuit. Eor the total energy +we want the sum of all such pairs. TÍ, instead of keeping track of the pairs, we +take the complete sum over all the fñlaments, we would be counting the energy +twice (we saw a similar efect in electrostatics), so the total energy can be written +U= tj2 - AdV. (15.20) +--- Trang 185 --- +'This formula corresponds to the result we found for the electrostatic energy: +U= 1 móat (15.21) +So we may if we wish think of A as a kind of potential for currents in magne- +tostatics. Unfortunately, this idea is not too useful, because 1t 1s true only for +static fñelds. In fact, neither of the equations (15.20) and (15.21) gïves the correcb +energy when the fields change with time. +15-4 Ö versus A +In this section we would like to discuss the following questions: Is the vector +potential merely a device which is useful in making calculations—as the scalar +potentfial is useful in electrostatics——or is the vector potential a “real” fñeld? Isn't +the magnetic feld the “real” field, because ït is responsible for the force on a +moving particle? Pirst we should say that the phrase “a real feld” is not very +meaningful. For one thing, you probably don't feel that the magnetic field is very +“real” anyway, because even the whole idea of a field is a rather abstract thing. +You cannot put out your hand and feel the magnetic fñeld. Furthermore, the value +of the magnetic field is not very defñnite; by choosing a suitable moving coordinate +system, for instance, you can make a magnetic fñeld at a given point disappear. +What we mean here by a “real” field is this: a real feld is a mathematical +function we use for avoiding the idea of action at a distance. If we have a charged +particle at the position 7, ¡t is affected by other charges located at some distance +from . One way to describe the interaction is to say that the other charges +make some “condition”—whatever it may be—in the environment at P. If we +know that condition, which we describe by giving the electric and magnetic fields, +then we can determine completely the behavior of the particle—with no further +reference to how those conditions came about. +In other words, if those other charges were altered in some way, but the +conditions at ?? that are described by the electric and magnetic feld at remain +the same, then the motion of the charge will also be the same. A “real” field is +then a set of numbers we speclfy in such a way that what happens øẺ œ po¿n£ +depends only on the numbers đÝ (hø‡ pozn. We do not need to know any more +about what's going on at other places. Ït is in this sense that we will discuss +whether the vector potential is a “real” feld. +You may be wondering about the fact that the vector potential is not unique—— +that it can be changed by adding the gradient of any scalar with no change at +all in the forces on particles. That has not, however, anything to do with the +question of reality in the sense that we are talking about. For instance, the +magnetic field is in a sense altered by a relativity change (as are also # and 4). +But we are not worried about what happens If the feld can be changed in this +way. That doesn't really make any diference; that has nothing to do with the +question of whether the vector potential is a proper “real” field for describing +magnetic efects, or whether ïÈ is Jjust a useful mathematical tool. +W©e should also make some remarks on the usefulness of the vector potential A. +We have seen that it can be used in a formal procedure for calculating the +magnetic fields of known currents, just as ý can be used to fñnd electric ñelds. In +electrostatics we saw that ó was given by the scalar integral +1 /ø(2) +2(1) = m5. (15.22) +tHrom this ó, we get the three components of # by three diferential operations. +This procedure is usually easier to handle than evaluating the three integrals in +the vector formula I (9) +Ø\2)€12 +E)= 1reo J _~ đV:. (15.23) +Pirst, there are three integrals; and second, each integral is in general somewhat +more difficult. +--- Trang 186 --- +The advantages are much less clear for magnetostatics. The integral for A is +already a vector integral: +I 72) d1 +A(q)= TT (15.24) +which is, of course, three integrals. Also, when we take the curl of A to get Ö, +we have six derivatives to do and combine by pairs. Ít is not immediately obvious +whether in most problems this procedure is really any easier than computing +directly from +B_)= >> .= TP đỰy, (15.25) +4eoc2 r# +Using the vector potential is often more difficult for simple problems for the +following reason. Suppose we are interested only in the magnetic fñeld Ö at one +point, and that the problem has some nice symmmetry—say we want the field at a +point on the axis 0Ÿ a ring of current. Because of the symmetry, we can easily +get by doïng the integral of Eq. (15.25). If, however, we were to ñnd A first, +we would have to compute from đeriuafiues oŸ Á, so we must know what A is +at all points in the neighborhood of the point of interest. And most of these points +are off the axis of symmetry, so the integral for A gets complicated. In the ring +problem, for example, we would need to use elliptic integrals. In such problems, +A ¡is clearly not very useful. lt is true that in many complex problems iÈ is easier +to work with A, but it would be hard to argue that this ease of technique would +Justify making you learn about one more vector field. +We have introduced 4 because it đoes have an important physical signiicance. +Not only is it related to the energies of currents, as we saw in the last section, but +1 is also a “real” physical ñeld in the sense that we described above. In classical +mechanies it is clear that we can write the force on a particle as +F=q(E+oxĐ), (15.26) +so that, given the forces, everything about the motion is determined. In any +region where Ö = 0 even if is not zero, such as outside a solenoid, there is +no discernible efect of A. Therefore for a long time it was believed that Á was +not a “real” fñeld. It turns out, however, that there are phenomena involving +quantum mechanics which show that the field A is in fact a “real” ñeld in the +sense we have defñned it. In the next section we will show you how that works. +1ã-5ã The vector potential and quantum mechanics +There are many changes in what concepts are Important when we go from +classical to quantum mechanics. We have already discussed some of them in +Vol. I. In particular, the force concept gradually fades away, while the concepts +of energy and momentum become of paramount importance. You remember +that instead of particle motions, one deals with probability amplitudes which +vary in space and time. In these amplitudes there are wavelengths related to +mmomenta, and frequencies related to energies. The momenta and energies, which +determine the phases of wave functions, are therefore the Important quantities in +quantum mechanics. Instead of forces, we deal with the way interactions change +the wavelength of the waves. The idea of a force becomes quite secondary——If 1t +1s there at all. When people talk about nuelear forces, for example, what they +usually analyze and work with are the energies of interaction of two nucleons, +and not the force between them. Nobody ever diferentiates the energy to fnd +out what the force looks like. In this section we want to describe how the vector +and scalar potentials enter into quantum mechanies. Ït is, in fact, Just because +mmomentum and energy play a central role in quantum mechanics that A and ¿ +provide the most direct way of introducing electromagnetic efects into quantuun +descriptions. +We must review a little how quantum mechanics works. We will consider +again the imaginary experiment described in Chapter 3/ of Vol. I, in which +--- Trang 187 --- +Ñ DETECTOR +SOURCE .. - ` .Ằ +SE VN TƯ, _ +S===_ Ñ —=” ++—————L +Fig. 15-5. An interference experiment with electrons (see +also Chapter 37 of Vol. l). +electrons are difracted by two slits. The arrangement is shown again in Eig. 15-5. +Electrons, all of nearly the same energy, leave the source and travel toward a wall +with two narrow slits. Beyond the wall is a “backstop” with a movable detector. +The detector measures the rate, which we call 7, at which electrons arrive at a +small region of the backstop at the distance #z from the axis of symmetry. The +rate Is proportional to the probability that an individual electron that leaves +the source will reach that region of the backstop. This probability has the +complicated-looking distribution shown in the fgure, which we understand as due +to the interference of two amplitudes, one from each slit. The interference of the +two amplitudes depends on their phase diference. 'That ïs, if the amplitudes are +C+c?”! and Œse?®?, the phase diference ổ = ®+ — ®s determines their interference +pattern [see Eq. (29.12) in Vol. T]. If the distance between the screen and the +slits is b, and if the diference in the path lengths for electrons going through +the two slits is ø, as shown In the figure, then the phase diference of the bwo +waves 1s given by : +ỗ= h (15.27) +As usual, we let À = À/27z, where À is the wavelength of the space variation of +the probability amplitude. For simplicity, we will consider only values of z much +less than L; then we can set +ô= TÀ (15.28) +When zø is zero, ổ is zero; the waves are in phase, and the probability has a +maximum. When ð is 7, the waves are out of phase, they interfere destructively, +and the probability is a minimum. So we get the wavy function for the electron +1ntensity. +Now we would like to state the law that for quantum mechanics replaces +the force law È! = gu x . It will be the law that determines the behavior of +quantum-mechanical particles in an electromagnetic feld. Since what happens +1s determined by amplitudes, the law must tell us how the magnetic inÑuences +affect the amplitudes; we are no longer dealing with the acceleration of a particle. +'The law is the following: the phase of the amplitude to arrive via any traJectory +is changed by the presence of a magnetic ñeld by an amount equal to the integral +of the vector potential along the whole trajectory times the charge of the particle +over Planck”s constant. That 1s, +Magnetic change in phase = n J A:-das. (15.29) +trajectory +--- Trang 188 --- +TÍ there were no magnetic ñeld there would be a certain phase of arrival. If there +1s a magnetic field anywhere, the phase of the arriving wave is increased by the +integral in Eq. (15.29). +Although we will not need to use it for our present discussion, we mention +that the efect of an electrostatic feld is to produce a phase change given by the +negafiue of the tữme integral of the scalar potential ở: +Eilectric change in phase —= =" J o dt. +These two expressions are correct not only for static ñelds, but together give the +correct result for øm+ electromagnetic field, static or dynamic. “This is the law +that replaces #' = g(E + o x B). Woe want now, however, to consider only a +static magnetic ñeld. +Suppose that there is a magnetic fñeld present in the two-slit experiment. We +want to ask for the phase of arrival at the sereen of the bwo waves whose paths +pass through the bwo slits. Their interference determines where the maxima In +the probability will be. We may call ® the phase of the wave along trajectory (1). +If ®ị(B = 0) is the phase without the magnetic field, then when the field is +turned on the phase will be +8; =8¡(B=0)+ li A-ds. (15.30) +Similarly, the phase for trajectory (2) is +8; = 82(B =0) + li A-ds. (15.31) +'The interference of the waves at the detector depends on the phase diference +3S ®ị(B= 0) = #4(B= 0) + 7 | A-de=n | A-ds. (15.32) +h Jạ) h J@) +The no-field diference we will call đ( = 0); it is just the phase difference we +have calculated above in Eq. (15.28). Also, we notice that the two integrals can +be written as ơne integral that goes forward along (1) and back along (2); we +call this the closed path (1-2). Šo we have +ö=ð(B=0)+ Ñ; A-ds. (15.33) +h Jas) +'This equation tells us how the electron motion is changed by the magnetic field; +with it we can fnd the new positions of the intensity maxima and minima at the +backstop. +Before we do that, however, we want to raise the following interesting and +Important point. You remember that the vector potential function has some +arbitrariness. Two different vector potential funetions 4 and 4“ whose diference +1s the gradient of some scalar function Vụ, both represent the same magnetic +fñeld, since the curl of a gradient is zero. They give, therefore, the same classical +force gu x . lfin quantum mechanics the efects depend on the vector potential, +tuhích of the many possible A-functions is correct? +'The answer is that the same arbitrariness in 4 continues to exist for quantum +mechanies. lfin Ðq. (15.33) we change A to A' = A + Vụ, the integral on A +becomes +1 Afsds= ÿ Ads+ J Vụ - da. +(1-3) (1-2) (1-2) +The integral of Wj is around the closed path (1-2), but the integral of the +tangential component of a gradient on a closed path is always zero, by Stokes” +theorem. Therefore both A and A” give the same phase diferences and the same +quantum-mechanical interference efects. In both classical and quantum theory +it is only the curl of A that matters; any choice of the function of 4 which has +the correct curl gives the correct physics. +--- Trang 189 --- +'The same conclusion is evident if we use the results of Section 14-1. There we +found that the line integral of A around a closed path is the ñux of Ö through +the path, which here is the fux bebween paths (1) and (2). Equation (15.33) can, +1Í we wish, be written as +ð=ð(B =0) + n [fux of between (1) and (2)Ì, (15.34) +where by the Ñux of we mean, as usual, the surface integral of the normal com- +ponent of Ö. 'The result depends only on Ö, and therefore only on the curl of A. +NÑow because we can write the result in terms of as well as in terms of A, +you might be inclined to think that the Ö holds its own as a “real” feld and that B +the A can still be thought of as an artificial construction. But the definition of +“real” fñeld that we originally proposed was based on the idea that a “real” fñeld +would not act on a particle from a distance. We can, however, give an example 4N? +in which #Ö is zero—or at least arbitrarily small—at any place where there is II +some chance to fnd the particles, so that it is not possible to think oŸ it acting +đirectử on them. ti +You remember that for a long solenoid carrying an electric current there is a +b-feld inside but none outside, while there is lots of A circulating around outside, ti +as shown in Eig. 15-6. If we arrange a situation in which electrons are to be +found only øu#side of the solenoid—only where there is A——there will still be an /\ +infuence on the motion, according to Eq. (15.33). Classically, that is impossible. k-„ 11 +Classically, the force depends only on #Ö; ¡in order to know that the solenoid is m—T +carrying current, the particle must go through it. But quantum-mechanically you Q2) +can find out that there 1s a magnetic field inside the solenoid by going øround | | | +it —without ever going close to itl +Suppose that we put a very long solenoid of small diameter just behind the Fig. 15-6. The magnetic field and vector +wall and bebween the two slits, as shown in Eig. 15-7. The diameter of the potential of a long solenoid. +solenoid is to be mụuch smaller than the distance đ between the two slits. In these +circumstances, the difÑfraction of the electrons at the slit gives no appreciable +probability that the electrons will get near the solenoid. What will be the efect +on our interference experiment? v +SOURCE x __—— Ầ...... +— ¬— ` No) +SOLENOID +LINES OF B +L —————>~* +Fig. 15-7. A magnetic field can influence the motion of electrons even though +It exists only in regions where there ¡is an arbitrarily small probability of finding the +electrons. +W©e compare the situation with and without a current through the solenoid. +Tf we have no current, we have no Ö or A and we get the original pattern of +electron intensity at the backstop. IÝ we turn the current on in the solenoid and +build up a magnetic field Ö inside, then there is an A outside. 'There is a shift +in the phase diference proportional to the cireulation of A outside the solenoid, +which will mean that the pattern of maxima and minima is shifted to a new +position. In fact, since the ñux of #Ö inside is a constant for any pair of paths, +so also is the circulation of A. For every arrival point there is the same phase +--- Trang 190 --- +change; this corresponds to shifting the entire pattern in z by a constant amount, +say #o, that we can easily calculate. The maximum intensity will occur where the +phase difference between the two waves is zero. Using Eq. (15.33) or Eq. (15.34) +for ô and Eq. (15.28) for z, we have +zọ =-zAy£ A:ds, (15.35) +đc hJq» +#u=—T Ầ h [ux of Ö between (1) and (2)]. (15.36) +The pattern with the solenoid in place should appearÝ as shown in Pig. 15-7. At +least, that is the prediction of quantum mechanics. +Precisely this experiment has recently been done. Ït is a very, very dificult +experiment. Because the wavelength of the electrons 1s so small, the apparatus +must be on a tiny scale to observe the interference. The slits must be very close +together, and that means that one needs an exceedingly small solenoid. It turns +out that in certain cireumstances, iron crystals will grow in the form of very long, +microscopically thin flaments called whiskers. When these iron whiskers are +magnetized they are like a tiny solenoid, and there is no ñeld outside except near +the ends. “The electron interference experiment was done with such a whisker +between two slits, and the predicted displacement in the pattern of electrons was +observed. +In our sense then, the 4-field is “real” You may say: “But there øøœs a +magnetic field.” There was, but remember our original idea—that a fñeld is “real” +1f it is what must be specified ø# the position oŸ the particle in order to get the +motion. The #-field in the whisker acts at a distance. If we want to describe its +inuence not as action-at-a-distance, we must use the vector potential. +This subject has an interesting history. The theory we have described was +known from the beginning of quantum mechanics in 1926. The fact that the +vector potential appears in the wave equation of quantum mechanics (called the +Schrödinger equation) was obvious from the day it was written. That it cannot +be replaced by the magnetic fñeld in any easy way was observed by one man afÍter +the other who tried to do so. “Phis is also clear from our example of electrons +moving in a region where there is no ñeld and being affected nevertheless. But +because in classical mechanics A did not appear to have any direct importance +and, furthermore, because it could be changed by adding a gradient, people +repeatedly said that the vector potential had no direct physica]l signiicance—that +only the magnetic and electric fñelds are “right” even in quantum mechanics. Ït +seems strange in retrospect that no one thought of discussing this experiment +until 1956, when Bohm and Aharonov frst suggested it and made the whole +question crystal clear. 'Phe implication was there all the time, but no one paid +attention to it. PThus many people were rather shocked when the matter was +brought up. PThat's why someone thought iÿ would be worth while to do the +experiment to see that it really was right, even though quantum mechanics, which +had been believed for so many years, gave an unequivocal answer. Ït is interesting +that something like this can be around for thirty years but, because of certain +prejudices of what is and is not significant, continues to be ignored. +Now we wish to continue in our analysis a little further. We will show the +connection between the quantum-mechanical formula and the classical formula—— +to show why it turns out that if we look at things on a large enouph scale it will +look as though the particles are acted on by a force equal to gu x the curl of A. +To get classical mechanics from quantum mechanics, we need to consider cases +in which all the wavelengths are very small compared with distances over which +external conditions, like fields, vary appreciably. We shall not prove the result in +great generality, but only in a very simple example, to show how it works. Again +we consider the same slit experiment. But instead of putting all the magnetic +* Tf the fñeld #Ö comes out of the plane of the fñigure, the fux as we have defined it is positive +and since g for electrons is negative, #o is positive. +--- Trang 191 --- +"¬- Ax | `2 +SOURCE __ ". <<”? +~“ _—” TÑI...- _._—. “———_-_-___— +TNN TS N|--_z-Z*%- =“-------” +` —-l Ñ . ` ¬>—— +Ñ[S- TLNEs oF B ` +Ñ ¬ › S +NI. 2 +Ñ ———'———— ` +Fig. 15-8. The shift of the interference pattern due to a strip of magnetic field. +ñeld in a very tiny region between the slits, we imagine a magnetic ñeld that +extends over a larger region behind the slits, as shown in EFig. 15-8. We will take +the idealized case where we have a magnetic fñeld which is uniform in a narrow +strip of width , considered small as compared with E. (That can easily be +arranged; the backstop can be put as far out as we want.) In order to calculate +the shift in phase, we must take the two integrals of A along the two trajecbories +(1) and (2). They differ, as we have seen, merely by the fux of between the +paths. To our approximation, the ñux is Bud. The phase diference for the Ewo +paths is then +öð=ð(B=0)+ ¡ Bud, (15.37) +W© note that, to our approximation, the phase shift is independent of the angle. +So again the efect will be to shift the whole pattern upward by an amount Az. +Using Eq. (15.35), +LÀ LÀ +Azm=——— Aô=——— lỗ - ö(B =0)|. +z=—= “ lä~ ð(B =0) +Using (15.37) for ö — ð(B = 0), " +A#= —LÀ n Du. (15.38) ¬ +Such a shift is equivalent to deflecting all the trajectories by the small angle œ ¬—. == +(see Eig. 15-S§), where Am Pn +Am À ¬ += —=_—x~qBu. 15.39 "¬. +œ==T p.1Bu (15.39) ¬. +Now classically we would also expect a thin strip of magnetic ñeld to defect " s ` _ T—LINES OF +all trajectories through some small angle, say œ', as shown in Eig. 15-9(a). As ¬ +the electrons go through the magnetic ñeld, they feel a transverse force gu x ¬ +which lasts for a tỉme +0/ø. The change in their transverse momentum is just +equal to this impulse, so “ (a) +ADbx = —quB. (15.40) +The angular deflection [Eig. 15-9(b)] is equal to the ratio of this transverse +mmomentum to the total momentum ø. We get that +A B Apx +ai ÂÐD _ _ 10B. (15.41) —_—a +: : : : : (@b) +W© can compare this result with Eq. (15.39), which gives the same quantity +computed quantum-mechanically. But the connection between classical mechanics Fig. 15-9. Deflection of a particle due to +and quantum mechanics is this: A particle of momentum ø corresponds to a passage through a strip of magnetic field. +--- Trang 192 --- +quantum amplitude varying with the wavelength À = ñ/p. With this equality, œ +and o/ are identical; the classical and quantum calculations give the same result. +trom the analysis we see how it is that the vector potential which appears In +quantum mechanics in an explicit form produces a classical force which depends +only on its derivatives. In quantum mechanics what matters is the interference +between nearby paths; it always turns out that the efects depend only on +how much the feld A chønges from point to point, and therefore only on the +derivatives of 4 and not on the value itself. NÑevertheless, the vector potential A +(together with the scalar potential ó that goes with it) appears to give the most +direct description of the physics. 'Phis becomes more and more apparent the +more deeply we go into the quantum theory. In the general theory of quantum +electrodynamics, one takes the vector and scalar potentials as the fundamental +quantities in a set of equations that replace the Maxwell equations: # and +are slowly disappearing from the modern expression of physical laws; they are +being replaced by A and ó. +15-6 What is true for statics is false for dynamics +W© are now at the end of our exploration of the subject of static ñelds. Already +in this chapter we have come perilously close to having to worry about what +happens when fields change with time. We were barely able to avoid It in our +treatment of magnetic energy by taking refuge in a relativistic areument. Even +so, our treatment of the energy problem was somewhat artificial and perhaps +even mysterious, because we ignored the fact that moving coils must, in fact, +produce changing fields. It is now time to take up the treatment of time-varying +fields——the subJect of electrodynamics. We will do so in the next chapter. First, +however, we would like to emphasize a few poinfs. +Although we began this course with a presentation of the complete and correct +equations oŸ electromagnetism, we immediately began to study some incomplete +pieces—because that was easier. There is a great advantage in starting with the +simpler theory of static fields, and proceeding only later to the more complicated +theory which includes dynamic fñields. There is less new material to learn all at +onee, and there is tỉme for you %o develop your intellectual muscles in preparation +for the bigger task. +But there is the danger in this process that before we get to see the complete +story, the incomplete truths learned on the way may become ingrained and taken +as the whole truth—that what is true and what is only sometimes true will +become confused. 5o we give in Table 15-1 a summary of the important formulas +we have covered, separating those which are true in general from those which are +true for statics, but false for dynamics. 'This summary also shows, in part, where +we are going, since as we treat dynamics we will be developing in detail what we +must Just state here without proof. +Tt may be useful to make a few remarks about the table. First, you should +notice that the equations we started with are the #rue equations—we have not +misled you there. The electromagnetic force (often called the Ùoren‡z ƒorce) +F=q(E+ox) šs truc. It is only Coulomb°s law that is false, to be used +only for statics. The four Maxwell equations for and #Ö are also true. The +cequations we took for statics are false, of course, because we left of all terms +with time derivatives. +Gauss' law, V - E = ø/co, remains, but the curÌ oŸ # is no‡ zero in general. +So #2 cannot always be equated to the gradient of a scalar—the electrostatic +potential. We will see that a scalar potential still remains, but it is a time- +varying quantity that must be used together with vector potentials for a complete +description oÊ the electric fñeld. The equations governing this new scalar potential +are, necessarily, also new. +W©e must also give up the idea that # is zero in conductors. When the felds +are changing, the charges in conduectors do not, in general, have time to rearrange +themselves to make the feld zero. They are set in motion, but never reach +equilibrium. “The only general statement is: electric fñelds in conductors produce +--- Trang 193 --- +Table 15-1 +FALSE IN GENER.AL (true only for statics) TRUE ALWAYS +_— 1 đị@ : —— +#=——_— (Coulomb's law) F=q(E+oxĐ) (Lorentz force) +4meg_ r2 +V.E=f (Gauss” law) +VxE=0 —= VxE-= _ (Faraday”s law) +E — — kL = — — — +Vớ Vớ PP +1 0(2)©1a2 +E(1)=—— | —>—d +) 47g J r% & +For conductors, # = 0, ô = constant. Q—= CV In a conductor, # makes currents. +>=V.Bb-=U0 (Ño magnetic charges) +BöB=VxA +: . E +cVxbB=? (Ampère's law) > VxB=7+ lên +€0 €0 ØF +1 72) X C12 +Đ(Œ)=—— | ———ởd +ú) 4mcgc2 J r4» & +Vˆ2¿ = _# (Poisson”s equation) Vˆ?¿— lợn =_-*# +€0 c2 8:2 €0 +3 1Ø?A 3 +VˆA=-—--“> Ý?2A_- _““_—_ J- +cọc? c2 Ø2 cọc2 +with with +V:A=0 c2V:A+ S2 =0 +1 fø0) ".-~ +1)=—— | _——d 1)=—— | ——ú +4 ) 47€o J T12 V %4 ; ) 47cg T12 V +1 (2 1 (2, +Au)= am Ati0= SG | #2 ám +47coc2 T13 47coc2 T12 +1 1y €0 cọc? +The equations marked with (—®) are Maxwells equations. +--- Trang 194 --- +currents. So in varying fields a conduectfor is nø‡ an equipotential. It also follows +that the idea of a capacitance is no longer precise. +Since there are no magnetic charges, the divergence 0Ÿ is akua/s 2zero. So +can always be equated to Ÿ x A. (Everything doesn't changel) But the generation +of B ¡s not only from currents; V x Ö ïs proportional to the current density pÏus +a new tem ØE/Ø. Thịis means that A is related to currents by a new cquation. +Tt is also related to ó. T we make use of our Íreedom to choose V - A for our +own convenience, the equations for A or ó can be arranged to take on a simple +and elegant form. We therefore make the condition that c?V : A = —02/ô, and +the diferential equations for Á or ó appear as shown in the table. +The potentials A4 and ø can still be found by integrals over the currents and +charges, but not the samne integrals as for statics. Most wonderfully, though, the +true integrals are like the static ones, with only a small and physically appealing +modification. When we do the integrals to fñnd the potentials at some point, say +point (1) in Fig. 15-10, we must use the values of 7 and ø at the point (2) a‡ an +carlier từme tÍ = t — ria/c. As you would expect, the influences propagate from +poïnt (2) to point (1) at the speed c. With this small change, one can solve for +the fields of varying currents and charges, because once we have Á and ó, we +get tữom W x A, as before, and # from —Wó — 8A/ði. +H2 +Fig. 15-10. The potentials at point (1) +and at the time f are given by summing Lm, +the contributions from each element of the Ix +source at the roving point (2), using the +currents and charges which were present at +the earlier time £ — na/c. +Finally, you will notice that some results—for example, that the energy density +in an electric feld is eo#?2/2—are true for electrodynamics as well as for statics. +You should not be misled into thinking that this is at all “natural” The validity +oŸ any formula derived in the static case must be demonstrated over again for the +dynamic case.  contrary example is the expression for the electrostatic energy +in terms oŸ a volume integral of øø. 'This result is true on for statics. +We will consider all these matters in more detail in due time, but it will +perhaps be useful to keep in mind this sunmary, so you will know what you can +forget, and what you should remember as always true. +--- Trang 195 --- +I6 +Xrnclreoel ẤtrrrioretÉs +16-1 Motors and generators +The discovery in 1820 that there was a close connection between electricity and 16-1 Motors and generators +magnetism was very exciting——until then, the two subjects had been considered as 16-2 'Transformers and inductances +quite independent. The fñrst discovery was that Currents in Wires make magnetic 16-3 Forces on induced currents +fñelds; then, in the same year, iÿ was found that wires carrying current in a 16-4 Electrical technology +magnetic field have forces on them. +One of the excitements whenever there is a mechanical force is the possibility +Of using it in an engine to do work. Almost immediately after their discovery, +people started to design electric motors using the forces on current-carrying wires. +The principle of the electromagnetic motor is shown in bare outline in Fig. 16-1. +A permanent magnet——usually with some pieces of soft iron——is used to produce +a magnetic fñeld in two slots. Across each slot there is a north and south pole, as +shown. Á rectangular coil of copper is placed with one side in each slot. When a +current passes through the coil, it Ñows in opposite directions in the two sÌots, so +the forces are also opposite, producing a torque on the coil about the axis shown. +Tí the coil is mounted on a shaft so that it can turn, it can be coupled to pulleys +or gears and can do work. +The same idea can be used for making a sensitive instrument for electrical +measurements. Thus the moment the force law was discovered the precision of +electrical measurements was greatly increased. First, the torque of such a motor =— ——— +can be made much greater Íor a given current by making the current go around +many turns instead of just one. hen the coil can be mounted so that it turns with +very little torque—either by supporting its shaft on very delicate jewel bearings +or by hanging the coil on a very ñne wire or a quartz2 fiber. Then an exceedingly N +small current will make the coïl turn, and for small angles the amount of rotation =WY +will be proportional to the current. The rotation can be measured by gluing a 3 ề +pointer to the coil or, for the most delicate instruments, by attaching a small £ 4 " R +mirror to the coil and looking at the shift of the image ofa scale. Such instruments & CC, +are called galvanometers. Voltmeters and ammeters work on the same principle. NG đề Ị +The same ideas can be applied on a large scale to make large motors Íor `N < » Sw +providing mechanical power. The coil can be made to go around and around by = è +arranging that the connections to the coil are reversed each half-turn by conbacts mm +mounted on the shaft. "hen the torque is always in the same direction. Small MAGNET +DC motors are made just this way. Larger motors, DC or AC, are often made BERMANENT +by replacing the permanent magnet by an electromagnet, energized from the +electrical power source. +With the realization that electric currents make magnetic fields, people Fig. 16-1. Schematic outline of a simple +immediately suggested that, somehow or other, magnets might also make electric electromagnetic motor. +fields. Varlous experiments were tried. For example, two wires were placed +parallel to each other and a current was passed through one of them in the +hope of ñnding a current in the other. 'Phe thought was that the magnetic fñeld +might in some way drag the electrons along in the second wire, giving some +such law as “likes prefer to move alike.” With the largest available current and +the most sensitive galvanometer to detect any current, the result was negative. +Large magnets next to wires also produced no observed efects. Finally, Earaday +discovered in 1840 the essential feature that had been missed——that electric effects +exist only when there is something chøngứng. Tf one of a pair of wires has a +changing current, a current is induced in the other, or iŸ a magnet is moued near +am electric circuit, there is a current. We say that currents are /nmduced. This was +--- Trang 196 --- +the induction efect discovered by Faraday. It transformed the rather dull subject +of static fields into a very exciting dynamic subject with an enormous range of +wonderful phenomena. 'This chapter ¡is devoted to a qualitative description of +some of them. As we will see, one can quickly get into fairly complicated situations +that are hard to analyze quantitatively in all their details. But never mỉnd, our +main purpose in this chapter is frst to acquaint you with the phenomena involved. +We will take up the detailed analysis later. +We can easily understand one feature of magnetic induction from what we +already know, although it was not known in Fầraday”s time. It comes from the +0 x B force on a moving charge that is proportional to its velocity in a magnetic +field. Suppose that we have a wire which passes near a magnet, as shown in +Fig. 16-2, and that we connect the ends of the wire to a galvanometer. lf we +move the wire across the end of the magnet the galvanometer pointer moves. +'The magnet produces some vertical magnetic field, and when we push the wire +across the field, the electrons in the wire feel a sdeuøs force—at right angles to +the ñeld and to the motion. The force pushes the electrons along the wire. But +why does this move the galvanometer, which is so far from the force? Because +when the electrons which feel the magnetic force try to move, they push——by +electric repulsion—the electrons a little farther down the wire; they, in turn, repel +the electrons a little farther on, and so on for a long distance. An amazing thing. +lt was so amazing to Gauss and Weber—who first built a galvanometer—that +they tried to see how far the forces in the wire would go. They strung a wire +all the way across their city. Mr. Gauss, at one end, connected the wires to +a battery (batteries were known before generators) and Mr. Weber watched +the galvanometer move. They had a way of signaling long distances—it was +the beginning of the telegraphl Of course, this has nothing directly to do with +induction—it has to do with the way wires carry currents, whether the currents +are pushed by induction or not. +Now suppose in the setup of Eig. 16-2 we leave the wire alone and move +the magnet. We still see an efect on the galvanometer. Às Earaday discovered, +moving the magnet under the wire—one way——has the same efect as moving +the wire over the magnet—the other way. But when the magnet is moved, we no +longer have any 0 x Ö force on the electrons in the wire. 'Phis is the new efect that +Faraday found. 'Today, we might hope to understand it from a relativity argument. +We already understand that the magnetic ñeld of a magnet comes from its +internal currents. So we expect to observe the same effect 1ƒ instead of a magnet +in Fig. 16-2 we use a coil oŸ wire in which there is a current. If we move the wire +past the coil there will be a current through the galvanometer, or also iŸ we move +the coil past the wire. But there is now a more exciting thing: IÝ we change the +magnetic ñeld of the coil no by moving it, but by chøng?ng ?ts current, there 1s +again an efect in the galvanometer. For example, if we have a loop of wire near +a coil, as shown in Eig. 16-3, and if we keep both of them stationary but switch +of the current, there is a pulse of current through the galvanometer. When we +switch the coil on again, the galvanometer kicks in the other direction. +'Whenever the galvanometer in a situation such as the one shown in Fig. 16-2, +or in Eig. 16-3, has a current, there is a net push on the electrons in the wire in one +direction along the wire. There may be pushes in diferent directions at diferent +places, but there is more push in one direction than another. What counfs is +the push integrated around the complete circuit. We call this net integrated +push the clectromotiue ƒorce (abbreviated emf) in the circuit. More precisely, the +emf is defned as the tangential force per unit charge in the wire integrated over +length, once around the complete circuit. EFaraday's complete discovery was that +emfs can be generated in a wire in three diferent ways: by moving the wire, by +moving a magnet near the wire, or by changing a current in a nearby wire. +Let's consider the simple machine of Fig. 16-1 again, only now, instead of +putting a current through the wire to make it turn, let”s turn the loop by an +external force, for example by hand or by a waterwheel. When the coil rotates, +its wires are moving in the magnetic ñeld and we will ñnd an emf in the circuit +of the coïil. The motor becomes a generatOor. +--- Trang 197 --- +< ¬Pff”C +ST <È2 ¡ 3g +— „BC - LÝ Ñ +Z BATTERY +O©O O +| GALVANOMETER +GALVANOMETER +Fig. 16-2. Moving a wire through a magnetic field pro- Fig. 16-3. A coil with current produces a current +duces a current, as shown by the galvanometer. In a second coil if the first coil is moved or If its +current ¡s changed. +The coil of the generator has an induced emf from its motion. he amount of +the emf is given by a simple rule discovered by Faraday. (We will just state the +rule now and wait until later to examine it in detail.) The rule is that when the +magnetic ux that passes through the loop (this ñux is the normal component +oŸ integrated over the area of the loop) is changing with time, the emfis equal to +the rate of change of the Ñux. We will refer to this as “the Ñux rule.” You see that +when the coil of Fig. 16-1 is rotated, the ñux through it changes. At the start some +ñux goes through one way; then when the coil has rotated 1802 the same ñÑux goes +through the other way. IÝ we continuously rotate the coïil the fux is frs positive, +then negative, then positive, and so on. “The rate of change of the ux must +alternate also. 5o there is an alternating emfin the coil. If we connect the Ewo ends +of the coil to outside wires through some sliding contacts——called slip-rings——(Just +so the wires wonˆt get twisted) we have an alternating-current generator. +Or we can also arrange, by means of some sliding contacts, that after every +one-half rotation, the connection between the coil ends and the outside wires 1s +reversed, so that when the emf reverses, so do the connections. 'Phen the pulses of +emf will always push currents in the same direction through the external circuit. +We have what is called a direct-current generator. +The machine of Fig. 16-1 is either a motor or a generator. The reciprocity +between motors and generators is nicely shown by using two identical DG “motors” +of the permanent magnet kind, with their coils connected by two copper wires. +'When the shaft of one is turned mechanically, it becomes a generator and drives +the other as a motor. IÝ the shaft of the second is turned, it becomes the generator +and drives the frst as a motor. So here is an interesting example of a new kind +: . : "- THIN IRON SOUND PRESSURE +of equivalence of nature: motor and generator are equivalent. 'Phe quantitative DISC | +equivalence 1s, in fact, not completely accidental. It is related to the law of +conservation oŸ energy. Ñ SOET \y ` Ñ +Another example of a device that can operate either to generate emf?s or to IRON ` COPPER COIL +respond to emf's is the receiver of a standard telephone—that is, an “earphone.” N ĐN +The original telephone of Bell consisted of two such “earphones” connected by ⁄ +two long wires. The basie principle is shown in Eig. 16-4. Á permanent magnet 2⁄2 +produces a magnetic ñeld in bwo “yokes” of soft iron and in a thin diaphragm TP NET BẠR +that is moved by sound pressure. When the diaphragm moves, it changes the +amount of magnetie feld in the yokes. 'Therefore a coil of wire wound around one Fig. 16-4. A telephone transmitter or +of the yokes will have the ñux through it changed when a sound wave hits the receiver. +--- Trang 198 --- +diaphragm. So there is an emf in the coil. If the ends of the coil are connected +to a circuit, a current which is an electrical representation of the sound is set up. +T the ends of the coïil of Fig. 16-4 are connected by bwo wires to another +identical gadget, varying currents will ow in the second coïl. These currents will +produce a varying magnetic ñeld and will make a varying attraction on the iron +diaphragm. "The diaphragm will wiggle and make sound waves approximately +similar to the ones that moved the original diaphragm. With a few bits of iron +and copper the human voice is transmitted over wiresl +(The modern home §elephone uses a receiver like the one described but uses an +improved invention to get a more powerful transmitter. It is the “carbon-button +microphone,” that uses sound pressure to vary the electric current from a battery.) +16-2 Transformers and inductances +One of the most interesting features of Earaday”s discoveries is not that an +emf exists in a moving coil—which we can understand in terms of the magnetic +force gu x —but that a changing current in one coil makes an emf in a second +coil. And quite surprisingly the amount of emf induced in the second coi] is +given by the same “ñux rule”: that the emf is equal to the rate of change of +the magnetic ñux through the coil. Suppose that we take two coils, each wound +around separate bundles of iron sheets (these help to make stronger magnetic 2 +felds), as shown in EFig. 16-5. Now we connect one of the coils——coil (a)—to 6) E2 LIGHT +an alternating-current generator. The continually changing current produces a B L? BULB +continuously varying magnetic field. 'Phis varying fñeld generates an alternating +emf in the second coil—coil (b). This emf can, for example, produce enough +power to light an electric bulb. +The emf alternates in coil (b) at a frequency which is, oŸ course, the same as X1 Z⁄ +the frequenecy of the original generator. But the current in coil (b) can be larger ¿1 +or smaller than the current in coil (a). The current in coil (b) depends on the ]= +emf induced in it and on the resistance and inductance of the rest of its circuit. |—E] +The emf can be less than that of the generator iÍ, say, there is little Ñux change. —+2 (~) GENEBATOR +Ór the emf in coil (b) can be made mụch larger than that ín the generator by —E +winding coil (b) with many turns, sỉnce in a given magnetic field the ux through — 2 +the coil is then greater. (Or if you prefer to look at it another way, the emf is the Amư +same in each turn, and since the total emf is the sum of the emf”s of the separate +turns, many turns in series produce a large emÍ.) +Such a combination of two coils——usually with an arrangement of iron sheets +to guide the magnetic felds—is called a transƒformer. Tt can “transform” one emf Fig. 16-5. Two coils, wrapped around +(also called a “voltage”) to another. bundles of iron sheets, allow a generator to +'There are also induction efects in a single coil. Eor instance, in the setup in light a bulb with no direct connection. +Eig. 16-5 there is a changing flux not only through coil (b), which lights the bulb, +but also through coïil (a). The varying current in coïil (a) produces a varying +magnetic field inside itself and the fux of this field is continually changing, so +there is a se[f-?nduccd emf in coil (a). There is an emf acting on any current +when ï§ is building up a magnetic fñeld——or, in general, when its field is changing +in any way. The efect is called self-inductance. +'When we gave “the fux rule” that the emf is equal to the rate of change of +the ñux linkage, we didn't specify the direction of the emf. There is a simple +rule, called Lenzˆs rule, for figuring out which way the em goes: the em £r¿es +to oppose any fñux change. That is, the direction of an induced emf is always +such that if a current were to Ñow ín the direction of the emf, it would produce a +ñux of that opposes the change in Ö that produces the emf. Lenz's rule can +be used to fñnd the direction of the emf in the generator of Fig. 16-1, or in the +transformer winding of Eig. 16-3. +In particular, 1f there is a changing current in a single coil (or in any wire) +there is a “back” emfin the circuit. 'Phis emf acts on the charges fowing in +coil (a) of Fig. 16-5 to oppose the change in magnetic feld, and so in the direction +to oppose the change in current. lt tries to keep the current constant; it is +opposite to the current when the current is increasing, and it is in the direction +--- Trang 199 --- +“.mmmaaan,, +“—(t ——] ©) LAMP j„ Fig. 16-6. Circuit connections for an elec- +í qQ ——T—† BATTERY“ tromagnet. The lamp allows the passage of +—nng current when the switch ¡is opened, prevent- +_S< ¡ng the appearance of excessive emf”s. +of the current when it is decreasing. Á current in a selfinductance has “inertia,” +because the inductive efects try to keep the ow constant, just as mechanical +inertia tries to keep the velocity of an object constant. +Any large electromagnet will have a large selfinductance. Suppose that a +battery is connected to the coil of a large electromagnet, as in Eig. 16-6, and that +a strong magnetic feld has been built up. (The current reaches a steady value +determined by the battery voltage and the resistance of the wire in the coil.) But +now suppose that we try to disconnect the battery by opening the switch. IÝ we +really opened the circuit, the current would go to zero rapidly, and in doïng so it +would generate an enormous emf. In most cases this emf would be large enough +to develop an arc across the opening contacts of the switch. The high voltage +that appears might also damage the Insulation of the coil—or you, iÝ you are +the person who opens the switchl For these reasons, electromagnets are usually +connected ïn a circuit like the one shown ín EFig. 16-6. When the switch is opened, +the current does not change rapidly but remains steady, Ñowing instead through +the lamp, being driven by the emf from the self-inductance of the coil. +16-3 Eorces on induced currents +You have probably seen the dramatic demonstration of Lenz's rule made +with the gadget shown in Eig. 16-7. It is an electromagnet, just like coil (a) of +Eig. 16-5. An aluminum ring is placed on the end of the magnet. When the coïl +is connected to an alternating-current generator by closing the switch, the rỉng +flies into the air. The force comes, of course, from the induced currents in the +ring. The fact that the ring fies away shows that the currents in it oppose the +change of the field through it. When the magnet is making a north pole at its +top, the induced current in the ring is making a downward-pointing north pole. +'The ring and the coil are repelled Just like two magnets with like poles opposite. +Tí a thin radial cut is made in the ring the force disappears, showing that it does +indeed come from the currents in the ring. +CONDUCTING RING +IRON CC) ⁄⁄ +CORE c2) / = +`2 | ⁄⁄ T +¡=2 ".. ⁄Z +2 m___> 42 +2D SWITCH — ` +tmmZ 0E EU +PERFECTLY CONDUCTING PLATE +Fig. 16-7. A conducting ring ¡s strongly repelled by Fig. 16-8. An electromagnet near a perfectly con- +an electromagnet with a varying current. ducting plate. +--- Trang 200 --- +Tf, instead of the ring, we place a disc of aluminum or copper across the end +of the electromagnet of Eig. 16-7, it is also repelled; induced currents circulate in +the material of the disc, and again produce a repulsion. +An interesting effect, similar in origin, occurs with a sheet of a perfect sà 2 +conductor. In a “perfect conductor” there is no resistance whatever to the +current. So IŸ currents are generated in it, they can keep going forever. In fact, ÀAN +the sljgh#esứ emf would generate an arbitrarily large current—which really means +that there can be no emfs at all. Any attempt to make a magnetic ñux go Fig. 16-9. A bar magnet is suspended +through such a sheet generates currents that create opposite Ö fields—all with above a superconducting bowl, by the repul- +Infnitesimal emf's, so with no Ñux entering. sion of eddy currents. +Tf we have a sheet of a perfect conductor and put an electromagnet next to ïf, +when we turn on the current in the magnet, currents called eddy currents appear +in the sheet, so that no magnetic ñux enters. The feld lines would look as shown +in Eig. 16-8. 'Phe same thing happens, of course, if we bring a bar magnet near +a perfect conductor. Since the eddy currents are creating opposing fields, the PIVOT +magnets are repelled from the conductor. 'Phis makes it possible to suspend a bar +magnet in air above a sheet of perfect conductor shaped like a dish, as shown in +Fig. 16-9. The magnet is suspended by the repulsion oŸ the induced eddy currents +in the perfect conductor. 'Phere are no perfect conductors at ordinary tempera- +tures, but some materials become perfect conductors at low enough temperatures. COPPER +For instance, below 3.8°K tin conducts perfectly. It is called a superconduetor. PLATE À Ñ +Tf the conduector in Eig. 16-8 is not quite perfect there will be some resistance B Ñ +to fow of the eddy currents. The currents will tend to die out and the magnet — +will slowly settle down. The eddy currents in an imperfect conductor need an => +emf to keep them goïng, and to have an emf the Ñux must keep changing. 'Phe ` =1 +ñux of the magnetic fñeld gradually penetrates the conductor. Í +In a normail conductor, there are not only repulsive forces from eddy currents, Z +but there can also be sidewise forces. For instance, if we move a magnet sideways =E _—] +along a conducting surface the eddy currents produce a force of drag, because the —— +induced currents are opposing the changing of the location of Ñux. Such forces +are proportional to the velocity and are like a kind oŸ viscous force. l +These efects show up nicely in the apparatus shown in EFig. 16-10. A square +sheet of copper is suspended on the end of a rod to make a pendulum. The — +copper swings back and forth between the poles of an electromagnet. When the Le e SWITCH +magnet is turned on, the pendulum motion is suddenly arrested. As the metal +plate enters the gap of the magnet, there is a current induced in the plate which Fig. 16-10. The braking of the pendulum +acts to oppose the change in fux through the plate. If the sheet were a perfect shows the forces due to eddy currents. +conductor, the currents would be so great that they would push the plate out +again—it would bounce back. With a copper plate there is some resistance in the +plate, so the currents at fñrst bring the plate almost to a dead stop as it starts to +enter the field. 'Phen, as the currents die down, the plate slowly settles to rest In +the magnetic feld. +'The nature of the eddy currents in the copper pendulum is shown In FEig. 16-11. <2 v +The strength and geometry of the currents are quite sensitive to the shape of the +plate. TỸ, for instance, the copper plate is replaced by one which has several narrow EDDY +slobs cut in it, as shown in Fig. 16-12, the eddy-current efects are drastically CURRENTS +reduced. “The pendulum swings through the magnetic fñeld with only a small (S)) +retarding force. 'Phe reason is that the currents in each section of the copper +have less fux to drive them, so the efects of the resistance of each loop are +greater. The currents are smaller and the drag is less. The viscous character of +the force is seen even more clearly if a sheet of eopper is placed between the poles +of the magnet of Fig. 16-10 and then released. It doesn't fall; it just sinks slowly +downward. The eddy currents exert a strong resistance to the motion—just like +the viscous drag in honey. +TÍ, instead of dragging a conductor past a magnet, we try to rotate ib in a +magnetic feld, there will be a resistive torque from the same effects. Alternatively, +1f we rotate a magnet—end over end——near a conducting plate or ring, the rỉng +1s dragged around; currents in the ring will create a torque that tends to rotate Fig. 16-11. The eddy currents in the +the ring with the magnet. copper pendulum. +--- Trang 201 --- +2 3 2 3 2 3 +sÉ l||}P 'lx⁄}" -E S\} +6 5 6 5 6 5 +(a) () (c) +2 3 2 3 2 3 +sÉ lll}P 2)" -ÍSy}= +6 5 6 5 6 5 +(4) (e) @) +Fig. 16-12. Eddy-current effects are drastically Fig. 16-13. Making a rotating magnetic field. +reduced by cutting slots in the plate. +A field just like that of a rotating magnet can be made with an arrangement of +coils such as is shown in Fig. 16-13. We take a torus oŸ iron (that is, a rỉng of iron +like a doughnut) and wind six coils on it. TỶ we put a current, as shown in part (a), +through windings (1) and (4), there will be a magnetic field in the direction shown +in the fgure. IÝ we now switch the current to windings (2) and (5), the magnetic +ñeld will be in a new direction, as shown in part (b) of the fñgure. Continuing the +process, we get the sequence of fields shown in the rest of the figure. If the process +is done smoothly, we have a “rotating” magnetic field. We can easily get the [D +required sequence of currents by connecting the coils to a three-phase power line, +which provides just such a sequence of currents. “'Phree-phase power” is made +in a generator using the principle of Eig. 16-1, except that there are #hree loops +fastened together on the same shaft in a symmmetrical way——that is, with an angle - - +of 120° rom one loop to the next. When the coils are rotated as a unit, the emf is |=C— * ¬ +a maximum in one, then in the next, and so on in a regular sequence. There are À%e___ +many practical advantages of three-phase power. One of them is the possibility | | +of making a rotating magnetic fñeld. 'Phe torque produced on a conductor by ÀN œ<=a ÀX +such a rotating fñeld is easily shown by standing a metal ring on an insulating +table just above the torus, as shown in Eig. 16-14. The rotating ñeld causes the Fig. 16-14. The rotatng field of +ring to spin about a vertical axis. The basic elements seen here are quite the Fig. 16-13 can be used to provide torque on +same as those at play in a large commercial three-phase induction motor. a conducting ring. +Another form of induction motor is shown in Eig. 16-15. The arrangement +shown is not suitable for a practical high-efficiency motor but will illustrate the +principle. 'Phe electromagnet M, consisting of a bundle of laminated iron sheets +wound with a solenoidal coil, is powered with alternating current from a generator. +The magnet produces a varying ñux of Ö through the aluminum disc. If we have +Jusb these two components, as shown in part (a) oŸ the figure, we do not yet have +a motor. “There are eddy currents in the disc, but they are symmetric and there is +no torque. (There will be some heating of the disc due to the induced currents.) If +we now cover only one-half of the magnet pole with an aluminum plate, as shown +in part (b) of the fñgure, the dise begins to rotate, and we have a motor. The opera- +tion depends on #oø eddy-current efects. Eirst, the eddy currents in the aluninun +plate oppose the change of ñux throuph it, so the magnetic ñeld above the plate +always lags the field above that half of the pole which is not covered. 'Phis so-called +--- Trang 202 --- +ALUMINIUM +PLATE +| |lllÌ HIÙÙ +1sa<, PIIMMMM 15a& EU +II @) \/IHMMI ø9 +IIIIMHMTT T5 JIHHIMI +Fig. 16-15. A simple example of a shaded-pole induction motor. +“shaded-pole” efect produces a feld which in the “shaded” region varies mụuch like +that in the “unshaded” region except that it is delayed a constant amount in time. +'The whole efect is as If there were a magnet only half as wide which is continually +being moved from the unshaded region toward the shaded one. 'Then the varying +fñelds interact with the eddy currents in the disc to produce the torque on it. +16-4 Electrical technology +When Faraday first made public his remarkable discovery that a changing +magnetic fux produces an emf, he was asked (as anyone is asked when he discovers +a new facb of nature), “What is the use of it?” AII he had found was the oddity +that a tiny current was produced when he moved a wire near a magnet. Of what +possible “use” could that be? His answer was: “What is the use of a newborn +baby?” +Yet think of the tremendous practical applications his discovery has led to. +What we have been describing are not just toys but examples chosen in most +cases to represent the principle of some practical machine. Eor instance, the +rotating ring in the turning fñeld is an induction motor. 'Phere are, Of cOUrse, +some diferences bebween it and a practical induction motor. The ring has a +very small torque; it can be stopped with your hand. For a good motor, things +have to be put together more intimately: there shouldn't be so mụuch “wasted” +magnetic fñeld out in the air. Pirst, the field is concentrated by using iron. We +have not discussed how iron does that, but iron can make the magnetic fñeld tens +of thousands of times stronger than copper coils alone could do. Second, the gaps +between the pieces of iron are made small; to do that, some iron is even built +into the rotating ring. Everything is arranged so as to get the greatest Íorces +and the greatest efficiency——that is, conversion of electrical power to mechanical +power-—until the “ring” can no longer be held still by your hand. +This problem of closing the gaps and making the thing work in the most +practical way is engineering. It requires serious study of design problems, although +there are no new basie prineiples from which the forces are obtained. But there is +a long way to go from the basic principles to a practical and economic design. Yet +1È is just such careful engineering design that has made possible such a tremendous +thing as Boulder Dam and all that goes with ït. +What is Boulder Dam? A huge river is stopped by a concrete wall. But what +a wall it is Shaped with a perfect curve that is very carefully worked out so that +the least possible amount of concrete will hold back a whole river. It thickens at +the bottom in that wonderful shape that the artists like but that the engineers +can appreciate because they know that such thickening is related to the increase +oŸ pressure with the depth of the water. But we are getting away from electricity. +Then the water of the river is diverted into a huge pipe. That”s a nice engineer- +ing accomplishment in itself. The pipe feeds the water into a “waterwheel”—a, +huge turbine—and makes wheels turn. (Another engineering feat.) But why turn +wheels? They are coupled to an exquisitely intricate mess of copper and ïron, all +--- Trang 203 --- +twisted and interwoven. With ©wo parts—one that turns and one that doesn't. +AII a complex intermixture of a few materials, mostly iron and copper but also +some paper and shellac for insulation. Á revolving monster thing. A generator. +Somewhere out of the mess of copper and iron come a few special pieces of cODDer. +The dam, the turbine, the iron, the copper, all put there to make something +special happen to a few bars of copper—an emf. Then the copper bars go a little +way and cirele for several times around another piece of iron in a transformer; +then theïr job is done. +But around that same piece of iron curls another cable of copper which has +no direct connection whatsoever to the bars from the generator; they have just +been inÑuenced because they passed near it—to get their emf. 'Phe transÍformer +converts the power from the relatively low voltages required for the eficient design +of the generator to the very hiph voltages that are best for efficient transmission +of electrical energy over long cables. +And everything must be enormously efficientthere can be no waste, no +loss. Why? 'Phe power for a metropolis is going through. lf a small fraction +were lost——one or two percent——think of the energy left behindl If one percent of +the power were left in the transformer, that energy would need to be taken out +somehow. lÝ it appeared as heat, it would quickly melt the whole thing. There is, +of course, some small inefficiency, but all that is required are a few pumps which +circulate some oil through a radiator to keep the transformer from heating up. +Out of the Boulder Dam come a few dozen rods of copper——long, long, long +rods of copper perhaps the thickness of your wrist that go for hundreds of miles In +all directions. Small rods of copper carrying the power of a giant river. 'Then the +rods are split to make more rods.... then to more transformers .... sometimes +to great generators which recreate the current in another form ... sometimes +to engines turning for big industrial purposes ... to more transformers.... then +more splitting and spreading... until ñnally the river is spread throughout the +whole city—turning motors, making heat, making light, working gadgetry. The +miracle of hot lights tom cold water over 600 miles away——all done with specially +arranged pieces of copper and iron. Large motors for rolling steel, or tiny mofors +for a dentist's drill. Thousands of little wheels, turning in response to the turning +of the big wheel at Boulder Dam. Stop the big wheel, and all the wheels stop; +the lights go out. They really are connected. +Yet there is more. The same phenomena that take the tremendous power of +the river and spread it through the countryside, until a few drops of the river +are running the dentist's drill, come again into the building of extremely fne +instruments.... for the detection ofincredibly small amounts of current... for the +transmission of voices, music, and pictures.... for computers.... for automatic +machines of fantastic precision. +AII this is possible because of carefully desiened arrangements of copper and +Iron——efficiently created magnetic fñields.... blocks of rotating iron six feet in +diameter whirling with clearaneces of 1/16 oŸ an inch... careful proportions of +copper for the optimum efficiency ... strange shapes all serving a purpose, like +the curve of the dam. +TÝ some future archaeologist uncovers Boulder Dam, we may guess that he +would admire the beauty of its curves. But also the explorers from some great +future civilizations will look at the generators and transformers and say: “Notice +that every iron piece has a beautifully efiecient shape. Think of the thought that +has gone into every piece of copper!” +This is the power of engineering and the careful design of our electrical +technology. “There has been created in the generator something which exists +nowhere else in nature. lt is true that there are forces of induction in other +places. Certainly in some places around the sun and stars there are efects of +electromagnetic induction. Perhaps also (though it”s not certain) the magnetic +ñeld of the earth is maintained by an analog of an electric generator that operates +on circulating currents in the interior of the earth. But nowhere have there been +pleces put together with moving parts to generate electrical power as is done in +the generator—with great efficiency and regularity. +--- Trang 204 --- +You may think that designing electric generators is no longer an interesting +subject, that i% is a dead subject because they are all designed. Almost perfect +generators or motors can be taken from a shelf. ven ïf this were true, we can +admire the wonderful accomplishment of a problem solved to near perfection. But +there remain as many unfñnished problems. Even generators and transformers are +returning as problems. It is likely that the whole fñeld of low temperatures and +superconductors will soon be applied to the problem of electric power distribution. +With a radically new factor in the problem, new optimum designs will have to +be created. Power nebworks of the future may have little resemblance to those of +today. +You can see that there is an endless number of applications and problems +that one could take up while studying the laws of induction. The study of the +design of electrical machinery is a life work in itself. We cannot go very far in +that direction, but we should be aware of the fact that when we have discovered +the law of induction, we have suddenly connected our theory to an enormous +practical development. We must, however, leave that subJect to the engineers +and applied scientists who are interested in working out the details of particular +applications. Physics only supplies the base—the basic principles that apply, +no matter what. (WS have not yet complebed the base, because we have yet to +consider in detail the properties of iron and of copper. Physics has something to +say about these as we will see a little laterl) +Modern electrical technology began with Faradayˆs discoveries. The useless +baby developed into a prodigy and changed the face of the earth in ways is +proud father could never have imagined. +--- Trang 205 --- +I7 +TĩĨìo L{á(tŒ-s oŸ ÍrteÏtreff©ort +17-1 The physics of induction +In the last chapter we described many phenomena which show that the efects 17-1 The physics of induction +of induction are quite complicated and interesting. Now we want to discuss 17-2 Exceptions to the “fux rule” +n ¬.- Pamoipies which gayem lo neo » ¬ nhoady ni 17-3 Particle acceleration by an +the emfÍ In a conducting circult as the total accumulated force on the charges h h . +throughout the length of the loop. More specifically, ¡it is the tangential component neo clectric Rold; the +of the force per unit charge, integrated along the wire once around the circuit. 17-4 A paradox +'This quantity is equal, therefore, to the total work done on a single charge that . +travels onee around the cireuit. 17-5 Alternating-current generator +We have also given the “ñux rule,” which says that the emf is equal to the 17-6 Mutual inductance +rate at which the magnetic Ñux through such a conducting circuit is changing. 17-7 SelEFinductance +Let”s see if we can understand why that might be. First, we”ll consider a case in 17-8 Inductance and magnetic energy +which the Ñux changes because a circuit is moved in a steady ñeld. +In Fig. 17-I we show a simple loop of wire whose dimensions can be changed. +The loop has 0wo parts, a ñxed U-shaped part (a) and a movable crossbar (b) +that can slide along the two legs of the . “There is always a complete circuit, but +1ts area is variable. Suppose we now place the loop in a uniform magnetic feld +with the plane of the Ú perpendicular to the feld. According to the rule, when +the crossbar is moved there should be in the loop an emf that is proportional to +the rate of change of the ñux through the loop. This em will cause a current in +the loop. We will assume that there is enough resistance in the wire that the +currents are small. 'Phen we can neglect any magnetic ñeld from this current. +The ñux through the loop is 0E, so the “Ñux rule” would give for the ¬ +cmf—which we write as É— ........ 1. +|. TT”... (6Ì ... +where 0 is the speed of translation of the crossbar. ~ == = = +NÑow we should be able to understand this result from the magnetic 0 x Ö " L——— r——X~—¬. _. +forces on the charges in the moving crossbar. These charges will feel a Íorce, "¬ LINES of B +tangential to the wire, equal to ø per unit charge. Tt 1s constant along the Eig. 17-1. An emf is induced in a loop if +length +0 of the crossbar and zero elsewhere, so the integral is the flux is changed by varying the area of +.) the circuit. +which is the same result we got from the rate of change of the ñÑux. +The argument just given can be extended to any case where there is a fñxed +magnetic ñeld and the wires are moved. Ône can prove, in general, that Íor any +circuit whose parts move in a fñxed magnetic fñeld the emf is the time derivative +of the ñux, regardless of the shape of the circuit. +On the other hand, what happens ïf the loop is stationary and the magnetic +field is changed? We cannot deduce the answer to this question from the same +argument. It was Earadays discovery—fom experiment—that the “fux rule” is +still correct no matter why the flux changes. he force on electric charges is given +in complete generality by #' = q(E~+ 0 x ); there are no new special “forces due +to changing magnetic fields” Any forces on charges at rest in a stationary wire +come from the # term. Earadayˆs observations led to the discovery that electric +and magnetic fñelds are related by a new law: in a region where the magnetic +ñeld is changing with time, electric fñelds are generated. It is this electric feld +--- Trang 206 --- +which drives the electrons around the wire—and so is responsible for the emfin +a sbationary circuit when there is a changing magnetic Ñux. +The general law for the electric fñeld associated with a changing magnetic +ñeld is 2B +VxE=-g: (17.1) +W© will call this Faradays law. It was discovered by Earaday but was frst written +in diÑerential form by Maxwell, as one of his equations. Let's see how this +cequation gives the “fux rule” for circults. +Using 5tokesˆ theorem, this law can be written in integral form as +{Pcds= | (VxE) nan =— [ TT, múa (17.2) +P S S +where, as usual, Ï` is any closed curve and Š is any surface bounded by ït. Here, +remember, Ï` is a rmathematöcal curve fixed in space, and Š is a fixed surface. +Then the time derivative can be taken outside the integral and we have +#s==ã | +t-ds——— | B-nda +lệ dt Js += —ux through ®%). (17.3) +Applying this relation to a curve ` that follows a ƒized circuit of conductor, we +get the “ñux rule” once again. "The integral on the left is the emf, and that on the +right is the negative rate of change of the fux linked by the cireuit. So Eq. (17.1) +applied to a fxed circuit is equivalent to the “fux rule.” +So the “fux rule”—that the emf in a circuit is equal to the rate of change of +the magnetic Ñux through the circuit—applies whether the Ñux changes because +the field changes or because the circuit moves (or both). The two possibilities— +“eircuit moves” or “fñeld changes”—are not distinguished in the statement of the +rule. Yet in our explanation of the rule we have used two completely distinct +laws for the two cases— x Ö for “circuit moves” and W x E = —ØB/ôt for +“feld changes.” +'W© know of no other place in physics where such a simple and accurate general +principle requires for its real understanding an analysis in terms of to đijfferent +phenomena. sually such a beautiful generalization is found to stem from a +single deep underlying principle. Nevertheless, in this case there does not appear +to be any such profound implication. We have to understand the “rule” as the +combined effects of two quite separate phenomena. +We must look at the “ñux rule” in the following way. In general, the force +per unit charge is F'⁄q= E + x Ö. In moving wires there is the force from the +second term. Also, there is an #-field if there is somewhere a changing magnetic +fñeld. They are independent efects, but the emf around the loop of wire is always +cqual to the rate of change of magnetic fux through it. +17-2 Exceptions to the “fux rule” +We will now give some examples, due in part to Faraday, which show the +importance of keeping clearly in mind the distinction between the two effects +responsible for induced emf?s. Qur examples involve situations to which the “ñux +rule” cannot be applied——either because there is no wire at all or because the +pa‡h taken by induced currents moves about within an extended volume of a +conductor. +We begin by making an Important point: The part of the emf that comes +from the #-field does not depend on the existence of a physical wire (as does the +®x B part). The E-field can exist in free space, and its line integral around any +Imaginary line fñxed in space is the rate of change of the fux of Ö through that +line. (Note that this is quite unlike the E-field produced by static charges, for in +that case the line integral of E around a closed loop is always zero.) +--- Trang 207 --- +| BAR : +MAGNET +'tth<” +<<” b COPPER DISC — Fig. 17-2. When the disc rotates there is +—y ⁄ T7] an emf from v x B, but with no change In +GALVANOMETER the linked flux. +Now we will describe a situation in which the ñux through a circuit does not +change, but there is nevertheless an emf. Figure 17-2 shows a conducting disc +which can be rotated on a fxed axis in the presence of a magnetic feld. One +contact is made to the shaft and another rubs on the outer periphery of the disc. +A circuit is completed through a galvanometer. As the disc rotates, the “cireuit,” +in the sense of the place in space where the currents are, is always the same. But +the part of the “circuit” in the disc is in material which is moving. Although the +ñux through the “circuit” is constant, there is still an emf, as can be observed by +the deflection of the galvanometer. Clearly, here is a case where the x Ö force In +the moving disc gives rise to an emf which cannot be equated to a change of ñux. +NÑow we consider, as an opposite example, a somewhat unusual situation in COPPER PLATES +which the fux through a “cireuit” (again in the sense oŸ the place where the : ) +current is) changes but where there is øo emf. Imagine two metal plates with Z —À II" ¬ +slightly curved edges, as shown in Fig. 17-3, placed in a uniform magnetic fñeld Ị « | NGỘ ' +perpendicular to their surfaces. Each plate is connected to one of the terminals lu \ TRE +of a galvanometer, as shown. The plates make contact at one point ?, so there is /ZZ1®.. h s7 _ TÀN +a complete circuit. IÝ the plates are now rocked through a small angle, the point ( NI \ cŒ +of contact will move to ?”. IÝ we imagine the “eircuit” to be eompleted through " ¡ TÌP\ _- +the plates on the dotted line shown in the figure, the magnetic ñux through this +circuit changes by a large amount as the plates are rocked back and forth. Yet +the rocking can be done with small motions, so that 0 x is very smaill and +there is practically no emf. The “fux rule” does not work in this case. It must be nn +applied to circuits in which the ?mafer7al of the circuit remains the same. When +the material of the circuit is changing, we must return to the basic laws. The +correct physics is always given by the two basic laws GALVANOMETER +Pˆ.=q(E+oxB), Fig. 17-3. When the plates are rocked +In a uniform magnetic field, there can be a +ÿxE- _ØB large change In the flux linkage without the +Ôt` generation of an emf. +17-3 Particle acceleration by an induced electric ñeld; the betatron +We have said that the electromotive force generated by a changing magnetic +field can exist even without conduectors; that is, there can be magnetic induction +without wires. We may still imagine an electromotive force around an arbitrary +mathematical curve in space. It is defned as the tangential component of E +integrated around the curve. Faraday”s law says that this line integral is equal to +mỉnus the rate oŸ change of the magnetic ux through the closed curve, Bq. (17.3). +As an example of the efect of such an induced electric field, we want now +to consider the motion of an electron in a changing magnetic fñeld. We imagine +a magnetic fñeld which, everywhere on a plane, points in a vertical direction, +as shown in Eig. 17-4. 'Phe magnetic fñeld is produced by an electromagnet, +but we will not worry about the details. Eor our example we will imagine that +the magnetic fñeld is symmetric about some axis, i.e., that the strength of the +magnetic fñeld will depend only on the distance from the axis. The magnetic ñeld +is also varying with time. We now imagine an electron that is moving in this ñeld +--- Trang 208 --- +lQ *„E +B ° ° +lQ ° cS lQ +° ®LINES OF B +SIDE VIEW 'TÓP VIEW +Fig. 17-4. An electron accelerating in an axially symmetric, +Increasing magnetic field. +on a path that is a circle of constant radius with its center at the axis of the field. +(We will see later how this motion can be arranged.) Because of the changing +magnetic fñeld, there will be an electric ñeld # tangential to the electron”s orbit +which will drive it around the circle. Because of the symmetry, this electric fñeld +will have the same value everywhere on the circle. Iƒ the electron's orbit has the +radius z, the line integral of # around the orbit is equal to minus the rate of +change of the magnetic ñux through the circle. 'Phe line integral of # is just its +magnitude times the circumference of the circle, 2rr. The magnetic ux must, in +general, be obtained from an integral. For the moment, we let ạy represent the +average magnetic fñeld in the interior of the circle; then the Ñux is this average +magnetic field times the area of the circle. We will have +2mr = ai - 712). +Since we are assuming 7 is constant, # is proportional to the time derivative +of the average field: +E=_._—_-. 17.4 +2 di ) +The electron will feel the electric force g# and will be accelerated by ít. Re- +membering that the relativistically correct equation of motion is that the rate of +change of the momentum is proportional to the force, we have +E=_—. 17.5 +gE= (17.5) +For the circular orbit we have assumed, the electric force on the electron is +always in the direction of its motion, so its total momentum will be increasing at +the rate given by Eq. (17.5). Combining Bqs. (17.5) and (17.4), we may relate +the rate of change of momentum to the change of the average magnetic field: +d rẻdB +_¬.. ` —=-~ (17.6) +dt 2_ di +Integrating with respect to £, we find for the electron��s momentum +Ð=po+ ạ ABav, (17.7) +where Øøo is the momentum with which the electrons start out, and A„y, is +the subsequent change in ạy. The operation of a Öefafron—a machine for +accelerating electrons to high energies——is based on this idea. +To see how the betatron operates in detail, we must now examine how the +electron can be constrained to move on a circle. We have discussed in Chapter l1 +of Vol. I the principle involved. IÝ we arrange that there is a magnetic field +--- Trang 209 --- +at the orbit of the electron, there will be a transverse force gu x Ö which, for a +suitably chosen #Ö, can cause the electron to keep moving on its assumed orbit. +In the betatron this transverse force causes the electron to move in a circular +orbit of constant radius. We can fñnd out what the magnetic field at the orbit +must be by using again the relativistice equation of motion, but this time, for the +transverse component of the force. In the betatron (see Eig. 17-4), is at right +angles to 0, so the transverse force 1s gu. Thus the force is equal to the rate of +change of the transverse component ø¿ of the momentum: +quB = an (17.8) +When a particle is moving in a c/rcle, the rate of change of is transverse +momentum ¡is equal to the magnitude of the total momentum tỉimes œ, the +angular velocity of rotation (following the arguments of Chapter II, Vol. l): +em =0, (17.9) +where, since the motion is circular, +U Si: (17.10) +Setting the magnetic force equal to the transverse acceleration, we have +qUĐgrbit =P "= (17.11) +where „rp¡( is the field at the radius r. +As the betatron operates, the momentum of the electron grows in proportion +to ạv, according to Ba. (17.7), and iŸ the electron is to continue to move ïn is +proper circle, Eq. (17.11) must continue to hold as the momentum of the electron +Increases. The value of E¿„u¡¿ musÈ increase in proportion to the momentum ø. +Comparing Eq. (17.11) with Bq. (17.7), which determines p, we see that the +following relation must hold between ạy, the average magnetic field #wszde the +orbit at the radius r, and the magnetic fñeld ‹p¡¿ at the orbit: +ABxv = 2AHw. (17.12) +"The correct operation oŸ a betatron requires that the average magnetic field inside +the orbit increases at twice the rate of the magnetic ñeld at the orbit itself. In +these circumstances, as the energy of the particle is increased by the induced +electric fñeld the magnetic fñeld at the orbit increases at just the rate required to +keep the particle moving in a circle. +'The betatron is used to accelerate electrons to energies of tens of millions of +volts, or even to hundreds of millions of volts. However, it becomes impractical for +the acceleration of electrons to energies much higher than a few hundred million +volts for several reasons. One of them is the practical difculty of attaining the +required high average value for the magnetic field inside the orbit. Another is that +Eq. (17.6) is no longer correct at very hiph energies because it does not include +the loss of energy from the particle due to its radiation of electromagnetic energy +(the so-called synchrotron radiation discussed in Chapter 36, Vol. I). For these +reasons, the acceleration of electrons to the highest energies—to many bïllions of +electron volts—is accomplished by means of a diferent kind of machine, called a +sụnchrotron. +17-4 Á paradox +W©e would now like to describe for you an apparent paradox. A paradox is a +situation which gives one answer when analyzed one way, and a different answer +when analyzed another way, so that we are left in somewhat of a quandary as +to actually what should happen. Of course, in physics there are never any real +paradoxes because there is only one correct answer; at least we believe that nature +--- Trang 210 --- +will act in only one way (and that is the r/ght 0a, naturally). So in physics a +paradox is only a confusion in our own understanding. Here is our paradox. +TImagine that we construct a device like that shown in Fig. 17-5. There is a +thín, circular plastic disc supported on a concentric shaft with excellent bearings, +so that it is quite free to rotate. Ôn the dise is a coil of wire in the form o a +short solenoid concentric with the axis of rotation. This solenoid carries a steady 44 +current 7 provided by a small battery, also mounted on the disc. Near the edge +of the disc and spaced uniformly around its cireumference are a number of small NETAL S2HERES COIL OF WIRE +metal spheres insulated from each other and from the solenoid by the plastic +material of the disc. Each of these small conduecting spheres is charged with the é z3 E è +same electrostatic charge Q. Everything is quite stationary, and the disc is at © = = ¬ ® +rest. Suppose now that by some accident—or by prearrangerment—the current in © => 29 P +the solenoid is interrupted, without, however, any intervention from the outside. À @ `. ATTEXY @ j +So long as the current continued, there was a magnetic Ñux through the solenoid \ © _” © ⁄ +more or less parallel to the axis of the disc. When the current is interrupted, this nh ® F3 ® # +fñux must go to zero. There will, therefore, be an electric fñeld induced which ` T————x T2 +will circulate around in cireles centered at the axis. The charged spheres on" s¡as+tc pisc : +the perimeter of the disc will all experience an electric field tangential to the +perimeter of the disc. 'This electric force is in the same sense for all the charges ) +and so will result in a net torque on the disc. trom these arguments We would Fig. 17-5. Will the disc rotate if the cur- +expect that as the current in the solenoid disappears, the disc would begin to rent Í is stopped? +rotate. If we knew the moment of inertia of the dise, the current in the solenoid, +and the charges on the small spheres, we could compute the resulting angular +velocity. +But we could also make a diferent argument. sing the principle of the +conservation of angular momentum, we could say that the angular momentum of +the disc with all its equipment is initially zero, and so the angular momentum of +the assermbly should remain zero. 'Phere should be no rotation when the current +1s stopped. Which argument is correct? WIlI the dise rotate or will ít not? We +will leave this question for you to think about. +'We should warn you that the correct answer does not depend on any nonessen- +tial feature, such as the asymmetrie position of a battery, for example. In fact, +you can imagine an ideal situation such as the following: The solenoid is made of +superconducting wire through which there is a current. After the disc has been +carefully placed at rest, the temperature of the solenoid is allowed to rise slowly +'When the temperature of the wire reaches the transition temperature between +superconduectivity and normal conductivity, the current in the solenoid will be +brought to zero by the resistance of the wire. The ñux will, as before, fall to zero, +and there will be an electric fñeld around the axis. We should also warn you that +the solution is not easy, nor is i% a trick. When you fñgure it out, you will have +discovered an important principle of electromagnetism. +17-5 Alternating-current generator +In the remainder of this chapter we apply the principles of Section 17-1 to +analyze a number of the phenomena. discussed in Chapter 16. We first look ___ +in more detail at the alternating-current generator. Such a generator consists —ị +basically of a coïil of wire rotating in a uniform magnetic field. 'Phe same result ——— : LOAD +can also be achieved by a fñxed coil in a magnetic fñeld whose direction rotates —— +in the manner described in the last chapter. We will consider only theformer ———N +case. Suppose we have a circular coil of wire which can be turned on an axis lỆ +along one of its diameters. Let this coil be located in a uniform magnetic fñeld +perpendicular to the axis of rotation, as in Eig. 17-6. We also imagine that the +two ends of the coil are brought to external connections through some kind of Fig. 17-6. A coil of wire rotating in a +sliding contacts. uniform magnetic field——the basic idea of +Due to the rotation of the coil, the magnetic ñux through it will be changing. the AC generator. +'The circuit of the coil will therefore have an emf ïn it. Let 5 be the area of the +coil and Ø the angle between the magnetic ñeld and the normal to the plane of +--- Trang 211 --- +the coil.* The ñux through the coïl is then +BS cos0. (17.13) +T the coil is rotating at the uniform angular velocity œ, Ø varies with time +as Ø = wÝ. +lach turn of the coil will have an emf equal to the rate oŸ change of this fux. +Tf the coil has ) turns of wire the total emf will be / times larger, so +c=_-N aiLÐScoswt) = NBSusinut. (17.14) +Tf we bring the wires from the generator to a poinÈ some distance from the +rotating coïl, where the magnetic feld is zero, or at least is not varying with time, +the curl of in this region will be zero and we can defne an electric potential. +In fact, if there is no current being drawn from the generator, the potential +diference W between the two wires will be equal to the emf in the rotating coil. +'That is, +V = NBSusin u = Vg sin œ‡. +The potential diference between the wires varies as sinưý. Such a varying +potential diference is called an alternating voltage. +Since there is an electric fñeld between the wires, they must be electrically +charged. It is clear that the emf of the generator has pushed some excess charges +out to the wire until the electric ñeld from them is strong enough to exactÌy +counterbalance the induction force. Seen from outside the generator, the bwo +wires appear as though they had been electrostatically charged to the potential +diference V, and as though the charge was being changed with time to give +an alternating potential diference. “There is also another diference from an —”= +electrostatic situation. If we connect the generator to an external circuit that +permits passage oŸ a current, we find that the emf does not permit the wires tO AC. Ð +be discharged but continues to provide charge to the wires as current is drawn Generator +from them, attempting to keep the wires always at the same potential diferenee. +Tí, in fact, the generator is connected in a circuit whose total resistance is #, the =—v +current through the circuit wïll be proportional to the emf of the generator and I=== _ sinœt£ +Inversely proportional to #. Since the emf has a sinusoidal time variation, so +also does the current. There is an alternating current Fig. 17-7. A circuit with an AC generator +Ề T and a resistance. +IT=—=_- sinut. +'The schematic diagram oŸ such a circuit is shown in Fig. 17-7. +W© can also see that the emf determines how much energy is supplied by the +generator. Each charge in the wire is receiving energy at the rate f'-ø, where #' +is the force on the charge and œ is its velocity. Now let the number of moving +charges per unit length of the wire be ø%; then the power being delivered into any +element đs of the wire is +F'-n da. +FOr a wire, 0 is always along đs, so we can rewrite the DOWeT as +nuŸÈ' - ds. +The total power being delivered to the complete circuit is the integral of this +expression around the complete loop: +Power = lo - đ8. (17.15) +Now remember that gwu is the current ƒ, and that the emf ¡is defned as the +integral of FJ⁄q around the circuit. We get the result +Power from a generator = €Ï. (17.16) +* Now that we are using the letter A for the vector potential, we prefer to let Š stand for a +surface area. +--- Trang 212 --- +When there is a current in the coil of the generator, there will also be +mnechanical forces on it. In fact, we know that the torque on the coïl is proportional +to its magnetic moment, to the magnetic feld strength , and to the sine of the +angle between. 'Phe magnetic moment is the current in the coil times is area. +'Therefore the torque 1s +T= N]ISBsin0. (17.17) +'The rate at which mechanical work must be done to keep the coil rotating is the +angular velocity œ times the torque: +n. =œwT =œN]ISBsin0. (17.18) +Comparing this equation with Eq. (17.14), we see that the rate oŸ mechanical +work required to rotate the coil against the magnetic forces 1s just equal to €Ï, +the rate at which electrical energy is delivered by the emf of the generator. All +of the mechanical energy used up ¡in the generator appears as electrical energy In +the circuit. +As another example of the currents and forces due to an induced emf, let”s +analyze what happens in the setup described in Section 17-1, and shown in +Fig. 17-1. Thhere are two parallel wires and a sliding crossbar located in a uniform +magnetic field perpendicular to the plane of the parallel wires. NÑow let's assume +that the “bottom” of the U (the left side in the figure) is made of wires of high +resistance, while the two side wires are made of a good conductor like copper—— +then we don”t need to worry about the change oŸ the circuit resistance as the +crossbar is moved. As before, the emf in the circuit is +€ = 0uBu. (17.19) +'The current in the circuit is proportional to this emf and inversely proportional +to the resistance of the circuit: +ŠC 0Bu +T]= R n” (17.20) +Because of this current there will be a magnetic force on the crossbar that is +proportional to i§s length, 6o the current ín it, and to the magnetic feld, such +'= Blu. (17.21) +Taking 7 rom Eq. (17.20), we have for the force +2.2 +m= —= 0. (17.22) +W© see that the force is proportional to the velocity of the crossbar. The direction +of the force, as you can easily see, is opposite to 10s velocity. Such a “velocity- +proportional” force, which is like the force of viscosity, is found whenever induced +currents are produced by moving conduectors in a magnetic fñeld. 'The examples of +cddy currents we gave in the last chapter also produced forces on the conductors +proportional to the velocity of the conductor, even though such situations, in +general, give a complicated distribution of currents which ¡is dificult to analyze. +Tt is often convenient in the design of mechanical systems to have damping +forces which are proportional to the velocity. Eddy-current forces provide one of +the most convenient ways of getting such a velocity-dependent force. An example +of the application of such a force is found in the conventional domestic wattmeter. +In the wattmeter there is a thin aluminum disc that rotates between the poles +of a permanent magnet. This disc is driven by a small electric motor whose +torque is proportional to the power being consumed in the electrical cireuit of the +house. Because of the eddy-current forces in the disc, there is a resistive Íorce +proportional to the velocity. In equilibrium, the velocity is therefore proportional +to the rate oŸ consumption of electrical energy. By means of a counter attached +to the rotating disc, a record is kept of the number of revolutions it makes. +--- Trang 213 --- +'This count is an indication of the total energy consumption, ï.e., the number of +watthours used. +We may also poin out that ad. (17.22) shows that the force from induced +currents—that is, any eddy-current force—is inversely proportional to the resis- +tance. The force will be larger, the better the conductivity of the material. The +reason, of course, is that an emf produces more current if the resistance is low, +and the stronger currents represent greater mechanical forces. +W© can also see from our formulas how mechanical energy is converted into +electrical energy. As before, the electrical energy supplied to the resistance of +the circuit is the product €ïÏ. "The rate at which work ¡is done in moving the +conducting crossbar is the force on the bar times its velocity. Using Eq. (17.21) +for the force, the rate of doïng work is +dW — u2P2u? +cEAwmw—m +We see that thís is indeed equal to the product €TÏ we would get rom Eqs. (17.19) +and (17.20). Again the mechanical work appears as electrical energy. +17-6 Mutual inductance “nỉ +W©e now want to consider a situation in which there are fxed coils of wire but \ | h +changing magnetic ñelds. When we described the produection of magnetic fields == COIL 1 +by currents, we considered only the case of steady currents. But so long as the ——R=”; +currents are changed slowly, the magnetic ñeld will at each instant be nearly the =^) +same as the magnetic fñeld of a steady current. We will assume in the discussion «ì— % +OŸ this section that the currents are always varying sufficiently slowly that this is COIL 2 =2 +true. `——2 ) +In Eig. 17-8 is shown an arrangement of two coils which demonstrates the `¬—” +basic efects responsible for the operation of a transformer. Coil 1 consists of a Tp \—< +conducting wire wound in the form of a long solenoid. Around this coil—and ` —— +insulated from it——is wound coil 2, consisting of a few turns of wire. lf now a +current is passed through coïil 1, we know that a magnetic fñeld will appear inside ị \ +it. This magnetic fñeld also passes through coïil 2. As the current in coil 1 is +varied, the magnetic Ñux will also vary, and there will be an induced emf in coïl 2. +W©e will now calculate this induced emf. Fig. 17-8. A current in coil 1 produces a +W© have seen in Section 13-5 that the magnetic field inside a long solenoid is magnetic fiel d through coil 2. +uniform and has the magnitude +B= ”nn (17.23) +where /) ¡is the number of turns in coil 1, 7¡ is the current throuph ït, and Í is its +length. Let°s say that the cross-sectional area of coil 1 is Š; then the fux of is +1s magnitude times Š. If coil 2 has N2 turns, this ñux links the coil NÑs times. +'Therefore the emfin coïl 2 is given by +Ca=—N›S _ (17.24) +The only quantity in q. (17.23) which varies with time is 7¡. The emf is therefore +given by Si +ÁMIN¿ 1 +Ca= “que di” (17.25) +W©e see that the emf in coil 2 is proportional to the rate of change of the +current in coil 1. The constant of proportionality, which is basically a geometric +factor of the two coils, is called the mmu#ual inductønce, and 1s usually designated +9i. Equation (17.25) is then written +Ca = at pc (17.26) +--- Trang 214 --- +Suppose now that we were to pass a current through coïil 2 and ask about +the emf in coïil 1. We would compute the magnetic field, which is everywhere +proportional to the current ĩ¿. The fux linkage throupgh coil 1 would depend on +the geometry, but would be proportional to the current ñ¿. "The emfin coïl 1 +would, therefore, again be proportional to đĨa/di: We can write +€¡ =lúa —“. 17.27 +1 12 Tứ ( ) +'The computation of 9¿ would be more difficult than the computation we have +Just done for 9Jt¿¡. We will not carry through that computation now, because we +will show later in this chapter that 9Ÿ; is necessarily equal to 9a. +Since for ømw coil its ñeld is proportional to its current, the same kind of result +would be obtained for any ©wo coils of wire. The equations (17.26) and (17.27) +would have the same form; only the constants 9†¿¡ and 9J:¿ would be diferent. +Theïr values would depend on the shapes of the coils and their relative positions. +Fig. 17-9. Any two colls have a mutual +Inductance #† proportional to the integral 1 +of l¬i * ds/na. +uppose that we wish to ñnd the mutual inductance between any two arbitrary +coils—for example, those shown in Fig. 17-9. We know that the general expression +for the emf in coil 1 can be written as +€1 — -xJ B.- m da, +where Ö is the magnetic fñeld and the integral is to be taken over a surface +bounded by circuit 1. We have seen in Section 14-1 that such a surface integral +of B can be related to a line integral of the vector potential. In particular, +J B-ndn = ÿ A-dsì, +where A represents the vector potential and đs is an element of circuit 1. The +line integral is to be taken around circuit 1. The emf in coil 1 can therefore be +written as : +Cị= -xẾ A -ds). (17.28) +đt Jay +Now let's assume that the vector potential at circuit 1 comes from currents +in circuit 2. 'Then i% can be written as a line integral around circuit 2: +1 Tạ ds +A= mÍ — (17.29) +47co€ (2) T12 +where lạ is the current in circuit 2, and rs is the distance from the element of +the circuit đs¿ to the point on circuit 1 at which we are evaluating the vector +potential. (See Fig. 17-9.) Combining Eqs. (17.28) and (17.29), we can express +the emf in cireuit 1 as a double line integral: +1 d Tạ ds +tì== sa 1 D52 a +47cogđ dt (1) Ở(2) T12 +In this equation the integrals are all taken with respect to stationary circuits. The +only variable quantity is the current ?¿, which does not depend on the variables +--- Trang 215 --- +of integration. We may therefore take it out of the integrals. The emf can then +be written as AI +E¡=9ta ^ +1 12 Tp) +where the coeflicient 9JÏs is +1 đ$s - ds +9a — "=.aÍ 1 _ (17.30) +47co€ () (2) T12 +W© see from this integral that ›¿ depends only on the circuit geometry. lt +depends on a kind of average separation of the two circuits, with the average +weighted most for parallel segments of the two coils. Qur equation can be used +for calculating the mutual inductance of any two circuits of arbitrary shape. Also, +1t shows that the integral for 9ffqa is identical to the integral for 9f¿¡. We have +therefore shown that the two coeficients are identical. Eor a system with only +two coils, the coefficients 9a and 9s are often represented by the symbol 9 +without subscripts, called simply the rmu‡uadl ínductance: +3i = 3Jta¡ = 2). +17-7 SelfFinductance +In discussing the induced electromotive forces in the two coils of Figs. 17-8 +or 17-9, we have considered only the case in which there was a current in one +coil or the other. lÝ there are currents in the two coils simultaneously, the +magnetic Ñux linking either coil will be the sum of the ©wo fuxes which would +exist separately, because the law of superposition applies for magnetic fields. The +emf in either coil will therefore be proportional not only to the change of the +current in the other coil, but also to the change in the current of the coil itself. +Thus the total emf in coil 2 should be writtenX +đh d1› +&a = lai —— +9a¿ ——. 17.31 +2 21 + b2 mn ( ) +Similarly, the emf in coil 1 will depend not only on the changing current in coil 2, +but also on the changing current in itself: +đla dđh +&i =ĐØfạ —“ +Øi —-. 17.32 +1 12 vụ + HH1 di ( ) +'The coefficients †a¿ and 9†qi are always negative numbers. Ït is usual to write +9lìi —T—1, 9laa — —fa, (17.33) +where £ and 6s are called the self-?nductances of the two coils. +The selfinduced emf will, of course, exist even if we have only one coil. Any +coil by itself will have a self-inductance £. Thhe emf will be proportional to the +rate of change of the current in it. Eor a single coil, it is usual to adopt the +convention that the emf and the current are considered positive if they are in the +same direction. With this convention, we may write for the emf of a single coïil +C=_-£Ê—. 17.34 +The negative sign indicates that the emf opposes the change in current——it is +often called a “back emf” +Since any coil has a self-inductance which opposes the change in current, +the current in the coil has a kind of inertia. In fact, if we wish to change the +current in a coil we must overcome this inertia by connecting the coil to some +external voltage source such as a battery or a generator, as shown in the schematie +* 'The sign of %†ia and s1 in Eqs. (17.31) and (17.32) depends on the arbitrary choices +for the sense of a positive current in the two coils. +--- Trang 216 --- +diagram of Eig. 17-10(a). In such a circuit, the current 7 depends on the voltage / +according to the relation _——> +V=.%®—. 17.35 +n (17.35) = +This equation has the same form as Newton's law of motion for a particle ¬ +in one dimension. We can therefore study it by the principle that “the same = +equations have the same solutions.” Thus, if we make the externally applied +voltage ? correspond to an externally applied force #', and the current Ï in a coïl @) +correspond to the velocity 0 of a particle, the inductance £ of the coi eorresponds +to the mass mw of the particle.* See Fig. 17-10(b). We can make the following +table of corresponding quantities. +Particle Codl +Œ' (force) Ý (potential diference) ————>~x m +0 (velocity) T (current) ++ (displacement) q (charge) (b) +F=m= W=£— | ¬ +dt dt Fig. 17-10. (a) A circuit with a voltage +mù (momentum) £l Source and an inductance. (b) An analogous +3m? (kinetic energy) 3,12 (magnetic energy) mechanical system. +17-8 Inductance and magnetic energy +Continuing with the analogy of the preceding section, we would expect that +corresponding to the mechanical momentum ø = m0, whose rate of change is the +applied force, there should be an analogous quantity equal to £©†, whose rate of +change is V. We have no right, of course, to say that 6T is the real momentum of +the circuit; in fact, it isn't. The whole circuit may be standing still and have no +mmomentum. lt is only that £©T is analogous to the momenbum ?w0 in the sense of +satisfying corresponding equations. In the same way, to the kinetic energy sinuŸ, +there corresponds an analogous quantity Hơi 2, But there we have a surprise. +This 3£1 2 is really the energy in the electrical case also. 'This is because the rate +of doïing work on the inductance is Vĩ, and in the mechanical system it is Pu, +the corresponding quantity. Therefore, in the case of the energy, the quantities +not only correspond mathematically, but also have the same physical meaning as +We may see this in more detail as follows. As we found in Bq. (17.16), the +rate of electrical work by induced forces is the product of the electromotive Íorce +and the current: đW +—— = CỈ. +Replacing Ê by its expression in terms of the current from Eq. (17.34), we have +———=_-&Ïl—. 17.36 +đt dt ( ) +Integrating this equation, we find that the energy required ữom an external +Source to overcome the emf in the selfinductance while building up the current† +(which must equal the energy stored, Ù) is +—W =U=$£1. (17.37) +'Therefore the energy stored in an inductanece is 321 2, +Applying the same arguments to a pair of coils such as those in Eigs. 17-8 +or 17-9, we can show that the total electrical energy of the system is given by +U = 5211? + 56213 + 9H lạ. (17.38) +* 'This is, incidentally, not the onlÏ way a correspondence can be set up between mechanical +and electrical quantities. +† We are neglecting any energy loss to heat from the current in the resistance of the coil. +Such losses require additional energy from the source but do not change the energy which goes +into the inductance. +--- Trang 217 --- +For, starting with 7ƒ = 0n both coils, we could first turn on the current ï¡ in +coil 1, with 1; =0. The work done is Just s2 1Ÿ. But now, on turning up lạ, we +not only do the work 32213 against the emf in cireuit 2, but also an additional +amount 9†1:ï¿, which is the integral of the emf [WẦ(d1z/đ#)] im circuit 1 times +the now consføn£ current Ïị in that circuit. +Suppose we now wish to fñnd the force between any two coils carrying the +currents í¡ and 1¿. We might at first expect that we could use the principle +of virtual work, by taking the change in the energy of Eq. (17.38). We must +remember, of course, that as we change the relative positions of the coils the +only quantity which varies is the mutual inductance 9. We might then write +the equation of virtual work as +—FE.Az = AU = I1I¿A9fẲ (wrong). +But this equation is wrong because, as we have seen earlier, it includes only the +change in the energy of the two coils and not the change in the energy of the +sources which are maintaining the currents Ï and ĩ¿ at their constant values. We +can now understand that these sources must supply energy against the induced +emfs in the coils as they are moved. lf we wish to apply the principle of virtual +work correctly, we must also include these energies. Âs we have seen, however, we +may take a short cut and use the prineiple of virtual work by remembering that +the total energy 1s the negative of what we have called mecn; the “mechanical +energy.” We can therefore write for the force +— FˆAz = AUmeen = —ÂU. (17.39) +The force between two coils is then given by +FPAxz=T 1t 2 AØ. +Equation (17.38) for the energy of a system oŸ two coils can be used to show +that an interesting inequality exists between mutual inductance † and the self- +inductances 6 and 6s of the two coils. It is clear that the energy of two coils +must be positive. If we begin with zero currents in the coils and increase these +currents to some values, we have been adding energy to the system. Tf not, the +currents would spontaneously increase with release of energy to the rest of the +world—an unlikely thing to happenl Ñow our energy equation, Eq. (17.38), can +cqually well be written in the following form: +1 Ø% NỔ 1 9% +U=;c#i|h+—I s|®s— — JH. 17.40 +2 ín E¡ ›) to(£ HE ( ) +That is just an algebraic transformation. 'This quantity must always be positive +for any values of ƒị and ïl¿. In particular, it must be positive if Tạ should happen +to have the special value +lạ= “8y h. (17.41) +But with this current for ï¿, the ñrst term in Eq. (17.40) is zero. IÝ the energy is +to be positive, the last term in (17.40) must be greater than zero. We have the +requirement that +#1822 > 9WẺ. +We have thus proved the general result that the magnitude of the mutual induc- +tance 9t of any two coils is necessarily less than or equal to the geometric mean +of the two sel-inductances. (9W itself may be positive or negative, depending on +the sign conventions for the currents 7 and 7a.) +|Jtl < V214a. (17.42) +The relation between 9# and the self-inductances is usually written as +9t —= kw 19a. (17.43) +--- Trang 218 --- +The constant & is called the coefficient of coupling. If most of the Ñux from one +coil links the other coil, the coeficient of coupling is near one; we say the coils +are “tightly coupled.” If the coils are far apart or otherwise arranged so that there +1s very little mutual ñux linkage, the coeficient of coupling is near zero and the +mutual inductanee is very small. +Eor calculating the mutual inductance of two coils, we have given in Eq. (17.30) +a formula which is a double line integral around the ©wo circuits. We might think +that the same formula could be used to get the self-inductance of a single coil by +carrying out both line integrals around the same coil. 'Phis, however, will no +work, because the denominator ra of the integrand will go to zero when the bwo +line elements đs and đsạ are at the same point on the coil. The self-inductance +obtained from this formula is infnite. "The reason is that this formula is an +approximation that is valid only when the cross sections of the wires of the +two cireuits are small compared with the distance from one circuit to the other. +Clearly, this approximation doesn't hold for a single coil. It is, in fact, true that +the inductance of a single coil tends logarithmically to inÑnity as the diameter of +1ts wire is made smaller and smaller. +W© must, then, look for a diferent way of calculating the self-inductance oŸ a +single coil. It is necessary to take into account the distribution of the currents +within the wires because the size of the wire is an important parameter. We +should therefore ask not what ¡is the inductance of a “circuit,” but what is the +inductance of a đisfr?bulöon of conductors. Perhaps the easiest way to fnd +this inductance is to make use of the magnetic energy. We found earlier, in +Section 15-3, an expression for the magnetic energy of a distribution of stationary +Currents: +U= tị2 - AdV. (17.44) +l we know the distribution of current density 7, we can compute the vector +potential A and then evaluate the integral of Eq. (17.44) to get the energy. This +energy ¡is equal to the magnetic energy of the self£-inductanece, 321 ?, Equating +the two gives us a formula for the inductance: +t= Jj:AdV (17.45) +We expect, of course, that the inductance is a number depending only on the +geometry of the cireuit and not on the current ƒ in the circuit. The formula of +Eq. (17.45) will indeed give such a result, because the integral in this equation is +proportional to the square of the current——the current appears once through 7 +and again through the vector potential A. The integral divided by 7 will depend +on the geometry of the circuit but not on the current Ï. +Equation (17.44) for the energy of a current distribution can be put in a quite +diferent form which is sometimes more convenient for calculation. Also, as we +will see later, it is a form that is important because it is more generally valid. In +the energy equation, Eq. (17.44), both A and 7 can be related to Ö, so we can +hope to express the energy in terms of the magnetic fñeld—just as we were able +to relate the electrostatic energy to the electric fñeld. We begin by replacing 7 +by cạc2V xÖ. We cannot replace A so easily, since = W x A cannot be +reversed to give A in terms of Ở. Anyway, we can write +U= t~ |IVxB)-Aat (17.46) +The interesting thing is that——with some restrictions——this integral can be +written as +U= SẼ [B.(Vx A)dt (17.47) +To see this, we write out in detail a typical term. Suppose that we take the +term (VW x B);A; which occurs in the integral of Eq. (17.46). Writing out the +--- Trang 219 --- +components, we get +0B, ðB, +“==—~ — -== ]A4; dz dụ dz. +l ( 3z Øụ ) _ +(There are, oŸ course, two more integrals of the same kind.) We now integrate +the frst term with respect to z——integrating by parts. hat is, we can say +3B 84; +Now suppose that our system——meaning the sources and fñields—is fnite, so that +as we go to large distances all fñelds go to zero. Then If the integrals are carried +out over all space, evaluating the term „4z; at the limits will give zero. We have +left only the term with Ø„(9A„/9z), which is evidently one part of Ð,(V x A)y +and, therefore, of 8 -(VW x 4). H you work out the other fve terms, you will see +that Bq. (17.47) is indeed equivalent to Bq. (17.46). +But now we can replace (W x 4) by Ö, to get +U= ma - BdV. (17.48) +We have expressed the energy of a magnetostatic situation in terms of the +magnetic ñeld only. The expression corresponds closely to the formula we found +for the electrostatic energy: +U= $9 [E- bar (17.49) +One reason for emphasizing these two energy formulas is that sometimes they +are more convenient to use. More important, it turns out that for dynamic fields +(when and Ö are changing with time) the two expressions (17.48) and (17.49) +remain true, whereas the other formulas we have given for electric or magnetic +energies are no longer correct—they hold only for static fñelds. +T we know the magnetic field of a single coil, we can find the self-inductance +by equating the energy expression (17.48) to 3@!2. Leb's see how this works +by fnding the self-inductance of a long solenoid. We have seen earlier that the +magnetic feld inside a solenoid is uniform and #Ö outside is zero. The magnitude +of the field inside is = ø%I/coe2, where ø is the number of turns per unit length +in the winding and ƒ is the current. If the radius of the coil is r and its length +is Ù (we take L very long, so that we can neglect end effects, i.e., Ù 3 r), the +volume inside is zr2L. The magnetic energy is therefore +coC“ sa m^Ï 2 +U = ——_— B“‹(Vol) = ~—- mrˆL +2 (Vol) 2coc2 Thiên +which is equal to 387. Or, +£E= —— L. (17.50) +--- Trang 220 --- +Tĩ:o IWqxearoll Eqrretff©orts +18-1 Maxwell's equations +In this chapter we come back to the complete set of the four Maxwell equations 18-1 Maxwells equations +that we took as our starting point in Chapter 1. Until now, we have been studying 18-2 How the new term works +Maxwell*s equations in bits and pieces; it is time to add one fnal piece, and to 18-3 AII of classical physics +put them all together. We will then have the complete and correct story for R +. . SH ốc . . 18-4 A travelling ñeld +electromagnetic fields that may be changing with time in any way. Anything said . +in this chapter that contradicts something said earlier is true and what was said 18-ã The speed of light +earlier is false—because what was said earlier applied to such special situations 18-6 Solving Maxwell's equations; the +as, for instance, sbeady currents or fñxed charges. Although we have been very potentials and the wave equation +careful to point out the restrictions whenever we wrote an equation, it is easy to +forget all of the qualiications and to learn too well the wrong equations. Ñow +we are ready to give the whole truth, with no qualifications (or almost none). +'The complete Maxwell equations are written in Table 18-1, in words as well as +in mathematical symbols. "The fact that the words are equivalent to the equations +should by this time be familiar—you should be able to translate back and forth +from one form to the other. +The frst equation—that the divergence of #/ is the charge density over eg—ÌS +true in general. In dynamic as well as in static fñields, Gauss' law is always valid. +The ñux of E through any closed surface is proportional to the charge inside. +The third equation is the corresponding general law for magnetic fields. Since +there are no magnetic charges, the ñux of Ö through any closed surface is aÌlways +zero. The second equation, that the curl of E is —ØB/6t, is Earaday?s law and +was discussed in the last two chapters. It also is generally true. The last equation +has something new. We have seen before only the part of it which holds for +steady currents. In that case we said that the curl of Ö is 7/coc?, but the correcE +general equation has a new part that was discovered by Maxwell. +Until Maxwells work, the known laws of electricity and magnetism were those +we have studied in Chapters 3 through 17. In particular, the equation for the +magnetic field of steady currents was known only as +VxbB=.?, (18.1) +Maxwell began by considering these known laws and expressing them as dier- +ential equations, as we have done here. (Although the V notation was not yet +invented, it is mainly due to Maxwell that the importance of the combinations of +derivatives, which we today call the curl and the divergence, first became appar- +ent.) He then noticed that there was something strange about Eq. (18.1). If one +takes the divergence of this equation, the left-hand side will be zero, because the +divergence of a curl is always zero. So this equation requires that the divergence +of 7 also be zero. But if the divergence of 7 is zero, then the total Ñux of current +out of any closed surface is also zero. +'The fñux of current from a closed surface is the decrease of the charge inside +the surface. 'Phis certainly cannot in general be zero because we know that the +charges can be moved from one place to another. The equation +Ý:7j= 2 (18.2) +has, in fact, been almost our defnition of ÿ. This equation expresses the very +fundamental law that electric charge is conserved——=any Ñow of charge must come +--- Trang 221 --- +Table 18-1 Classical Physics +Maxwell's equations +L W.E=f (Flux of E through a closed surface) = (Charge inside) /eo +0B H. d +IL. VxE= —g. (Line integral of E around a loop) = —lux of through the loop) +II V-.=0 (Flux of through a closed surface) = 0 +2 J 9E z +IV. cẤẦVxb=T+ tap (Integral of around a loop) = (Current through the loop)/eo ++ + tlux of E through the loop) +Conservation of charge +V-7= “a. (Flux of current through a closed loop) = —a(Charge inside) +Force law +t=q(E+oux®) +Law of motion +dư = h kiểu (Newton's law, with Einstein's modification) +—(p) = È, where =—————p ewton's law, wi instein's modification +di P , v1— 12/c2 +Gravitation +FƑ=-G —” cự +from some supply. Maxwell appreciated this dificulty and proposed that it could +be avoided by adding the term Ø/Ô to the right-hand side of Eq. (18.1); he +then got the fourth equation in Table 16-1: +Mww%." +IV cÓVxB=T+—. +€0 ØF +lt was not yet customary in Maxwells time to think in terms of abstract +fields. Maxwell discussed his ideas in terms of a model in which the vacuum was +like an elastic solid. He also tried to explain the meaning of his new equation in +terms of the mechanical model. 'There was much reluctance to accept his theory, +first because of the model, and second because there was at fñirst no experimental +Jjustifcation. Today, we understand better that what counts are the equations +themselves and not the model used to get them. We may only question whether +the equations are true or false. This is answered by doïng experiments, and untold +numbers of experiments have confrmed Maxwells equations. IÝ we take awawy +the scafolding he used to build it, we ñnd that Maxwell's beautiful edifce stands +on its own. He brought together all of the laws of electricity and magnetism and +made one complete and beautiful theory. +Let us show that the extra term is just what is required to straighten out +the difculty Maxwell discovered. Taking the divergence of his equation (TV in +Table 18-1), we must have that the divergence of the right-hand side is zero: +V.—=+V:—_—=0. 18.3 +€0 + ðt ( ) +In the second term, the order of the derivatives with respect to coordinates and +--- Trang 222 --- +time can be reversed, so the equation can be rewritten as +V-7~+co——V:E=0. (18.4) +But the first of Maxwells equations says that the divergence of E is ø/co. +Inserting this equality in Eq. (18.4), we get back Eq. (1§.2), which we know is +true. Conversely, if we accept Maxwells equations—and we do because no one +has ever found an experiment that disagrees with them——we must conclude that +charge is always conserved. +'The laws of physics have no answer to the question: “What happens If a charge +1s suddenly created at this point—what electromagnetic efects are produced?” +No answer can be given because our equations say it doesn't happen. Ïlf it ere +to happen, we would need new laws, but we cannot say what they would be. We +have not had the chance to observe how a world without charge conservation +behaves. According to our equations, iŸ you suddenly place a charge at some +point, you had to carry it there from somewhere else. In that case, we can say +what would happen. +'When we added a new term to the equation for the curl of #7, we found that +a whole new class of phenomena was described. We shall see that Maxwell”s little +addition to the equation for V x Ö also has far-reaching consequences. We can +touch on only a few of them in this chapter. +18-2 How the new term works +As our first example we consider what happens with a spherically symmetriec ` E / j +radial distribution of current. Suppose we imagine a little sphere with radioactive \ / +material on it. This radioactive material is squirting out some charged particles. N / E +(Or we could imagine a large block of jello with a small hole in the center into \ h +which some charge had been injected with a hypodermic needle and from which r +the charge is slowly leaking out.) In either case we would have a current thatis ~.. Bề _* +everywhere radially outward. We will assume that it has the same magnitude in ~ 7 <Ấ +all directions. +Let the total charge inside any radius r be Q(z). IÝ the radial current density @ +at the same radius is 7(z), then Eq. (18.2) requires that Q decreases at the rate x⁄ wy +P6” ¬ +0Q0) = —4mr2/(r). (18.5) “ +Øt › % +'W© now ask about the magnetie fñeld produced by the currents in this situation. h N E +Suppose we draw some loop ψ on a sphere of radius r, as shown in Fig. 18-1. ý \ +'There is some current through this loop, so we might expect to fnd a magnetic \/ +fñeld circulating ¡in the direction shown. "¬ +But we are already in dificulty. How can the Ö have any particular direction F1g. 18-1, What is the magnetfc field of +on the sphere? A diferent choice of I' would allow us to conclude that its direction a spherlcally symmetric current? +1s exactly opposite to that shown. So how cưn there be any circulation of +around the currents? +W© are saved by Maxwell's equation. The circulation of depends not only +on the total current through I` but also on the rate of change with time of the +clectric ƒfuz through ït. It must be that these two parts just cancel. Let”s see 1f +that works out. +The electric field at the radius z must be Q(z)/4eor?—so long as the charge +is symmetrically distributed, as we assume. ÏIt is radial, and is rate of change is +0E__ 1L 0G (18.6) +ÔÈ 4mcor2 Ô‡ +Comparing this with Eq. (1S.5), we see +=— (18.7) +--- Trang 223 --- +LOOP T¡ lS +_ ` : ⁄ +| B NN Ỹ “⁄ +§ ) ày ` EIEEEEEEEEEEEEEEEEETEEEESEERES, ứ ⁄ +q8 _—.......~ +KÑyyyy °p nN “g +¿ () ⁄ ) +Fig. 18-2. The magnetic field near a charging capacitor. +In Eq. IV the two source terms cancel and the curl of Ö is always zero. There is +no magnetic field in our example. +As our second example, we consider the magnetic feld of a wire used to +charge a parallel-plate condenser (see Fig. 18-2). IÝ the charge Q on the plates is +changing with time (but not too fast), the current in the wires is equal to đQ/dt. +W©e would expect that this current will produce a magnetic fñeld that eneircles the +wire. Surely, the current close to the plate must produce the normal magnetic +fñeld—it cannot depend on where the current is going. +Suppose we take a loop [1+ which is a circle with radius z, as shown in part (a) +of the fñgure. The line integral of the magnetic field should be equal to the +current 7 divided by cọc2. We have +2mrB ự 18.8 +mr = m2 (18.8) +Thịs is what we would get for a steady current, but it is also correct with Maxwell”s +addition, because If we consider the plane surface ,Š9 inside the circle, there are +no electric fields on it (assuming the wire to be a very good conductor). The +surface integral of ØE/Ø is zero. +Suppose, however, that we now slowly move the curve I` downward. We get +always the same result until we draw even with the plates of the condenser. Then +the current Ï goes to zero. Does the magnetic fñeld disappear? hat would be +quite strange. Let”s see what Maxwells equation says for the curve ¿, which is a +circle of radius z whose plane passes between the condenser plates [Eig. 1S-2(b)]. +The line integral of around ` is 2xr. 'This must equal the time derivative oŸ +the fux of # through the plane circular surface S2. Thịis fux of #, we know from +Gauss” law, must be equal to 1/eo times the charge @ on one oŸ the condenser +plates. We have +2 dđ(@ +cẰ2nrB = at () (18.9) +That is very convenient. It is the same result we found in Eq. (18.8). In- +tegrating over the changing electric fñeld gives the same magnetic feld as does +integrating over the current in the wire. Of course, that is just what Maxwells +cquation says. I% is easy to see that this must always be so by applying our +same arguments to the two surfaces 5¡ and 51 that are bounded by the same +circle E in Fig. 18-2(b). Through Š5¡ there is the current 7, but no electric flux. +Through 5} there is no current, but an electric Ñux changing at the rate Ï/e. +'The same #Ö is obtained ïf we use Eq. IV with either surface. +trom our discussion so far of Maxwell's new term, you may have the impression +that it doesn't add much—that it just fñxes up the equations to agree with what +we already expect. It is true that if we just consider Eq. IV bụ #sejf, nothing +particularly new comes out. The words “bự ?£sejƒ” are, however, all-important. +Maxwells small change in Eq. IV, when combzncd tuíth the other equations, does +--- Trang 224 --- +indeed produce much that is new and important. Before we take up these matters, +however, we want to speak more about Table 18-1. +18-3 All of classical physics +In Table 18-1 we have all that was known of fundamental cilasszcal physics, +that 1s, the physics that was known by 1905. Here ï$ all is, in one table. With +these equations we can understand the complete realm of classical physics. +Flirst we have the Maxwell equations—written in both the expanded form and +the short mathematical form. Then there is the conservation of charge, which is +even written in parentheses, because the moment we have the complete Maxwell +cequations, we can deduce from them the conservation of charge. So the table is +even a little redundant. Next, we have written the force law, because having all +the electric and magnetic fields doesn't tell us anything until we know what they +do to charges. Knowing # and Ö, however, we can find the force on an object +with the charge g moving with velocity 0. EFinally, having the force doesn't tell +us anything until we know what happens when a force pushes on something: we +need the law of motion, which is that the force ¡is equal to the rate of change +of the momentum. (Remember? We had that in Volume I.) We even include +relativity efects by writing the momentum as ø = ?mo0/4/1— 02/2. +Tf we really want to be complete, we should add one more law—Newton”s law +of gravitation—so we put that at the end. +Therefore in one small table we have all the fundamental laws of classical +physics—even with room to write them out in words and with some redundaney. +Thịs is a great moment. We have climbed a great peak. We are on the top of K-2—— +we are nearly ready for Mount Everest, which is quantum mechanics. We have +climbed the peak of a “Great Divide,” and now we can go down the other side. +W© have mainly been trying to learn how to understand the equations. NÑow +that we have the whole thing put together, we are going to study what the +equations mean—what new things they say that we havent already seen. Weˆve +been working hard to get up to this point. It has been a great efÑfort, but now we +are going to have nice coasting downhill as we see all the consequences of our +accomplishment. +18-4 A travelling ñeld +Now for the new consequences. Phey come from putting together all of +Maxwells equations. First, let”s see what would happen ïn a circumstance which +we pick to be particularly simple. By assuming that all the quantities vary only in +one coordinate, we will have a one-dimensional problem. The situation is shown In +Fig. 18-3. We have a sheet of charge located on the øz-plane. “The sheet is ñrst at +rest, then instantaneously given a velocity œ in the -direction, and kept moving +with this constant velocity. You might worry about having such an “infnite” +⁄ MOVING BOUNDARY +OF FIELDS +SHEET QF dứa +«‹ j ⁄ l +L< Ñ sử x⁄ HN +-_ XS `. +B `, T- ¡ +`> ` ____LŸ _ế +B ì S AM Ị +E ` eN E ` ÝE "¬ +Z. | ) `... Fig. 18-3. An infinite sheet of charge Is +àề ` _~“NO EIELDS suddenly set into motion parallel to itself. +N ⁄ _- X E=B=0 There are magnetic and electric fields that +"Ta. propagate out from the sheet at a constant +q vt 2) speed. +x=0 x=X%o 18-5 +--- Trang 225 --- +acceleration, but it doesn”$ really matter; just imagine that the velocity is brought +to very quickly. So we have suddenly a surface current .j (27 is the current per +unit width in the z-direction). 'TTo keep the problem simple, we suppose that there +1s also a statilonary sheet of charge of opposite sign superposed on the zz-plane, so +that there are no electrostatic efects. Also, although in the fñigure we show onlÌy +what is happening in a fñnite region, we imagine that the sheet extends to inÑnity +in + and +z. In other words, we have a situation where there is no currenf, +and then suddenly there is a uniform sheet of current. What will happen? +'Well, when there is a sheet of current in the plus -direction, there is, as we +know, a magnetic feld generated which will be in the minus z-direction for ø > 0 +and in the opposite direction for z < 0. We could fnd the magnitude of Ð by BorE +using the fact that the line integral of the magnetic fñeld will be equal to the +current over cọc”. We would get that = J/2eoc” (since the current 7 in a strip v +of width +0 is J+ and the line integral of Ð is 2u). +This gives us the fñeld next to the sheet—for small ø—but sỉnce we are - +Imagining an infnite sheet, we would expect the same argument to give the vụ ———— +magnetic fñeld farther out for larger values ofz. However, that would mean that the +moment we turn on the current, the magnetic fñeld ¡is suddenly changed from zero +to a ñnite value everywhere. But waitl If the magnetic fñeld ¡is suddenly changed, &) +it will produce tremendous electrical efects. (If it changes in øng/ way, there are BorE +electrical efects.) So because we moved the sheet of charge, we make a changing +magnetic ñeld, and therefore electric ñelds must be generated. If there are electric +fñelds generated, they had to start from zero and change to something else. 'There vự=T) — +will be some ØE/ðØt that will make a contribution, together with the current Ở, x +to the production of the magnetic field. So through the various equations there ” +is a big intermixing, and we have to try to solve for all the ñelds at once. +By looking at the Maxwell equations alone, it is not easy to see directly () +how to get the solution. So we will ñrst show you what the answer is and then BorE +verify that i% does indeed satisfy the equations. The answer is the following: The +fñeld Ö that we computed is, in fact, generated right next to the current sheet (for v +small z). It must be so, because if we make a tiny loop around the sheet, there +is no room for any electric ñux to go through it. But the field Ö out farther—for +larger ø-—ls, at first, zero. It stays zero for awhile, and then suddenly turns on. - vĩ ¬ ĩ +In short, we turn on the current and the magnetic fñeld immediately next to it +turns on to a constant value #Ö; then the turning on of spreads out from the (c) +source region. After a certain time, there is a uniform magnetic field everywhere +out to some value zø, and then zero beyond. Because of the symmetry, it spreads Fig. 18-4. (a) The magnitude of B (or E) +in both the plus and minus z-directions. asa function oŸ x at time £ after the charge +The E-field does the same thing. Before ý = 0 (when we turn on the current), sheet is set in motion. (b) The fields for +the fñeld is zero everywhere. Then after the time , both # and #Ö are uniform H charge sheet Set In motlon, toward ne9a- +- : ive y at £ = T. (c) The sum of (a) and (b). +out to the distance ø = 1, and zero beyond. The fñelds make their way forward +like a tidal wave, with a front moving at a uniform velocity which turns out to +be c, but for a while we will just call it ø. Á graph of the magnitude of E or +versus #, as they appear at the time ý, is shown in Eig. 1S-4(a). Looking again +at Fig. 18-3, at the time f, the reglon between ø = +ý ¡is “fñlled” with the fñelds, +but they have not yet reached beyond. We emphasize again that we are assuming +that the current sheet and, therefore the fields # and #Ö, extend infinitely far +in both the #- and z-directions. (We cannot draw an infinite sheet, so we have +shown only what happens in a finite area.) +We want now to analyze quantitatively what is happening. To do that, we +want to look at two cross-sectional views, a top view looking down along the +u-axis, as shown in Fig. 18-5, and a side view looking back along the z-axis, as +shown in Fig. 1S-6. Suppose we start with the side view. WWe see the charged +sheet moving up; the magnetic fñeld points into the page for +z, and out of the +page for —z, and the electric field is downward everywhere—out to # = -+ui. +Let”s see if these felds are consistent with Maxwell's equations. Let”s first draw +one of those loops that we use to calculate a line integral, say the rectangle a +shown in EFig. 18-6. You notice that one side of the rectangle is in the reglon +where there are fields, but one side is in the region the fñelds have still not +--- Trang 226 --- +TOP VIEW yẠ_ SIDEVIEW +Ixl¡ [xxx lxÌ -J*1JxIxIxIs ' +| | mˆ E | +Lxl xi x xỦ ¡ /11 . xe x|d|x|x ¡ /1? +I lộ | le | r L +(šI]!151 "111 *lƑ ⁄ +SHEET . . 2 SHEET 2 +lx | | x | x | x x ` . x x x x +† † Fxx vt 1 T—VAt F†xI« vt T—vAt +- Xi Xi Xi Xa TT” Ix|x|x|x| ¡ +x=0 X=Xo 0 Xo +Fig. 18-5. Top view of Fig. 18-3. Fig. 18-6. Side view of Fig. 18-3. +reached. “There is some magnetic Ñux through this loop. lf it is changing, there +should be an emf around it. TỶ the wavefront is moving, we will have a changing +magnetic ñux, because the area in which #Ö exists is progressively increasing +at the velocity ø. “The ñux inside 2 is times the part of the area inside l`¿ +which has a magnetic fñeld. 'Phe rate of change of the Ñux, since the magnitude +oŸ is constamt, is the magnitude tỉimes the rate of change of the area. “The rate +of change of the area is easy. lÝ the width of the rectangle Ùạ is b, the area in +which exists changes by ø Af in the time Af. (See Eig. 18-6.) The rate of +change of fux is then j0. According to Earaday”s law, this should equal minus +the line integral oŸ # around 2, which is just #⁄L. We have the equation +=8. (18.10) +So if the ratio of # to Ö is 0, the fñelds we have assumed will satisfy Faradayˆs +equation. +But that is not the only equation; we have the other equation relating +and Ö: - +cvxp-=J+°E, (18.11) +€0 lôI) +To apply this equation, we look at the top view in Eig. 18-5. We have seen that +this equation will give us the value of next to the current sheet. Also, for any +loop drawn outside the sheet but behind the wavefront, there is no curl of nor +any 7 or changing #, so the equation is correct there. Now let”s look at what +happens for the curve Eị that intersects the wavefront, as shown In Fig. 18-5. +Here there are no currents, so 2q. (18.11) can be written——in integral form——as +Œ 1 B-ds= — J +E-n da. (18.12) +inside + +The line integral of is just times L. The rate of change of the ñux of # is +due only to the advancing wavefront. The area inside ị, where # is not zero, 1s +increasing at the rate 0E. The right-hand side of Eq. (18.12) is then 0L. That +equation becomes +c?B = Eu. (18.13) +W© have a solution in which we have a constant Ö and a constant behind +the front, both at right angles to the direction in which the font is moving and +at right angles to each other. Maxwell's equations specify the ratio of ⁄ to B. +Erom Eqs. (18.10) and (18.13), +E=ubB, and tb= " hB. +But one momentl We have found #wo đjƒeren‡ conditions on the ratio #/B. Can +such a fñeld as we describe really exist? 'Phere is, of course, only one velocity 0 +--- Trang 227 --- +for which both of these equations can hold, namely = c. The wavefront must +travel with the velocity c. We have an example in which the electrical inluence +from a current propagates at a certain fñnite veloclty e. +Now let°s ask what happens if we suddenly stop the motion of the charged +sheet after it has been on for a short time 7'. We can see what will happen by +the principle of superposition. We had a current that was zero and then was +suddenly turned on. We know the solution for that case. Now we are going to +add another set of fields. We take another charged sheet and suddenly start I% +moving, in the opposite direction with the same speed, only at the time 7' after +we started the first current. 'Phe total current of the two added together is first +zero, then on for a time 7', then of again——because the two currents cancel. We +have a square “pulse” of current. +'The new negative current produces the same fields as the positive one, onÌy +with all the signs reversed and, of course, delayed in time by 7'. A wavefront again +travels out at the velociby c. At the tỉme £ it has reached the distance z = +c(£—7), +as shown in Fig. 18-4(b). So we have two “blocks” of ñeld marching out at the +speed é, as in parts (a) and (b) of Eig. 18-4. The combined fñelds are as shown +in part (c) of the ñgure. The fields are zero for ø > cý, they are constant (with +the values we found above) between z = c(£ — 7} and z = cứ, and again zero +for z < c(£— T). +In short, we have a little piece of fñeld——a block of thiekness đƒ—which has +left the current sheet and is travelling through space all by itself. The ñelds have +“taken off”; they are propagating freely through space, no longer connected in +any way with the source. The caterpillar has turned into a butterfyl +How can this bundle of electric and magnetic fñelds maintain itself? "The +answer is: by the combined efects of the EFaraday law, W x E = —ØB/Ôt, and +the new term of Maxwell, cẰÀV x = ôE/ôt. They cannot help maintaining +themselves. Suppose the magnetic field were to disappear. 'Phere would be a +changing magnetic field which would produce an electric fñeld. Tf this electric +field tries to go away, the changing electric feld would create a magnetic feld +back again. 5o by a perpetual interplay——by the swishing back and forth om +one field to the other——they must go on forever. ÏIt is impossible for them to +disappear.* They maintain themselves in a kind of a dance—one making the +other, the second making the frst—propagating onward through space. +18-5 The speed of light +We have a wave which leaves the material source and goes outward at the +velocity c, which ¡is the speed of light. But let's go back a moment. From a +historical point of view, it wasnt known that the coefficient c in Maxwell”s +equations was also the speed of light propagation. There was just a constant in +the equations. We have called it e from the beginning, because we knew what it +would turn out to be. We didn'$ think it would be sensible to make you learn +the formulas with a different constant and then go back to substitute c wherever +1 belonged. From the point of view of electricity and magnetism, however, we +just start out with two constants, cạ and e2, that appear in the equations of +electrostatics and magnetostatics: +Wg.E=P (18.14) +VxbBb-= sục” (18.15) +Tf we take any arbiraru delñnition of a unit of charge, we can determine exper- +imentally the constant eo required in Eq. (18.14)—say by measuring the force +between two unit charges at rest, using Coulomb”s law. We must also determine +* Well, not quite. They can be “absorbed” if they get to a region where there are charges. +By which we mean that other felds can be produced somewhere which superpose on these +fields and “cancel” them by destructive interference (see Chapter 31, Vol. I). +--- Trang 228 --- +experimentally the constant coc2 that appears in E4q. (18.15), which we can do, +say, by measuring the force bebtween two unit currents. (Á uni current means +one unit of charge per second.) The ratio of these wo experimental constants +is c”—just another “electromagnetic constant.” +Notice now that this constant e? is the same no matter what we choose +for our unit of charge. lÝ we put bwice as much “charge”——say twice as many +proton charges—in our “unit” oŸ charge, co would need to be one-fourth as large. +'When we pass two of these “unit” currents through two wires, there will be twice +as much “charge” per second in each wire, so the force between EWo Wires is +four times larger. The constant cgc2 must be reduced by one-fourth. But the +ratio coc2/co is unchanged. +So just by experiments with charges and currents we fnd a number c2 which +turns out to be the square of the velocity of propagation of electromagnetic +inluences. From static measurements—by measuring the forces between two unit +charges and between two unit currents—we find that e = 3.00 x 10Š meters/sec. +When Maxwell first made this calculation with his equations, he said that +bundles of electric and magnetic felds should be propagated at this speed. He +also remarked on the mysterious coincidence that this was the same as the speed +of light. “We can scarcely avoid the inference,” said Maxwell, “that light consists +in the transverse undulations of the same medium which is the cause of electric +and magnetic phenomena.” +Maxwell had made one of the great unifications of physics. Before his time, +there was light, and there was electricity and magnetism. The latter two had +been unified by the experimental work of Earaday, Oersted, and Ampère. Then, +all of a sudden, light was no longer “something else,” but was only electricity and +magnetism in this new form——little pieces of electric and magnetic fields which +propagate through space on theïir own. +We have called your attention to some characteristics of this special solution, +which turn out to be true, however, for œnww/ electromagnetic wave: that the +magnetic feld is perpendicular to the direction of motion of the wavefront; +that the electric field ¡is likewise perpendicular to the direction of motion of +the wavefront; and that the two vectors # and #Ö are perpendicular to each +other. Eurthermore, the magnitude of the electric ñeld # is equal to e times the +magnitude of the magnetic ñeld #. 'These three facts—that the t6wo fields are +transverse to the direction of propagation, that Ö is perpendicular to #, and +that # = cB—are generally true for any electromagnetic wave. Ôur special case +is a good one—it shows all the main features of electromagnetic waves. +18-6 Solving Maxwell?s equations; the potentials and the wave equation +Now we would like to do something mathematical; we want to write Maxwell's +equations in a simpler form. You may consider that we are complicating them, but +1f you will be patient a little bit, they will suddenly come out simpler. Although +by this time you are thoroughly used to each of the Maxwell equations, there are +many pieces that must all be put together. That's what we want to do. +We begin with V - = 0—the simplest of the equations. We know that it +Iimplies that # is the curl of something. So, if we write +BöB=VxA, (18.16) +we have already solved one of Maxwells equations. (Incidentally, you appreciate +that it remains true that another vector A“ would be just as good if 4” = A+Vj— +where ÿ is any scalar fñield——because the curl of Wøj is zero, and #Ö ¡s still the +same. We have talked about that before.) +W© take next the Faraday law, W x E = —ØB/6ðt, because it doesnˆt involve +any currents or charges. lÝ we write Ö as V x A and diferentiate with respect +to É, we can write Faraday”s law in the form +VxE=-_—VxA. +--- Trang 229 --- +Since we can diferentiate either with respect to time or to space first, we can +also write this equation as +Vx (z + r) =0. (18.17) +W© see that E + ØA/6f is a vector whose curl is equal to zero. Therefore that +vector is the gradient of something. When we worked on electrostatics, we had +VxE=(0, and then we decided that # itself was the gradient of something. +W© took it to be the gradient of —ở (the minus for technical convenience). We +do the same thing for + ØA/Ôf; we set +E+—_—=-V¿. (18.18) +W© use the same symbol ó so that, in the electrostatic case where nothing changes +with time and the ØA/Ô term disappears, # will be our old —Wø. So Faraday”s +cequation can be put in the form +b=-Vó¿———. 18.19 +ó— (13.19) +W© have solved two of Maxwell's equations already, and we have found that +to describe the electromagnetic fields # and ?Ö, we need four potential functions: +a scalar potential ø and a vector potential A, which is, of course, three functions. +Now that A determines part of E, as well as , what happens when we +change A to A“= A+ Vú? In general, E would change if we didn”t take some +special precaution. We can, however, still allow 4 to be changed in this way +without afecting the ñelds # and —that is, without changing the physics—If +we always change 4 and óð £ogether by the rules +A'=A+ Vụ, @=ó_— Sc: (18.20) +Then neither Ö nor , obtained from Eq. (18.19), is changed. +Previously, we chose to make V - A =0, to make the equations of statics +somewhat simpler. We are not going to do that now; we are going to make a +diferent choice. But we'll wait a bit before saying what the choice is, because +later it will be clear ø¿ the choice is made. +Now we return to the two remaining Maxwell equations which will give us +relations between the potentials and the sources ø and 7. Ônce we can determine +A and ó from the currents and charges, we can always get E and Ö from Eqs. +(18.16) and (18.19), so we will have another form of Maxwell's equations. +We begin by substituting Eq. (18.19) into Ð - E = ø/co; we get +YK.|-V¿--“^|=T +( Ụ lô? ) SIN +which we can write also as +-W24- W.A=f, (18.21) +This is one equation relating j and A to the sources. +Our fñnal equation will be the most complicated. We start by rewriting the +fourth Maxwell equation as +cẦVxB- =a = 7, +ÔF €0 +and then substitute for and 7 in terms of the potentials, using qs. (18.16) +and (18.19): +8 3A 3 +Vx(Vx4)-- |-Vø—-—]=_—. +, * ( * ) ØF ( % Øi ) €0 +--- Trang 230 --- +The first term can be rewritben using the algebraic identity: W x(Wx A4) = +V(V: A) —- V2A; we get +8 32A j +-cV?A+c?V(V-A)+ | +We© can calculate the kinetic energy minus the potential energy and integrate for ị +such a path... or for any other path we want. The miracle is that the true path +1s the one for which that integral 1s least. +“Let's try it out. Eirst, suppose we take the case of a free particle for which +there is no potential energy at all. Then the rule says that in going from one \ | +point to another in a given amount of time, the kinetic energy integral is least, : v3 +So iÿ musE go at a uniform speed. (We know that”s the right answer—to go at a +, tị +uniform speed.) Why is that? Đecause if the particle were to go any other way, +the velocities would be sometimes higher and sometimes lower than the average. +The average velocity is the same for every case because it has to get from “here” +to “there” in a given amount of time. +“As an example, say your job is to sbart from home and get to school in a +given length of time with the car. You can do it several ways: You can accelerate +like mad at the beginning and slow down with the brakes near the end, or you +can øo at a uniform speed, or you can go backwards for a while and then go “ +forward, and so on. The thing is that the average speed has got to be, of course, Đông +the total distance that you have gone over the time. But ïf you do anything but + ; +go at a uniform speed, then sometimes you are going too fast and sometimes ; 8g Ị +you are going too slow. Now the mean sguøare of something that deviates around Ị +an average, as you know, is always greater than the square of the mean; so the +kinetic energy integral would always be higher if you wobbled your velocity than +1Í you went at a uniform velocity. So we see that the integral is a minimum ïf the *e~ ï +velocity is a constant (when there are no forces). The correct path is like this. l +—— +, ++ + +“NÑow, an objJect thrown up ïn a gravitational field does rise faster first and +then slow down. 'That is because there is also the potential energy, and we must +have the least đ¿fference of kinetic and potential energy on the average. Because +the potential energy rises as we go up in space, we will get a lower đjƒƒerence 1Ÿ n1 yrtưx£ +we can get as soon as possible up to where there is a high potential energy. Then ~PE. —” +we can take that potential away from the kinetic energy and get a lower average. +So iÈ is better to take a path which goes up and gets a lot of negative stuf from \ +the potential energy. + RE +“Ơn the other hand, you can”t go up too fast, or too far, because you will +then have too much kinetic energy involved—you have to go very fast to get way +up and come down again in the fñxed amount of time available. So you don'$ +want to go too far up, but you want %o øo up some. So it turns out that the h ị +solution is some kind of balance between trying to get more potential energy with ” 5 +the least amount of extra kinetic energy——trying to get the diference, kinetic ' _= +minus the potential, as small as possible. +--- Trang 233 --- +“hat is all my teacher told me, because he was a very good teacher and knew +when to stop talking. But I don't know when to stop talking. So instead of leaving +1ÿ as an interesting remark, Ï am goïing to horrify and disgust you with the complex- +1ties of life by proving that ït is so. The kind of mathematical problem we will have +1s very dificult and a new kind. We have a certain quantity which ¡s called the +acfion, ŠS. Tt 1s the kinetic energy, minus the potential energy, inteprated over time. +Action = SŠ= J (KE— PE) di. +Remember that the PE and KE are both functions of time. For each diferent +possible path you get a diferent number for this action. Our mathematical +problem is to fnd out for what curve that number is the least. +“You say—Oh, that's Just the ordinary calculus of maxima and minima. You +calculate the action and Just diferentiate to fnd the minimum. +“But watch out. Ordinarily we Just have a function of some variable, and we +have to fnd the value of that 0arzøœble where the function is least or most. For +instance, we have a rod which has been heated in the middle and the heat is spread +around. For each point on the rod we have a temperature, and we must find the +point at which that temperature is largest. But now for cach pa‡h ïn spacc we have +a number—quite a diferent thing—and we have to find the path n space for which +the number is the minimum. That is a completely diferent branch of mathematics. +lt is not the ordinary calculus. In fact, it is called the calculus oƒ 0uariations. +“There are many problems in this kind of mathematics. EFor example, the +cirele is usually defned as the locus of all points at a constant distance from a +fñxed point, but another way of defning a circle is this: a circle is that curve øƒ +giuen length which encloses the biggest area. Any other curve encloses less area +for a given perimeter than the circle does. So If we give the problem: fnd that +curve which encloses the greatest area for a given perimeter, we would havc BC „L +problem of the calculus of variations—a diferent kind of caleulus than you re tri +used to. +“So we make the calculation for the path of an object. Here is the way we % +are going to do ït. The idea is that we imagine that there is a true path and that Z„a +any other curve we draw is a false path, so that if we calculate the action for the tra +false path we will get a value that is bigger than if we calculate the action for S.1èS +the true path. |——— + : +“Problem: Find the true path. Where is it? One way, of course, is to calculate +the action for millions and millions of paths and look at which one is lowest. : +'When you fnd the lowest one, that°s the true path. +“hat 's a possible way. But we can do it better than that. When we have a +quantity which has a minimum——for instance, in an ordinary function like the +temperature—one of the properties of the minimum is that iŸ we go away from the +minimum in the #zs order, the deviation of the function from its minimum value Lương +is only second order. At any place else on the curve, iŸ we move a small distance +the value of the function changes also in the first order. But at a minimum, a tỉny AT Ax +motion away makes, in the first approximation, no dierence. m—— +“That is what we are goỉng to use to calculate the true path. lf we have ¬ ATec lAx)” +the true path, a curve which difers only a little bit from it will, in the first xeeem /ˆ +approximation, make no diference in the action. Any diference will be in the +second approximation, I1f we really have a minimum. XAY-NG +“hat is easy to prove. lÝ there is a change in the first order when I deviate +the curve a certain way, there is a change in the action that is proportional to +the deviation. "The change presumably makes the action greater; otherwise we +haven”t got a minimum. But then ï1f the change is proportonal to the deviation, +reversing the sign oŸ the deviation will make the action less. We would get the +action to increase one way and to decrease the other way. The only way that it +could really be a minimum is that in the frs‡ approximation it doesn't make any +change, that the changes are proportional to the square of the deviations from +the true path. +--- Trang 234 --- +“So we work it this way: We call z(£) (with an underline) the true path—the +one we are trying to ñnd. We take some trial path #(£) that differs from the true + +path by a small amount which we will call z(£) (eta of £). +|————— +“NÑow the idea is that if we calculate the action Š for the path #(£), then the +điference bebween that 9 and the action that we calculabed for the path #(£)—to +simplify the writing we can call it S——the diference of Š and Š must be zero in x(©) (€) +the first-order approximation of small ạ. t can differ in the second order, but in ` K +the frst order the diference must be Zero. \ %(t) +“And that must be true for any + at all. Well, not quite. The method doesnˆt +mean anything unless you consider paths which all begin and end at the same +two points—each path begins at a certain point at ứ¡ and ends at a certain other % +point at ‡¿, and those points and times are kept fñxed. 5o the deviations in our 7 +have to be zero at each end, ?(#¡) = 0 and (4s) =0. With that condition, we +have specified our mathematical problem. +“If you didn'® know any calculus, you might do the same kind of thing to +fñnd the minimum of an ordinary function ƒ(+). You could discuss what happens +1 you take ƒ(#z) and add a small amount b to # and argue that the correction +to ƒ() in the frst order in h must be zero at the minimum. You would substitute +œ-+ h for z and expand out to the first order in h... just as we are going to do +with 1. +“The idea is then that we substitute #(#) = z{) + n(#) im the formula for the +action: +m (dzŠ” +s= =| |] TY đ‡ +(0) =re|e +where I call the potential energy WV(z). The derivative đø/dt is, of course, the +derivative of z(£) plus the derivative of ?(£), so for the action I get this expression: +m(dxz dn +s= —=..¬£ dt. +lì BH) &+n) +“Now I must write this out in more detail. For the squared term ÏI get +dxzÝ\? dz dn đn ; +(5) " đt dị " t +But wait. m not worrying about higher than the first order, so I will take all +the terms which involve ?ˆ and higher powers and put them in a little box called +“second and higher order. Erom this term Ï get only second order, but there will +be more from something else. 5o the kinetic energy part is +dựz\”. de d +5 (#) +m T mn + (second and higher order). +“NÑow we need the potential V at z +. I consider r small, so Ï can write +WV(+) as a Taylor series. IÈ is approximately V{(z); in the next approximation +(rom the ordinary nature of derivatives) the correction is ?; times the rate of +change of V with respect to ø, and so on: +Ví +) = VỆ) +1ỊV (#) + Vˆ(#) +: +lI have written Ví“ for the derivative of V with respect to + in order bo save +writing. The term in 7 and the ones beyond fall into the 'second and higher +orderˆ category and we donˆt have to worry about them. Putting ít all together, +m ( dz d+z dn +S— —| — —V —— —_ +/ HÀ (2) + ma +— ?V”(z) + (second and higher order)| dt. +--- Trang 235 --- +Now Iƒ we look carefully at the thing, we see that the first two terms which I +have arranged here correspond to the action 5 that I would have calculated with +the true path z. The thing I want to concentrate on is the change in S——the +diference between the Š and the Š that we would get for the right path. This +diference we will write as ổ5, called the variation in Š. Leaving out the “second +and higher orderˆ terms, I have for ôS +2[. da dn , +5s= | mộ a —TV (2)|a: +“Now the problem is this: Here is a certain integral. I don'® know what +the z is yet, but I do know that mo matter that †ị 1s, thĩs integral must be zero. +Well, you think, the only way that that can happen is that what multiplies + +must be zero. But what about the first term with dđn/đ? WGelIl, after all, if +can be anything at all, its derivative is anything also, so you conclude that the +coefficient of dự/dt must also be zero. That isn't quite right. It isn't quite right +because there is a connection between ? and its derivative; they are not absolutely +independent, because ?(#) must be zero at both íq and ‡a. +“he method of solving all problems in the calculus of variations always uses +the same general principle. You make the shift in the thing you want to vary (as +we đỉd by adding ?); you look at the frst-order terms; #Öen you always arrange +things in such a form tha% you get an integral of the form “some kind of stuf tỉmes +the shift (?),` but with no other derivatives (no đ?/đ£). It must be rearranged so +1t is always 'something' times ?. You will see the great value of that in a minute. +(There are formulas that tell you how to do this in some cases without actually +calculating, but they are not general enough to be worth bothering about; the +best way is to calculate it out this way.) +“How can I rearrange the term in đn/đf to make it have an ở? Ï can do +that by integrating by parts. It turns out that the whole trick of the calculus +Of variations consists oŸ writing down the variation of 5 and then integrating +by parts so that the derivatives of r; disappear. It is always the same in every +problem in which derivatives appear. +“You remember the general principle for integrating by parts. l you have +any function ƒ times đj/đứ integrated with respect to ý, you write down the +derivative of ?†ƒ: : : n +a0) =1 tờ: +The integral you want is over the last term, so +Hrimm=nmr= [nipe +“In our formula for ở, the function ƒ is rm times đ+/đí; therefore, I have the +following formula for ở5. +da t2 ta d d bn ta , +ôS=m ¬ 0|) l n (» x0 dt / V'(z) n() dt. +'The frst term must be evaluated at the t©wo limits f¡ and £¿. Then Ï must have +the integral from the rest of the integration by parts. The last term is brought +down without change. +“NÑow comes something which always happens——the integrated part disappears. +(In fact, 1ƒ the integrated part does not disappear, you restate the principle, adding +conditions to make sure it doesl) We have already said that z must be zero at both +ends of the path, because the principle is that the action is a minimum provided +that the varied curve begins and ends at the chosen points. The condition is that +n‹ti) =0, and ?(t¿) = 0. So the integrated term is zero. We collect the other +terms together and obtain this: +ta d2 bn , +ôS= l |—m 1E— V 6) n(t) dt. +--- Trang 236 --- +The variation in Š is now the way we wanted it—there is the sbuff in brackets, +say Ƒ, all multiplied by ?{#) and integrated from í to ‡a. +“We have that an integral of something or other tìmes ?J(£) is always 2ero: +[ru m(£) dt = 0. +Ih . ¬ . . Tr(©) +ave some function of ý; I multiply it by ?{£); and T integrate it from one end +to the other. And no matter what the ? is, Ï get zero. That means that the +function #'{#) is zero. That`s obvious, but anyway PH show you one kind oŸ proof. +“Suppose that for ?(f) I took something which was zero for all ¿ except right +near one particular value. It stays zero until it gets to this ứ, +then it blips up for a moment and blips right back down. When we do the integral Tì = +of this r times any function #', the only place that you get anything other than +zero was where ?(£) was blipping, and then you get the value of #' at that place +times the integral over the blip. The integral over the blip alone isn”t zero, but +when multiplied by #! it has to be; so the function #! has to be zero whore the +blip was. But the blip was anywhere Ï wanted to put it, so #' must be zero +everywhere. +“We see that 1ƒ our integral is zero for any ?, then the coeflicient of ?; must +be zero. 'Phe action integral will be a minimum for the path that satisfies this +complicated diferential equation: +|—m q5 v0) =0. +It°s not really so complicated; you have seen it before. It is Just # = ma. "The +first term is the mass times acceleration, and the second is the derivative of the +potential energy, which is the force. +“So, for a conservative system at least, we have demonstrated that the principle +of least action gives the right answer; it says that the path that has the minimum +action is the one satisfying Newton's law. +“One remark: I did not prove it was a ?min#nưmn—maybe 1t”s a maximum. +In fact, it doesnˆt really have to be a minimum. lt is quite analogous to what +we found for the “principle of least time” which we discussed in optics. Thhere +also, we said at first it was “least” time. It turned out, however, that there were +situations In which it wasnt the /eøsf time. The fundamental principle was that +for any firsỉ-order 0uariation away from the optical path, the chønge in tỉme was +zero; 1È 1s the same story. What we really mean by “least” is that the first-order +change ïn the value of S, when you change the path, is zero. It is not necessarily +a 'minimum. +“Next, I remark on some generalizations. In the first place, the thing can be +done in three dimensions. Instead of Just z, [ would have zø, , and z as functions +of £; the action is more complicated. Eor three-dimensional motion, you have to +use the complete kinetic energy——(m/2) times the whole velocity squared. That +m[( dz\? dụ ? dz\Ÿ +Ke= 9 |(7) t[m) tÂm) | +Also, the potential energy is a function of #, , and z. And what about the path? +The path is some general curve in space, which is not so easily drawn, but the +idea is the same. And what about the ?? Well, ạ can have three components. +You could shift the paths in z, or in , or in z—or you could shift in all three +directions simultaneously. So r; would be a vector. 'This doesn”t really complicate +things too much, though. Since only the frsi-order varlation has to be zero, +we can do the caleulation by three successive shifts. We can shift r only in the +z-direction and say that coefficient must be zero. We get one equation. Then +we shift i% in the ¿-direction and get another. And in the z-direction and get +another. Ôr, oŸ course, in any order that you want. Anyway, you get three +--- Trang 237 --- +cquations. And, of course, Newton”s law is really three equations in the three +dimensions—one for each component. I think that you can practically see that it +is bound to work, but we will leave you to show for yourself that it will work for +three dimensions. Incidentally, you could use any coordinate system you want, +polar or otherwise, and get Newton's laws appropriate to that system right of +by seeing what happens if you have the shift zin radius, or in angle, etc. +“Similarly, the method can be generalized to any number of particles. If you +have, say, two particles with a force between them, so that there is a mutual +potential energy, then you just add the kinetic energy of both particles and take +the potential energy of the mutual interaction. And what do you vary? You vary +the paths of bofh particles. Then, for bwo particles moving In three dimensions, +there are six equations. You can vary the position of partiele 1 in the z-direction, +in the ø-direction, and in the z-direction, and similarly for particle 2; so there +are sỉix equations. And that?s as i§ should be. There are the three equations +that determine the acceleration of particle 1 in terms of the force on ït and three +for the acceleration of particle 2, from the force on it. You follow the same +game through, and you get Newton”s law in three dimensions for any number of +particles. +“[ have been saying that we get Newton's law. 'That is not quite true, because +Newton's law includes nonconservative forces like friction. Newton said that ma +1s equal to any #'. But the principle of least action only works Íor conseruafiue +systems——where all forces can be gotten from a potential function. You know, +however, that on a microscopic level—on the deepest level of physics—there +are no nonconservative forces. Nonconservative forces, like friction, appear only +because we neglect microscopic complications—there are just too many particles +to analyze. But the ƒfundamental laws can be put In the form of a prineiple of +least action. +“Let me generalize still further. Suppose we ask what happens If the particle +moves relativistically. We did not get the right relativistic equation of motion; +†}Ẻ =ma is only right nonrelativistically. The question is: Is there a corresponding +principle of least action for the relativistic case? 'There is. The formula in the +case of relativity is the following: +SK= —muẻ2 | v1I— 02/2 di — vị [0(+, 9, z,) — 0 - A(+,0,z, Đ)| di. +The frst part of the action integral is the rest mass mọ tỉmes c2 tỉmes the +integral of a function of velocity, 4/1 — ø^/c2. Then instead of just the potential +energy, we have an integral over the scalar potential ó and over times the +vector potential A. Of course, we are then including only electromagnetic forces. +AII electric and magnetic fields are given in terms of ô and A. This action +function gives the complete theory oÝ relativistic motion oŸ a single particle in an +electromagnetic field. +“Of course, wherever I have written ø, you understand that before you try +to figure anything out, you must substitute đz/đf for „ and so on for the other +components. Also, you put the point along the path at time £, z(f), (#), z(#) +where Ï wrote simply zø, , z. Properly, it is only after you have made those +replacements for the 0's that you have the formula for the action for a relativistic +particle. I will leave to the more ingenious of you the problem to demonstrate that +this action formula does, in fact, give the correct equations of motion for relativity. +May Ï suggest you do it first without the A, that is, for no magnetic feld? Then +you should get the components of the equation of motion, đp/đt = —q Vọ, where, +you remember, øØ = rngo/4/1 — 02/2. +“HE is much more dificult to include also the case with a vector potential. The +variatlons get mụch more complicated. But in the end, the force term does come +out equal to g(E + ø x ), as ¡9 should. But I will leave that for you to play +“I would like to emphasize that in the general case, for instance in the rela- +tivistic formula, the action integrand no longer has the form of the kinetic energy +--- Trang 238 --- +minus the potential energy. That”s only true in the nonrelativistic approximation. +Eor example, the term ?moc24/1 — 02/c2 is not what we have called the kinetic +energy. The question of what the action should be for any particular case must +be determined by some kind of trial and error. Ít is just the same problem as +determining what are the laws of motion in the frst place. You just have to +fñddle around with the equations that you know and see if you can get them into +the form of the principle of least action. +“One other poïint on terminology. The function that is integrated over time +to get the action ®Š is called the Eagrøngian, Ö, which 1s a function only of the +velocities and positions of particles. So the principle of least action is also written +s=Ï %(¿, 0¿) dt, +where by z; and ¿ are meant all the components of the positions and velocities. +So if you hear someone talking about the “Lagrangian,' you know they are +talking about the function that is used to ñnd ®Š. For relativistic motion in an +electromagnetic feld +2Ö = -moc2v1— 02/c2 — q(¿ — o- A). +“Also, I should say that Š is not really called the “action” by the most precise +and pedantic people. It ¡is called 'Hamilton's first principal functionˆ Now I hate +to give a lecture on “the-principle-of-least-Hamiltonˆs-first-principal-function” So +T call ít 'the action” Also, more and more people are calling it the action. You +see, historically something else which is not quite as useful was called the action, +but I think it's more sensible to change to a newer defnition. 5o now you too +will call the new function the action, and pretty soon everybody will call it by +that simple name. +“Ñow I want to say some things on this subject which are similar to the +discussions I gave about the principle of least time. 'There is quite a diference +in the characteristic of a law which says a certain integral from one place to +another is a minimum——which tells something about the whole path—and of a +law which says that as you go along, there is a force that makes it accelerate. +The second way tells how you inch your way along the path, and the other is a +grand statement about the whole path. In the case of light, we talked about the +connection of these two. Now, I would like to explain why it is true that there are ~ +diferential laws when there is a least action principle of this kind. 'The reason 1s +the following: Consider the actual path in space and time. As before, let?s take +only one dimension, so we can plot the graph of z as a function of ý. Along the +true path, Š is a minimum. Let's suppose that we have the true path and that it %Ö +goes through some point ø in space and time, and also through another nearby ^ Ï +poiïnt b. +Now ïf the entire integral from + to #¿ is a minimum, ï§ 1s also necessary that : + € +the integral along the little section from œø to 0 is also a minimum. I§ can t be « * +that the part from ø to b ïs a little bi more. Otherwise you could just fñddle +with just that piece of the path and make the whole integral a little lower. +“5o every subsection of the path must also be a minimum. And this is truc +no matter how short the subsection. 'Pherefore, the principle that the whole path +gives a minimum can be stated also by saying that an Iinfinitesimal section of +path also has a curve such that it has a minimum action. Now 1 we take a short +enouph section of path—between ÿwo points œ and 0 very close together—=how the +potential varies from one place to another far away is not the Important thing, +because you are staying almost in the same place over the whole little piece of +the path. The only thing that you have to discuss is the first-order change in the +potential. The answer can only depend on the derivative of the potential and +not on the potential everywhere. So the statement about the gross property of +the whole path becomes a statement of what happens for a short section of the +path—a diferential statement. And this diferential statement only involves the +--- Trang 239 --- +derivatives of the potential, that is, the force at a point. That”s the qualitative +explanation of the relation between the gross law and the diferential law. +“In the case of light we also discussed the question: How does the particle +fnd the right path? From the diferential point of view, it is easy to understand. +lvery moment it gets an acceleration and knows only what to do at that instant. +But all your instincts on cause and efect go haywire when you say that the +particle decides to take the path that is goïng to give the minimum action. Does +1t “smell" the neighboring paths to fñnd out whether or not they have more action? +In the case of light, when we put blocks in the way so that the photons could +not test all the paths, we found that they couldn't figure out which way to go, +and we had the phenomenon of difÑfraction. +“Is the same thing true in mechanics? Is it true that the particle doesn”t just +“take the right pathˆ but that it looks at all the other possible trajectories? And +1f by having things in the way, we don'$ let it look, that we will get an analog +of diÑfraction? “The miracle of it all is, of course, that it does Just that. That”s +what the laws of quantum mechanics say. So our principle of least action 1s +ineompletely stated. It isn't that a particle takes the path of least action but that +it smells all the paths in the neighborhood and chooses the one that has the least +action by a method analogous to the one by which light chose the shortest time. +You remember that the way light chose the shortest time was this: lÝ it went on +a path that took a diferent amount of time, it would arrive at a diferent phase. +And the total amplitude at some point is the sum of contributions of amplitude +for all the diferent ways the light can arrive. All the paths that give wildly +diferent phases don't add up to anything. But ïf you can fnd a whole sequence +of paths which have phases almost all the same, then the little contributions will +add up and you get a reasonable total amplitude to arrive. 'Phe important path +becomes the one for which there are many nearby paths which give the same +phase. +“E is just exactly the same thing for quantum mechanics. 'Phe complete +quantum mechanics (for the nonrelativistic case and neglecting electron spin) +works as follows: The probability that a particle starting at point 1 at the time +will arrive at point 2 at the time #¿ is the square of a probability amplitude. The +total amplitude can be written as the sum of the amplitudes for each possible +path—for each way of arrival. For every z(f) that we could have—for every +possible imaginary trajectory—we have to calculate an amplitude. 'Phen we +add them all together. What do we take for the amplitude for each path? Our +action integral tells us what the amplitude for a single path ought to be. The +amplitude is proportional to some constant times e?5/”, where Š is the action +for that path. 'Phat is, if we represent the phase of the amplitude by a complex +number, the phase angle is S/h. The action 9 has dimensions of energy tỉmes +time, and Planck”s constant Ö has the same dimensions. lt is the constant that +determines when quantum mechanies is important. +“Here is how it works: Suppose that for all paths, Š is very large compared +to ñ. One path contributes a certain amplitude. For a nearby path, the phase is +quite diferent, because with an enormous Š even a small change in Š means a +completely diferent phase—because ñ is so tiny. 5o nearby paths will normally +cancel their efects out in taking the sun——except for one region, and that is when +a path and a nearby path all give the same phase in the first approximation (more +precisely, the same action within Ö). Only those paths will be the important ones. +So in the limiting case in which Planck”s constant ñ goes to zero, the correcE +quantum-mechanical laws can be summarized by simply saying: “Eorget about +all these probability amplitudes. “The particle does go on a special path, namely, +that one for which Š does not vary in the frst approximation That”s the relation +between the principle of least action and quantum mechanics. 'The fact that +quantum mechanics can be formulated in this way was discovered in 1942 by a +student of that same teacher, Bader, I spoke of at the beginning of this lecture. +[Quantum mechanics was originally formulated by giving a diferential equation +for the amplitude (Schrödinger) and also by some other matrix mathematics +(Heisenberg).] +--- Trang 240 --- +“NÑow I want to talk about other minimum prineiples in physics. Thhere are +many very interesting ones. I will not try to list them all now but will only +describe one more. Later on, when we come to a physical phenomenon which has +a nice minimum principle, I will tell about it then. Ï want now to show that we +can describe electrostatics, not by giving a diferential equation for the fñeld, but +by saying that a certain integral is a maximum or a minimum. First, let°s take +the case where the charge density is known everywhere, and the problem is to +fnd the potential ó everywhere in space. You know that the answer should be +V”ó = —p/eo. +But another way of stating the same thing is this: Calculate the integral U*, +U*=S [(Vd)?dV— [ nöat +which is a volume integral to be taken over all space. 'This thíng is a minimum +for the correct potential distribution ð(z, 9, 2). +“We can show that the §wo sbatements about electrostatics are equivalent. +Let”s suppose that we pick any function ó. We want to show that when we take +for ó the correct potential ó, plus a small deviation ƒ, then in the first order, the +change in U is zero. So we write +¿=ó+ƒ. +The ở is what we are looking for, but we are making a variation of it to find +what it has to be so that the variation of Ư* is zero to first order. Eor the first +part of Ư”, we need +(Vớ)” = (Vó)”“+2V¿- Vƒ + (VỰ)”. +'The only first-order term that will vary is +2V¿- VỸ. +In the second term of the quantity *, the integrand is +0Q = 0Ó + 0ƒ, +whose variable part is øƒ. 5o, keeping only the variable parts, we need the +integral +AU*= JtaYo-Vf~øf) dV. +“Now, following the old general rule, we have to get the darn thing all clear +of derivatives of ƒ. Let's look at what the derivatives are. 'Phe dot product is +ÔÒ ðƒ„ Đó 0ƒ „ 90 0ƒ +9z Ôxz ØụÔy— Ôz Ôz' +which we have to integrate with respect to ø, to , and to z. Now here is the +trick: to get rid of ؃/Ø+ we integrate by parts with respect to z. That will +carry the derivative over onto the Ộ. Tt”s the same general idea we used to get +rid of derivatives with respect to ý. We use the equality +Đó ôð lôi 82 +lồng cth=1np— [Tan +3z Ô+z Øz 8z2 +The integrated term is zero, since we have to make ƒ zero at infnity. (That +corresponds to making 7 zero at ý and f¿. So our principle should be more +accurately siated: is less for the true ó than for any other ó(z, , z) having +the same values at infnity.) Then we do the same thing for and 2z. Šo our +integral AU* is +AU*= Jtcev?o — p)ƒ dV. +--- Trang 241 --- +In order for this variation to be zero for any ƒ, no matter what, the coefficient +of ƒ must be zero and, therefore, +V”¿ = —0/so. +W©e get back our old equation. So our “minimum” proposition is correctf. +“We can generalize our proposition if we do our algebra in a little diferent +way. Let's go back and do our integration by parts without taking components. +W© start by looking at the following equality: +V:(ƒVó) = Vƒ: Vó+ ƒ V2. +Tf I diferentiate out the left-hand side, I can show that it is just equal to the +ripht-hand side. Now we can use this equation to integrate by parts. In our +integral AU*, we replace Vó - Vƒ by V - (ƒ Vớỏ) — ƒ V”ó, which gets integrated +over volume. The divergence term integrated over volume can be replaced by a +surface integral: +Jv -(ƒ Wð) dV = | 2V: nàn +Since we are integrating over all space, the surface over which we are integrating +1s at Infinity. There, ƒ is zero and we get the same answer as before. +“Only now we see how to solve a problem when we đon?£ know where all the +charges are. Suppose that we have conductors with charges spread out on them +in some way. We can still use our minimum principle if the potentials of all the +conductors are ñxed. We carry out the integral for Ứ only in the space outside +of all conductors. Then, since we can” vary ó on the conductor, ƒ is zero on all +those surfaces, and the surface integral +J ƑVọ-nda +1s still zero. The remaining volume integral +AU*= Jco V?ó@— p)ƒ dV +1s only to be carried out in the spaces between conductors. Of course, we get +Poissonˆs equation again, +V°ó = —j/sạ. +So we have shown that our original integral Ư is also a minimum if we evaluate +it over the space outside of conductors all at ñxed potentials (that is, such that +any trial ó(z, , z) must equal the given pobential of the conductors when (%, , 2) +is a point on the surface oŸ a conductor). +“There is an interesting case when the only charges are on conductors. Then +U*= 3 J(Vỏ)?dt, +Our minimum principle says that in the case where there are conductors set +at certain given potentials, the potential between them adjusts itself so that +integral U is least. What is this integral? The term Wóộ is the electric feld, +so the integral is the electrostatic energy. The true field is the one, of all those =3} +coming from the gradient of a potential, with the minimum total energy. +“I would like to use this result to calculate something particular to show you +that these things are really quite practical. Suppose I take ©wo conductors in the +form of a cylindrical condenser. — +The inside conductor has the potential V, and the outside is at the potential +zero. Let the radius of the inside conductor be a and that of the outside, b. Now +we can suppose ønw distribution of potential between the two. lÝ we use the +correct ó, and caleulate eo/2 ƒ (Vớ)? dV, it should be the energy of the system, +--- Trang 242 --- +3CV}. So we can also calculate Œ by our principle. But iŸ we use a wrong +distribution of potential and try to calculate the capacity Œ by this method, we +will get a capacity that is too big, since V is specifed. Any assumed potential ¿ +that is not the exactly correcE one will give a fake Œ that is larger than the +correct value. But if my false ở is any rough approximation, the Œ will be a good +approximation, because the error in is second order in the error in ở. +“Suppose I don 't know the capacity of a cylindrical condenser. Ï can use +this principle to fnd it. Ijust guess at the potential function ø until Ï get the +lowest Œ. 5uppose, for instance, I pick a potential that corresponds to a constant +feld. (You know, of course, that the fñeld isnˆt really constant here; i% varies +as l/r.) A field which is constant means a potential which goes linearly with +distance. 'To ñt the conditions at the two conductors, it must be +¿=WV ( b— 3 : +This function 1s W at rz = aø, zero at z = ð, and in between has a constant sÌlope +equal to —V/(b— a). So what one does to find the integral U* is multiply the +square of this gradient by co/2 and integrate over all volume. Let?s do this +calculation for a cylinder of unit length. A volume element at the radius r +1s 2r dr. Doing the integral, I ñnd that my frst try at the capacity gIves +1 2 €0 k V2 +5 ŒCV“(first try) = 3J (b—a)? 2mr dừ. +'The integral is easy; it is jus$ +V2 (5) - +So Ï have a formula for the capacity which is not the true one but is an approximate +Job: +2mco 2(b— a)' +It is, naturally, diferent from the correct answer Œ = 27co/ln(b/a), but its not +too bad. Let”s compare it with the right answer for several values of b/a. I have +computed out the answers in this table: +b Ctrue C(frst approx.) +bề) 27€o 27€o +2 1.4423 1.500 +4 0.721 0.833 +10 0.434 0.612 +100 0.217 0.51 +1.5 2.4662 2.50 +1.1 10.492059 10.500000 +ven when b/ø is as big as 2—which gives a pretty big variation in the fñield +compared with a linearly varying field—I get a pretty fair approximation. 'Phe +answer is, of course, a little too high, as expected. The thing gets much worse If +you have a tỉny wire inside a big cylinder. Then the fñeld has enormous variations +and iÝ you represent it by a constant, youre not doïng very well. With b/a = 100, +we re off by nearly a factor of two. Things are much better for small b/ø. To take +the opposite extreme, when the conductors are not very far apart—say b/a = 1.1— +then the constant field is a pretty good approximation, and we get the correct +value for Œ to within a tenth of a percent. +“Ñow I would like to tell you how to improve such a calculation. (Of course, +you &nou the right answer for the cylinder, but the method is the same for some +other odd shapes, where you may not know the right answer.) The next step is +--- Trang 243 --- +to try a better approximation to the unknown true ó. Eor example, we might try +a constant plus an exponential ó, etc. But how do you know when you have a +better approximation unless you know the true @? Answer: You calculate Œ; the +lowest Œ is the value nearest the truth. Let us try this idea out. Suppose that +the potential is not linear but say quadratic in r—that the electric field is not +constant but linear. The most generøl quadratic form that fits = 0 atr—b +and ¿= V atr=ais +r—d r—a +¿=vli+a(—) -u+a(—) | +b—wœ b—wœ +where œ is any constant number. 'Phis formula is a little more complicated. lt +involves a quadratic term in the potential as well as a linear term. Ït is very easy +to get the field out of it. The field is Just +đó œVW (r— a)V +E= dự — Am. (b— a)2ˆ +Now we have to square this and integrate over volume. But wait a moment. +'What should I take for œ? I can take a parabola for the ở; but what parabola? +Here's what I do: Calculate the capacity with an arbftraru œ. What T get is +lÕi a [b(o? 2œ 1 . +.-.. nh )tạ° tả +It looks a little complicated, but it comes out of integrating the square of the +fñeld. Now I can pick my ơ. I know that the truth lies lower than anything that +Tam going to caleulate, so whatever Ï put ïn for œ is goïng to give me an answer +too bịg. But ïf I keep playing with œ and get the lowest possible value I can, that +lowest value is nearer to the truth than any other value. So what I do next is +to pick the œ that gives the minimum value for Œ. Working ït out by ordinary +calculus, I get that the minimum Œ occurs for œ = —2b/(b + ø). Substituting +that value into the formula, I obtain for the minimum capacity +lồi b2 + 4ab + a2 +2m 3(b2— a2) ` +“[ve worked out what this formula gives for Œ for various values of b/a. I call +these numbers C(quadratic). Here is a table that compares C(quadratic) with +the true C. +b Ctrue C(quadratic) +a 27cp 27g +2 1.4423 1.444 +4 0.721 0.733 +10 0.434 0.475 +100 0.217 0.346 +1.5 2.4662 2.4667 +1.1 10.492059 10.492065 +“For example, when the ratio of the radii is 2 to 1, I have 1.444, which is a +very good approximation to the true answer, 1.4423. Even for larger b/a, it stays +pretty good——it is much, much better than the frst approximation. ϧ is even +fairly good—only of by 10 percent—when Ö/a is 10 to 1. But when it gets to be +100 to 1—well, things begin to go wild. I get that Œ is 0.346 instead of 0.217. +Ơn the other hand, for a ratio of radii of 1.5, the answer is excellent; and for +a b/a of 1.1, the answer comes out 10.492065 instead of 10.492059. Where the +answer should be good, it is very, very good. +“I have given these examples, first, to show the theoretical value of the +principles of minimum action and minimum principles in general and, second, +--- Trang 244 --- +to show their practical utility—not Just to calculate a capacity when we already +know the answer. For any other shape, you can guess an approximate fñeld with +some unknown parameters like œ and adjust them to get a minimum. You will +get excellent numerical results for otherwise intractable problems.” +19-2 A note added after the lecture +“TI should like to add something that I didn't have tỉme for in the lecture. (I +always seem to prepare more than I have time to tell about.) As I mentioned +earlier, Ï got interested in a problem while working on this lecture. Ï want to +tell you what that problem is. Among the minimum principles that I could +menfion, Ï noticed that most of them sprang in one way or another from the +least action prineiple of mechanics and electrodynamics. But there is also a class +that does not. As an example, 1Ý currents are made to go through a piece of +material obeying Ohm's law, the currents distribute themselves inside the piece +so that the rate at which heat is generated is as little as possible. Also we can +say (If things are kept isothermal) that the rate at which energy is generated +1s a minimum. Now, this principle also holds, according to classical theory, in +determining even the distribution of velocities of the electrons inside a metal +which is carrying a current. “The distribution of velocities is not exactly the +cquilibrium distribution [Chapter 40, Vol. I, Eq. (40.6) because they are drifting +sideways. 'The new distribution can be found from the principle that ï§ is the +distribution for a given current for which the entropy developed per second by +collisions is as smaill as possible. 'The true description of the electrons” behavior +ought to be by quantum mechanics, however. The question is: Does the same +principle of minimum entropy generation also hold when the situation is described +quantum-mechanically? I havenˆt found out yet. +“The question is interesting academically, of course. Such principles are +fascinating, and it is always worth while to try to see how general they are. +But also from a more practical point of view, Ï øøn‡ to know. Ï, with some +colleagues, have published a paper in which we calculated by quantum mechanics +approximately the electrical resistance felt by an electron moving through an ionie +crystal like ÑaCl. [Feynman, Hellwarth, Iddings, and Platzman, “Mobility of Slow +Electrons in a Polar Crystal,” Phụs. Reo. 127, 1004 (1962).] But if a minimum +principle existed, we could use it to make the results much more accurate, Just as +the minimum principle for the capacity of a condenser permitted us 0o get such +accuracy for that capacity even though we had only a rough knowledge of the +electric ñeld” +--- Trang 245 --- +Seœlrrfforts @œŸ /Weveee©ollˆs Eqrretffores íre Froo +Speree© +20-1 Waves ỉn free space; pÏane waves +In Chapter 1S we had reached the point where we had the Maxwell equations 20-1 Waves in free space; plane waves +in complete form. All there is to know about the classical theory of the electric 20-2 Three-dimensional waves +and magnetic ñelds can be found in the four equations: 20-3 Scientifc imagination +p 9B 20-4 Spherical waves +1. V BS TL VxE=—n +„ (20.1) +HIL W-B=0 IV. ƒýWwxp-1+° +'When we put all these equations together, a remarkable new phenomenon ocCurS: Refcrences: Chapter 17, Vol. Ï: Sownd: +fñelds generated by moving charges can leave the sources and travel alone through The Waue Equation +space. We considered a special example in which an infnite current sheet 1s Chapter 28, Vol. I: Bilec- +suddenly turned on. After the current has been on for the time ý, there are tromagnetic Radiation +uniform electric and magnetic fñelds extending out the distance cý from the source. +Suppose that the current sheet lies in the z-plane with a surface current density +going toward positive . The electric ñeld will have only a -component, and the IE| = c|B| +magnetic field, only a z-component. “The field components are given by +1 =cB„ = Đcạc" (20.2) +for positive values of z less than cý. For larger # the fñelds are zero. There are, +Of course, similar felds extending the same distance from the current sheet in " «đx +the negative zø-direction. In Fig. 20-1 we show a graph of the magnitude of the Fig. 20-1. The electric and magnetic field +fields as a function of z at the instant ý. Às time goes on, the “wavefront” at cÝ asia function Of x at the time £ after the +moves outward in ø at the constant velocity e. current sheet is turned on. +Now consider the following sequence of events. We turn on a current of unit +strength for a while, then suddenly increase the current strength to three units, +and hold it constant at this value. What do the fields look like then? We can see E +what the fñelds will look like in the following way. First, we imagine a current 2 +of unit strength that is turned on at ý = 0 and left constant forever. The fñelds +for positive ø are then given by the graph in part (a) of Eig. 20-2. Next, we ask : +what would happen 1Ý we turn on a steady current of 6wo units at the time . °ọ ta) cEx +The fields in this case will be t©wice as high as before, but will extend out E +in # only the distance c(£ — #1), as shown in part (b) of the fñgure. When we add +these two solutions, using the principle of superposition, we ñnd that the sum of ? +the Ewo sources is a current of one unit for the time from zero to #‡ and a current 1 +of three units for times greater than ¡. At the time # the fields will vary with ø 0 cŒ-h) = +as shown in part (c) of Fig. 20-2. E œ) +Now let's take a more complicated problem. Consider a current which is 3 +turned on to one unit for a while, then turned up to three units, and later turned 2 +of to zero. What are the fñelds for such a current? We can ñnd the solution in 1 +the same way——by adding the solutions of three separate problems. First, we 0 cứcn) HT: +fñnd the fields for a step current of unit strength. (We have solved that problem (c) +already.) Next, we ñnd the fields produced by a step Current of two units. Finally, Fig. 20-2. The electric field of a current +we solve for the fields of a step current of mznus three units. When we add the . +. . ¬^ l sheet. (a) One unit of current turned on +three solutions, we will have a current which is one unit strong from ý = 0 to at £ = 0; (b) TWO units of current turned on +some later time, say í¡, then three units strong until a still later time #2, and at £ = t¡; (c) Superposition of (a) and (b). +--- Trang 246 --- +0 ñ f› t 0 c(t—ta) c(—#i) ct x +(a) () +Fig. 20-3. lf the current source strength varies as shown in (a), then at the time £ +shown by the arrow the electric field as a function of x is as shown in (b). +then turned of—that is, to zero. À graph of the current as a function of tỉme is +shown in EFig. 20-3(a). When we add the three solutions for the electric field, we +fñnd that its variation with #, at a given instant ý, is as shown in Fig. 20-3(b). +The field is an exact representation of the current. 'Phe fñeld distribution in space +is a nice graph of the current variation with time—only drawn backwards. Às +time goes on the whole picture moves outward at the speed e, so there is a little +blob of fñeld, travelling toward positive z, which contains a completely detailed +memory of the history of all the current variations. If we were to stand miles +away, we could tell from the variation of the electric or magnetic fñeld exactly +how the current had varied at the source. +You will also notice that long after all activity at the source has completely +stopped and all charges and currents are zero, the block of feld continues to +travel through space. We have a distribution of electric and magnetic ñelds that +exist independently of any charges or currents. That is the new effect that comes +from the complete set of Maxwell°s equations. If we want, we can give a complete +mathematical representation of the analysis we have just done by writing that +the electric field at a given place and a given time is proportional to the current +ab the source, only not at the sưme tìme, but at the eøarljer tỉme £— #/c. We can +wrIt© 1 /2) +1(#) = __——x (20.3) +W©e have, believe it or not, already derived this same equation from another +point of view in Vol. I, when we were dealing with the theory of the Index of +refraction. 'Then, we had to fñgure out what fñelds were produced by a thin +layer of oscillating dipoles in a sheet of dielectric material with the dipoles set in +motion by the electric fñeld of an incoming electromagnetic wave. Qur problem +was to calculate the combined felds of the original wave and the waves radiated +by the oscillating dipoles. How could we have calculated the fields generated by +moving charges when we didn't have Maxwells equations? At that time we took +as our starting point (without any derivation) a formula for the radiation fñelds +produced at large distances from an accelerating point charge. lf you will look in +Chapter 31 of Vol. I, you will see that Eq. (31.9) there is just the same as the +Eq. (20.3) that we have just written down. Although our earlier derivation was +correct only at large distances from the source, we see now that the same result +continues to be correct even right up to the source. +W© want now to look in a general way at the behavior of electric and magnetiec +fields in empty space far away from the sources, i.e., from the currents and charges. +Very near the sources—near enough so that during the delay in transmission, the +source has not had time to change much—the fñelds are very much the same as +we have found in what we called the electrostatic or magnetostatic cases. If we go +out to distances large enough so that the delays become important, however, the +nature of the fñelds can be radically diferent from the solutions we have found. +In a sense, the fields begin to take on a character of their own when they have +gone a long way from all the sources. So we can begin by discussing the behavior +of the fields in a region where there are no currents or charges. +--- Trang 247 --- +Suppose we ask: What kind of fñelds can there be in regions where ø and 7 +are both zero? In Chapter 18 we saw that the physics of Maxwell's equations +could also be expressed in terms of differential equations for the scalar and vector +potentials: +1 Ø2 0 +W?2¿—-- =-h 20.4 +ở c2 9:2 sọ” ( ) +1 8A 3 +V?A--=—===-—-—. 20. +c2 Ø2 cọc2 (20-5) +TÝ ø and 7 are zero, these equations take on the simpler form +V¿— =—=0 20.6 +; c2 012 : (20.6) +1 ؈A +A—-=—==Ô0. 20. +V 5 0p 0 (20.7) +Thus in free space the scalar potential ¿ and each component of the vector +potential A all satisfy the same mathematical equation. Suppose we let ÿ (psi) +stand for any one of the four quantities ó, A„, Á„, Áz; then we want to investigate +the general solutions of the following equation: +VẺụ~— —s =0. 20. +ú c2 Ø2 (20.8) +'This equation is called the three-dimensional wave equatilon——three-dimensional, +because the function may depend in general on ø, , and z, and we need to +worry about variations in all three coordinates. 'Phis is made clear if we write +out explicitly the three terms of the Laplacilan operatOor: +ý 0U Ø0 1 Ø0 +———_—=(0. 20.9 +9+2 + Øụ2 + 9z2 c2 Ô12 ( ) +In free space, the electric felds # and #Ö also satisfy the wave equation. For +example, since Ö = V x A, we can get a diferential equation for Ö by taking +the curl of Ðq. (20.7). Since the Laplacian is a scalar operator, the order oŸ the +Laplacian and curl operations can be interchanged: +V x(V?A) = V”(V x A) = Vˆ?B. +Similarly, the order of the operations curl and Ø/Ø£ can be interchanged: +1 ØA 1 Ø2 1 0%B +Vx=—-s=_-.-z(Vx4)=-=—. +c2 Ø2 c2 Ø2 ) c7 ði2 +Using these results, we get the following diferential equation for Ö: +1 0°B +V?B—- — —_—=U0. 20.10 +c2 Ø12 ) +So each component of the magnetic field Ö satisfes the three-dimensional wave +cquation. Similarly, using the fact that = —Wó — ØA/ðt, ¡it follows that the +electric feld # in free space also satisfies the three-dimensional wave equation: +1 Ø*E +V?E— s—s =0. 20.11 +c2 Ø2 (20.11) +AlI of our electromagnetic fields satisfy the same wave equation, Eq. (20.8). +W©e might well ask: What ¡is the most general solution to this equation? However, +rather than tackling that dificult question right away, we will look fñrst at what +can be said in general about those solutions in which nothing varies in # and z. +(Always do an easy case first so that you can see what is goïng to happen, and then +you can go to the more complicated cases.) Let's suppose that the magnitudes of +the fñelds depend only upon ø—that there are no 0øar?af2ons oŸ the fñelds with +--- Trang 248 --- +and z. W© are, of course, considering plane waves again. We should expect to get +results something like those in the previous section. In fact, we will fnd precisely +the same answers. You may ask: “Why do it all over again?” It is important to +do it again, fñrst, because we did not show that the waves we found were the most +general solutions for plane waves, and second, because we found the fields only +from a very particular kind of current source. We would like to ask now: What +is the most general kind of one-dimensional wave there can be in free space? We +cannot fñnd that by seeing what happens for this or that particular source, but +must work with greater generality. Also we are going to work this time with +diferential equations instead of with integral forms. Although we will get the +same results, i% is a way of practicing back and forth to show that it doesn 6 make +any diference which way you go. You should know how to do things every which +way, because when you get a hard problem, you will often fnd that only one of +the various ways is tractable. +WS could consider directly the solution of the wave equation for some elec- +tromagnetic quantity. Instead, we want to start right from the beginning with +Maxwell's equations in free space so that you can see their close relationship to +the electromagnetic waves. So we start with the equations in (20.1), setting the +charges and currents equal to zero. 'Phey become +1. V.:E=0 +II. _VxE= _= +(20.12) +HIL V:B=0 +IV. cẪẦVxB= = +We write the first equation out in components: +9l„ Ô0h, 0E, +V.E= Đa + ðy + 2z =0. (20.13) +W© are assuming that there are no variations with and z, so the last two terms +are zero. Phis equation then tells us that +=— 0. (20.14) +lts solution is that !„, the component of the electric field in the zø-direction, is a +constant in space. If you look at TV in (20.12), supposing no -variation in +and z either, you can see that ly is also constant in time. Such a fñield could be +the steady DC field from some charged condenser plates a long distance away. We +are not interested now in such an uninteresting static ñeld; we are at the moment +interested only in dynamically varying fields. For dựụngmiức fields, !„ = 0. +W© have then the important result that for the propagation of plane waves In +any direction, the electric field must be a‡ right angles to the dieclion oj propa- +gation. Tt can, of course, sfill vary in a complicated way with the coordinate z. +'The transverse E-feld can always be resolved into bwo components, say the +u-component and the z-component. So let”s first work out a case in which the +electric fñeld has only one transverse component. We'll take frst an electric fñeld +that is always in the #-direction, with zero z-component. Evidently, if we solve +this problem we can also solve for the case where the electric fñeld is always in the +z-direction. The general solution can always be expressed as the superposition of +two such fields. +How easy our equations now get. The only component of the electric field +that is not zero is #„, and all derivatives——except those with respect to #—are +zero. The rest of Maxwells equations then become quite simple. +--- Trang 249 --- +Let”s look next at the second oŸ Maxwell's equations [II of Eq. (20.12)]. Writing +out the components of the curl , we have +8E 8E +WxE)„=—ˆ- “=0, +( ) li Øz +9l„ ÔE, +VxE),=——_—-—=——=0 +(V x9), 9z 3z Í +8E 8E 8E +WxE),= .„“”—--_ =-.., +( ) 3z ỡy Øz +The z-component of V x # is zero because the derivatives with respect to +and z are zero. The -component is also zero; the first term is zero because the +derivative with respect to z is zero, and the second term is zero because „ is +zero. The only components of the curÌ of that is not zero is the z-component, +which is equal to Ø#⁄„/Øz. Setting the three components of V x # equal to the +corresponding components of —Ø/6Ô, we can conclude the following: +3B 9B +— =0 — =0. 20.15 +ðt l ðt ' ) +8B, ðEy +———=_———.. 20.16 +lôIU Øz ' ) +Since the z-component of the magnetic fñeld and the -component of the magnetic +field both have zero time derivatives, these two components are just constant +fñelds and correspond to the magnetostatic solutions we found earlier. Somebody +may have left some permanent magnets near where the waves are propagating. +We will ignore these constant fields and set „ and Ö„ cqual to zero. +Incidentally, we would already have concluded that the z-component of +should be zero for a different reason. Since the divergence of Ö is zero (from the +thiưd Maxwell equation), applying the same arguments we used above for the +electric fñeld, we would conclude that the longitudinal component of the magnetic +ñeld can have no variation with z. 5ince we are ignoring such uniform fñelds in +our wave solutions, we would have set ö„ equal to zero. In plane electromagnetic +waves the Ö-field, as well as the E-field, must be directed at right angles to the +direction of propagation. +Equation (20.16) gives us the additional proposition that if the electric field +has only a -component, the magnetic field will have only a z-component. So E2 +and Ð are a‡ right angles to each other. 'This is exactly what happened in the +special wave we have already considered. +W© are now ready to use the last of Maxwell's equations for free space [IV of +Eq. (20.12)]. Writing out the components, we have +8B 3B 8E +2 2 zZ 2 1ụ “ +VxB);y=c-~“-đ-==—~.. +cq ) “ Øy “ øz lôIU +3B 8B 8E +2 2 b5 2 z 1U +WxB),=cˆ—*“-c°—`-=-._” 20.17 +CWWxBìy=C Tp TU Tân = Tẩy 0.7) +3B 3B 8E +2 2a du 2a Gz z +VxB);=cˆ——-c——=_—. +at }z= gy —“ ray — ấy +Of the six derivatives of the components of Ö, only the term ØÖ;/Øz is not equal +to zero. 5o the three equations give us simply +8B 8E +2 0z ụ +— = : 20.18 +Ý "8m Øt ' ) +The result of all our work is that only one component each of the electric and +magnetic fields is not zero, and that these components must satisfy Eqs. (20.16) +and (20.18). The two equations can be combined into one iƒ we diferentiate the +first with respect to #z and the second with respect to ; the left-hand sides of +--- Trang 250 --- +the two equations will then be the same (except for the factor c2). So we find +that „ satisles the equation +0E 1 3F +=s Tra am =0. (20.19) +9x2 c2 012 +'W©e have seen the same diferential equation before, when we studied the propa- +gatlon of sound. It is the wave equation for one-dimensional waves. +You should note that in the process of our derivation we have found something +more than 1s contained in Bq. (20.11). Maxwell's equations have given us the +further information that electromagnetic waves have fñeld components only at +right angles to the direction of the wave propagation. +Let's review what we know about the solutions oŸ the one-dimensiona] wave +cquation. If any quantity j satisfies the one-dimensional wave equation +82 1 Ø +0w _ L6 ự =0, (20.20) +9x2 c2 Ø12 +then one possible solution is a funection (+, £) of the form +that is, some function oŸ the s/ngle variable (œ — cf). The function ƒ(+ — c£) +represents a “rigid” pattern in z which travels toward positive ø at the speed e +(see Eig. 20-4). For example, if the function ƒ has a maximum when its argument +is zero, then for ý = 0 the maximum of +, will occur at ø =0. A% some later +tỉme, say = 10, will have its maximum at # = 10c. As time goes on, the +maximum moves toward positive ø at the speed e. f c+ Ị +Sometimes it is more convenient to say that a solution of the one-dimensional | +wave equation is a function oŸ (£ — #/c). However, this is saying the same thing, II YN +because any function of (£ — #/c) is also a function of ( — cÊ): ‹ = +_— £ [8 `_—” No” Xx +PŒ~ z/e) = tr. = ƒ(z— et). +ẹ Fig. 20-4. The function f(x — ct) repre- +Let”s show that ƒ(œ — c#) is indeed a solution of the wave equation. Since it is _ 3 conetan S1ape nhạt travels toward +a function of only one variable—the variable (œ — c£)—we will let ƒ” represent the pOSIIVS XU °9PSSS C- +derivative of ƒ with respect to its variable and ƒ” represent the second derivative +of ƒ. Diferentiating Eq. (20.21) with respect to z, we have +——= jJ(z-—ci), +since the derivative of (œ — c#) with respect to ø is 1. The second derivative oŸ ÿ, +with respect to ø 1s clearly +3 = ƒ “(z - et). (20.22) +'Taking derivatives of ý with respect to ý, we find +S =ffœ~ đ)(—9), += = +€?ƒ”(œ — e). (20.23) +We see that ý does indeed satisfy the one-dimensional wave equation. +You may be wondering: “If [ have the wave equation, how do Ï know that I +should take ƒ(œ— c£#) as a solution? T don't like this backward method. Isn't there +some ƒorard way to fñnd the solution?” Well, one good forward way is to know +the solution. It is possible to “cook up” an apparently forward mathematical +argument, especially because we know what the solution is supposed to be, but +with an equation as simple as this we don't have to play games. 5oon you will get +--- Trang 251 --- +so that when you see lq. (20.20), you nearly simultaneously see = ƒ(% — #£) +as a solution. (Just as now when you see the integral of #2 dz, you know right +away that the answer is #Ở/3.) +Actually you should also see a little more. NÑot only is any function oŸ (œ — c£) +a solution, but any function of (4 - œ£) is also a solution. Since the wave equation +contains only c2, changing the sign of e makes no diference. In fact, the mosf +general solution of the one-dimensional wave equation is the sum of two arbitrary +functions, one oŸ (œ — c#) and the other of (œ + c£): +q = ƒ(œ — cÈ) + g(+ + cl). (20.24) +The first term represents a wave travelling toward positive zø, and the second +term an arbitrary wave travelling toward negative ø. The general solution is the +superposition of two such waves both existing at the same tỉme. +We will leave the following amusing question for you to think about. Take a +function ÿ of the following form: +4 = cos kz# cos kct. +This equation isn't in the form of a function of (œ — e£) or of (+ c£). Yet you can easily +show that this function is a solution of the wave equation by direct substitution into +Eq. (20.20). How can we then say that the general solution is of the form of Eq. (20.24)? +Applying our conclusions about the solution of the wave equation to the +-component of the electric field, „, we conclude that 2y can vary with # in any +arbitrary fashion. However, the felds which do exist can always be considered +as the sum of two patterns. One wave is sailing through space in one direction +with speed c, with an associated magnetic feld perpendicular to the electric +field; another wave is travelling in the opposite direction with the same speed. +Such waves correspond to the electromagnetic waves that we know about—light, +radiowaves, infrared radiation, ultraviolet radiation, x-rays, and so on. We have +already discussed the radiation of light in great detail in Vol. I. 5ince everything +we learned there applies to any electromagnetic wave, we donˆt need to consider +in great detail here the behavior of these waves. +'W© should perhaps make a few further remarks on the question of the polar- +1zation of the electromagnetic waves. In our solution we chose to consider the +special case in which the electric ñeld has only a -component. 'Phere is clearly +another solution for waves travelling in the plus or minus z-direction, with an +electric field which has only a z-component. Since Maxwell's equations are linear, +the general solution for one-dimensional waves propagating in the z-direction is +the sum of waves oŸ „and waves of #„. Thịs general solution is summarized in +the following equations: +t= (0, đưy, E,) +Tụ = ƒ(œ — c£) + g(œ + ct‡) +1y = F(œ — cÈ) + G(z + c‡) +: (20.25) +B= (0, Dụ, B,) +cB; = ƒ(œ — cÈ) — g(z + c‡) +cñy = —F(z — ct) + G(œ + ct). +Such electromagnetic waves have an #-vector whose direction is not constant but +which gyrates around in some arbitrary way in the zz-plane. At every point the +magnetic field is always perpendicular to the electric field and to the direction of +propagation. +Tí there are only waves travelling in one direction, say the positive ø-direction, +there is a simple rule which tells the relative orlentation of the electric and +--- Trang 252 --- +magnetic felds. The rule is that the cross product # x ——which is, of course, +a vector at right angles to both and ——points in the direction in which the +wave is travelling. If is rotated into Ö by a right-hand screw, the screw points +in the direction of the wave velocity. (We shall see later that the vector E x +has a special physical signifcanee: i% is a vector which describes the ow of energy +in an electromagnetic feld.) +20-2 Three-dimensional waves +We want now to turn to the subject of three-dimensional waves. We have +already seen that the vector # satisfies the wave equation. Ït is also easy to arrive +at the same conclusion by arguing directly from Maxwells equations. Suppose +we start with the equation +VxE-=-— +and take the curl of both sides: +Vx(VWxE)==2.(V x ). (20.26) +You will remember that the curl of the curl of any vector can be written as the +sum of two terms, one involving the divergence and the other the Laplacian, +Vx(VxE)=YV(V-E) - V°E. +In free space, however, the divergence of # is zero, so only the Laplacian term +remains. Also, from the fourth of Maxwells equations in free space [Eq. (20.12)] +the time derivative of c2 W x is the second derivative of E with respect to ¿: +2ỡ (VxB)= nE +ˆ Ø — 0` +Equation (20.26) then becomes +V?ˆE=— ——, +which is the three-dimensional wave equation. Written out ín all its glory, this +cequation is, Of course, +0?E 60?3E 0E 1ØE +—s + >„x+—>-x_—:z az=U. (20.27) +8x2 — Øy? 8z? c2 02 +How shall we find the general wave solution? "The answer is that all the +solutions of the three-dimensional wave equation can be represented as a superpo- +sition of the one-dimensional solutions we have already found. We obtained the +equation for waves which move in the #z-direction by supposing that the fñeld did +not depend on and z. Obviously, there are other solutions in which the fields +do not depend on #z and z, representing waves going in the ¿-direction. hen +there are solutions which do not debpend on z and ø, representing waves travelling +in the z-direction. Or in general, since we have written our equations in vector +form, the three-dimensional wave equation can have solutions which are plane +waves moving in any direction at all. Again, since the equations are linear, we +may have simultaneously as many plane waves as we wish, travelling in as many +diferent directions. Thus the most general solution of the three-dimensional +wave equation is a superposition of all sorts of plane waves moving in all sorts of +directions. +Try to imagine what the electric and magnetic ñelds look like at present in +the space in this lecture room. Pirst of all, there is a steady magnetic field; +it comes from the currents in the interior of the earth—that 1s, the earth”s +steady magnetic fñeld. 'Phen there are some irregular, nearly static electric ñelds +produced perhaps by electric charges generated by fiction as various people move +--- Trang 253 --- +about in their chairs and rub their coat sleeves against the chair arms. hen +there are other magnetic fields produced by oscillating currents in the electrical +wiring—fñelds which vary at a Írequency of 60 cycles per second, in synchronism +with the generator at Boulder Dam. But more interesting are the electric and +magnetic felds varying at much higher frequencies. Eor instance, as light travels +from window to foor and wall to wall, there are little wiggles of the electrie and +magnetic ñelds moving along at 186,000 miles per second. Then there are also +infrared waves travelling from the warm foreheads to the cold blackboard. And +we have forgotten the ultraviolet light, the x-rays, and the radiowaves travelling +through the room. +Flying across the room are electromagnetic waves which carry music 0Ÿ a jaZ2 +band. 'Phere are waves modulated by a series of impulses representing pictures +of events going on in other parts of the world, or of iImaginary aspirins dissolving +in imaginary stomachs. To demonstrate the reality of these waves i% is only +necessary to turn on electronic equipment that converts these waves Into pictures +and sounds. +TÍ we go into further detail to analyze even the smallest wiggles, there are tỉny +electromagnetic waves that have come into the room from enormous distances. +'There are now tiny oscillations of the electric ñeld, whose crests are separated by +a distance of one foot, that have come from millions of miles away, transmitted to +the earth from the Mariner IĨI space craft which has Just passed Venus. Its signals +carry summaries oŸ information it has picked up about the planets (information +obtained from electromagnetic waves that travelled from the planet to the space +craft). +There are very tiny wiggles of the electric and magnetic ñelds that are waves +which originated billions of light years away—from galaxies in the remotest +corners of the universe. 'Phat this is true has been found by “6ñlHing the room with +wires”——by building antennas as large as this room. Such radiowaves have been +detected from places in space beyond the range of the greatest optical telescopes. +ven they, the optical telescopes, are simply gatherers of electromagnetic waves. +'What we call the stars are only inferences, inferences drawn from the only physical +reality we have yet gotten from them——from a careful study of the unendingly +complex undulations of the electric and magnetic fñelds reaching us on earth. +There is, of course, more: the fields produced by lightning miles away, the +fñelds of the charged cosmic ray particles as they zip through the room, and more, +and more. What a complicated thing is the electric fñeld in the space around +youl Yet it always satisfies the three-dimensional wave equation. +20-3 Scientific imagination +T have asked you to imagine these electric and magnetic fields. What do +you do? Do you know how? How do Ï imagine the electric and magnetic fñeld? +What do ƒ actually see? What are the demands of scientifc imagination? Is +it any diferent from trying to imagine that the room is full of invisible angels? +No, ït is not like Imagining invisible angels. It requires a much higher degree of +imagination to understand the electromagnetic ñeld than to understand invisible +angels. Why? Because to make invisible angels understandable, all I have to do is +to alter their properties ø jiie bi—I make them slightly visible, and then I can +see the shapes of their wings, and bodies, and halos. Once Ï succeed in imagining +a visible angel, the abstraction required——which is to take almost invisible angels +and imagine them completely invisible—is relatively easy. So you say, “Professor, +please gïve me an approximate description of the electromagnetic waves, even +thouph it may be slightly inaccurate, so that I too can see them as well as Ï +can see almost invisible angels. Then I will modify the picture to the necessary +abstraction.” +Tm sorry I can't do that for you. I don '® know how. I have no picture of +this electromagnetic fñeld that is in any sense accurate. I have known about +the electromagnetic feld a long time——ÏI was in the same position 25 years ago +that you are now, and I have had 2ð years more of experience thinking about +--- Trang 254 --- +these wigsgling waves. When I start describing the magnetic field moving through +space, Ï speak of the #- and Ö-fields and wave my arms and you may imagine +that I can see them. Il] tell you what Ï see. I see some kind of vague shadowy, +wiggling lines—here and there is an # and written on them somehow, and +perhaps some of the lines have arrows on them——an arrow here or there which +disappears when I look too closely at it. When I talk about the fields swishing +throuph space, I have a terrible confusion between the symbols I use to describe +the objects and the objects themselves. Ï cannot really make a picture that is +even nearly like the true waves. 5o if you have some difficulty in making such a +picture, you should not be worried that your difficulty is unusual. +Our science makes terriic demands on the imagination. “The degree of +imagination that is required is much more extreme than that required for some +of the ancient ideas. The modern ideas are much harder to imagine. We use a lot +of tools, though. We use mathematical equations and rules, and make a lot of +pictures. What I realize now is that when I talk about the electromagnetic fñeld in +space, Ï see some kind oŸ a superposition of all of the diagrams which Ïve ever seen +drawn about them. I don't see little bundles of fñeld lines running about because +1ÿ worries me that ïf I ran at a diferent speed the bundles would disappear, Ï +donˆt even always see the electric and magnetic fñelds because sometimes I think +T should have made a picture with the vector potential and the scalar potential, +for those were perhaps the more physically signifcant things that were wigeling. +Perhaps the only hope, you say, is to take a mathematical view. Now what is +a mathematical view? From a mathematical view, there is an electric fñeld vector +and a magnetic field vector at every point in space; that is, there are six numbers +associated with every point. Can you imagine six numbers associated with each +point in space? 'Phat°s too hard. Can you imagine even øwe number associated +with every point? I cannotl Ï can imagine such a thing as the temperature at +every point in space. That seems to be understandable. There ¡is a hotness and +coldness that varies from place to place. But I honestly do not understand the +idea of a rwmber at every point. +So perhaps we should put the question: Can we represent the electric field +by something more like a temperature, say like the displacement of a piece of +jello? Suppose that we were to begin by imagining that the world was fñlled with +thin jello and that the fields represented some distortion——say a stretching or +twisting——of the jello. TThen we could visualize the feld. After we “see” what it is +like we could abstract the jello away. For many years that's what people tried to +do. Maxwell, Ampère, Faraday, and others tried to understand electromagnetism +this way. (Sometimes they called the abstract jello “ether.”) But it turned out +that the attempt to imagine the electromagnetic fñeld in that way was really +standing in the way oŸ progress. We are unfortunately limited to abstractions, to +using instruments to detect the fñeld, to using mathematical symbols to describe +the fñield, etc. But nevertheless, in some sense the felds are real, because after we +are all ñnished ñddling around with mathematical equations—with or without +making pictures and drawings or trying to visualize the thing—we can still make +the instruments detect the signals from Mariner II and ñnd out about galaxies a +bilion miles away, and so on. +'The whole question of Imagination in science is often misunderstood by people +in other disciplines. They try to test our imagination in the following way. They +say, “Here is a picture of some people in a situation. What do you imagine will +happen next?” When we say, “ÏI can't imagine,” they may think we have a weak +imagination. They overlook the fact that whatever we are đÌloued to imagine in +science must be consistent tuïth cueruthing else tue knou: that the electric fields +and the waves we talk about are not just some happy thoughts which we are +free to make as we wish, but ideas which must be consistent with all the laws +of physics we know. We can t allow ourselves to seriously imagine things which +are obviously in contradiction to the known laws of nature. And so our kind of +imagination is quite a difficult game. One has to have the imagination to think +of something that has never been seen before, never been heard of before. At +the same time the thoughts are restricted in a strait jacket, so to speak, limited +--- Trang 255 --- +by the conditions that come from our knowledge of the way nature really is. +The problem of creating something which is new, but which is consistent with +everything which has been seen before, is one of extreme dificulty. +'While m on this subject Ï want to talk about whether it will ever be possible +to imagine beautu that we can”t see. It is an interesting question. When we look +at a rainbow, it looks beautiful to us. Everybody says, “Ooh, a rainbow.” (You +see how scientifc l am. I am afraid to say something is beautiful unless Ï have +an experimental way of delning it.) But how would we describe a rainbow If we +were blind? We are blind when we measure the infrared reflection coefficient +of sodium chloride, or when we talk about the frequency of the waves that are +coming om some galaxy that we can”% see—we make a diagram, we make a plot. +For instance, for the rainbow, such a plot would be the intensity of radiation +vs. wavelength measured with a spectrophotometer for each direction in the sky. +Gencrally, such measurements would give a curve that was rather flat. hen +some day, someone would discover that for certain conditions of the weather, and +a% certain angles in the sky, the spectrum of intensity as a function of wavelength +would behave strangely; it would have a bump. AÄs the angle of the instrument +was varied only a little bit, the maximum of the bump would move om one +wavelength to another. 'Phen one day the physical review of the blind men might +publish a technical article with the title “The Intensity of Radiation as a Function +of Angle under Certain Conditions of the Weather.” In this article there might +appear a graph such as the one in Fig. 20-5. The author would perhaps remark +that at the larger angles there was more radiation at long wavelengths, whereas +for the smaller angles the maximum in the radiation came at shorter wavelengths. +(From our point oŸ view, we would say that the light at 409 is predominantly +green and the light at 42° is predominantly red.) +> s` . +5 XS b s“ Fig. 20-5. The intensity of electromag- +# sf netic waves as a function of wavelength for +. three angles (measured from the direction +opposite the sun), observed only with cer- +ZZ tain meteorological conditions. +\Wavelength +Now do we fñnd the graph of Fig. 20-5 beautiful? It contains much more detail +than we apprehend when we look at a rainbow, because our eyes cannot see the +exact details in the shape of a spectrum. “The eye, however, finds the rainbow +beautiful. Do we have enough imagination to see in the spectral curves the same +beauty we see when we look directly at the rainbow7? I don't know. +But suppose I have a graph of the refection coefficient of a sodium chloride +crystal as a function of wavelength in the inữared, and also as a function oŸ angle. +T would have a representation of how it would look to my eyes if they could see in +the infrared—perhaps some glowing, shiny “green,” mixed with refections tom +the surface in a “metallic red.” That would be a beautiful thing, but T don't know +whether I can ever look at a graph of the reflection coefficient of NaCl measured +with some instrument and say that it has the same beauty. +On the other hand, even iŸ we cannot see beauty in particular measured +results, we cơn already claim to see a certain beauty in the equations which +describe general physical laws. For example, in the wave equation (20.9), there's +something nice about the regularity of the appearance of the ø, the , the z, and +the ứ. And this nice symmetry in appearance of the z, , z, and ý suggests to +the mind still a greater beauty which has to do with the four dimensions, the +possibility that space has four-dimensional symmetry, the possibility of analyzing +that and the developments of the special theory of relativity. So there is plenty +of intellectual beauty associated with the equations. +--- Trang 256 --- +20-4 Spherical waves +W© have seen that there are solutions of the wave equation which correspond +to plane waves, and that any electromagnetic wave can be described as a su- +perposition of many plane waves. Ín certain special cases, however, it is more +convenient to describe the wave field in a diÑerent mathematical form. We would +like to discuss now the theory of spherical waves—waves which correspond to +spherical surfaces that are spreading out from some center. When you drop a +stone into a lake, the ripples spread out in circular waves on the surface—they +are two-dimensional waves. AÁ spherical wave is a similar thing except that it +spreads out in three dimensions. +Before we start describing spherical waves, we need a little mathematics. +Suppose we have a function that depends only on the radial distance r from a +certain origin—=in other words, a function that is spherically symmetric. Let”s +call the function (z), where by r we mean ++ = \/x2 -+ Ủng -+ z3, +the radial distance from the origin. In order to fnd out what functions j(r) +satisfy the wave equation, we will need an expression for the Laplacian of . So +we want to fñnd the sum of the second derivatives of with respect to ø, , and z. +W©e will use the notation that //(r) represents the derivative of with respect +to r and ÿ”{r) represents the second derivative of with respect to 7. +Pirst, we fnd the derivatives with respect to ø. The first derivative is +Ø(r) Ør +TS. =0) +'The second derivative of with respect tO # 1s +Ø? CIÊN Ør +ụ — ” — -+E ự ¬: +Øz Øz Øz +W©e can evaluate the partial derivatives of r with respect to ø from +ðr — # Ø?r 1 1 + +Ôxz_ rỶ Ôxz2_ + r3j. +So the second derivative of ý with respect %O # is +9U z2 „1 +2 +—=>ồ==. -|1— -z lự. 20.28 +9x2 — r2 + T r2 W ( ) +Likewise, +Ø8 ` „1 Dã +—=s= . -|1i- š+lự 20.29 +9W z2 „1 z2 +—====.= -|1— = ]ử. 20.30 +8z2 r3 + T lo W ( ) +The Laplacian is the sum of these three derivatives. Remembering that +#2 + 2 + z2 = rỶ, we get +V?U(r) = U”() + — /ứ). (20.31) +Tt is often more convenient to write this equation in the following form: +V20) = ~ T5(rU) (20.32) +r) =—- —s(r). : +TÍ you carry out the diferentiation indicated in Eq. (20.32), you will see that the +right-hand side is the same as in Eq. (20.31). +T we wish to consider spherically symmetric fñelds which can propagate as +spherical waves, our field quantity must be a function of both z and #. ŠSuppose +--- Trang 257 --- +we ask, then, what functions (z,£) are solutions oŸ the three-dimensional wave +equation +V20(.1) — s 2g 00,1) =0 (20.33) +r — — _— T = Ù. . +Í c2 82 Í +Since (, £) depends only on the spatial coordinates through z, we can use the +cquation for the Laplacian we found above, Eq. (20.32). To be precise, however, +since # is also a function of ý, we should write the derivatives with respect to ? +as partial derivatives. Then the wave equation becomes +1 Ø2 1 Ø2 += mạ (n0) — 5 agU=0, +rÔr c2 6t +'W©e must now solve this equation, which appears to be mụch more complicated +than the plane wave case. But notice that if we multiply this equation by r, we +82 1 Ø +This equation tells us that the function rý satisfes the one-dimensional wave +equation ¡in the variable r. Ủsing the general principle which we have emphasized +so often, that the same equations always have the same solutions, we know that +]Í rủ is a function only of (r — c£) then it will be a solution o£ Ðq. (20.34). So we +know that spherical waves must have the form +rú(,£) = ƒ(r— e‡). +Ór, as we have seen before, we can equally well say that rj can have the form +rụ = ƒ( — r/e). +Dividing by z, we fnd that the fñeld quantity ÿ (whatever it may be) has the +following form: +Ÿ— TC += ƒH ro) (20.35) +Such a function represents a general spherical wave travelling outward from the +origin at the speed c. If we forget about the r in the denominator for a moment, +the amplitude of the wave as a function of the distance from the origin at a +given time has a certain shape that travels outward at the speed c. 'Phe facbor r +in the denominator, however, says that the amplitude of the wave decreases in +proportion to 1/z as the wave propagates. In other words, unlike a plane wave +in which the amplitude remains constant as the wave runs along, in a spherical +wave the amplitude steadily decreases, as shown in Eig. 20-6. 'This efect is easy +to understand from a simple physical argument. +% À " +— 1/r > +tị —T—_ S— = rị +v=c T"T———~_—_ __ +k ta " 12 +0 1 Ta r 0 tị ta t +|‡——— c(Œ— ñ) ———>l +(a) (b) +Fig. 20-6. A spherical wave + = f(f — r/c)/r. (a) as a function of r for £ = f¡ and the same wave +for the later time ta. (b) as a function of £ for r = r¡ and the same wave seen at ứa. +--- Trang 258 --- +W© know that the energy density in a wave depends on the square of the wave +amplitude. Äs the wave spreads, its energy is spread over larger and larger areas +proportional to the radial distance squared. If the total energy is conserved, the +energy density must fall as 1/zŸ, and the amplitude of the wave must decrease +as l/r. So Eq. (20.35) is the “reasonable” form for a spherical wawe. +W© have disregarded the second possible solution to the one-dimensiona] wave +equation: +rỷ — g(t + r/©), +'This also represents a spherical wave, but one which travels 7nard from large r +toward the origin. +We are now going to make a special assumption. We say, without any +demonstration whatever, that the waves generated by a source are only the waves +which go ou#uard. Since we know that waves are caused by the motion of charges, +we want to think that the waves proceed outward from the charges. It would be +rather strange to imagine that before charges were set in motion, a spherical wave +started out from infinity and arrived at the charges just at the time they began +to move. That is a possible solution, but experience shows that when charges +are accelerated the waves travel outward from the charges. Although Maxwell”s +equations would allow either possibility, we will put in an ødditional ƒfact+—based +on experience—that only the outgoing wave solution makes “physical sense.” +'W©e should remark, however, that there is an interesting consequence to this +additional assumption: we are removing the symmetry with respect to time that +exists in Maxwells equations. “The original equations for and #Ö, and also +the wave equations we derived from them, have the property that if we change +the sign of ý, the equation is unchanged. 'These equations say that Íor every +solution corresponding to a wave goïng in one direction there is an equally valid +solution for a wave travelling in the opposite direction. Our statement that we will +consider only the outgoing spherical waves is an important additional assumption. +(A formulation of electrodynamics in which this additional assumption is avoided +has been carefully studied. Surprisingly, in many cireumstances it does øœø lead +to physically absurd conclusions, but it would take us too far astray to discuss +these ideas Just now. We will talk about them a little more in Chapter 28.) +Wc must mention another important point. In our solution for an outgoing +wave, q. (20.35), the function ÿ is infinite at the origin. That is somewhat +peculiar. We would like to have a wave solution which is smooth everywhere. +Our solution must represent physically a situation in which there is some source +at the origin. In other words, we have inadvertently made a mistake. We have +not solved the free wave equation (20.33) eueryuhere; we have solved Eq. (20.33) +with zero on the right everywhere, except at the origin. Qur mistake crept in +because some of the steps in our derivation are not “legal” when r = 0. +Let's show that it is easy to make the same kind of mistake in an electrostatic +problem. Suppose we want a solution of the equation for an electrostatie potential +in free space, V2ø = 0. The Laplacian is equal to zero, because we are assuming +that there are no charges anywhere. But what about a spherically symmetric +solution to this equation——that is, some function ở that depends only on z. Ủsing +the formula of Eq. (20.32) for the Laplacian, we have +r dr2 (rộ) =0. +Multiplying this equation by r, we have an equation which is readily integrated: +dị? (rø) =0. +Tf we integrate once with respect to r, we fnd that the first derivative of rộ is a +--- Trang 259 --- +constant, which we may call a: +m (rỏ) =a. +Integrating again, we fnd that rọó is of the form +rộ = ar + Ù, +where b is another constant of integration. So we have found that the following ở +1s a solution for the electrostatic potential in free space: +Something is evidently wrong. In the region where there are no electric +charges, we know the solution for the electrostatic potential: the potential is +everywhere a constant. hat corresponds to the fñrst term in our solution. But +we also have the second term, which says that there is a contribution to the +potential that varies as one over the distance from the origin. We know, however, +that such a potential corresponds to a poïint charge at the origin. So, although +we thought we were solving for the potential in free space, our solution also øïves +the fñeld for a point source at the origin. Do you see the similarity between what +happened now and what happened when we solved for a spherically symmetric +solution to the wave equation? If there were really no charges or currents at +the origin, there would not be spherical outgoing waves. 'The spherical waves +must, of course, be produced by sources at the origin. In the next chapter we +will investigate the connection between the outgoing electromagnetic waves and +the currents and voltages which produce them. +--- Trang 260 --- +Seœlrff©œrts @œŸ£ IWœvere©ollˆs F[qrrerfếf@res triểh: +(tarr-oretÉs (rae[ Ế Ïrerrgj©os +21-1 Light and electromagnetic waves +W© saw in the last chapter that among their solutions, Maxwells equations 21-1 Light and electromagnetic waves +have waves of electricity and magnetism. These waves correspond to the phe- 21-2 Spherical waves from a point +nomena of radio, light, x-rays, and so on, depending on the wavelength. We Source +have already studied light in great detailin Vol. I. In this chapter we want to 21-3 The general solution of Maxwell's +tỉe together the two subjects—we want 0o show that Maxwells equations can equations +indeed form the base for our earlier treatment of the phenomena of light. 21-4 The fields of an oscillating dipole +'When we studied light, we began by writing down equations for the electric l . +and magnetic felds produced by a charge which moves in any arbitrary way. 21-5 The potentials of a moving +Those equations were charge; the general solution of +Liếnard and Wiechert +)n... l# + ” x() + 1 Lai sơ, (21.1) 21-6 The potentials fora charge +4reo|r2 c đt\r?2 c2 di2 moving with constant velocity; +and the Lorentz formula +cB= y: X k. +[See Eqs. (28.3) and (28.4), Vol. I. As explained below, the signs here are the +negatives of the old ones.] +Tí a charge moves in an arbitrary way, the electric feld we would ñnd no at +some point depends only on the position and motion of the charge not now, but +a% an earl2er time—at an instant which is earlier by the time it would take light, +going at the speed é, to travel the distance ?“ from the charge to the ñeld point. +In other words, if we want the electric field at point (1) at the tỉme #, we must Reuieu: Chapter 28, Vol. Lj Electromag- +calculate the location (2) of the charge and its motion at the time (# — r//e), metic Radiation +where ?? is the distance to the point (1) om the position of the charge (27) at Chapter 31, Vol. l, The Origin +the time (£ — rˆ/c). The prime is to remind you that rÝ is the so-called “retarded 0ƒ the Refractiue Indez +distance” from the point (27) to the point (1), and not the actual distance bebtween Chapter 34, Vol. I, Relatiuistic +point (2), the position of the charge at the time ý, and the fñeld point (1) (see kEJfects in Radiation +Eig. 21-1). Note that we are using a different convention now for the đữecfion oŸ +the unit vector e„. In Chapters 28 and 3⁄4 of Vol. l it was convenient to take 7 +(and hence ez) pointing #øouard the source. Now we are following the definition +we took for Coulomb's law, in which r is directed from the charge, at (2), £otuard +the field point at (1). The only diference, of course, is that our new 7 (and e) +are the negatives of the old ones. +We© have also seen that if the velocity 0 of a charge is always much less than é, +and if we consider only points at large distances from the charge, so that only +the last term of Eq. (21.1) is important, the fields can also be written as +pm... Ban of the charge at (# — r/ 9Ì (91.1) : r — +— 4megc2r: |projected at right angles to r l Gà . +and nụ j +Position at +cB = e„. x E. t—r/c +Let”s look at what the complete equation, Ðq. (21.1), says in a little more Ị +detail. The vector e„¿ is the unit vector to poïnt (1) from the retarded position (27). Posnen ae +'The first term, then, is what we would expect for the Coulomb field of the charge +aE its retarded position—we may call this “the retarded Coulomb field” The Fig. 21-1. The fields at (1) at the time £ +electric field depends inversely on the square of the distance and is directed away depend on the position (2) occupied by the +from the retarded position of the charge (that is, in the direction of ez›). charge q at the time (£ — r'/€). +--- Trang 261 --- +But that is only the fñrst term. “The other terms tell us that the laws of +electricity do noøứ say that all the fields are the same as the static ones, but just +retarded (which is what people sometimes like to say). To the “retarded Coulomb +field” we must add the other two terms. The second term says that there is a +“correction” to the retarded Coulomb field which is the ra#e oƒ change of the +retarded Coulomb feld multiplied by ?//c, the retardation delay. In a way of +speaking, this term tends to cornpensoœte for the retardation in the frst term. The +first #ưuo terms correspond to computing the “retarded Coulomb field” and then +extrapolating it toward the future by the amount r7“/c, that is, righ‡ up to the +tzme tt The extrapolation is linear, as iŸ we were to assume that the “retarded +Coulomb fñeld” would continue to change at the rate computed for the charge +at the point (27). TÝ the field is changing slowly, the efect of the retardation is +almost completely removed by the correction term, and the two terms together +give us an electric fñield that is the “instantaneous Coulomb fñeld”——that is, the +Coulomb feld of the charge at the point (2)—to a very good approximation. +Einally, there is a third term in Eq. (21.1) which is the second derivative of the +unit vector ez:. Eor our study oŸ the phenomena of light, we made use of the fact +that far away from the charge the fñrst two terms went inversely as the square of +the distance and, for large distances, became very weak in comparison to the last +term, which decreases as l/r. So we concentrated entirely on the last term, and +we showed that it is (again, for large distances) proportional to the component of +the acceleration of the charge at right angles to the line of sight. (Also, for most +of our work in Vol. l, we took the case in which the charges were moving nonrela- +tivistically. We considered the relativistic efects in only one chapter, Chapter 34.) +Now we should try to connect the b6wo things together. We have the Maxwell +cquations, and we have Eq. (21.1) for the field of a point charge. We should +certainly ask whether they are equivalent. If we can deduce Eq. (21.1) rom +Maxwells equations, we will really understand the connection between light and +electromagnetism. 'To make this connection is the main purpose of this chapter. +lt turns out that we wont quite make it—that the mathematical details get +too complicated for us to carry through ín all their gory details. But we will come +close enough so that you should easily see how the connection could be made. +The missing pieces will only be in the mathematical details. Some of you may +fnd the mathematics in this chapter rather complicated, and you may not wish +to follow the argument very closely. We think it is important, however, to make +the connection between what you have learned earlier and what you are learning +now, or at least to indicate how such a connection can be made. You will notice, +1f you look over the earlier chapters, that whenever we have taken a statement as +a starting point for a discussion, we have carefully explained whether it is a new +“assumption” that is a “basic law,” or whether it can ultimately be deduced from +some other laws. We owe it to you in the spirit of these lectures to make the +connection between light and Maxwells equations. If it gets dificult in places, +well, that's life—there is no other way. +21-2 Spherical waves from a poỉnt source +In Chapter 1S we found that Maxwells equations could be solved by letting +Eb=-Vó— Đr (21.2) +BöB=VxA, (21.3) +where ó and Á must then be solutions of the equations +2 1 Ø2 Ð +V⁄“¿ 2ØP. ae (21.4) +5 1 82A 3 +VW^ˆA ® 0B ae (21.5) +--- Trang 262 --- +and must also satisfy the condition that +V.A=_-—--—.. 21.6 +c2 ði (21) +NÑow we will ñnd the solution of Eqs. (21.4) and (21.5). To do that we have +to fnd the solution , of the equation +where s, which we call the source, is known. Of course, s corresponds to Ø/eo +and to ó for Bq. (21.4), or s is 7„/coc2 if is A„, ete., but we want to solve +Eq. (21.7) as a mathematical problem no matter what and s are physically. +In places where ø and j are zero—in what we have called “free” space—the +potentials ø and A, and the felds and ?Ö, all satisfy the three-dimensional +wave equation without sources, whose mathematical form is +V?”ụ— ->- — =0. 21.8 +In Chapter 20 we saw that solutions of this equation can represent waves of +various kinds: plane waves in the zø-direction, = ƒ( — #/c); plane waves in the +ụ- or z-direction, or in any other direction; or spherical waves of the form +(The solutions can be written in still other ways, for example cylindrical waves +that spread out from an axis.) +W© also remarked that, physically, Bq. (21.9) does not represent a wave in +Íree space—that there must be charges at the origin to get the outgoing wave +sbarted. In other words, Eq. (21.9) is a solution of Eq. (21.8) everywhere except +right near r = 0, where it must be a solution oŸ the complete equation (21.7), +ineluding some sources. Let's see how that works. What kind of a source sin +Eq. (21.7) would give rise to a wave like Eq. (21.9)? +Suppose we have the spherical wave of Eq. (21.9) and look at what is happening +for very small z. TThen the retardation —r/cin ƒ(— r/c) can be neglected— +provided ƒ is a smooth function——and becomes +q= BẦU) (r — 0). (21.10) +So 0 is just like a Coulomb fñeld for a charge at the origin that varies with time. +That is, ifƒ we had a little lump oŸ charge, limited to a very small region near the +origin, with a density ø, we know that +¿= 9n, +where Q= ƒ odV. Now we know that such a ở satisfies the equation +V?¿=—, +Following the same mathematics, we would say that the j of Eq. (21.10) +satisfles +V?u=-—s (r0), (21.11) +where s is related to ƒ by +__= J sdV. +--- Trang 263 --- +The only diference is that in the general case, s, and therefore Š, can be a +function of time. +Now the important thing is that if ý, satisles q. (21.11) for small z, it also +satisfes Eq. (21.7). As we go very close to the origin, the 1/z dependence oŸ +causes the space derivatives to become very large. But the time derivatives keep +their same values. [They are just the time derivatives of ƒ(f).] So as r goes to +zero, the term Ø2J/Øf? in Eq. (21.7) can be neglected in comparison with V2, +and Eq. (21.7) becomes equivalent to Eq. (21.11). +To sunmarize, then, if the source function s(#) of Eq. (21.7) is localized at +the origin and has the total strength +S() = J s(1) dV, (21.12) +the solution of Eq. (21.7) is +1 S(t— r/c) +†#)= ———. 21.13 +0,0,2.) =1 ——- (21.13) +The only efect of the term 62/Ø? in Eq. (21.7) is to introduce the retarda- +tion (É — r/e) in the Coulomb-like potential. +21-3 The general solution of Maxwell?s equations +W©e have found the solution of Bq. (21.7) for a “point” source. "The next +question is: What is the solution for a spread-out source? 'Phat”s easy; we can +think oŸ any source s(Z, , z,#) as made up oŸ the sum oŸ many “point” sources, +one for each volume elemnent đV, and each with the source strength s(z, , z, #) đV. +Since Eq. (21.7) is linear, the resultant field is the superposition of the felds from +all of such source elements. +Using the results of the preceding section [Eq. (21.13)] we know that the +fñeld dự at the point (#, \; zi)—or (1) for short—at the tỉme £, from a source +element sđV at the poïnt (za, a2, za2)—or (2) for short—is given by +s(2,£— ria/c) dVa +đụ(1,#)=————————— +0(,9 on +where ra; is the distance from (2) to (1). Adding the contributions from all +the pieces of the source means, of course, doing an integral over all regions +where s # Ú; so we have +s(2,£— ria/c) +1,#)= j ——————dÙ:. 21.14 +ð00= | SST— 2P mà (21.14) +That is, the field at (1) at the time # is the sum of all the spherical waves which +leave the source elements at (2) at the times (£ — r12/c). Thịs is the solution of +our wave equation for any set Of sources. +W©e see now how to obtain a general solution for Maxwells equations. TỶ +for we mean the scalar potential ó, the source function s becomes ø/eo. Ôr we +can let represent any one of the three components of the vector potential A, +replacing s by the corresponding component of 7/cọc?. Thus, if we know the +charge density ø(z, , z, £) and the current density 7(z, , z, £) everywhere, we can +immediately write down the solutions of Eqs. (21.4) and (21.5). They are +ó(1,f£) = na (21.15) +47€0T13 +J(2,t— +A(.0®)= J 7Ó. — naƒ€) dụ, (21.16) +47€oc271a +The fñelds # and #Ö can then be found by diferentiating the potentials, using +Eqs. (21.2) and (21.3). [Incidentally, it is possible to verify that the ô and 4 +obtained from Eqs. (21.15) and (21.16) do satisfy the equality (21.6).] +--- Trang 264 --- +We have solved Maxwell's equations. Given the currents and charges in any +circumstance, we can find the potentials directly from these integrals and then +diferentiate and get the felds. So we have fñnished with the Maxwell theory. Also +this permits us to close the ring back to our theory of light, because to connect with +our earlier work on light, we need only calculate the electric ñeld from a moving +charge. All that remains is to take a moving charge, calculate the potentials from +these integrals, and then diferentiate to fnd E from —Wó — ØA/Ô. We should +get Eq. (21.1). It turns out to be lots of work, but that”s the principle. +So here is the center of the universe of electromagnetism——the complete theory +of electricity and magnetism, and of light; a complete description of the fields +produced by any moving charges; and more. It is all here. Here is the structure +built by Maxwell, complete in all its power and beauty. Et is probably one of the +greatest accomplishments of physics. 'To remind you of its Importance, we will +put It all together in a nice frame. +Maxwellˆs equations: +W.E=f Ý:B=0 +C0 +9B ` 9E +VxE=_- ‹ỄẰÀVxB=J7+— +lôi? €0 lôi? +Theïr solutions: 2A +E=-Vð- +bB=VxA +2£—r 12/C +ó(1,£) = [2 —n gụ, +47€ogr 12 +J(2,£—r 12/C +A(L)= =— d— H9 nụ, +47€o€C2r1s +21-4 The fields of an oscillating dipole +W© have still not lived up to our promise to derive Eq. (21.1) for the electric +fñeld of a point charge in motion. Even with the results we already hawe, it is a +relatively complicated thing to derive. We have not found Eq. (21.1) anywhere in +the published literature except in Vol. I of these lectures.* So you can see that it is +not easy to derive. (The felds ofa moving charge have been written in many other +forms that are equivalent, oŸ course.) We will have to limit ourselves here just to +showing that, in a few examples, Eqs. (21.15) and (21.16) give the same results +as Eq. (21.1). Pirst, we will show that Eq. (21.1) gives the correct fields with only +the restriction that the motion of the charged particle is nonrelativistic. (Just this +special case will take care of 90 percent, or more, of what we said about light.) +W© consider a situation in which we have a blob of charge that is moving +about in some way, in a small region, and we will fñnd the fñelds far away. To +put it another way, we are ñnding the field at any distance from a point charge +that is shaking up and down in very small motion. 5ince light is usually emitted +from neutral objects such as atoms, we will consider that our wiggling charge q +1s located near an equal and opposite charge at rest. If the separation between +the centers of the charges is đ, the charges will have a dipole moment ø = gở, +* The formula was first published by Oliver Heaviside in 1902. It was independently +discovered by R. P. Feynman, in about 1950, and given in some lectures as a good way of +thinking about synchrotron radiation. +--- Trang 265 --- +which we take to be a function of time. Now we should expect that if we look +at the fields close bo the charges, we won't have to worry about the delay; the +electric fñeld wïll be exactly the same as the one we have calculated earlier for +an electrostatic đipole—using, of course, the insbantaneous dipole moment ø(Ÿ). +But if we go very far out, we ought to nd a term in the feld that goes as l/z +and depends on the acceleration of the charge perpendicular to the line of sight. +Let°s see iŸ we get such a result. +We begin by calculating the vector potential A, using Eq. (21.16). Suppose +that our moving charge is in a small blob whose charge density is given by ø(z, 9, 2), +and the whole thing is moving at any instant with the velocity ø. Then the +current density 7(z,,z) will be equal to ø/ø(z,,z). It will be convenient to Z +take our coordinate system so that the z-axis is in the direction of 0; then the +geometry of our problem is as shown in Fig. 21-2. We want the integral +J(2,t— +TK = da. (21.17) đ) +T12 F +Now ïf the size of the charge-blob is really very small compared with r1a, we A ƒ +can set the 71a term in the denominator equal to r, the distance to the center of +the blob, and take r outside the integral. Next, we are also going to set r1a =7 4 y +in the numerator, although that is not really quite right. It is not right because p(x, y,Z) +we should take j at, say, the top of the blob at a slightly diferent time than we +used for 7 at the bottom of the blob. When we set ra = r in 7(f— r12/c), we are +taking the current density for the whole blob at the same time (£ — rz/c). That is * +an approximation that will be good only if the velocity ø of the charge is much Fig. 21-2. The potential at (1) are given +less than c. So we are making a nonrelativistic calculation. Replacing j by ø0, by integrals over the charge density ø. +the integral (21.17) becomes +— | 00(2,t— r/c) đM. +Since all the charge has the same velocity, this integral is just ø/z times the total +charge g. But go is just Øp/Ôt, the rate of change of the dipole moment——which +is, Of course, to be evaluated at the retarded tỉme (# — r/c). W©e will wribe it +as Đ(£ — r/c). So we get for the vector potential +1 D(t — +Aq,0p=——_ PH=?/°), (21.18) +4megc2 T +Our result says that the current in a varying dipole produces a vector potential +in the form of spherical waves whose source strength is ø/coc2. +W©e can now get the magnetic feld from Ö = V x A. 5ince ø is totally in the +z-direction, A has only a z-component; there are only wo nonzero derivatives in +the curl. So Ö;y = ØA;/ðØ and Öyạ = —ÔA„/Ø+z. Let)s first look at „: +84; 1 Ø p(t—-r/ec) +B.= —= ——=t>_—_—* ~, 21.19 +` Øụ 4xcoc2 Øụ r ) +To carry out the diferentiation, we must remember that ? = 4⁄22 + 2 + zŸ, so +1 Ø (1 1 1ô +B„=———j(t— ¬= ——>_���---p(t— : 21.20 +” Amegc2 Dự — r/©) Øụ () + 4mxegc2 r Øụ Dự — r/e) ( ) +Remembering that Ør/Ø = g/r, the first term gives +1 )(È — +¬.. U0N — rịc): (21.21) +47coc2 rỏ +which drops of as 1/z2 like the potential of a static dipole (because #/7 is constant +for a given direction). +The second term in Bq. (21.20) gives us the new efects. Carrying out the +diferentiation, we get +———;s-sj(t— 21.22 +. n. (21.33) +--- Trang 266 --- +where ø means, of course, the second derivative of p with respect to f. This +term, which comes from diferentiating the numerator, is responsible for radiation. +Eirst, it describes a feld which decreases with distance only as 1/r. Second, it +depends on the øcceleraiion oŸ the charge. You can begin to see how we are going +to get a result like Eq. (21.17), which describes the radiation of light. +Let”s examine ïn a little more detail how this radiation term comes about——1t +is such an interesting and important result. We start with the expression (21.18), +which has a 1/7 dependence and is therefore like a Coulomb potential, except for +the delay term in the numerator. Why ¡is it then that when we differentiate with +respect to space coordinates to get the fields, we donˆt just get a 1/r2 field—with, +Of course, the corresponding time delays? +W© can see why in the following way: Suppose that we let our dipole oscillate +up and down in a sinusoidal motion. 'Then we would have +Ð= Đ¿ = Đo sin uÈ +1 œøpogcosu(£É — r/c) +A¿=——>s_————.. +4mxegc2 r +Tf we plot a graph of Á; as a function of z at a given instant, we get the curve A, +shown in Fig. 21-3. The peak amplitude decreases as 1/7, but there is, in addition, +an oscillation in space, bounded by the 1/z envelope. When we take the spatial N 1/ư +derivatives, they will be proportional to the slope of the curve. Erom the fñgure N⁄ +we see that there are slopes much steeper than the slope of the 1/7 curve itself. Ề XS +lt is, in fact, evident that for a given frequency the peak sÌopes are proportional “>=/À ~~>x---z>>v-—- +to the amplitude of the wave, which varies as 1/z. So that explains the drop-of \A⁄/_->—~⁄--->~" +rate of the radiation term. TT” +Tt all comes about because the variations th time at the source are translated J +Into variations #w spœce as the waves are propagated outward, and the magnetic / +fields depend on the spøf7a[ derivatives of the potential. / +Let's go back and fñnish our calculation of the magnetic ñeld. We have for „ ! +the two terms (21.21) and (21.22), so Eig. 21-3. The z-component of A as a +- - function of r at the instant £ for the spheri- +Đ„= _1 _,=h — m HỊ, cal wave from an oscillating dipole. ' +4meoc2 rỏ cr2 +'With the same kind of mathematics, we get +By = 1 — + 8E, +47g rỏ er2 +Or we can put it all together in a nice vector formula: +1 lp+ (r/©)Ð]¡—r/e —— “. `... +. (a) Nunh @) +Fig. 21-5. (a) A “point” charge—considered as a small cubical distribution of +charge—moving with the speed v toward point (1). (b) The volume element AV/ used +for calculating the potentials. +where 0z, is the component of the velocity of the charge parallel to r12—namely, +toward point (1). We will now show you why. To make the argument easier to +follow, we will make the calculation frst for a “point” charge which is in the +form oŸ a little cube of charge moving toward the point (1) with the speed 0, as +shown in Fig. 21-5(a). Let the length of a side of the cube be ø, which we take +to be much, much less than 71a, the distance from the center of the charge to the +point (1). +Now to evaluate the integral of Đq. (21.28), we will return to basic principles; +we will write it as the sum +» ÔN, (21.30) +W 1Ƒ ls[ lề +where 7¿ is the distance from point (1) to the ?th volume element AVW; and ø; is IiIIIIIIIIIII Œ) +the charge density at AV; at the tỉme f¿ = £ — r;/c. Since 7¿ 3> a, always, it will (a) IIIIIIIIIIIIIIII +be convenient to take our AV; in the form of thin, rectangular sÌices perpendicular l1i DI +to 71a, as shown in Fig. 21-5(b). HỊ 1 ĐÀ +Suppose we start by taking the volume elements AVW; with some thickness + „Ị + +much less than a. The individual elements will appear as shown in Fig. 21-6(a), ®) tú ' n () +where we have put in more than enough to cover the charge. But we have nöf lỊ +shown the charge, and for a good reason. Where should we draw it? Eor each 1 +volume element AW; we are to take ø at the time ý; = (£ — r;/c), but since the m ' +charge is moưing, it 1s in a djfferent place for cách 0olưme element A(V;l Mã + +Let°s say that we begin with the volume element labeled “1” in Eig. 21-6(a), 1 (q) +chosen so that at the tỉme #q = (£ — r1/c) the “back” edge of the charge occu- CÓ, H ~—”^~—> +pies AVI, as shown in Fig. 21-6(b). Then when we evaluate ø¿ AV2, we must use + +the position of the charge at the slightly ia#er tỉme f¿ = (£ — r¿/c), when the pH ' +charge will be in the position shown in Eig. 21-6(e). And so on, for AV3, AV, " + +etc. NÑow we can evaluate the sum. + +Since the thickness of each AV; is , its volume is a2. Then each volume (q), " —R_* +element that overlaps the charge distribution contains the amount of charge +0a2ø, 1 +where ø is the density of charge within the cube—which we take to be uniform. 1 +'When the distance rom the charge to point (1) is large, we will make a negligible ~— vAt a—m +error by setting all the z;'s in the denominators equal to some average value, say +the retarded position r“ of the center of the charge. Then the sum (21.30) is (e) “mm... +» cm Ở #———b———— +where AV¿y is the last AVW; that overlaps the charge distributions, as shown in Elg. 21-6. Integrating ø(£ — r /c) đV for +. : a moving charge. +Fig. 21-6(e). The sum is, clearly, +N PUUẺ _ pdÌ () | +r rí g +Now øaŸ is just the total charge q and W is the length b shown in part (e) of +the ñgure. 5o we have +¿= T1) (21.31) +4mcorˆ\Aa +--- Trang 270 --- +What is 0? It is the length of the cube of charge ?wcreased by the distance +moved by the charge between q = (f — r1/c) and £w = (f— rw/c)—which is the +distance the charge moves in the time +Af =†N — tị = (rìạ —TN)Í/c = ÙỤc. +Since the speed of the charge is ø, the distance moved is ò Af£ = 0b/c. But the +length b is this distance added to a: +b=a+ —b. +Solving for b, we get +_ 1=(0/e) +Of course by ø we mean the velocity at the retarded time #“ = (£ — rˆ/c), which +we can indicate by writing [1 — 0/c]ze:, and Eq. (21.31) for the potential becomes +ø(1,f) =——————. +ú,9) Ameor” [L— (0/€)]xet +This result agrees with our assertion, Eq. (21.29). Thhere is a correction term +which comes about because the charge is moving as our integral “sweeps Over +the charge.” When the charge is moving toward the point (1), its contribution +to the integral is increased by the ratio b/œ. Therefore the correct integral is +g/+ˆ multiplied by b/a, which is 1/[1 — 0/c]set. +Tf the velocity of the charge is not directed toward the observation point (1), +you can see that what matters is the compønen‡ of its velocity toward point (1). +Calling this velocity component „, the correction factor is 1/[1 — 0;z/c]xe:. Also, +the analysis we have made goes exactly the same way for a charge distribution of +am shape—it doesn't have to be a cube. Finally, since the “size” of the charge g +doesn”t enter into the fñnal result, the same result holds when we let the charge +shrink to any size—even to a point. The general result ¡is that the scalar potential +for a point charge moving with any velocity 1s +Ø(1,f)=———————. 21.32 +(IỦ” 1a] = m./9)e co»? +'This equation 1s often written in the equivalent form +4⁄)=——— Ta (21.33) +4rco|r — (0Ð - r/€)]set +where ?' is the vector from the charge to the point (1), where ó is being evaluated, +and all the quantities in the bracket are to have their values at the retarded +time # = £ — r!/e. +The same thing happens when we compute A for a point charge, from +Eq. (21.16). The current density is ø and the integral over ø is the same as we +found for ý. The vector potential is +A(1,f)= ——>=————-: 21.34 +9 4mcoc?[r — (0 - r/€)]set ) +'The potentials for a point charge were frst deduced in this form by Liếnard +and Wiechert and are called the Liénard- Wiechert potentials. +To close the ring back to Ed. (21.1) it is only necessary to compute # and +from these potentials (using = W x A4 and E = —Vó — ØA/0t). lt is now +only arithmetic. 'Phe arithmetic, however, is fairly involved, so we will not write +out the details. Perhaps you will take our word for it that Eq. (21.1) is equivalent +to the Liếnard-Wiechert potentials we have derived.* +* Tf you have a lot of paper and tỉme you can try to work it through yourself. We would, +then, make two suggestions: First, donˆt forget that the derivatives of z“ are complicated, since +it is a function of £#“. Second, don't try to đeriue (21.1), but carry out all of the derivatives in it, +and then compare what you get with the obtained from the potentials (21.33) and (21.34). +--- Trang 271 --- +21-6 The potentials for a charge moving with constant velocity; the Lorentz +formula +'W©e want next to use the Liénard-Wiechert potentials for a special case—to fnd +the fields of a charge moving with uniform velocity in a straight line. We will do it +again later, using the principle of relativity. We already know what the potentials +are when we are standing in the rest ame of a charge. When the charge is +moving, we can fgure everything out by a relativistic transformation from one +system to the other. But relativity had its origin in the theory of electricity and +magnetism. "The formulas of the Lorentz transformation (Chapter 15, Vol. ]) were +discoveries made by Lorentz when he was studying the equations of electricity and +magnetism. So that you can appreciate where things have come from, we would +like to show that the Maxwell equations do lead to the Lorentz transformation. +W© begin by calculating the potentials of a charge moving with uniform velocity, +directly from the electrodynamics of Maxwell's equations. We have shown that +Maxwell's equations lead to the potentials for a moving charge that we got in +the last section. 5o when we use these potentials, we are using Maxwell”s theory. +(x, y2) +“RETARDED” POSITION +(Attf=t— r/c) r ï +Fig. 21-7. Finding the potential at P ofa ——— vÝ —— CN ứ +charge moving with uniform velocity along TEEN ỶÝỶ~ tư/ +the x-axis. ¬—_.. +E4 +Suppose we have a charge moving along the zø-axis with the speed 0. We +want the potentials at the point P{z, , z), as shown in Eig. 21-7. IÝ£ =0 ¡s the +mmoment when the charge is at the origin, at the time ý the charge is at # = 0, +=z=0. What we need to know, however, is Its position at the retarded time +ứ=t——, (21.35) +where ?“ is the distance to the point from the charge œ the retarded time. At +the earlier tìme #“, the charge was at ø = 0f”, so +rˆ = VW(& — 0)2 + 2+ Z2. (21.36) +To find ?ˆ or £ˆ we have to combine thìs equation with Eq. (21.35). Eirst, we +eliminate rˆ by solving Đq. (21.35) for rˆ and substituting in Eq. (21.36). Then, +squaring both sides, we get +c( — 2 — (œ — 0#)? + g7 + z2, +which is a quadratic equation in f“. Expanding the squared binomials and +collecting like terms in #, we get +(02 — c2)3 — 2(œ — c®t) + z2 + 2+ z2 — (eП =0. +Solving for f', +Đ 0z 1 Đ +1—-š|#=-=_—-_-\|(œ-tf+ |1— = ]|(w?+2?). 21.37 +(-5)='-#- s0 “5)02+z2). — (147 +--- Trang 272 --- +To get rzˆ we have to substitute this expression for # into +rˆ = c(t — #). +Ñow we are ready to find ó from Eq. (21.33), which, since is constant, +becomes 1 +f#=——— .r.—. 21.38 +ð(œ, ụ, 2, ) 4meo r! — (0 -r/c) ) +The component oŸ ø in the direction of r? is 0 x (œ — 0È)/r', so 0 -r” 1s just +0x (œ — 0£), and the whole denominator is +(~f)— “@œ—ø#)=clt- [1 DẦN, +c(f— È)— —(œ— 0È) =c|t— =—|l—--s . +lô c c2 +Substituting for (1 — 02/c?)#' from Ea. (21.37), we get for ở +Đ)=T————_, +Œ= 99+ (=5 )09+z) +This equation is more understandable if we rewrite i as +†#=———————_——————— -nnnz-. 21.39 +-š|[Eän) +] +C 1— 02/c2 +The vector potential A is the same expression with an additional factor of 0/c2: +A =-ø. +In Eq. (21.39) you can clearly see the beginning of the Lorentz transformation. +Tf the charge were at the origin in its own rest frame, its potential would be +Ó(#, 9, 2) — 47g [x2 + ự2 + z2)1⁄2 ì +W© are seeing it in a moving coordinate system, and it appears that the coordinates +should be transformed by +Ø — UẺ +=> —————pDp +v1_— 032/c2 +Ụ-?U, +zZ->Z. +That is Jjust the Lorentz transformation, and what we have done is essentially +the way Lorentz discovered ït. +But what about that extra factor 1/4/1 — 02/c2 that appears at the front of +Eq. (21.39)? Also, how does the vector potential A appear, when it is everywhere +zero in the rest rame of the particle? We will soon show that A and ó fogether +constitute a four-vector, like the momentum ø and the total energy of a particle. +The extra 1/4/1— 02/c2 in Bq. (21.39) is the same factor that always comes +in when one transforms the components of a four-vector——just as the charge +density ø transforms to Ø/4/1— 02/c2. In fact, it is almosb apparent from Eqs. +(21.4) and (21.5) that A and ð are components of a four-vector, because we have +already shown in Chapter 13 that 7 and ø are the components of a four-vector. +Later we will take up In more detail the relativity of electrodynamies; here +we only wished to show how naturally the Maxwell equations lead to the Lorentz +transformation. You will not, then, be surprised to fnd that the laws of electricity +and magnetism are already correct for Einstein's relativity. We will not have to +“ñx them up,” as we had to do for Newton's laws of mechanics. +--- Trang 273 --- +A€ fïrewr£s +22-1 Impedances +Most of our work in this course has been aimed at reaching the complete 22-1 Impedances +equations of Maxwell. In the last two chapters we have been discussing the 22-2 Qenerators +consequences of these equations. We have found that the equations contain all 22-3 Networks of ideal elements; +the static phenomena we had worked out earlier, as well as the phenomena. of Kirchhoffs rules +electromagnetic waves and light that we had gone over in some detail in Volume I. : ¬- +The Maxwell equations give both phenomena, depending upon whether one 22-4 Equivalent circuits +computes the ñelds close to the currents and charges, or very far rom them. 225 Enersy +There is not much interesting to say about the intermediate region; no special 22-6 A ladder network +phenomena appear there. 22-7 Filters +There still remain, however, several subJects in electromagnetism that we 22-8 Other circuit elements +want to take up. We want to discuss the question of relativity and the Maxwell +equations—what happens when one looks at the Maxwell equations with respect +to moving coordinate systems. There is also the question of the conservation of +energy in electromagnetic systems. Then there is the broad subject of the elec- +tromagnetic properties of materials; so far, except for the study of the properties +of dielectrics, we have considered only the electromagnetic fñelds in free space. +And although we covered the subject of light in some detail in Volume I, there Rcuieu: Chapter 22, Vol. l, Algebra +are still a few things we would like to do again om the point of view of the fñeld Chapter 23, Vol. l, Resonance +cequations. Chapter 25, Vol. lj Lưnear Sụs- +In particular, we want to take up again the subJect of the index of refraction, tems and Reuieu +particularly for dense materials. Finally, there are the phenomena associated +with waves confined in a limited region of space. We touched on this kind of +problem briefly when we were studying sound waves. Maxwell's equations lead +also %o solutions which represent confined waves of the electric and magnetic +fields. We will take up this subJect, which has important technical applications, +in some of the following chapters. In order to lead up to that subject, we will +begin by considering the properties of electrical circuits at low frequencies. We +will then be able to make a comparison bebween those situations in which the +almost static approximations of Maxwells equations are applicable and those +situations in which high-frequency efects are dominant. +So we descend from the great and esoteric heights of the last few chapters +and turn to the relatively low-level subject of electrical circuits. We will see, +however, that even such a mundane subject, when looked at in sufficient detail, +can contain great complications. +We have already discussed some of the properties of electrical circuits in +Chapters 23 and 25 of Vol. I. NÑow we will cover some oŸ the same material again, +but in greater detail. Again we are going to deal only with linear systems and +with voltages and currents which all vary sinusoidally; we can then represent +all voltages and currents by complex numbers, using the exponential notation +described in Chapter 28 of Vol. I. Thus a time-varying voltage V(£) will be +written +VỤ) = Ÿe*', (22.1) +where Ữ represents a complex number that ¡is independent of . It is, of course, +understood that the actual time-varying voltage V(£) is given by the real part of +the complex function on the right-hand side of the equation. +--- Trang 274 --- +Similarly, all of our other time-varying quantities will be taken to vary sinu- +soidally at the same frequency œ. So we write +I= Íc““ (curent), +€=êe”““ (emf, (22.2) +E=Êc““" (eleetrie feld), +and so on. +Most of the time we will write our equations in terms of V, ï, €,... (instead +of in terms of Ÿ, Ỉ › Ê, ...), remembering, though, that the time variations are +as given in (22.2). +In our earlier discussion of circuits we assumed that such things as inductances, La +capacitances, and resistances were familiar to you. We want now to look ïn a ~— +little a more detail at what is meant by these idealized circuit elements. We +begin with the inductance. +An inductance is made by winding many turns of wire in the form of a coil and +bringing the two ends out to terminals at some distance from the coil, as shown +in Fig. 22-1. We want to assume that the magnetic ñeld produced by currents in V +the coil does not spread out strongly all over space and interact with other parts +of the circuit. This is usually arranged by winding the coil in a doughnut-shaped +form, or by confning the magnetic fñeld by winding the coïl on a suitable iron +core, or by placing the coil in some suitable metal box, as indicated schematically +in Eig. 22-1. In any case, we assume that there is a negligible magnetic fñeld in T* P +the external region near the terminals ø and 0. We are also going to assume that +we can neglect any electrical resistance in the wire of the coïl. Einally, we will +assume that we can neglect the amount of electrical charge that appears on the Fig. 22-1. An inductance. +surface of a wire in building up the electric fields. +With all these approximations we have what we call an “ideal” inductance. +(We will come back later and discuss what happens in a real inductance.) Eor an +ideal inductance we say that the voltage across the terminals is equal to E(đ1T/đ9). +Let's see why that is so. When there ¡is a current through the inductance, a +magnetic fñeld proportional to the current is built up inside the coil. If the current +changes with time, the magnetic field also changes. In general, the curl of J is +cqual to —Ø/ðt; or, put diferently, the line integral of E all the way around any +closed path is equal to the negative of the rate of change of the fux of Ö through +the loop. Now suppose we consider the following path: Begin at terminal a and +go along the coil (staying always inside the wire) to terminal b; then reburn rom +terminal b to terminal ø through the air in the space outside the inductance. The +line integral of #/ around this closed path can be written as the sum of Ewo parts: +{E-ds= | E-ds+ J E- da. (22.3) +va outside +As we have seen before, there can be no electric felds inside a perfect conduector. +(The smallest fields would produce infnite currents.) Therefore the integral from +ø to Ö via the coil is zero. The whole contribution to the line integral of # comes +from the path outside the inductance from terminal b to terminal a. Since we +have assumed that there are no magnetic fñelds in the space outside of the “box,” +this part of the integral is independent of the path chosen and we can defñne the +potentials of the ©wo terminals. The diference of these two potentials is what we +call the voltage difference, or simply the voltage V, so we have +v=-Ï '`ẤN... +The complete line integral is what we have before called the electromotive +force € and is, of course, equal to the rate of change of the magnetic ñux in the +--- Trang 275 --- +coil. We have seen earlier that this emf is equal to the negative rate of change of +the current, so we have " +V=-Ê=h 1" +where E is the inductance of the coil. Since đĨ/d£ = 2Ï, we have +V = iuLT. (22.4) +The way we have described the ideal inductance illustrates the general ap- +proach to other ideal circuit elements——usually called “lumped” elements. 'Phe +properties of the element are described completely in terms of currents and +voltages that appear at the terminals. By making suitable approximations, it +1s possible to ignore the great complexities of the fields that appear inside the +object. A separation is made bebween what happens inside and what happens +outside. +Eor all the circuit elements we will nd a relation like the one in Eq. (22.4), in Ị +which the voltage is proportional to the current with a proportionality constant TC sa +that is, in general, a complex number. 'This complex coefficient of proportionality +is called the #mpedøance and is usually written as z (not to be confused with the +z-coordinate). It is, in general, a function of the frequenecy œ. So for any lumped +element we write ˆ +TT Z. (22.5) V +For an inductance, we have +z (nductance) = zr, = i1. (22.6) +Now let's look at a capacitor from the same point of view.* A capacitor —C b +consists of a pair of conducting plates from which two wires are brought out to ! +suitable terminals. The plates may be of any shape whatsoever, and are often +separated by some dielectric material. We illustrate such a situation schematically Fig. 22-2. A capacitor (or condenser). +in Eig. 22-2. Again we make several simplifying assumptions. We assume that the +plates and the wires are perfect conductors. We also assume that the insulation +between the plates is perfect, so that no charges can ow across the insulation +from one plate to the other. Next, we assume that the two conduectfors are close +to each other but far from all others, so that all fñeld lines which leave one plate +end up on the other. Then there are always equal and opposite charges on the +two plates and the charges on the plates are much larger than the charges on +the surfaces of the lead-in wires. Fìinally, we assume that there are no magnetic +fñields close to the capacitor. +Suppose now we consider the line integral of # around a closed loop which +starts at terminal a, goes along inside the wire to the top plate of the capacitor, +Jjumps across the space bebween the plates, passes from the lower plate to +terminal b through the wire, and returns to terminal ø in the space outside the +capacitor. Since there is no magnetic fñeld, the line integral of E around this +closed path is zero. The integral can be broken down into three parts: +{Ea | B.ds+ J E‹ds+ | E- da. (22.7) +along between outeide +Wires plates +The integral along the wires is zero, because there are no electric fñelds inside +perfect conductors. The integral from ö to ø outside the capacitor ¡is equal to the +negative of the potential diference between the terminals. Since we imagined +* 'There are people who say we should call the objec#s by the names “inductor” and +“capacitor” and call their properties “inductance” and “capacitance” (by analogy with “resistor” +and “resistance”). We would rather use the words you will hear in the laboratory. Most people +still say “inductance” for both the physical coil and its inductance L. The word “capacitor” +seems to have caught on—although you will still hear “condenser” fairly often—and most people +still prefer the sound of “capacity” to “capacitance.” +--- Trang 276 --- +that the two plates are in some way isolated from the rest of the world, the total +charge on the two plates must be zero; 1ƒ there is a charge Q on the upper plate, +there is an equal, opposite charge —Œ on the lower plate. We have seen earlier +that if two conductors have equal and opposite charges, plus and minus @, the +potential difference between the plates is equal to Q/Œ, where C is called the +capacity of the two conductors. From E4q. (22.7) the potential difference between +the terminals œ and ở is equal to the potential diference between the plates. We +have, therefore, that +The electric current Ï entering the capacitor through terminal ø (and leaving ~ a +through terminal ð) is equal to đQ/d£, the rate of change of the electric charge +on the plates. Writing đỰ/(đf as 2V, we can put the voltage current relationship +for a capacitor in the following way: +" 1 V +uV = Gi +V= mọi (22.8) j +'The impedance z of a capacitor, is then TP +z (capacitor) = zœ = ai (22.9) +Fig. 22-3. A resistor +'The third element we want to consider is a resistor. However, since we have +not yet discussed the electrical properties of real materials, we are not yet ready +to talk about what happens inside a real conductor. We will just have to accept +as fact that electric fñelds can exist inside real materials, that these electric fields +give rise to a ñow of electric charge—that is, to a current—and that this current +1s proportional to the integral of the electric ñeld from one end of the conductor +to the other. We then imagine an ideal resistor constructed as in the diagram +of Eig. 22-3. 'IWwo wires which we take to be perfect conductors go from the +terminals œ and ð to the two ends oŸ a bar of resistive material. Following our +usual line of argument, the potential diference between the terminals ø and b +1s equal to the line integral of the external electric fñeld, which is also equal to +the line integral of the electric ñeld through the bar of resistive material. It then +follows that the current 7 through the resistor is proportional to the terminal +voltage V: +R (a) (b) (c) (4) +where # is called the resistance. We will see later that the relation bebween +the current and the voltage for real conducting materials is only approximately +linear. We will also see that this approximate proportionality is expected to be \ Ỉ +independent of the frequency of variation of the current and voltage only If the 4+) V L C R +frequency is not too high. For alternating currents then, the voltage across a J | +resistor is in phase with the current, which means that the impedance is a real ủ +number: P +z (resistance) = zp = Ï. (22.10) „_Vv juL củ R +Our results for the three lumped circuit elements—the inductor, the capaecitor, +and the resistor——are summarized in Eig. 22-4. In this ñgure, as well as in the Fig. 22-4. The ideal lumped circuit ele- +preceding ones, we have indicated the voltage by an arrow that is directed om — ments (passwe). +one terminal to another. If the voltage is “positive”—that is, if the terminal ø is +at a higher potential than the terminal b—the arrow indicates the direction of a +positive “voltage drop.” +Although we are talking about alternating currents, we can of course include +the special case of circuits with steady currents by taking the limit as the +Írequency œ goes to zero. Eor zero frequency—that is, for DG—the impedance of +an inductance gøoes to zero; it becomes a short circuit. For DC, the impedance of +--- Trang 277 --- +a condenser goes to infinity; it becomes an open circuit. Since the impedance of +a resistor is independent oŸ frequency, it is the only element left when we analyze +a circuit for DC. +In the circuit elements we have described so far, the current and voltage are +proportional to each other. IÝ one is zero, so also is the other. We usually think in +terms like these: An applied voltage is “responsible” for the current, or a current +“gives rise to” a voltage across the terminals; so in a sense the elements “respond” +to the “applied” external conditions. For this reason these elerments are called +øass?ue clemen#s. Thhey can thus be contrasted with the active elements, such as +the generators we will consider in the next section, which are the sowrces of the +oscillating currents or voltages in a circuit. INN 3 +22-2 Generators N | +Now we want to talk about an øc#?ue circuit element——one that is a source of À +the currents and voltages in a circuit—namely, a generator. =—bI = V +Suppose that we have a coil like an inductance except that it has very few +turns, so that we may neglect the magnetic field of its own current. This coil, = +however, sits in a changing magnetic fñeld such as might be produced by a rotating +magnet, as sketched in Eig. 22-5. (We have seen earlier that such a rotating ° ú +magnetic fñeld can also be produced by a suitable set of coils with alternating +currents.) Again we must make several simplifying assumptions. The assumptions ° +we will make are all the ones that we described for the case of the inductanece. In +particular, we assume that the varying magnetic field is restricted to a deÑnite Fig. 22-5. A generator consisting of a +region in the vicinity of the coil and does not appear outside the generator in the fixed coil and a rotating magnetic field. +space between the terminals. +Following closely the analysis we made for the inductance, we consider the line +integral of #/ around a complete loop that starts at terminal ø, goes through the +coil to terminal b and returns to its starting point in the space between the two +terminals. Again we conclude that the potential diference between the terminals +is equal to the total line integral of # around the loop: +V=_— ‡ E- da. +'This line integral is equal to the emf in the circuit, so the potential diference V ` +across the terminals of the generator is also equal to the rate of change of the V +magnetic ñux linking the coil: 7 +V=-£Ê= đc ux). (22.11) . +For an ideal generator we assume that the magnetic fux linking the coil is deter- Fig. 22-6. Symbol for an ideal generator. +mined by external conditions—such as the angular velocity of a rotating magnetic +ñeld and is not inÑuenced in any way by the currents through the generator. +Thus a generator—at least the jdeal generator we are considering—is not an +impedance. “The potential diference across its terminals is determined by the ar- +bitrarily assigned electromotive force €(f). Such an ideal generator is represented +by the symbol shown in Fig. 22-6. The little arrow represents the direction oŸ the +emf when it is positive. A positive emf in the generator of Fig. 22-6 will produce +a voltage W = €, with the terminal a at a higher potential than the terminal 0. +There is another way to make a generator which is quite diferent on the +inside but which is indistinguishable from the one we have just described insofar +as what happens beyond its terminals. Suppose we have a coil of wire which is +rotated in a ƒized magnetic ñeld, as indicated in Fig. 22-7. We show a bar magnet +to indicate the presence of a magnetic field; ít could, of course, be replaced by +any other source oŸ a steady magnetic fñeld, such as an additional coil carrying a +steady current. As shown in the figure, connections from the rotating coil are +made to the outside world by means of sliding contacts or “slip rings” Again, we +are interested in the potential diference that appears across the bwo terminals +--- Trang 278 --- +Fig. 22-7. A generator consisting of a S j +coil rotating in a fixed magnetic field. b +ø and 0, which is of course the integral of the electric ñeld from terminal ø to +terminal Ò along a path outside the generator. +Now in the system of Fig. 22-7 there are no changing magnetic fñelds, so we +might at fñrst wonder how any voltage could appear at the generator terminals. In +fact, there are no electric felds anywhere inside the generator. We are, as usual, +assuming for our ideal elements that the wires inside are made of a perfectly +conducting material, and as we have said many times, the electric field inside a +perfect conduector is equal to zero. But that is not true. It is not true when a +conductor is moving in a magnetic fñeld. 'Phe true statement is that the total +ƒorce on any charge inside a perfect conductor must be zero. Otherwise there +would be an infinite flow of the free charges. So what is always true is that the +sum o the electric field # and the eross product of the velocity of the conductor +and the magnetic fñield ——which is the total force on a unit charge—must have +the value zero inside the conductor: +F'/unit chargee = E+uxiB=0. (in a perfect conductor), (22.12) +where ® represents the velocity of the conductor. Our earlier statement that +there is no electric ñeld inside a perfect conductor is all right if the velocity of +the conductor is zero; otherwise the correct sbatement is given by Bq. (22.12). +Returning to our generator of Eig. 22-7, we now see that the line integral of +the electric fñeld # om terminal œø to terminal b through the conducting path of +the generator must be equal to the line integral of x Ö on the same path, +J E-ds—=— J (o xÐ) - ds. (22.13) +condevor condevor +lt is still true, however, that the line integral of E around a complete loop, +including the return om ö to œø outside the generator, must be zero, because +there are no changing magnetic fields. So the first integral in Eq. (22.13) is +also equal to W, the voltage between the two terminals. It turns out that the +right-hand integral of Eq. (22.13) is Just the rate of change of the fux linkage +through the coil and is therefore—by the Ñux rule—equal to the emf in the coil. +So we have again that the potential diference across the terminals is equal to +the electromotive force in the circuit, in agreement with Eq. (22.11). So whether +we have a generator in which a magnetic fñeld changes near a fñxed coil, or one +in which a coil moves in a fxed magnetic feld, the external properties of the +generators are the same. 'here is a voltage diference V across the terminals, +which is independent of the current in the cireuit but depends only on the +arbitrarily assigned conditions inside the generator. +So long as we are trying to understand the operation of generators from +the point of view of Maxwell's equations, we might also ask about the ordinary +chemical cell, like a fashlight battery. It is also a generator, i.e., a voltage source, +although it will of course only appear in DC circuits. The simplest kind of cell +--- Trang 279 --- +to understand is shown in Fig. 22-8. We imagine ©wo metal plates immersed +in some chemical solution. We suppose that the solution contains positive and +negative ions. We suppose also that one kind of ion, say the negative, is mụuch +heavier than the one of opposite polarity, so that its motion through the solution +by the process of difusion ¡is mụuch slower. We suppose next that by some means +or other ¡% is arranged that the concentration of the solution is made to vary +from one part of the liquid to the other, so that the number of ions of both +polarities near, say, the lower plate is much larger than the concentration of ions : =— an ! +near the upper plate. Because of their rapid mobility the positive ions will drift ¬ nh, mm. “a2 +more readily into the region of lower concentration, so that there will be a slight ý HQ un Tài +excess of positive charge arriving at the upper plate. The upper plate will become " l "¬ +positively charged and the lower plate will have a net negative charge. -k . +As more and more charges difuse to the upper plate, the potential of this x ¬ +plate will rise until the resulting electrie ñeld between the plates produces forces cà ¬ + ¬ v +on the ions which just compensate for their excess mobility, so the two plates of ha +the cell quickly reach a potential diference which is characteristic of the internal " 4<. L. ị +construetion. ¬- ¬ +Arguing just as we did for the ideal capacitor, we see that the potential difer- . ¬ - b +ence between the terminals ø and ở is just equal to the line integral of the electric ¬ ¬_ +fñeld between the two plates when there is no longer any net difusion of the ions. +'There is, of course, an essential diference between a capacitor and such a chemical Fig. 22-8. A chemical cell. +cell. If we short-circuit the terminals of a condenser for a moment, the capacitor +is discharged and there is no longer any potential diference across the terminals. +In the case of the chemical cell a current can be drawn from the terminals con- +tinuously without any change in the emf—=until, of course, the chemicals inside +the cell have been used up. In a real cell it is found that the potential diference +across the terminals decreases as the current drawn from the cell increases. In +keeping with the abstractions we have been making, however, we may imagine an +ideal cell in which the voltage across the terminals is independent of the current. +A real cell can then be looked at as an ideal cell in series with a resistor. +22-3 Networks of ideal elements; Kirchhoff?s rules +As we have seen in the last section, the description of an ideal circuit element +in terms of what happens outside the element is quite simple. The current and the š P +voltage are linearly related. But what is actually happening inside the element is > —>_— / +quite complicated, and ït is quite dificult to give a precise description in terms of \ & / +Maxwell's equations. Imagine trying to give a precise description of the electric { +and magnetic fields of the inside of a radio which contains hundreds oŸ resistOrs, 5. +capacitors, and inductors. It would be an impossible task to analyze such a / \ 5 +thing by using Maxwell's equations. But by making the many approximations g ⁄ \ +we have described in Section 22-2 and summarizing the essential features of the ` +real circuit elements in terms of idealizations, it becomes possible to analyze an ` \ +electrical circuit in a relatively straightforward way. We will now show how that }⁄ ⁄ ' +is done. ⁄⁄ rV +Suppose we have a circuit consisting of a generator and several impedances f +connected together, as shown in Eig. 22-9. According to our approximations N † +there is no magnetic feld in the region outside the individual cireuit elements. \ ú |” +Therefore the line integral of # around any curve which does not pass through | M | +any of the elements is zero. Consider then the curve I' shown by the broken line L : \ +which goes all the way around the circuit in Fig. 22-9. 'The line integral of E / _Ằ—- ` +around this curve is made up of several pieces. Each piece is the line integral đ ` +from one terminal of a circuit element to the other. 'This line integral we have š W +called the voltage drop across the circuit element. The complete line integral is +then just the sum of the voltage drops across all of the elements in the circuit: Fig. 22-9. The sum of the voltage drops +around any closed path Is zero. +‡ E-da= ` Vị, +Since the line integral is zero, we have that the sum of the potential diferences +--- Trang 280 --- +around a complete loop of a circuit is equal to zero: +` t,=0. (22.14) +any loop +Thịs result follows from one of Maxwell's equations—that in a region where there +are no magnetic fñelds the line integral of # around any complete loop is 2ero. +uppose we consider now a circuit like that shown in Fig. 22-10. “The horizontal 2 b c d +line joining the terminals ø, Ù, c, and đ is intended to show that these terminals +are all connected, or that they are joined by wires of negligible resistance. In là +any case, the drawing means that terminals ø, b, c, and ở are all at the same / +potential and, similarly, that the terminals e, ƒ, g, and h are also at one common V 6€) Z Z Z +potential. 'Then the voltage drop W across each of the four elements is the same. +Now one of our idealizations has been that negligible electrical charges accu- \ +mulate on the terminals of the impedances. We now assume further that any ụ‹ +electrical charges on the wires joining terminals can also be neglected. Then the e f k h +conservation of charge requires that any charge which leaves one circuit element Eig. 22-10. The sum of the currents into +Immediately enters some other circuit element. Ôr, what is the same thing, we . +. . . . . . any node Is zero. +require that the algebraic sum of the currents which enter any given junction +must be zero. By a junction, of course, we mean any set of terminals such as +ø, Ù, c, and đ which are connected. Such a set of connected terminals is usually +called a “node.” “The conservation of charge then requires that for the circuit of +Eig. 22-10, +TH — lạ— lạ — lạ =0. (22.15) +The sum of the currents entering the node which consists of the four terminals +©, ƒ, g, and h must also be zero: +— + lạ~+ Ts + Tạ = 0. (22.16) +Thịs is, of course, the same as Bq. (22.15). The two equations are not independent. +The general rule is that the sưma oƒ the currents tnio ơn node rmust be zero: 3 b c +À3 1,=0. (22.17) +a node (9 +Our earlier conclusion that the sum of the voltage drops around a closed loop +is zero must apply to any loop in a complicated circuit. Also, our result that the +sum of the currents into a node is zero must be true for any node. These two d œ® +cequations are known as l{rchhofƒs rules. With these two rules it is possible %o +solve for the currents and voltages in any network whatever. +uppose we consider the more complicated circuit of Fig. 22-11. How shall we +ñnd the currents and voltages in this circuit? We can ñnd them in the following z Z6 +straiphtforward way. We consider separately each of the four subsidiary closed +loops, which appear in the circuit. (Eor instance, one loop goes Írom terminal a +to terminal b to terminal e to terminal đ and back to terminal a.) For each of +the loops we write the equation for the first of Kirchhoffˆs rules—that the sum 9 +of the voltages around each loop is equal to zero. We must remember to coun$ +the voltage drop as positive if we are going 7n the direction of the current and Fig. 22-11. Analyzing a circuit with Kirch- +negative if we are going across an element in the direction øpposite to the current; hoff”s rules. +and we must remember that the voltage drop across a generator is the negøtue +of the emf in that direction. Thus If we consider the small loop that starts and +ends at terminal a we have the equation +21h + zaÏa + 241 — €1 =0. +Applying the same rule to the remaining loops, we would get three more equations +of the same kind. +Next, we must write the current equation for each of the nodes in the circuit. +For example, summing the currents into the node at terminal b gives the equation +— lạ — lạ =0. +--- Trang 281 --- +Similarly, for the node labeled e we would have the current equation +Tạ — lị + lạ — lạ =0. +For the circuit shown there are fve such current equations. Ït turns out, however, +that any one of these equations can be derived from the other four; there are, +therefore, only four independent current equations. We thus have a total of eight +independent, linear equations: the four voltage equations and the four current +cquations. With these eight equations we can solve for the eight unknown currents. +Once the currents are known the circuit is solved. 'Phe voltage drop across any +element is given by the current through that element times its impedance (or, in +the case of the voltage sources, it is already known). +We have seen that when we write the current equations, we get one equation +which is not independent of the others. Generally it is also possible to write down +too many voltage equations. Eor example, in the circuit of EFig. 22-11, although we +have considered only the four small loops, there are a large number of other loops +for which we could write the voltage equation. There is, for example, the loop +along the path abcƒeda. 'Phere is another loop which follows the path œbcƒehgda. +You can see that there are many loops. In analyzing complicated circults it is +very easy to get too many equations. There are rules which tell us how to proceed +so that only the minimum number of equations is written down, but usually with +a little thought it is possible to see how to get the ripght number of equations +in the simplest form. Besides, writing an extra equation or two doesn”t do any +harm. They will not lead to any wrong answers, only perhaps a littÏe unnecessary +algebra. +In Chapter 25 of Vol. Ï we showed that 1f the two Impedances z¡ and za are +in series, they are equivalent to a single impedance z; given by +zz=zi+2a. (22.18) ©) R „ » +W© also showed that if the two impedances are connected in parailel, they are +cquivalent to the single impedance z„ given by +1 Z1Z22 +” /4)+(0/2) +22) (2219) Fig. 22-12. A circuit which can be ana- +lyzed in terms of series and parallel combi- +Tf you look back you will see that in deriving these results we were In effect nations. +making use of Kirchhoffs rules. It is often possible to analyze a complicated +circuit by repeated application of the formulas for series and parallel impedaneces. +For instance, the circuit of Fig. 22-12 can be analyzed that way. First, the +impedaneces z4 and zz can be replaced by theïr parallel equivalent, and so aÌso can +zs and z;. Then the impedance z¿ can be combined with the parallel equivalent +of zs and z; by the series rule. Proceeding in this way, the whole cireuit can be +reduced to a generator in series with a single impedance Z. 'Phe current through +the generator is then just €/Z. Then by working backward one can solve for the +currents in each of the impedaneces. +There are, however, quite simple circuits which cannot be analyzed by this +method, as for example the circuit of Fig. 22-13. 'To analyze this circuit we must +|*= — (h+la) +Fig. 22-13. A circuit that cannot be ana- +lyzed in terms of series and parallel combi- +natlons. +--- Trang 282 --- +write down the current and voltage equations from Kirchhoffs rules. Let”s do it. +'There is just one current equation: +hạ +ls+ la =0, +so we know immediately that +Tạ = —(h + 1a). +W© can save ourselves some algebra if we immediately make use of this result in +writing the voltage equations. For this circuit there are ©wo independent voltage +cequations; they are +—ẾI + l2z2 — TịZn =0 +&2 — (h + 12)za — z2 =0. +'There are Ewo equations and two unknown currents. Solving these equations for +lị and l¿, we get › +h— 262— Ea + 2)ểt (22.20) +ZI(Za + Z4) + Z2Z4 +=—..... (22.21) +Z1 (z2 + Z3) + 2223 G +The third current is obtained from the sum of these two. mx +Another example of a circuit that cannot be analyzed by using the rules for +series and parallel impedanee is shown in Fig. 22-14. Such a circuit is called a SN ⁄4 +“bridge.” It appears in many instruments used for measuring impedances. With +such a circuit one is usually interested in the question: How must the various +impedaneces be related if the current through the impedance zs is to be zero? We +leave it for you to ñnd the conditions for which this is so. +Fig. 22-14. A bridge circuit. +22-4 bquivalent circuits +Suppose we connect a generator Ê to a circuit containing some complicated +interconnection of impedances, as indicated schematically in Eig. 22-15(a). AI +of the equations we get from Kirchhof?s rules are linear, so when we solve them l +for the current 7 through the generator, we will get that Ï is proportional to €. —" ¿§ +We can write +T= ¬ ( Any +Zcf (a) V cưa +where now ze£ is some complex number, an algebraic function of all the elements \ zs +in the circuit. (Tf the circuit contains no generators other than the one shown, +there is no additional term independent of Ê.) But this equation is just what H +we would write for the circuit of Fig. 22-15(b). So long as we are interested +only in what happens ứø ứhe leƒft of the two terminals ø and b, the two circuits +of Eig. 22-15 are cguzualent. We can, therefore, make the general statement l +that an two-terminal nebwork of passive elements can be replaced by a single —> £ +impedance zeg# without changing the currents and voltages in the rest of the +circuit. 'Phis statement is of course, jus a remark about what comes out of +Kirchhoffs rules—and ultimately from the linearity of Maxwell's equations. li ứ-) Zefr +The idea can be generalized to a circuit that contains generators as well as +impedances. Suppose we look at such a circuit “from the point oŸ view” of one of +the impedances, which we will call z„, as in Fig. 22-16(a). IÝ we were to solve h +the equation for the whole circuit, we would fnd that the voltage V„ between +the two terminals ø and b is a linear function of Ï, which we can write Fig. 22-15. Any two-terminal network of +passive elements is equivalent to an effective +V„y=A_— Bl,, (22.22) impedance. +where 44 and depend on the generators and impedances in the circuit to the +--- Trang 283 --- +left of the terminals. For instance, for the circuit of Eig. 22-13, we ñnd VỊ = Tqz\. I, +This can be written (by rearranging Eq. (22.20)] as a—>= +W= I. — êi mm. (22.23) Any l +Z2 + Z3 2a + Z3 (a) Circuit W Zn +of z's +The complete solution is then obtained by combining this equation with the one and #'s \ +for the impedance z¡, namely, VỊ = Ïlqz¡, or in the general case, by combining +Eq. (22.22) with b +Vi = laza. +Tf now we consider that z„ is attached to a simple series circuit of a generator lạ +and a current, as in Eig. 22-15(b), the equation corresponding to Eq. (22.22) is ạ TT +tạ = lợn — laZet, +which is identical to Eq. (22.22) provided we set Ể¿ø = 4 and zeq = Ö. So if we Ze +are interested only in what happens fo ứhe rúgh‡ of the terminals ø and b, the ®œ) W, Fn +arbitrary circuit of Fig. 22-16 can always be replaced by an equivalent combination +OŸ a generator in series with an impedanece. \ +22-5 Energy +We have seen that to build up the current ƒ in an inductanece, the energy = b +3L]? must be provided by the external circuit. When the current falls back to Fig. 22-16. Any two-terminal network +zero, this energy is delivered back to the external circuit. 'There is no energy-Ìoss can be replaced by a generator in series with +mnechanism in an ideal inductance. When there is an alternating current through an impedance. +an inductance, energy fows back and forth between it and the rest of the circuit, +but the auerage rate at which energy is delivered to the circuit is zero. We say that +an inductance is a nondissipatiue element; no electrical energy is dissipated—that +1s, “lost”=—in it. +Similarly, the energy of a condenser, Ữ = sCV3, is returned to the external +circuit when a condenser is discharged. When a condenser is in an AC circuit +energy flows in and out of it, but the net energy flow in each cycle is zero. An +ideal condenser is also a nondissipative element. +W© know that an emf is a source of energy. When a current 7 Ñows in the direc- +tion of the emf, energy is delivered to the external cireuit at the rate đU “dt = 6T. +TÍ current is driven agøns the emf—by other generators in the cireuit—the emf +will absorb energy at the rate €1; since Ï is negative, đŨ /dt will also be negative. +T a generator is connected to a resistor #, the current through the resistor +is Ï = €/R. The energy being supplied by the generator at the rate €T is being +absorbed by the resistor. This energy goes into heat in the resistor and is los§ +from the electrical energy of the circuit. We say that electrical energy is đissipated +in a resistor. The rate at which energy is dissipated in a resistor is đU/dt = R12. +In an AC circuit the average rate of energy lost to a resistor is the average +of I!2 over one cycle. Since Ï = h e”“f—by which we really mean that T varies +as œos¿—the average of I2 over one eycle is |Í|2/2, sinee the peak current is |Í] +and the average of cos2 œf is 1/2. E +'What about the energy loss when a generator is connected to an arbitrary —— +impedance z? (By “loss” we mean, of course, conversion oŸ electrical energy into z — +thermal energy.) Any impedance z can be written as the sum of its real and +imaginary parts. That is, +z=l+:x, (22.24) +where and X are real numbers. From the point of view of equivalent circuits +we can say that any impedance is equivalent to a resistance in series with a pure +imaginary impedance—called a reactance——as shown ïn Fig. 22-17. Fig. 22-17. Any impedance is equivalent +W© have seen earlier that any circuit that contains only 7s and C”s has an to a series combination of a pure resistance +impedance that is a pure imaginary number. Since there is no energy loss Into any and a pure reactance. +ofthe 's and Œ”s on the average, a pure reactance containing only ˆs and C”s will +have no energy loss. We can see that this must be true in general for a reactanee. +--- Trang 284 --- +lÝ a generator with the emf Ê is connected to the impedance z of Fig. 22-17, +the emf must be related to the current 7 from the generator by +€=I(R+¿¡X). (22.25) +To ñnd the average rate at which energy is delivered, we want the average of +the product €ï. NÑow we must be careful. When dealing with such products, we +must deal with the real quantities Ê(£) and 7(£). (The real parts of the complex +functions will represent the actual physical quantities only when we have Ìmeør +cquations; now we are concerned with produecis, which are certainly not linear.) +Suppose we choose our origin of # so that the amplitude Ï is a real number, +let°s say Tọ; then the actual time variation Ï is given by +T = locOsưt. +The emf of Eq. (22.25) is the real part oŸ +lọạe“"(R + ¡X) +€ = loRcosut — lọX sinut. (22.26) +The two terms in Eq. (22.26) represent the voltage drops across χ and X in +Hig. 22-17. We see that the voltage drop across the resistance is 7" phase with +the current, while the voltage drop across the purely reactive part is ou‡ oƒ phase a a +with the current. +The aerage rate oŸ energy loss, (P)zv, from the generator is the integral oŸ (y = Z2 —=Zi +2 +the produect €ƒ over one cycle divided by the period 7; in other words, +1V 1V 2 2 1V 2 : +(P)av= Clảt =~ 1 R cos“ ¡‡ đt — — 1 X cos œ‡ sin „‡ dự. +TJo TJo To 3 rz “1z | +[Z4 | [z: | 1 +The first integral is sIãR, and the second integral is zero. 5o the average +energy loss in an impedance z = + ¿X depends only on the real part of z, and +is IỂR/2, which is in agreement with our carlier result for the energy loss in a b b +resistor. 'Phere is no energy loss in the reactive part. +22-6 A ladder network +W© would like now to consider an interesting circuit which can be analyzed = (4) = (@) +in terms of series and parallel combinations. Suppose we start with the circuit b b +of Eig. 22-18(a). We can see right away that the impedance from terminal ø to +terminal ð is simply zị + z¿. Now let's take a little harder circuit, the one shown 11,1 z =zi tay +in Eig. 22-18(b). We could analyze this circuit using Kirchhoffs rules, but it is ZA. Z2 Z2 +also easy to handle with series and parallel combinations. We can replace the Fig. 22-18. The effective impedance of +two impedances on the right-hand end by a single impedance zs = z¡ + z2, as in a ladder. +part (c) of the figure. Then the two impedances z2 and zs can be replaced by +their equivalent parallel impedance z4, as shown in part (d) of the fñgure. Einally, +z¡ and z4 are equivalent to a single impedance zs, as shown in part (©). +Now we may ask an amusing question: What would happen I1f in the network +of Fig. 22-IS(b) we kept on adding more sections ƒoreuer——as we indicate by +the dashed lines in Fig. 22-19(a)? Can we solve such an infinite network? Well, +(a) etc. (b) = +b — b b +Fig. 22-19. The effective impedance of an infinite ladder. +--- Trang 285 --- +that 's not so hard. First, we notice that such an infnite network is unchanged ïf +we add one more section at the “front” end. Surely, if we add one more section +to an infnite network it is still the same infinite network. Suppose we call the +Impedance between the bwo terminals ø and ö of the infnite network zọo; then +the impedance of all the stuf to the right of the wo terminals e and đ is also Zọ. +Therefore, so far as the front end is concerned, we can represent the network +as shown in Eig. 22-19(b). Forming the parallel combination of za¿ with zo and +adding the result in series with z¡, we can immediately write down the impedance +Of this circuit: +1 Z2Z0 +221 172)+ (1/20) OF mm... +But this impedance is also equal to zọ, so we have the equation +Z2Z0 +Zo = Z1 + z2 +20” +W© can solve for zọ to get +z0 = 5 + \/(zŸ/4) + z1za. (22.27) +So we have found the solution for the impedance of an infnite ladder of repeated +series and parallel impedances. “The impedance zọ is called the characteristic +tmpedance of such an infinite network. 2 ung _ _ Ly +Let's now consider a specifc example in which the series element is an +inductance 7 and the shunt element is a capacitance Ở, as shown in Fig. 22-20(a). “ b : : _~ +In this case we fnd the impedance of the inñnite network by setting z¡ = ?œ TT” +and 2a = 1/2. Notice that the first term, z‡/2, in Eq. (22.27) is jusE one-half L/2 1/2 L/2 L/2 +the impedance of the first element. It would therefore seem more natural, or kển vi Á À & ù Á TÀ _-- +at least somewhat simpler, IÝ we were to draw our infnite network as shown in +Eig. 22-20(b). Looking at the inũnite network from the terminal a” we would see &) b ° : cac +the characteristic Impedance = +zo = V(L/Œ) — (u2L2/4). (22.28) Fig. 22-20. An L-C ladder drawn in two +equlivalent ways. +Now there are two interesting cases, depending on the frequeney œ2. IÝ¿JÊ is less +than 4/LƠ, the second term in the radical will be smaller than the frst, and the +impedanee zọ will be a real number. On the other hand, if ¿2 is greater than 4/EŒ +the impedance zọ will be a pure imaginary number which we can write as +Zo = ‡VW(œ2L2/4) — (L/C). +We© have said earlier that a circuit which contains only imaginary impedances, +such as inductances and capacitances, will have an impedance which is purely +imaginary. How can i% be then that for the circuit we are now studying—which +has only 's and C”s—the impedance is a pure resistance for frequencies be- +low 4⁄4/EŒ? Eor higher frequencies the impedance is purely imaginary, in +agreement with our earlier statement. Eor lower frequencies the impedance is +a pure resistance and will therefore absorb energy. But how can the circuit +continuously absorb energy, as a resistance does, ïÝ it is made only of inductances +and capacitances? Anseer: Because there is an infinite number of inductances +and capacitances, so that when a source is connected to the circuit, it supplies +energy to the first inductance and capacitance, then to the second, to the third, +and so on. In a circuit of this kind, energy is continually absorbed from the +generator a% a constant rate and fows constantly out into the network, supplying +energy which is stored in the inductances and capacitances down the line. +This idea suggests an interesting poin about what is happening ïn the circuit. +We would expect that IÝ we connect a source to the front end, the efects of +this source will be propagated through the nebwork toward the infnite end. +The propagation of the waves down the line is mụch like the radiation from an +antenna which absorbs energy from its driving source; that is, we expect such +a propagation to occur when the impedance is real, which occurs 1Ý œ is less +than 4⁄4/TŒ. But when the impedance is purely imaginary, which happens for +ằœ greater than 4⁄4/TŒ, we would not expect to see any such propagation. +--- Trang 286 --- +22-7 Eilters +We saw in the last section that the infñnite ladder nebwork of Eig. 22-20 +absorbs energy continuously if it is driven at a frequency below a certain critical +frequency 4⁄4/LC, which we will call the cutofƑ frequenecw œg. We suggested +that this efect could be understood in terms oŸ a continuous transport of energy +down the line. Ôn the other hand, at high frequencies, for œ > œọ, there is no +continuous absorption of energy; we should then expect that perhaps the currents +donˆt “penetrate” very far down the line. Let's see whether these ideas are right. +Suppose we have the front end of the ladder connected to some AC generator +and we ask what the voltage looks like at, say, the 754th section of the ladder. +Since the network is infnite, whatever happens to the voltage from one section +to the next is always the same; so let”s just look at what happens when we go +from some section, say the r+th to the next. We will defñne the currents ?„ and +voltages Vạ„ as shown in Fig. 22-21(a). +l lạ la +—> —> —> In ln+1 +__— —»> —»> +Ỉ T I T — +' ' ; j +Fig. 22-21. Finding the propagation factor of a ladder. +W© can get the voltage V„++ om + by remembering that we can always re- +place the rest of the ladder after the „th section by its characteristic impedance zọ; +then we need only analyze the circuit of Fig. 22-21(b). First, we notice that +any Mạ, sỉnce iÈ is across zo, must equal ï„zọo. Also, the điference between V„ +and W¿i is Just Ï„z1: +Z1 +My — „+ = TđZ1 = Vy ——, +Z0 +So we get the ratio +n1 _j_Ÿ1_2—Z1 +ặ› Z0 20 l +W© can call this ratio the propagafion ƒactor for one section of the ladder; we'll +call it œ. It is, of course, the same for all sections: +Zo — Z +œ= “—^. (22.29) +Z0 +The voltage after the ?+th section is then +V„ = œ”€. (22.30) +You can now fñnd the voltage after 754 sections; it is just œ to the 754th power +tỉmes Ê. +Suppose we see what œ is like for the -Œ ladder of Eig. 22-20(a). Using zọ +from Eq. (22.27), and z¡ = iœ, we get +TL/Œ) — (2L^2/4) — ¡(uL/2 +¿— VI/G) = 8T5J) — ï(øLJ2) 6331) +V{(L/G) — (u2L2/4) + i¡(uL/2) +Tf the driving frequency is below the cutoff requency œo = 4⁄4/LEC, the radical +1s a real number, and the magnitudes of the complex numbers in the numerator +and denominator are equal. Therefore, the magnitude of œ is one; we can write +--- Trang 287 --- +which means that the magnitude of the voltage is the same at every section; +only its phase changes. The phase change ổ is, in fact, a negative number and +represents the “delay” of the voltage as it passes along the network. +For frequencies above the cutof frequeney œ 1% is better to factor out an ¿ +from the numerator and denominator of Eq. (22.31) and rewrite it as +2T2/A) — — œ +„=Y = /4)~ Q/@) - (@LJ2) (22.32) " +(2215/4) = (EJG) + (1/2) +The propagation factor œ is now a reøl number, and a number ess fhan, one. +That means that the voltage at any section is always less than the voltage at the +preceding section by the factor œ. For any frequency above œọ, the voltage dies | +away rapidly as we go along the network. A plot of the absolute value oŸ œ as a Nộ tp mm +function of frequenecy looks like the graph in Fig. 22-22. +W© see that the behavior of œ, both above and below œ, agrees with our Flg. 22-22. The propagation factor of a +Interpretation that the network propagates energy for œ¿ < œ and bloecks 1E section of an [-C ladder. +for œ > œạ. W©e say that the network “passes” low frequencies and “rejects” or +“filters out” the high frequencies. Any network designed to have its characteristics +vary in a prescribed way with frequency is called a “ñlter” We have been analyzing +a “low-pass filter.” +You may be wondering why all this discussion of an infinite network which C Ï e N +obviously cannot actually occur. The point is that the same characteristics are | F—T- +found in a fnite network If we finish it of at the end with an impedance equal +to the characteristic impedance zọ. Now in practice it is not possible to ezøctl r r r r +reproduce the characteristic impedance with a few simple elements—like s, Ù,”s, +and Œ?s. But it is often possible to do so with a faïir approximation for a certain _ +range of Írequencies. In this way one can make a fñnite filter network whose &) +properties are very nearly the same as those for the infnite case. For instance, lai +the L-Œ ladder behaves much as we have described 1t If it is terminated in the +pure resistance Jd= /L/Œ. +lf in our L-Œ ladder we interchange the positions of the Ƒs and C”s, to make 1 +the ladder shown in Fig. 22-23(a), we can have a filter that propagates ¿0h +frequencies and rejects iou frequencies. Ït is easy to see what happens with this +network by using the results we already have. You will notice that whenever we ' +change an Ù to a Ở and 0iee 0ersa, we also change every 2œ to 1/2. So whatever 0 +happened at œ before will now happen at 1/œ. In particular, we can see how œ 1/0p 1/0 +will vary with frequency by using Fig. 22-22 and changing the label on the axis @Œ) +to 1/0, as we have done in Eig. 22-23(b). Eig. 22-23. (a) A high-pass filter; (b) its +The low-pass and high-pass filters we have described have various technical : . +mm" - : - propagation factor as a function of 1/0. +applications. An I~-Œ low-pass filter is often used as a “smoothing” filter in a +DC power supply. If we want to manufacture DC power from an AC source, we +begin with a rectiler which permits current to fow only in one direction. Erom +the rectifier we get a series of pulses that look like the function V(£) shown in +Fig. 22-24, which is lousy DC, because it wobbles up and down. Suppose we +would like a nice pure DC, such as a battery provides. We can come close to that +by putting a low-pass filter between the rectifier and the load. +W©e know from Chapter 50 of Vol. I that the time function in Eig. 22-24 can +be represented as a superposition of a constant voltage plus a sine wave, plus a +higher-frequency sine wave, plus a still higher-frequency sine wave, etc.—by a V() +Fourier series. IÝ our filter is linear (ïf, as we have been assuming, the s and Œ”s +donˆt vary with the currents or voltages) then what comes out of the filter is +the superposition of the outputs for each component at the input. lÝ we arrange +that the cutoff frequenecy œọ of our flter is well below the lowest frequency in the ụ T f +function V{), the DC (for which œ = 0) goes through ñine, but the amplitude of Eig. 22-24. The output voltage of a full- +the ñrst harmonic will be cụt down a lot. And amplitudes of the higher harmonics wave rectifier. +will be cut down even more. So we can get the output as smooth as we wish, +depending only on how many flter sections we are willing to buy. +A high-pass filter is used iŸ one wants to reject certain low frequencies. For +Instance, in a phonograph amplifer a high-pass filter may be used to let the +--- Trang 288 --- +music through, while keeping out the low-pitched rumbling from the motor of +the turntable. +Tlt 1s also possible to make “band-pass” filters that reject frequencies below +some Írequency œ and above another frequenecy œs (greater than œ1), but pass +the frequenecies between œ¿¡ and œ¿. This can be done simply by putting together +a high-pass and a low-pass filter, but it is more usually done by making a ladder in +which the impedaneces z¡ and z¿ are more complieated——being each a combination Lai +of Es and C”s. Such a band-pass filter might have a propagation constant like (a) Bi ị +that shown in Eig. 22-25(a). It might be used, for example, in separating signals h +that occupy only an interval of frequencies, such as each of the many voice T1 +channels in a high-frequency telephone cable, or the modulated carrier of a radio mm “ +transmission. Ị +W©e have seen in Chapter 25 of Vol. I that such filtering can also be done N_Ị +using the selectivity of an ordinary resonance curve, which we have drawn for (@) Ị ' +comparison in Fig. 22-25(b). But the resonant filter is not as good for some CIẮN- +purposes as the band-pass filter. You will remember (Chapter 48, Vol. I) that _ Ị np _ +when a carrier of Ífrequenecy œ„ is modulated with a “signal” frequenecy œ„, the +total signal contains not only the carrier frequenecy but also the two side-band Fig. 22-25. (a) A band-pass filter. (b) A +frequencies œ¿„ + œ@; and œ¿ — œ;¿. With a resonant filter, these side-bands are simple resonant filter. +always attenuated somewhat, and the attenuation is more, the higher the signal +Ífrequency, as you can see from the figure. So there is a poor “fequency response.” +The higher musical tones don't get through. But if the fñltering is done with a +band-pass filter desipgned so that the width œ¿ — œ is at least twice the highest +signal frequeney, the frequenecy response will be “fat” for the signals wanted. +We want to make one more point about the ladder filter: the L-Œ ladder +of Eig. 22-20 is also an approximate representation of a transmission line. Tf +we have a long conductor that runs parallel to another conductor—such as a +wire in a coaxial cable, or a wire suspended above the earth—there will be some +capacitance between the two conductors and also some inductance due to the +magnetic fñeld between them. IÝ we imagine the line as broken up into small +lengths A⁄, each length will look like one section of the -C ladder with a series +inductance A”, and a shunt capacitance AC. We can then use our results for the +ladder filter. If we take the limit as Aý goes to zero, we have a good description +of the transmission line. Notice that as A/ is made smaller and smaller, both AT +and AC decrease, but in the same proportion, so that the ratio AL/AC remains +constant. So if we take the limit of Eq. (22.28) as AE and AC go to zero, we +fnd that the characteristic impedance zọ is a pure resistance whose magnitude l; +is /AL/AC. WG can also write the ratio AU/AC as Lo/Cc, where Do and Œg h ——- +are the inductance and capacitance of a unit length of the line; then we have —= +Zo = Z : (22.33) +You will also notice that as AE and AC go to zero, the cutof frequency œg = +V4/LC goes to infinity. There is no cubof frequeney for an ideal transmission +—> _-— +22-8 Other circuit elements +We have so far defned only the ideal cireuit impedances—the inductance, the ¬ ă +capacitance, and the resistance—as well as the ideal voltage generator. We want +now to show that other elements, such as mutual inductances or transisbOrs Or +vacuum tubes, can be described by using only the same basic elements. Suppose +that we have two coils and that on purpose, or otherwise, some Ñux om one of +the coils links the other, as shown in Fig. 22-26(a). Then the two coils will have +. xẰ- : (b) +a mutual inductance MỸ such that when the current varies in one of the coils, +there will be a voltage generated in the other. Can we take into account such an Fig. 22-26. Equivalent circuit of a mutual +effect in our equivalent circuits? We can in the following way. We have seen that inductance. +--- Trang 289 --- +the induced emf's in each of two interacting coils can be written as the sum of +twO DartS: +€¡i=_—HLỊ n. +1 c, +h h (22.34) +Ca = —ha q +M " +The first term comes from the self-inductance of the coil, and the second term +comes from its mutual inductance with the other coil. "The sign of the second term +can be plus or minus, depending on the way the ñux from one coil links the other. +Making the same approximations we used in describing an ideal inductance, we +would say that the potential diference across the terminals of each coil is equal +to the electromotive force in the coil. Then the two equations of (22.34) are the +same as the ones we would get from the cireuit of Fig. 22-26(b), provided the +electromotive force in each of the two cireuits shown depends on the current in +the opposite circuit according to the relations +E)=+iuMls, — Êy=+iaMH,. (22.35) l Ẹ +So what we can do is represent the efect of the selEinductance in a normal way ÝỶ TY YÝT +but replace the efect of the mutual inductance by an auxiliary ideal voltage E——~r-—r—r n +generator. We must in addition, of course, have the equation that relates this emf &) CC } ] 2) +to the current in some other part of the circuit; bu so long as this equation is ¿2 j +linear, we have Just added more linear equations to our circuit equations, and all —_——_—-=_— _=-— +of our earlier conclusions about equivalent circuits and so forth are still correct. +In addition to mutual inductances there may also be mutual capacitances. So C D +far, when we have talked about condensers we have always imagined that there +were only two electrodes, but in many situations, for example in a vacuum tube, +there may be many electrodes close to each other. IÝ we put an electric charge on A B +any one of the electrodes, its electric ñeld will induce charges on each of the other +electrodes and affect its potential. As an example, consider the arrangement of +four plates shown in Eig. 22-27(a). Suppose these four plates are connected to (b) +external cireuits by means of the wires A, Ö, Œ, and D. So long as we are only +worried about electrostatic efects, the equivalent circuit of such an arrangement +of electrodes is as shown in part (b) of the figure. The electrostatic interaction €C D +of any electrode with each of the others is equivalent to a capacity between the : ¬ +two eleetrodes. F19. 22-27. Equivalent circuit of mutual +Finally, let°s consider how we should represent such complicated devices as capacitance. +transistors and radio tubes in an AC circuit. We should point out at the start +that such devices are often operated ¡in such a way that the relationship between +the currents and voltages is not at all linear. In such cases, those statements we +have made which depend on the linearity of equations are, of course, no longer +correct. Ôn the other hand, in many applications the operating characteristics +are sufliciently linear that we may consider the transistors and tubes to be linear +devices. By this we mean that the alternating currents in, say, the plate of +a vacuum tube are linearly proportional to the voltages that appear on the +other electrodes, say the grid voltage and the plate voltage. When we have such OPLATE P +linear relationships, we can incorporate the device into our equivalent circuit +represenftation. +As in the case of the mutual inductance, our representation will have to GRIÐ Ỷ +include auxiliary voltage generators which describe the inÑuence of the voltages W +or currents in one part of the device on the currents or voltages in another part. XS +For example, the plate circuit of a triode can usually be represented by a resistance Ỏ +in series with an ideal voltage generator whose source strength is proportional to CATHODE ¬ +the grid voltage. We get the equivalent cireuit shown in Fig. 22-28.* Similarly, — —“M +the collector circuit of a transistor is convenientÌy represented as a resistOr in Fig. 22-28. A low-frequency equivalent +series with an ideal voltage generator whose source strength is proportional to the circuit of a vacuum triode. +* The equivalent circuit shown is correct only for low frequencies. For high frequencies the +equivalent circuit gets mụuch more complicated and will include various so-called “parasitic” +capacitances and inductances. +--- Trang 290 --- +_..u E c +Fig. 22-29. A low-frequency equivalent mà +circuit of a transIstor. BASE B +Ê =Kl¿ +current from the emitter to the base of the transistor. The equivalent circult is +then like that in Fig. 22-29. 5o long as the equations which describe the operation +are linear, we can use such representations for tubes or transistors. Then, when +they are incorporated in a complicated network, our general conclusions about +the equivalent representation of any arbitrary connection of elements is still valid. +There is one remarkable thing about transistor and radio tube circuits which +is diferent from circuits containing only impedances: the real part of the efective +Impedance z¿g can become negative. We have seen that the real part oŸ z +represents the loss oŸ energy. But it is the important characteristic of transistors +and tubes that they sưppiy energy to the circuit. (Of course they don”t just +“make” energy; they take energy from the DC circuits of the power supplies and +convert it inio AC energy.) So it is possible bo have a circuit with a negative +resistance. Such a circuit has the property that iƒ you connect i to an impedance +with a positive real part, i.e., a positive resistance, and arrange matters so that +the sum of the two real parts is exactly zero, then there is no dissipation in +the combined circuit. HÝ there is no loss of energy, any alternating voltage once +started will remain forever. 'This is the basic idea behind the operation of an +oscillator or signal generator which can be used as a source of alternating voltage +at any desired frequency. +--- Trang 291 --- +(ttrtfyy lïcseortcrfOr-S +23-1 Real circuit elements +'When looked at from any one pair of terminals, any arbitrary circuit made 23-1 Real circuit elements +up of ideal Impedances and generators is, at any given Írequency, equivalent to a 23-2 A capacitor at hỉgh frequencies +generator € in series with an impedance z. 'That COImes about because if we put 23-3 A resonant cavity +a voltage V across the terminals and solve all the equations to ñnd the current T, -4 Cavit des +we must get a linear relation between the current and the voltage. 5ince all the 24-4 aụ y mọ ¬¬ +cequations are linear, the result for ƒ must also depend only linearly on V. "The 23-5 Cavities and resonant circuits +mmost general linear form can be expressed as +T=-(V-€©). (23.1) +: Reuicu: Chapter 23, Vol. Ï, esonance +In general, both z and Ê may depend in some complicated way on the frequency 0. Chapter 49, Vol. l, Modes +Equation (28.1), however, is the relation we would get if behind the two terminals +there was Just the generator €(œ) in series with the impedance z(0). +There is also the opposite kind of question: Tf we have any electromagnetic +device at all with two terminals and we measure the relation between ƒ and V to +determine € and z as functions of frequeney, can we find a combination of our ideal +elements that is equivalent to the internal impedance z? 'The answer is that for any +reasonable—that is, physically meaningful—function z(0), it is possible to øpp7oz- +#mate the situation to as high an accuracy as you wish with a circuit containing +a finite set of ideal elements. We don't want to consider the general problem now, h +but only look at what might be expected from physical arguments for a Íew cases. +Tf we think of a real resistor, we know that the current throuph ït will produce c +a magnetic field. So any real resistor should also have some inductance. Also, R +when a resistor has a potential diference across it, there must be charges on +the ends of the resistor to produce the necessary electric fñelds. As the voltage +changes, the charges will change in proportion, so the resistor will also have some +capacitance. We expect that a real resistor might have the equivalent circuit +shown in Fig. 23-1. In a well-designed resistor, the so-called “parasitic” elements Fig. 23-1. Equivalent circuit of a real +Land Ở are small, so that at the frequencies for which ït is intended, œ is mụch resistor. +less than ?#, and 1/Œ is much greater than ??. It may therefore be possible to +nepglect them. As the frequency is raised, however, they will eventually become +Important, and a resistor begins to look like a resonant circuit. +A real inductance is also not equal to the idealized inductance, whose impe- +danee is 2œ. A real coil of wire will have some resistance, so at low frequencies the +coil is really equivalent to an inductance in series with some resistance, as shown +in Fig. 23-2(a). But, you are thinking, the resistance and inductance are together +in a real coil—the resistance is spread all along the wire, so it is mixed in with the +inductance. We should probably use a circuit more like the one in Fig. 23-2(b), +which has sevcral little ƒs and Ƒs in series. But the total impedance of such a +circuit is just 3) 7+3 `7, which is equivalent to the simpler diagram of part (a). +As we go up in frequency with a real coil, the approximation of an inductance +plus a resistance is no longer very good. The charges that must build up on the +wires to make the voltages will become important. It is as iŸ there were little +condensers across the turns of the coil, as sketched in Eig. 23-3(a). We might try +to approximate the real coil by the circuit in Eig. 23-3(b). At low frequencies, (2) (b) +this circuit can be imitated fairly well by the simpler one in part (c) oŸ the figure +(which is again the same resonant circuit we found for the high-frequency model Fig. 23-2. The equivalent circuit of a real +Of a resistor). For higher frequencies, however, the more complicated circuit of inductance at low frequencies. +--- Trang 292 --- +Hig. 23-3(b) is better. In fact, the more accurately you wish to represent the +actual impedance of a real, physical inductance, the more ideal elements you will +have to use in the artificial model of it. ‹C== ` +Let”s look a little more closely at what goes on in a real coil. The impedance <—— +of an inductance goes as œ, so it becomes zero at low frequeneies—it is a “short S= +circuit”: all we see is the resistance of the wire. Ás we go up in frequenecy, œÙ, lC +soon becomes mụch larger than #, and the coil looks pretty much like an ideal +inductance. As we go still higher, however, the capacities become important. (a) +Theïr impedance is proportional to 1/(UŒ, which is large for small œ¿. Eor small +enough frequencies a condenser is an “open circuit,” and when ït is in parallel +with something else, it draws no current. But at high frequencies, the current +prefers to fow into the capacitance between the turns, rather than through +the inductance. So the current in the coil jumps from one turn to the other +and doesnt bother to go around and around where it has to buck the emf. So +although we may have ?n#ended that the current should go around the loop, i% +will take the easier path—the path of least impedanee. +Tf the subJect had been one of popular interest, this efect would have been +called “the high-frequenecy barrier,” or some such name. 'Phe same kind of thing +happens in all subjects. In aerodynamics, if you try to make things go faster +than the speed of sound when they were designed for lower speeds, they don'$ +work. It doesn”t mean that there is a great “barrier” there; it just means that the +object should be redesigned. So this coil which we designed as an “inductance7” @œ) (©) +1s nob going to work as a good inductance, but as some other kind of thing at +very hiph frequencies. For high frequencies, we have to fnd a new design. Fig. 23-3. The equivalence circuit of a +real inductance at higher frequencies. +23-2 A capacitor at high frequencies +Now we want to discuss in detail the behavior of a capacitor—a geometrically +ideal capacitor—as the fÍrequency gets larger and larger, so we can see the +transition of its properties. (We prefer to use a capacitor instead of an inductance, +because the geometry of a pair of plates is much less complicated than the geometry +of a coil.) We consider the capacitor shown in Fig. 23-4(a), which consists oŸ two +parallel circular plates connected to an external generator by a pair of wires. If +we charge the capacitor with DGC, there will be a positive charge on one plate and +a negative charge on the other; and there will be a uniform electric ñeld bebween +the plates. +Now suppose that instead of DC, we put an AC of low frequenecy on the plates. +(We will ñnd out later what is “low” and what is “high”.) Say we connect the +capacitor to a lower-frequency generator. AÄs the voltage alternates, the positive +charge on the top plate is taken off and negative charge is put on. While that +1s happening, the electric fñeld disappears and then bưilds up in the opposite +| | SURFACE +\ v4 lun T77 +C__——D R ® ® ® 9 2 j| ® +-_=†=—=Ƒ—=E--j +Le EEEEml cc19//V1 +TSLEE-EETNee | J2 4/21 +c7 CURVE / | /Ả /2 CURVE Ta +\ ÿ ————————— “MT. +| LINÈS OF B8 +LINES ÓF E_ (a) (b) +Fig. 23-4. The electric and magnetic fields between the plates of a capacitor. +--- Trang 293 --- +direction. As the charge sloshes back and forth slowly, the electric field follows. +At each instant the electric field is uniform, as shown in Eig. 23-4(b), except for +some edge efects which we are goïing to disregard. We can write the magnitude +of the electric ñeld as +E= Eoe“", (23.2) +where 2g 1s a constant. +Now will that continue to be right as the frequency goes up? No, because as +the electric field is goïng up and down, there is a ñux of electric feld through +any loop like Vị in Eig. 23-4(a). And, as you know, a changing electric field acts +to produce a magnetic field. One of Maxwell's equations says that when there is +a varying electric field, as there is here, there has got to be a line integral of the +magnetic ñeld. 'Phe integral of the magnetic fñeld around a closed ring, multiplied +by c?, is equal to the time rate-of-change of the electric ñux through the area +inside the ring (ïf there are no currents): +c0 Beds— T J E-nda. (23.3) +inside +So how much magnetic field is there? That's not very hard. Suppose that we +take the loop L1, which is a circle of radius r. We can see from symmetry that +the magnetic fñeld goes around as shown in the fñgure. Then the line integral +of B is 2xr. And, since the electric feld is uniform, the fux of the electric ñeld +is simply # multiplied by xz2, the area of the circle: +2 li 2 +cẴB-2nr = — H- nr“. (23.4) +The derivative of #⁄ with respect to time is, for our alternating field, sim- +ply /œEpe*“t. So we fnd that our capacitor has the magnetie field +B= s2 Eue°et, (23.5) +In other words, the magnetic field also oscillates and has a strength proportional +'What is the efect of that? When there is a magnetic fñeld that is varying, +there will be induced electric ñelds and the capacitor will begin to act a little bit +like an inductance. Äs the frequency goes up, the magnetic feld gets stronger; it +1s proportional to the rate of change of #, and so to œ. The Impedanece of the +capacitor will no longer be simply 1/2Œ. +Let's continue to raise the frequency and to analyze what happens more +carefully. We have a magnetic field that goes sloshing back and forth. But then +the electric fñeld cannot be uniform, as we have assumedl When there is a varying +magnetic field, there must be a line integral of the electric fñeld——because of +Faraday”s law. So If there is an appreciable magnetic feld, as begins to happen +at hiph frequencies, the electric field cannot be the same at all distances from +the center. The electric fñeld must change with r so that the line integral of the +electric ñeld can equal the changing ñux of the magnetic ñeld. +Let's see If we can fgure out the correct electric fñeld. We can do that by +computing a “correction” to the uniform field we originally assumed for low +frequencies. Let”s call the uniform field E¡, which will still be oe?”“!, and write +the correct fñeld as +1= hị + Ea, +where 2 is the correction due to the changing magnetic field. For any ¿ we will +write the field at the center of the condenser as #oe“ (thereby deñning Ep), so +that we have no correction at the center; Fạ = 0 at =0. +To ñnd 2 we can use the integral form of Faraday”s law: +‡ E-ds = ——_(fux of PB). +--- Trang 294 --- +The integrals are simple iŸ we take them for the curve Ƒ'ạ, shown in Fig. 23-4(b), +which goes up along the axis, out radially the distance r along the top plate, +down vertically to the bottom plate, and back to the axis. The line integral of +around this curve is, of course, zero; so only 2 contributes, and its integral is +Jusb —⁄2(r)-h, where h is the spacing between the plates. (We call # positive if it +points upward.) This is equal to minus the rate oŸ change oŸ the flux of Ö, which +we have to get by an integral over the shaded area. Š inside Ùs in Eig. 25-4(b). +The Hux through a vertical strip of width đứ is B(r)h dờ, so the total ñux is +h J DB() dr. +Setting —Ø/Øt of the Hux cqual to the line integral of F2, we have +E2(r) = — | Bữ) dr. (23.6) +Notice that the h cancels out; the felds dont depend on the separation of the +plates. +Using Eq. (23.5) for Ö(r), we have +Ø iur? ++ => Eạe““!. +›ữ) ðt 4c2 “9 +The time derivative just brings down another factor ?J; we get = +ằ„2r2 — TT —= =c +#a(r) —= — T3. Ee?*t, (23.7) +4c ⁄ N +As we expect, the induced field tends to reduce the electric field farther out. The ⁄ Eì + E 2N +corrected field # = ¡ + ; is then Ị Ị +1 /2r2 +E=Ei+Es= |[1—-— — |Eoue*!. 23.8 Ị +i2 ( -= ) 0đ ( ) ö so 1T +The electric field in the capacitor is no longer uniform; i§ has the parabolie Fig. 23-5. The electric field between the +shape shown by the broken line in Eig. 23-5. You see that our simple capacitor capacitor plates at high frequency. (Edge +is getting slightly complicated. effects are neglected.) +W©e could now use our results to calculate the impedance of the capacitor +at híph frequencies. Knowing the electric field, we could compute the charges +on the plates and fnd out how the current through the capacitor depends on +the frequency œ, but we are not interested in that problem for the moment. We +are more interested in seeing what happens as we continue to go up with the +frequency——to see what happens at even higher frequencies. Arenˆt we already +fnished? No, because we have corrected the electric fñeld, which means that +the magnetic ñeld we have calculated is no longer right. The magnetic ñeld of +Eq. (28.5) is approximately right, but it is only a frst approximation. So let°s +call it Øị. We should then rewrite Eq. (23.5) as +1T đuot +Bị = 2e Fne . (23.9) +You will remember that this fñield was produced by the variation of Eị. Now the +correct magnetic ñeld will be that produced by the total electric fñeld #) + ba. IÍ +we write the magnetic fñeld as 8 = Bị + Ba, the second term is Just the additional +ñeld produced by #2. To ñnd Ö;¿ we can go through the same arguments we +have used to fnd ị; the line integral of Ö¿ around the curve Ủ¡ is equal to the +rate of change of the Ñux oŸ #2 through Dị. We will just have Eq. (23.4) again +with Ö replaced by ; and # replaced by l2: +ŒB› - 2mr = aifux of # through T)). +--- Trang 295 --- +Since 2 varies with radius, to obtain its ñux we must integrate over the circular +surface inside ị. Using 27mr dr as the element of area, this integral is +J a(r) - 2mr dứ. +So we get for a(r) +Ba(r) = = my | Fatrhr dự. (23.10) +Using F2(r) from Eq. (23.7), we need the integral of r3 dr, which is, of course, +r*/4. Qur correction to the magnetic feld becomes +ju3rŠ : +B =———— Eue"“!. 23.11 +2(r) TP ng ( ) +But we are still not ñnishedl Tf the magnetic ñeld is not the same as we +first thought, then we have incorrectly computed 2. We must make a further +correction to !, which comes from the extra magnetic ñeld . Let's call this +additional correction to the electric ñeld lZz. It is related to the magnetic ñeld +in the same way that + was related to Øị. We can use Eq. (23.6) all over again +Just by changing the subscripts: +Js(r) = Pợ Đa(r) dr. (23.12) +Using our result, Eq. (23.11), for ạ, the new correction to the electric field is +t =+—— Eoe“!. 23.1 +s(r) T gợi 0€ ( bì 3) +'Writing our doubly corrected electric field as = j + lạ + l3, we get +E= Euesli- _(*“ m CAN (23.14) += € — —> —— —— * * +°ọ 22 22-42 e +The variation of the electric field with radius is no longer the simple parabola we +drew in Eig. 23-5, but at large radii lies slightly above the curve (E) + 2). +W© are not quite through vyet. The new electric feld produces a new correction +to the magnetic fñeld, and the newly corrected magnetic feld will produce a further +correction to the electric fñeld, and on and on. However, we already have all the +formulas that we need. For ạ we can use Eq. (23.10), changing the subscripts +of B and # from 2 to 3. +'The next correction to the electric field is +1 DIẦW 2uut +F4 =—35.12.gP (2) q© . +So to this order we have that the complete electric field is given by +1 @rN 1 ằrN 1 ằ@rNŠ +E= Eoe““|1———=|— ——=l—l -—zl—] +&---|. (2315 +b“lt=an(E) +) —np(M) #°:|- 8) +where we have written the numerical coeficients in such a way that it is obvious +how the series is to be continued. +Our fñnal result is that the electric field bebween the plates of the capacitor, +for any frequeney, is given by Foe”“f times the infinite series which contains only +the variable œr/c. T we wish, we can defne a special function, which we will +call Jo(z), as the infinite series that appears in the brackets of Eq. (23.15): +1 z\ể 1 z\! 1 LẦU +J, =l-_—:|- =xsl] -z.zl=l #--- 23.16 +d2=1= (3) +ap(5) — apl5) _ +--- Trang 296 --- +Then we can write our solution as Epe”“f times this function, with # = œr/€: +E= Eoe'“tJ (#) . (23.17) +The reason we have called our special function ởọ 1s that, naturally, this 1s +not the first time anyone has ever worked out a problem with oscillations in a +cylinder. The function has come up before and is usually called Jọ. It always +comes up whenever you solve a problem about waves with cylindrical symmetry. +The funection Jọ is to cylindrical waves what the cosine function is to waves on a +straight line. So 1È is an Important funection, invented a long time ago. Then a +man named Bessel got his name attached to it. 'Phe subscript zero means that +Bessel invented a whole lot of diferent functions and this is just the first of them. +'The other functions of Bessel—.J, J¿, and so on—=have to do with cylindrical +waves which have a variation of their strength with the angle around the axis of +the cylinder. +The completely corrected electric fñeld between the plates of our circular +capacitor, given by Eq. (23.17), is plotted as the solid line in Eig. 23-5. Eor +frequencies that are not too hiph, our second approximation was already quite +good. The third approximation was even better—so good, ¡in fact, that if we had +plotted it, you would not have been able to see the diference between ¡t and the +solid curve. You will see in the next section, however, that the complete series 1s +needed to get an accurate description for large radil, or for high frequencies. +23-3 A resonant cavity +W©e want to look now at what our solution gives for the electric ñeld bebween +the plates of the capacitor as we continue to go to higher and higher frequencies. +Eor large œ, the parameter # = œr/c also gets large, and the first few terms in +the series for jọ of will increase rapidly. That means that the parabola we have +drawn in Eig. 23-5 curves downward more steeply at higher frequencies. In fact, +1t looks as though the fñeld would fall all the way to zero at some high frequency, +perhaps when é/œ is approximately one-half of a. Let?s see whether 7o does +indeed go through zero and become negative. We begin by trying z = 2: +J(2)=1—1+j”—;zsz=0.2. +'The funection is still not zero, so let°s try a higher value of z, say, z = 2.5. Putting Jo(x) +in numbers, we write 1.0 +2Jo(2.5) = 1— 1.56 + 0.61 — 0.11 = —0.06. n5 +The function Jọ has already gone through zero by the time we get to ø = 2.5. 2.405 Z +Comparing the results for z = 2 and øz = 2.ð, it looks as though do goes through 0 $ t—t —x +zero at one-ffth of the way from 2.5 to 2. We would guess that the zero OCCUTS W N9 ⁄ +for z approximately equal to 2.4. Let's see what that value of ø gives: -0B 732 +Jo(2.4) =1— 1.44 + 0.52 — 0.08 = 0.00. +Fig. 23-6. The Bessel function .Jo(x). +We get zero to the accuracy of our §wo decimal places. If we make the calculation +more accurate (or since Jo is a well-known function, if we look it up in a book), +we fnd that it goes through zero at ø = 2.405. We have worked it out by hand +to show you that you too could have discovered these things rather than having +to borrow them from a book. +As long as we are looking up 2o in a book, iÈ is interesting to notice how ï§ +goes for larger values of ø; it looks like the graph in Fig. 23-6. As ø increases, +Jo(#) oscillates between positive and negative values with a decreasing amplitude +of oscillation. +W©e have gotten the following interesting result: If we go high enough in +frequency, the electric feld at the center of our condenser will be one way and +the electric fñeld near the edge will point in the opposite direction. For example, +--- Trang 297 --- +suppose that we take an œ hiph enough so that # = ằœr/c at the outer edge of +the capacitor is equal to 4; then the edge of the capacitor corresponds to the +abscissa œ = 4n Eig. 23-6. 'Phis means that our capacitor is being operated at +the frequency œ = 4c/ø. At the edge of the plates, the electric feld will have +a rather high magnitude opposite the direction we would expect. 'Phat is the +terrible thing that can happen to a capacitor at high frequencies. lÝ we go to +very high frequencies, the direction of the electric fñeld oscillates back and forth +many times as we go out from the center of the capacitor. Also there are the +magnetic fñelds associated with these electric fields. It is not surprising that our +capacitor doesn”t look like the ideal capacitance for high frequencies. WWe may +even start% to wonder whether it looks more like a capacitor or an inductance. +'W©e should emphasize that there are even more complicated efects that we have +neplected which happen at the edges of the capacitor. Eor instance, there will be +a radiation of waves out past the edges, so the fields are even more complicated +than the ones we have computed, but we will not worry about those efects now. +W© could try to fñgure out an equivalent circuit for the capacitor, but perhaps +1E is better iƒ we just admit that the capacitor we have designed for low-frequency +fñelds is just no longer satisfactory when the requency is too hiph. TỶ we want to +treat the operation of such an object at hiph frequencies, we should abandon the +approximations to Maxwell's equations that we have made for treating circuits +and return to the complete set of equations which describe completely the fields +in space. Instead of dealing with idealized cireuit elements, we have to deal with +the real conductors as they are, taking into account all the fñelds in the spaces in +between. For instance, if we want a resonant circuit at high frequencies we will +not try to design one using a coil and a parallel-plate capacitor. LINES OF B +W©e have already mentioned that the parallel-plate capacitor we have been [` TTYYITVT TT. | +analyzing has some of the aspects of both a capacitor and an inductance. With ||© © © @ ® & +the electric field there are charges on the surfaces of the plates, and with the lÌ / 7Ì +magnetic felds there are back emf?s. Is it possible that we already have a resonant | | +senit2 . . . . Ì © © ©) ®@ &$ S| +circuit? We do indeed. Suppose we pick a frequency for which the electric fñeld \. : 1b b A : Ì +pattern falls to zero at some radius inside the edge of the disc; that is, we ` —===========esees +choose œ@ø/c greater than 2.405. Everywhere on a circle coaxial with the plates (a) +the electric fñeld will be zero. Now suppose we take a thin metal sheet and cut +a strip just wide enough to ft between the plates of the capacitor. 'Phen we +bend it into a cylinder that will go around at the radius where the electric ñeld E, +1s zero. Since there are no electric fields there, when we put this conducting +cylinder in place, no currents will ñow in it; and there will be no changes in 1.0 +the electric and magnetic fields. We have been able to put a direct short circuit +. . . . (b) Ị +across the capacitor without changing anything. And look what we have; we +have a complete cylindrical can with electrical and magnetic fñelds inside and +no connection at all to the outside world. 'Phe fñelds inside won'ˆt change even if +we throw away the edges of the plates outside our can, and also the capacitor r +leads. All we have left ¡is a closed can with electric and magnetic felds inside, 2.405c/ +as shown in Eig. 23-7(a). The electric fields are oscillating back and forth at cBạ +the frequency œ——which, don”$ forget, determined the diameter of the can. he 1.0 +amplitude of the oscillating # fñeld varies with the distance from the axis of the +can, as shown in the graph of Fig. 23-7(b). This curve is just the ñrst arch ofthe () +Bessel function oŸ zero order. 'There is also a magnetic field which goes in circles +around the axis and oscillates in time 90 out of phase with the electric field. +W© can also write out a series for the magnetic field and plot it, as shown in r +the graph of Eig. 23-7(c). +How is it that we can have an electric and magnetic field inside a can with Fig. 23-7. The electric and magnetic +no external connections? It is because the electric and magnetic fñelds maintain fields in an enclosed cylindrical can. +themselves, the changing makes a Ö and the changing Ö makes an #—all +according to the equations of Maxwell. The magnetic fñeld has an inductive +aspect, and the electric fñeld a capacitive aspect; together they make something +like a resonant circuit. Notice that the conditions we have described would only +happen if the radius of the can is exactly 2.405c/œ. For a can oŸ a given radius, +the oscillating electric and magnetic fields will maintain themselves—in the way +--- Trang 298 --- +we have described—only at that particular equency. 5o a cylindrical can of +radius 7 is resonøn‡ at the frequency +œo = 2.405 " (23.18) +We© have said that the fñelds continue to oscillate in the same way after the can +is completely closed. 'Phat is not exactly right. Iý would be possible if the walls of +the can were perfect conductors. Eor a real can, however, the oscillating currents +which exist on the inside walls of the can lose energy because of the resistance +of the material. The oscillations of the fñields will gradually die away. We can +see from Elig. 23-7 that there must be strong currents associated with electric +and magnetic fñelds inside the cavity. Because the vertical electrical fñeld stops +suddenly at the top and bottom plates of the can, ¡it has a large divergence there; +so there must be positive and negative electric charges on the inner surfaces of +the can, as shown in Fig. 23-7(a). When the electric fñeld reverses, the charges +must reverse also, so there must be an alternating current between the top and <+ [I¡ 1Ð +bottom plates of the can. These charges will ñow in the sides of the can, as shown +in the fñgure. We can also see that there must be currents in the sides of the can InpUT L1] | OUTPUT +by considering what happens to the magnetic field. The graph of Fig. 23-7(c) LOOP 2†<1115[x LOOP +tells us that the magnetic feld suddenly drops to zero at the edge of the can. +Such a sudden change in the magnetic field can happen only if there is a current Bmmimil +in the wall. This current is what gives the alternating electric charges on the top CÍ !l| 5 +and bottom plates of the can. +You may be wondering about our discovery of currents in the vertical sides Fig. 23-8. Coupling into and out of a +of the can. What about our earlier statement that nothing would be changed resonant cavity. +when we introduced these vertical sides in a regilon where the electric field was +zero? Remember, however, that when we first put in the sides of the can, the top +and bottom plates extended out beyond them, so that there were also magnetic +fñelds on the outside oŸ our can. It was only when we threw away the parts of +the capacitor plates beyond the edges of the can that net currents had to appear +on the insides of the vertical walls. +Although the electric and magnetic fields in the completely enclosed can will SN AL +gradually die away because of the energy losses, we can stop this from happening GENERATOR +1ƒ we make a little hole in the can and put in a little bit of electrical energy to +make up the losses. We take a small wire, poke it through the hole in the side of ĐNMPLIEIER - +the can, and fasten ït to the inside wall so that it makes a small loop, as shown In @ œ® _— ọ +Fig. 23-8. HÍ we now connect this wire to a source of high-frequency alternating \ =2 +current, this current will couple energy into the electrie and magnetic fñelds of CAVITY +the cavity and keep the oscillations goïng. This will happen, of course, only If Fig. 23-9. A setup for observing the cav- +the frequency of the driving source is at the resonant fÍrequency of the can. ITf ity resonance. +the source is at the wrong frequenecy, the electric and magnetic fñelds will not +resonate, and the fñelds in the can will be very weak. +The resonant behavior can easily be seen by making another small hole in the +can and hooking in another coupling loop, as we have also drawn in Fig. 23-8. The +changing magnetic ñeld through this loop will generate an induced electromotive L +force in the loop. TỶ this loop is now connected to some external measuring ñ +circuit, the currents will be proportional to the strength of the fields in the cavity. & Ị +Suppose we now connect the input loop of our cavity to an RE signal generator, v | +as shown in EFig. 23-9. 'Phe signal generator contains a source of alternating z +current whose frequency can be varied by varying the knob on the front of the b ——Aw = wạ/Q +generator. 'Phen we connect the output loop of the cavity to a “detector,” which ° +1s an instrument that measures the current from the output loop. Ït gives a meter +reading proportional to this current. IÝ we now measure the output current as a » f requency +function of the frequency of the signal generator, we ñnd a curve like that shown in Fig. 23-10. The frequency response curve +Fig. 23-10. The output current is small for all requencies except those very near of a resonant cavity. +the frequency œọ, which is the resonant frequency of the cavity. The resonance +curve is very much like those we described in Chapter 23 of Vol. I. The width of +the resonance is however, much narrower than we usually ñnd for resonant circuits +made of inductances and capacitors; that is, the Q of the cavity is very hiph. lt +--- Trang 299 --- +1s not unusual to fnd Q}s as hiph as 100,000 or more 1Ý the inside walls of the +cavity are made of some material with a very good conductivity, such as siÌver. +23-4 Cavity modes r +Suppose we now try to check our theory by making measurements with an hở 3050 +actual can. We take a can which is a cylinder with a diameter of 3.0 inches and ủ 3300 3820 +a heipht of about 2.5 inches. 'The can is ftted with an input and output loop, as E +shown in Eig. 23-8. lf we calculate the resonant frequency expected for this can † +according to Eq. (23.18), we get that ƒo = œo/2z = 3010 megacycles. When we ö +set the frequency of our signal generator near 3000 mmegacycles and vary it slightly 3000 3500 4000 +until we fñnd the resonance, we observe that the maximum output currenf Ooccurs (2/2 (Megacycles per second) +for a frequency of 3050 megacycles, which is quite close to the predicted resonant Fig. 23-11. Observed resonant frequen- +frequency, but not exactly the same. 'Phere are several possible reasons for the cies of a cylindrical cavity. +discrepancy. Perhaps the resonant frequency is changed a little bit because oŸ the +holes we have cut to put in the coupling loops. A little thought, however, shows +that the holes should lower the resonant frequency a little bit, so that cannot be +the reason. Perhaps there is some slight error in the frequenecy calibration of the +sipnal generator, or perhaps our measurement of the diameter of the cavity is in +not accurate enough. Anyway, the agreement is fairly close. +Much more important is something that happens if we vary the frequency of +T» +get a clue from Fig. 23-6. Although we have been assuming that the first zero +of the Bessel function occurs at the edge of the can, i% could also be that the +second zero of the Bessel function corresponds to the edge of the can, so that (a) +there is one complete oscillation of the electric field as we move from the center +of the can out to the edge, as shown in Fig. 23-12. 'This is another possible mode +for the oscillating fñelds. We should certainly expect the can to resonate in such E +a mode. But notice, the second zero of the Bessel function occurs at # = 5.52, Eo +which is over bwice as large as the value at the fñrst zero. The resonant frequency +of this mode should therefore be higher than 6000 megacycles. We would, no r = 5.52c/0| +doubt, fnd ït there, but it doesnt explain the resonance we observe at 3300. Ị Ị +The trouble is that in our analysis of the behavior of a resonant cavity we +have considered only one possible geometric arrangement of the electric and +magnetic fields. We have assumed that the electric fñelds are vertical and that ‹ +the magnetic fñields lie in horizontal circles. But other fñelds are possible. 'Phe +only requirements are that the felds should satisfy Maxwells equations inside +the can and that the electric fñeld should meet the wall at right angles. We have +considered the case in which the top and the bottom of the can are fẨat, but (b) +things would not be completely diferent If the top and bottom were curved. In : : +bact, how 1s the can supposed to know which is i§s top and bottom, and which Lg. 23-12. A higher-frequency mode. +are 1s sides? It is, in fact, possible to show that there is a mode of oscillation +of the fields inside the can in which the electric fñelds go more or less across the +diameter of the can, as shown in Fig. 23-13. +Tt is not too hard to understand why the natural equency oŸ this mode +should be not very diferent rom the natural frequency oŸ the first mode we have +considered. Suppose that instead of our cylindrical cavity we had taken a cavity +which was a cube 3 inches on a side. It is clear that this cavity would have three +diferent modes, but all with the same frequency. A mode with the electric ñeld +goïng more or less up and down would certainly have the same frequency as the +mode in which the electric ñeld was directed right and left. IÝ we now distort +the cube into a cylinder, we wiïll change these frequencies somewhat. We would +still expect them not to be changed too much, provided we keep the dimensions +of the cavity more or less the same. So the frequency of the mode of Fig. 23-13 +should not be too diÑferent tom the mode of Eig. 25-8. We could make a detailed Fig. 23-13. A transverse mode of the +calculation of the natural frequeney of the mode shown in Fig. 23-13, but we will cylindrical cavity. +--- Trang 300 --- +not do that now. When the calculations are carried through, iE is found that, for +the dimensions we have assumed, the resonant frequency comes out very close to +the observed resonance at 3300 megacycles. +By similar calculations it is possible to show that there should be still another +mode at the other resonant frequency we found near 3800 megacycles. For this +mode, the electric and magnetic fields are as shown in Fig. 23-14. 'Phe electric +feld does not bother to go all the way across the cavity. I% goes from the sides +to the ends, as shown. C2 x Ô +As you will probably now believe, if we go higher and higher in frequency we +should expect to ñnd more and more resonances. There are many diferent modes, c———Ss +cach of which will have a diferent resonant frequency corresponding to some +particular complicated arrangement of the electric and magnetic fñelds. Each of +these fñeld arrangements is called a resonant rmode. The resonance frequency of +each mode can be calculated by solving Maxwell's equations for the electric and ZẦ_—_—— Z^ +magnetic fields in the cavity. =%X_ „ +When we have a resonance at some particular frequency, how can we know C tr 3 +which mode is being excited? One way is to poke a little wire into the cavity +through a small hole. TÝ the electric field is along the wire, as in Fig. 23-15(a), Fig. 23-14. Another mode of a cylindrical +there will be relatively large currents in the wire, sapping energy from the cavity. +fields, and the resonance will be suppressed. lf the electric field is as shown in +Eig. 23-15(b), the wire will have a much smaller efect. We could fnd which +way the fñeld points in this mode by bending the end of the wire, as shown in +Hig. 23-15(c). Then, as we rotate the wire, there will be a big efect when the +end of the wire is parallel to # and a smaill efect when it is rotated so as to be +at 90° to E. +ï 8ï ìnng +Sy =k RN SN Sầ S +Euni KS) +(a) () (c) +Fig. 23-15. A short metal wire inserted Into a cavity will disturb the +resonance much more when it is parallel to E than when ¡t is at right angles. +23-5 Cavities and resonant circuits +Although the resonant cavity we have been describing seems to be quite +diferent from the ordinary resonant circuit consisting of an inductance and a +capacitor, the two resonant systems are, of course, closely related. They are both +members of the same family; they are just bwo extreme cases of electromagnetiec +resonators—and there are many intermediate cases between these two extremes. +Suppose we start by considering the resonant circuit of a capacitor in parallel +with an inductance, as shown in Fig. 23-16(a). This circuit will resonate at the +Írequency œo = l/ VLŒ. TỶ we want to raise the resonant frequeney of this cireuit, +we can do so by lowering the inductance Ù. One way is to decrease the number +oŸ turns in the coil. We can, however, go only so far in this direction. Eventually +we will get down to the last turn, and we will have just a piece oŸ wire joining the +top and bottom plates of the condenser. We could raise the resonant frequency +still further by making the capacitance smaller; however, we can also continue to +decrease the inductance by putting several inductances in parallel. Two one-turn +inductances in parallel will have only half the inductance of each turn. So when +our inductance has been reduced to a single turn, we can continue to raise the +--- Trang 301 --- +LINES OF B TT +r——————¬ =——————— +3 SN l | +¬ #*~#⁄NA 79) J2 ° +3 đ\ | mm +=ẽ= lẤ AI lÍ E | +tất =itịic | *Nà đ IlK SÂỖ || | }® ® | +\ \ NN r4 ÿ SP +\VÀN———<⁄^<⁄ +St ®@ @ ¬. ® 6$ +L_.___—===Ù „>> +(a) (b) (c) +Fig. 23-16. Resonators of progressively higher resonant frequencles. +resonant frequency by adding other single loops rom the top plate to the bottom +plate of the condenser. For instance, Eig. 23-16(b) shows the condenser plates +connected by six such “single-turn inductances.” IÝ we continue to add many such +pieces of wire, we can make the transition to the completely enelosed resonant +system shown in part (c) of the fñgure, which is a drawing of the cross section of a +cylindrically symmetrical object. Our inductance is now a cylindrical hollow can +attached to the edges of the condenser plates. 'The electric and magnetic fields +will be as shown in the ñgure. Such an object is, of course, a resonant cavity. It +is called a “loaded” cavity. But we can still think oŸ it as an L-C circuit in which +the capacity section is the region where we fñnd most of the electric ñeld and the +inductance section is that region where we fnd most of the magnetic feld. +T we want to make the Írequency of the resonator in Eig. 23-16(c) still higher, +we can do so by continuing to decrease the inductance b. To do that, we must +decrease the geometric dimensions of the inductance section, for example by +decreasing the đimension h in the drawing. As h5 ¡is decreased, the resonant +Írequency will be increased. E;ventually, of course, we will get to the situation in +which the height ñh is just equal to the separation between the condenser plates. +W©e then have Just a cylindrical can; our resonant circuit has become the cavity +resonator of Eig. 23-7. +You will notice that in the original L-C resonant circuit of Fig. 23-16 the +electric and magnetic fñelds are quite separate. As we have gradually modified the +resonant system to make higher and higher frequencies, the magnetic fñeld has +been brought closer and closer to the electric field until in the cavity resonator —— +the two are quite intermixed. +Althouph the cavity resonators we have talked about in this chapter have been +cylindrical cans, there is nothing magic about the cylindrical shape. Ä can of any +shape will have resonant frequencies corresponding to various possible modes of . +oscillations of the electric and magnetic ñelds. For example, the “cavity” shown +in Eig. 23-17 will have 1ts own particular set of resonant frequencies—although Fig. 23-17. Another resonant cavity. +they would be rather difficult to calculate. +--- Trang 302 --- +M/œt+ogrrielos +24-1 The transmission line +In the last chapter we studied what happened to the lumped elements of 24-1 The transmission line +circuits when they were operated at very high frequencies, and we were led to see 24-2 The rectangular waveguide +that a resonant circuit could be replaced by a cavity with the fields resonating 24-3 The cutoff frequency +inside. Another interesting technical problem is the connection of one objecf R +to another, so that electromagnetic energy can be transmitted between them. 244 The speed o[ the guided waves +In low-frequency circuits the connection is made with wires, but this method 24-5 Observing guided waves +doesn't work very well at high frequencies because the circuits would radiate 24-6 Waveguide plumbing +energy into all the space around them, and it is hard to control where the energy 24-7 Waveguide modes +will go. 'Phe fields spread out around the wires; the currents and voltages are not 24-8 Another way of looking at the +“guided” very well by the wires. In this chapter we want to look into the ways guided waves +that objects can be interconnected at high frequencies. At least, that”s one way +of presenting our subject. +Another way is to say that we have been discussing the behavior oŸ waves in +free space. Now it is time to see what happens when oscillating fields are confined +in one or more dimensions. We will discover the interesting new phenomenon +when the fñelds are confned in only two dimensions and allowed to go free in +the third dimension, they propagate in waves. Thhese are “guided waves”——the +subject of this chapter. +We© begin by working out the general theory of the fransmission line. The or- +dinary power transmission line that runs from tower to tower over the countryside +radiates away some of its power, but the power frequencies (50-60 cycles/sec) are +so low that this loss is not serious. The radiation could be stopped by surrounding +the line with a metal pipe, but this method would not be practical for power lines +because the voltages and currents used would require a very large, expensive, +and heavy pipe. So simple “open lines” are used. +For somewhat higher Írequencies—say a few kilocycles—radiation can already +be serious. However, it can be reduced by using “twisted-pair” transmission lines, +as is done for short-run telephone connections. At higher frequencies, however, +the radiation soon becomes intolerable, either because of power losses or because +the energy appears in other cireuits where ït isn't wanted. Eor frequencies from a +few kilocycles to some hundreds of megacycles, electromagnetic signals and power +are usually transmitted via coaxial lines consisting of a wire inside a cylindrical A +“outer conductor” or “shield.” Although the following treatment will apply to a —=x.——=—=—=—=—=—=———- +transmission line of two parallel conductors of any shape, we will carry it out „H1 bồ — +referring to a coaxial line. mas—__————————— +W©e take the simplest coaxial line that has a central conductor, which we +suppose is a thin hollow cylinder, and an outer conductor which is another thin +cylinder on the same axis as the inner conductor, as in Fig. 24-1. We begin by Fig. 24-1. A coaxial transmission line. +fñguring out approximately how the line behaves at relatively low frequencies. VWWe +have already described some of the low-frequency behavior when we said earlier +that two such conductors had a certain amount of inductance per unit length or +a certain capacity per unit length. We can, in fact, describe the low-frequency +behavior oŸ any transmission line by giving is inductance per unit length, họ and +1ts capacity per unit length, Co. Then we can analyze the line as the limiting case +of the L-C filter as discussed in Section 22-6. We can make a filter which imitates +the line by taking small series elements g Az and small shunt capacities Œo Az, +where Az is an element of length of the line. Ủsing our results for the infinite +filter, we see that there would be a propagation of electric signals along the line. +--- Trang 303 --- +Rather than following that approach, however, we would now rather look at the +line from the point of view of a diferential equation. +Suppose that we see what happens at two neighboring points along the +transmission line, say at the distances z and # + Az from the beginning of the +line. Lets call the voltage diference between the two conductors V(z), and the +current along the “hot” conductor Ï(+) (see Fig. 24-2). If the current in the line +is varying, the inductance will give us a voltage drop across the small section of +line from z to # + Az in the amount +AV = V(z + Az) — V(z) = —họụ Az a +Ór, taking the limit as Az —> 0, we get +9V Ø1 +——=_—-hÙo—.. (24.1) +ởz ði WIRE 1 = =1) +The changing current gives a gradient of the voltage. P N +Referring again to the fñgure, if the voltage at + is changing, there must VÉ) | | VỆx + Ax) +be some charge supplied to the capacity in that region. lỶ we take the small +piece of line between z+ and z + Az, the charge on it is g= Œọ AzVW. The time WIRE 2 \ ự +rate-oEchange of this charge is Œo Az đV/dit, but the charge changes only iŸ the X x+ Ax +current T(z) into the element is diferent from the current ƒ(œ + Az) out. Calling Fig. 24-2. The currents and voltages of +the diference AT, we have a transmission line. +ATI=-ŒgoAz—-. +Taking the limit as Az —> Ö, we get +——==_—€Œa—. 24.2 +3z °" .Øy (24.2) +So the conservation of charge implies that the gradient of the current is propor- +tional to the time rate-of-change of the voltage. +Equations (24.1) and (24.2) are then the basic equations of a transmission +line. If we wish, we could modify them to include the efÑfects of resistance in the +conductors or of leakage of charge through the insulation between the conductfors, +but for our present discussion we will just stay with the simple example. +'The two transmission line equations can be combined by diferentiating one +with respect to ý and the other with respect to z and eliminating either W or Ï,. +Then we have either a2 a2 +=> =Coho => 24.3 +aạs = CoÈo oa (24.3) +62T 62T +—s=Coho—= 24.4 +2ạs = Cubo 2g (24.4) +Once more we recognize the wave equation in ø. For a uniform transmission +line, the voltage (and current) propagates along the line as a wave. The voltage +along the line must be of the form V{(z,£) = ƒ(œ — 0É) or V{(#,t) = g(+ + 0£), or +a sum of both. Now what is the velocity ø? We know that the coefficient of the +63/012 term is just 1/0, so +U=—=—=—. (24.5) +W©e will leave i% for you to show that the voltage ƒor cach uaue in a line 1s +proportional to the current of that wave and that the constant of proportionality +is just the characteristic impedance zọ. Calling V and 7. the voltage and current +for a wave goïng in the plus z-direction, you should get ++ = Zol+. (24.6) +Similarly, for the wave going toward minus z the relation is +V_= zoÏl_. +--- Trang 304 --- +The characteristic impedance—as we found out from our flter equations—is +given by +Zo =\|>—; 24.7 +"Nr (2417) +and is, therefore, a pure resistance. +To fnd the propagation speed œ and the characteristic impedance zọ of a +transmission line, we have to know the inductance and capacity per unit length. +We can calculate them easily for a coaxial cable, so we will see how that goes. +For the inductance we follow the ideas of Section 17-8, and set sL1 2 cqual to the +magnetic energy which we get by integrating coc2B2/2 over the volume. Suppose +that the central conduetor carries the current 7; then we know that Ö = 1/2mcoc2r, +where ? is the distance from the axis. Taking as a volume element a cylindrical +shell of thickness dr and of length ?, we have for the magnetic energy +2 b ] 2 +U= =“— J ——- | Ì2nr dĩ, +2 J¿ \27coc2r +where œø and ö are the radii of the inner and outer conductors, respectively. +Carrying out the integral, we get +T2] b +U=——al-. 24.8 +4eoc2 nà ( ) +Setting the energy equal to $L1?, we find +L=———Ìn-. 24.9 +2zcoc2 nạ ( ) +Tt 1s, as it should be, proportional to the length ƒ of the line, so the inductance +per unit length họ is +In(b/a) +TỦọ = ———<. 24.10 +Uˆ” 2cpc2 ( ) +W© have worked out the charge on a cylindrical condenser (see Section 12-2). +Now, dividing the charge by the potential diference, we get +2zcoÏl +In(b/a) +The capacity per unit length Œc is C/l. Combining this result with Eq. (24.10), +we see that the produet LoCŒo is just cqual to 1/c2, so = 1/V(boC§ is equal +toc. The wave travels down the line with the speed of light. We point out +that this result depends on our assumptions: (a) that there are no dielectrics +or magnetic materials in the space between the conduectors, and (b) that the +currents are all on the surfaces of the conductors (as they would be for perfect +conductors). We will see later that for good conductors at high frequencies, +all currents distribute themselves on the surfaces as they would for a perfect +conductor, so this assumption is then valid. +Now it is interesting that so long as assumptions (a) and (b) are correct, the +produet LoCŒo is equal to 1/c? for any parallel pair of conductors—even, say, for a +hexagonal inner conductor anywhere inside an elliptical outer conductor. So long +as the cross section is constant and the space between has no material, waves are +propagated at the velocity of light. +No such general statement can be made about the characteristic impedance. +For the coaxial line, it 1s +In(b/a) +=———.. 24.11 +r9 27€oG ( ) +The factor 1/coc has the dimensions of a resisbtance and is equal to 1207 ohms. +The geometric factor In(b/a) depends only logarithmically on the dimensions, so +for the coaxial line—and most lines—the characteristic Impedance has typical +values oŸ from 50 ohms or so to a few hundred ohms. +--- Trang 305 --- +24-2 The rectangular waveguide ¬ +The next thing we want to talk about seems, at first sight, to be a striking lồ +phenomenon: ïf the central conduector is removed from the coaxial line, it can still +carry electromagnetic power. In other words, at high enough frequenecies a hollow À +tube will work just as well as one with wires. It is related to the mysterious +way in which a resonant circuit of a condenser and inductance gets replaced by ¬ ___"x +nothing but a can at high frequencies. TT” +Although it may seem to be a remarkable thing when one has been thinking ¬ +in terms of a transmission line as a distributed inductance and capacity, we all ¬ ` +know that electromagnetic waves can travel along inside a hollow metal pipe. Tf ¬ ` +the pipe is straight, we can see through itl So certainly electromagnetic waves go ¬ +through a pipe. But we also know that it is not possible to transmit low-frequency ¬ +waves (power or 6elephone) through the inside of a single metal pipe. So it must À\ +be that electromagnetic waves will go through ïif their wavelength is short enough. ` +Therefore we want to discuss the limiting case of the longest wavelength (or the +lowest frequency) that can get through a pipe of a given size. Since the pipe is Nz +then being used to carry waves, it is called a œeguide. +We will begin with a rectangular pipe, because it is the simplest case to FÍg. 24-3. Coordinates chosen for the +analyze. We will fñrst give a mathematical treatment and come back later to look rectangular waveguide. +at the problem in a much more elementary way. The more elementary approach, +however, can be applied easily only to a rectangular guide. “The basic phenomena y +are the same for a general guide of arbitrary shape, so the mathematical argument a +is fundamentally more sound. +Our problem, then, is to find what kind of waves can exist inside a rectangular +pipe. Let's first choose some convenient coordinates; we take the z-axis along mm +the length of the pipe, and the z- and -axes parallel to the two sides, as shown E b +in Fig. 24-3. +We know that when light waves go down the pipe, they have a transverse +electric field; so suppose we look first for solutions in which # is perpendicular (a) * +to z, say with only a -component, l⁄„. This electric fñeld will have some variation Ey +across the guide; in fact, it must go to zero at the sides parallel to the -axis, +because the currents and charges in a conductor always adjust themselves so that +there is no tangential ecomponent of the electric ñeld at the surface of a conductor. +So F„ will vary with # in some arch, as shown in Eig. 24-4. Perhaps it is the +Bessel function we found for a cavity? No, because the Bessel function has to +do with cylindrical geometries. For a rectangular geometry, waves are usually Œœ) : ĩ +simple harmonic functions, so we should try something like sin k„z. Fig. 24-4. The electric field in the wave- +Since we want waves that propagate down the guide, we expect the field to guide at some value of z. +alternate between positive and negative values as we go along in z, as in Eig. 24-5, +and these oscillations will travel along the guide with some velocity 0. lỶ we +have oscillations at some defnite frequency œ, we would guess that the wave Lư +might vary with z like cos (# — &;z), or to use the more convenien© mathematical Ị +form, like ef«—Èz2), 'This z-dependence represents a wave travelling with the +speed = œ/k; (see Chapter 29, Vol. ]). ¡lol lỗ | |® @| |@® +So we might guess that the wave in the guide would have the following ø[ l@ F [| l@ ø[ le +mmathematical form: +Eụ = Eoe'€f—E22) sìn k„ứ:, (24.12) —*z @) +Let's see whether this guess satisfies the correct feld equations. Eirst, the , +electric field should have no tangential components at the conductors. Our field +satisfies this requirement; it is perpendicular to the top and bottom faces and —fm +1s zero at the two side faces. Well, it is if we choose k„ so that one-half a cycle gi +of sin k„z+ just fts in the width of the guide—that is, if +k„a — T. (24.13) œ) +There are other possibilities, like k„œ = 2m, 3r,..., or, in general, ___L1g. 24FŠ. The z-dependence of the field +in the waveguide. +k„@ — TT, (24.14) +--- Trang 306 --- +where mø is any integer. These represent various complicated arrangements of the +feld, but for now let”s take only the simplest one, where &„ = 7/ø, where ø is +the width of the inside of the guide. +Next, the divergence of must be zero in the free space inside the guide, +since there are no charges there. Qur # has only a -component, and it doesn'$ +change with , so we do have that V - = 0. +Pinally, our electric fñeld must agree with the rest of Maxwell's equations in +the free space inside the guide. 'That is the same thing as saying that it must +satisfy the wave equation +9?°Ðy, , 0°E, 0 ?*E, 1 0°Fy +0a Troy l7 ø@ 0Ð SU (24.15) +We have to see whether our guess, Eq. (24.12), will work. The second derivative +of Fụ with respect to ø is jusi —k?F„. The second derivative with respect to 1 +1s zero, since nothing depends on #. “The second derivative with respect to z +is —k? E„, and the second derivative with respect to £ is =2. Equation (21.15) +then says that : +k}E,, + k}Ey, — s Ey„ =0. +Unless #„ is zero everywhere (which is not very interesting), this equation is +correct if +k + k}— =0. (24.16) +W© have already fxed &k„, so this equation tells us that there can be waves of the +type we have assumed iŸ &; is related to the frequenecy œ so that Eq. (24.16) is +satisfed——in other words, 1Ý +ky; = W(œ2/c2) — (x2/a2). (24.17) x +The waves we have described are propagated in the z-direction with this value +OŸ kz. % +The wave number k; we get from Eaq. (24.17) tells us, for a given frequency 0œ, T +the speed with which the nodes of the wave propagate down the guide. “The *\ +phase velocity is K> = +b=, (24.18) “ THRI¿¿ +ky ` Í +. . ¬- MAX vI +You will remember that the wavelength À of a travelling wave is given by À = ch—=———^ +270/6, so k„ is also equal to 2r/À¿, where À¿ is the wavelength of the oscillations ` +along the z-direction—the “guide wavelength.” The wavelength in the guide is +diferent, of course, from the free-space wavelength of electromagnetic waves +of the same frequency. If we call the free-space wavelength Ào, which ¡is equal : +to 2me/œ, we can write Bq. (24.17) as Fig. 24-6. The magnetic field in the +Waveguide. +À;= —=——ễễ. (24.19) +Besides the electric fñelds there are magnetic felds that will travel with the +wave, but we will not bother to work out an expression for them right now. +Since c2V x B = 0/ôt, the lines of will cireulate around the regions in +which ØE/ðt is largest, that is, halfway between the maximum and minimum +of . The loops of will lie parallel to the zz-plane and between the crests and +troughs of #, as shown in Eig. 24-6. +24-3 The cutoff frequency +In solving Eq. (24.16) for k;z, there should really be two roots—one plus and +one minus. We should write +k; = +w(œ2/c2) — (x2/a2). (24.20) +--- Trang 307 --- +The two signs simply mean that there can be waves which propagate with a +negative phase velocity (toward —z), as well as waves which propagate in the +positive direction in the guide. Naturally, ¡it should be possible for waves to go +in either direction. Since both types of waves can be present at the same time, +there will be the possibility of standing-wave solutions. +Our equation for k; also tells us that higher frequencies give larger values +of k;, and therefore smaller wavelengths, until in the limit of large œ, & becomes +cqual to œ¿/c, which is the value we would expect for waves in Íree space. The +light we “see” through a pipe still travels at the speed c. But now notice that if +we go toward low frequencies, something strange happens. At first the wavelength +gets longer and longer, but if œ gets too small the quantity inside the square root +of Eq. (24.20) suddenly becomes negative. This will happen as soon as œ gets tO +be less than e/œ—or when Ào becomes greater than 2ø. In other words, when +the frequency gets smaller than a certain critical frequency œ¿ = c/a, the wave +number &; (and also À„) becomes imaginary and we haven't got a solution any +more. Or do we? Who said that k; has to be real? What if it does come out +Imaginary? Our field equations are still satisied. Perhaps an imaginary &; also +Tepresenfs a Wave. +Suppose œ is less than œ„; then we can write +ky = +ik!, (24.21) +where kÝ is a positive real number: +k = W(2/42) — (œ^2/c2). (24.22) +Tf we now go back to our expression, Eq. (24.12), for #⁄„, we have +Eụ = EoeltẰefff'2) sìn ke, (24.23) +which we can write as +Eụ = Eoe**Ze*“t sin k„a:, (2424) 7 +This expression gives an -field that oscillates with tỉme as e”“ but which +varies with z as e?*”, It decreases or increases with z smoothly as a real +exponential. In our derivation we didnt worry about the sources that started the +waves, but there must, of course, be a source someplace In the guide. The sign +that goes with k“ must be the one that makes the feld decrease with increasing +distance from the source of the waves. +So for frequencies below œ¿ = c/ø, waves do not propagate down the guide; +the oscilating felds penetrate into the guide only a distance of the order oŸ 1/kf. +For this reason, the frequency ¿ö; is called the “cutof frequency” of the guide. +Looking at Bq. (24.22), we see that for frequencies just a little below œ¿, the +number &“ is small and the felds can penetrate a long distance into the guide. +But ïf œ is mụch less than œ„, the exponential coefficient k is equal to zø/œand o a 2a a - +the fñeld dies of extremely rapidly, as shown in Eig. 24-7. 'Phe fñeld decreases ” TL +by 1/ein the distance g/1, orin only about one-third of the guide width. The Fig. 24-7. The variation of E„ with z +fields penetrate very little distance from the source. for @ < ức. +We want to emphasize an interesting feature of our analysis of the guided +waves—the appearance of the imaginary wave number &;. Normally, if we +solve an equation in physics and get an imaginary number, it doesn't mean +anything physical. Eor œøœues, however, an imaginary wave number đøes mean +something. 'Phe wave equation ïs still satisfied; ít only means that the solution +gives exponentially decreasing fñelds instead of propagating waves. 5o in any +wave problem where k becomes imaginary for some frequeney, it means that the +form of the wave changes——the sine wave changes into an exponential. +24-4 The speed of the guided waves +'The wave velocity we have used above is the phase velocity, which is the speed +of a node of the wave; it is a function of frequency. TỶ we combine qs. (24.17) +--- Trang 308 --- +and (24.18), we can write +: (24.25) +Đphase — —————————. : +phase 4 — (ø./„)2 — (œ„/œ)2 +Eor frequencies above cutoff——where travelling waves exist—¿¿ /œ is less than one, +and 0pnase is real and greafer than the speed of light. We have already seen in +Chapter 48 oŸ Vol. I that phase velocities greater than light are possible, because +1E is just the nodes of the wave which are moving and not energy or information. +In order to know how fast s¿ønais will travel, we have to calculate the speed of +pulses or modulations made by the interference of a wave of one frequency with +one or more waves of slightly diferent frequencies (see Chapter 48, Vol. I). We +have called the speed of the envelope of such a group of waves the group velocity; +it is not œj/k but đu /dk: +Ugroup FT (24.26) +Taking the derivative of Eq. (24.17) with respect to œ and inverting to get dư /dk, +we fnd that +Ugroup — CV 1— (œe/6)2, (24.27) +which is less than the speed of light. +The geometric mean 0Ÿ 0pnase and 0group iS just c, the speed of light: +ĐphaseÙgroup “ cẺ. (24.28) +'This is curious, because we have seen a similar relation in quantum mechanics. For +a particle with any velocity——even relativistic—the momentum ø and energy +are related by +U2 = p?c2 + m'2c†. (24.29) +But in quantum mechanics the energy is ñœ, and the momentum is Ö/À, which +is equal to ñ5&; so Bq. (24.29) can be written +,UỐ a , THẾC +ca” k^“+ _x (24.30) +k = V(2/c2) — (m2c2/h2), (24.31) +which looks very much like Eq. (24.17)... Inmberestingl +The group velocity ofthe waves is also the speed at which energy is transported +along the guide. If we want to ñnd the energy flow down the guide, we can get it +from the energy density times the group velocity. If the root mean square electric +fñeld is Fọ, then the average density of electric energy is co#2/2. There is also +some energy associated with the magnetic ñeld. We will not prove i% here, but +in any cavity or guide the magnetic and electric energies are equal, so the total +electromagnetic energy density is co. The power đỮ/dt transmitted by the +guide is then +nền co E8ab0sxoup- (24.32) +(We will see later another, more general way of getting the energy flow.) +24-5 Observing guided waves +tEnergy can be coupled into a waveguide by some kind of an “antenna.” For +example, a little vertical wire or “stub” will do. The presence of the guided waves +can be observed by picking up some oŸ the electromagnetic energy with a little +receiving “antenna,” which again can be a little stub of wire or a small loop. In +Fig. 24-8, we show a guide with some cutaways to show a driving stub and a +pickup “probe”. 'Phe driving stub can be connected to a signal generator via a +coaxial cable, and the pickup probe can be connected by a similar cable to a +detector. It is usually convenient to insert the pickup probe via a long thin slot +--- Trang 309 --- +SIGNAL _„TO DETECTOR +GENERATOR ⁄ +\ Ủ = +\ —Hi— k1 +stub and a pickup probe. « +in the guide, as shown in Fig. 24-8. 'Phen the probe can be moved back and forth +along the guide to sample the fñelds at various positions. +T the signal generator is set at some Írequency œ greater than the cutoff +frequency œ¿, there will be waves propagated down the guide from the driving +stub. These will be the only waves present if the guide is inñnitely long, which +can efectively be arranged by terminating the guide with a carefully designed +absorber in such a way that there are no refections from the far end. Then, since +the detector measures the time average of the fields near the probe, it will pick +up a signal which is independent of the position along the guide; its output will +be proportional to the power being transmitted. +Tf now the far end of the guide is ñnished of in some way that produces a +reflected wave—as an extreme example, if we closed it of with a metal pÌate— +there will be a refected wave in addition to the original forward wave. 'These +two waves will interfere and produce a standing wave in the guide similar to the +standing waves on a string which we discussed in Chapter 49 of Vol. I. "Then, +as the pickup probe is moved along the line, the detector reading will rise and +fall periodically, showing a maximum in the fields at each loop of the standing +wave and a minimum at each node. 'Phe distance bebween ÿwo successive nodes +(or loops) is just À¿/2. This gives a convenient way of measuring the guide +wavelength. lf the requency is now moved closer %o œ¿, the distances between +nodes increase, showing that the guide wavelength increases as predicted by +Eq. (24.19). +Suppose now the signal generator is set at a frequency just a little below œạ. +'Then the detector output wiïll decrease gradually as the pickup probe is moved +down the guide. If the frequency is set somewhat lower, the field strength will +fall rapidly, following the curve of Fig. 24-7, and showing that waves are not +propagated. +24-6 Waveguide plumbiỉng +An important practical use of waveguides is for the transmission of high- +frequency power, as, for example, in coupling the high-frequency oscillator or +output amplifier of a radar set to an antenna. In fact, the antenna itself usually +consists of a parabolie reflector fed at its focus by a waveguide flared out at the +end to make a “horn” that radiates the waves coming along the guide. Although +hiph frequencies can be transmitted along a coaxial cable, a waveguide is better +for transmitting large amounts of power. First, the maximum power that can be +transmitted along a line is limited by the breakdown of the insulation (solid or +gas) between the conductors. For a given amount of power, the field strengths in +a guide are usually less than they are in a coaxial cable, so hipgher powers can be +transmitted before breakdown occurs. Second, the power losses in the coaxial +cable are usually greater than in a waveguide. In a coaxial cable there must be +insulating material to support the central conductor, and there is an energy loss +in this material—particularly at high frequencies. Also, the current densities on +the central conductor are quite high, and since the losses go as the sguøre of the +current density, the lower currents that appear on the walls of the guide result in +lower energy losses. lo keep these losses to a minimum, the inner surfaces of the +guide are often plated with a material of high conductivity, such as silver. +--- Trang 310 --- +tực X : "—_ CAVHY +` ` ì b +: : x Ả mm +Fig. 24-9. Sections of waveguide connected with Fig. 24-10. A low-loss connection between two +flanges. sections of waveguide. +The problem of connecting a “circuit” with waveguides is quite different +from the corresponding circuit problem at low frequencies, and is usually called +microwave “plumbing.” Many special devices have been developed for the purpose. +For instance, two sections of waveguide are usually connected together by means % +of langes, as can be seen in Fig. 24-9. Such connections can, however, cause / +Serlous energy losses, because the surface currents must fow across the joint, _ CN +which may have a relatively high resistance. Ône way to avoid such losses 1s tO __€ — “ +make the fanges as shown in the cross section drawn in Fig. 24-10. A small space _ 7 .., +1s left between the adJacent sections of the guide, and a groove is cut in the face of Á +one of the flanges to make a small cavity of the type shown in Fig. 23-16(c). The & ; +dimensions are chosen so that this cavity is resonant at the frequency being used. _=^ +'This resonant cavity presents a high “impedance” to the currents, so relatively > ~ +little current flows across the metallic joints (at œ in Fig. 24-10). The high guide +currents simply charge and discharge the “capacity” of the gap (at bin the figure), Fig. 24-11. A waveguide “T.” (The +where there is little dissipation of energy. flanges have plastic end caps to keep the +uppose you want to stop a waveguide in a way that won't result in reflected inside clean while the “T” is not being used. +waves. Then you must put something at the end that imitates an infnite length +of guide. You need a “termination” which acts for the guide like the characteristic +Impedance does for a transmission line—something that absorbs the arriving ———_————~ r-.- mm r +waves without making reflections. Then the guide will act as thouph it went on +forever. Such terminations are made by putting inside the guide some wedges of F +resistance material carefully designed to absorb the wave energy while generating +almost no reflected waves. ~v- _.v> +TÝ you want to connect hree things together——for instance, one source to EWO +diferent antennas—then you can use a “” like the one shown in Eig. 24-11. l +Power fed in at the center section of the “Ƒ” will be split and go out the two side +arms (and there may also be some refected waves). You can see qualitatively () +from the sketches in Eig. 24-12 that the fields would spread out when they get to +the end of the input section and make electric felds that will start waves going .e6eầœỒó ru +out the two arms. Depending on whether electric fñelds in the guide are parallel © © S © © © @ © +or perpendicular to the “top” of the “T,” the fields at the junction would be +roughly as shown in (a) or (b) of Fig. 24-12. ~— — +Finally, we would like to describe a device called an “unidirectional coupler,” ữ °©Ọ ế +which is very useful for telling what is going on after you have connected a +. . . O.©) +complicated arrangement of waveguides. Suppose you want to know which way Í +the waves are going in a particular section of guide—you might be wondering, +for instance, whether or not there is a strong reflected wave. The unidirectional 6) °© +coupler takes out a small fraction of the power of a guide if there is a wave going +one way, but none if the wave is going the other way. By connecting the output Fig. 24-12. The electric fields in a wave- +of the coupler to a detector, you can measure the “one-way” power in the guide. — guide “T” for two possible field orientations. +--- Trang 311 --- +Figure 24-13 is a drawing of a unidirectional coupler; a piece of waveguide A7? +has another piece of waveguide €7 soldered to it along one face. The guide C1 +1s curved away so that there is room for the connecting fanges. Before the +guides are soldered together, two (or more) holes have been drilled in each guide +(matching each other) so that some of the fields in the main guide 4Ø can be +coupled into the secondary guide C1. Each of the holes acts like a little antenna +that produces a wave in the secondary guide. If there were only one hole, waves +would be sent in both directions and would be the same no matter which way +the wave was goïing in the primary guide. But when there are #œo holes with a +separation space equal to one-quarter of the guide wavelength, they will make +two sources 90” out oŸ phase. Do you remember that we considered in Chapter 29 _Z“ › +of Vol. I the interference of the waves from two antennas spaced À/4 apart and +excited 90° out of phase in time? We found that the waves subtract in one | ===—Z +direction and add in the opposite direction. The same thing will happen here. ⁄ +The wave produced in the guide ỞÐ will be going in the same direction as the +wave in APH. Kế. +Tf the wave in the primary guide is travelling from A toward ?, there will ° +be a wave at the output Ð of the secondary guide. If the wave in the primary +guide goes from Ö toward A, there will be a wave goïing toward the end Œ of the Fig. 24-13. A unidirectional coupler. +secondary guide. 'This end is equipped with a termination, so that this wave is +absorbed and there is no wave at the output of the coupler. +24-7 Waveguide modes +The wave we have chosen to analyze is a special solution of the fñield equations. ⁄ +'There are many more. Each solution is called a waveguide “mode.” Eor example, +our #-dependence of the fñeld was just one-half a cycle of a sine wave. 'There is an +cqually good solution with a full cycle; then the variation of l„ with # is as shown +in Fig. 24-14. The k„ for such a mode is twice as large, so the cutoff frequency is E +much higher. Also, in the wave we studied # has only a -component, but there +are other modes with more complicated electric fields. If the electric fñeld has +components only in z and —so that the total electric fñeld is always at right (a) x +angles to the z-direction—the mode is called a “transverse electric” (or TE) mode. +The magnetic fñeld of such modes will always have a z-component. It turns out Ey +that 1Ÿ E has a component in the z-direction (along the direction of propagation), +then the magnetic feld will always have only transverse components. 5o such +felds are called transverse magnetic (TM) modes. Eor a recbangular guide, all +the other modes have a higher cutoff frequency than the simple 'EE mode we have +described. It is, therefore, possible—and usual—to use a guide with a frequency x +Just above the cutoff for this lowest mode but below the cutof frequenecy for all +the others, so that Just the one mode is propagated. Otherwise, the behavior +gets complicated and dificult to control. @®) +24-8 Another way of looking at the guided waves Fig. 24-14. Another possible variation of +W©c want now to show you another way of understanding why a waveguide Ey with x +attenuates the fields rapidly for frequenecies below the cutoff frequeney œ¿¿. hen +you will have a more “physical” idea of why the behavior changes so drastically +between low and high frequencies. We can do this for the rectangular guide by +analyzing the fñelds in terms of refections—or images—in the walls of the guide. +The approach only works for rectangular guides, however; that's why we started +with the more mathematical analysis which works, in principle, for guides of any +shape. +Eor the mode we have described, the vertical dimension (ïn ) had no efect, +So we can ipgnore the top and bottom oŸ the guide and imagine that the guide is +extended indefnitely in the vertical direction. We imagine then that the guide +Just consists of two vertical plates with the separation a. +Let's say that the source of the fields is a vertical wire placed in the middle +of the guide, with the wire carrying a current that oscillates at the Írequency 0. +In the absence of the guide walls such a wire would radiate cylindrical waves. +--- Trang 312 --- +Now we consider that the guide walls are perfect conductors. Then, just as +in electrostatics, the conditions at the surface will be correct if we add to the +field of the wire the field of one or more suitable image wires. The image idea +works just as well for electrodynamies as it does for electrostatics, provided, of +course, that we also include the retardations. We know that is true because we %a- +have often seen a mirror producing an image of a light source. And a mirror is +Just a “perfect” conductor for electromagnetic waves with optical frequencies. +Now let's take a horizontal cross section, as shown in Eig. 24-15, where VW Ss+ IMAGE +and W2 are the two guide walls and ,%g is the source wire. We call the direction ` +of the current in the wire positive. Now if there were only one wall, say Mì, Sis= +we could remove it if we placed an image source (with opposite polarity) at the Mó. +position marked 5¡. But with both walls in place there will also be an image S2 SOURCE a "mo +of S%g in the wall W›, which we show as the image Š+. This source, too, will have +an Iimage in W7, which we call 5s. Now both S%¡ and ŠSs will have Images in W2 " Wz +at the positions marked 5¿ and S%s, and so on. For our ©wo plane conductors IMAGE +with the source halfway between, the fields are the same as those produced by ` +an infnite line of sources, all separated by the distance œ. (It is, in facb just sasr +what you would see if you looked at a wire placed halfway between two parallel +mirrors.) For the fields to be zero at the walls, the polarity of the currents in the S6s— +images must alternate from one image to the next. In other words, they oscillate : : +180 out of phase. The waveguide field is, then, just the superposition of the F18. Z+1ồ. The line source So between +fñelds of such an infnite set of line sources the conducting plane wals tự and 2. The +„ ' - - walls can be replaced by the infinite sequence +WSe know that iŸ we are close to the sources, the fñeld is very much like the Of image sources. +siatic ñelds. We considered in Section 7-5 the static ñeld of a grid of line sources +and found that ït is like the field of a charged plate except for terms that decrease +exponentially with the distance from the grid. Here the average source strength +is zero, because the sign alternates from one source to the next. Any fields which +exist should fall of exponentially with distance. Close to the source, we see the +fñeld mainly of the nearest source; at large distances, many sources contribute +and theïr average efect 1s zero. So now we see why the waveguide below cutoff +frequenecy gives an exponentially decreasing field. At low frequencies, in particular, +the static approximation is good, and it predicts a rapid attenuation of the fields +with distance. +Now we are faced with the opposite question: Why are waves propagated ` h ọN "5 +at all? That is the mysterious partl "The reason is that at hipgh frequenecies the `Y `Y ` +retardation of the fñelds can introduce additional changes in phase which can $S;s- N N ) N +cause the fñelds of the out-of-phase sources to add instead of cancelling. In fact, N ` +in Chapter 29 of Vol. Ï we have already studied, just for this problem, the fields Ssk+ ` h “© ` +generated by an array of antennas or by an optical grating. 'There we found that ` N +when several radio antennas are suitably arranged, they can gïve an interference `" ` +pattern that has a strong signal in some direction but no signal in another. si®= : < 7 ` N +Suppose we go back to Fig. 24-15 and look at the fields which arrive at a ` Q +large distance rom the array of image sources. The fields will be strong onlyin s§ ` x22 X8 +certain directions which depend on the requency——only in those directions for HS ` si ` +which the felds from all the sources add in phase. At a reasonable distance from . ` 3» ` +the sources the field propagates in these special directions as plane waves. We ®ÝAs/2 ` N ` ` +have sketched such a wave in Fig. 24-16, where the solid lines represent the wave ` ` ` +crests and the dashed lines represent the troughs. The wave direction will be the 5** N ` N +one for which the diference in the retardation for two neighboring sources to the `Y ` ¬ +crest of a wave corresponds to one-half a period of oscillation. In other words, the Sa ` N ọ ` +diference between rz and ro in the fgure is one-half of the Íree-space wavelength: ` ` ` +Ào ° x SN ` ` +T2 — T0 — —. +The angle Ø is then given by ? : Fig. 24-16. One set of coherent waves +N rom an array of line sources. +sin Ø = 2. (24.33) +'There is, of course, another set of waves travelling downward at the symmetric +angle with respect to the array of sources. The complete waveguide field (not +--- Trang 313 --- +too close to the source) is the superposition oŸ these Ewo sets of waves, as shown +in Eig. 24-17. The actual fñelds are really like this, oŸ course, only between the +two walls of the waveguide. +At points like A4 and Œ, the crests of the two wave patterns coincide, and the +fñeld will have a maximum; at points like , both waves have their peak negative +value, and the fñeld has its minimum (largest negative) value. As time goes on the +field in the guide appears to be travelling along the guide with a wavelength À¿, Sse+ +which is the distance from A to Œ. That distance is related to Ø by ` ZÀ ZA CA Z/N +Ào Sé- Mế ` vé »í +cosØ = `”, (24.34) MÀ SN NZ NZ xZNZN +, ;. VVX»XšŠX +Using Eq. (24.33) for Ø, we get that SN 2N N ZNHH N +À À SH X N X»ZX xX Xx +".....ẽ. (24.35) 1à -® 20a +cosØ ../⁄I1~— (Ào/2a)2 /\ ZÈNƯN NÃN/Z +which is Just what we found in Eq. (24.19). ¬ +Now we see why there is only wave propagation above the cutof frequency œọ. . Flg. 24 17. The Wwaveguide field can be +. . viewed as the superposition of two trains of +Tí the free-space wavelength is longer than 2ø, there is no angle where the waves plane waves +shown in Eig. 24-16 can appear. The necessary constructive interference appears : +suddenly when Ao drops below 2ø, or when œ goes above œạọ = 76/a. +Tf the requenecy is high enough, there can be two or more possible directions +in which the waves will appear. For our case, this will happen IÝ Àọ < ¡a. In +general, however, ¡% could also happen when Ào < ø. 'Phese additional waves +correspond to the higher guide modes we have mentioned. +lt has also been made evident by our analysis why the phase velocity of the +guided waves is greater than c and why this velocity depends on œ. Ás œ is +changed, the angle of the free waves of Fig. 24-16 changes, and therefore so does +the velocity along the guide. +Although we have described the guided wave as the superposition of the fields +oŸ an infÑnite array of line sources, you can see that we would arrive at the same +result if we imagined ©wo sets of free-space waves being continually refected back +and forth between two perfect mirrors—remembering that a refection means +a reversal of phase. 'Phese sets of reflecting waves would all cancel each other +unless they were going at just the angle Ø given in Eq. (24.33). There are many +ways of looking at the same thing. +--- Trang 314 --- +Mgiocfroclyrteatrtaics ra lo Ï(fftfsếfC 'Voferffort +25-1 Eour-vectors +W© now discuss the application of the special theory of relativity to electrody- 25-1 Eour-vectors +namics. Since we have already studied the special theory of relativity in Chapters 25-2 The scalar produc +1ð throueh 17 of Vol. l, we will just review quickly the basic ideas. 25-3 The four-dimensional gradient +Tt is found experimentally that the laws of physics are unchanged if we move vs. +. . . › . "= . . . 25-4 blectrodynamics in +with uniform velocity. You can't tell if you are inside a spaceship moving with : R : +: và. : . : . four-dimensional notation +uniform velocity in a straight line, unless you look outside the spaceship, or at . . +least make an observation having to do with the world outside. Any true law of 25-5 The four-potential of a moving +physics we write down must be arranged so that this fact of nature is built ín. charge +The relationship bebween the space and time of two systems of coordinates, 25-6 The invariance of the equations of +one, ®”, in uniform motion in the #-direction with speed 0 relative to the other, electrodynamics +S, is given by the Loren#z transformation: +; Ÿ — U# ; +tứ = VI-u5 Ụ =Ú, . +—Ð (25.1) In this chapter: e = 1 +; ø — UỶ ; +# = ———, zZ =z. +vV1— 2 +The laws of physics must be such that after a Lorentz transformation, the new +form of the laws looks just like the old form. 'This is just like the principle that Reuieu: Chapter 15, Vol. L, The Special +the laws of physics dont depend on the ør?entafion of our coordinate system. Theoru oƒ Relatiutụ +In Chapter II of Vol. Ij we saw that the way to describe mathematically the Chapter 16, Vol. lj Relatiistic +invariance of physics with respect to rotations was to write our equations in terms Energu and Momentum +Of U0ec‡oTrs. Chapter 17, Vol. l 6Space- +For example, If we have two vectOrs Time +Chapter 13, Vol. II, Mœgneto- +A =(A¿, 4, A;) and B = (P,, Bụ,P,), statics +we found that the combination +A-B=A,PB„+ A,By+ A,DB, +was not changed If we transformed to a rotated coordinate system. So we know +that if we have a scalar product like A - Ở on both sides of an equation, the +cequation will have exactly the same form in all rotated coordinate systems. We +also discovered an operator (see Chapter 2), +8 Ø8 Ô +X — a_?s¬ 0a _ ]› +9z Øụ Ôz +which, when applied to a scalar function, gave three quantities which transform +Just like a vector. With this operator we defñned the gradient, and in combination +with other vectors, the divergence and the Laplacian. Finally we discovered that +by taking sums of certain produects of pairs of the componenfs of two vectors we +could get three new quantities which behaved like a new vector. We called it the +cross product of two vectors. sing the cross product with our operator V we +then defned the curl of a vector. +Since we will be referring back to what we have done in vector analysis, we +have put in Table 25-1 a summary of all the Important vector operations in three +dimensions that we have used in the past. The point is that it must be possible to +write the equations of physics so that both sides transform the same way under +--- Trang 315 --- +rotations. If one side is a vector, the other side must also be a vector, and both +sides will change together in exactly the same way if we rotate our coordinate +system. Similarly, if one side is a scalar, the other side must also be a scalar, so +that neither side changes when we rotate coordinates, and so on. Table 25-1 +Now in the case of special relativity, time and space are inextricably mixed, The important quanfities and operations +and we must do the analogous things for four dimensions. We want our equations of vector analysis in three dimensions +to remain the same not only for rotations, but also for an inertial frame. That +means that our equations should be invariant under the Lorentz transformation Defnition of a +of equations (25.1). The purpose of this chapter is to show you how that can be vector A=(4;,4¿, A;) +done. Before we get started, however, we want to do something that makes our Scalar producb A.Db +work a lot easier (and saves some confusion). And that is to choose our units of Diferential vect +l - : l - 1ferential vector +length and time so that the speed of light e is equal to 1. You can think oÝ it as operator v +taking our unit of tìme to be he time that ? takes líght to go one mmeter (which : +is about 3 x 10” sec). We can even call this tỉme unit “one meter.” Using this Gradient Vỏ +unit, all oŸ our equations will show more clearly the space-time symmetry. Also, Divergence V:A +all the đs will disappear from our relativistic equations. (Tf this bothers you, you Laplacian V.V=V? +can always put the đs back into any equation by replacing every £ by cứ, or, in Cross produet AxB +general, by sticking in a c wherever it is needed to make the dimensions of the Cun VxA +cquations come out right.) With this groundwork we are ready to begin. Qur „ Ễ +program is to do ïn the four dimensions of space-time all of the things we did +with vectors for three dimensions. It is really quite a simple game; we just work +by analogy. The only real complications is the notation (we ve already used up +the vector symbol for three dimensions) and one slight twist of signs. +tirst, by analogy with vectors in three dimensions, we defne a ƒour-0ector +as a set of the four quantities œ¿, ø„, a„, and ø;, which transform like ứ, #, 1, +and z when we change to a moving coordinate system. There are several different +notations people use for a four-vector; we will write ø„, by which we mean the +group of four numbers (đ¿, đ„, đ„, ø„)——in other words, the subscript can take +on the four “values” ứ, ø, , z. It will also be convenient, at times, to indicate +the three space components by a three-vector, like this: a„ = (œ¿, ). +W© have already encountered one four-vector, which consists of the energy +and momentum of a particle (Chapter 17, Vol. l): In our new notation we write +Đụ = (E,p), (25.2) +which means that the four-vector ø„ is made up of the energy #2 and the three +components of the three-vector ø of a particle. +Tt looks as though the game is really very simple—for each three-vector in +physics all we have to do is ñnd what the remaining component should be, and +we have a four-vector. To see that this is not the case, consider the velocity +vector with components +d+z dụ đz +Uy = Tn: ——. Uy = TT +The question is: What is the time component? Instinct should give the right +answer. Since four-vectors are like Ý, ø, , z, we would guess that the time +componenf 1s +— đi — 1 +Th¿s ?s rong. The reason 1s that the # in each denominator is not an invari- +ant when we make a Lorentz transformation. 'Phe numerators have the right +behavior to make a four-vector, but the đ£ in the denominator spoils things; it is +unsymmetric and is not the same in two different systems. +lt turns out that the four “velocity” components which we have written down +will become the components of a four-vector IŸ we Just divide by V1— 02. We +can see that that is true because If we start with the momentum four-vector +To TnoÐ +1= (ÉP) = (Ong) (25.3) +--- Trang 316 --- +and divide it by the rest mass rmọ, which is an invariant scalar in four dimensions, +we have +. c= T=) (25.4) +mo VI-— 2` vV1— 02 +which must still be a four-vector. (Dividing by an ?moariœmt scalar doesn”t change +the transformation properties.) So we can define the “uelocitu ƒour-uector” u„ by +tuy — ———————, tuy —————D, +v1— t2 v1— t2 +(25.5) +U„y Uz ++ = ———: +L„ — ———: +” w1—ø2 “ v1—ø2 +'The four-velocity 1s a useful quantity; we can, for instance, write +Đụ — Tngtu,. (25.6) +'This is the typical sort of form an equation which is relativistically correct must +have; cach side is a four-vector. (The right-hand side is an invariant times a +four-vector, which is still a four-vector.) +25-2 The scalar product +lt is an accident of life, if you wish, that under coordinate rotations the +distance of a point rom the origin does not change. 'This means mathematically +that r2 = z2 + 12 + z2 is an invariant. In other words, after a rotation r2 = r2, +a2 + g2 + z2 — g2 + g2 + 22, +Now the question is: Is there a similar quantity which is Iinvariant under the +Lorentz transformation? There is. Erom Eaq. (25.1) you can see that +2 Tạ =2 T— „È, +'That is pretty nice, except that it debends on a particular choice of the z-direction. +We can fx that up by subtracting z2 and z2. Then any Lorentz transformation +pÏus a rotation will leave the quantity unchanged. So the quantity which is +analogous to z2 for three dimensions, in four dimensions is +tĐ — „2T g2 — 23, +lt is an invariant under what ¡is called the “complete Lorentz group”—=which +means for transformation of both translations at constant velocity and rotations. +Now since this invariance is an algebraic matter depending only on the +transformation rules of Eq. (25.1)—plus rotations—it is true for any Íour-vector +(by deñnition they all transform the same). 5o for a four-vector a„ we have that +để — địt — đu — để = đệ — dạ — dạ — d2. +We will call this quantity the square of “the length” of the four-vector đ„. +(Sometimes people change the sign of all the terms and call the length a2 + d2 + +g2 — đ¿, so you”]l have to watch out.) +Now 1Í we have #uo vectors ø„ and b„ theiïr corresponding components trans- +form in the same way, so the combination +db; — đ„D„ — dub„ — dazbz„ +is also an invariant (scalar) quantity. (We have in fact already proved this in +Chapter L7 of Vol. I.) Clearly this expression is quite analogous to the dot product +for vectors. We will, in fact, call it the do product or scalar produc‡ oŸ bwo +four-vectors. Ït would seem logical to write it as a„, - b„, so it would look like a dot +product. But, unhappily, it's not done that way; 1t is usually written without the +--- Trang 317 --- +dot. 5o we will follow the convention and write the dot product sỉimply as a„b,„. +So, 0U đeftmition, +qub„ = diÙ¿ — g„Ù„ — quDy — xÙ. (25.7) +Whenever you see two identical subscripts together (we will occasionally have to +use or some other letber instead oŸ ) it means that you are to take the four +products and sum, remembering the minus sign for the produects of the space +components. With this convention the invariance of the scalar product under a +Lorentz transformation can be written as +Ƒ, 1Ự +a,ÐD, = dụ. +Since the last three terms in (25.7) are just the scalar dot product in three +dimensions, it is often more convenient to write +aubu — dịÙy —G-b. +Tlt is also obvious that the four-dimensional length we described above can be +wrltten as a„d„: +đu, = đệ — độ — 0y — dộ = d; — Œ-Œ, (25.8) +Tt will also be convenient to sometimes write this quantity as dạ, +d, = qud,. +W©e will now give you an illustration of the usefulness oŸ four-vector dot +products. Antiprotons (P) are produced in large accelerators by the reaction +P+P-P+P+P+P. +That is, an energetic proton collides with a proton at rest (for example, in a +hydrogen target placed in the beam), and ïŸ the incident proton has enough +energy, a proton-antiproton pair may be produced, in addition to the two original +protons.* The question is: How much energy must be given to the incident proton +to make this reaction energetically possible? +The easiest way to get the answer is 0o consider what the reaction looks +like in the center-of-mass (CM) system (see Eig. 25-1). We'”ll call the incident +BEFORE AFTER +là 4 b C +s Độ Dạ Đụ +° Im e> ———o +Fig. 25-1. The reaction P+P -› 3P+P FC ======Ẩ= +viewed in the laboratory and CM systems. È +The incident proton Is supposed to have just S= 2? b' c¡ +: Eui Đụ Pụ Pụ +barely enough energy to make the reaction á E| ®“———> ° +go. Protons are denoted by solid circles; Sà +antiprotons by open circles. 4 +* You may well ask: Why not consider the reactions +P+P-P+P+P, +OT ©€Ve€eI _— +P+PDEP+P +which clearly require less energy? The answer is that a principle called conseruation oŸ barons +tells us the quantity “number of protons minus number of antiprotons” cannot change. 'This +quantity is 2 on the left side of our reaction. Therefore, if we want an antiproton on the right +side, we must have also #hree protons (or other baryons). +--- Trang 318 --- +proton ø and its four-momentum 7. 5Similarly, we”ll call the target proton Ù +and its four-momentum Dị: Tf the ineident proton has 7usf barel/ enough energy +to make the reaction go, the final state—the situation after the collision——will +consist of a glob containing three protons and an antiproton at rest in the CM +system. IỶ the incident energy were slightly higher, the fñnal state particles would +have some kinetic energy and be moving apart; if the incident energy were slightly +lower, there would not be enough energy to make the four particles. +TỶ we call ø, the total four-momentum of the whole glob im the fñnal state, +conservation oŸ energy and momentum tells us that +ph+p=Pp', +Et“+ E°= E'. +Combining these two equations, we can write that +DỤ» + Độ, = Độ: (25.9) +Now the important thing is that this is an equation among four-vectors, and +1s, therefore, true In any inertial frame. We can use this fact to simplify our +calculations. We start by taking the “length” of each side of Bq. (25.9); they are, +of course, also equal. We get +(bự + pr)(Đ,, + Đ,) — Độ: (25.10) +Since Ø7, is Invariant, we can evaluate it in any coordinate system. In the CM +system, the tỉme cormponent oŸ 7, is the rest energy of Íour protons, namely 4M, +and the space part Ø is zero; so ø, = (4A, 0). Wo have used the fact that the +rest mass of an antiproton equals the rest mass of a proton, and we have called +this common mass /M. +Thus, Eq. (25.10) becomes +PyD, + 200), + p„p, = 16MỔ. (25.11) +Now p„ and D„D,, are very easy, since the “length” of the momentum four-vector +of any particle is just the mass of the particle squared: +ĐụÐụ = E2 — pˆ= M}?. +This can be shown by direct calculation or, more cleverly, by noting that for a +particle aÝ zest p„ = (M, 0), so p„p„ = M2. But since it is an invariant, it is +equal to M2 in øngy frame. sing these results in Eq. (25.11), we have +20,0, = 14M +DĐ, = TM”. (25.12) +Now we can also evaluate D„.DP, = Đ ph in the laboratory system. The +four-vector ø#ˆ can be written (E“”,p°), while ph = (M,0), since it describes +a proton at rest. Thus, ĐR ph must also be equal to Ä#!“; and since we know +the scalar product is an invariant this must be numerically the same as what we +found in (25.12). Š5o we have that +E“'=TM, +which is the result we were after. The #oal energy of the initial probon must +be at least 7A (about 6.6 Gev since ă = 938 MeV) or, subtracting the rest +mass #ứ, the kinefic energy must be at least 6 (about 5.6 Gev). The Bevatron +accelerator at Berkeley was designed to give about 6.2 Gev of kinetic energy to +the protons it accelerates, in order to be able to make antiprotons. +Since scalar products are invariant, they are always interesting to evaluate. +What about the “length” of the four-velocity w,„u,,? +— „2 2— — +Mu, M1 u81 cu TS h +Thus, „, is the unớt ƒour-uector. +--- Trang 319 --- +25-3 The four-dimensional gradient +The next thing that we have to discuss is the four-dimensional analog of the +gradient. WWe recall (Chapter 14, Vol. I) that the three diferential operators +9/9z, 9/Øụ, Ø/Ôz transform like a three-vector and are called the gradient. The +same scheme ought to work in four dimensions; that is, we might guess that the +four-dimensional gradient should be (0/6, 9/9z,Ø/9ụ,Ð/Øz). Thịs ts turong. +'To see the error, consider a scalar function @ which depends only on z and ý. +The change in ó, if we make a small change Af ¡in £ while holding z constant, is +Ad= —_- Ai. 25.13 +ó= ` (25.13) +On the other hand, according to a moving observer, +Ad==—,Az+— A. +; 0p —” + Øtứ +W©e can express Az/ and Af in terms oŸ A£ by using Eq. (25.1). Remembering +that we are holding z constant, so that Az+ = 0, we write +Az'=—————At Af=—=—. +v1~ 2 v1ì—›2 +'Thus, +9ó Đ đó At +A2=-—| - ————A¿ —;| — +0= vì Tấm = = 3) +— (9 đó At +— \Ø# ° Đạp? ⁄1—w>2ˆ +Comparing this result with Eq. (25.13), we learn that +đó 1 9ó 9ó +—-—= ——— | -,_—-U—-„]. 25.14 +ðt "HL. 7Ð (5.14) +A similar calculation gives +đó 1 9ó 9ó +— —= — | _-_-_-°—_]. 25.15 +Ôz "=.- “øm (5.15) +Now we can see that the gradient is rather strange. The formulas for ø and £ +in terms oŸ zø“ and # [obtained by solving Eaq. (25.1)] are: +: +uz + + 0É += —=ễm,Ụ #=——p. +v1—%ˆ2 v1—02 +Thịs is the way a Íour-vecbor rmusứ transform. But Bqs. (25.14) and (25.15) have +a couple of signs wrongl +The answer is that instead of the ?mcorrect (0/0t,W), we must define the +Jour-dimensional gradien‡ operator, which we will call Vị,, by +8 8 8 8 8 +VW„=[=c:—V]Ì=|-=.:-=-:—-=-:—=-]- 25.16 +, lá: ) lạ: 9z` Øụ 5) ) +With this defñnition, the sign dificulties encountered above go away, and Vj, +behaves as a four-vector should. (It”s rather awkward to have those minus signs, +but that”s the way the world is.) Of course, what it means to say that V,„ “behaves +like a four-vector” is simply that the four-gradient of a scalar is a four-vector. +Tf @ is a true scalar invariant field (Lorentz invariant) then V,,ở is a four-vector +AII right, now that we have vectors, gradients, and dot products, the next +thing is to look for an invariant which is analogous to the divergence oŸ three- +dimensional vector analysis. Clearly, the analog is to form the expression Vj,b„, +--- Trang 320 --- +where b„ is a four-vector field whose components are functions of space and time. +We đefine the điuergence of the four-vector b„ = (b¿,b) as the dot product of Vị, +and b„: +8 8 8 8 +sexZx-C25- C4)» C2} +Øt 3z Øyj ” Øz +5 ụ (25.17) +=.b¿+V-b +Ôt £ + ) +where V - b ¡is the ordinary three-divergence of the three-vector b. Note that +one has to be careful with the signs. Some of the minus signs come from the +defnition of the scalar produect, Eq. (25.7); the others are required because the +space components of V,„ are —Ø/Øz, etc., as in Eq. (25.16). The divergence as +defined by (25.17) is an invariant and gives the same answer in all coordinate +systems which difer by a Lorentz transformation. +Let°s look at a physical example in which the four-divergence shows up. We +can use i% to solve the problem of the felds around a moving wire. We have already +seen (Section 13-7) that the electric charge density ø and the current density 7 +form a four-vector 7„ = (ø,7). lf an uncharged wire carries the current 7„, then +in a frame moving past it with velocity ø (along +), the wire will have the charge +and current density |obtained from the Lorentz transformation Eqs. (25.1)] as +follows: : : +gj= —U2+ j2 = Jz +v1— 02` ”v1-02 +These are just what we found in Chapter 13. We can then use these sources +in Maxwell's equations in he rmmouing sụstem to ñnd the fñelds. +'The charge conservation law, Section 13-2, also takes on a simple form in the +four-vector notation. Consider the four divergence oŸ 7„: +Wuj,= S + V-j. (25.18) +The law of the conservation of charge says that the outlow of current per unit +volume must equal the negative rate of increase of charge density. In other words, +that ô +V-7=_—_—.. +Putting this into Eq. (25.18), the law of conservation of charge takes on the +simple form +Vu7„ = 0. (25.19) +Since V„7„ is an invariant scalar, ïf it is zero in one frame it is zero in all frames. +W©e have the result that if charge is conserved in one coordinate system, it is +conserved in all coordinate systems moving with uniform velocity. +As our last example we want to consider the scalar product of the gradient +operator Vị, with itself. In three dimensions, such a product gives the Laplacian +8? 8? 82 +VỶ=V-V=-—=+-s+-= +8z2 + Øụ2 + 8z2 +What do we get in four dimensions? That's easy. Following our rules for dot +products and gradients, we get +g8 Ø8 8 8 8 8 8 8 +VWu„Wu=z=zl- =-ll =-]- | - =>] -=]- | - => |-= +Øt Ôt 3z 3z Øụ Øy Øz Øz += ø — V2 +'This operator, which 1s the analog of the three-dimensional Laplacian, ¡is called +the DAlembertian and has a special notation: +2 ỡ? 2 +--- Trang 321 --- +trom ïits definition it is an invariant scalar operator; 1ƒ it operates on a four-vector +fñeld, it produces a new fÍour-vector field. (Some people define the D°Alembertian +with the opposite sign to Eq. (25.20), so you will have to be careful when reading +the literature.) +W© have now found four-dimensional equivalents of most of the three-dimen- +sional quantities we had listed in Table 25-1. (We do not yet have the equivalents +of the eross product and the curl operation; we won't get to them until the next +chapter.) It may help you remember how they go if we put all the important +defñnitions and results together in one place, so we have made such a summary +in Table 25-2. +Table 25-2 +The important quantities of vector analysis in three and four dimensions. +'Three dimensions tFour dimensions +Vector A = (A¿, Ay,A,) đụ — (d¿; đy, dụ; dy) —= (d¿, Œ) +Scalar product A-P—A;b„+ AyByạ+ A,B, đuDu — dib¿ — g„b„ — quÐby — a;Ðy —= d¿Ö¿ —g-b +Vector operator © = (Ôô/Ôz,Ô/Ôụ, 9/82) Vụ = (0/0t,—8/8z,—Ð/Øụ,—8/8z) = (8/ôt,—V) +: Ø0) Đụ Đụ 9Ø _Øø _Ø¿ _Ởý Đụ +dient ˆ ^^. =| “ _-“__*“-_- =|Ì=-|_- +Gradien Vụ tn_ V#— (2p' 0z! Dụ' Đa DI Á, +. A4Ay ĐA öAz Øœ Øa„y Ôa Øay — Ôq¿ +D . A— _=*“®+_—_ 1 +-_—_Z =—= — + — 1+." 1+ _—_`"—_—_- - +Ivergence v 0m Ì 0y ` 'Ðz Vua, = rủi nạ, Tny Tra; =røp TV a4 +Laplacian and —— 82? 8? Ø2? 2 —— lu Ø2 Ø2 Ø2 Ø? 2 2 +D'Alembertian V.W= na pz=V VuVu = 8 — 0g 0g 020p. VU +25-4 Electrodynamics in four-dimensional notation +W©e have already encountered the DˆAlembertian operator, without giving +19 that name, in Section 18-6; the diferential equations we found there for the +potentials can be written in the new notations as: +H2¿=#, r?A=7. (25.21) +The four quantities on the right-hand side of the two equations in (25.21) are +Ø: ?z› J„› 2z divided by cọ, which is a universal constant which will be the same +in all coordinate systems if the same unit of charge is used in all frames. 5o the +four quantities ø/€o, jz/€o. jy/€o, jx/eo also transform as a four-vector. We can +write them as 7„/cọ. The D'Alembertian doesn't change when the coordinate +system is changed, so the quantities ó, Á„, Áy, Á; rmust also transform like a +four-vector—which means that they are the components of a four-vector. Ïn +short, +A„ — (ó, A) +1s a four-vector. What we call the scalar and vector potentials are really diferent +aspects of the same physical thing. They belong together. And ïf they are +kept together the relativistic invariance of the world is obvious. We call 4, the +ƒour-potential. +In the four-vector notation Eqs. (25.21) become simply +2A _ nu +LAu=“=, (25.22) +--- Trang 322 --- +The physics of this equation is just the same as Maxwell's equations. But there is +some pleasure in beïng able to rewrite them in an elegant form. The pretty form +is also meaningful; it shows directly the invariance of electrodynamics under the +Lorentz transformation. +Remember that Eqs. (25.21) could be deduced from Maxwell's equations onÌy +1ƒ we imposed the gauge condition +Sẽ +V.A-=(0, (25.23) +which just says V„u.4,„ = 0; the gauge condition says that the divergence of the +four-vecbor A„ is zero. Thịis condition is called the Lorenz condijtion. TW is very +convenient because it is an invariant condition and therefore Maxwell”s equations +sbay in the form o£ Eq. (25.22) for all frames. +25-5 The four-potential of a moving charge +Although it is implicit in what we have already said, let us write down the +transformation laws which give ó and . in a moving system in terms of j and A +in a stationary system. 5Since 4„ = (ở, Ả) is a four-vector, the equations must +look just like Eqs. (25.1), except that £ is replaced by ó, and ø is replaced by A. +'Thus, y +dc _ A,= Aụ, s ⁄ +— % S +(25.24) +jm-.¬.n.. ` P +v1 s2 ; | +'This assumes that the primed coordinate system is moving with speed øœ in the ¬ +positive z-direction, as measured in the unprimed coordinate system. Z ¬¬ +'W© will consider one example of the usefulness of the idea of the four-potential. z TT. v +What are the vector and scalar potentials of a charge g moving with speed 0 +along the zø-axis? The problem is easy in a coordinate system moving with the Fig. 25-2. The frame S” moves with ve- +charge, since in this system the charge is standing still. Let's say that the charge locity v (in the x-direction) with respect +is at the origin of the S/-rame, as shown in Fig. 25-2. The scalar potential in to S. A charge at rest at the origin of S” is +the moving system is then given by at x = vt in S. The potentials at P can be +computed ¡in either frame. +ý =——,Ụ (25.25) +47eogr/ +r7“ being the distance from g to the feld point, as measured in the moving system. +The vector potential AÍ is, of course, zero. +Now it is straipghtforward to ñnd ø and A, the potentials as measured in the +sbationary coordinates. The inverse relations to Eqs. (25.24) are +/Ƒ A/ +=<<= - Au = Aụ, +" , (25.26) +A„= Ảu tuổc A.=A/. +v1—ˆ2 : +Using the #/ given by Eq. (25.25), and A“ = 0, we get +_ 4mco r'V1 — 02 += 4mco 1— u24/+2 + ự2 + x2. +This gives us the scalar potential @ we would see in Š, but, unfortunately, +expressed in terms of the Š” coordinates. We can get things in terms OÏ Ý, #, , Zz +by substituting for f', ø', ', and z”, using (25.1). We get +=————————————, (25.27) +#0 vVI~— 02 Íl(œ— øt)/VT1— 0]2 + g2 + z2 +--- Trang 323 --- +Following the same procedure for the components of A, you can show that +A = bọ. (25.28) +These are the same formulas we derived by a diferent method in Chapter 21. +25-6 The invariance of the equations of electrodynamics +We have found that the potentials ó and 4 taken together form a four- +vector which we call A,„, and that the wave equations—the full equations which +determine the 4„ in terms of the 7„—can be written as in Eq. (25.22). Thịis +cquation, together with the conservation of charge, bq. (25.19), gives us the +fundamental law of the electromagnetic field: +2 1. ; +LˆÁu = —ƒ„; Vu7„ =0. (25.29) +'There, in one tỉny space on the page, are all of the Maxwell equations—beautiful +and simple. Did we learn anything from writing the equations this way, besides +that they are beautiful and simple? In the first place, is it anything diferent from +what we had before when we wrote everything out in all the various components? +Can we from this equation deduce something that could not be deduced from +the wave equations for the potentials in terms of the charges and currents? The +answer is defñnitely no. 'Phe only thing we have been doing is changing the +names of things—using a new notation. We have written a square symbol to +represent the derivatives, but it still means nothing more nor less than the second +derivative with respect to ý, minus the second derivative with respect to #, minus +the second derivative with respect to , minus the second derivative with respect +to 2. And the means that we have four equations, one each for ứ = Ý, #, #, +or z. What then is the signiflcance of the fact that the equations can be written +in this simple form? Erom the point of view of deducing anything directly, 1% +doesn't mean anything. Perhaps, though, the simplicity of the equations means +that nature also has a certain simplicity. +Let us show you something interesting that we have recently discovered: Ai +0ƒ the latus oƒ phụsics can be contained ín one cquation. hat equation is +U =0. (25.30) +'What a simple equation! Of course, it is necessary to know what the symbol means. +U is a physical quantity which we will call the “unworldliness” of the situation. +And we have a formula for it. Here is how you calculate the unworldliness. You +take all of the known physical laws and write them in a special form. For example, +Suppose you take the law of mechanics, #” = ma, and rewrite it as E' — mœ = 0. +Then you can call (E' — rza)—which should, of course, be zero——the “mismatch” +of mechanics. Next, you take the sguare of this mismatch and call ít U, which +can be called the “unworldliness of mechanical efects.” In other words, you take +U¡ = (F'— ma)Š. (25.31) +NÑow you write another physical law, say, W - E = ø/co and deñne +U› = (v .E~ ˆ) +which you might call “the gaussian unworldliness of electricity.” You continue to +write Úx, Ủa, and so on——=one for every physical law there is. +Finally you call the £o#øl unworldliness Ù of the world the sum oŸ the various +unworldlinesses U; from all the subphenomena that are involved; that is, U = 3” Ú;¿. +Then the great “law of nature” is +--- Trang 324 --- +This “law” means, of course, that the sum of the squares of all the individual +mmismatches is zero, and the only way the sum oŸ a lo oŸ squares can be Zero is +for each one of the terms to be zero. +So the “beautifully simple” law in Eq. (25.32) is equivalent to the whole +series Of equations that you originally wrote down. It is therefore absolutely +obvious that a simple notation that Just hides the complexity in the defnitions +of symbols is not real simplicity. Tứ ¡s 7usf a tríck. he beauty that appears in +Eq. (25.32)—just from the fact that several equations are hidden within it—is +no more than a trick. When you unwrap the whole thing, you get back where +you were before. +However, there 7s more to the simplicity of the laws of electromagnetism +written in the form oŸ Eq. (25.29). Ib means more, just as a theory oŸ vector +analysis means more. The fact that the electromagnetic equations can be written +in a very particular notation t0húch tuas designed for the four-dimensional geometry +of the Lorentz transformations——in other words, as a vector equation in the four- +space—means that it is invariant under the Lorentz transformations. Ït is because +the Maxwell equations are invariant under those transformations that they can +be written in a beautiful form. +Tt is no accident that the equations of electrodynamies can be written in the +beautifully elegant form of Eq. (25.29). The theory of relativity was developed +because tt tuas ƒound exzperimentallu that the phenomena predicted by Maxwells +cquations were the same in all inertial systems. And it was precisely by studying +the transformation properties of Maxwell's equations that Lorentz discovered his +transformation as the one which left the equations invariant. +There is, however, another reason for writing our equations this way. It has +been discovered——after Hinstein guessed that it might be so—that ai of the laws +of physics are invariant under the Lorentz transformation. 'That is the principle +of relativity. Therefore, if we invent a notation which shows immediately when a +law is written down whether iE is invariant or not, we can be sure that in trying +to make new theories we will write only equations which are consistent with the +principle of relativity. +'The fact that the Maxwell equations are simple in this particular notation is +not a miracle, because the notation was invented with them in mind. But the +interesting physical thing is that euer lau of physics—the propagation of meson +waves or the behavior of neutrinos in beta decay, and so forth—must have this +same invariance under the same transformation. Then when you are moving at a +uniform velocity in a spaceship, all of the laws of nature transform together in +such a way that no new phenomenon will show up. lt is because the principle of +relativity is a fact of nature that in the notation of four-dimensional vectors the +equations of the world will look simple. +--- Trang 325 --- +XVLoroméÉs ÏT-etrtsfOr'rttdrffO@rts @œŸ flìo Frol‹ls +26-1 The four-potential of a moving charge +We saw in the last chapter that the potential A„ = (ø, A) is a four-vector. 26-1 The four-potential of a moving +'The time component is the scalar potential ó, and the three space componenfs are charge +the vector potential A. We also worked out the potentials of a particle moving 26-2 The fields of a point charge with +with uniform speed on a straight line by using the Lorentz transformation. (We a constant velocity +had already found them by another method in Chapter 21.) Eor a point charge 26-3 Relativistic transformation of the +whose position at the tỉme # is (£,0,0), the potentials at the point (z, 0, 2z) are felds +1 q 20-4 The equations of motion ỉin += ——————— TT cAOD "1⁄2 relativistic notation +——s |(— 0£) +4mcoV1 — 02 '== +#2+ 2| +———m |(— 0£) +4ceoV1 — 02 '== +z2+ 2| +Ay=4A; =0. +Reuieu: Chapter 20, Vol. II, Solutlion +Equations (26.1) give the pobentials at z, , and z at the time ý, for a charge 0ƒ Mazuell's Equations ín Free +whose “present” position (by which we mean the position aý the time f) is Space +ab œ = 0É. Notice that the equations are in terms of (œ — 0£), , and z which are +the coordinates measured from the current position P of the moving charge (see +Eig. 26-1). The actual inÑuence we know really travels at the speed é, so iÈ is the +behavior of the charge back at the retarded position ? that really counts.X The +point f? is at z = ơfˆ (where £ = £— r//e is the retarded time). But we said that +the charge was moving with uniform velocity in a straight line, so naturally the +behavior at P and the current position are directly related. In fact, iÏ we make +the added assumption that the potentials depend only upon the position and the y +velocity at the retarded moment, we have in equations (26.1) a complete formula +for the potentials for a charge moving ø1w way. It works this way. Suppose that +you have a charge moving in some arbitrary fashion, say with the trajectory in +Fig. 26-2, and you are tryïng to ñnd the potentials at the poïnt (z,,z). First, (x.y.Z) +you find the retarded position “ and the velocity œ at that point. 'Then you RETARDED | +imagine that the charge would keep on moving with this velocity during the delay PGRIION r lý +time (/ — £), so that ít would then appear at an imaginary position ;;o¡, which | PRESENT | +we can call the “projected position,” and would arrive there with the velocity œ. P ZZ POSITION, +(Of course, it doesn”t do that; its real position at £ is at P.) Then the potentials v_vt Ị x +ab (2,0, z) are jusi what equations (26.1) would give for the imaginary charge vt +at the projected position Đrzoj. What we are saying is that since the potentials +depend only on what the charge is doing at the refarded time, the potentials +will be the same whether the charge contimued moving ata Consbat velocity or Eig. 26-1. Finding the fields at (x, y, z) +whether it changed its velocity after £——that is, after the potentials that were due to a charge g moving along the x-axis +going to appear at (z,, 2) at the time ý were already determined. with the constant speed v. The field "now" +You know, of course, that the moment that we have the formula for the at the point (x, y,z) can be expressed in +potentials from a charge moving in any manner whatsoever, we have the complete terms of the “present” position P, as well +electrodynamies; we can get the potentials of any charge distribution by superpo- as in terms of P”, the “retarded” position +sition. Therefore we can summarize all the phenomena of electrodynamics either (at f =t— r/e). +* The primes used here to indicate the re£arded positions and times should not be confused +with the primes referring to a Lorentz-transformed frame in the preceding chapter. +--- Trang 326 --- +by writing Maxwell's equations or by the following series of remarks. (Remember +them in case you are ever on a desert island. Erom them, all can be reconstructed. +You will, of course, know the Lorentz transformation; you will never forget hat +on a desert island or anywhere else.) +list, A, 1s a four-vecbor. Second, the Coulomb potential for a stationary +charge is g/4meor. Thứ, the potentials produced by a charge moving in any (x,y,Z) +way depend only upon the velocity and position at the retarded time. With +those three facts we have everything. Erom the fact that Á„ is a four-vector, we +transform the Coulomb potential, which we know, and get the potentials for a ⁄ +constant velocity. Then, by the last statement that potentials depend only upon ự “ +the past velocity at the retarded time, we can use the projected position game to +fnd them. It is not a particularly useful way of doing things, but it is interesting RE 'AROro " ˆ +to show that the laws of physics can be put in so many diferent ways. an \2Ro JECTED" +Tlt is sometimes said, by people who are careless, that all of electrodynamiecs q POSITION +can be deduced solely from the Lorentz transÍíormation and Coulomb”s law. P_— “PRESENT" +Of course, that is completely false. Pirst, we have to suppose that there is a TRAJECTORY vx POSHION +scalar potential and a vector potential that together make a four-vector. That +tells us how the potentials transform. 'Phen why is it that the efects at the Fig. 26-2. A charge moves on an arbitrary +retarded time are the only things that count? Better yet, why is it that the — trajectory. The potentials at (x, y, z) at the +potentials depend only on the position and the velocity and not, for instance, time £ are determined by the position 7 +on the acceleration? The fields E and Ö do depend on the acceleration. lf you and velocity v' at the retarded time £ — r /c. +try to make the same kind of an argument with respect to them, you would They are convenientÌy ©xpressed h terms of +" . . the coordinates from the “projected” posi- +say that they depend only upon the position and velocity at the retarded time. tion Pzo;. (The actual position at £ is P.) +But then the felds from an accelerating charge would be the same as the felds mai : +from a charge at the projected position——which is false. The ƒ#elds depend not +only on the position and the velocity along the path but also on the acceleration. +So there are several additional tacit assumptions in this great statement that +everything can be deduced from the Lorentz transformation. (Whenever you see +a sweeping statement that a tremendous amount can come from a very small +number of assumptions, you always ñnd that it is false. There are usually a large +number of implied assumptions that are far from obvious if you think about them +sufficiently carefully.) +26-2 The fields of a point charge with a constant velocity +Now that we have the potentials from a point charge moving at constant +velocity, we ought to fnd the fields—for practical reasons. There are many +cases where we have uniformly moving particles—for instance, cosmic rays going +through a cloud chamber, or even slow-moving electrons in a wire. So let”s at least +see what the felds actually do look like for any speed——even for speeds nearly +that of light—assuming only that there is no acceleration. ÏIt is an interesting +question. +We get the felds from the potentials by the usual rules: +E=-V- TỐ, bö=VYxA. +Pirst, for Fy +p._ 80 0A: +3z lôI2 +But 4; is zero; so differentiating ở in equations (26.1), we get +E,=——T— TA. (26.2) +4zceogV1 — 02 l=D +?+ 2| +Similarly, for l¿, +#„=———————aaz (26.3) +4ceoV1 — 02 '== +?+ 2| +--- Trang 327 --- +'The z-component is a little more work. The derivative of ó is more complicated +and 4z is not zero. Eirst, +ô — 0f)/(1— 0? +3z (œ — f£)2 3⁄2 +4coV 1 — 02 TICm 1+2 +Then, diferentiating 4Á; with respect to £, we fnd +3A —U2(z — 0) /(1— 02 +— "ôp ““.... (26.5) +4mcoVv1 — u2 '== + ? + 2| +And finally, taking the sum, +Ø — UÈ +đ„ — —"*— — —————..` (26.6) +(œ — £)2 Ey E +4meoVl— 2 |>————+ụ?+z? +1—0 (x,y,Z) E, +We'll look at the physics of # in a minute; let?s frst ñnd . FEor the z- nOSHION | +component, ⁄ X +B.= ĐẦu Đ4z PRESENT ' +z— _ — "øp” bị pZZ P9STION +Since 4¿ is zero, we have just one derivative to get. Notice, however, that Ảz is vÉ x= v£ | " +just øó, and Ø/Øw of oỏ is just —u„. So “ +By = uhy,. (26.7) +Simllarly, Fig. 26-3. For a charge moving with con- +94y 9A, 9ó stant speed, the electric field points radially +Bụ= 'Ôz ôm =+U Ôz' from the “present” position of the charge. +DBụ = —-uF;¿. (26.8) +Einally, „ is zero, since A„ and 4; are both zero. We can write the magnetic E +fñeld simply as +bB=uxE. (26.9) +Now let”s see what the fñelds look like. We will try to draw a picture of the +field at various positions around the present position of the charge. ÏIt is true that +the infuence of the charge comes, in a certain sense, rom the retarded position; (a) v=0 +but because the motion 1s exactly specified, the retarded position is uniquely +given in terms of the present position. Eor uniform velocitlies, it”s nicer to relate +the felds to the current position, because the feld components at (#, , z2) depend +only on ( — 0£), , and z—which are the components of the displacement rom +the present position to (2, , z) (see Fig. 26-3). +Consider frst a point with z = 0. Then has only z- and 9-components. E +trom Eqs. (26.3) and (26.6), the ratio of these components is Just equal to the +ratlo of the ø- and -components of the displacement. That means that # is in +the saưme đircction as m, as shown in Fig. 26-3. Since Hy is also proportional to z, +1t 1s clear that this result holds in three dimensions. In short, the electric fñeld is v +radial tom the charge, and the feld lines radiate directly out of the charge, just —- +as they do for a stationary charge. OÝ course, the field isn't exactly the same 6) v=0.%c +as for the stationary charge, because of all the extra facbors of (1 — 2). But we +can show something rather interesting. The difference is just what you would +get If you were to draw the Coulomb field with a peculiar set of coordinates in +which the scale oŸ z was squashed up by the factor W1 — 02. If you do that, the +fñeld lines will be spread out ahead and behind the charge and will be squeezed Fig. 26-4. The electric field of a charge +together around the sides, as shown in Fig. 26-4. moving with constant speed v = 0.9c, +Tf we relate the strength of # to the density of the field lines in the conventional part (b), compared with the field of a charge +way, we see a stronger field at the sides and a weaker field ahead and behind, — 3t rest, part (a). +--- Trang 328 --- +which is just what the equations say. First, if we look at the strength of the field +at right angles to the line of motion, that is, for (+ — 0ý) = 0, the distance rom +the charge is 4⁄22 + z2. Here the total fñeld strength is ,/ H + E2, which is +E 1roVì =3 2+z (26.10) +The field is proportional to the inverse square of the distance—Just like the +Coulomb feld except increased by the constant, extra factor 1/1 — 02, which is +always greater than one. 5o at the s7des of a moving charge, the electric field is +stronger than you get from the Coulomb law. In fact, the field in the sidewise +direction 1s bigger than the Coulomb potential by the ratio of the energy of the +particle to its rest mass. +Ahead of the charge (and behind), ¿ and z are zero and +g(1— 0”) += E„ = 1rco(œ— 0Š” (26.11) +'The field again varies as the inverse square of the distance from the charge but +is now reduced by the facbor (1 — 02), in agreement with the picbure of the fñeld +lines. If 0/e is small, ø2/cŸ is still smaller, and the efect of the (1 — 02) terms is +very small; we get back to Coulomb's law. But if a particle is moving very close +to the speed of light, the field in the forward direction is enormously reduced, +and the feld in the sidewise direction is enormouslÌy increased. +Our results for the electric feld of a charge can be put this way: Suppose +you were to draw on a piece of paper the field lines for a charge at rest, and +then set the picture to travelling with the speed o. 'Phen, of course, the whole +picture would be compressed by the Lorentz contraction; that is, the carbon +granules on the paper would appear in diÑferent places. The miracle of it is that +the picture you would see as the page fies by would still represent the ñeld lines +of the point charge. The contraction moves them closer together at the sides and +spreads them out ahead and behind, just in the right way to give the correct +line densities. We have emphasized before that feld lines are not real but are +only one way of representing the field. However, here they almost seem to be +real. In this particular case, 1Ÿ you make the mistake of thinking that the fñield +lines are somehow really there in space, and transform them, you get the correct +fñield. That doesnˆt, however, make the field lines any more real. All you need +do to remind yourself that they aren”t real is to think about the electric fñelds +produced by a charge together with a magnet; when the magnet moves, new B +electric felds are produced, and destroy the beautiful picture. So the neat idea +of the contracting picture doesnt work in general. It is, however, a handy way . +to remember what the fields from a fast-moving charge are like. _—_ | kr 1Ì +The magnetic field is ø x [from Ed. (26.9)]|. TÝ you take the velocity crossed ở +into a radial E-fñeld, you get a Ö which circles around the line of motion, as +shown in Eig. 26-5. lf we put back the cs, you will see that is the same result +we had for low-velocity charges. Á good way to see where the đs must go is to Fig. 26-5. The magnetic field near a +refer back to the force law, moving charge is v x E. [Compare with +Fig. 26-4.] +t+'=q(E+ox Bì). +You see that a velocity times the magnetic field has the same dimensions as an +electric ñeld. So the right-hand side of Eq. (26.9) must have a factor 1/cŸ: +b= _ (26.12) +Eor a slow-moving charge (0 < c), we can take for the Coulomb field; then +q ®xrT +b= "~¬ (26.13) +--- Trang 329 --- +This formula corresponds exactly to equations for the magnetic ñeld of a current +that we found in Section 14-7. +W© would like to point out, in passing, something interesting for you to think qe— —>~ qa +about. (We will come back to discuss it again later.) Imagine two protons with (a) ì +velocities at right angles, so that one will cross over the path of the other, but in +front of it, so they don”t collide. At some instant, their relative positions will be +as in Eig. 26-6(a). We look at the force on g¡ due to ga and vice versa. Ôn q2 +there is only the electric force from g¡, since gi makes no magnetic ñeld along its FL vi xBì +line of motion. Ôn gi, however, there is again the electric force but, in addition, xa | +a magnetic force, since it is moving in a -feld made by qs. The Íorces are as Mi đEc = F2 +drawn in Fig. 26-6(b). The electric forces on g¡ and gs are equal and opposite. 0) ai @B vạ +However, there is a sidewise (magnetic) force on gị œnd no sideuñse ƒorce on qa. +Does action not equal reaction? We leave it for you to worry about. +Fig. 26-6. The forces between two mov- +26-3 Relativistic transformation of the ñelds Mu charges are not always equal and oppo- +site. lt appears that “action” ¡is not equal to +In the last section we calculated the electric and magnetic fñelds from the “reaction.” +transformed potentials. "Phe fields are mmportant, oŸ course, in spite of the +arguments given earlier that there is physical meaning and reality to the potentials. +'The fields, too, are real. It would be convenient for many purposes to have a way +to compute the fñelds in a moving system if you already know the fñelds in some +“rest” system. We have the transformation laws for ó and Á, because Á„ is a +four-vector. Now we would like to know the transformation laws of E and Ö. +Given # and in one frame, how do they look in another Íframe moving past? lt +is a convenient transformation to have. We could always work back through the +potentials, but it is useful sometimes to be able to transform the fields directly. +W©e will now see how that goes. +How can we fnd the transformation laws of the fields? We know the transfor- +mation laws of the ø and A, and we know how the fields are given in terms of ở +and A—it should be easy to find the transformation for the and #. (You might +think that with every vector there should be something to make it a four-vector, +so with # there”s got to be something else we can use for the fourth component. +And also for Ö. But it)s not so. It?s quite diferent from what you would expect.) +To begin with, let°s take just a magnetic field Ö, which is, of course V x A. Now +we know that the vector potential with its z-, -, and z-components is only a +piece of something; there is also a f-component. Also we know that for derivatives +like V, besides the z, , z parts, there is also a derivative with respect to ‡. So +let's try to ñgure out what happens if we replace a “” by a “f”, or a “z” by a “,” +or something like that. +First, notice the form of the terms in V x 4 when we write out the components: +""..¬..ẽ.ẽ. ẽ.ẽ +Øụ Øz Øz 3z 3z Øụ +The z-component is equal to a couple of terms that involve only ø- and z- +components. Suppose we call this combination of derivatives and components a +“zu-thing,” and give it a shorthand name, #z„. We simply mean that +_ ĐA, QÁy +Tu = Đụ 2” (26.15) +Similarly, „ is equal to the same kind of “thing,” but this tỉme it is an “zz-thing” +And Ö; ¡s, of course, the corresponding “yz-thing” We have +By = Fỳụ, Bụ = F„., By = HLụz. (26.16) +Now what happens if we simply try to concoct also some “f”-type things like +T„¿ and Fị; (since nature should be nice and symmetric in #, , z, and £)? Eor +instance, what 1s F;¿;? It is, of course, +3A, ÔA, +Ôz ÔtL ` +--- Trang 330 --- +But remember that 4; = ở, so it is also +9ó ÔAz +9z ỐtL ` +You've seen that before. It is the z-component of #/. Well, almost—there is a +sign wrong. But we forgot that in the four-dimensional gradient the f-derivative +comes with the opposite sign from z, , and z. So we should really have taken +the more consistent extension oŸ l;;, as +8A; ÔA, +Hự„y = +” 26.17 +0z + Øt ) +Then it is exactly equal to —;. Trying also F;¿„ and ¿„, we fnd that the three +possibilities give +H„ =—E„, Ty = —Ey, Hy =—E,. (26.18) +What happens ïif both subscripts are #? Or, for that matter, if both are z? +We get things like +3A, ÔA; +Tụ =———nng› +3A 3A +đà ===— —› +which give nothing but zero. +W©e have then six of these #-things. 'There are six more which you get by +reversing the subscripts, but they give nothing really new, since +đTòy — —Fưz, +and so on. 5o, out oŸ sixteen possible combinations oŸ the four subscripts taken +in pairs, we get only six diferent physical obJects; and the are the componen‡s +øƒ B and E. +'To represent the general term of #', we will use the general subscripts and 1, +where each can stand for 0, 1, 2, or 3—meaning in our usual four-vector notation +È, ø, , and z. Also, everything will be consistent with our four-vector notation 1Ý Table 26-1 +we defne F;„ by +Fụy — W.A, —— WAu, (26.19) The components of È',,„;„ +remembering that V, = (9/؇,—9/9z,—0/Øụ, —Ø/9z) and that A„ = (ở, A„, Ây, Fyy = —Eụy +What we have found is that there are six quantities that belong together in Tu, = 0 +nature—that are diferent aspects of the same thing. The electrie and magnetic Fyuu=—B, — Fạ„¿= E, +fñelds which we have considered as separate vectors in our slow-moving world +(where we don” worry about the speed of light) are not vectors in four-space. tụz==Hz Hạ = Eụ +They are parts of a new “thing” Our physical “ñeld” is really the six-component F¿„ =—B, F..,=E, +object #;„„. That is the way we must look at it for relativity. We summarize our +results on #;„ in Table 26-1. +You see that what we have done here 1s to generalize the cross product. We +began with the curl operation, and the fact that the transformation properties +of the curl are the same as the transformation properties of #o vectors—the +ordinary three-dimensional vector Á and the gradient operator which we know +also behaves like a vector. Let's look for a moment at an ordinary cross product +in three dimensions, for example, the angular momentum of a particle. When an +object is moving in a plane, the quantity (uy — 0„) is important. For motion +in three dimensions, there are three such important quantities, which we call the +angular momentum: +Tuy = m(®0y — 0y), ly = Tm(WU; — Z0y), T;y„ —= (Z0, ��� #0;). +Then (although you may have forgotten by now) we discovered in Chapter 20 +of Vol. I the miracle that these three quantities could be identifed with the +--- Trang 331 --- +components of a vector. In order to do so, we had to make an artifcial rule with +a right-hand convention. It was just luck. It was luck because ;; (with ¿ and j7 +cqual to #, , or z) was an antisymmetrie object: +TỦ = —hj¡, l¿ = Ú. +Of the nine possible quantities, there are only three independent numbers. And +1t just happens that when you change coordinate systems these three objects +transform in exactly the same way as the components of a vector. +The same thing lets us represent an element of surface as a vector. Á surface +element has two parts—say đz+ and dụ —which we can represent by the vector đœ +normal to the surface. But we can't do that in four dimensions. What is the +“normal” to dz+ đụ? Is it along z or along f? +In short, for three dimensions it happens by luck that after you ve taken a +combination of two vectors like Ù¿;, you can represent it again by another vector +because there are Jjust three terms that happen to transform like the components +of a vector. But in four dimensions that is evidently impossible, because there +are six independent terms, and you can't represent six things by four things. +ven in three dimensions it is possible to have combinations of vectors that +can”t be represented by vectors. Suppose we take any two vectors œ = (đ„, đụ, đ;) +and b = (b„,b„,b„), and make the various possible combinations of components, +like ø„b„, a„b„, etc. There would be nine possible quantities: +đ„Đạ, q„Ðu, đ„Dz„, +đ, bự, quD„, quD„ l +dzb„, az„b„, daxbz. +'We might call these quantities 7¿;. +TÝ we now go to a rotated coordinate system (say rotated about the z-axis), +the components of œ and b are changed. In the new system, a„x, for example, +gets replaced by +d„ — dạ cOs + d„ sỉn 0, +and b„ gets replaced by +by = bạ cos Ø — bạ sìn 0. +And similarly for other components. The nine components of the product quan- +tity T¿; we have invented are all changed too, of course. For instance, 7x = ø„Ðb„ +gets changed to +Ty = a„by(cos” 9) — a„b„(cos Øsỉn 8) + aub„(sin Ø cos 9) — aub„(sin” 6), +Ty = Ty cos” Ø — 77„ cos Øsỉn Ø + T„„ sỉn Ø eos Ø — 7„ sin” 0. +Each component of 77, is a linear combination oŸ the components oŸ Tả;. +So we discover that it is not only possible to have a “vector product” like œ x b +which has three components that transform like a vector, but we can—artificially—— +also make another kind of “product” of two vectors 17;; with nne components +that transform under a rotation by a complicated set of rules that we could ñgure +out. Such an object which has two indices to describe it, instead of one, is called +a fensor. TW 1s a tensor of the “second rank,” because you can play this game +with three vectors too and get a tensor of the third rank—or with four, to get a +tensor oŸ the fourth rank, and so on. À tensor of the first rank is a vector. +The poïint of all this is that our electromagnetic quantity #,„ is also a tensor +of the second rank, because it has two indices in it. It is, however, a tensor in +four dimensions. ÏIt transforms in a special way which we will work out in a +moment-—it is just the way a product of vectors transforms. For #;,„ it happens +that if you change the indices around, #;„ changes sign. That's a special case——it +is an antisummetric tensor. 5o we say: the electric and magnetic felds are both +part of an antisymmetric tensor of the second rank in four dimensions. +--- Trang 332 --- +You ve come a long way. Remember way back when we delñned what a velocity +meant? Now we are talking about “an antisymmetric tensor of the second rank +in four dimensions.” +Now we have to fnd the law of the transformation of H„. lb isn't at all +difficult to do; it”s just laborious—the brains involved are nïl, but the work is +not. What we want is the Lorentz transformation of V„A„ — Vụ 4„. Since VỤ +is Just a special case oŸ a vector, we will work with the general antisymmetric +vector combination, which we can call ŒG„„: +Guu = dub„ — qubụ,. (26.20) +(Eor our purposes, ø„ will eventually be replaced by V„ and ö„ will be replaced +by the potenial A„.) The components of z„ and b„ transform by the Lorentz +formulas, which are +..ha.. — Ủy — UÙy, +£ 1— 2 — ụ2 , £ 1—g2 — Ụ2 +m — đạ„ — Đũ‡ bự — Ủy — ĐŨ¡ +°VI-=u” vi? (26.21) +dụ = qy, lủp = bụ, +d; = dạ, b =Ùb¿. +Now let's transform the components of Œ,„„. VWe start with G„„: +Gi„ = a(Ù,, — a„Ù +1E) ) Em) +v1—ˆ2 v]— 2 v1—ˆ2 w1- 02 += dÖ„ — dựby. +But that is just G¿x; so we have the simple result +Gì, — Gụ. +W©e will do one more. +G= dị — 08, — : by — ĐÙy — (abụ — ayb¿) — 0(a„b„ — ayb��) +#- V1-w ” “vI-—w2 v1—0? l +So we get that +Œ — Gụụ — UỚ„yw +W v1—ˆ2 +And, of course, in the same way, +leu — G;„ — UGz +tz 4_— „2 — ụ2 +lt is clear how the rest will go. Let”s make a table of all six terms; only now we +may as well write them for Fj„: +Tyu — ĐUứt +Tị, = đầu Ty TW và +đị — UF, +fụ= =S ' đực ấm (26.22) +pha Tz— UÍáz Ƒ — FzxT— 0Èại +EÓ VICu S8 vì cu +Of course, we still have J„ = —F7„„ and FJ„ = 0. +--- Trang 333 --- +So we have the transformation of the electric and magnetic fields. All we have +to do is look at Table 26-1 to find out what our grand notation in terms of F¿„„ +means in terms of # and #Ö. It's just a matter of substitution. So that we can +see how it looks in the ordinary symbols, we'll rewrite our transformation of the +fñeld components in Table 26-2. +Table 26-2 +The Lorentz transformation of the electric and magnetic 8elds (Note: c —= 1) +T„ = E„ Đ,=bB, +..Ặ Eu— 0B: Bỉ _ Đụ + 0È: +“ v1=0w2 “ v1=02 +.._ Pz + 0B, H _ Ð: ~ 0y +: v1i—02 ữ v1i—02 +The equations in 'Table 26-2 tell us how # and ?Ö change if we go from one +inertial frame to another. If we know # and Ö in one system, we can fnd what +they are in another that moves by with the speed 0. +W© can write these equations in a form that is easier to remember iŸ we notice +that since 0 is in the z-direction, all the terms with 0 are components of the cross +products 0 x and ø x Ö. So we can rewrite the transformations as shown In +Table 26-3. +Table 26-3 +An alternative form for the 8eld transformations (Note: c — 1) +E„, = Eu B, = B, +œ— (E+ox B), pg._ (B-oxE), +” v1 — 02 ⁄ v1 — 02 +._ (E+oxB): g._(B-0xE): +ữ v1—+2 : v1—- 2 +lt is now easier to remember which components go where. In fact, the +transformation can be written even more simply if we defne the fñield components +along ø as the “parallel” eomponents Ej and BỊ (because they are parallel to +the relative velocity of 9 and Š”), and the total transverse components—the +vector sums of the ø- and z-components—as the “perpendicular” components +T#¡ and Bị. Then we get the equations in Table 26-4. (WSe have also put back +the đs, so i9 will be more convenient when we want to refer back later.) +Table 26-4 +Still another form for the Lorentz transformation of and +HỊ = EỊ BỊ = BỊ +( p_.9 S) +.— (E+oxB). B.= Œ -L +: A⁄1— 02/2 : A⁄1— 02/2 +The fñeld transformations give us another way of solving some problems we +have done before—for Instance, for finding the fields of a moving point charge. +W©e have worked out the fñelds before by diferentiating the potentials. But we +could now do it by transforming the Coulomb ñeld. If we have a point charge at +rest in the S-frame, then there is only the simple radial E-field. In the S”-frame +we will see a point charge moving with the velocity œ, If the S”-frame moves by the +--- Trang 334 --- +S-frame with the speed 0 = —u. We will let you show that the transformations +of Tables 26-3 and 26-4 give the same electric and magnetic fields we got in +Section 26-2. +The transformation of Table 26-2 gives us an interesting and simple answer for +what we see if we move past am system of fixed charges. For example, suppose +we want to know the fields in ouz frame 5” if we are moving along between the +plates of a condenser, as shown in Eig. 26-7. (It is, of course, the same thing +1Ý we say that a charged condenser is moving past 1s.) What do we see? The +transformation is easy in this case because the jÖ-feld in the original system + |†+ |7 + |+ |7 +1s zero. Suppose, first, that our motion is perpendicular to #; then we will see E +an E' = E/1-— %2/c2 which is still completely transverse. We will see, in +addition, a magnetic field B” = —ò x E/c?. (The w/1 — 02/c2 doesn't appear in || +our formula for because we wrote it in terms of #“ rather than #; but it's the =—-= L—~ =—-—__—~ +same thing.) So when we move along perpendicular to a static electric fñeld, we , +see a reduced # and an added transverse Ö. Tf our motion is not perpendicular _ Llg. 26-7. The coordinate frame 5 mov- +to E, we break into E and E¡. The parallel part is unchanged, lì = PI; Ing through a statlc electric field. +and the perpendicular component does as just described. +Let”s take the opposite case, and imagine we are moving through a pure static +magnetic feld. 'This tìme we would see an elecfrie fñeld E7 equal to ø x B/, and +the magnetic fñeld changed by the facbor 1/4/1 — 02/c2 (assuming it is transverse). +So long as 0 is much less than c, we can neglect the change in the magnetic fñeld, +and the main efect is that an electric fñield appears. Ás one example of this efect, +consider this once famous problem of determining the speed of an airplane. It's no +longer famous, since radar can now be used to determine the air speed from ground +reflections, but for many years it was very hard to ñnd the speed oŸ an airplane in +bad weather. You could not see the ground and you didn't know which way was +up, and so on. Yet iÿ was important to know how fast you were moving relative to +the earth. How can this be done without seeing the earth? Many who knew the +transformation formulas thought of the idea of using the fact that the airplane +moves in the magnetic feld of the earth. Suppose that an airplane is ying where +there is a magnetic fñeld more or less known. Let's just take the simple case +where the magnetic fñeld is vertical. If we were fying through it with a horizontal +velocity , then, according to our formula, we should see an electric fñeld which +1s 0 x , ¡.e., perpendicular to the line of motion. If we hang an insulated wire +across the airplane, this electric fñeld will induce charges on the ends of the wire. +'That is nothing new. From the point of view of someone on the ground, we are +moving a wire through a feld, and the ø x #Ö force causes charges to move to +the ends of the wire. 'Phe transformation equations just say the same thing in +a diferent way. (The fact that we can say the thing more than one way doesn't +mean that one way is better than another. We are gctting so many diferent +methods and tools that we can usually get the same result in 65 diferent waysl) +So to measure 0, all we have to do is measure the voltage between the ends of +the wire. We can't do it with a voltmeter because the same fields will act on the +wires in the voltmeter, but there are ways of measuring such fields. We talked +about some of them when we discussed atmospherie electricity in Chapter 9. So +19 should be possible to measure the speed of the airplane. +This important problem was, however, never solved this way. The reason 1s +that the electric fñeld that is developed is of the order of millivolts per meter. +It is possible to measure such fields, but the trouble 1s that these fields are, +unfortunately, not any diferent from any other electric ñelds. 'Phe fñeld that is +produced by motion through the magnetic fñeld can't be distinguished from some +electric field that was already in the air tom another cause, say from electrostatic +charges in the aïr, or on the clouds. We described in Chapter 9 that there are, +typically, electric felds above the surface of the earth with strengths of about +100 volts per meter. But they are quite irregular. So as the airplane fies through +the air, it sees Ñuctuations of atmospherie electric fñelds which are enormous ïn +comparison to the tiny felds produced by the x Ö term, and ït turns out for +practical reasons to be impossible to measure speeds of an airplane by its motion +through the earth's magnetic feld. +--- Trang 335 --- +26-4 The equations of motion ỉn relativistic notation* +Tt doesn't do much good to fnd electric and magnetic ñelds rom Maxwells +cquations unless we know what the fñelds do when we have them. You may +remember that the fñelds are required to fnd the forces on charges, and that +those forces determine the motion of the charge. So, of course, part of the theory +of electrodynamics is the relation between the motion of charges and the Íorces. +For a single charge in the felds and ?Ö, the force is +t=q(E+ox Đ). (26.23) +This force is equal to the mass times the acceleration for low velocities, but +the correct law for any velocity is that the force is equal to dp/dt. Writing +Ð = mo/vw1— 02/c2, we ñnd that the relativistically correct equation of motion +d Tnrọˆ ) +—| ———_ =F =u(E+ùx Đ). 26.24 +di cm. ) C649 +'W©e would like now to discuss this equation from the point of view of relativ- +1ty. Since we have put our Maxwell equations in relativistic form, it would be +interesting to see what the equations of motion would look like in relativistic +form. Let”s see whether we can rewrite the equation in a four-vector notation. +W© know that the momentum is part of a four-vector ø„ whose tỉme component +is the energy ?noc2/4/1 — 02/c2. So we might think to replace the left-hand side +of Eq. (26.24) by dp„/df. Then we need only ñnd a fourth component to go +with #'. This fourth component must equal the rate-of-change of the energy, or +the rate of doing work, which is f“-ø. We would then like to write the right-hand +side of Eq. (26.24) as a four-vector like (E'- 0,>,Fy,F;). But this does not +make a four-vector. +The £Zme derivative of a four-vector is no longer a four-vector, because the d/đự +requires the choice of some special ame for measuring ý. We got into that trouble +before when we tried to make ø into a four-vector. Our first guess was that the +tỉme component would be cd£/đ# = c. But the quantities +dz dụ dz +—:—›:——l|(œ°® 26.25 +“.n. (26.35) +are no the components of a four-vector. We found that they could be made into +one by multiplying each component by 1/4/1— 02/c2. The “four-velocity” u„, is +the four-vector +( = ° ) (26.26) +tu = | —=———=—=: ————— |: : +š v1I—02/c2 wW1-— 02/c +So it appears that the trick is to multiply đ/đ£ by 1/4/1— 02/c2, if we want the +derivatives to make a four-vector. +Our second guess then is that +— 26.27 +should be a four-vector. But what is 0? It is the velocity of the particle—not of +a coordinate framel Then the quantity ƒ„ deñned by +F'.0 + +đụ = —= — Ea.) (26.28) +vVI—-03/c2. V1-— 02/c +1s the extension into four dimensions of a force—we can call it the “four-force.” lt +is Indeed a four-vector, and its space components are not the components of Ƒ* +but of F//1— 02/2. +The question is—why is ƒ„ a four-vector? It would be nice to get a little +understanding of that 1/4/1— 02/c2 factor. Since it has come up twice now, iÈ +* In this section we will put back all of the €'s +--- Trang 336 --- +1s tìme to see why the đ/đ# can always be ñxed by the same factor. The answer +is in the following: When we take the time derivative oŸ some function #, we +compute the increment Az in a small interval A# in the variable ý. But in another +frame, the interval At might correspond to a change in both #ˆ and z', so IÝ we +vary only #, the change in # will be diferent. We have to find a variable for our +diferentiation that is a measure of an “interval” in space-time, which will then +be the same in all coordinate systems. When we take Az for that interval, it will +be the same for all coordinate frames. When a particle “moves” in four-space, +there are the changes A¿, Az, A2, Az. Can we make an invariant interval out of +them? Well, they are the components of the four-vecbor #„ = (cf, +, , z) sO iŸ we +defñne a quantity As by +SN: | ,2A¿2 2 2 2 +(As)“ = = Azu„Az„ = =(cˆAf“ = Az2 = Awˆ = A2) (26.29) +——which is a four-dimensional dot product—we then have a good four-scalar to +ulse as a measure of a four-dimensiona] interval. Erom As——or its limit đs——we +can delne a parameter s = ƒ ds. And a derivative with respect to s, đ/ds, is a +mice four-dimensional operation, because it is invariant with respect to a Lorentz +transformation. +lt is easy to relate ds to d for a moving particle. Eor a moving point particle, +da = 0„ đt, đụ = 0ụ đt, dz = 0; dt, (26.30) +ds = \( (df2/c?)(cŸ — 0ệ — 02 — 0) = d1 — 02/cŸ, (26.31) +So the operator +v1— 02/c3 đf +1s an #nwuariœn‡ operator. ]Ý we operate on any four-vector with it, we get another +four-vector. For instance, iŸ we operate on (c, z, , 2), we get the four-velocity tụ: +—— — tt. +W© see now why the factor 4⁄1 — 02/c2 fixes things up. +The invariant variable s is a useful physical quantity. It is called the “proper +time” along the path of a particle, because đs is always an Iinterval of time +in a frame that is moving with the particle at any particular instant. (Then, +Az = Ay= Az=0,and As = A(.) TÝ you can imagine some “clock” whose rate +doesn't depend on the acceleration, such a clock carried along with the partiele +would show the tỉme s. +W© can now go back and write Newton”s law (as correcbed by Einstein) in +the neat form : +——= 26.32 +ds l ( ) +where /„ is given in Eq. (26.28). Also, the momentum ø„ can be written as +Đụ Tgt = TRọ CT—: (26.33) +where the coordinates #„ = (cứ, ø,, z) now describe the trajectory of the particle. +Pinally, the four-dimensional notation gives us this very simple form of the +equations of motion: +— d2, 26.34 +J„ — nọ _ds2 2 (26.34) +which is reminiscent of #' = ma. It is important to notice that Eq. (26.34) is nof +the same as #' = ma, because the four-vector formula Eq. (26.34) has in it the +relativistic mechanics which are different from Newton”s law for high velocities. +Tt is unlike the case of Maxwell's equations, where we were able to rewrite the +--- Trang 337 --- +equations in the relativistic form t+0#hout an change ím the meaning at aÌÌ—but +with just a change of notation. +Now let”s return to Eq. (26.24) and see how we can write the right-hand side in +four-vector notation. The three components—when divided by 4⁄1 — 02/c?—are +the components oŸ ƒ„, SO +ƒ —=“—= Ặăẽ="..an-. căn. (96.35) +` v1— 12/c2 V1i—-12/2 1—%2/c2 1—02/c2] +NÑow we must put all quantities in their relativistic notation. First, e/4/1 — 02/c2 +and 0/v/1— 02/c2 and 0;/ 1— 02/c? are the í-, -, and z-components of the +four-velocity u„. And the components of # and Ö are components of the second- +rank tensor of the fields F„. Looking back in Table 26-1 for the components +of #„ that correspond to #„, Ö;, and , we getŠ +t — q(uyF>¡ — thụ Fụu -= wzF>;), +which begins to look interesting. Every term has the subscript z, which is +reasonable, since we re finding an z-component. 'Phen all the others appear in +palirs: #‡, , zz—except that the zz-term is missing. So we just stick ït in, and +y = q(uiF>ị( — ty F>„ — 0y F>y — 0y F„„). (26.36) +We haven't changed anything because #?„ is antisymmetric, and ÈFzx„ is zero. +The reason for wanting to put in the #z-term is so that we can write Eq. (26.36) +in the short-hand form +f„ = quu„F}„. (26.37) +Thịỉs equation is the same as q. (26.36) if we make the ruÏe that whenever any +subscript occurs #ữee (as does here), you automatically sum over terms in the +same way as for the scalar product, usứng the same conuenlion [or the sign4. +You can easily believe that (26.37) works equally well for = or = z, but +what about = #? Let's see, for fun, what it says: +t!ằ= q(u¿1 — MựF‡„ — thụ Ứtụ — 0y Et„). +Now we have to translate back to 7s and ”s. We get +Uy Uụ Uy +=q|{0+———— '; + ————— !, =r-LỒ) 26.38 +⁄ ( V1I—02/© ”Ô W1—0u2/c© ” V1-u2/2 7 ( ) +qu-: E +Jì = ———=ằ.: +v1— 02/c2 +But from Eq. (26.28), ƒ; is supposed to be +t‹o —qE+uxB)-o +v1— 12/c2 V1i—-02/c2 - +This is the same thing as Eq. (26.38), since (0 x ) - 0 is zero. So everything +comes out all right. +Summarizing, our equation of motion can be written in the elegant form +mọ mm = „ = qu„F„. (26.39) +Although it is nice to see that the equations can be written that way, this form is +not particularly useful. It's usually more convenient to solve for particle motions +by using the original equations (26.24), and that's what we will usually do. +* When we put the đs back in Table 26-1, all components of Ï?„„, corresponding to +components of # are multiplied by 1/e. +--- Trang 338 --- +Mioldl FEreorgạy «rtel Fiolcl Wortt©rtftrrtt +27-1 Local conservation +Tt is clear that the energy of matter is not conserved. When an object radiates 27-1 Local conservation +light it loses energy. However, the energy lost is possibly describable in some 27-2 Energy conservation and +other form, say in the light. Therefore the theory of the conservation of energy electromagnetism +is incomplete without a consideration of the energy which is associated with 27-3 Energy density and energy fow +the light or, in general, with the electromagnetic ñeld. We take up now the law in the electromagnetic field +of conservation of energy and, also, oŸ momentum for the fields. Certainly, we ¬- +cannot treat one withoat the other, because in the relativity theory they are 27-4 The ambigulty of the ñeld energy +diferent aspects of the same four-vector. 27-5 Examples of energy flow +Very early in Volume I, we discussed the conservation of energy; we said then 27-6 Eield momentum +merely that the total energy in the world is constant. NÑow we want to extend +the idea of the energy conservation law in an important way—in a way that +says something in đe#ai about hou energy is conserved. The new law will say +that iŸ energy goes away Írom a region, iÈ is because i ƒfious away through the +boundaries of that region. It is a somewhat stronger law than the conservation +of energy without such a restriction. +'To see what the statement means, let°s look at how the law of the conservation +of charge works. We described the conservation of charge by saying that there is +a current density 7 and a charge density ø, and that when the charge decreases +a% some place there must be a fow of charge away from that place. We call that +the conservation of charge. The mathematical form of the conservation law is +V:7= DTẾ (27.1) +TThis law has the consequence that the tobal charge in the world is always constant— °) @) +there is never any net gain or loss of charge. However, the total charge in the world ⁄ 2 +could be constant in another way. Suppose that there is some charge Œ near 2 Z +some point (1) while there is no charge near some point (2) some distance away G@ Q +(Fig. 27-1). Ñow suppose that, as time goes on, the charge Q¡ were to gradually &) +fade away and that sữnultanecousiu with the decrease of Q some charge Q+ would +appear near point (2), and in such a way that at every instant the sum of Q1 +and Qs was a constant. In other words, at any intermediate state the amount of +charge lost by Q¡ would be added to Q¿. Then the total amount of charge in ⁄ +the world would be conserved. 'That°s a “world-wide” conservation, but not what T7 : Z2 +we will call a “local” conservation, because in order for the charge to get from ⁄ , ⁄ +(1) to (2), it didn't have to appear anywhere in the space between point (1) and @ & +point (2). Locally, the charge was Just “lost.” +'There ¡is a dificulty with such a “world-wide” conservation law ¡in the theory of (Œ) +relativity. The concept of “simultaneous moments” at distant points is one which : +. . l . l . Fig. 27-1. Two ways to conserve charge: +1s not equivalent in diferent systems. 'wo events that are simultaneous in one (a) Q¡ + Q; is constant; (b) dQ¡/dt = +system are not simultaneous for another system moving past. For “world-wide” ~ ƒj-nda = ~dQ›/dt. : +conservation oŸ the kind described, it is necessary that the charge lost from +should appear simultanecousiu in Q2. Otherwise there would be some moments +when the charge was not conserved. “There seems to be no way to make the +law of charge conservation relativistically invariant without making it a “local” +conservation law. As a matter of fact, the requirement of the Lorentz relativistic +invariance seems to restrict the possible laws of nature in surprising ways. In +modern quantum field theory, for example, people have often wanted to alter the +theory by allowing what we call a “nonlocal” interaction—where something here +--- Trang 339 --- +has a direct efect on something #here—but we get in trouble with the relativity +principle. +“Local” conservation involves another idea. It says that a charge can get +from one place to another only if there is something happening in the space +between. To describe the law we need not only the density of charge, ø, but also +another kind of quantity, namely 7, a vector giving the rate of low of charge +across a surface. Then the ñow ¡s related to the rate of change of the density +by E4q. (27.1). Thịs is the more extreme kind of a conservation law. It says that +charge is conserved in a special way——conserved “locally.” +lt turns out that energy conservation is also a local process. There is not only +an energy density in a given region of space but also a vector to represent the +rate of fow of the energy through a surface. For example, when a light source +radiates, we can fñnd the light energy moving out from the source. If we imagine +some mathematical surface surrounding the light source, the energy lost from +inside the surface is equal to the energy that fows out through the surface. +27-2 Energy conservation and electromagnetism +We want now to write quantitatively the conservation of energy for electro- +magnetism. 'To do that, we have to describe how much energy there is in any +volume element of space, and also the rate of energy flow. Suppose we think +ñrst only of the electromagnetic fñeld energy. We will let œ represent the energ +đensity 1n the field (that is, the amount of energy per unit volume in space) and +let the vector 9 represent the energ fluz of the field (that is, the fow of energy +per unit tỉme across a unit area perpendicular to the fow). Then, in perfect +analogy with the conservation of charge, Eq. (27.1), we can write the “local” law +of energy conservation in the fñeld as +Diên V.S. (27.2) +Of course, this law is not true in general; it is not true that the feld energy is +conserved. Suppose you are in a dark room and then turn on the light switch. AlI +öŸ a sudden the room is full of light, so there is energy in the fñeld, although there +wasn”t any energy there before. Equation (27.2) is not the complete conservation +law, because the field energy aÏone is not conserved, only the total energy in the +world—there is also the energy of matter. The fñeld energy will change if there is +some work being done by matter on the fñield or by the field on matter. +However, If there is matter inside the volume of interest, we know how much +energy it has: Each particle has the energy ?noc2/v/1 — 02/c2. The total energy +of the matter is Just the sum of all the particle energies, and the ñow of this +energy through a surface is just the sum of the energy carried by each particle +that crosses the surface. We want now to talk only about the energy of the +electromagnetic field. So we must write an equation which says that the total +ƒield energy in a given volume decreases e#her because fñeld energy fÑows out oŸ +the volume ør because the field loses energy to matter (or gains energy, which is +Just a negative loss). The field energy inside a volume V is +J udV, +and its rate of decrease is minus the time derivative of this integral. The fow of +fñeld energy out of the volume V is the integral of the normal component of S +over the surface 3 that encloses V, +l S-nda. +-xjJ udV = l S -mt da + (work done on matter inside V). (27.3) +--- Trang 340 --- +We have seen before that the fñeld does work on each unit volume oŸ matter +at the rate # - 7. [The force on a particle is E' = q(E + o x B), and the rate of +doïng work is È- —g-o. lf there are Ñ particles per unit volume, the rate +of doiïng work per unit volume is Mq#-ø, but Nựou = 7.] So the quantity E - 7 +must be equal to the loss of energy per unit time and per unit volume ủ the +fñeld. Equation (27.3) then becomes +—— | udV= | S-nda+ | E-17dV. (27.4) +dt Jv » V +This is our conservation law for energy in the field. We can convert it inbo a +điferential equation like Eq. (27.2) if we can change the second term to a volune +integral. hat is easy to do with Gauss' theorem. “The surface integral of the +normal component of Š is the integral of its divergence over the volume inside. +So Bq. (27.3) is equivalent to +— | TT dV= | V-SdV+ | E-7dV, +v Ời V V +where we have put the time derivative of the first term inside the integral. 5ince +this equation is true for any volume, we can take away the integrals and we have +the energy equation for the electromagnetic fields: +—S =V-S+E'J. (27.5) +Now this equation doesn't do us a bịt of good unless we know what and ®S +are. Perhaps we should just tell you what they are in terms of E and #Ö, because +all we really want is the result. However, we would rather show you the kind of +argument that was used by Poynting in 1884 to obtain formulas for Š and ö, +So you can see where they come from. (You won't, however, need to learn this +derivation for our later work.) +27-3 Energy density and energy fow in the electromagnetic field +The idea is to suppose that there is a feld energy density and a ñux S +that depend only upon the fñelds and Ö. (Eor example, we know that in +electrostatics, at least, the energy density can be written seo - #.) Of course, +the u and Š might depend on the potentials or something else, but let”s see what +we can work out. We can try to rewrite the quantity #/- 7 in such a way that it +becomes the sum of two terms: one that is the time derivative oŸ one quantity +and another that is the divergence of a second quantity. The frst quantity would +then be ¡ and the second would be 5 (with suitable signs). Both quantities musb +be written in terms of the fields only; that is, we want to write our equality as +t-?=-—-—=-—V-S. 27.6 +J=— (27.6) +The left-hand side must first be expressed in terms of the felds only. How can +we do that? By using Maxwell's equations, of course. From Maxwells equation +for the curl of Ö, += VxbBb-‹«ạ—... +Mj €ọC €0 ôt +Substituting this in (27.6) we will have only E?s and %5: +E-Jj=cocE:(VxB)- cọạE: 2r” (27.7) +W©S are already partly ñnished. “The last term is a time derivative—it is +(0/0(5eof- E). So seo - E is at least one part of u. It's the same thỉng we +found in electrostatics. Now, all we have to do is to make the other term into +the divergence of something. +--- Trang 341 --- +Notice that the frst term on the right-hand side oŸ (27.7) is the same as +(Vx B):E. (27.8) +And, as you know from vector algebra, (œ x b) - e is the same as ø - ( x €); sO +our term is also the same as +W:(BxE) (27.9) +and we have the divergence of “something,” just as we wanted. Ônly that”s wrongl +W©e warned you before that V is “like” a vector, but not “exactly” the same. 'Phe +reason it is not is because there is an additional conuention from calculus: when +a derivative operator is in Íront of a produect, it works on everything to the right. +In Eq. (27.7), the W operates only on Ö, not on . But in the form (27.9), the +normal convention would say that W operates on both Ö and #. 5o its no¿ the +same thing. In fact, if we work out the components of V - ( x E) we can see +that it is equal to E-(W x ) pius some other terms. Its like what happens +when we take a derivative of a product in algebra. For instance, +đ đƒ dụ +— = -— + ——., +1x9) =0 +Rather than working out all the components of V - ( x E), we would like +to show you a trick that is very useful for this kind of problem. It is a trick +that allows you to use all the rules of vector algebra on expressions with the +V operator, without getting into trouble. The trick is to throw out——for a while +at least——the rule of the calculus notation about what the derivative operator +works on. You see, ordinarily, the order of terms is used for #wøo separate Durposes. +One is for calculus: ƒ(đ/dz+)g is not the same as ø(đ/d+) ƒ; and the other is for +vectors: œ x b is diferent from b x ø. We can, If we want, choose to abandon +mmomentarily the calculus rule. Instead of saying that a derivative operates on +everything to the right, we make a øeu rule that doesnt depend on the order +in which terms are written down. “hen we can juggle terms around without +WOITying. +Here is our new convention: we show, by a subscript, what a diferential +operator works on; the order has no meaning. Suppose we let the operator l) +stand for Ø/Øz. Then 7Ï); means that only the derivative of the variable quantity ƒ +is taken. hen ôƒ +DrƑƒ ==—_-. +But if we have J)r ƒg, it means +D =[|a- l0. +rJg ( 2x ) g +But notice now that according to our new rule, ƒD¿ø means the same thing. We +can write the same thing any which way: +Drfg = gD¡ƒ = ƒDịg= fgÐi. +You see, the 2; can even come øƒ#er everything. (Its surprising that such a +handy notation is never taught in books on mathematics or physics.) +You may wonder: What if Ï uan£ to write the derivative of ƒg? I uanm£ the +derivative of bo£h terms. That”s easy, you just say so; you write D¿(ƒø)+ Dg(ƒ9). +That is just g(0ƒ/9z) + ƒ(Øg/9+), which is what you mean in the old notation +by 0(g)/Øz. +You will see that it is now goïing to be very easy to work out a new expression +for V-(B x E). We start by changing to the new notation; we write +WV:(BxẮE)=Vp-(BxĂẮE)+Vr-(Bx E). (27.10) +'The moment we do that we don'ˆt have to keep the order straight any more. WWe +always know that Ÿg operates on # only, and pg operates on Ö only. In these +--- Trang 342 --- +circumstances, we can use V as though it were an ordinary vector. (Of course, +when we are fnished, we will want to return to the “standard” notation that +everybody usually uses.) So now we can do the various things like interchanging +dots and crosses and making other kinds of rearrangements of the terms. FOor +instance, the middle term of Eq. (27.10) can be rewritten as - W x Ö. (You +remember that œ-bxœ= b-ex ø.) And the last term is the same as - E x Vp. +Tt looks freakish, but it is all right. Now If we try to go back to the ordinary +convenftion, we have to arrange that the V operates only on its “own” variable. +The frst one is already that way, so we can just leave of the subscript. The +second one needs some rearranging to put the V in front of the #, which we can +do by reversing the cross product and changing sign: +B-(ExÄVe) =—B-(Vg x E). +Now it is in a conventional order, so we can return to the usual notation. Equa- +tion (27.10) is equivalent to +WV:.(BxẮE)=E-(VxbB)—-B.(VxE'). (27.11) +(A quicker way would have been to use components in this special case, but it +was worth taking the time to show you the mathematical trick. You probably +wont see it anywhere else, and it is very good for unlocking vector algebra from +the rules about the order of terms with derivatives.) +W©e now return to our energy conservation discussion and use our new result, +Eq. (27.11), to transform the W x Ö term of Eq. (27.7). That energy equation +becomes +E-j=cạcV-(BxE)+cạcB-(VxE)— ar(eoÐ -#). (27.12) +NÑow you see, we re almost fñnished. We have one term which is a nice derivative +with respect to ý to use for œ and another that ¡is a beautiful divergence to +represent ,Š. Ủnfortunately, there is the center term left over, which is neither +a divergence nor a derivative with respect to ý. So we almost made it, but not +quite. After some thought, we look back at the diferential equations of Maxwell +and discover that W x is, fortunately, equal to —Ø/ðt, which means that we +can turn the extra term into something that is a pure time derivative: +9B 3(/B:bB +B-(VxE)=bB-|-._Ì=--..|--—]- +Now we have exactly what we want. Qur energy equation reads +. 3 Ô (oc2 €0 +E-7=V-(cạ¿c£Bx E)- —| —B-bB¬--_E-E]|. (27.13) +®%\_ 2 2 +which is exactly like Eq. (27.6), If we make the definitions +u=S B.E+ TC B-B (27.14) +S=cạcEx B. (27.15) +(Reversing the cross product makes the signs come out right.) +Our program was successful. We have an expression for the energy density +that is the sum of an “electric” energy density and a “magnetic” energy density, +whose forms are just like the ones we found in statics uhen ue tuorked out the +cnergu ïn terms oƒ the felds. Also, we have found a formula for the energy flow +vector of the electromagnetic ñeld. This new vector, 9 = egc?2E x Ö, is called +“Poynting's vector,” after its discoverer. It tells us the rate at which the field +energy moves around in space. 'Phe energy which flows through a small area. da +per second is Š -? da, where ?w is the unit vector perpendicular to da. (Ñow that +we have our formulas for and ,Š, you can forget the derivations iŸ you want.) +--- Trang 343 --- +27-4 The ambiguity of the field energy +Before we take up some applications of the Poynting formulas [Eqs. (27.14) +and (27.15)], we would like to say that we have not really “proved” them. AII +we did was to fnd a øoss?ble “u” and a possible “S” How do we know that by +Jjuggling the terms around some more we couldn”$ ñnd another formula for “w” +and another formula for “Š”? "The new ,Š and the new would be diferent, +but they would still satisfy Eq. (27.6). It's possible. It can be done, but the +forms that have been found always involve various đer?øfiues of the ñeld (and +always with second-order terms like a second derivative or the square of a frst +derivative). There are, in fact, an infinite number of diferent possibilities for +and , and so far no one has thought of an experimental way to tell which one is +rightl People have guessed that the simplest one is probably the correct one, but +we must say that we do not know for certain what is the acbual location in space +of the electromagnetic field energy. So we too will take the easy way out and say +that the field energy is given by Eq. (27.14). Then the fow vector S must be +given by Eq. (27.15). +]t is interesting that there seems to be no unique way to resolve the indefnite- +ness in the location oŸ the field energy. It is sometimes claimed that this problem +can be resolved by using the theory oŸ gravitation in the following argument. In +the theory of gravity, all energy is the source of gravitational attraction. 'Pherefore +the energy density of electricity must be located properly 1Ÿ we are to know in +which direction the gravity force acts. As yet, however, no one has done such +a delicate experiment that the precise location of the gravitational inÑuence of +electromagnetic fñelds could be determined. That electromagnetic fields alone +can be the source of gravitational force is an idea i% is hard to do without. It has, +in fact, been observed that light is delected as it passes near the sun—we could +say that the sun pulls the light down toward it. Do you not want to allow that +the light pulls equally on the sun? Anyway, everyone always accepts the simple +expressions we have found for the location of electromagnetic energy and its ñow. +And although sometimes the results obtained from using them seem strange, +nobody has ever found anything wrong with them——that is, no disagreement with +experiment. So we will follow the rest of the world——besides, we believe that it is +probably perfectly right. +We should make one further remark about the energy formula. In the fñrst +place, the energy per unit volume in the field is very simple: It is the electrostatic +energy plus the magnetic energy, ¿' we write the electrostatic energy in terms +of E2 and the magnetic energy as 2. We found two such expressions as øossible +expressions for the energy when we were doïng static problems. We also found +a number of other formulas for the energy in the electrostatic field, such as øở, +which is egual to the integral of # - E in the electrostatic case. However, in an +electrodynamic fñeld the equality failed, and there was no obvious choice as tO +which was the right one. NÑow we know which is the right one. S5imilarly, we have +found the formula for the magnetic energy that is correct in general. The right +formula for the energy density oŸ dựngamm¿e fields is Eq. (27.14). +27-5 Examples of energy fÑow E +Our formula for the energy fow vector Š is something quite new. We want +now to see how it works in some special cases and also to see whether it checks +out with anything that we knew before. The frst example we will take is light. In S +a light wave we have an # vector and a Ö vector at right angles to each other and +to the direction of the wave propagation. (See Eig. 27-2.) In an electromagnetic ⁄ +wave, the magnitude of Ö ¡s equal to 1/e times the magnitude oŸ E, and since B DIRECTION OF WAVE +they are at right angles, PROPAGATION +IE * BỊ — [ Fig. 27-2. The vectors E, B, and S for +C a light wave. +'Therefore, for light, the fow of energy per unit area per second is +S8 = cạcE°. (27.16) +--- Trang 344 --- +Eor a light wave in which # = Fo cosœ(£ — #/c), the average rate of energy flow +per unit area, (5)av—which is called the “intensity” of the light—is the mean +value of the square of the electric feld times cọc: +Intensity = (S)¿v = coc(E?)„v. (27.17) +Believe it or not, we have already derived this result in Section 31-5 of Vol. l, +when we were studying light. We can believe that it is right because it also checks +against something else. When we have a light beam, there is an energy density +in space given by Eq. (27.14). Using cÖ = E for a light wave, we get that +€0 „ma , co (E 2 +=—È“+ —-|-cj |] =coŸ.. +u= + 2 ( P ) €0 +But # varies in space, so the average energy density is +(1)av — co(E®)„v. (27.18) +Now the wave travels at the speed c, so we should think that the energy that +goes through a square meter in a second is c times the amount oŸ energy in one +cubic meter. 5o we would say that +(9)av = coc(E®),v. +And iÊ”s right; ¡ is the same as Eq. (27.17). I +Now we take another example. Here is a rather curious one. We look at the l +energy fow in a capacibor that we are charging slowly. (We don” want Írequencies | +so hiph that the capacitor is beginning to look like a resonant cavity, but we +don” want DƠ either.) Suppose we use a circular parallel plate capacitor oŸ our ï +usual kind, as shown in Fig. 27-3. 'Phere is a nearly uniform electric ñeld inside +which is changing with time. At any instant the total electromagnetic energy ¬...., +Inside is œ times the volume. TỶ the plates have a radius œ and a separation h, \ `x +the total energy between the plates is +_ ( Ê0 m2 2 +U= (§z )m h). (27.19) NC Z/ +This energy changes when # changes. When the capacitor is beïng charged, the H +volume between the plates is receiving energy at the rate Ủ+ +dUỮ 2 . Fig. 27-3. Near a charging capacitor, the +rn = ca “hEb. (27.20) Poynting vector Š points inward toward the +So there must be a fow of energy into that volume from somewhere. OÝ course +you know that it must come in on the charging wires—not at alll It can't enter +the space between the plates from that direction, because # is perpendicular to +the plates; # x Ö must be øarailel to the plates. +You remember, of course, that there is a magnetic feld that circles around +the axis when the capacitor is charging. We discussed that in Chapter 23. Ủsing +the last of Maxwells equations, we found that the magnetic feld at the edge of +the capacitor is given by +2xac?2B = E - naẺ, +B=—-F. +Its direction is shown in Fig. 27-3. So there is an energy ow proportional +to E x that comes in all around the edges, as shown in the fñigure. The +energy isn't actually coming down the wires, but from the space surrounding the +capacItor. +Let”s check whether or not the total amount of ñow through the whole surface +between the edges of the plates checks with the rate of change of the energy +--- Trang 345 --- +inside—it had better; we went through all that work proving Eq. (27.15) to make +sure, but let's see. The area of the surface is 2rah, and 9 = coc2E x B isin +magnitude +cọc? (5 £) , +so the total Ñux of energy is +ma ˆhegE2E. ° +It does check with Eq. (27.20). But it tells us a peculiar thing: that when we are +charging a capacitor, the energy is not coming down the wires; i is coming in +through the edges of the gap. That's what this theory saysl ————————— +How can that be? That's no‡ an easy question, but here is one way of thinking +about it. Suppose that we had some charges above and below the capacitor TT +and far away. When the charges are far away, there is a weak but enormousÌy +spread-out fñeld that surrounds the capacitor. (See Eig. 27-4.) Then, as the +charges come together, the field gets stronger nearer t%o the capacitor. So the VN +ñeld energy which is way out moves toward the capacitor and eventually ends up +between the plates. +As another example, we ask what happens in a piece oŸ resistance wire when +1b is carrying a current. Since the wire has resistance, there is an electric field +along it, driving the current. Because there is a potential drop along the wire, : : : +there is also an electric fñeld just outside the wire, parallel to the surface. (See Hlg. 27-4. The fields outside 2 capacItor +Hig. 27-5.) There is, in addition, a magnetic field which goes around the wire chan đo s larde detanee Dringing bwo +because of the current. The # and #Ö are at right angles; therefore there is a : +Poynting vector directed radially inward, as shown in the ñgure. There is a Ñow +of energy into the wire all around. It is, of course, equal to the energy being lost +in the wire in the form of heat. 5o our “crazy” theory says that the electrons are +getting their energy to generate heat because of the energy Ñowing into the wire +from the fñeld outside. Intuition would seem to tell us that the electrons get their +energy from being pushed along the wire, so the energy should be fowing down +(or up) along the wire. But the theory says that the electrons are really being +pushed by an electric field, which has come from some charges very Íar away, +and that the electrons get their energy for generating heat from these fields. The h h +energy somehow fows from the distant charges into a wide area of space and + „ +then inward to the wire. : 5 '— k +Pinally, in order to really convince you that this theory is obviously nuts, +we wiïll take one more example—an example in which an electric charge and a +magnet are ø‡ resử near each other——both sitting quite still. Suppose we take the +example of a point charge sitting near the center of a bar magnet, as shown in +Fig. 27-6. Ðverything is at rest, so the energy is not changing with time. Also, ++; and are quite static. But the Poynting vector says that there is a fow of Fig. 27-5. The Poynting vector S near a +energy, because there is an x that is not zero. If you look at the energy Wire Carrying a current. +fow, you fnd that it just circulates around and around. There isn't any change +in the energy anywhere—everything which Ñows into one volume flows out again. +Tt is like incompressible water owing around. So there is a circulation of energy +in this so-called static condition. How absurd it getsl ST : +Perhaps it isnˆ so terribly puzzling, though, when you remember that what we SN /Z~ +called a “static” magnet is really a circulating permanent current. Ïn a permanent == Í B +magnet the electrons are spinning permanently inside. So maybe a circulation of +the energy outside isn't so queer after all. “SN +You no doubt begin to get the impression that the Poynting theory at least s +partially violates your intuition as to where energy is located in an electromagnetiec +field. You might believe that you must revamp all your intuitions, and, therefore Fig. 27-6. A charge and a magnet pro- +have a lot of things to study here. But it seems really not necessary. You dont — đuce a Poynting vector that circulates in +need to feel that you will be in great trouble if you forget once in a while that closed loops. +the energy in a wire is Ñowing into the wire from the outside, rather than along +the wire. It seems to be only rarely of value, when using the idea of energy +conservation, to notice in detail what path the energy is taking. The circulation +of energy around a magnet and a charge seems, in most circumstances, to be quite +--- Trang 346 --- +unimportant. lt is not a vital detail, but it is clear that our ordinary intuitions +are qulte wrong. +27-6 Field momentum +Next we would like to talk about the mornentưm ïn the electromagnetic fñeld. +Just as the field has energy, it will have a certain momentum per unit volume. +Let us call that momentum density g. Of course, momentum has various possible +directions, so that g must be a vector. Let's talk about one component at a time; +first, we take the z-component. 5ince each component of momentum is conserved +we should be able to write down a law that looks something like this: +Ø (momentumì\ _ Øz momentum +— Ø£ ( of matter ). _— ( outflow )- +The left side is easy. The rate-of-change of the momentum of matter is just the +force on it. For a particle, it is E! = gq(E + o x B); for a distribution of charges, +the force per unit volume is (øE + ÿ x Ö). The “momentum outfow” term, +however, is strange. It cannot be the divergence of a vector because it is not a +scalar; it is, rather, an #-component of some vector. Anyway, it should probably +look something like +9x 0b Ôc +0z 0y 0z) +because the z-momentum could be flowing in any one of the three directions. +In any case, whatever ø, Ò, and e are, the combination is supposed to equal the +outow of the z-momentum. +Now the game would be to write ø + 7 x B in terms only of E and B—— +eliminating ø and 7 by using Maxwells equations—and then to juggle terms and +make substitutions to get it into a form that looks like +Øgy Ôa Ôb_ Ôc +0E 0x 0y 0z +'Then, by identifying terms, we would have expressions for ø„, ø, Ò, and c. lt's a +lot of work, and we are not going to do it. Instead, we are only going to ñnd an +expression for g, the momentum density——and by a diferent route. +There is an Important theorem in mechanics which is this: whenever there +is a fow oŸ energy in any circumstance at all (fñeld energy or any other kind of +energy), the energy flowing through a unit area per unit từme, when multiplied +by 1/2, is equal to the momentum per unit volume in the space. In the special +case of electrodynamies, this theorem gives the result that g is 1/c2 tỉimes the +Poynting vector: +g= = S. (27.21) +So the Poynting vector gives not only energy flow but, if you đivide by e2, also +the momentum density. 'Phe same result would come out of the other analysis +we suggested, but it is more interesting to notice this more general result. We +will now give a number of interesting exarmples and arguments to convince you +that the general theorem is true. +First example: Suppose that we have a lot of particles in a box——let”s say +per cubic meter—and that they are moving along with some velocity 0. Now +let's consider an imaginary plane surface perpendicular to ø. 'Phe energy flow +through a unit area of this surface per second is equal to /oø, the number which +fow through the surface per second, times the energy carried by each one. The +energy in each particle is moc2/4/1 — 02/c2. So the energy flow per second is +v1— 032/c2 +--- Trang 347 --- +But the momentum of each particle is no0/4/1 — 02/c2, so the đensifg oŸ mo- +1nentum 1s mạp +N "ma. .^` +V1— 05/2 +which is just 1/c2 tỉimes the energy fow—as the theorem says. So the theorem is +true for a bunch of particles. +lt is also true for light. When we studied light in Volume I, we saw that when +the energy is absorbed from a light beam, a certain amount of momentum is +delivered to the absorber. We have, in fact, shown in Chapter 3⁄44 of Vol. I that +the momentum is 1/c times the energy absorbed [Eq. (3.24) of Vol. I]. If we +let Ủo be the energy arriving at a unit area per second, then the momentum +arriving at a unit area per second is Ứe/c. But the momentum is travelling at +the speed e, so its densiy in front of the absorber must be Ứo/c?. So again the +theorem is right. A]"¬>—____ +Pinally we will give an argument due to Einstein which demonstrates the +same thỉng once more. Suppose that we have a railroad car on wheels (assumed +frictionless) with a certain big mass Ä⁄. At one end there is a device which will U +shoot out some particles or light (or anything, it doesnˆt make any diference what +1t is), which are then stopped at the opposite end of the car. There was some +energy originally at one end—say the energy indicated in Eig. 27-7(a)—and +then later it is at the opposite end, as shown in Fig. 27-7(c). The energy has (o) M Co) | +been displaced the distance L, the length of the car. Now the energy has (a) +the mass Ứ/e, so if the car stayed still, the center of gravity of the car would | +be moved. Einstein didn”$ like the idea that the center of gravity of an object +could be moved by fooling around only on the inside, so he assumed that it is R Ị +impossible to move the center oŸ gravity by doing anything inside. But if that is = | +the case, when we moved the energy from one end to the other, the whole car - +must have recoiled some distance #, as shown in part (c) of the ñgure. You can ụ | +see, in fact, that the total mass of the car, times z, must equal the mass of the +energy moved, /c2 tỉimes Ù (assuming that U/c2 is much less than M): co) Mã co) | +Max = _ h. (27.22) (b) | +Let's now look at the special case of the energy being carried by a light flash. | +(The argument would work as well for particles, but we will follow Einstein, +who was interested in the problem of light.) What causes the car to be moved? U Ì +tHinstein argued as follows: When the light is emitted there must be a recoil, | +some unknown recoll with momentum ø. lt is this recoil which makes the car +roll backward. "The recoil velocity ø of the car will be this momentum divided by Ị +the mass of the car: p Co) (o) -x ¬ +U—= M_ (c) +The car moves with this velocity until the light energy gets to the opposite Fig. 27-7. The energy U in motion at the +end. 'Then, when it hits, 1E 81VCS back its momentum and StODS the car. lÍ ø is speed c carries the momentum U/c. +small, then the time the car moves is nearly equal to Ù/œ; so we have that +, h p h +m.—.—-= +Putting this z in Eq. (27.22), we get that +Again we have the relation of energy and momentum for light. Dividing by e to +get the momentum density øg = p/c, we get once more that +g= ¬ (27.23) +You may well wonder: What is so important about the center-of-gravity +theorem? Maybe ? is wrong. Perhaps, but then we would also lose the con- +servation of angular momentum. Suppose that our boxcar is moving along a +--- Trang 348 --- +track at some speed ø and that we shoot some light energy from the #øp to the +bottom of the car—say, from A to in Eig. 27-8. Now we look at the angular +momentum of the system about the point . Before the energy leaves A, it has +the mass rm = Ư/c2 and the velocity 0, so it has the angular momentum 074. +'When it arrives at , it has the same mass and, if the iZzeør momentum of the +whole boxcar is not to change, it must still have the velocity 0. It's angular mo- +mentum about ?? is then mrpg. The angular momentum will be changed wøless +the right recoil momentum was given to the car when the light was emitted—that +is, unless the light carries the momentum Ù/c. I% turns out that the angular n +mmomentum conservation and the theorem of center-of-gravity are closely related +in the relativity theory. 5o the conservation of angular momentum would also ø ìc —v> +be destroyed i1f our theorem were not true. Â% any rate, it does turn out to be ``ứ +a true general law, and in the case of electrodynamics we can use i% to get the B +mmomentum in the feld. +W© will mention two further examples of momentum in the electromagnetic Co) co) +fñeld. We pointed out in Section 26-2 the failure of the law of action and reaction rA +when ÿwo charged particles were moving on orthogonal trajectories. The forces rp +on the two particles donˆt balance out, so the action and reaction are not equal; +therefore the net momentum of the matter must be changing. It is not conserved. "mm. ^ +But the momentum in the fñeld is also changing in such a situation. lÝ you : +work out the amount of momentum given by the Poynting vector, i% is not Flg. 27-8. The energy Ư must carry the +constant. However, the change of the particle momenta is just made up by the momentum U/c ̓ the angular momentum +. . about P is to be conserved. +field momentum, so the total momentum of particles plus field is conserved. +Finally, another example is the situation with the magnet and the charge, +shown in Fig. 27-6. We were unhappy to fnd that energy was fowing around +in cireles, but now, since we know that energy fow and momentum are pro- +portional, we know also that there is momentum circulating in the space. But +a crculating momentum means that there is angular momentum. So there is +angular momentum in the fñeld. Do you remember the paradox we described in +Section 17-4 about a solenoid and some charges mounted on a disc? It seemed +that when the current turned of, the whole disc should start to turn. 'Phe puzzle +was: Where did the angular momentum come from? 'Phe answer is that if you +have a magnetic ñeld and some charges, there will be some angular momentum +in the fñeld. It must have been put there when the feld was built up. When +the fñeld is turned of, the angular momentum is given back. So the disc in the +paradox øould start rotating. This mystic circulating fow of energy, which at +frst seemed so ridiculous, is absolutely necessary. There is really a momentum +fow. It is needed to maintain the conservation of angular momentum in the +whole world. +--- Trang 349 --- +Mglocfrorntcrgraoffc JWVẪœss +28-1 The field energy of a poïint charge +In bringing together relativity and Maxwells equations, we have finished our 28-1 The field energy ofa poiỉnt charge +main work on the theory of electromagnetism. 'There are, of course, some details 28-2 The ñeld momentum of a moving +we have skipped over and one large area that we will be concerned with in the charge +future—the interaction of electromagnetic fields with matter. But we want to 28-3 Electromagnetic mass +stop for a moment to show you that this tremendous edifce, which is such a : +beautiful success in explaining so many phenomena, ultimately falls on its face. 28-4 The force of an electron øn iiself +'When you follow any of our physics too far, you ñnd that it always gets into some 28-ã Attempts to modify the Maxwell +kind of trouble. Now we want to discuss a serious trouble—the failure of the theory +classical electromagnetic theory. You can appreciate that there is a failure of all 28-6 The nuclear force field +classical physies because of the quantum-mechanical efects. Classical mechanics +is a mathematically consistent theory; it just doesn't agree with experience. +lt is interesting, though, that the classical theory of electromagnetism is an +unsatisfactory theory all by itself. "There are difficulties associated with the +zdeas of Maxwell*s theory which are not solved by and not directly associated +with quantum mechanics. You may say, “Perhaps there's no use worrying about +these difculties. 5ince the quantum mechanics is going to change the laws +of electrodynamics, we should wait to see what dificulties there are after the +modification.” However, when electromagnetism is joined to quantum mechanics, +the dificulties remain. So it will not be a waste of our time now to look at what +these difficulties are. Also, they are of great historical importance. Purthermore, +you may get some feeling of accomplishment from being able to go far enough +with the theory to see everything——including all of its troubles. +'The dificulty we speak of is associated with the concepts of electromagnetic +mmomentum and energy, when applied to the electron or any charged particle. The +concepts of simple charged particles and the electromagnetic feld are In some +way inconsistent. 'Io describe the dificulty, we begin by doïing some exercises +with our energy and momentum concepts. +Flirst, we compute the energy of a charged particle. Suppose we take a simple +model of an electron in which all of its charge g is uniformly distributed on the +surface of a sphere of radius ø, which we may take to be zero for the special case +of a point charge. Now let”s calculate the energy in the electromagnetic ñeld. If +the charge is standing still, there is no magnetic fñeld, and the energy per unit +volume is proportional to the square of the electric field. The magnitude of the +electric feld is g/4zeor2, and the energy density is +€0 m2 q7 +_—= h= 3272cgr1+` +To get the total energy, we must integrate this density over all space. sing the +volume element 4zr2 đr, the total energy, which we will call 2q„e, is +ae. = J H d. +'This 1s readily integrated. 'The lower limit is a, and the upper limit is oo, so +1 g2 1 +sliec — 3 đme a (28.1) +--- Trang 350 --- +Tf we use the electronic charge g. for g and the symbol e2 for g2/4zeo, then +is = Am (28.2) +Tlt is all fne until we set œø equal to zero for a point charge—there's the great +difculty. Because the energy of the fñeld varies inversely as the fourth power of +the distance from the center, its volume integral is inñnite. 'There is an infinite +amount of energy in the fñeld surrounding a point charge. +What's wrong with an infnite energy? If the energy can't get out, but must +stay there forever, is there any real dificulty with an infnite energy? OÝ course, +a quantity that comes out infinite may be annoying, but what really matters is +only whether there are any øbseruable physical efects. To answer that question, +we must turn to something else besides the energy. Suppose we ask how the +energy changes when we rnoue the charge. Then, if the chønges are infnite, we +will be in trouble. +28-2 The field momentum of a moving charge +Suppose an electron is moving at a uniform velocity through space, assuming +for a moment that the velocity is low compared with the speed oflight. Associated +with this moving electron there is a momentum——even ïf the electron had no +mass before 1t was charged——because of the momentum in the electromagnetic TH +fñeld. We can show that the fñeld momentum is in the direction of the velocity ® —. F +of the charge and is, for small veloeities, proportional to ø. For a point P at the ⁄l'Ð r ñ ' +distance z from the center of the charge and at the angle Ø with respect to the F=< - +. . . R - R > x | —4 Ị +line of motion (see Fig. 28-1) the electric field is radial and, as we have seen, the S.I.x +magnetic feld is ø x #/c?. The momentum density, Eq. (27.21), is $ +SPHERICAL +g= cọ x B. ELERON +Tt is directed obliquely toward the line of motion, as shown in the fñgure, and has Fig. 28-1. The fields E and B and the +the magnitude momentum density g for a positive electron. +g= = E2sin 9. For a negative electron, E and B are re- +C versed but g Is not. +The felds are symmetric about the line of motion, so when we integrate over +space, the transverse components will sum to zero, giving a resultant momentun +parallel to ø. 'Phe component of g in this direction is gsinØ, which we must +Integrate over all space. We take as our volume element a ring with its plane +perpendicular to , as shown in Fig. 2§-2. Its volume is 2mr2sinØ đ0dr. The +total momentum is then +p= J - E2 sin? Ø9 2mrŸ sin 0 d6 dr. r d6 +=:' ˆ / ` +Since #7 is independent of Ø (for < c), we can immediately integrate over 6; LTHErsn9 +the integral is +8 2 cos” Ø j +Jén 0 d0 = -Ja — cos” Ø) d(cos Ø) = — cos Ø + _ +và . . Fig. 28-2. The volume element +The limits of Ø are 0 and ø, so the Ø-integral gives merely a factor of 4/3, and 2mr2 sin 8 độ dr used for calculating the field +ÑT cụU J 3. momentum. += ��_—> | E“rˆdr. +B3 œ +The integral (for 0 < c) is the one we have just evaluated to ñnd the energy; it +is g2/162cáa, and +_— 2 q2? +nh» 4meg úc2` +=z—U. 28.3 +P=s (28.3) +--- Trang 351 --- +The momentum in the field——the electromagnetic momentum——is proportional +to 0. Ib is just what we should have for a particle with the mass equal to the +coefficient of ø. We can, therefore, call this coeficient the clecfromagnetic mmass, +m„¡ec, and write it as +Thelec — 3 ac2” (28.4) +28-3 blectromagnetic mass +'Where does the mass come from? In our laws of mechanics we have supposed +that every obJecb “carries” a thing we call the mass——which also means that I§ +“carries” a momentum proportional to its velocity. Now we discover that it is +understandable that a charged particle carries a momentum proportional to its +velocity. It might, in fact, be that the mass is Just the efect of electrodynamiecs. +The origin of mass has until now been unexplained. We have at last in the +theory of electrodynamics a grand opportunity to understand something that we +never understood before. lIt comes out of the blue—or rather, from Maxwell and +Poynting—that any charged particle will have a momentum proportional to its +velocity just from electromagnetic inẦuences. +Let's be conservative and say, for a moment, that there are ©wo kinds of +mmass—that the total momentum of an object could be the sum of a mechanical +mmomentum and the electromagnetic momentum. The mechanical momentum +1s the “mechanical” mass, mecu, tmes 0. ÏÍn experiments where we measure +the mass of a particle by seeing how much momentum it has, or how 1% swings +around in an orbit, we are measuring the total mass. We say generally that the +momentum is the total mass (7neeh + ?nølec) times the velocity. So the observed +mass can consist of 6wo pieces (or possibly more iŸ we include other fields): a +mechanical piece plus an electromagnetic piece. We know that there is deflnitely +an electromagnetic piece, and we have a formula for it. And there is the thrilling +possibility that the mechanical piece is not there at all—that the mass is all +electromagnetic. +Let's see what size the electron must have ïf there is to be no mechanical +mass. W©e can fnd out by setting the electromagnetic mass of Eq. (28.4) equal +to the observed mass ?=„ of an celectron. We find +¬—- (28.5) +'The quantity +is called the “classical electron radius”; it has the numerical value 2.82 x 10~13 em, +about one one-hundred-thousandth of the diameter of an atom. +'Why is rọ called the electron radius, rather than our ø? Because we could +equally well do the same calculation with other assumed distributions of charges—— +the charge might be spread uniformly through the volume of a sphere or it might be +smeared out like a fuzzy ball. For any particular assumption the factor 2/3 would +change to some other fraction. Eor instance, for a charge uniformly distributed +throughout the volume of a sphere, the 2/3 gets replaced by 4/5. Rather than +to argue over which distribution is correct, it was decided to defñne rọ as the +“nominal” radius. Then diferent theories could supply their pet coefficients. +Let”s pursue our electromagnetic theory ofmass. Our calculation was Íor 0 < Œ; +what happens if we go to high velocities? Early attempts led to a certain amount +of confusion, but Lorentz realized that the charged sphere would contract into a +ellipsoid at high velocities and that the felds would change in accordance with +the formulas (26.6) and (26.7) we derived for the relativistic case in Chapter 26. +T you carry through the integrals for ø in that case, you ñnd that for an arbitrary +velocity ®, the momentum is altered by the factor 1/4/1— 02/c2: +2. c? Đ +p= _= _ỬỦhm.- (28.7) +--- Trang 352 --- +In other words, the electromagnetic mass rises with velocity inversely as +v1— 02/c2—a discovery that was made before the theory of relativity. +lBarly experiments were proposed to measure the changes with velocity in +the observed mass of a particle in order to determine how much of the mass was +mechanical and how much was electrical. I9 was believed at the time that the +electrical part œould vary with velocity, whereas the mechanical part would no. +But while the experiments were being done, the theorists were also at work. Soon +the theory of relativity was developed, which proposed that no matter what the +origin of the mass, iE øÏÏ should vary as rnmo/4/1— 02/c2. Equation (28.7) was +the beginning of the theory that mass depended on velocity. +Let's now go back to our calculation of the energy in the fñeld, which led +to Eq. (28.2). According to the theory of relativity, the energy Ứ will have the +mass Ư/c?; Ðq. (28.2) then says that the feld of the electron should have the +mass U 1c +elec € +Ttlee = _t* = 2 qc2) (28.8) +which is not the same as the electromagnetic mass, ?n2Jec, of Eq. (28.4). In fact, +1ƒ we just combine Eqs. (28.2) and (2§.4), we would write +Uclcc — ". +This formula was discovered before relativity, and when Einstein and others +began to realize that it must always be that = rmc2, there was great confusion. +28-4 The force of an electron on itself +The discrepancy between the two formulas for the electromagnetic mass +1s especially annoying, because we have carefully proved that the theory of +electrodynamies is consistent with the principle of relativity. Yet the theory of +relativity Implies without question that the momentum must be the same as the +energy tỉmes 0/c2. So we are ỉn some kind of trouble; we must have made a +mistake. We did not make an algebraic mistake in our calculations, but we have +left something out. +In deriving our equations for energy and momentum, we assumed the conser- +vation laws. We assumed that all forces were taken into account and that any +work done and any momentum carried by other “nonelectrical” machinery was +included. Now 1ƒ we have a sphere of charge, the electrical forces are all repulsive +and an electron would tend to fy apart. Because the system has unbalanced +forces, we can get all kinds of errors in the laws relating energy and momen- +tum. To get a cons¿stent picture, we must imagine that something holds the +electron together. The charges must be held to the sphere by some kind of rubber +bands——something that keeps the charges from fying of. It was first pointed +out by Poincaré that the rubber bands—or whatever it is that holds the electron +together——must be included in the energy and momentum calculations. For this +reason the extra nonelectrical forces are also known by the more elegant name +“the Poincaré stresses.” If the extra forces are included in the calculations, the +masses obtained in 0wo ways are changed (in a way that depends on the detailed +assumptions). And the results are consistent with relativity; ¡.e., the mass that +comes out from the momentum calculation is the same as the one that comes +from the energy calculation. However, both of them contain #øo contributions: +an electromagnetic mass and contribution from the Poincaré stresses. Only when +the two are added together do we get a consistent theory. +lt is therefore impossible to get all the mass to be electromagnetic in the +way we hoped. It is not a legal theory If we have nothing but electrodynamics. +Something else has to be added. Whatever you call them——“rubber bands,” or +“Poincaré stresses,” or something else—there have to be other forces in nature to +make a consistent theory of this kind. +Clearly, as soon as we have to put forces on the inside of the electron, the +beauty of the whole idea begins to disappear. Things get very complicated. You +--- Trang 353 --- +would want to ask: How strong are the stresses? How does the electron shake? +Does it oscilate? What are all its internal properties? And so on. It might +be possible that an electron does have some complicated internal properties. +lÝ we made a theory of the electron along these lines, it would predict odd +properties, like modes of oscillation, which havenˆt apparently been observed. We +say “apparently” because we observe a lot of things in nature that still do not +make sense. We may someday ñnd out that one of the things we don” understand +today (for exarmnple, the muon) can, in fact, be explained as an oscillation of the +Poincaré stresses. It doesnt seem likely, but no one can say for sure. 'here +are so many things about fundamental particles that we still donˆt understand. +Anyway, the complex structure implied by this theory is undesirable, and the +attempt to explain all mass in terms of electromagnetism——at least in the way +we have described——has led to a blind alley. +W©e would like to think a little more about why we say we have a mass when +the momentum ¡in the field is proportional to the velocity. Easyl The mass is +the coefficient between momentum and velocity. But we can look at the mass in +another way: a particle has mass if you have to exert a force in order 0o accelerate +it. So it may help our understanding if we look a little more closely at where the +forces come from. How do we know that there has to be a force? Because we +have proved the law of the conservation of momentum for the felds. If we have a +charged particle and push on ït for awhile, there will be some momentum in the +electromagnetic field. Momentum must have been poured into the ñeld somehow. +Therefore there must have been a force pushing on the electron in order to get it +going—a force in addition to that required by its mechanical inertia, a force due +to its electromagnetic interaction. And there must be a corresponding force back +on the “pusher.” But where does that force come from? +_ = dF _= d2F _ = dF +— — / — — +— — \ A — — +(a) (@b) (c) +Fig. 28-3. The self-force on an accelerating electron is not zero because of the retardation. +(By dF we mean the force on a surface element da; by d?F we mean the force on the surface +element da„ from the charge on the surface element đaa. +The picture is something like this. We can think of the electron as a charged +sphere. When it is at rest, each piece of charge repels electrically each other piece, +but the forees all balance in pairs, so that there is no sœe£ force. [See Eig. 28-3(a).] +However, when the electron is being accelerated, the forces will no longer be in +balance because of the fact that the electromagnetic inÑuences take time to go +from one piece to another. Eor instance, the force on the piece œ in Fig. 28-3(b) +from a piece on the opposite side depends on the position of Ø at an earlier +time, as shown. Both the magnitude and direction of the force depend on the +motion of the charge. IÝ the charge is accelerating, the forces on various parÈs of +the electron might be as shown in Eig. 28-3(c). When all these forces are added +up, they don't cancel out. 'They would cancel for a uniform velocity, even though +1t looks at frst glance as though the retardation would give an unbalanced force +even for a uniform velocity. But it turns out that there is no net force unless the +electron is being accelerated. With acceleration, if we look at the forces between +the various parts of the electron, action and reaction are not exactly equal, and +the electron exerts a force ønw 2£sejƒf that tries to hold back the acceleration. l§ +holds itself back by its own bootstraps. +--- Trang 354 --- +lt is possible, but dificult, to calculate this self-reaction force; however, we +dont want to go into such an elaborate calculation here. We will tell you what the +result is for the special case of relatively uncomplicated motion in one dimension, +say ø. hen, the self-force can be written in a series. The first term in the series +depends on the acceleration #, the next term is proportional to #, and so on.* +The result is 2 3e? 2 +in g na thun (28.9) +where œ and + are numerical coeficients of the order of 1. "The coeficient œ +of the # term depends on what charge distribution is assumed; ïif the charge is +distributed uniformly on a sphere, then œ = 2/3. So there is a term, proportional +to the acceleration, which varies inversely as the radius ø of the electron and agrees +exactly with the value we got in Eq. (28.4) for rmajec. TỶ the charge distribution +is chosen to be diferent, so that œ is changed, the fraction 2/3 in Eq. (28.4) +would be changed in the same way. The term in # is ?ndependent of the assumed +radius ø, and also of the assumed distribution of the charge; its coeffcient is +ahuays 2/3. The next term is proportional to the radius ø, and its coeflicient + +depends on the charge distribution. You will notice that if we let the electron +radius ø go to zero, the last term (and all higher terms) will go to zero; the second +term remains constant, but the first term——the electromagnetic mass—goes tO +infinity. And we can see that the infnity arises because of the force of one part of +the electron on another——because we have allowed what is perhaps a silly thing, +the possibility of the “point” electron acting on itself. +28-5. Attempts to modify the Maxwell theory +W©e would like now to discuss how it might be possible to modify Maxwell”s +theory of electrodynamiecs so that the idea of an electron as a simple point charge +could be maintained. Many attempts have been made, and some of the theories +were even able to arrange things so that all the electron mass was electromagnetic. +But all of these theories have died. It is still interesting to discuss some of the +possibilities that have been suggested——to see the struggles of the human mỉnd. +W© started out our theory of electricity by talking about the interaction of +one charge with another. hen we made up a theory of these interacting charges +and ended up with a field theory. We believe it so mụuch that we allow it to tell +us about the force of one part of an electron on another. Perhaps the entire +dificulty is that electrons do not act on themselves; perhaps we are making too +great an extrapolation from the interaction of separate electrons to the idea that +an electron interacts with itself. 'Therefore some theories have been proposed +in which the possibility that an electron acts on itself is ruled out. Then there +is no longer the infinity due to the selEaction. Also, there is no longer any +electromagnetic mass associated with the particle; all the mass is back to being +mmechanical, but there are new difficulties in the theory. +We must say immediately that such theories require a modifcation of the +idea of the electromagnetic feld. You remember we said at the start that the +force on a particle at any poïint was determined by just two quantities—E and 8Ö. +Tf we abandon the “self-force” this can no longer be true, because 1f there is an +electron in a certain place, the force isn'$ given by the total and #Ö, but by +only those parts due to o¿her charges. So we have to keep track always of how +much of # and #Ö ¡s due to the charge on which you are calculating the force +and how much is due to the other charges. This makes the theory much more +elaborate, but it gets rid of the difficulty of the inñnity. +So we can, jƒ te tuanf to, say that there is no such thing as the electron acting +upon itself, and throw away the whole set of forces in Eq. (28.9). However, we +have then thrown away the baby with the bathl Because the second term in +Eaq. (28.9), the term in #, is needed. That force does something very defnite. +T you throw it away, youre in trouble again. When we accelerate a charge, +* W© are using the notation: # = dœ/dt, # = d2z/d2, # = d3+/diẺ, etc. +--- Trang 355 --- +1 radiates electromagnetic waves, so it loses energy. Therefore, to accelerate +a charge, we musf require more force than is required to accelerate a neutral +object of the same mass; otherwise energy wouldn”t be conserved. 'Phe rate at +which we do work on an accelerating charge must be equal to the rate of Ìoss +of energy by radiation. We have talked about this efect before—it is called the +radiation resistance. We still have to answer the question: Where does the extra +force, against which we must do this work, come from? When a big antenna is +radiating, the forces come from the inÑuence of one part of the antenna current +on another. For a single accelerating electron radiating into otherwise empty +space, there would seem to be only one place the force could come from——the +action of one part of the electron on another part. +W© found back in Chapter 32 of Vol. I that an oscillating charge radiates +energy at the rate 32a +đH 2c). (28.10) +dị 3 c3 +Let's see what we get for the rate of doing work øn an electron against the +bootstrap force of Eq. (28.9). The rate of work is the force times the velocity, +or F; 3V 2 x +Tản. an... (28.11) +The first term is proportional to đ#2/đf, and therefore just corresponds to the rate +of change of the kinetic energy smu2 associated with the electromagnetic mass. +The second term should correspond to the radiated power in Eq. (28.10). But it +is diferent. The discrepancy comes from the fact that the term in Bq. (28.11) is +generally true, whereas Eq. (28.10) is right only for an oscillating charge. We +can show that the two are equivalent if the motion of the charge is periodic. 'lo +do that, we rewrite the second term of Eq. (28.11) as +22 d 2c? „- +—g 2 0 +5 a8) +which is just an algebraic transformation. lf the motion of the electron is periodic, +the quantity ## returns periodically to the same value, so that if we take the +duerage of its time derivative, we get zero. The second term, however, is aÌlways +positive (is a square), so its average is also positive. Thhis term gives the net +work done and is just equal to Eq. (28.10). +The term in # of the bootstrap Íorce is required in order to have energy +conservation in radiating systems, and we can t throw it away. It was, in fact, +one of the triumphs of Lorentz to show that there is such a force and that ït +comes from the action of the electron on itself. We must believe in the idea of +the action of the electron on itself, and we øeed the term in z. The problem is +how we can get that term without getting the first term in Eq. (2§.9), which +gives all the trouble. We don”t know how. You see that the classical electron +theory has pushed itself into a tight corner. +There have been several other attempts to modify the laws in order to +straiphten the thing out. Ône way, proposed by Born and Infeld, is to change +the Maxwell equations in a complicated way so that they are no longer linear. +Then the electromagnetic energy and momentum can be made to come out ñnite. +But the laws they suggest predict phenomena which have never been observed. +Their theory also sufers from another dificulty we will come to later, which is +common to all the attempts to avoid the troubles we have described. +The following peculiar possibility was suggested by Dirac. He said: Let's +admit that an electron acbs on itself through the secøond term in Eq. (28.9) but +not throuph the frst. He then had an ingenious idea for getting rid of one but not +the other. Look, he said, we made a special assumption when we took only the +retardcd wawve solutions of Maxwells equatlions; If we were to take the aduanced +waves instead, we would get something diferent. The formula for the self-force +would be 2 x 2 +Án on thun (28.12) +--- Trang 356 --- +This equation is just like Eq. (28.9) except for the sign of the second term——and +some higher terms——of the series. [Changing from retarded to advanced waves is +Just changing the s¿øn of the delay which, it is not hard to see, is equivalent to +changing the sign of £ everywhere. The only efect on Eq. (28.9) is to change the +sien of all the odd time derivatives.| So, Dirac said, let?s make the new rule that +an electron acts on itself by one-half the đjƒerence oŸ the retarded and advanced +felds which it produces. The difference of Eqs. (28.9) and (28.12), divided by +two, is then +F'=—„->z+higher terms. +In all the higher terms, the radius ø appears to some positive power in the +numerator. 'Pherefore, when we go to the limit of a point charge, we get only the +one term——just what is needed. In this way, Dirac got the radiation resistance +force and none of the inertial forces. There is no electromagnetic mass, and the +classical theory is saved——but at the expense of an arbitrary assumption about +the self-force. +The arbitrariness of the extra assumption of Dirac was removed, to some +extent at least, by Wheeler and Feynman, who proposed a still stranger theory. +They suggest that point charges interact on with other charges, but that the +interaction is half through the advanced and half through the retarded waves. lt +turns out, most surprisingly, that in most situations you wont see any efects of +the advanced waves, but they do have the efect of producing just the radiation +reaction force. “The radiation resistance is no due to the electron acting on itself, +but from the following peculiar efect. When an electron is accelerated at the +time ý, it shakes all the other charges in the world at a iafer time £ = £ + r/c +(where r is the distance to the other charge), because of the zefarded waves. +But then these other charges react back on the original electron through their +aduanced waves, which will arrive at the time #”, equal to £ˆ mánus r/c, which is, oŸ +course, just ý. (They also react back with their retarded waves too, but that just +corresponds to the normal “reflected” waves.) The combination of the advanced +and retarded waves means that at the instant it is accelerated an oscillating +charge feels a force from all the charges that are “going to” absorb its radiated +waves. You see what tight knots people have gotten into in trying to get a theory +of the electronl +'W©']ll describe now still another kind of theory, to show the kind of things +that people think of when they are stuck. This is another modification of the +laws of electrodynamies, proposed by Bopp. You realize that once you decide +to change the equations of electromagnetism you can start anywhere you want. +You can change the force law for an electron, or you can change the Maxwell +cquations (as we saw in the examples we have described), or you can make a +change somewhere else. One possibility is to change the formulas that give the +potentials in terms of the charges and currents. One of our formulas has been +that the potentials at some poïnt are given by the current density (or charge) +at each other point at an earlier time. sing our four-vector notation for the +potentials, we write +A,(1,1) = — . “=Ặ- (28.13) +47cod T12 +Bopp's beautifully simple idea is that: Maybe the trouble is in the 1/z factor in +the integral. Suppose we were to start out by assuming only that the potential +at one point depends on the charge density at any other point as some function +of the distance between the points, say as ƒ(r1a). The total potential at point (1) +will then be given by the integral of 7„ times this function over all space: +A,(L,Ð = [23,+— na/e)fína) đi, +'Thats all. No diferential equation, nothing else. Well, one more thing. We also +ask that the result should be relativistically invariant. 5o by “distance” we should +--- Trang 357 --- +take the invariant “distance” between two points in space-time. 'This distance +squared (within a sign which doesnt matter) is +S12 = c( —tfs)Ÿ— r'o += c1 — tạ)? — (#1 — m2)? — lDn — 9a)? — (z1 — z2)Ÿ. (28.14) +So, for a relativistically invariant theory, we should take some function of the +magnitude of sa, or what is the same thing, some function of s24. So Bopps +theory is that F(s?) +A,(1,f1i)= [0.503 dỤ da. (28.15) +(The integral must, of course, be over the four-dimensional volume đứa da đụa đza.) +AII that remains is to choose a suitable function for #'. We assume only one +thing about #—that it is very smaill except when its argumenf is near Zero—§O +that a graph of ' would be a curve like the one in Eig. 28-4. lb is a narrow +spike with a fnite area centered at s2 = 0, and with a width which we can say is +roughly a2. W© can say, crudely, that when we calculate the potential at point (1), +only those points (2) produce any appreciable efect if s?¿ = cÊ(fq — fa)” — ra is +within +a2 of zero. We can indicate this by saying that #' is important only for +sa = c (hị — tạ)? — rịa +dŸ. (28.16) ò = +You can make iÿ more mathematical if you want to, but that”s the idea. +Now suppose that ø is very small in comparison with the size of ordinary +objects like motors, generators, and the like so that for normal problems ra 3 đ. +Then Eq. (28.16) says that charges contribute to the integral of Eq. (28.15) only 1 +when #ị — #¿ is in the small range ra +/ a3 ` +c{H — tạ) r2 +22 =a 1+->. NI +TỊa @®) +Since a2/r?s < 1, the square root can be approximated by 1 + a2/2r2s, so Fig. 28-4. The function F(s”) used in +the nonlocal theory of Bopp. +hạ -tạ= 2 1+ -112, #4. +€ 2r1s € 2r1s€ +What is the significance? This result says that the only #mes ta that are +important in the integral of 4, are those which differ from the time í¡, at which +we want the potential, by the delay r1s/c—with a negligible correction so long +as r1a >> ø. In other words, this theory of Bopp approaches the Maxwell theory— +so long as we are far away ữom any particular charge—in the sense that it gives +the retarded wave effects. +W© can, ín fact, see approximately what the integral of Eq. (28.15) is going +to give. lf we integrate first over ứa from —oo to +oo——keeping r¿ fxed—then +s‡s is also going to go from —oœ to +oo. The integral will all come from s”s in +a small interval of width Af£¿ = 2 x a2/2rae, centered at fq — r1a2/c. Say that +the funetion (s2) has the value at s2 = 0; then the integral over f¿ gÌves +approximately #7„Ai1s, or +KaŠ j„ +é T12 l +We should, of course, take the value oŸ 7„ at f¿ = fq — rias/e, so that Eq. (28.15) +becomes : +ta lu(2,t1 — Tia/€ +Au(ýH) = —— J ĐHỂnH = ngụ, +é T12 +TÝ we pick K = 1/4meoca2, we are right back to the retarded potential solution of +Maxwells equations——including automatically the 1/z dependencel And it all +came out of the simple proposition that the potential at one point in space-time +depends on the current density at all other points in space-time, but with a +--- Trang 358 --- +weighting factor that is some narrow function of the four-dimensional distance +between the two points. This theory again predicts a ñnite electromagnetic mass +for the electron, and the energy and mass have the right relation for the relativity +theory. They must, because the theory is relativistically invariant from the start, +and everything seems to be all right. +There is, however, one fundamental objection to this theory and to all the +other theories we have described. All particles we know obey the laws of quantum +mnechanies, so a quantum-mechanical modification of electrodynamies has to be +made. Light behaves like photons. It isnt 100 percent like the Maxwell theory. +So the electrodynamic theory has to be changed. We have already mentioned +that it might be a waste oŸ time to work so hard to straighten out the classical +theory, because it could turn out that in quantum electrodynamics the difficulties +will disappear or may be resolved in some other fashion. But the difficulties +do not disappear in quantum electrodynamics. Thhat is one of the reasons that +people have spent so much efort trying to straighten out the classical dificulties, +hoping that if they could straighten out the classical dificulty and #hen make +the quantum modifications, everything would be straightened out. The Maxwell +theory still has the dificulties after the quantum mechanics modifications are +The quantum efects do make some changes—the formula for the mass 1S +modified, and Planck's constant appears——but the answer still comes out infinite +unless you cut of an integration somehow—just as we had to stop the classical +integrals at r = a. And the answers depend on how you stop the integrals. We +cannot, unfortunately, demonstrate for you here that the dificulties are really +basically the same, because we have developed so little of the theory of quantum +mmechanics and even less of quantum electrodynamics. 5o you must just take our +word that the quantized theory of Maxwell's electrodynamies gives an infnite +mass for a point electron. +lt turns out, however, that nobody has ever succeeded in making, a seÏJ- +consistent quantum theory out oŸ an of the modifed theories. Born and Infeld”s +ideas have never been satisfactorily made into a quantum theory. The theories +with the advanced and retarded waves of Dirac, or of Wheeler and Feynman, +have never been made into a satisfactory quantum theory. The theory of Bopp +has never been made into a satisfactory quantum theory. So today, there is no +known solution to this problem. We do not know how to make a consistent +theory-—including the quantum mechanics—which does not produce an infÑnity +for the selfenergy of an electron, or any point charge. And at the same tỉme, +there is no satisfactory theory that describes a non-point charge. It's an unsolved +problem. +In case you are deciding to rush off to make a theory in which the action +of an electron on itself is cormpletely removed, so that electromagnetic mass is +no longer meaningful, and then to make a quantum theory of it, you should be +warned that you are certain to be in trouble. “There is defnite experimental +evidence of the existence of electromagnetic inertia—there is evidence that some +of the mass of charged particles is electromagnetie in origin. +lt used to be said in the older books that since Nature will obviously not +present us with two particles—one neutral and the other charged, but otherwise +the same——we will never be able to tell how much of the mass is electromagnetic +and how much is mechanical. But it turns out that Nature høs been kind enough +to present us with Just such objects, so that by comparing the observed mass of +the charged one with the observed mass of the neutral one, we can tell whether +there is any electromagnetic mass. For example, there are the neutrons and +protons. They interact with tremendous forces—the nuclear forces—whose origin +is unknown. However, as we have already described, the nuclear forces have one +remarkable property. So far as they are concerned, the neutron and proton are +exactly the same. "The clear forces between neutron and neutron, neutron and +proton, and proton and proton are all identical as far as we can tell. Only the +little electromagnetic forces are different; electrically the proton and neutron are +as diferent as night and day. This is just what we wanted. 'There are two particles, +--- Trang 359 --- +identical from the point of view of the strong interactions, but diferent electrically. +And they have a small diference in mass. 'The mass difference between the proton +and the neutron—expressed as the điference in the rest-energy zmc? in units of +MeV——is about 1.3 MeV, which is about 2.6 times the electron mass. 'Phe classical +theory would then predict a radius of about Š to 3 the classical electron radius, +or about 10~†3 em. Of course, one should really use the quantum theory, but by +some strange accident, all the constants—2zs and ψs, etc.—come out so that +the quantum theory gives roughly the same radius as the classical theory. "The +only trouble is that the siøn is wrongl The neutron is heawier than the proton. +Table 28-1 +Particle Masses +. Charge Mass Am! +n (neutron) 0 939.5 +p (proton) +1 938.2 | —1.3 +7 (-meson) 0 135.0 ++1 139.6 | +4.6 +K (K-meson) 0 497.8 ++1 493.9 | —3.9 +> (sigma) 0 1191.5 ++] 1189.4 —2.1 +—1 1196.0 +4.5 +1 Am = (mass of charged) — (mass of neutral). +Nature has also given us several other pairs——or triplets——of particles which +appear to be exactly the same except for their electrical charge. Thhey interact +with protons and neutrons, through the so-called “strong” interactions of the +nuclear forces. In such interactions, the particles of a given kind—say the - +mesons—behave in every way like one object ezcep‡ for their electrical charge. In +'Table 28-1 we give a list of such partieles, together with their measured masses. +The charged z-mesons—positive or negative—have a mass of 139.6 MeV, but +the neutral x-meson is 4.6 MeV lighter. We believe that this mass diference is +electromagnetic; it would correspond to a particle radius of 3 to 4 x 10~12 em. +You will see from the table that the mass diferences of the other particles are +usually of the same general size. +Now the size of these particles can be determined by other methods, for ¬- ¬— +instance by the diameters they appear to have in high-energy collisions. So the "_—--.--ẮẮẳẮẶ_Ắ +electromagnetic mass seems to be in general agreement with electromagnetic ¬ +theory, if we stop our integrals of the fñeld energy at the same radius obtained by " : " tư nNGG: +these other methods. 'Phat's why we believe that the diferences do represent vàn ` ` »" _ _ +electromagnetic mass. " +You are no doubt worried about the diferent signs of the mass differences ¬ PROTON +in the table. It is easy to see why the charged ones should be heavier than ¬ +the neutral ones. But what about those pairs like the proton and the neutron, +where the measured mass comes out the other way? Well, it turns out that FÍg. 28-5. Á neutron may exist, at times, +these particles are complicated, and the computation of the electromagnetic mass 2S a proton surrounded by a negatlve 7- +must be more elaborate for them. For instance, although the neutron has no 0œ meson, +charge, it does have a charge distribution inside it—it is only the ne# charge that +1s zero. In fact, we believe that the neutron looks——at least sometimes——like a +proton with a negative r-meson in a “cloud” around 1$, as shown in Eig. 28-5. +Although the neutron is “neutral,” because its total charge is zero, there are still +electromagnetic energies (for exarmnple, it has a magnetic moment), so it”s no +easy to tell the sign of the electromagnetic mass difference without a detailed +theory of the internal structure. +--- Trang 360 --- +W© only wish to emphasize here the following points: (1) the electromagnetic +theory predicts the existence of an electromagnetie mass, but it also falls on its +face in doïing so, because it does not produce a consistent theory——and the same +is true with the quantum modifications; (2) there is experimental evidence for +the existence of electromagnetic mass; and (3) all these masses are roughly the +same as the mass of an electron. 5o we come back again to the original idea of +Lorentz—maybe all the mass of an electron is purely electromagnetic, maybe the +whole 0.511 MeV ¡s due to electrodynamies. Ïs it or isn't it? We haven't got a +theory, so we cannot say. +W© must mention one more piece of information, which is the most annoying. +There is another particle in the world called a zmuon—or /imeson—which, so far +as we can tell, difers in no way whatsoever from an electron except Íor its mass. +lt acts in every way like an electron: it interacts with neutrinos and with the +electromagnetic field, and it has no nuclear forces. It does nothing diferent from +what an electron does—at least, nothing which cannot be understood as merely +a consequence of its higher mass (206.77 times the electron mass). "Therefore, +whenever someone finally gets the explanation of the mass of an electron, he will +then have the puzzle of where a muon gets its mass. Why? Because whatever the +electron does, the muon does the same——so the mass ought to come out the same. +There are those who believe faithfully in the idea that the muon and the electron +are the same particle and that, in the ñnal theory of the mass, the formula for the +mass will be a quadratic equation with two roots——one for each particle. 'There +are also those who propose it will be a transcendental equation with an infnite +number of roots, and who are engaged in guessing what the masses of the other +particles in the series must be, and why these particles haven”t been discovered +28-6 The nuclear force field +We would like to make some further remarks about the part of the mass +of nuclear particles that is not electromagnetic. Where does this other large +fraction come from? There are other forces besides electrodynamics—like nuclear +forces—that have their own field theories, although no one knows whether the +current theories are right. These theories also predict a fñeld energy which gives +the nuclear particles a mass term analogous to electromagnetic mass; we could +call it the “m-mesic-field-mass.” It is presumably very large, because the forces +are great, and it is the possible origin of the mass of the heavy particles. But +the meson field theories are still in a most rudimentary state. Even with the +well-developed theory of electromagnetism, we found it impossible to get beyond +first base in explaining the electron mass. With the theory of the mesons, we +strike out. +We may take a moment to outline the theory of the mesons, because of its +interesting connection with electrodynamics. In electrodynamics, the field can +be described in terms of a four-potential that satisies the equation +L].A„ = sources. +Now we have seen that pieces of the field can be radiated away so that they +exist separated from the sources. These are the photons of light, and they are +described by a diferential equation without sources: +L”A„ =0. +People have argued that the field of nuclear forces ought also to have its own +“photons”—they would presumably be the -mesons—and that they should be +described by an analogous diferential equation. (Because of the weakness of the +human brain, we can't think of something really new; so we argue by analogy +with what we know.) So the meson equation might be +Ll¿ =0, +--- Trang 361 --- +where ở could be a diferent four-vector or perhaps a scalar. lt turns out that the +pion has no polarization, so ở should be a scalar. With the simple equation L]?¿ = +0, the meson field would vary with distance from a source as 1/72, just as the +electric fñeld does. But we know that nuclear forces have much shorter distances +of action, so the simple equation wont work. 'There is one way we can change +things without disrupting the relativistic invariance: we can add or subtract +from the DˆAlembertian a constant, times ø. So Yukawa suggested that the free +quanta. of the nuclear force feld might obey the equation +—[]j — u?¿ = 0, (28.17) +where /2 is a constant—that is, an invariant scalar. (Since L] is a scalar +diferential operator in four dimensions, its invariance is unchanged if we add +another scalar to it.) +Let's see what Eq. (2§.17) gives for the nuclear force when things are no +changing with time. We want a spherically symmetric solution of +V°— uŠó =0 +around some point source at, say, the origin. If ó depends only on z, we know +that a8 +V*¿ = - __—.(rỏ). +So we have the equation +=2 (r9) — uêó =0 +—— (r@)— = +r Ôr2 ự +„5 9) = H (rộ). +Thinking oŸ (rở) as our dependent variable, this is an equation we have seen ? +many times. lIts solution is +rộ = Kec”!”", \ +Clearly, ó cannot become infinite for large r, so the -+ sign in the exponent is \ +ruled out. The solution is —g \ +¿=K“—. (28.18) \ +This function is called the Yukœœ potential. Eor an attractive Íorce, # is a ` l/r +negative number whose magnitude must be adjusted to ft the experimentally X e—Mr +observed strength of the forces. à < "xa +The Yukawa potential of the nuclear forces dies of more rapidly than 1/z SN +by the exponential factor. The potential—and therefore the force—falls to zero TT—___ ¬ +much more rapidly than 1/z for distances beyond 1/0, as shown in Eig. 28-6. ö +'The “range” of nuclear forces is much less than the “range” of electrostatic forces. 0 1/u 2/u 3/u r +Tt is found experimentally that the nuclear forces do not extend beyond about +1013 em, so ø 2 1015 m—1, Fig. 28-6. The Yukawa potential e “/r, +Einally, let?s look at the free-wave solution of Eq. (28.17). IÝ we substitute compared with the Coulomb potential 1/r. +Ộ — óoc?(et£—kZ) +into Eq. (28.17), we get that +“=—k?—? =0. +Relating frequency to energy and wave number to momentum, as we did at the +end of Chapter 344 of Vol. I, we get that +E2 += TĐ=/h +which says that the Yukawa “photon” has a mass equal to /u/c. TỶ we use Íor ð +the estimate 1012 m—!, which gives the observed range of the nuclear forces, the +--- Trang 362 --- +mass comes out to 3 x 10725 g, or 170 MeV, which is roughly the observed mass +of the z-meson. 5o, by an analogy with electrodynamics, we would say that the +7-meson 1s the “photon” of the nuclear force fñeld. But now we have pushed the +ideas of electrodynamics into regions where they may not really be valid——we +have gone beyond electrodynamies to the problem of the nuclear forces. +--- Trang 363 --- +Tĩ:o Woffore of Ấ hetrggos ri EÍoecfr-c (rre‹Ï +IMÑagyreofic Firolcis +29-1 Motion in a uniform electric or magnetic ñeld +W© want now to describe—mainly in a qualitative way——the motions oÊ charges 29-1 Motion in a uniform electric or +in various circumstances. Most of the interesting phenomena in which charges are magnetic feld +moving in fields occur in very complicated situations, with many, many charges 29-2 Momentum analysis +all interacting with each other. Eor instance, when an electromagnetic wave 29-3 An electrostatic lens +goes through a block of material or a plasma, billions and billions of charges are 29-4 A magnetic lens +interacting with the wave and with each other. We will come to such problems . +later, but now we Just want to discuss the mụch simpler problem of the motions of 29-5 The electron mỉcroscope +a single charge in a ø0en fñeld. We can then disregard all other charges——except, 29-6 Accelerator guide fields +of course, those charges and currents which exist somewhere to produce the fields 29-7 Alternating-gradient focusing +we will assume. 29-8 Motion in crossed electric and +W©e should probably ask first about the motion of a particle in a uniform magnetic 8elds +electric fñeld. At low velocities, the motion is not particularly interesting——it is just +a uniform acceleration in the direction of the field. However, if the particle picks +up enough energy to become relativistic, then the motion gets more complicated. +But we will leave the solution for that case for you to play with. +Next, we consider the motion in a uniform magnetic feld with zero electric +fñeld. We have already solved this problem——one solution is that the particle goes . . . +in a circle. The magnetic force gu x ¡is always at right angles to the metion, Reuien: Chapter ở0, Vol 1, Dijruclion +so đp/đf is perpendicular to ø and has the magnitude øp/R, where #? is the +radius oŸ the cirele: +F=quB= `. +The radius of the circular orbit is then F +R= mà (29.1) ~— +That is only one possibility. If the particle has a component of its motion +along the field direction, that motion is constant, since there can be no component ` +of the magnetie force in the direction of the ñeld. “The general motion of a particle +in a uniform magnetic fñeld is a constant velocity parallel to Ö and a circular vị I> +motion at right angles to —the traJectory is a cylindrical helix (Eig. 29-1). The ` +radius of the helix is given by Eq. (29.1) if we replace ø by ø¡, the component of —_ +mmomentum at right angles to the feld. +29-2 Momentum analysis "T2 +A uniform magnetic field is often used in making a “momentum analyzer,” or | +“momentum spectrometer,” for high-energy charged particles. Suppose that (a) (b) +charged particles are shot into a uniform magnetic fñeld at the poiny 4 in . ¬ +Fig. 29-2(a), the magnetic feld being perpendicular to the plane of the drawing. : Flg. 29-1. Mu of a particle In a uni- +BEach particle will go into an orbit which is a circle whose radius is proportional orm magnetlc Iieid. +to its momentum. Tf all the particles enter perpendicular to the edge of the field, +they will leave the field at a distance # (rom 4) which is proportional to their +mmomentum ø. Á counter placed at some point such as Œ will detect only those +particles whose momentum is in an interval Ấp near the momentum p = g8z/2. +Tt 1s, of course, not necessary that the particles go through 180° before they +are counted, but the so-called “180 spectrometer” has a special property. Ït is not +--- Trang 364 --- +necessary that all the particles enter at right angles to the fñeld edge. Eigure 29-2(b) +shows the trajectories of three particles, all with the sœme momentum but entering l +the field at diferent angles. You see that they take diferent trajecbories, but ⁄ UNIEORM MAGMETIC SIELD +all leave the fñeld very close to the point Œ. WSe say that there is a “fÍocus.” +Such a focusing property has the advantage that larger angles can be accepted “7 +at A—although some limit is usually imposed, as shown in the fñgure. A larger ⁄Z 4% ⁄2 +angular acceptance usually means that more particles are counted in a given lđ. +time, decreasing the time required for a gïven measurement. T2) ⁄) ⁄) ⁄ +By varying the magnetic ñeld, or moving the counter along in #, or by using £ 1 SNWổ R 2 +many counters to cover a range of #, the “spectrum” of momenta in the incoming v +beam can be measured. [By the “momentum spectrumn” ƒ(p), we mean that (a) +the number of particles with momenta bebween ø and (p + đp) is ƒ(p) dp.| Such +mmeasurements have been made, for example, to determine the distribution of Z +energies in the Ø-decay of various nuclei. ⁄ 5 +'There are many other forms of momentum spectrometers, but we will describe r7 +Jjust one more, which has an especially large soljd angle of acceptance. lt is based Z2 +on the helical orbits in a uniform field, like the one shown in Fig. 29-1. Let”s ⁄ “_ “ Ị Ị Tm—.__ __—ễ Ị Ị Ị +j I j I I +I I I I I I +I I I I I I +CIRCULAR CIRCULAR CIRCULAR +ORBIT B ORBIT B ORBIT B +„4ã —— NNỊ +i | | | | | +Fig. 29-11. Radial motion of a particle In Fig. 29-12. Radial motion of a particle In Fig. 29-13. Radial motion of a particle In +a magnetic field with a large positive slope. a magnetic field with a small negative slope. a magnetic field with a large negative slope. +One would, at frst, guess that radial focusing could be provided by making a +magnetic field which increases with increasing distance from the center of the +design path. Then iŸ a particle goes out to a large radius, it will be in a stronger +ñeld which will bend it back toward the correct radius. If it goes to too small a +radius, the bending will be less, and it will be returned toward the design radius. +lÝ a particle is once started at some angle with respect to the ideal cirele, 1% +will oscillate about the ideal circular orbit, as shown in Eig. 29-11. 'Phe radial +focusing would keep the particles near the circular path. +Actually there is still some radial focusing even with the opposite field slope. +This can happen I1f the radius of curvature of the trajectory does not increase +more rapidly than the increase in the distance of the particle rom the center of +the fñeld. The particle orbits will be as drawn in Eig. 29-12. If the gradient of +the fñeld is too large, however, the orbits will not reburn to the design radius but +will spiral inward or outward, as shown in Fig. 29-13. +W© usually describe the slope of the ñeld in terms of the “relative gradient” s +or ield indez, m: 1z +dB/B —> +nh — Tp « (29.2) TO CENTER * +drír OF ORBIT +-——— --_-- —---r +A guide field gives radial focusing If this relative gradient is greater than —1. CENTRAL +. . . . . ORBIT +A radial fñeld gradient will also produce 0erfical forces on the particles. +Suppose we have a field that is stronger nearer to the center of the orbit and +weaker at the outside. Á vertical cross section of the magnet at right angles to N +the orbit might be as shown in Eig. 29-14. (Eor protons the orbits would be +coming out oŸ the page.) If the ñeld is to be stronger to the leftƠ and weaker to +the right, the lines of the magnetic fñeld must be curved as shown. W©e can see Fig. 29-14. A vertical guide field as seen +that this must be so by using the law that the circulation of is zero in free in a cross section perpendicular to the orbits. +space. IÝ we take coordinates as shown in the fñgure, then +9B, ôB, +VxP),=—-—=—=0 +Ũ 9z 3z Í +3B 8B +—-.. (29.3) +Since we assume that Ø; /Øz is negative, there must be an equal negative 9B„ /9z. +Tf the “nominal” plane of the orbit is a plane of symmetry where ö„ = 0, then +the radial component „ will be negative above the plane and positive below. +The lines must be curved as shown. +--- Trang 368 --- +Such a field will have vertical focusing properties. Imagine a proton that 1s +travelling more or less parallel to the central orbit but above ït. "The horizontal +component of will exert a downward force on ï§. TỶ the proton is below the +central orbit, the force is reversed. So there is an elfective “restoring force” toward +the central orbit. From our arguments there will be vertical focusing, provided +that the 0ertical field decreases with increasing radius; but if the field gradient +is positive, there will be “vertical defocusing.” 5o for vertical focusing, the fñeld +Index ø must be less than zero. We found above that for radial focusing øœ had +to be greater than —1. 'Phe bwo conditions together give the condition that +—=l<#m„<0 +1f the particles are to be kept in stable orbits. In cyclotrons, values very near zero +are used; in betatrons and synchrotrons, the value „ = —0.6 is typically used. +29-7 Alternating-gradient focusing +Such smaill values oŸ n give rather “weak” focusing. It is clear that much more +effective radial focusing would be given by a large positive gradient (œ 3 1), but +then the vertical forces would be strongly defocusing. Similarly, large negative +slopes (m << —1) would give stronger vertical forces but would cause radial +defocusing. lI§ was realized about 10 years ago, however, that a force that +alternates between strong focusing and strong defocusing can still have a net +focusing force. +To explain how alfernating-gradient [ocusing works, we will fñrst deseribe the +operation of a quadrupole lens, which is based on the same principle. Imagine +that a uniform negative magnetic fñeld is added to the ñeld of Fig. 29-14, with +the strength adJjusted to make zero field at the orbit. The resulting field——for +small displacements from the neutral point—would be like the feld shown in +Jig. 29-15. Such a four-pole magnet is called a “quadrupole lens” A positive +particle that enters (from the reader) to the right or left of the center is pushed +back toward the center. I the particle enters above or below, iÈ is pushed ad +from the center. 'Phis is a horizontal focusing lens. If the horizontal gradient +is reversed—as can be done by reversing all the polarities—the signs of all the +forces are reversed and we have a vertical focusing lens, as in Fig. 29-16. Eor such +lenses, the fñield strength——and therefore the focusing forces——increase linearly +with the distance of the lens rom the axis. +Now imagine that t6wo such lenses are placed in series. If a particle enters +with some horizontal displacement from the axis, as shown in Fig. 29-17(a), it +ÍS ,. SN ỀỒN ` ——Ì +== = y +Ñ ⁄ ⁄ +Fig. 29-15. A horizontal focusing quad- Fig. 29-16. A vertical focusing quadru- +rupole lens. pole lens. +--- Trang 369 --- +HORIZONTAL VERTICAL +DISPLACEMENT DISPLACEMENT +FROM AXIS FROM AXIS +DISTANCE DISTANCE +HORIZONTAL HORIZONTAL VERTICAL VERTICAL +FOCUSING DEFOCUSING DEFOCUSING FOCUSING +FIELD FIELD FIELD FIELD +(a) (b) +Fig. 29-17. Horizontal and vertical focusing with a pair of quadrupole lenses. +will be defected toward the axis in the frst lens. When it arrives at the second “TƯỜNG +lens it is closer to the axis, so the force outward is less and the outward deflection C l ) +is less. There is a net bending toward the axis; the øøerage efect is horizontally NY Z +focusing. Ôn the other hand, If we look at a particle which enters of the axis in —_ +the vertical direction, the path will be as shown in Fig. 29-17(b). The particle is 1% +first deflected auø¿ from the axis, but then it arrives at the second lens with a ”% +larger displacement, feels a stronger force, and so is bent toward the axis. Again ïm +the net efect is focusing. Thus a pair of quadrupole lenses acts independently ï +for horizontal and vertical motion—very much like an optical lens. Quadrupole ï +lenses are used to form and control beams of particles in much the same wawy cị +that optical lenses are used for light beams. | Ï——_ ⁄ +We should point out that an alternating-gradient System does not alauaJs Co 8 ñ +produce focusing. TÝ the gradients are too large (in relation to the particle — +momentum or to the spacing between the lenses), the net efect can be a defocusing Lˆ - 4 +one. You can see how that could happen ïŸ you imagine that the spacing between b) +the two lenses of Fig. 29-17 were increased, say, by a factor of three or four. +Let's return now to the synchrotron guide magnet. We can consider that it " lJZ +consists of an alternating sequence of “positive” and “negative” lenses with a (——————————————] +superimposed uniform field. 'Phe uniform field serves to bend the partieles, on +the average, in a horizontal circle (with no efect on the vertical motion), and Fig. 29-18. A pendulum with an oscillat- +the alternating lenses act on any particles that might tend to go astray—pushing ing pivot can have a stable position with the +them always toward the central orbit (on the average). bob above the pivot. +There is a nice mechanical analog which demonstrates that a force which +alternates between a “focusing” force and a “defocusing” force can have a net +“focusing” efect. Imagine a mechanical “pendulum” which consists of a sol2đ +rod with a weight on the end, suspended from a pivot which is arranged to be +moved rapidly up and down by a motor driven crank. Such a pendulum has +tuo equilibrium positions. Besides the normal, downward-hanging position, the +pendulum ïs also in equilibrium “hanging upward”——with Its “bob” abooe the TT +pivot! Such a pendulum is drawn in Eig. 29-18. ì +By the following argument you can see that the vertical pivot motion is _ “ +cquivalent to an alternating focusing force. When the pivot is accelerated ` +downward, the “bob” tends to move inward, as indicated in Eig. 29-19. When \ \ +the pivot is accelerated upward, the efect ¡is reversed. “The force restoring the \ \ +“bob” toward the axis alternates, but the average effect is a force toward the axis. \ +So the pendulum will swing back and forth about a neutral position which is just N +opposite the normal one. +There is, of course, a much easier way of keeping a pendulum upside down, +and that is by baÏønc7ng it on your ñngerl But try to balance #ưo ¿ndependen‡ +sticks on the sœme fngerL Ör one stick with your eyes closedl Balancing involves ụ ` | +making a correction for what is going wrong. And this is not possible, in general, xế +1ƒ there are several things going wrong at onee. In a synchrotron there are billions Fig. 29-19. A downward acceleration of +OŸ particles going around together, each one of which may start out with a diferent the pivot causes the pendulum to move to- +“error.” The kind of focusing we have been describing works on them all. ward the vertical. +--- Trang 370 --- +29-8 Motion in crossed electric and magnetic fields +So far we have talked about particles in electric fñelds only or in magnetic +fñelds only. 'There are some interesting efects when there are both kinds of +fields at the same time. Suppose we have a uniform magnetic field Ö and an +electric fñeld # at right angles. Particles that start out perpendicular to Ö will +move in a curve like the one in Eig. 29-20. (The figure is a pỈanwe curve, of a — +helix!) We can understand this motion qualitatively. When the particle (assumed vọ +positive) moves in the direction oŸ , it picks up speed, and so it is bent less E +by the magnetic ñeld. When it is goïing against the -field, it loses speed and | +1s continually bent more by the magnetic fñeld. The net efect is that it has an @ +average “drift” in the direction of E x Ö. B +W©e can, in fact, show that the motion 1s a uniform circular motion super- +imposed on a uniform sidewise motion at the speed uạ = !/B——the trajectory Fig. 29-20. Path of a particle in crossed +in Eig. 29-20 is a cycloid. Imagine an observer who is moving to the right at electric and magnetic fields. +a constant speed. In his rame our magnetic fñeld gets transformed to a new +magnetic ñeld pius an electric ñeld in the dotm+uard direction. Tf he has just the +right speed, his total electric ñeld will be zero, and he will see the electron going +in a circle. So the motion 0e see is a circular motion, plus a translation at the +drift speed ạ = /B. The motion of electrons in crossed electric and magnetic +fields is the basis of the zmagnetron tubes, 1.e., oscillators used for generating +InICTOWaAVe€ GIGTBV. +There are many other interesting examples of particle motions in electric and +magnetic felds——such as the orbits of the electrons and protons trapped in the +Van Allen belts—but we do not, unfortunately, have the time to deal with them +--- Trang 371 --- +Tĩìo Inéorrtcrl Ấ©ormteofrtg ©Ÿ ẤTggsế(xÏs +30-1 The internal geometry of crystals +We have fñnished the study of the basic laws of electricity and magnetism, 30-1 The internal geometry of crystals +and we are now going to study the electromagnetic properties of matter. We 30-2 Chemical bonds in crystals +begin by describing solids—that is, crystals. When the atoms of matter are not 30-3 The growth of crystals +moving around very much, they get stuck together and arrange themselves in a R +. : : ; : 30-4 Crystal lattices +confguration with as low an energy as possible. If the atoms in a certain place ¬ . . +have found a pattern which seems to be of low energy, then the atoms somewhere 30-5 5ymmetries in two dimensions +else will probably make the same arrangement. EFor these reasons, we have in a 30-6 5ymmetries in three dimensions +solid material a repetitive pattern of atoms. 30-7 The strength of metals +In other words, the conditions in a crystal are this way: The environment of 30-8 Dislocations and crystal growth +a particular atom in a crystal has a certain arrangement, and ïf you look at the 30-9 The Bragg-Nye crystal model +same kind of an atom at another place farther along, you will ñnd one whose +surroundings are exactly the same. If you pick an atom farther along by the +same distance, you will ñnd the conditions exactly the same once more. 'Phe +pattern is repeated over and over again——and, of course, in three dimensions. +TImagine the problem of designing a wallbaper—or a cloth, or some geometric +desizn for a plane area—in which you are supposed to have a design element Reference: C. Kittel, Infroduelion to +which repeats and repeats and repeats, so that you can make the area as large as . - +you want. This is the two-dimensional analog of a problem which a crystal solves goi ¡4 Sfate Phụsics, John +: . . . . Wiley and Sons, Inc., New +in three dimensions. Eor example, Fig. 30-1(a) shows a common kind oŸ wallpaper York 2nd ed.. 1956 +design. There is a single element repeated in a pattern that can go on forever. The Í ` l +geometric characteristics of this wallbaper design, considering only its repetition +properties and not worrying about the geometry of the fower itself or its artistic +merit, are contained in Fig. 30-1(b). I you start at any point, you can fnd the +corresponding point by moving the distance ø along the direction of arrow 1. You _Ñ, +can also get to a corresponding poiïnt if you move the distance b in the direction Z v24 +of the other arrow. 'There are, of course, many other directions. You can go, for +example, from point œ to point Ø and reach a corresponding position, but such a đổ -ấp ae s -ấg „5 +step can be considered as a combination of a step along direction 1, followed by a +step along direction 2. Ône of the basic properties of the pattern can be described S +by the two shortest steps to nearby equal positions. By “equal” positions we mean +that if you were to stand in any one of them and look around you, you would +see exactly the same thing as If you were to stand in another one. 'Phat's the (a) +fundamental property of a crystal. The only diference is that a crystal is a three- +dimensional arrangement instead of a two-dimensional arrangement; and naturally, +instead of owers, cach element of the lattice is some kind of an arrangement œ œ œ œ +of atoms——perhaps six hydrogen atoms and ©wo carbon atoms——in some kind of +pattern. 'Phe pattern of atoms in a crystal can be found out experimentally by Lý j +x-ray diÑraction. We have mentioned this method briefy before, and won”t say œ +any more now except that the precise arrangement of the atoms in space has +been worked out for most simple crystals and also for some fairly complex ones. +The internal pattern of a crystal shows up in several ways. First, the binding = +strength of the atoms in certain directions is usually stronger than in other +directions. 'Phis means that there are certain planes through the crystal where it œ +is more easily broken than others. 'Phey are called the cleœuaøe planes. lÝ you +crack a crystal with a knife blade it will often split apart along such a plane. Œœ) +Second, the internal structure often appears at the surface because of the way the +crystal was formed. Imagine a crystal being deposited out of a solution. 'There Fig. 30-1. A repeating pattern in two +are the atoms floating around in the solution and finally settling down when they dimensions. +--- Trang 372 --- +fnd a position of lowest energy. (It”s as if the wallpaper got made by fowers N +drifting around until one drifted accidentally into place and got stuck, and then “ : +the next, and the next so that the pattern gradually grows.) You can appreciate “ Ấ. Gia" +that there will be certain directions in which it will grow at a diferent speed "` +than in other directions, thereby growing into some kind of geometrical shape. \: +Because of such efects, the outside surfaces of many crystals show some of the +character of the internal arrangement of the atoms. xố ä +Eor example, Fig. 30-2(a) shows the shape of a typical quartz crystal whose ) rẻ ; +internal pattern is hexagonal. If you look closely at such a crystal, you will notice ' #) ‹ +that the outside does not make a very good hexagon because the sides are not Ễ * , +all of equal length—they are, in fact, often very unequal. But in one respect I§ : H ằ Fị & +is a very good hexagon: the øngles between the faces are exactly 120”. Clearly, ': @? (- +the size of any particular face is an accident of the growth, but the angles are a cụ +representation of the internal geometry. So every crystal of quartz has a diferent h lỗ +shape, even though the angles between corresponding faces are always the same. +The internal geometry oŸ a crystal of sodium chloride is also evident from is (a) +external shape. Figure 30-2(b) shows the shape of a typical grain of salt. Again +the crystal is not a perfect cube, but the faces are exactly at right angles to one +another. +A more complicated crystal is mica, which has the shape shown in Fig. 30-2(c). +Tt is a highly anisotropic crystal, as is easily seen from the fact that it is very tough , vả! VI. +1Ý you try to pull it apart in one direction (horizontally in the fñgure), but very 1É. TY +easy to split by pulling apart in the other direction (vertically). It has commonly Sô{ +- sẻ. +been used to obtain very tough, thin sheets. Mica and quartz are two examples b +of natural minerals containing silica. A third example of a mineral with silica is @œ) +asbestos, which has the interesting property that it is easily pulled apart in Ewo +directions but not ïn the third. Ht appears to be made oŸ very strong, lzneør fbers. +30-2 Chemical bonds in crystals ẹ bà *'# A% KG +4 zd0 16.” +'The mechanical properties of crystals clearly depend on the kind of chemical s ` “ +bindings between the atoms. The strikingly diferent strength of mica along 4®) lÀ S” +diferent directions depends on the kinds of interatomic binding in the diferent ÑW NX.`\|' +đirections. You have already learned in chemistry, no doubt, about the different xu +kinds of chemiecal bonds. First, there are ionic bonds, as we have already discussed +for sodium chloride. Roughly speaking, the sodium atoms have lost an electron () +and become positive lons; the chlorine atoms have gained an electron and become Fig. 30-2. Natural crystals: (a) quartz, +negative ions. The positive and negative ions are arranged in a three-dimensional (b) sodium chloride, (c) mica. +checkerboard and are held together by electrical forces. +"The covalent bond——in which electrons are shared between two atoms——is more +common and is usually very strong. In a diamond, for example, the carbon atoms +have covalent bonds in all four directions to the nearest neighbors, so the crystal +is very hard indeed. There is also covalent bonding between silicon and oxygen +in a quartz crystal, but there the bond is really only partially covalent. Because +there is not complete sharing of the electrons, the atoms are partly charged, and +the crystal is somewhat ionic. Nature is not as simple as we try to make it; there +are really all possible gradations between covalent and ionic bonding. +A sugar crystal has still another kind of binding. In it there are large molecules +in which the atoms are held strongly together by covalent bonds, so that the +molecule is a tough structure. But since the strong bonds are completely satis- +fñed, there are only relatively weak attractions between the separate, individual +molecules. In such rmolecular crystals the molecules keep theïr individual identity, +so to speak, and the internal arrangement might be as shown in Fig. 30-3. Since +the molecules are not held strongly to each other, the crystals are easy to break. Fig. 30-3. The lattice ofa molecular crystal. +They are quite diferent from something like diamond, which is really one giant +molecule that cannot be broken anywhere without disrupting strong covalent +bonds. Parafin is another example of a molecular crystal. +An extreme example of a molecular crystal occurs in a substance like solid +argon. There is very little attraction bebween the atoms——each atom is a com- +--- Trang 373 --- +pletely saturated monatomie molecule. But at very low temperatures, the thermal +motion is very small, so the slight interatomic Íorces can cause the atoms to +settle down into a regular array like a pile of closely packed spheres. +The metals form a completely diferent class of substances. The bonding is of +an entirely diÑerent kind. In a metal the bonding is not between adjacent atoms +but is a property of the whole crystal. 'The valence electrons are not attached +to one atom or to a pair of atoms but are shared throughout the crystal. Each +atom contributes an electron to a universal pool oŸ electrons, and the atomic +positive ions reside in the sea of negative electrons. 'Phe electron sea holds the +ions together like some kind of glue. +In the metals, since there are no special bonds in any particular direction, +there is no strong directionality in the binding. “They are still crystalline, however, +because the total energy is lowest when the atomic ions are arranged in some +defnite array——although the energy of the preferred arrangement is not usually +much lower than other possible ones. 'To a first approximation, the atoms of +many metals are like small spheres packed in as tightly as possible. +30-3 The growth of crystals +Try to imagine the natural formation of crystals in the earth. In the earth”s +surface there is a big mixture of all kinds of atoms. They are being continually +churned about by volcanic action, by wind, and by water—continually being +moved about and mixed. Yet, by some trick, silicon atoms gradually begin to +ñnd each other, and to ñnd oxygen atoms, to make silica. One atom at a time +is added to the others to build up a crystal—the mixture gets unmixed. And +somewhere nearby, sodium and chlorine atoms are ñnding each other and building +up a crystal of salt. +How does it happen that once a crystal is started, it permits only a particular +kind of atom to joïn on? lt happens because the whole system is working toward +the lowest possible energy. A growing crystal will accept a new atom IÍ it is +going to make the energy as low as possible. But how does it knou that a +silicon—or an oxygen—atom at some particular spot is goiỉng to result in the (a) +lowest possible energy? I§ does it by trial and error. In the liquid, all of the +atoms are In perpetual motion. Each atom bounces against its neighbors about +10!3 tỉimes every second. If it hits against the right spot of growing crystal, it +has a somewhat smaller chance of jumping of again if the energy is low. By +continually testing over periods of millions of years at a rate of 1013 tests per +second, the atoms gradually build up at the places where they fnd their lowest +energy. Eventually they grow into big crystals. +30-4 Crystal lattices ⁄)3<< 2 +The arrangement of the atoms in a crystal—the crystal /af#ce—can take on +many geometric forms. We would like to describe frst the simplest lattices, which +are characteristic of most of the metals and of the solid form of the inert gases. C ) () +They are the cubic lattices which can occur in two forms: the body-centered +cubic, shown in Eig. 30-4(a), and the face-centered cubic shown in Eig. 30-4(b). œ) ® +The drawings show, of course, only one cube of the lattice; you are to imagine +that the pattern is repeated indefinitely in three dimensions. Also, to make the C) +drawing clearer, only the “centers” of the atoms are shown. In an actual crystal, +the atoms are more like spheres in contact with each other. 'Phe dark and light +spheres in the drawings may, in general, stand for different kinds of atoms or Fig. 30-4. The unit cell of cubic crystals: +may be the same kind. For instanee, iron has a body-centered cubic lattice at (a) body-centered, (b) face-centered. +low temperatures, but a face-centered cubic lattice at higher temperatures. The +physical properties are quite difÑferent in the bwo crystalline forms. +How do such forms come about? Imagine that you have the problem of packing +spherical atoms together as tightly as possible. One way would be to start by +making a layer in a “hexagonal close-packed array,” as shown in EFig. 30-5(a). +'Then you could buïild up a second layer like the first, but displaced horizontally, +--- Trang 374 --- +\è NN \\ Z ` NÓ ` +À À a¬. ` ì ì) +\nÑ . : an . +` À\ ` ở +(a) y ` ` (b) `. `. NÓ ỜN, +\»nny 2) +nà ayc an . +⁄ ⁄ ⁄ ` +ầ Nà 8 3 +Fig. 30-5. Building up a hexagonal close-packed lattice. +as shown in Fig. 30-5(b). Next, you can put on the third layer. But noticel +There are #uo distinct ways of placing the £hørd layer. IÝ you start the third layer +by placing an atom at 4 in Eig. 30-5(b), each atom in the third layer is directly +above an atom of the bottom layer. Ôn the other hand, if you start the third +layer by putting an atom at the position , the atoms of the third layer will be +centered at points exactly in the middle of a triangle formed by three atoms of +the bottom layer. Any other starting place is equivalent to A or Ö, so there are +only two ways of placing the third layer. +Tf the third layer has an atom at point Ö, the crystal lattice is a face-centered +cubic—but seen at an angle. It seems funny that starting with hexagons you +can end up with cubes. But notice that a cube looked at from a corner has a +hexagonal outline. For instance, Fig. 30-6 could represent a plane hexagon or a +cube seen in perspectivel +T a third layer ¡is added to Eig. 30-5(b) by starting with an atom at A, there +is no cubical structure, and the lattice has instead only a hexagonal symmetry. +Tt is clear that both possibilities we have described are equally close-packed. +Some metals—for example, copper and silver——choose the fñrst alternative, the Eig. 30-6. Is this a hexagon or a cube +face-centered cubic. Others—for example, beryllium and magnesiun—choose the seen from one corner? +other alternatives; they form hexagonal crystals. Clearly, which crystal lattice +appears cannot depend only on the packing of little spheres, but must also be +determined in part by other factors. In particular, it depends on the slight +remaining angular dependence of the interatomic forces (or, in the case oŸ the +metals, on the energy of the electron pool). You will, no doubt, learn all about +such things in your chemistry €OUrses. +30-5 Symmetries in two dimensions +W©e would now like to discuss some of the properties of crystals from the poiïnt +of view of their internal symmetries. The main feature of a crystal is that if you +start at one atom and move to a corresponding atom one lattice unit away, you are +again in the same kind of an environment. That”s the fundamental proposition. +But if you were an atom, there would be another kind of change that could +take you again to the same environment——that is, another possible “syrmmmetry.” +Eigure 30-7(a) shows another possible “wallpaper-type” design (though one you +have probably never seen). Suppose we compare the environments for points A +and . You might, at first, think that they are the same——but not quite. Points +Œ and D are equivalent to 4, but the environment of Ö ïs like that of A only if +the surroundings are reversed, as in a mirror refection. +--- Trang 375 --- +y.ỀR R +£ òlj »olé ðlx j sjé , | olé — è|lœ +⁄ s|é _ è|x_ so œ_, slé joœ »oló +CÓ XS CC ta Ngàn +E---=-=-¬_--—¬ —¬—=-l=-=------l—-R +òlx — xjóé — è óé } òlo soió — ò|œ +slé _ lv sjóé , le ; sjó — èlx — »|é +2“ I|° ?^ ¡ 2|°_, 7?“ ZIE 9= +y..R R +(a) (œ) +Fig. 30-7. A pattern of high symmetry. +'There are other kinds of “equivalent” points in the pattern. Eor instance, the +points # and #' have the “same” environments except that one is rotated 90° +with respect to the other. The pattern is quite special. A rotation of 90°——or any +multiple of it —about a vertex such as 4 gives the same pattern all over again. +A crystal with such a structure would have square corners on the outside, but +Iinside it is more complicated than a simple cube. +Now that we have described some special examples, let's try to ñgure out all +the possible symmetries a crystal can have. First, we consider what happens In +a plane. A piane lattice can be defined by the two so-called prữmiiiue vectors +that go from one point of the lattice to the two øœearest equivalent points. "The +two vectors 1 and 2 are the primitive vectors of the lattice of Fig. 30-1. The two +vectors ø and b of Fig. 30-7(a) are the primitive vectors of the pattern there. VWe +could, of course, equally well replace œ by —ø, or by —b. Since ø and Ö are z +cqual in magnitude and at right angles, a rotation of 90° turns ø into b, and b Z +into —œ, giving the same lattice once again. D 1% C +We see that there are lattices which have a “four-sided” symmetry. Andwe " “` pZ +have described earlier a close-packed array based on a hexagon which could have p\ 4 +a six-sided symmetry. A rotation of the array of circles in Fig. 30-5(a) by an _ 6m +angle of 60° about the center of any cirele brings the pattern back to itself. ————— 3 NT +'What other kinds of rotational symmetry are there? Can we have, for example, A) +a fvefold or an eightfold rotational symmetry? It is easy to see that they are +impossible. The onlụ sựmmetru tuíth more sides than ƒour ts a siz-sided sựmmetrg. +Jtirst, let's show that more than sixfold symmetry is impossible. Supposewe 7C +try to imagine a lattice with two equal primitive vectors with an enclosed angle +less than 609, as in Eig. 30-8(a). We are to suppose that points and Œ are D b +cquivalent to 4, and that œ and b are the two shortest vectors from A to its 72 +cequivalent neighbors. But that is clearly wrong, because the distance between ụ h +and C is shorter than from either one to A. There must be a neighbor at 2 ; - 729 +equivalent to A which is closer than Ö or Œ. We should have chosen b“ asone —--¿~—_—-~ nh —— +.- ¬- Ẽ Ạ 3 B +of our primitive vectors. So the angle between the two primitive vectors musf +be 60” or larger. Octagonal symmetry is not possible. œ) +What about fvefold symmetry? If we assume that the primitive vectors œ Fig. 30-8. (a) Rotational symmetries +and b have equal lengths and make an angle of 2z/5 = 729, as in Eig. 30-5(b), greater than sixfold are not possible. +then there should also be an equivalent lattice point at D, at 72° from Œ. But the (b) Fivefold rotational symmetry is not +vector b' rom #2 to D is then less than Ð, so b is not a primitive vector. There can possible. +be no fñvefold symmetry. The only possibilities that do not get us into this kind +of dificulty are Ø = 609, 902, or 120”. Zero or 1802 are also clearly possible. Ône +way of stating our result is that the pattern can be left unchanged by a rotation +of one full turn (no change at all), one-half oŸ a turn, one-third, one-fourth, or +one-sixth of a turn. And those are all the possible rotational symmetries in a +plane—a total of five. TỶ Ø = 2z/n, we speak of an “ø-fold” symmetry. We say +--- Trang 376 --- +ế ế ế ế ế ế +ề ề ề l l l +(a) (b) +ế ế ế ế ế ế +ề ề ề ề ế ế +ề ề ề ề ề ề +? ? ? ? ? ? +(c) (d) +Fig. 30-9. Symmetry under inversion. Pattern (b) is unchanged if R —> —R, but pattern (a) is +changed. In three dimensions pattern (d) ¡s symmetric under an inversion but (c) is not. +that a pattern with œ equal to 4 or to 6 has a “higher symmetry” than one with +m cqual to 1 or to 2. +Returning to Pig. 30-7(a), we see that the pattern has a fourfold rotational +symmetry. We have drawn in Fig. 30-7(b) another design which has the same +symmetry properties as part (a). The little comma-like figures are asymunetric +objects which serve to defne the symmetry of the design inside of each square. +Notice that the commas are reversed in alternate squares, so that the unit cell is +larger than one of the small squares. lf there were no commas, the pattern would +siil have fourfold symmetry, but the unit cell would be smaller. The patterns of +Fig. 30-7 also have other symmetry properties. Eor instance, a refection about +any of the broken lines f-Ï reproduces the same pattern. +The patterns of Fig. 30-7 have still another kind of symmetry. If the pattern +is refected about the line Y-Y ”. znd shifted one square to the right (or left), we +get back the original pattern. The line Y-Y is called a “glide” line. +These are all the possible symmetries in two dimensions. There is one more +spatial symmetry operation which ¡is equivalent ¿n fuo đimensions to a 1800 rota- +tion, but which is a quite distinct operation in three dimensions. Ít 1s ?uers?on. +By an inversion we mean that any point at the vector displacement iề from some +origin [for instance, the point A in Eig. 30-9(b)] is moved to the point at — F. +An inversion of pattern (a) of Eig. 30-9 produces a new pattern, but an +inversion of pattern (b) reproduces the same pattern. For a two-dimensional +pattern (as you can see from the figure), an inversion of the pattern (b) through +the point A is equivalent to a rotation of 180° about the same point. Suppose, +however, we make the pattern in Eig. 30-9(b) three dimensional by imagining +that the little 6's and 9?s each have an “arrow” poinmting ou‡ oƒ the pagạc. After +an inversion in three đimensions all the arrows will be reversed, so the pattern +1s no‡ reprodueced. If we indicate the heads and tails of the arrows by dots and +crosses, respectively, we can make a three-đữmensional pattern, as in Eìig. 30-0(c), +which is nø# symmetric under an inversion, or we can make a pattern like the +one shown in(d), which đoes have such a symmetry. Notice that it is no£ possible +to imitate a three-dimensional inversion by any combination of rotations. +Tf we characterize the “symmetry” of a pattern—or lattice—by the kinds of +symmetry operations we have been describing, it turns out that for two dimensions +17 distinct patterns are possible. We have drawn one pattern of the lowest possible +--- Trang 377 --- +symmetry in Eig. 30-1, and one oŸ high symmetry in Fig. 30-7. We will leave you +with the game of trying to fñgure out all of the 17 possible patterns. +lt is peculiar how few of the 17 possible patterns are used in making wallpaper +and fabrics. One always sees the same three or four basic patterns. Is this because +of a lack of imagination of designers, or because many of the possible patterns +are not pleasing to the eye? +30-6 Symmetries in three dimensions +So far we have talked only about patterns in two dimensions. What we are ~” mm +really interested in, however, are patterns of atoms in three dimensions. First, it is 1 k h +clear that a three-dimensional crystal will have #hree primitive vectors. If we then %p h _=----- cử +ask about the possible symmetry operations in three dimensions, we fnd that 5 ng +there are 230 diferent possible symmetriesl For some purposes, these 230 types TRICLINIC +can be grouped into seven classes, which are drawn in Fig. 30-10. “The lattice with +the least symmetry is called the #r¿clm+c. Its unit cell is a parallelepiped. “The _ : _ +primitive vectors are of diferent lengths, and no two of the angles between them . M4 ˆ +are equal. There is no possibility of any rotational or reflection symmetry. There 62 “2 +are, however, still two possible symmetries—the unit cell is, or ¡is not, changed by E) ⁄ +an inversion through the vertex. (By an inversion in three dimensions, we again TRIGONAL +mean that spatial displacements are replaced by —#—in other words, that “.Ặ.—ẮẮ—.‹< +(z,,Zz) goes into (—z,—,—2)). So the triclinic lattice has only two possible Trrrrrrrr +symmetries, unless there is some special relation among the primitive vectors. +For example, 1f all the vectors are equal and are separated by equal angles, one c +has the #rigonal lattice shown in the Ñgure. This ñgure can have an additional là nh "¬ +symmetry; it may be unchanged by a rotation about the long, body diagonal. E ⁄ +Tf one of the primitive vectors, say e, is at right angles to the other two, we get MONOCLINIC +a monoclinic unit cell. A new symmetry is possible—a rotation by 180° about e. "———.Ố +The hezagonal cell is a special case in which the vectors ø and b are cqual and "“« "` +the angle between them is 607, so that a rotation of 60°, or 120”, or 1809 about __ " +the vecbor œ repeats the same lattice (for certain internal symmetries). c à h ¬ ọỊ +TÝ all three primitive vectors are at right angles, but of diferent lengths, we ¬=< +get the orfhorhomjbic cell. 'Phe fñgure is symmetric for rotations of 1809 about S5 . +the three axes. Higher-order symmetries are possible with the #efragonal cell, HEXAGONAL +which has all right angles and two equal primitive vectors. Pinally, there is the ——....... +cubic cell, which is the most symmetric of all. “1 x4 +The poïnt of all this discussion about symmetries is that the internal sym- an Tí 1 +metries of the crystals show up——sometimes in subtle ways——in the macroscopic ' +physical properties of the crystal. For instance, a crystal will, in general, have a , 4... +tensor electric polarizability. If we describe the tensor in terms of the ellipsoid of tớ +polarization, we should expect that some of the crystal symmetries should show 3 +: ¬ : . ¬ ORTHORHOMBIC +up also in the ellipsoid. Eor example, a cubic crystal is symmetric with respect ¬ +to a rotation of 909 about any one of three orthogonal directions. Clearly, the ằ Í +only ellipsoid with this property is a sphere. A4 cubic crustal must be ơn isotropic mm - nã +điclectric. l ¬ „Ị +On the other hand, a tetragonal crystal has a fourfold rotational symmetry. . ⁄ Lo ⁄ +Its ellipsoid must have two of its principal axes equal, and the third must be L“ +parallel to the axis of the crystal. Similarly, since the orthorhombie crystal has TETRAGONAL +twofold rotational symmetry about three orthogonal axes, its axes must coincide Ar===== gi +with the axes of the polarization ellipsoid. In a like manner, øne of the axes of a y4] ⁄ Ị +monoclinie crystal must be parallel to øne of the principal axes of the ellipsoid, ~--3‡-—- lã ị +though we can't say anything about the other axes. Since a triclinic crystal has Ị ị +no rotational symmetry, the ellipsoid can have any orientation at all. M1... +As you can see, we can make a big game of ñguring out the possible symmetries ¬- +and relating them to the possible physical tensors. We have considered only the 5 4 +polarization tensor, but things get more complicated for others—for instance, for CUBIC +the tensor of elasticity. There 1s a Dranch of mathematies called “group theory” Eig. 30-10. The seven classes of crystal +that deals with such subjects, but usually you can fgure out what you want with lattices. +COmIOH S€nS€. +--- Trang 378 --- +1 2 3 2 +(a) (@b) +Fig. 30-11. Slippage of crystal planes. +30-7 The strength of metals +WS have said that metals usually have a simple cubic crystal strucbure; we +want now to discuss their mechanical properties—which depend on this structure. +Metals are, generally speaking, very “soft,” because it is easy to slide one layer of ` +the crystal over the next. You may think: “Phats ridiculous; metals are strong.” ề +Not so, a s?ngle crustal of a metal can be distorted very easily. +3uppose we look at two layers of a crystal subjecbed to a shear force, as shown +in the diagram of Fig. 30-11(a). You might at frst think the whole layer would ` +resist motion until the force was big enough to push the whole layer “over the N +hump,” so that it shif#ted one notch to the left. Althouph slipping does occur along +a plane, it doesn't happen that way. (Ifit did, you would calculate that the metal ầ +is much stronger than it really is.) What happens is more like one atom going N +at a time; first the atom on the left makes its jump, then the next, and so on, as +indicated in Eig. 30-11(b). In efect it is the vacant space between two atoms that ` +quickly travels to the right, with the net result that the whole second layer has ` +moved over one atomic spacing. The slipping goes this way because i% takes much À +less energy to lIÍt one atom at a time over the hump than to lit a whole row. N +Once the force is enough to start the process, it goes the rest of the way very fast. +lt turns out that in a real crystal, slipping will occur repeatedly at one plane, +then will stop there and start at some other plane. The details of why it starts +and stops are quite mysterious. Ít is, in fact, quite strange that successive regions Fig. 30-12. A photograph of a small crys- +of slip are often fairly evenly spaced. Figure 30-12 shows a photograph ofatiny — tai of copper after stretching. [Courtesy +thin copper crystal that has been stretched. You can see the various planes where of 5. S. Brenner, Senior Scientist, United +slipping has occurred. States Steel Research Center, Monroeville, +'The sudden slipping of individual crystal planes is quite apparent if you take Pa.] +a piece of tin wire that has large crystals in ¡% and stretch it while holding ¡it +next to your ear. You can hear a rush of “tieks” as the planes snap to their new +positions, one after the other. +The problem of having a “missing” atom in one row is somewhat more difficult +than it might appear from Eig. 30-11. When there are more layers, the situation +must be something like that shown in EFig. 30-13. Such an ñmmnperfection in a AÁNư^—~œx—ư— +crystal is called a. đislocafion. Tt is presumed that such dislocations are either CGX:XXXX) +present when the crystal was formed or are generated at some notch or crack at C(CXXXXXY) +the surface. Once they are produced, they can move relatively freely through the Ạ N rà v2 +crystal. 'Phe gross distortions result from the motions of many of such dislocations. @® /X:X.X') ® +Dislocations can move freely—that is, they require little extra energy——sO cCXSŠ C} C) C) (% ) +long as the rest of the crystal has a perfect lattice. But they may get “stuck” ) G) : \ @ @® +1f they encounter some other kind of imperfection in the crystal. If it takes a C)  (X) ụ +lot of energy for them to pass the Imperfection, they will be stopped. “This is X) @) CI) ) `) +precisely the mechanism that gives strength to #nperƒect metal crystals. Pure @®)@@ CX )C) +iron crystals are quite soft, but a small concentration of impurity atoms may CXXXXXXYX) +cause enough imperfections to efectively immobilize the dislocations. As you A2) n Ạ m ơ c2 +know, steel, which is primarily iron, is very hard. 'To make steel, a small amount GXXXXX) @® +of carbon is dissolved in the iron melt; ¡f the melt is cooled rapidly, the carbon Ị ' ' Ị ' Ị +precipitates out in litde grains, making many microscopic disbortions in the Fig. 30-13. A dislocation in a crystal. +lattice. The dislocations can no longer move about, and the metal is hard. +Pure copper is very soft, but can be “work-hardened.” 'This is done by ham- +mering on it or bending i§ back and forth. In this case, many new dislocations +of various kinds are made which interfere with one another, cutting down their +--- Trang 379 --- +mobility. Perhaps you've seen the trick of taking a bar of “dead soft” copper and +gently bending it around someoneˆs wrist as a bracelet. In the process, it becomes <<< +work-hardened and cannot easily be unbent again! A work-hardened metal like <<<<<<<<> +copper can be made soft again by annealing at a high temperature. The thermal <<<<===<<=» +motion of the atoms “irons out” the dislocations and makes large single crystals S»<> +it should ultimately get. With so many bonds lacking, its energy is not very Ìow. X4 K<> +It would be better of at position 7, where it already has one-half of its quota +of bonds. Crystals do indeed grow by attaching new atoms at places like 7Ö. +'What happens, though, when that line is fñnished? 'To start a new line, an < <<» +atom must come to rest with only ©wo sides attached, and that is again not very << +likely. BEven i1f it did, what would happen when the layer was fñnished? How ` +could a new layer get started? One answer is that the crystal prefers to grow +at a dislocation, for instance around a screw dislocation like the one shown in +Eig. 30-14. As blocks are added to this crystal, there is always some place where `Z +there are three available bonds. 'Phe crystal prefers, therefore, to grow with a +dislocation built in. Such a spiral pattern of growth is shown in Fig. 30-16, which +1s a photograph of a single crystal of paraffin. Fig. 30-15, Crystal growth. +: ⁄ — vn » +` MS : - ⁄ Fig. 30-16. A paraffin crystal which has +ẫ XS. grown around a screw dislocation. [From +-. > ` 4 _ | Charles Kittel, Introduction to Solid State +K.... ẹ c7... % Physics, John Wiley and Sons, Inc., New +ï .&.. York, 2nd ed., 1956.] +30-9 The Bragg-Nye crystal model +W© cannot, of course, see what goes on with the individual atoms in a crystal. +Also, as you realize by now, there are many complicated phenomena that are not +easy to treat quantitatively. Sir Lawrence Bragg and J. F. Nye have devised a +scheme for making a model of a metallic crystal which shows In a striking way +many of the phenomena. that are believed to occur in a real metal. In the following +pages we have reproduced theïir original article, which describes their method +and shows some of the results they obtained with it. (The article is reprinted +from the Procecdings oƒ the Rouadl SocietU oƒ London, Vol. 190, September 1947, +DĐ. 474-481—with the permission of the authors and oŸ the Royal Society.) +--- Trang 380 --- +Œ ——j HR +A dynamical model of a crystal structure «&&___3 Ễ +By SIR LAWRENCE BRAGG, F.R.S. AND /J. EF. NYE Z—¬EE +Cauendish Laboratoru, Un¿uersitụ oƒ Cambridge T +(Recceiued 9 Januar 1947—Read 19 Jưne 1947) é_ LÔ ¬ +S===<< +[Plates 8 to 21] +JIGURE 3. Apparatus for producing bubbles of small size. +The crystal structure of a metal is represented by an assemblage +of bubbles, a millimetre or less in diameter, foating on the surface +of a soap solution. The bubbles are blown from a fne pipette be- OÊ pressure. Unwanted bubbles can easily be destroyed by playing a +neath the surface with a constant air pressure, and are remarkably small fame over the surface. Figure l shows the apparatus. We have +uniform in size. They are held together by surface tension, either found it of advantage to blacken the bottom of the vessel, because +in single layer on the surface or in a three-dimensional mass. An details of structure, such as grain boundaries and dislocations, then +assemblage may contain hundreds of thousands of bubbles and per- +sists for an hour or more. The assermblages show structures which show up more clearly. +have been supposed to exist in metals, and simulate efects which Eigure 2, plate 8, shows a portion oŸ a raft or two-dimensional +have been observed, such as, grain boundaries, dislocations and crystal of bubbles. Its regularity can be judged by looking at the +other types of fault, slip, recrystallization, annealing, and strains fgure in a glancing direction. "The size of the bubbles varies with the +due to “foreign” atoms. aperture, but does not appear to vary to any marked degree with the +pressure or the depth of the orifice beneath the surface. The main +1. THE BUBBLE MODEL effect of increasing the pressure 1s tO Ìncrease the rate of issue of the +bubbles. As an example, a thick-walled jet of 49 bore with a pressure +Models of crystal structure have been described from time to time of 100cm. produced bubbles of 1-2 mm. in diameter. A thin-walled +in which the atoms are represented by small foating or suspended jet of 27w diameter and a pressure of 180cm. produced bubbles of +magnets, or by circular disks foating on a water surface and held 0-6mm. diameter. Ït is convenient to refer to bubbles of 2-0 to 1-0 mm. +together by the forces of capillary attraction. 'These models have điameter as “large` bubbles, those from 0-8 to 0-6mm. diameter as +certain disadvantages; for instance, in the case of foating objects in “medium' bubbles, and those from 0-3 to 0-1 mm. diameter as “small' +contact, frictional forces impede their free relative movement. A more bubbles, since their behaviour varies with their size. +serious disadvantage is that the number of components is limited, 'With this apparatus we have not found it possible to reduce the size +for a large number of componentfs is required in order to approach of the jet and so produce bubbles of smaller diameter than 0-6mm. As +the state of affairs in a real crystal. The present paper describes the it was desired to experiment with very small bubbles, we had recourse +behaviour of a model in which the atoms are represented by small to placing the soap solution in a rotating vessel and introducing a fne +bubbles from 2-0 to 0-1mm. in diameter floating on the surface oŸ jet as nearly as possible parallel to a stream line. The bubbles are +a soap solution. “These small bubbles are sufficiently persistent for swept away as they form, and under steady conditions are reasonably +experiments lasting an hour or more, they slide past each other without uniform. They issue at a rate of one thousand or more per second, +friction, and they can be produced in large numbers. Some of the giving a high-pitched note. The soap solution mounts up in a steep +illustrations in this paper were taken from assemblages of bubbles wall around the perimeter of the vessel while it is rotating, but carries +numbering 100,000 or more. The model most nearly represents the back most of the bubbles with it when rotation ceases. With this +behaviour of a metal structure, because the bubbles are of one type đevice, illustrated in ñgure 3, bubbles down to 0-12 mm. in diameter +only and are held together by a general capillary attraction, which can be obtained. As an example, an orifice 38 across in a thin-walled +represents the binding force of the free electrons in the metal. A brief jet, with a pressure of 190cm. of water, and a speed of the Ñuid of +description of the model has been given in the Jourznal oƒ Sc¿entific 180 cm./sec. past the orifice, produced bubbles of 0-14mm. diameter. +Tnstrumnents (Bragg 1942Ù). In this case a dish of diameter 9-5 cm. and speed of 6rev./sec. was +used. Eigure 4, plate 8, is an enlarged picture of these “small' bubbles +\ and shows their degree of regularity; the pattern is not as perfect +with a rotating as with a stationary vessel, the rows being seen to be +slightly irregular when viewed in a glancing direction. +'These two-dimensional crystals show structures which have been +2 ‹‹“ supposed to exist in metals, and simulate efects which have been +be T Ÿ observed, such as grain boundaries, dislocations and other types of +Tu mg... ụ fault, slip, recrystallization, annealing, and strains due to “foreign" +atoms. +PIGURE 1. Apparatus for producing rafts of bubbles. 3. QRAIN BOUNDARIBS +2 METHOD OF FORMATION Figures 5ø, 5b and ðc, plates 9 and 10, show typical grain bound- +aries for bubbles of 1-87, 0-76 and 0-30mm. diameter respectively. +"The bubbles are blown from a fne orifce, beneath the surface The width of the disturbed area at the boundary, where the bubbles +of a soap solution. We have had the best results with a solution the have an irregular distribution, is in general greater the smaller the +formula of which was given to us by Mr Green of the Royal Institution. bubbles. In fñgure 5ø, which shows portions of several adjacent grains, +15-2c.c. of oleic acid (pure redistilled) is well shaken in 50c.c. of bubbles at a boundary between two grains adhere definitely to one +distilled water. TThis is mixed thoroughly with 73c.c. of 10% solution crystalline arrangement or the other. In fgure ðc there is a marked +of tri-ethanolamine and the mixture made up to 200c.c. To this is “Beilby layer” between the two grains. The smaill bubbles, as will be +added 164c.c. of pure glycerine. It is left to stand and the clear liquid seen, have a greater rigidity than the large ones, and this appears to +is drawn off from below. In some experiments this was diluted in three give rise to more irregularity at the interface. +times its volume of water to reduce viscosity. The orifice of the jet is Separate grains show up distinctly when photographs of polycrys- +about 5 mm. below the surface. A constant air pressure of 50 to 200 em. talline rafts such as fgures 5ø to 5c, plates 9 and 10, and figures 12ø +of water is supplied by means of two Winchester fasks. Normally the to 12e, plates 14 to 16, are viewed obliquely. With suitable lighting, +bubbles are remarkably uniform in size. Occasionally they issue in the foating raft of bubbles itself when viewed obliquely resembles a +an irregular manner, but this can be corrected by a change of jet or polished and etched metal in a remarkable way. +--- Trang 381 --- +Tt often happens that some “impurity atoms', or bubbles which that recrystallization may be expected. The boundaries approach and +are markedly larger or smaller than the average, are found in a poly- the strip is absorbed into a wider area of perfect crystal. +crystalline raft, and when this is so a large proportion of them are Figures 11a to 11ø, plates 13 and 14 are examples of arrangements +situated at the grain boundaries. It would be incorrect to say that which frequently appear in places where there is a local deficiency of +the irregular bubbles make their way to the boundaries; it is a defect bubbles. While a dislocation is seen as a dark stripe in a general view, +of the model that no đifusion of bubbles through the structure can these structures show up in the shape of the letter V or as triangles. +take place, mutual adjustments of neighbours alone being possible. A typical V structure is seen in fgure 11ø. When the model is being +It appears that the boundaries tend to readjust themselves by the đistorted, a V structure is formed by two dislocations meeting at an +growth of one crystal at the expense of another till they pass through inclination of 609; it is destroyed by the dislocations continuing along +the irregular atoms. their paths. Figure 11 shows a small triangle, which also embodies a +đislocation, for it will be noticed that the rows below the fault have +4. DISLOCATIONS one more bubble than these below. Tí a mild amount of “thermail +mmovement) is imposed by gentle agitation of one side of the crystal, +When a single crystal or polycrystalline raft is compressed, ex- such faulty places disappear and a perfect structure is formed. +tended, or otherwise deformed it exhibits a behaviour very similar Here and there in the crystals there is a blank space where a bubble +to that which has been pictured for metals subjected to strain. Úp is missing, showing as a black dot in a general view. Examples occur +to a certain limit the model is within its elastic range. Beyond that in fñgure 11g. Such a gap cannot be closed by a local readjustment, +point it yields by slip along one of the three equally inclined directions since filling the hole causes another to appear. Such holes both appear +of closely packed rows. Slip takes place by the bubbles in one row and disappear when the crystal is “cold-worked”. "These structures +moving forward over those in the next row by an amount equal to in the model suggest that similar local faults may exist in an actual +the distance between neighbours. It is very interesting to watch this metal. They may play a part in processes such as difusion or the order- +prOcess taking place. "The movement is not simultaneous along the đisorder change by reducing energy barriers in their neighbourhood, +whole row but begins at one end with the appearance of a “dislocation', and act as nuclei for crystallization in an allotropic change. +where there is locally one more bubble in the rows on one side of +the slip line as compared with those on the other. 'This đislocation 6. RECRYSTALLIZATION AND ANNEALING +then runs along the slip line from one side of the crystal to the other, Figures 12ø to 12e, plates 14 to 16, show the same raft of bubbles +the ñnal result being a slip by one “inter-atomic” distance. Such a at successive times. A raft covering the surface of the solution was +process has been invoked by Orowan, by Polanyi and by Taylor to given a vigorous stirring with a glass rake, and then left to adjust +explain the small forces required to produce plastic gliding in metal itself. Pigure 12ø shows is aspect about 1 sec. after stirring has ceased. +structures. The theory put forward by Taylor (1934) to explain the The raft is broken into a number of small “crystallites'; these are in a +mechanism of plastic deformation of crystals considers the mutual high state of non-homogeneous strain as is shown by the numerous +action and equilibrium of such đislocations. The bubbles afford a very dislocations and other faults. The following photograph (fgure 12) +striking picture of what has been supposed to take place in the metal. shows the same raft 32sec. later. The small grains have coalesced +Sometimes the dislocations run along quite slowly, taking a matter of to form larger grains, and much of the strain has disappeared in the +seconds to cross a crystal; stationary dislocations also are to be seen process. Recrystallization takes place right through the series, the +in crystals which are not homogeneously strained. 'They appear as last three photographs of which show the appearance of the raft 2, 14 +short black lines, and can be seen in the series of photographs, figures and 25 min. after the initial stirring. Ít is not possible to follow the +12ø to 12e, plates 14 to 16. When a polycrystalline raft is compressed, rearrangement for much longer times, because the bubbles shrink after +these dark lines are seen to be dashing about ¡in all directions across long standing, apparently due to the difusion of air through their +the crystals. walls, and they also become thin and tend to burst. No agitation +Eigures 6a, 6b and 6c, plates 10 and 11, show examples of disloca- was given to the model during this process. An ever slower process +tỉions. In fgure 6a, where the diameter of the bubbles is 1-9 mm., the of rearrangement goes on, the movement of the bubbles in one part +dislocation is very local, extending over about six bubbles. In fñgure 6b of the raft setting up strains which activate a rearrangement in a +(diameter 0-76 mm.) it extends over twelve bubbles, and in figure 6e neighbouring part, and that in its turn still another. +(diameter 0-30 mm.) its influence can be traced for a length of about A number of interesting points are to be seen in this series. Note +fñfty bubbles. The greater rigidity of the small bubbles leads to longer the three small grains at the points indicated by the co-ordinates AA, +đislocations. "The study of any mass of bubbles shows, however, that BB,CC. A persists, though changed in form, throughout the whole +there is not a standard length of dislocation for cach size. The length series. Ö is still present after 14min., but has disappeared in 25 mỉn., +depends upon the nature of the strain in the crystal. A boundary leaving behind it four dislocations marking internal strain in the grain. +between two crystals with corresponding axes at approximately 30 Grain Ở shrinks and finally disappears in fgure 12d, leaving a hole +(the maximum angle which can occur) may be regarded as a series of and a V which has disappeared in fgure 12c. At the same time the +đislocations in alternate rows, and in this case the dislocations are ill-defned boundary in fñgure 12đ at DD has become a defnite one in +very short. As the angle between the neighbouring crystals decreases, figure 12e. Note also the straightening out of the grain boundary in +the dislocations occur at wider intervals and at the same time become the neighbourhood of 2 in fgures 12b to 12e. Dislocations of various +longer, tỉ one fñnally has single dislocations in a large body of perfect lengths can be seen, marking all stages between a slight warping of the +structure as shown in figures 6ø, 6b and 6c. structure and a defñnite boundary. Holes where bubbles are missing +Eigure 7, plate 11, shows three parallel dislocations. TỶ we call them show up as black dots. Some of these holes are formed or flled up by +positive and negative (following Taylor) they are positive, negative, movements of dislocations, but others represent places where a bubble +positive, reading from left to right. The strip between the last two has has burst. Many examples of V”s and some of triangles can be seen. +three bubbles in excess, as can be seen by looking along the rows in a Other interesting points will be apparent from a study of this series +horizontal direction. Figure 8, plate 12, shows a đislocation projecting of photbographs. +from a grain boundary, an efect often observed. Eigures 13ø, 13b and 13c, plate 17, show a portion of a raft +Figure 9, plate 12, shows a place where two bubbles take the 1sec., 4sec. and 4min. after the stirring process, and is interesting +place of one. 'This may be regarded as a limiting case of positive and as showing two successive stages in the relaxation towards a more +negative dislocations on neighbouring rows, with the compressive sides perfect arrangement. 'The changes show up well when one looks in a +of the dislocations facing cach other. The confrary case would lead to glancing đirection across the page. "The arrangement 1S VeOTV broken in +a hole in the structure, one bubble being missing at the point where figure 13a. In figure 13 the bubbles have grouped themselves in rows, +the dislocations met. but the curvature of these rows indicates a high degree of internal +strain. In fgure l1ä3c this strain has been relieved by the formation of +5. OTHER TYPES OF FAULT a new boundary at A-A, the rows on either side now being straight. +It would appear that the energy of this strained crystal is greater +Figure 10, plate 12, shows a narrow strip between bwo crystals of than that of the intercrystalline boundary. We are indebted to Messrs +parallel orientation, the strip being crossed by a number of fault lines Kodak for the photographs of fgure 13, which were taken when the +where the bubbles are not in close packing. It is in such places as these cinematograph film referred to below was produced. +--- Trang 382 --- +7. EFFECT OF IMPURITY ATOM boundaries when single crystal and polycrystalline rafts are sheared, +. . . compressed, or extended. Moreover, if the soap solution is placed in a +. Pigure 14, plate 18, shows the widespread effect of a bubble which glass vessel with a fat bottom, the model lends itself to projection on +is of the wrong size. If this figure is compared with the perfect rafts a large scale by transmitbed light. Since a certain depth is required +shown in fgures 2 and 4, plate 8, it will be seen that three bubbles, one for producing the bubbles, and the solution is rather opaque, it is +larger and two smaller than normal, disturb the regularity of the rows desirable to make the projection through a glass block resting on the +over the whole of the fñgure. As has been mentioned above, bubbles bottom of the vessel and just submerged beneath the surface. +of the wrong size are generally found in the grain boundaries, where In conclusion, we wish to express our thanks to Mr Ơ. E. Harrold, +holes of irregular size occur which can accommodate them. of King's College, Cambridge, who made for us some of the pipetbes +which were used to produce the bubbles. +8. MECHANICAL PROPERTIES OF THE TWO-DIMENSIONAL MODEL +The mechanical properties of a two-dimensional perfect raft have R.EFERENCES +been described in the paper referred to above (Đragg 1942”). The raft +lies between two parallel springs dipping horizontally in the surface Bragg, W. L. 19424 Nature, 149, 511. +of the soap solution. The pitch of the springs is adjusted to ft the Bragg, W. L. 1942b J. Sc¿. Instrum., 19, 148. +spacing of the rows of bubbles, which then adhere firmly to them. One Taylor, G. I. 1934 Proc. Roy. Soc. A, 145, 362. +spring can be translated parallel to itself by a micrometer screw, and +the other is supported by ©wo thin vertical glass fbres. The shearing +stress can be measured by noting the deflexion of the glass fibres. +When subjected to a shearing strain, the raft obeys Hooke”'s law of +elasticity up to the point where the elastic limit is reached. It then +slips along some intermediate row by an amount equal to the width of +one bubble. The elastic shear and slip can be repeated several times. +"The elastic limit is approximately reached when one side of the raft +has been sheared by an amount equal to a bubble width past the other +side. 'Phis feature supports the basic assumption made by one of us in +the calculation of the elastic limit of a metal (Bragg 1942a), in which +it is supposed that each crystallite in a cold-worked metal only yields +when the strain in it has reached such a value that energy is released +by the slip. +A calculation has been made by M. M. NÑicolson of the forces +between the bubbles, and will be published shortly. I% shows two +interesting points. “The curve for the variation of potential energy +with distance between centres is very similar to those which have +been plotted for atoms. It has a minimum for a distance between +centres slightly less than a free bubble diameter, and rises sharplÌy +for smaller distances. Further, the rise is extremely sharp for bubbles +of 0-1mm. diameter but much less so for bubbles of 1 mm. diameter, +thus confirming the impression given by the model that the small +bubbles behave as if they were much more rigid than the large ones. +9. THREE-DIMENSIONAL ASSEMBLAGES +Tf the bubbles are allowed to accumulate in multiple layers on the +surface, they form a mass of three-dimensional “crystals` with one of +the arrangements of closest packing. Figure l5, plate 18, shows an +oblique view of such a mass; its resemblance to a polished and etched +metal surface is noticeable. In fgure 16, plate 20, a similar mass is +seen viewed normally. Parts of the structure are definitely in cubic +closest packing, the outer surface being the (111) face or (100) face. +Pigure 17a, plate 19, shows a (111) face. The outlines of the three +bubbles on which each upper bubble rests can be clearly seen, and the +next layer of these bubbles is faintly visible in a position not beneath +the uppermost layer, showing that the packing of the (111) planes +has the well-known cubic succession. Figure 17, plate 19, shows a +(100) face with each bubble resting on four others. The cubic axes +are of course inclined at 45° to the close-packed rows of the surface +layer. Figure 17c, plate 19, shows a twin in the cubic structure across +the face (111). The uppermost faces are (111) and (100), and they +make a small angle with each other, though this is not apparent in the +figure; it shows up in an oblique view. Eigure 17d, plate 19, appears +to show both the cubic and hexagonal succession of closely packed +planes, but it is difficult to verify whether the left-hand side follows +the true hexagonal close-packed structure because it is not certain +that the assemblage had a depth of more than two layers at this point. +Many instances of twins, and of intercrystalline boundaries, can be +seen in figure 16, plate 20. +Eigure 18, plate 21, shows several dislocations in a three-dimen- +sional structure subjected to a bending strain. +10. DEMONSTRATION OF THE MODEL +'With the co-operation of Messrs Kodak, a 16 mm. cinematograph +film has been made of the movements of the dislocations and grain +--- Trang 383 --- +` —s——-———~- ` `Z `Z + 1^`Z `Z `Z `* `Z ` xxx 1xx 1xx TH 1xx xxx x1 "xứ x* +®I919181919 QC) 3 3 3 2€ 2k 2€ 2k 2L 2 22 2.2L 2L 2( 22 2L 2 3 3 3 3 3 3 3 34 +. 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S.C c c °¿e°¿€9,€6 7e °g( +xxx và th vào về si lv eS9 6946669069499, 6°°. +TL KG 1 e1 k9 v1 sSss seo cceveteisssS +x62 x c1 3 6566 c s9 ccctetsẺ +lttcccccG C22 44995 ve 66 S6 cá 395 (910 c°46ce°(¿€6/s %c$9(€,- +4 CÁ SN ST Qs4S ve 3 99 vest 52s S62 s €9 e°€ 2e 56%: +` 4 XS». XXXXN © ŠS©S®, { “4 X ˆ%9.. 995.6 %9 y $9 6949 e9°¿9-9.. +N YLS-À^ SA ^vóCX {XS vv (424 S9. 69/9. 6 96/4:9 +%S%%%%%%‹ xxx 4 lề, Na Xu 4« X Xác Xót bỆi chà St sS4ss9s 463. +VS^Ä, vệ ` 4 ¿4 h s..°. +Vi vt4944446369/ 0c+6961L4%9/9 6966666696956 56 69 €) +EIGURE 5a. Diameter 1-87mm. +FIGURE 5b. Diameter 0-76mm. +--- Trang 385 --- +(C>‹ (>t C>‹ C>C>t (>({>(>(>(c) ( ( +( ( { { 4 4 L v, 1Í { { ( Ì í ( +( ( ( ( { { (3> ( ( ( ( ( +{ c{Cc(C£{c + ¿€Cet ( { ( { ( F +è -.1 { ` I › { { { › ( › { ` ( › ( Ầ { +( C*C>{c`c€ct+c€ccct€ecccc‹ +Á Ạ í ` í ( › : { › , { › { ` { › { : +Œ1 c C c ¿q ( CcCrCe+4 cÁ{©c<: Cổẩ{Cc‹ +cCccC>zcC>c>c`{(c*c‹c<‹cvsc»:£k*£tvch(t C>‹« € '@›¬@:~ ( h ( ( +tí ` ` z ` { ` { › { * f ` ¿ ` { ` { : í t ` +4 va, \ ` C ®›—‹ (C>x‹ C>‹ ( ( 1 +CA csc ao an® ( c€c£€C£{cˆ+' ( F +£Ẳ«€£ece‹c ( ®( C3 cC‹ C*c©c' L3 ‹ +( ch —- tự Š©VCccc< ( ( +>(C>(C>({C>(>c>c*c©kscv‹ * ` 4 (z>(«( +chsccsSc~-9›-~- 8>~ @>—: { Á Ử { { 2C r¬=..r +{ C>({€>(€đ( { ( { ( { ({ `›#^ (C7 .(Y~e‹\ +{ ( '@>^ X >‹ Đ—‹ { ( 0 a2. 9 41 ay“Ñc‹ +( 9›<9-‹®>-8›-e;—. ( F tỳ sô” 9a'..r.9 e4 4y.“ Œ, ( +Cz€C é { ( ( ( F c>O£Z© {` Ctr+z ( ¡ +Ề { { ụ { ( Foy, ( `. é ¿ 9 q3 y2 SC +c©£©cecc' C4 C*w© Cy '97e>eove®7@>aye SỬ @Ỳ, CC: +C-.c>c>‹ t 98. S3 4C CO #vZ v(CXYX ?z.({vẻ5xxX( +CS tCv cxz©‹ { z\_( { >\_( { . tĐưzt ' cự cz +Cr©S*{,©Cc££©Cct£zCcc¿z‹<{ccerccczZzCctz©ce +C—-{€©^X Á xYz! 'Cfz by C£eZ¿vCc“^.‹cv-.-x. che +®; C3 ztC Cẽzt,X Crz{Ccy$¿C©Cct#,c©Ccv-Ẳ<( &¿C( +` ⁄ @ C* < ( (€Œ z Ñ'.ư ` yv^‹ ¬*> ( - C +CrŠCCc€rcCcez©cz¿©Ct ŒC€crEcCtrCQG cCo%c +( 29 'ere..8: ' 1a 9)e'2..9eyay. 9 ey, ©CC£ez¿Cẽ CC 9 +CếZzC(Cczc (CÔ CtzS,Ccv¿S,CvyzScCrế5=<ốt: '®'exe ®! @}, +CC 242C CO zSG Co £zCcC£zCCt£zSÔO£>S.Cry>z<{©c>" +EIGURE ðc. A grain boundary. Diameter 0-30 mm. +CC r0 LWLWLWIQP(00(07(00|Á00100(Q01Q71009V//2@72 7/2269 xo xsv> xxx xxx +tv4v24v. +. 2. >> ».®..2L,2ÝNLÊN NA VNI LAN LAN LAN LAN Lá NT Tư xx¬ +VY MÔ VÀ VY Y VY Vy YY%YWYXYY%XWYYở +---S-.H4 NT. LÊN LVL2W-¿ẺAQ: £\ MA ÁtO ÁN LÀN ÁN Ai ~ =2 +)®'®®G®®66S8S8686982422-2-2--a-^2'“ bí N NÊN ' +ÁN NÊN ẤN LAN ÁN ỨC VN VQ CC CC: VY: ẢCQ AC ẤM LÁ CÁN ÁN Án +VY NV VY V X XYXY T2 v22 VY YY )®œ:®r NÓ › +“2z z2 ->-»-.ÁL2 L2 L2 ẤN ẤN AM ẢN CC AI LAI AC S2 +I®14@1414 1® 1® ®:®4®4@ s2 22asas^as^^^^®Ằ®®Ẳ®%: +2-22 c2 va .4 LOAN ÁN ẤN ÂN ẨN ẢN AI ĐÀ LÊN TS 2⁄2 +I@10'01®1®9®%®e®œooaaooosssoœeœ®%® +R.Yáe, Yá8,Y⁄68.Y⁄48.Yáe`YZS`YZ2`Y⁄a v⁄^ 7< 225 c2 C2 C22 V22 va 22 YZe`Y682Y8YAfYAfeYERY +JEIGURE 6ø. A dislocation. Diameter 1-9mm. +--- Trang 386 --- +Dislocations +` ¬¬¬¬¬¬¬¬ ¬¬¬ ¬ "` +¬¬^¬^¬^^¬^¬^¬¬¬¬ +bi Lb [E TE) l6: He Xe lập #T@T4@1/@1/f@ T61 T@I@&f&E@16 +t9 Tết l5 0u 1 X6 X6 1@ {6J@Ïf,Y&TÏE Y6 S166 +1¬ ¬¬^¬^¬^¬^¬¬^¬¬ +¬ 51.335 ^2^ 22» 1¬ ¬` +1 ^^ ^^ ^^ ờốẽ Tố +Ð16 16116161616 11@19141@1811®f%1 (611/161 +5 22^2^¬¬¬¬ +saieIeieTeïe1e1efe1e1e191e1e14e1616 i6 .1101) +TC Xe ae VoVoVeVeV-vV 2¬ v2.4 2 9-4 .V-V.V-YeVaVee' +FIGURE 6b. Diameter 0-76mm. +Y4 Y/ *í Xí Yý Y4 VỊ VÝ Y 1000000002110) TA TA A0 00000601000001/0(6100010/900/0/6019/0/96196096/ 361 v11 113 +00 00969606666090908902.6.4i, 'â:ê:ê:4 {4x vý xý ví lôtô'ẩ: rô 'ê :ê ‹ð rễ : Š:&:&: Ê ‹: cv +J/8:8:8'â:â:ã'&:ô†: '&:ô:: ví (V0 /VIA6.006, COI0/0/006/00600066 6 6.( ‡ +t0000000000/00900100 00010001000010000110000000110020011010111000001010010001100117001000005300.4.2,4.402/ +H011010000010100000104014110111041211/174 A00 10001001100014000001100011000010/401080100110002 đi AI 9/018/201018 +lề ô:§1ễ:â:â:š'ê:Ê¿'ê'ê 22020; Jớt 2200202020007 270000006100000000515 “ư “á¿ ệ #: +Z2 J(2 .À›4-9:4:Â.4:4:â:4:À:Á:Â-ê:4:â:á-á'á'á:ð'á'§ Ê'ổ ê 2220162010 Q01(24/(142172-031 ị tt 2Ì +J(JÍ 734 )(7t3Í TY Yí TỊ Ví 3ƒ YY VY v Y v/Y/vÝ vý v.v v2 v.v v2 v2 x2v22⁄)v)02()02())6À026202022016222)62071:234x \ : +J177()(J (Y3 Wypri .ê:4-à.§.Ê:ê:Ê:êrễ:â-ô- 8â: à:â-à-âzêrê-ê:š:à-Ê:â-À-ð4⁄8:6 4-6-4; 91091002-4:4:6:4, 7YÍ: +g04/4000008000/0000017 TẾ Yƒ ví vị + xi );ê;á:Ê:ô:Š:ê:ð:ễrễ:ô:âê 20/0701110101090/0000005)000100 T00 đD 1 +/#12:90:9:81( 'ˆ:4: À (GV /VGJ00V0 00060000). NA Ld.v .. : tí xf ý v 1 ñ +lổ:ê:â:â:â:â: HẤT $.$:8/8/4/8:8:4 IỆ/61Ê;8 Ý;4;Ê:â:ê:Ê: lê:â:8;Í 'â:Ê”aYẾ ! rễrế: 'ê:â:â:9:4:8:Í ~À ~Â. 222 v/v Yƒ :ánh:Ẻ-}:Ê +FIGURE 6c. Diameter 0-30mm. +.. 32C} X3 C)C)C)C): “ ẺJAVi v.v. 9/4, 9/419/v1V21ý947<9ý2XväyấĂ tung: ‹$. tư .yẽ +MX 3} ( X 9990900000000 00067016 70.70.7707... 700060001 +XOC X b R bà, XXX X 3X 3C) 3C) )C3C 3 3 C3) XX XX X33 3X X J3) 30 +Ê-9:9:9'9:4'0:á'4'4 4 vẻ 6 2220 .69909096600999099990 6990999690 .0:9 +CXX X3 X XS X X51, )VÈC% X 3X X X3 XXXXXXXXXXXXYYXXX XS) +XX XX X XS V2 )290XXXX XS» XX X3 X3 XS XX X X3 šX ý X3 X X3 +XXXXXXXXX VY k2) X XMX xXXX XS X XX X3 X xXXXx< XS CX3 X3) +MXX⁄XXXX%XX X Y XU} 9909690909909 0909099069000900009009/009099.6/90:/0. +XXXXXXX Y XNK X20 0X X XS XS XXX X3 3X X +CXX X XX X X X3 3 122XXXX XS X3 XX XX XS XXYX X3 X3 3X X) +XXXXXXX (3X X22 SKX XS XS X3 K3 X XY YY XX) +KXX X XS XS X X3 312⁄0CXX X3 XE 3X) XYX XU) XXXX3) +XXXX XX X XS X1 2VVYXXXXXY X3 X X3 XXXXXXXXx2 X X X2) +KỀ⁄CX X XNK XJ CC MX MS XXXX M) KXXXXSXY)9 x3 X⁄ÈÈXÈX⁄X⁄kšXXð) +MXX X )CRCX XCK X Y XNK XP CKMOCXX XX (3 Kỳ) kh XX X9 X X2 ) +MXXMXME.E.MMENRVXYKYVEKXRKE.XXEXYEYXXYXY)0 YLE)YPXFK.XKEXXKEXX*XKXk>x} +là. 0 9900000006000 060000696060 0.0/0006.0.0.00606 000.1096.000 020 .0.9.9, +ˆ.., XE CC} D CC} ME KCMXMXKXkKMXKk% <6 * h3} }⁄é bhXếh}*XXXY) +; 1190090000690 20190000 6200 09290609.0.09.0160. 006720909009. +k}} MO CN 2C CC» CMXKCKĐXMXKXXXXXĐWXXXb*R} kh} h3 +ì {3X CC) CV X3 XXN) KCM)() (X XkXY XYškYẺYYÈ})3 E}X}) +9090990020600 (009090996 909/096900209009909009900900200 0600906 +MXH X MCXCXCX XM )OOCKk)C KV XXšXXXXXKXKkX k3 X3 X2 +` < ` ` Yăn Yâm Y4 Ya Yận Y6 Yến Yân Yăn Yđ Yến ` X6 6a YẾG VN, Vựt Vận ` +FEIGURE 7. Parallel dislocations. Diameter 0-76 mm. +--- Trang 387 --- +twvè tY rên có +~~SSX TK v CC GG +y nò &, , SN xS*x +JIGURE 8. Dislocation projecting from a grain boundary. Diameter 0-30mm. +S2 2 2N NT SN NCT XWMN +KG TẠO VAT S0 2V 1À 2/2 2 v20 /2vÀA +TS SN SN SẺ VÀ A0 2 2VA/2Vv2VAV +TH NA Xz NNNNN@Wwv3 +NÀNG NA A1 NV; +N N N NNưN*x` hở ` 2222... +NA NA ƯA AI AC SN Á Á LÁ AI AÁ (2A (A- +7N SN NV NT 1 VN N X V Y Ỷ +D2 00000010 0V 0 V00 V00 S0 0À VÀU VÀỔ +TÊN NT NT 1N TƯ NV NV VN NZ +NA NA Lá áÁa-¬-^-¬-^- ^^... ^.X. +7N NV CN SN N N NZ N M CÁ VY XÝ XY X +NA NG uN ưN ua ưa... . `. +JHIGURE 9. Dislocations in adjacent rows. Diameter 1-9mm. +PIGURE 10. Series of fault lines bebtween bwo areas of parallel orientation. Diameter 0-30 mm. +--- Trang 388 --- +^^ ^X ^^ ^^ ¬¬¬ˆ ^^ ` .®:@.®)9: +^^ ^^¬^¬¬¬ 5 5= x40, 4, j +5 ^ ^ 02 ^¬^¬^¬^^^¬^^¬¬¬¬— ^^ ¡#@:®), +^^ 2n2^^¬^¬^ ¬^¬¬¬- ^^ +^^ 0 ^^^^¬¬¬¬5^ : H } „#4; +^^ ^ ^^ 2^^^^¬^¬^¬¬^^¬¬ ` ` ^ Ya'@e.®;®. +^^ ^^ ^^ ^^ ^^ ^^ ¬¬ ^^ Ta mm +^^ ^ ^^ ^^^^ ^^ ^^ ^ ¬ ˆ ;£/®, ) +^^ ^^ ^^ ^^ ^^ ^^ ^ ^^ ¬¬ ^^ ` Tư n 'j +^^ ^^^^^ ^^ ^^ ^ ^ ^^ ^ ^^ ^^ +^^ ^ ^^ Z~^¬¬¬¬^ ^^ ^^ ^^ : +^^^^^~XXŠ¬^¬^¬¬¬^ ¬¬¬¬¬ˆ ¬ˆ¬ “4a. ) +^^ ^^ ^^^~¿#Š ^^ ^ ^^ ^^ ^^ ^ +^^ ^ ^^~XX¬~ ^^ 3^¬^^^¬¬¬¬¬^^¬ Ỷ ; +^”^^^Á^^~^ .0:0/: _^^^^^^^^^^^ \/ +^2^^^ZX—~~ ^^ ¬^¬^^¬^¬¬^ +^^^^^^~XX~>~~ẽ==— ` +^^ ¬^¬¬¬¬¬ ^^^^^^¬^^ ; +^~ I®ia2innsiasasaaanamanaaAA' ` +^^ ^^ ¬¬ ^^ ^^ ^^ ^^ : ^^ +TP -2^^^^^^¬^¬^¬^¬¬¬¬¬~^~~ ^^» : +Mạn set 9<9)659:9.9.079-9.6.9.9.611.919.0: kia 07212 2YAY: +Diameter 0-68 mm. Diameter 0-68 mm. +¡ế'$¡#:918:4191919141474:§'£1'7'1§7£” +1w s99 9090®@9 09091904996 Ó(©Ò©Ó ®€ 4919 1% +z*⁄ TY Y TY YYCYYSYYÁvÁy, 1994/94/99 9/4 4/9 9:99 1 ??7/ +¬¬⁄¬ 2¬ ^^ ^^ ^^ ^^¬^ J9:9/9/9/9/9/4'4/9/4'6-9:6:9'%'9:9:9'9'9 +2¬ 2^2^^ ^^ v00 002520/0050/913/2012542/212 +CC) 2C. NA OÁ À UÀ À2 ý. 949e9€ 4944499696909 96940999 95696. +®I®I®ï®I®ï@I®ï@Iereïere1e) 1/21/3/5/3/3/1/3/5/3, 4 3/3/3/0/3/1/2/5/ 5:22 +l®Y®Y®Y®YeYYYYSYY.Y.Y, Jgie/e:4 lVie's ii”? gia7e/e:9:e:e:9:9'/ +[®I1®1®1®1®1®ïe1®1®1®I®I®© $/9/919'e:eis'e'e”/ 25s 4;e/e:9:0/9:ý19 +CC) €Oœ®_.¬^œ€œ€ I1) lv... D422, 15414 2 3(1/2/2/3/027)ƒ +CCOSccc©cccG l0501601100014010/01)/10/ +“—‹⁄ ⁄ 27^ S2 ; ; ,Ю:®.4 d Ủvv xý3/^5#y ;‹ „4 ) #)1 +®Í®11®1X 91% œ[®I®[®ï® (Á J0 (J10/0010010711)1)00)105 +J®I®I®1I® ©-.9j.%j/8;s IoYe1eïe:e1e1eieïeïe) +IĐTĐIĐIĐIĐIĐIĐIĐ1, “¬£@»Š® +Diameter 0-6mm. Diameter 0-6 mm. +JIGURE 11. Types of fault. +--- Trang 389 --- +F) - È = b ⁄ ¬ = È ` ^ 5 b = ¬ = k` 5 4 = : - È ` ) >` > +"—݈đỄ>đm®>»>4®>+r‹®>-‹a®>‹8, >®>‹<®>‹8®>-‹ . 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Mu. ta ^Ẻox92ug1x^' do. “9o “ao. 0y, lí, vì +ra d0u 90s ddoaa6di v98 coete00o e6, “ìaa 00.099 94g1gðuwWuevo) +e0 0u d0 9à v9 hết uôyu ng 2, NI đọ 0 bí y6, +S92 2-0 býd4á4„°6u9/ 209 lạ 002 nau? 924.9965601 0e ca 0á" NƯợề, +t Thù vêdd 0v, ˆ cw%‹ r #Ó Là N0 +SA 040 2061492490049. 3Ó vạ v 9. v04.69 20491 g3 seo v0xbbds0 005, +v94. da .01440edvaiS‹Saaed “Sá.a22 200 S0 2 v2 4V, CC CÁ» (6i +ø. After 1 sec. +š5u61""vygu¿ữg: F9 .“0”"g9~*c8*“8zïi lUuuux8uxi KH nh cá +1 qr? 0-6960 00-60-06 rờ S0 ạ Bọ By n0 500509500505 yt tự +vệ ¿ae 669900 a-d4 l1 nu u86 8u8w8uBuwBuddvw90grtoyw 9o g' +\ '9ựwW"u9.,.u 19% g0. vu vu ,N xưy ve ve +¿ 9500000099 -2ng6u56uBuMuyMudadswð 5x02 èi +S1 GA As say XY 7 ra tờ 0 u62 Aa#udfuubod xi SvA v2 9v +hờ nh TY #0 0 00469 2v g0. 0 0904 2ð ð tí lu VỀ, +M _^^^x Sư u21 v0 Su 2 vứt xổ, 2-2 axý sụt, vì ++1 CA TA rà 8u È ạy By ạ In “ À, Êtœ9-đu) Rg lu 4-0-0 212 0A vờ bạ 9 uy +“„g: — +lu d9. e0 v96 u80 80v È S:á-do8:á:ê-d-d. 05 v#ñ-d-ô-dd 4 da y0 +s00 6 lạ e 9, g9, lại ừw at .w6-aa d.d-d.d-ds Y KKXYX VY x/6yu 5 v9, +cv ra Sự‹ vụ 0 Bầu —— h <- hôi .n Uy, v +c1 hấu bi tà D-c1 là 80g A4 qê-t-0-0-ê-6-4-d-d-o- 4 đưê-ô-do 0A 40V sợi +- W TƯ« Ỷ—^ ` 'âu..W vu”, +sưu SẮu bí hạ tí Bu ạiÈ o ạ ly Xu rủ 3u ae e0 09 :6:9-6-9-0-4-0 tô dagfuyB„vuợ! +NV nh À Họ hạ Mr tờ gụ dành cụ at đệ gà tt 2i 2894 41050000n5/9194,0wvwxvSi +g kh se X ~ À-^ — +sâ09 1. vu ve 6 gi * rí “ó‹ dd dd du. +; vi Chu ựh " wW"”, rl*~®.wW^*S"" "Jvạaea. nh daac .,. Đ, W0 +„ c^á0 v05) bờ na nu 0 8/60 g exờ-gc9.9 4c 4 0 v9.9 9590.0.00005324:4 — +1n da u90) s0v/9uu9 ví aw00066)6 66909 iu cứ dau NV +ST Ặ TT rr - h— —-< +`4 + + 29949 1e+eee°o 9 bê t9: 960 0060902001099 1g. 0đđđ đa g yố +b. After 4 sec. +* 1 ~ˆ .„_.m=.m.“=< vÓ “: ~ .~ 4“ x~ ~— _— ⁄ +l0 g0 (0-0 -0-0-á:0-0-4.0-4-00 v10 Va ray 0w 0 cay vụ uy wyVM trời xù Dụyg bày ð cụ: +lv UV, V acád rat te te t5 vu, vờ, v9, u59, 0w9y0, 9,02, lu ð vờ CS +xiên VU, toà và dxvub se b9, 9 yW y9 “yờ, ụu,V SwyEð, 5,0, wð,u5g0, gồ. +TM Apcoe ro ưu vờ at uab wừW, vờ, wSuyW dưðy„0Syð ,d5#yY vờ vŠy +Tu Suy Sov uy 0S vu wð with ov vÝ wdlyy vờ, 5u SwyW Mê +xuX,v vvdv u29, vu, v%,vớ, g0 ch Sa yờ vu. p-) +nh ——— —— T— —— --. SH Số. - “ hd“ +HN Tư vư ty t”/,uy® cv vu“ayk hd tdtáu,, SẺ, nh -“‹ hy." +ch >> TT ằ hố TH ằ~ ——`————-. lv. +6 ——— ...* — +Tu vớ vớ ào Vu iu lu 0S ad dd va 00-0 san đê đu ww Đụ, +SỈ ke ri ~ "vư “.ị ^~ ~ `"... ¬đ nh WWN -Ó-4 .W_vw" +ST s——x~ —— TT NT“. +"1 — cu cay, rằ-ôád dc 'ứdd-g g0 â--0-â-9-40 2 gu bu ð: +—Ý— TT —-¬ RA ví 6)-đ¬ệ¬ { ` +và và a9 và. utv1ua9 900990946400 2924994944 4(4g 2u Vụ: +nh h h"h — —ằ — — —— xxx ` *-~= +và Ánh v.v. v.v u?% vut vv" vÀ. ——~ — Phụ, +KG Men poruuvvuvvboarbsuwakisia049006 4010069949444 ẤP: +-. HS - -¬——^—-»—-»-. 1 VÀ SÁ2-Ào£ 25 +—»-YÝŸ`yý—. nh —~ TT ——-.Ÿỷ.Ÿ-ớƑ._Ƒ.~..~.. 1`} < -—“ +ca adcoaodaL vn s29 v9 Ý và”. ut (v0 c6 dooaca đÔiddgad@ ad M.Á) +1.11 ' v.v (doi t0 0. 1w 6660909 20 (0010090909094 dg 0v. 09699 (đgý. +“ ` ráo đa e‹ -.suxa°*“.et19990902220 9.00991029604696 690900 9090x. +xo USH bóc: +ĐÀ vợ 22 ( +lIỆP SĐ2X/2/c2⁄⁄21.:L A +c. After 4 min. +JEIGURE 13. TWwo stages of recrystallization. Diameter 1-64mm. +--- Trang 393 --- +2A Ý ga T: ®T/®Y®Y/Y.®Yï®T/TYY®T/®T®Yâ®Y®Sï®Y®T1®Ya +' c )C)C)C)C)C CC C) CC CC) CC CC )C)C)C +CCC)CCCC CC CC CC C}C} +` ` Á CÀ ° AE Ác /Â< /Á« /Á« /Á« /Á« /Á+ . ` * ` +C)C)C)C)C)C)C)C CC )C)C)C CC CC CC )C +CA TA A À ÀO ÀO ÀC CC ÀC CÁC CÀ CÀ VÀ +` `ế ⁄ T42 Y2®Y@®Y2®®Y2œ ọ 2B Y 4 +®1@®1®1®1919181®181®1818181®18®18191®1@1@1 +£ £ £ £ F 3 YY“ mY 2SY TY ®Y đ®Y@® mm ^ r^ = ` 4đ +[8ï81@181®)®)®@1@r8i881@881®:®1®:81814 +%®1@1®181®I81®7@1@:91®1@1819181818181814 +C%)C%-X 6 C. ^^ Y NV NV VY C "ð ' 2B sả +Br@®I@1@Y cX. —®-‹( ẢC)C)C '®1@: r®Y ®Y 4ỀY. +@®181@®18:8181®Ý7w®.®I®:®/8/81®19:8:818 +sT®T®ï®7®Iï@ïr@i@)S^5 = 7 >~œT®Y®Y®Tï®®T/®T4Y. +53G. ` aaes08965666696686I +@1@181@I9®/5/` xa :era®i® 9.21918181891814 +Sï@7®r&r®r®@1I@1@19/0/S)) 75 ?ararxaraYa'2. +ĐÁ Ác Ás Ât ^~X⁄“ ~T⁄ C C € £ lÑ h “AC ( Á» (- SA +@iØ®/G//S.``a'a:e:.ai®/919191@181818ì4 +- ⁄ ⁄ ⁄ F £ £ & Ác h^ ^^ zZ^~s^~s—~2^- +®i9:9:81919191917);arar®i9:81818:818:8)4): +®%® ®®` ` xaaaa8é @18919181818)14 +⁄ rườY, "5 `” ` -`.‹.xe2e^ 2. “*⁄2^ +ĐI. `a:a:a ai®i®:8i6 {-))CX-)-)\ +f L ( ¬¬ ` `. .‹.ẻ2zY2ườ2ư t2 +CC) )c Xe Ác >2 CC )e)C 1@1@ Ệ 1@ Ệ 18 \ +ĐO xarxa:ai@®i8i8.81919191@114 ++)Á Xe Xe Äs ^s/^/S2S“ SG) 3) )te Js )(- )C( )(S +NV)” >»aaaaa®38®66° a8“ +HIGURE 14. Effect of atoms of impurity. Diameter of uniform bubbles about 1-3mm. +FEAV/i. - š //Hm ì : +.—~ tí LÁ S4 22⁄27227 222, 3y 7 7 +ý ¬ 3 T//,) ⁄ 722722 2⁄:/1159 +k > J⁄/////// V722 s2 , ⁄ (6% “Tfff +, __ 0 /277211THỀN S526 ` 2/ Ạ +NG _— E....... Ni Sa 4. cáo éc +—=. «xe 0 0227272 +SA: K= <<. +—. ‹ ` TRUY 5z đt += Ki. S223) 22227/ 72 <2 +` ` \\G k NA sa: Đ-- +Nà am ` Sinš: 28 gà +An M..^-:- : Ga là +` -...\(Ì bề: ì Xa = +° WAWWWNNWNNUOWK.. = +" «<5 `. = +% 2⁄72 đï VN 54 cà n +=s TT Sa NNN _=N= +—...... : ⁄ 7777777727) CN +EZ277/77: 1//77//Ì//ƒ(/; (7/77... +Nó "—.-. +}\ —- (/ +l4 —.- rz z2 s : +2< "“_. +EIGURE 15. Oblique view of three-dimensional raft. +--- Trang 394 --- +KNGả s. x2“4 _~ề -= „` "xu +` xã .\ _. . . : ¬z'` «- 4 — _” +xi v1 y ¿ri < TS. số +` : 2 ` ( ` “( _ỂNG VY v ví ¿ v +-““8 »- “ À „. W*. 1x. ' %» +` NĂNG VY YNG CC +_— 4 _-. 5 ` - `... “ : +r ` { —: ~.«* Ầ “ah- + F Sa Tờ T SUÀ Ý XOÀ Ý, : +` ắ Gv í vs “4 .”'> tì + ' \ M tr +` Ẳ ." : V. ì 1 1 ` -È 4 ˆ ` .ˆ q ` +\ “` No g `. tá T8 T2. ., “Sà` +>- LŠ + ¬ +`. p...Y „.. xi sẮ TY + w ww : _,# +` —“ + +. V Ầ +: \ "ð . ị ` “4 .. _“ `“ NT» Là +ˆ.' V* .~.. . . M« : ` +- - xấu 1À. v — x : : +` Xa w, + SG SvV “V VỀ V*V” +làn ` *^: { Vy ' tư 1 ` +` 4 ` “4£ pc v { `. N‹: Tử SÀ CC..." +‹ >- - «= « ..” “ + * . . uà: ,® +^ r h* : ÀN -' ^`s `” t ẹ° >. PF +>: ` .. e $ ° +” r ở VY ` +` " P .Y í “S44 ừ Xv< ` ‹v . .`› ;” `. +TH ` _ về “à SÁ Sư ”`YV Vư AC , ° +.. x3 > >- .VÝ +` F ˆ\ '4 SG. , L `. " St ." ` „ _ +, > >- = Ẳ - ...x*^. Z7 'saT +vNG í ` S ị ss ~ 2 V7, Nự ) t y : +` .. ` , ` Q4 2*VV. Vy _ +- † ~ ( x< v” "AW > > | +: \ : ` ` Ộ - #*%. .2Y*®% sua ` »" tý +: >>. `4 `” v.v. ế z ví \ứ + : v +` ` ` : M , ổL " _ ` “ˆ- .. " + +' . ` H `. `“... ˆ h ” z +` - ` ^-1.... xà ï Siờ —. x.” +NG ' =". " vt# tí 4 +ø. (111) face. b. (100) face. +Face-centered cubic structure. +).* ì : `: >3 x;i? ïx WEP +L "‹‹ 7 - 4!" + ) ` ; về X : . ` „` +` , ` 4 [ LÍ “ xF 4. ~..‹ { >-~<* { Thia 4 . +, ụ .” › « F - “4 v.< “4 ` “/ +. `. +»..x.. wé.. “' 3 \ Ẳ V so ˆ:\ +¿ } à /Ƒ > ọ j 1 L ¬ 4 ;..< <Ỉ “* >- si si» +-° 'Ậ F 'Š * . -< 3 “7 “4 " À, “( _* x s\ +ì 4 ` ^¬ Ẳ 1 ' { „. lŠ xà À » x. +CC 2< gà xe XS +1. +Ƒ ' `, ý tờ *+⁄⁄.# %`-" “4 ` - . +»-- * : ì S v. \ ^Á ` +, ,r Ậ r l ` ợ w..x ` .# &G +đC 3< »/ s.ổ . BS 0à C023 cx Dy +- ; > '. TS „# -., ¬ „ s..> `... wW.- “ +.s. x _»2-‹ n 4 ¿ { - © >>. \ . ` +# ˆ ` 7 ` “.z .x > lẢ ˆ xi .- - xi + +4 Sư +« ~ s.ằ... .“.‹ . x \ +” 'Ậ » ⁄/"ˆw '. W2 "Vu... v.v...“ 4 cà `" z4 +ị -4 ` “ r4 4 3 lễ <‹ s ®%¿ \ 1 x ` +ễ ` `4 ` .z -“ * `. &% ch ¬ q `« - X ị _ ` +“ - ° ° - 1 : ° ` +` C `6 X yí >> +` “ _ * : ` . `, P +'Ê \ .Z ` .“ -z ` .*xw.* N..“ “4 v >- “/ x. +»‹ »j - , ., ¬V Xlà= ¬ s 3 ` \ N +.. .. ; ÝY -v“ - “. +»- }.^ ¡ rị : x +, ' 4‹ xe ~ - L ` s⁄4 `: “4 : +- - Í 4 ` “4 " ' +.. \ , >5 * ` { 4 ` # ` +c. Twin across (111), cubic structure. đd. Possible example of hexagonal close- +Diameter 0-70 mm. packing. +EIGURE 17 +--- Trang 395 --- +->= "ga #e Séc &y ?@a TT +TT TT NGe-, Nˆx- +x _- sáu xÁ TT x +- v 'S<4 s seo se he 0G Sẻ, 'S`, SG Si n G.G., ta: +sảà 6© :®:©:©-e 6G G@e...xăs —— — — +c?< : S9 coc e6 '#8S:Q ==: +- 3 se SG. “.“aïs XS ` _“__ 2 +* : m Soc ca *&GS ~~ = Nho Qno 296:x +.” S^- Xe -#:&:G —~^~ 9:9-G re. == +c củ” TT 9:-G-tg:eS-Gộ- J te, = “%'#'ø +- x> l9 Go Go SA >:@-@-@rE tà Ấ x~ ni SS S0 vn +NT 2 SG 9-0 QC ge Tn G n +~S ` *6:G:Q-G ion Tà sẽ __ +cà S4: sáo XÃ. ` nuoc "@x3:tờ Z&:G sen ;e Ninh ƯNNế +K sò“ - : : R.oo:0:o- AA cx 7À — +n x ó h r©'©-@:©-@- ĐC 20:00 N nh n Go in +là 34 S045 0ù vn th ết cane :9:0-0-0 TT ST Ta An Go, +Ái Sùg lý là "y4 SS xo ~ nh nn An Che an | +- g2 àề:, #Q CÁ, >@œ: s: 99:00, TT x — +- - à w + —~” s NueT - ˆ” —. +c” —_ Xa.“ v20, > A5269 06 hit tên tây Q.6. » +r Sa 2 con tà, x.t Hé X22 4s: 96, - Ko kẽ hạn S:6-6- Ẳ +vn nh vn viec ch va g vườn cong n ng 00 E ece, +s9 2 S22. cÓ ha —————.- d4) St 6` XS x24 2:Ằ-. +— — n mo Sẽ: 22 9u 5à 2-5 v26 ti Sa 04)9 x- ion g0 G00 XÃ. k9 c +: HN Ty J ê Kế ve) Sự Hài C9 v5 +s s vu vn Sàn tEg) L6 cá êm t0. 5 le Tin in c0, +— 3222-4025 sẽ Ác x xxx v92, cẾ à - 9: ch độ, vớ +" " TH ~ CN Tế, ê 1.0, 8g: ——. Xà ác sj TH Tử) i0 d6 gi 1 +NI ng 294 014), — TH HH, i +HH s6), TH TH TÊH Tư Tân 2 Hệ, HH TS HỀ Hộ, Đo cty ¬ +: s4 0 4) 32 TH (9 gi So v24: “x An HH th H c th H ng Án kề Tết +lu 00 TS HH TT nn D5010 00g tện Š St tuc in n +* „ Tu 2⁄2 £ 4v TH yến . › X dt TU TP bực +6904250305462 lì À5 EUai cá vi to vệ ý gự 94) i62 15 ĐÓ TC TẾ Ai SỈ vi tá _ +“. Hi xì ca SG: -&N V4, 2 VỂ: Suy 28, TK Ho KG 25 Si» Ty chệy 4 vì, đi y¿4 +xo đo S0 45: 60g S, vˆ28: 89 v2 Thn thhệc BE, CV XiC, Bi Shyi àầ: +>> nh Ly đà S0 S02 y0 Vồ) Dy S55 Xe x gu +TY TY in nhớ) Si 0: Di n0 v0000600 ve: +SE nv E cHrhEr TP D0414: n4 4ys1S Me su G Bọ hưng +đt. 292006 0) se 214 ý 227 6:4 k3, 95. chà vì) 2 SN 2 se: tra +V274 SA 2x4: ~ kề Là Say hi 6i, DhEH khi 41 | +—— ra? ^ 2s `, cả ~~ ~ vn ~ ccc Xe, .. '. l Snc y§ b s ~ TH“ +ch TH 2= N nh tàn G1 xi2 Sinh An nhi s +TY TT 12/17 Tin 5 To HE yên sưng: +ầ-&:@:@'0 “..? >..% ˆ-* ca “xe ^.~ “2 ¬ Q sGẢ x Sí .—. — ~... +ni TT. ... sức +oan gi v2 CC 14 1g TẾ g3 g9 554 Ôn Tnhh 6, +~. "an, XS, 24+ 13- + 8/2 ko, TƯ A96 _—... CÁ AT] +“6:6:9:ê:0-g # „2x TT cSA2 Xu o9 V0 2 AC Sáng ấu .... ác“. MA XÓx +- Da 2“... kơ, ..”» .... “~^ c9, +? kưc 2x H2 MT ` +- TS .." mà @, + (U92 >..” TẾ tá s. .. có j,v 9 lẻ S2 rót +“% ha... , .v 9y xạ ® ~ k ở tá Xã >... xá * vu .“. +——— TVˆV ý 4 +. idx.Í a0, ẹ “ h ng 01 10104 21-12x5 ca 9G p*S ca +.~ c> _ưẻX "vê lung © _—~“*~ => c2 CC vu, Tớ, ky ....c sẻ `. +đ ng 62006 0:6 9t rrro e2 v2 c2 in Số nhu DỤC +- — SG TH TS“ b — % S. Vy +Tả ——— :@:ê:ê:@:& — ki 0í. lờ, 5. cố, .*, vu SỈ Đền àiểu Su lu) +„ Trên “ro.@ Thy Tê —.. +SYyvx „M.#. 5 tê dự vi, “ong Q6, vi x ĐEN pH Ơ 6 _ +các l2: ô-8 su Lhế ủy lề S07: 02276 g c.6 5 sêt ..>t.ị .... Hộ NA lẻ the > +TC cụ A0:  TH ly 2.67 yề2c^v2v v2 CÔ TH (No N iêN Xu š . +-— Xe) xxx lày ` sô, ^, ~À NV N Mục TT & +TS — lW w Xi vi,5cv, 2c nữ 29g à hy 420 >> NA bế À ¬ +K ụ Ác li ngà lu, S2 Say Êla lá 4 c2 — _— L M NÓNG PxŠ tệ vô tỷ NÊN +—N đà Xu ưu ty vi Šzàrê-à-ô-ô-âô 8 6 6 rà» 1 lấy V024 XU NUNG +” " &e ST ..... ~—~~ nh rttni NXNNG +—~ loxâg Xế K cụ S6 uy Bể x32 vê can mi 1 vài +SG lv it vo va vi tui va TS x0:à:â»@- XIN Vu Sa go Š k Su +"Xi TƯ K tt gen ước vờ AệN Nu kề đc s vu 1 +r..vV Tớ VU vu Xưv viyg MS Su +— SH" ~-~ tí Š XS ~ +KÀ và CN Tin TT, xx Nhan 0á vớ XšYY cv vs xxx) +là Về nà và 3 cù lăn hi, 1 Chị xì VšY ty về vyc +rà và 3 => và 2v ——¬~~- S;Q:0:0:0 96, vi Mi M uy * ,) Uy Y1 t +— "ô»ê:ô-Ê-ê:ê:ê-ê-ô; ^^? SN by hạ My xát) +ch Và nong cà ong +và Àc ài% Nà cà Xa hờ bàng ^^ 912142 Av' _) vo *v SN +—= Si vo ii cự ly: VN MS Su? s +: TT Ă Xin v22 2x22v2v^x lẻ —= 70 mm. +b TT NNĂ A2 y/2y2 : iameter 0- +-——— xo. .v 5x XXXXA lly. 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Diameter 0-70mm. +--- Trang 397 --- +Tortsors +31-1 The tensor of polarizability +Physicists always have a habit of taking the simplest example of any phe- 31-1 The tensor of polarizability +nomenon and calling it “physics,” leaving the more complicated examples to 31-2 Transforming the tensor +become the concern of other fñelds—say of applied mathematics, electrical engi- components +neering, chemistry, or crystallography. ven solid-state physics is almost only half 31-3 The energy ellipsoid +physics because it worries too mụuch about special substances. So in these lectures 31-4 Other tensors: the tensor of +we will be leaving out many interesting things. For instance, one of the important l R ? +. . l . co 1312 Inertia +properties of crystals—or of most substances—is that their electric polarizability +is diferent in diferent directions. If you apply a field in any direction, the atomie 3I-š The cross product +charges shift a little and produce a dipole moment, but the magnitude of the 31-6 The tensor oŸ stress +moment depends very much on the direction of the fñeld. 'Phat is, oŸ course, quite 31-7 Tensors of higher rank +a complication. But in physics we usually start out by talking about the special 31-8 The four-tensor of +case In which the polarizability is the same in all directions, to make life easier. electromagnetic momentum +W© leave the other cases to some other field. 'Pherefore, for our later work, we +will not need at all what we are going to talk about in this chapter. +The mathematics of tensors is particularly useful for describing properties of +substances which vary in direction—although that”s only one example of their use. +Since most of you are not going to become physicists, but are going to go into the +rea[ world, where things depend severely upon direction, sooner or later you will +need to use tensOrs. In order not to leave anything out, we are going to describe Reuieu: Chapter 11, Vol. 1, Vectors +tensors, although not in great detail. We want the feeling that our treatment of : +¬- có. Chapter 20, Vol. I, fofation +physics 1s complete. For example, our electrodynamics is complete—as complete m6 +as any electricity and magnetism course, even a graduate course. Qur mechanics _ +is not complete, because we studied mechanics when you didn't have a high level +of mathematical sophistication, and we were not able to discuss subjects like the +principle of least action, or Lagrangians, or Hamiltonians, and so on, which are +more clegan‡ uays of describing mechanics. Except for general relativity, however, +we do have the complete /as of mechanics. Our electricity and magnetism is +complete, and a lot of other things are quite complete. 'Phe quantum mechanics, +naturally, will not be—we have to leave something for the future. But you should +at least know what a t©enSor is. +We emphasized in Chapter 30 that the properties of crystalline substances +are diferent in diferent directions—we say they are ønisofropic. The variation +of the induced dipole moment with the direction of the applied electric feld is +only one example, the one we will use for our example of a tensor. Let's say +that for a given direction of the electric feld the induced dipole moment per +unit volume ? is proportional to the strength of the applied fñeld #. (This is +a good approximation for many substances if # is not too large.) We will call +the proportionality constant œ.* We want now to consider substances in which +œ depends on the direction of the applied field, as, for example, in crystals like +calcite, which make double images when you look through them. +Suppose, in a particular crystal, we fnd that an electric fñeld in the +-direction produces the polarization ??¡ ¡in the z-direction. 'Phen we fnd that +an electric fñeld #2; in the -direction, with the same sfrengfh, as E produces a +diferent polarization ?ạ in the -direction. What would happen if we put an +* In Chapter 10 we followed the usual convention and wrote P = cox# and called x (“khi”) +the “susceptibility.” Here, it will be more convenient to use a single letter, so we write œ ÍOr 0X. +for isotropic dielectrics, œ = ( — 1)co, where £ is the dielectric constant (see Section 10-4). +--- Trang 398 --- +electric fñeld at 45°? Well, that 's a superposition of two fields along ø and , so +the polarization will be the vector sum of ?ì and Đa, as shown in Eig. 31-1(a). +The polarization is no longer in the same direction as the electric fñeld. You can +see how that might come about. There may be charges which can move easily +up and down, but which are rather stif for sidewise motions. When a Íorce is +applied at 45°, the charges move farther up than they do toward the side. 'Phe +displacements are not in the direction of the external force, because there are +asymmetric internal elastic forces. Eạ +There is, oŸ course, nothing special about 45°. It is generaliu true that the +induced polarization of a crystal is nø# in the direction of the electric ñeld. In our +example above, we happened to make a “lucky” choice of our z- and -axes, for +which was along # for both the z- and -directions. If the crystal were rotated P› +with respect to the coordinate axes, the electric field #a ¡in the -direction would +have produced a polarization ?? with both an z- and a -component. Similarly, +the polarization due to an electric field in the z-direction would have produced a P Eìị +polarization with an #ø-component and a -component. “hen the polarizations (a) +would be as shown in Eig. 31-1(b), instead oŸ as in part (a). Things get more +complicated——=but for any field , the magnitude oŸ P is still proportional to the +magnitude o£ #. +W©e want now to treat the general case of an arbitrary orientation of a crystal +with respect to the coordinate axes. An electric feld in the z-direction will +produce a polarization ? with z-, ø-, and z-componenfs; we can write E; +Py=o„E, Py=ogwE, P.=oE,. (31.1) JZ +All we are saying here is that if the electric field is in the z-direction, the E +polarization does not have to be in that same direction, but rather has an z-, a (b) +u-, and a z-component——each proportional to #„. We are calling the constants of +proportionality œ„„, œ„„, and œz„, respectively (the fñrst letter to tell us which Fig. 31-1. The vector addition of polar- +component of is involved, the last to refer to the direction of the electric ñeld). izations in an anisotropic crystal. +Similarly, for a fñeld in the -direction, we can write +Ty = œ„yRy, Đụ = duy 2y, Ð = œzyEyy; (31.2) +and for a field in the z-direction, +Tụ — œ„z„F„, Tụ —= ằœyxl„, , = œzxF„. (31.3) +Now we have said that polarization depends linearly on the fields, so 1f there +is an electric fñeld # that has both an z- and a -component, the resulting +z-component of P will be the sum of the two f„'s of Edqs. (31.1) and (31.2). If +E2 has components along z, , and z, the resulting components of will be the +sum of the three contributions in Eqs. (31.1), (31.2), and (31.3). In other words, +P vill be given by +Tụ — O„ Jzy + Ou FRuụ + ằxzE/„, +Tụ = œyx E„ + uy Eụ + au„ E„, (31.4) +Tỷ = Gz„ E„ + œxu Eu + 0xx. +The dielectrie behavior of the crystal is then completely described by the nine +quantities (đ„„, đ„y, œ„;, œ„„, ...), which we can represent by the symbol œ¿¿. +(The subscripts 2 and 7 each stand for any one of the three possible letters #, g, +and z.) Any arbitrary electric fñeld # can be resolved with the components +Ty, Fụ, and F/„; from these we can use the œ; to ñnd y, „, and P;, which +together give the total polarization ?. 'The set of nine coefficients œ¿; is called a +tensor——in this instance, the fensor oƒ polarizabilifg. Just as we say that the three +numbers (E„, E„, E„) “form the vector #7,” we say that the nine numbers (dz„„, +œx„, ...) “form the tensor œ¿;.” +--- Trang 399 --- +31-2 Transforming the tensor components +You know that when we change to a different coordinate system +, /, and Z', +the components „, F2, and z of the vector will be quite diferent——as will +also £he cormponen#s of P. 5o all the coefficients œ; will be different for a +diferent set of coordinates. You can, ¡in fact, see how the œ's must be changed by +changing the components of # and ?P in the proper way, because if we describe +the same phụs¡ical electric field in the new coordinate system we should get the +same polarization. Eor any new set of coordinates, f2; is a linear combination of +D„, Pụ, and P¿: +Đụ = aF„ + bPụ + cP,, +and similarly for the other components. IÝ you substitute for ,, „, and P; in +terms of the 7s, using Eq. (31.4), you get +Đụ: — a(dx„E2„ + Ou FEuụ + œxz„ E„) ++ bD(dyz E„ + quy Eụ + œụ; È⁄„) ++ c(dz„ Ey + œxu Eụ + œ„„ E,). +Then you write Hy, „, and #; in terms of H„, h„/, and F;:; for instance, +Tưy — q Hà, + b bự + chà, +where đ', Ö, đ are related to, but not equal to, ø, Ð, e. So you have z;, expressed +in terms of the components 2, J2, and „/; that is, you have the new œ¿¿. lt +1s fairly messy, but quite straightforward. +When we talk about changing the axes we are assuming that the crystal +stays put #n space. TỶ the crystal were rotated 6h the axes, the œ's would not +change. Conversely, if the orientation of the crystal were changed with respect +to the axes, we would have a new set of œ's. But if they are known Íor ng one +orientation of the crystal, they can be found for any other orientation by the +transformation we have Just described. In other words, the dielectrie property of +a crystal is described cørmpletelu by giving the components of the polarization +tensor œ¿; with respect to any arbitrarily chosen set of axes. Jus as we can +associate a vector velocity Ø = (0x, 0y, 0x) with a particle, knowing that the three +components will change in a certain defnite way If we change our coordinate axes, +so with a crystal we associate its polarization tensor œ¿;, whose nine components +will transform in a certain defnite way if the coordinate system is changed. +The relation bebween and E written in Eq. (31.4) can be put in the more +compact notatfion: +Đ, = » G07 F27, (31.5) +where it is understood that ¿ represents either +, , or z and that the sum is +taken on j = zø, , and z. Many special notations have been invented for dealing +with tensors, but each of them is convenient only for a limited class of problems. +One common convention is to omit the sum sign (32) ín Eq. (31.5), leaving it +wnderstood that whenever the same subscript occurs bwice (here 7), a sum is to +be taken over that index. 5ince we will be using tensors so little, we will not +bother to adopt any such special notations or conventions. +31-3 The energy ellipsoid +We want now to get some experience with tensors. Suppose we ask the +interesting question: What energy is required to polarize the crystal (in addition +to the energy in the electric feld which we know is coZ2/2 per unit volume)? +Consider for a moment the atomic charges that are being displaced. The work +done in displacing the charge the distance đã is g⁄„ dz, and ifthere are / charges +per unit volume, the work done is g„/V da. But gÑN dz is the change đÐ;y ¡in the +dipole moment per unit volume. So the energy required per nở 0olume 1s +b„dP,. +--- Trang 400 --- +Combining the work for the three components of the field, the work per unit +volume is found to be +E..dP. +Since the magnitude of # is proportional to #, the work done per unit volune +in bringing the polarization from 0 to ? is the mmtegral of E - đP. Calling this +wOrk tp,*Š we Write +up=$3E-P= 3} ` E,P,. (31.6) +NÑow we can express in terms of # by Eq. (31.5), and we have that +up =šÿÀ ` œ¡jE,E). (31.7) +The energy density ứp is a number independent of the choice of axes, sO 1È is a +scalar. A tensor has then the property that when ï§ is summed over one index +(with a vector), it gives a new vector; and when it is summed over Öø#Ö indexes +(with two vectors), it gives a scalar. +The tensor œ¿; should really be called a “tensor of second rank,” because it +has two indexes. A vector—with ønwe index—is a tensor of the first rank, and +a scalar—with no Index——ls a tensor of zero rank. 5o we say that the electric +fñield # is a tensor of the first rank and that the energy density œp is a tensor of +zero rank. lt is possible to extend the ideas of a tensor to three or more indexes, +and so to make tensors of ranks higher than ©wo. +The subscripts of the polarization tensor range over three possible values—— +they are tensors in three dimensions. The mathematicians consider tensors in four, +five, or more dimensions. We have already used a four-dimensional tensor #}„ ỉn +our relativistic description of the electromagnetic fñeld (Chapter 26). +The polarization tensor œ¿; has the interesting property that it is sựwmectric, +that is, that œ„y = œ„„, and so on for any pair of indexes. (This is a phụsical +property of a real crystal and not necessary for all tensors.) You can prove Íor +yourself that this must be true by computing the change in energy of a crystal +throuph the following cycle: (1) Turn on a fñeld in the z-direction; (2) turn on +a field in the g-direction; (3) turn øƒƒ the z-feld; (4) turn of the ø-ñeld. The +crystal is now back where ït started, and the net work done on the polarization +must be back to zero. You can show, however, that for this to be true, œ„„ must +be equal to œ„„. The same kind of argument can, of course, be given for œ„z, +etc. So the polarization tensor is symmetric. +'This also means that the polarization tensor can be measured by Just measuring +the energy required to polarize the crystal in various directions. Suppose we apply +an E-field with only an z- and a #-component; then according to Ed. (31.7), +up = š|dz„ED + (day + ae) E„Ey + oyy E2]. (31.8) +'With an F„ alone, we can determine œ„„; with an #2 alone, we can determine œ„; +with both #„ and #„, we get an extra energy due to the term with (d„y + œyz). +Since the œ„„ and œ„„ are cqual, this term is 2œ„„ and can be related to the +©nergy. +The energy expression, Đq. (31.8), has a nice geometric interpretation. Sup- +pose we ask what fields H„ and !2„ correspond to some given energy density—— +say uạọ. That is Just the mathematical problem of solving the equation +da E2 + 20 E„ Ey + œyy E2 = 2u. +Thịis is a quadratic equation, so if we plot #„ and #2„ the solutions of this equation +are all the points on an ellipse (Fig. 31-2). (It must be an ellipse, rather than +a parabola or a hyperbola, because the energy for any field is always positive +and fñnite.) The vector with components „ and 1 can be drawn from the +* This work done in produc#ng the polarization by an electric feld is not to be confused +with the potential energy —øg - # of a permanent dipole moment Øạ. +--- Trang 401 --- +origin 0o the ellipse. 5o such an “energy ellipse” is a nice way of “visualizing” the +polarization tensor. +Tf we now generalize to include all three components, the electric vector # in +1 direction required to give a unit energy density gives a point which will be " +on the surface of an ellipsoid, as shown in Eig. 31-3. 'Phe shape of this ellipsoid +of constant energy uniquely characterizes the tensor polarizability. +Now an ellipsoid has the nice property that it can always be described simply +by giving the directions of three “principal axes” and the diameters of the ellipse +along these axes. The “principal axes” are the directions of the longest and +shortest diameters and the direction at right angles to both. 'They are indicated +by the axes a, ð, and cin Eig. 31-3. With respect to these axes, the ellipsoid has +the particularly simple equation +2 2 2— +Gaa§ + Ẳpp Rÿ + œ¿c = 2u. +. . . Fig. 31-2. Locus of the vector E = +So with respect to these axes, the dielectric tensor has only three componenfs (E.,E,) that gives a constant energy of +that are not zero: œ„a, œpp, and œ¿¿. Thhat is to say, no matter how complicated a polarization. +crystal is, it is always possible to choose a set of axes (not necessarily the crystal +axes) for which the polarization tensor has only three components. With such a +set of axes, Ðq. (31.4) becomes simply +, — Oqaa; In — Gpp.Ep, In — œsek. (31.9) +An electric field along any one of the principal axes produces a polarization along +the same axis, but the coefficients for the three axes may, of course, be diferent. b +Often, a tensor is described by listing the nine coefficients in a table inside of +a pair of brackets: +fyy Oyu O„z +Quy uy Oyz|- (31.10) +Ozr Ozu, @zz +For the principal axes ø, b, and e, only the diagonal terms are not zero; we say c +then that “the tensor is diagonal” "The complete tensor is Eig. 31-3. The energy ellipsoid of the +tạ, 0 0 polarization tensor. +0 Œb 0 ‹ (31.11) +0 Ú_ œ« +The important point is that any polarization tensor (in fact, œnmg sựmmetric +tensor o�� rank two in any number of đimensions) can be put in this form by +choosing a suitable set of coordinate axes. +Tí the three elements of the polarization tensor in diagonal form are all equal, +that is, if +Oqa„ — Opp — Q¿c = G, (31.12) +the energy ellipsoid becomes a sphere, and the polarizability is the same in all +directions. The material is isotropic. In the tensor notation, +@¿j = œỗ¿/ (31.13) +where ð¿; is the ni tensor +ổ„ = |0 I1 0Ị. (31.14) +'That means, of course, +ðy=1l, l ¿=j +` (31.15) +ðy =0, i77. +The tensor ổ;; is often called the “Kronecker delta.” You may amuse yourself +by proving that the tensor (31.14) has exactly the same form ïŸ you change +--- Trang 402 --- +the coordinate system to any other rectangular one. 'Phe polarization tensor of +Eq. (31.13) gives +which means the same as our old result for isotropic dielectrics: +P-=oE. +The shape and orientation of the polarization ellipsoid can sometimes be +related to the symmetry properties of the crystal. We have said in Chapter 30 that +there are 230 diferent possible internal symmetries of a three-dimensional lattice +and that they can, for many purposes, be conveniently grouped into seven cÌasses, +according to the shape of the unit cell. NÑow the ellipsoid of polarizability must +share the internal geometric symmetrles of the crystal. Eor example, a triclinic +crystal has low symmetry—the ellipsoid of polarizability will have unequal axes, +and is orlentation will not, in general, be aligned with the crystal axes. Ôn +the other hand, a monoclinic crystal has the property that its properties are +unchanged ïf the crystal is rotated 180° about one axis. So the polarization +tensor must be the same after such a rotation. It follows that the ellipsoid of +the polarizability must return to itself after a 180° rotation. Thhat can happen +only 1ƒ one of the axes of the ellipsoid is in the same direction as the symmetry +axis of the crystal. Otherwise, the orientation and dimensions of the ellipsoid are +unrestricbed. +For an orthorhombic crystal, however, the axes of the ellipsoid must correspond +to the crystal axes, because a 180 rotation about any one of the three axes +repeats the same lattice. If we go to a tetragonal crystal, the ellipse must have +the same symmetry, so it must have two equal diameters. Finally, for a cubic +crystal, all three diameters of the ellipsoid must be equal; it becomes a sphere, +and the polarizability of the crystal is the same ín all directions. +There is a big game of figuring out the possible kinds of tensors for all the +possible symmetries of a crystal. It is called a “group-theoretical” analysis. But +for the simple case of the polarizability tensor, it is relatively easy to see what +the relations must be. +31-4 Other tensors; the tensor of inertia +'There are many other examples of tensors appearing in physics. For example, +in a metal, or in any conductor, one often finds that the current density 7 is +approximately proportional to the electric fñeld #; the proportionality constant +1s called the conductivity ơ: +For crystals, however, the relation between 7 and # is more complicated; the +conductivity is not the same in all directions. The conductivity is a tensor, and +W© WTIẲ© +Ji. — » Ở;?7 bị. +Another example of a physical tensor is the moment of inertia. In Chapter 18 +of Volume I we saw that a solid object rotating about a fxed axis has an +angular momentum Ù proportional to the angular velocity œ¿, and we called the +proportionality factor 7, the moment of inertia: +h = lu. +For an arbitrarily shaped object, the moment of inertia depends on its orientation +with respect to the axis of rotation. Eor instance, a rectangular block will have +diferent moments about each of its three orthogonal axes. Now angular velocity œ +and angular momentum Ö are both vectors. For rotations about one of the axes +of symmetry, they are parallel. But if the moment of inertia is diferent for the +three principal axes, then œ and Ù are, in general, not in the same direction +--- Trang 403 --- +(see Eig. 31-4). They are related in a way analogous to the relation between ~— . +and . In general, we must write +T„y — đ„„U„ + đuyy(0u + 1yz(Jz, +Tụ —= Tyxúz + lyuúy Tyy0z, (31.16) +Ty = Tu „0z + Tuy + 1z. +The nine coefficients 1;; are called the tensor of inertia. Eollowing the analogy +with the polarization, the kinetic energy for any angular momentum must be +some quadratic form in the components œ„, ¿„, and ¿;: _+ +KE=i T0. 31.17 Fig. 31-4. The angular momentum L of +2 » 4) ( ) a solid object is not, in general, parallel to +J Its angular velocity œ. +We can use the energy to define the ellipsoid of inertia. Also, energy arguments +can be used to show that the tensor is symmetric—that 1; = l¡. +The tensor of inertia for a rigid body can be worked out if the shape of the +object is known. We need only to write down the total kinetic energy of all +the particles in the body. AÁ particle of mass mm and velocity has the kinetic +cenergy simuŸ, and the total kinetie energy is just the sum +over all of the particles of the body. The velocity of each particle is related to +the angular velocity œ of the solid body. Let's assume that the body is rotating +about its center of mass, which we take to be at rest. Thhen 1Ý ø is the displacement +of a particle from the center of mass, its velocity ® is given by œ x. So the +total kinetic energy is +KE= ` šm(œ x r)Ÿ. (31.18) +Now all we have to do is write œ x ? out in terms of the compOonenfS (0„, œ„, (0z, +and z, , z, and compare the result with Eq. (31.17); we find ï;; by identifying +terms. Carrying out the algebra, we write +(œ x r)” = (œ x r)2 + (œ@ x T)2 + (œ x T)Ÿ += (yz — 0zU)Ÿ + (0x# — œ„2)Ÿ + (0x — œy+)Ÿ += + (0127 — 2u0zZ1J -E (022 ++ 02#2 — 20;(„øz + 227 ++ DU TH — 2„@1# + (017. +Multiplying this equation by zm/2, summing over all particles, and comparing +with Eq. (31.17), we see that Ïz„, for instance, is given by +l„„ = À m(y? +2). +Thịs is the formula we have had before (Chapter 19, Vol. I) for the moment oŸ +inertia of a body about the z-axis. Since z? = #2 + #2 + z, we can also write +this term as +l„„ = » m(rŸ — #°). +Working out all of the other terms, the tensor of inertia can be written as +Sm(rẺ — +”) —`m+ —`mz+zz +1; = —À`mz Sm(r? — 02) —>)mụz |. (31.19) +—À`mzz —À`mz Sm(r? — z?) +T you wish, this may be written in “tensor notation” as +1; = Àm(r?ồi; — T¿Tj), (31.20) +--- Trang 404 --- +where the r¿ are the components (#, , 2) oŸ the position vector of a particle and +the ồ ` means to sum over all the particles. The moment of inertia, then, is a +tensor of the second rank whose terms are a property of the body and relate +to œ by +Lị =À ` 10. (31.21) +For a body of any shape whatever, we can find the ellipsoid of inertia and, +therefore, the three principal axes. Referred to these axes, the tensor will be +diagonal, so for any objJect there are always three orthogonal axes for which +the angular velocity and angular momentum are parallel. They are called the +principal axes of inertia. +31-5 The cross product +W© should poïnt out that we have been using tensors of the second rank since +Chapter 20 of Volume I. 'There, we defñned a “torque in a plane,” such as 7„„ by +Tựu = #Fụ — UFụ,. +Generalized to three dimensions, we could write +T7 =T¿È) — T7 hạ. (31.22) +The quantity 7¿; is a tensor of the second rank. One way to see that this is sO is +by combining 7¿; with some vector, say the unit vector e, according to +» Tạ? 7 € 7" +lí this quantity is a 0ecfor, then 7¿; must transÍíorm as a tensor—this is our +definition of a tensor. 5ubstituting for 7¿;, we have +» T¡¿jCj — » rịF)©j — » T;©jF¡ +=T4(EF - e) — (r- e)Fị. +Since the dot products are scalars, the two terms on the right-hand side are +vectors, and likewise their diference. So 7¿; is a tensor. +But 7¿; is a special kind of tensor; it is anfis/metric, that is, +Tj — —Tji; +so it has only three nonzero terms—7z„, 7„;, and 7;„. We were able to show in +Chapter 20 of Volume I that these three terms, almost “by accident,” transform +like the three components of a vector, so that we could đefine +T = (Tạ, Tụ; Tz) — (Tụz: Tzz; Tzụ) +W© say “by accident,” because it happens only in three dimensions. In four +dimensions, for instance, an antisymmetric tensor of the second rank has up +to s#z nonzero terms and certainly cannot be replaced by a vector with ƒour +components. +Just as the axial vector T = r x È' is a tensor, so also is every cross product +of two polar vectors—all the same arguments apply. By luck, however, they are +also representable by vectors (really pseudo vectors), so our mathematics has +been made easier for us. +Mathematically, if œ and b are any two vectors, the nine quantities a¿b; form +a tensor (although it may have no useful physical purpose). 'Thus, for the position +vector ?, r¿r; is a tensor, and since ổ;; is also, we see that the right side of +Eq. (51.20) is indeed a tensor. Likewise Eq. (31.22) is a tensor, since the bwo +terms on the right-hand side are tensors. +--- Trang 405 --- +31-6 The tensor of stress +The symmetric tensors we have described so far arose as coefficients in relating +one vector to another. We would like to look now at a tensor which has a diferent +physical signiñicance—the tensor of s‡ress. Suppose we have a solid object with +various forces on it. We say that there are various “stresses” inside, by which we ơ +mean that there are internal forces between neighboring parts of the material. ơ +W© have talked a little about such stresses in a two-dimensional case when we AEi +considered the surface tension in a stretched diaphragm in Section 12-3. We will +now see that the internal forces in the material of a three-dimensional body can lÍ +be described in terms of a tensor. +Consider a body of some elastic material—say a block of jello. If we make a +cut through the block, the material on each side of the cut will, in general, get +displaced by the internal forces. Before the cut was made, there must have been / +forces between the two parts of the block that kept the material in place; we can j +defñne the stresses in terms of these forces. Suppose we look at an imaginary — † ⁄Ữ = ~ +plane perpendicular to the z-axis—like the plane ø in Eig. 31-5——and ask about +the force across a small area A# Az in this plane. The material on the left of the +area exerts the force A' on the material to the right, as shown in part (b) of @) 6) +the fñgure. There is, of course, the opposite reaction foree —AF exerted on the Fig. 31-5. The material to the left of the +material to the left of the surface. If the area is small enough, we expect that plane ơ exerts across the area Ay Az the +AE} is proportional to the area A¿# Az. force AFi¡ on the material to the right of +You are already familiar with one kind oŸ stress—the pressure in a static the plane. +liquid. 'PThere the force is equal to the pressure times the area and is at right +angles to the surface element. Eor solids—also for viscous liquids in motion——the +force need not be normal to the surface; there are shear forces In addition to +pressures (positive or negative). (By a “shear” force we mean the tœngential +components of the force across a surface.) All three components of the force +must be taken into account. Notice also that if we make our cut on a plane with +some other orientation, the forces will be diferent. A complete description of the +internal stress requires a tensor. +Á 27 AFai +Ay Z2 Fig. 31-6. The force AF¡ across an el- +⁄ ement of area Ay Az perpendicular to the +x-axIs Is resolved into three components +We defne the stress tensor in the following way: Pirst, we imagine a cu +perpendicular to the z-axis and resolve the force A4 across the cut into is +components A2, AH¡+, AF;t, as in Fig. 31-6. The ratio of these forces to the +area A„ Az, we call S„„, S„„, and S„„. For example, +S AFmi +1 AyuAz' +The first index g refers to the direction force component; the second index # is +normal to the area. If you wish, you can write the area A¿# Az as Aa„, meaning +an element of area perpendicular to z. Then +Next, we think of an imaginary cut perpendicular to the g-axis. Across a small +--- Trang 406 --- +area Az Az there will be a force AFs. Again we resolve this force into three Af +components, as shown in Fig. 31-7, and define the three components of the stress, +S„ụ; Suy; Sx„, as the force per unit area in the three directions. Einally, we make +an imaginary cut perpendicular to z and defñne the three components S„;, S„;, +and Š;;. So we have the nine numbers AE +Sư S„y S»z +5S = |5 Sụy Swz|- (31.23) +Sz„ 5S» Szz +We want to show now that these nine numbers are sufficient to describe 7 Z77Z AEos +completely the internal state of stress, and that Š;¿; is indeed a tensor. Suppose 7 Z +we want to know the Íforce across a surface oriented at some arbitrary angle. Can k = +we fnd it rom $%;;? Yes, in the following way: We imagine a little solid figure +which has one face ín the new surface, and the other faces parallel to the +coordinate axes. If the face / happened to be parallel to the z-axis, we would +have the triangular piece shown in Eig. 31-8. (This is a somewhat special case, Ara +but will iHustrate well enough the general method.) Now the stress forces on +the hñttle solid triangle in Eig. 31-8 are in equilibrium (at least in the limit of Fig. 31-7. The íorce across an element +infinitesimal dimensions), so the total force on iÈ must be zero. We know the w ares Derpendcusr to y Is resolved Into +forces on the faces parallel to the coordinate axes directly from J5%;;. Their vector three rectangular components. +sum must equal the force on the face /, so we can express this force in terms +of Sỹ. +Our assumption that the surƒace forces on the small triangular volume are in +cquilibrium neglects any other boởy forces that might be present, such as gravity +or pseudo forces iÝ our coordinate system is not an inertial frame. Notice, however, Afýn +that such body forces will be proportional to the 0olwme of the little triangle AF, +and, therefore, to Az Aw Az, whereas all the surface forces are proportional to n +the areas such as Az A, Aw Az, etc. So if we take the scale of the little wedge 2 +small enough, the body forces can always be neglected in comparison with the 2⁄ ⁄ +surface forces. í / đà +Let's now add up the forces on the little wedge. We take frst the #z-component, Ay ⁄ ⁄⁄⁄ ` AFxn +which is the sum of five parts—one from each face. However, if Az is small Z7 +enough, the forces on the triangular faces (perpendicular to the z-axis) will be /6. +equal and opposite, so we can forget them. 'The z-component of the force on the Ax +bottom rectangle is Fig. 31-8. The force Fạ across the face M +AHya = S„ụ ÄAz Az. (whose unit normal is nm) is resolved into +The #z-component of the force on the vertical rectangle is Cormponents. +AFyl = S„„ AuU Az. +These two must be equal to the z-component of the force ou#ørd across the +face /. Let”s call rw the unit vector normal to the face /, and the force on 1t #ạ; +then we have +AFwn = S„„ AU Az + „vu Az A2. +The z-component S„„, of the stress across this plane is equal to A„„ divided +by the area, which is Az4/Az2 + A22, or +v⁄Az2+ A2 v'Az2 + A2 +Now Az/VWAz2 + A2 is the cosine of the angle Ø bebtween ø and the -axis, +as shown in Eig. 31-8, so it can also be written as mø, the ¿-component of øẹ. +Similarly, A/wWAz2 + A2 is sin Ø = nạ. We can write +Syn —= S„„1„ + S„w1,. +TÍ we now generalize to an arbitrary surface element, we would get that +Sàn — 5x + S„uTiu + S„zT1z +--- Trang 407 --- +or, in general, +Sïn = » Sij1J. (31.24) +We can find the force across any surface element in terms of the Š;;, so it does +describe completely the state of internal stress of the material. S +Equation (31.24) says that the tensor 5;; relates the stress Š„ to the unit +vector ?w, just as œ¿; relates Í? to #. 5ince ?+ and Š„ are vectors, the components +of 5%; must transform as a tensor with changes in coordinate axes. So É;; is % +indeed a tensor. +We can also show that Š¿; is a sựmmectric tensor by looking at the Íorces Sự +on a little cube of material. Suppose we take a little cube, oriented with its +faces parallel to our coordinate axes, and look at it in cross section, as shown Sxx +in Fig. 31-9. If we let the edge of the cube be one unit, the z- and -components Sxx +of the forces on the faces normal to the zø- and ø-axes might be as shown in the s +figure. If the cube is small, the stresses do not change appreciably from one side 7 +of the cube to the opposite side, so the force components are equal and opposite S +as shown. Now there must be no torque on the cube, or it would start spinning. +The total torque about the center is (5z — S„„) (times the unit edge of the cube), +and since the total is zero, S„„ is equal to 5„„, and the stress tensor is symmetric. +Since 5;; is a symmetric tensor, it can be described by an ellipsoid which Sư +will have three principal axes. For surfaces normal to these axes, the stresses Fig. 31-9. The x- and y-forces on four +are particularly simple—they correspond to pushes or pulls perpendicular to the faces of a small unit cube. +surfaces. 'There are no shear forces along these faces. Eor øn/ stress, we can +always choose our axes so that the shear components are zero. Tf the ellipsoid +1s a sphere, there are only normal forces in ønw direction. This corresponds to +a hydrostatic pressure (positive or negative). So for a hydrostatic pressure, the +tensor is diagonal and all three components are equal; they are, in fact, just equal +to the pressure ø. We can write +5 = Đồi. (31.25) +The stress tensor——and also its ellipsoid—will, in general, vary from poïint +to poïint in a block of material; to describe the whole block we need to give the +value of each component of %;; as a function oŸ position. 5o the stress tensor is a +field. WS have had scalar fields, like the temperature 7 (+, , z), which give one +number for each poïnt in space, and 0ecfor fields like E(z, 9, z), which give three +numbers for each point. Now we have a £ensor ficld which gives nine numbers +for each poïnt in space—or really six for the symmetric tensor Š;;. A complete +description of the internal forces in an arbitrarily distorted solid requires six +functions oŸ ø, , and z. +31-7 Tensors of higher rank +The stress tensor 5;; describes the internal ƒorces of matter. lf the material +1s elastic, 1 is convenient to describe the internal đ¿sforfion 1n terms of another +tensor 7¿;—called the s/rain tensor. Eor a simple object like a bar of metal, you +know that the change in length, A1, is approximately proportional to the force, +so we say it obeys Hooke”s law: +AL =+F. +For a solid elastic body with arbitrary distortions, the strain 7¿; is related to the +stress %¿; by a set of linear equations: +1ịy = »..-.- (31.26) +Also, you know that the potential energy of a spring (or bar) is +3FƑAL = š+F”. +--- Trang 408 --- +The generalization for the elastic energy đensitu in a solid body is +lastic — » 3ijkLSij Sr. (31.27) +The complete description of the elastic properties of a crystal must be given In +terms of the coefficients +;„¡. Thịs introduces us to a new beast. It is a tensor of +the ƒfourth rank. Since each index can take on any one of three values, #, , Or Z, +there are 3“ = 8§1 coefficients. But there are really only 21 đjƒerent numbers. +First, since 5%; is symmetric, it has only six different values, and only 36 điƒferenf +coefficients are needed in Eq. (31.27). But also, 5;; can be interchanged with S¡¡ +without changing the energy, so +;;„¡ must be symmetric if we interchange ¿7 +and kỉ. 'Phis reduces the number of diferent coefficients to 21. So to describe the +elastie properties of a crystal of the lowest possible symmetry requires 21 elastic +constantsl This number is, of course, reduced for crystals of higher symmetry. +For example, a cubic crystal has only three elastic constants, and an isotropic +substance has only two. +That the latter is true can be seen as follows. How can the components OŸ %;7z¡ +be independent of the direction of the axes, as they must be I1f the material is +isotropic? Ansuer: They can be independent on if they are expressible in terms +of the tensor ð;;. There are two possible expressions, ở;;ổ„; and ổ;gổ¿¡ + ổjổ¿z, +which have the required symmetry, so +;;x¡ must be a linear combination oŸ them. +'Therefore, for isotropic materials, +^ijki —= đ(ỗ¡jöki) + D(ỗ¡gỗjt + ỗn 7k), +and the material requires two constants, ø and ö, to describe Its elastic properties. +W©e will leave it for you to show that a cubic crystal needs only three. +As a fñnal example, this time of a third-rank tensor, we have the piezoelectrie +efect. Under stress, a crystal generates an electric fñield proportional to the stress; +hence, in general, the law is +đi — » Địjy5jk, +where #¿ is the electric fñield, and the ;„ are the piezoelectric coefflicients——or +the piezoelectric tensor. Can you show that if the crystal has a center of inversion +(invariant under #, 1, —> —#, —, —2) the piezoelectric coeflicients are all zero? +31-8 The four-tensor of electromagnetic momentum +All the tensors we have looked at so far in this chapter relate to the three +dimensions of space; they are defned to have a certain transformation property +under spatial rotations. In Chapter 26 we had occasion to use a tensor in the +four dimensions of relativistie space-time——the electromagnetic fñeld tensor F,„. +'The components of such a four-tensor transform under a Lorentz transformation +of the coordinates in a special way that we worked out. (Although we did not do +it that way, we could have considered the Lorentz transformation as a “rotation” +in a four-dimensional “space” called Minkowski space; then the analogy with +what we are doing here would have been clearer.) +As our last example, we want to consider another tensor in the four di- +mensions (¿, +, , z) of relativity theory. When we wrote the stress tensor, we +defñned 5%;; as a component of a force across a unit area. But a force is equal to +the time rate of change of a momentum. Therefore, instead of saying “%„„ is the +#-component of the force across a unit area perpendicular to ,” we could equally +well say, “S5„„ is the rate of ñow of the z-component of momentum through a +unit area perpendicular to 2.” In other words, each term of 6%; also represents +the fow of the ;-component of momentum through a unit area perpendicular +to the 7-direction. 'These are pure space components, but they are parts of a +“larger” tensor ,5,„ in four dimensions (u and 1 = £,z, ¿, 2) containing additional +components like 5;„, S„¿, 5;¿, etc. We will now try to fnd the physical meaning +of these extra components. +--- Trang 409 --- +We know that the space components represent fow of momentum. We can +get a clue on how to extend this to the time dimension by studying another kind +of “ñow”——the fow of electric charge. Eor the scalar quantity, charge, the rate +of flow (per unit area perpendicular to the flow) is a space øecfor——the current +density vector 7. We have seen that the time component of this ow vector is +the density of the stuf that is fowing. For instance, 7 can be combined with a +time component, 7¿ = ø, the charge density, to make the four-vector j„ = (0ø, 3); +that is, the in 7„ takes on the values ý, #, #, z to mean “density, rate of fow in +the zø-direction, rate of Ñow in ø, rate of flow in z” of the scalar charge. +Now by analogy with our statement about the time component of the ow of +a scalar quantity, we might expect that with Sz„, 5„„, and S„;, describing the +fow of the z-component of momentum, there should be a time component S„; +which would be the density of whatever is fowing; that is, S„;¿ should be the +density of z-mormentum. So we can extend our tensor horizontally to include a +f-component. We have +5„¿ — density of z-momentum, +5x — #-fow of z-momentum, +5x = u-fow of z-momentum, +SŠ„„ = 2-fow 0Í #z-momentum. +Similarly, for the -component of momentum we have the three components of +fow—Syz, 5y; Suz—to which we should add a fourth term: +Sự; = density of -momentum. +And, of course, to Š;„, 5„„, 5„„ we would add +Sx¡ — density of z-momentum. +In four dimensions there is also a f-component of momentum, which is, we +know, energy. So the tensor 5;; should be extended vertically with S‡;„, S%„, +and S%;;, where +Sy„ = #-fow O energy, +S¿;„ = -fow of energy, (31.28) +S¡„ — Z-fow OÍ energy; +that 1s, S;„ is the fow of energy per unit area and per unit time across a surface +perpendicular to the zø-axis, and so on. Finally, to complete our tensor we need 5%, +which would be the densit oŸ energ. We have extended our stress tensor Š;; oŸ +three dimensions to the four-dimensional s‡ress-energ tensor S„„„. The index +can take on the four values ứ, #ø, , and z, meaning, respectively, “density,” “Ñow +per unit area in the zø-direction,” “fow per unit area in the z-direction,” and +“fow per unit area in the z-direction.” In the same way, 1 takes on the four values +‡, ø, , z to tell us uha£ flows, namely, “energy,” “momentum ¡in the #-direction,” +“momentum ïn the -direction,” and “momentum in the z-direction.” +As an example, we will điscuss this tensor not in matter, but in a region of +free space in which there is an electromagnetic feld. We know that the fow of +energy is the Poynting vector $ = cọc2E x B. So the z-, ÿ-, and z-components +of Š are, from the relativistic point of view, the components Š;„, 5;„, and $%;z +of our four-dimensional stress-energy tensor. The symmetry of the tensor ®Š¡; +carries over into the tỉme components as well, so the four-dimensional tensor Š¿„„„ +1s symmme€trIC: +Suụ —= Su: (31.29) +In other words, the components S„;, S„¿, S;¿, which are the đensifies OŸ #, U, +and z mmøomentum, are also equal to the z-, -, and z-components of the Poynting +vector ,S%, the energu flou—as we have already shown in an earlier chapter by a +diferent kind oŸ argument. +--- Trang 410 --- +The remaining components of the electromagnetic stress tensor 5, can also +be expressed in terms of the electric and magnetic felds # and #Ö. That ïs to +say, we must admit stress or, to put it less mysteriously, Ñow of momentum in +the electromagnetic field. We discussed this in Chapter 27 in connection with +Eq. (27.21), but did not work out the details. +'Those who want to exercise their prowess In tensors in four dimensions might +like to see the formula for ®$„„ in terms of the fields: +Su, —= —€0 (= Thu tua — TỔ » FạuP +œ ằœ,8 +where sums on œ, ổ are on £, #, , z but (as usual in relativity) we adopt a special +meaning for the sum sign ồ ` and for the symbol ổ. In the sums the #, , z terms are +to be subtracted and ỗ¿¿ = +1, while ð„„ = ðyy = ðzy = —1 and ổ„¿„ = 0 ÍOT /t # 1⁄ +(c= 1). Can you verify that it gives the energy density % = (eo/9) (E2 + B3) +and the Poynting vector co x ? Can you show that in an electrostatic field +with = 0 the principal axes of stress are in the direction of the electric feld, +that there is a fension (eo/2)E2 along the direction of the feld, and that there is +an equal pressure in directions perpendicular to the fñield direction? +--- Trang 411 --- +Mửofrctcfit©o Irdiov ©Ÿ lÌoreso W(qforterls +32-1 Polarization of matter +W©e want now to discuss the phenomenon of the refraction of light——and also, 32-1 Polarization of matter +therefore, the absorption of light——by dense materials. In Chapter 31 of Volume Ï 32-2 Maxwells equations in a +we discussed the theory of the index of refraction, but because of our limited dielectric +_¬. 0c RE cán __. n to ` nao ves to nh ng à 32-3 Waves in a dielectrie +Index only lor matcrlals of low densIty, like gases. e physIcal prIinciples that R . +l . : 32-4 Th | d f refract +produced the index were, however, made clear. The electric feld of the light wave ° sanp x1 " 016aeuon +polarizes the molecules of the gas, producing oscillating dipole moments. “The 32-5 The index of a mixture +acceleration of the oscillating charges radiates new waves of the fñeld. 'This new 32-6 Waves in metals +fñeld, interfering with the old field, produces a changed field which is equivalent 32-7 Low-frequency and +to a phase shift of the original wave. Because this phase shift is proportional to high-frequency approximations; +the thickness of the material, the efect is equivalent to having a diferent phase the skin depth and the plasma +velocity in the material. When we looked at the subject before, we neglected the frequency +complications that arise from such efects as the new wave changing the fñelds at +the oscillating dipoles. We assumed that the forces on the charges in the atoms +came just from the zncomzng wave, whoreas, in fact, their oscillations are driven +not only by the incoming wave but also by the radiated waves of all the other +atoms. It would have been dificult for us at that time to include this efect, so +we studied only the rarefied gas, where such efects are not important. Teuieu: See Table 32-1. +Now, however, we will ñnd that i is very easy to treat the problem by the +use of diferential equations. This method obscures the physica]l origin of the +index (as coming from the re-radiated waves interfering with the original waves), +but it makes the theory for dense materials much simpler. “This chapter will +bring together a large number of pieces from our earlier work. We*ve taken up +practically everything we will need, so there are relatively few really new ideas +to be introduced. Since you may need to refresh your memory about what we +are going to need, we give in Table 32-1 a list of the equations we are going to +use, together with a reference to the place where each can be found. In most +instances, we will not take the time to give the physical arguments again, but +will just use the equations. +Table 32-1 +Our work in this chapter will be based on the following material, +already covered ỉin earlier chapters +Damped oscillations Vol. I, Chap. 25 m(& + +ä + uậ%) = F +Index oŸ gases Vol. I, Chap. 31 m= = — ức +2 co(uộ — 2) +?: = T.` — ?” +Mobility Vol. I, Chap. 41 m + na = F' +Electrical conductivity Vol. I, Chap. 43 I=c —; ơ= XdcT +Polarizability Vol. II, Chap. 10 Øpi =—W-P +Inside dielectrics Vol. II, Chap. 11 oca —= + = P +--- Trang 412 --- +We begin by recalling the machinery of the index of refraction for a gas. WWe +suppose that there are / particles per unit volume and that each particle behaves +as a harmonie oscillator. We use a model of an atom or molecule in which the +electron is bound with a force proportional to its displacement (as though the +electron were held in place by a spring). We emphasized that this was not a +legitimate cÍassical model oŸ an atom, but we will show later that the correct +quantum mechanical theory gives results equivalent to this model (in simple +cases). In our earlier treatment, we did not include the possibility of a damping +force in the atomic oscillators, but we will do so now. Such a force corresponds +to a resistance to the motion, that is, to a force proportional to the velocity of +the electron. 'Then the equation of motion is +P=qeE= m(ä + +ã + 083), (32.1) +where # is the displacement parallel to the direction of E. (We are assuming an +#sotropic oscillator whose restoring force is the same in all directions. Also, we +are taking, for the moment, a linearly polarized wave, so that # doesnt change +direction.) TÝ the electric fñeld acting on the atom varies sinusoidally with time, +we WIIbe +E= Eoe“!. (32.2) +'The displacement will then oscillate with the same frequency, and we can let ++ = xục*!, +Substituting # = + and # = —u2z, we can solve for # in terms of += — 8m E. (32.3) +—2 + iu + +lnowing the displacement, we can calculate the acceleration # and find the +radiated wave responsible for the index. This was the way we computed the +Index in Chapter 3l of Volume I. +Now, however, we want to take a different approach. The induced dipole +moment ø of an atom is ge# or, using Eq. (32.3), +p= —#m__ E. (32.4) +—“ + 1⁄0 -T U§ +Since ø is proportional to #7, we write +Ð= coo(0)E, (32.5) +where œ is called the œformic polarizabilit.X With thịs delnition, we have +"—..._.. (32.6) +—2 + iu + uậ +The quantum mechanical solution for the motions of electrons in atoms gives +a similar answer except with the following modifications. "The atoms have several +natural frequencies, each frequency with its own dissipation constant +. Also +the efective “strength” of each mode is diferent, which we can represent by +multiplying the polarizability for each frequency by a strength factor ƒ, which is +a number we expect to be of the order of 1. Representing the three parameters +œọ, +, and ƒ by œạy, +, and ƒ„ for each mode of oscillation, and summing over +the various modes, we modify Eq. (32.6) to read +LPP Jh +œ(0) = —— ——————-. 32.7 +(6) cụm » —2 + k0 - Uy, (327) +* Throughout this chapter we follow the notation of Chapter 31 of Volume I, and let œ +represent the ø#om%c polarizability as defned here. In the last chapter, we used œ to represent +the 0olưzne polarizability—the ratio of P to #. In the notation of #h2s chapter ? = Nœeo +(see Eq. 32.8). +--- Trang 413 --- +TẾ N is the number of atoms per unit volume in the material, the polarization +1s Just p = cogNằœ, and is proportional to F7; +ÐP = cạNaœ(U)E. (32.8) +In other words, when there is a sinusoidal electric fñeld acting ín a material, +there is an induced dipole moment per unit volume which is proportional to the +electric field——with a proportionality constant œ that, we emphasize, depends +upon the frequency. At very high frequencies, œ is small; there is not much +response. However, at low frequencies there can be a strong response. Also, the +proportionality constant is a complex number, which means that the polarization +does not exactly follow the electric feld, but may be shifted in phase to some +extent. At any rate, there is a polarization per unit volume whose magnitude is +proportional to the strength of the electric field. +32-2 Maxwell's equations ỉn a dielectric +The existence of polarization in matter means that there are polarization +charges and currents inside of the material, and these must be put into the +complete Maxwell equations in order to find the felds. We are goïng to solve +Maxwell's equations this time in a situation in which the charges and currents +are not zero, as in a vacuum, but are given implieitly by the polarization vector. +Our frst step is to fnd explicitly the charge density ø and current density 7, +averaged over a small volume of the same size we had in mind when we defñned . +'Then the ø and 7 we need can be obtained from the polarization. +W© have seen in Chapter 10 that when the polarization ? varies from place +to place, there is a charge density given by +0p =—V -P. (32.9) +At that time, we were dealing with static fields, but the same formula is valid +also for time-varying fñelds. However, when j? varies with time, there are charges +in motion, so there is also a polarization curren‡. Bach of the oscillating charges +contributes a current equal to its charge qe, times 1s velocity ø. With ÑN such +charges per unit volume, the current density 7 is +3 = Nqẹ0. +Since we know that 0 = d+/di, then j = NMq.(dz/díf), which is just đP/dt. +Therefore the current density from the varying polarization is +đpol = s (32.10) +Our problem is now direct and simple. We write Maxwell*s equations with +the charge density and current density expressed in terms of , using Eqs. (32.9) +and (32.10). (We assume that there are no other currents and charges in the +material.) We then relate to E with Eq. (32.S), and we solve the equation for ++t; and B—looking for the wave solutions. +Before we do this, we would like to make an historical note. Maxwell originally +wrote his equations in a form which was diferent from the one we have been using. +Because the equations were written in this diferent form for many years—and +are still written that way by many people—we will explain the difference. In the +early days, the mechanism of the dielectric constant was not fully and clearly +appreciated. "The nature of atoms was not understood, nor that there was a +polarization of the material. So people did not appreciate that there was a +contribution to the charge density ø from V- 7. 'Phey thought only in terms of +charges that were not bound to atoms (such as the charges that flow in wires or +are rubbed of surfaces). +Today, we prefer to let ø represent the foføŸ charge density, including the part +from the bound atomic charges. If we call that part øpoI, we can write +0—~ Ppol + other;› +--- Trang 414 --- +where Øother 1s the charge density considered by Maxwell and refers not bound to +individual atoms. We would then write +\v⁄ .Et— Øpol + other : +Substituting øpoi from Eq. (32.9), +O eFr 1 +và 2= n. +V. (coE + P) = fother- (32.11) +The current density in the Maxwell equations for V x #Ö also has, in general, +contributions from bound atomic currents. We can therefore write +7 — 2pol + đother› +and the Maxwell equation becomes +j loi , ÔE +cÓVxB~ deber „ To. CỬ, (32.12) +€0 €0 ðt +Using Eq. (32.10), we get +cọc Vxb-= 2other + areoE+ P). (32.13) +Now you can see that IfÍ we were to đefine a new vector Ù by +D-=‹coE+P, (32.14) +the two field equations would become +V-D= petney (32.15) +cọạc?V x B=7„.„ + 2r" (32.16) +'These are actually the forms that Maxwell used for dielectrics. His Ewo remaining +equations were +VxE=-— +V.B-=O0, +which are the same as we have been using. +Maxwell and the other early workers also had a problem with magnetic +materials (which we will take up soon). Because they did not know about the +circulating currents responsible for atomic magnetism, they used a current density +that was missing still another part. Instead of Eq. (32.16), they actually wrote +„ , gD +VxH=7+-_.., (32.17) +where HH difers from coc2B because it includes the efects of atomic currents. +(Then 7“ represents what is left of the currents.) So Maxwell had ƒouz feld +vectors—E, D, B, and H—the D and H were hidden ways of not paying +attention to what was going on inside the material. You will ñnd the equations +written this way in many places. +To solve the equations, 1t is necessary to relate 2 and H to the other fields, +and people used to write +D-=cE and B—uHhH. (32.18) +--- Trang 415 --- +However, these relations are only approximately true for some materials and +even then only if the fields are not changing rapidly with time. (For sinusoidally +varying fields one often cøn write the equations this way by making e and +complex functions of the frequency, but not for an arbitrary time variation of +the fields.) So there used to be all kinds of cheating in solving the equations. +W©e think the right way 1s to keep the equations in terms of the fundamental +quantities as we now understand them——and that°s how we have done it. +32-3 Waves in a dielectric +Wce want now to ñnd out what kind oŸ electromagnetic waves can exist in a +dielectric material in which there are no extra charges other than those bound +in atoms. 5o we take = —VW - P and j = ØP/ðt. Maxwells equations then +become +V.P 3g(/P +(a) W-E=-———— (b) 2W xB= ap( +E) +€0 lôI) €0 +(32.19) +(c) WxE=-—- (dd) V:B=0 +W© can solve these equations as we have done before. We start by taking the +curl of Eq. (32.19c): +Vx(VxE)=-a.V xỞ. +Next, we make use of the vector identity +Vx(VxE)=V(V:E) - V°E, +and also substitute for V x Ö, using Eq. (32.19b); we get +V(V:E)—V?E= 1 6ô?P 103E +_—��� cọc ôi c2 2` +Using Eq. (32.19a) for V - E, we get +1 3E 1 1 Ø@P +VỶE— =—s=-_V(V:P)+——- —.>. 32.20 +c2 Ø2 €0 ( )† cọc? Ø2 ( ) +So instead of the wave equation, we now get that the DˆAlembertian of is +cqual to ©wo terms involving the polarization . +Since depends on #, however, q. (32.20) can still have wave solutions. We +will now limit ourselves to 7sofropic dielectrics, so that PP ¡is always in the same +direction as #. Let”s try to ñnd a solution for a wave going in the z-direction. +Then, the electric feld might vary as e1f—#Z),We will also suppose that the wave +1s polarized in the z-direction——that the electric fñield has only an z-component. +We write +E„ = Epcl©tf—2), (32.21) +You know that any function of (z — 0£) represents a wave that travels with +the speed ø. The exponent of Bq. (32.21) can be written as +—?k|z— ~t +so, q. (32.21) represents a wave with the phase velocity +Đph = /R. +The index of refraction øw is defined (see Chapter 31, Vol. I) by letting +ĐUph — n +--- Trang 416 --- +Thus Eq. (32.21) becomes +E„= Eacl20—nz/©), +So we can find ø by fnding what value of k is required if Eq. (32.21) is to satisfy +the proper ñeld equations, and then using +=—. 32.22 +n=— (32.22) +In an isotropic material, there will be only an z-component of the polarization; +then ? has no variation with the z-coordinate, so V - P? =0, and we get rid of +the first term on the right-hand side of Eq. (32.20). Also, since we are assuming +a linear dielectric, f„ will vary as €”“!, and Ø?P,/Øt2 = —œ?P„. The Laplacian +in Eq. (32.20) becomes simply ô2„/Øz? = —k?„, so we get +—k?E„ + E,=— “SP. (32.23) +c2 cọc2 +Now let us assume for the moment that since # is varying sinusoidally, we +can set Ð proportional to #, as in Eq. (32.8). (W© ll come back to discuss thìs +assumnption later.) We write +Ty = cọoNoœH„. +Then 2z drops out of Eq. (32.23), and we fnd +k^= = (1+ Na). (32.24) +W©e have found that a wave like Bq. (32.21), with the wave number & given by +Ea. (32.24), will satisfy the ñeld equations. Using Eq. (32.22), the index ø is +given by +nẰ=1+ No. (32.25) +Let's compare this formula with what we obtained in our theory of the index +of a gas (Chapter 31, Vol. I). There, we got Eq. (31.19), which is +1 Wq 1 +=l+;—“—ssz. 32.26 +⁄ + 2 mneọ —2 + ưÿ ) +Taking œ from Ea. (32.6), Eq. (32.25) would give us +nh=1+—“—————n. (32.27) +Tnég_ —~U“ + 10 -T U§ +First, we have the new term in 2+, because we are including the dissipation +of the oscillators. Second, the left-hand side is ø instead of nŸ, and there is an +extra factor of 1/2. But notice that if Ñ is small enough so that ø is close to +one (as it is for a gas), then Eq. (32.27) says that n2 is one plus a small number: +n? =1 +c. W© can then write ø = 1 + 1 + c/2, and the two expressions +are equivalent. Thus our new method gives for a gas the same result we found +earlier. +NÑow you might think that Bq. (32.27) should give the index of refraction +for dense materials also. It needs to be modified, however, for several reasons. +First, the derivation of this equation assumes that the polarizing feld on each +atom is the field „. That assumption is no right, however, because in dense +materials there is also the ñeld produced by other atoms in the vicinity, which +may be comparable to +. We considered a similar problem when we studied +the static fields in dielectrics. (See Chapter I1.) You will remember that we +estimated the fñeld at a single atom by imagining that it sat in a spherical hole +in the surrounding dielectric. The fñeld in such a hole—which we called the iocal +feld—is inereased over the average field # by the amount P/3eo. (Remember, +--- Trang 417 --- +however, that this result is only strictly true in isotropic materials——including +the special case of a cubic crystal.) +The same arguments will hold for the electric fñeld in a wave, so long as the +wavelength of the wave is mụuch longer than the spacing between atoms. Limiting +ourselves to such cases, we write +luc = + —. (32.28) +Thịis local field is the one that should be used for in Eq. (32.3); that is, Eq. (32.8) +should be rewritten: +P = coNgư Husa. (32.29) +Using Eloeai rom Ba. (32.28), we fnd +P= co lô2 (z + XS.) +P=_————_- cử. 32.30 +1- (Na/3)"° (8230) +In other words, for dense materials is still proportional to # (for sinusoidal +fñelds). However, the constant oŸ proportionality is not eojœ, as we wrote below +Eq. (32.23), but should be eoWœ/[1 — (Nø/3)|. So we should correct Eq. (32.25) +to read N +=l+———--ax: 32.31 +⁄ 1— (WNa/3) 231) +Tt will be more convenient if we rewrite this equation as +n2 — 1 +3 —=——=N 32.32 +Ta, (32.32) +which is algebraically equivalent. 'This is known as the Clausius-Mossotti equation. +There is another complication in dense materials. Because neighboring atoms +are so close, there are strong interactions between them. 'Phe internal modes +of oscillation are, therefore, modified. 'The natural frequencies of the atomic +oscillations are spread out by the interactions, and they are usually quite heavily +damped——the resistance coefficient becomes quite large. So the œ's and +ˆs of the +solid wiïll be quite diferent from those of the free atoms. With these reservations, +we can still represent œ, at least approximately, by Eq. (32.7). We have then that +2_—_ 1 N 2 +3— = ` (32.33) +m2 +2 Tneo TT M” T 1k -T ấy, +One fñnal complication. If the dense material is a mixture of several compo- +nents, each will contribute to the polarization. "The total œ will be the sum of +the contributions from each component of the mixture [except for the inaccuracy +of the local fñeld approximation, Eq. (32.28), in ordered crystals—efects we +discussed when analyzing ferroelectricsl. Writing ; as the number of atoms of +cach component per unit volume, we should replace Eq. (32.32) by +n2 — 1 +where each œ; will be given by an expression like Eq. (32.7). Equation (32.34) +completes our theory of the index of refữraction. The quantity 3(n? — 1)/(n? +2) +1s given by some complex function of frequency, which ¡is the mean atomic +polarizability œ(œ). "The precise evaluation of œ(œ) (that is, fñnding ƒ», + +and œog) in dense substanees is a dificult problem of quantum mechanics. It has +been done from first principles only for a few especially simple substances. +--- Trang 418 --- +32-4 The complex index of refraction +We want to look now at the consequences of our result, Bq. (32.33). Pirst, +we notice that œ is complex, so the index ø is going to be a complex number. +What does that mean? Let”s say that we write ?ø as the sum of a real and an +lmaginary part: += Ti — ỨHỊ, (32.35) +where £p and ø are real functions of ¡. We write zn; with a minus sign, so +that mự will be a positive quantity in all ordinary optical materials. (In ordinary +Inactive materials—that are not, like lasers, light sources themselves—z+ is a +positive number, and that makes the imaginary part of + negative.) Qur plane +wave of Eq. (32.21) is written in terms of né as N +By = Byetet=nz/©), N v7 +Writing m= as in Eq. (32.35), we would have ÀV +Eụ = EoeTentZ/ccje=nnz/6), (32.36) TơNG +The term c2Œ~”z#Z/°) represents a wave travelling with the speed c/®=p, SO tạ _— ——" Z +represents what we normally think of as the index ofrefraction. But the amplitude ^ < +of this wave is x⁄ ecen2/ ¿„ the index is real, and the metal becomes transparent. +You know, of course, that metals are reasonably transparent to x-rays. But +--- Trang 423 --- +some metals are even transparent in the ultraviolet. In Table 32-3 we give +for several metals the experimental observed wavelength at which they begin +to become transparent. In the second column we give the calculated critical +wavelength À„ = 2ze/œ„. Considering that the experimental wavelength is not +too well defñned, the ft of the theory is fairly good. +You may wonder why the plasma frequency œ„ should have anything to do +with the propagation of electromagnetic waves in metals. 'Phe plasma frequency +came up in Chapter 7 as the natural frequency of đens#u oscillations of the free +electrons. (A clumnp of electrons is repelled by electric forces, and the inertia of Table 32-3 +the electrons leads to an oscillation of density.) So longitud¿nal pÌasma waves are Wavelengths below which the metal +resonant at œ„. But we are now talking about #ransuerse electromagnetic waves, becomes transparent” +and we have found that transverse waves are absorbed for frequencies below ứ;. +(Tt's an interesting and nøø£ accidental coincidence.) +Although we have been talking about wave propagation in metals, you appre- Li 1550  1550  +ciate by this time the universality of the phenomena of physics—that it doesn”t Na 2100 2090 +make any diference whether the free electrons are in a metal or whether they K 3150 2870 +are in the plasma. of the iIonosphere of the earth, or in the atmosphere of a +star. To understand radio propagation in the ionosphere, we can use the same : +expressions—using, of course, the proper values for W and 7. We can see now Erom: C. Kittel, Iniroduction to 5olid +; ; State Phụs¿cs, John Wiley and Song, +why long radio waves are absorbed or refected by the ionosphere, whereas short Inc., New York, 2nd ed., 1956, p. 266. +waves go ripht through. (Short waves must be used for communication with +sabellites.) +W© have talked about the high- and low-frequency extremes for wave DroP- +agation in metals. Eor the in-bebween frequencies the full-blown formula of +Ea. (32.42) must be used. In general, the index will have real and imaginary +parts; the wave is attenuated as it propagates into the metal. Eor very thin layers, +mmetals are somewhat transparent even at optical frequencies. As an example, +special goggles for people who work around high-temperature furnaces are made +by evaporating a thin layer of gold on glass. The visible light is transmitted fairly +well—with a strong green tỉinge—but the infared is strongly absorbed. +Finally, it cannot have escaped the reader that many of these formulas resemble +in some ways those for the dielectric constant œ discussed in Chapter 10. 'Phe +dielectric constant & measures the response of the material to a constant field, +that is, for œ = 0. If you look carefully at the defñnition of ø and & you see +that is simply the limit of nŸ as œ —> 0. Indeed, placing œ = 0 and n2 = +in equations of this chapter will reproduce the equations of the theory of the +dielectric constant of Chapter 11. +--- Trang 424 --- +Moflocffort frorm Srrrfere©s +33-1 Reflection and refraction of light +The subject of this chapter is the refection and refraction of light——or elec- 33-1 Reflection and refraction of light +tromagnetic waves in general—at surfaces. We have already discussed the laws 33-2 Waves in dense materials +of refection and refraction in Chapters 26 and 33 of Volume I. Here's what we 33-3 The boundary conditions +found out there: 33-4 The reflected and transmitted +1. The angle of reflection is equal to the angle of incidence. With the angles Wave©s +defned as shown in Eig. 33-], 33-5 Reflection from metals +0„ = Ú,. (33.1) 33-6 Total internal relection +2. The produet ø+sin Ø is the same for the incident and transmitted beams +(Snell's law): +1 sin Ổ¿ = 1a sin Úy. (33.2) +3. The intensity of the refected light depends on the angle of ineidence and +also on the direction of polarization. EFor # perpendicular to the plane of Reuicu: Chapter 33, Vol. Lý Polariza- +incidence, the refection coefficient Ï¡ is tion +Rị 1 - S (0i— 0/), (33.3) +1, sin2(6; + 6;) +Eor E parallel to the plane of incidence, the refection coefficient lđ\ is +Rị= T = tan (ái — 6u) ĐỒ (33.4) +ỉ tan (6; + 6,) +4. Eor normal incidence (any polarization, of coursel), T2 | ¬. " +Tụ - (5) : (33.5) ¬“ c* +1, Ta + T.Ị ` “at " ¬- Š +N `. Ẳ. kia 3$ +(Parlier, we used ¿ for the incident angle and r for the refracbed angle. Since ¬- Ó, - ` " 2À +we can” use ? for both “refracted” and “reflected” angles, we are now using ". +6Ø; = incident angle, ��„ = refected angle, and Ø¿ = transmitted angle.) N : +Our earlier discussion is really about as far as anyone would normally need to mm : <4NŠ So +go with the subject, but we are going to do it all over again a diferent way. Why? ¬ . ¬_. SURFACE +One reason is that we assumed before that the indexes were real (no absorption NT. _ +in the materials). But another reason is that you should know how to deal with " mm =a +what happens to waves at surfaces from the point of view of Maxwell's equations. "sô ` _. "¬ +'W©elll get the same answers as before, but now from a straightforward solution of "¬ s. n¬ ¬ : +the wave problem, rather than by some clever arguments. " +We want to emphasize that the amplitude of a surface reflection is not a Fig. 33-1. Reflection and refraction of +property of the rmøterzal, as is the Index of refraction. Ït is a “surface property,” light waves at a surface. (The wave direc- +one that depends precisely on how the surface is made. Á thin layer oŸ extraneous tions are normal to the wave crests.) +Jjunk on the surface between two materials oŸ indices + and nạ will usually change +the reflection. (There are all kinds of possibilities of interference here——like the +colors of oïl fñlms. Suitable thickness can even reduce the refected amplitude to +zero for a given frequency; that?s how coated lenses are made.) The formulas we +will derive are correct only if the change ofindex is sudden—within a distance very +small eompared with one wavelength. For light, the wavelength is about 5000 Ä, +so by a “smooth” surface we mean one in which the conditions change in goỉng +a distance of only a few atoms (or a few angstroms). Qur equations will work +--- Trang 425 --- +for light for highly polished surfaces. In general, ¡f the index changes gradually +over a distance of several wavelengths, there is very little refection at all. +33-2 Waves in dense materials +Pirst, we remind you about the convenient way of describing a sinusoidal y +plane wave we used in Chapter 34 of Volume I. Any field component in the wave +(we use # as an example) can be written in the form P +E= Ege@et-k). (33.6) ` +where #/ represents the amplitude at the point r (from the origin) at the time ứ. ý > +The vector & points in the direction the wave is travelling, and its magnitude |k| = +k = 2z/À is the wave number. The phase velocity of the wave is 0pụ = (/k; for ⁄) +a light wave in a material of index , 0pụ = C/n, SO ⁄Z2 +móng ⁄S<1 x^ +. (33.7) N +WAVE CRESTS +Suppose k is in the z-direction; then & - r is jusb kz, as we have often used it. + XS +For k in any other direction, we should replace z by r„, the distance from the +origin in the k-direction; that is, we should replace &kz by kr„, which is Just k -r. +(See Fig. 33-2.) So Bq. (33.6) is a convenient representation oŸ a wave in any Fig. 33-2. For a wave moving in the +đirection. direction k, the phase at any point +We must remember, of course, that IS (0£ — k-r). +k-r = k„# + kuU + kz;z, +where k„, k„, and k; are the components of & along the three axes. In fact, we +pointed out once that (œ, k„, kụ, k„) is a four-vector, and that its scalar product +with (f,z,,z) is an invariant. So the phase of a wave is an invariant, and +Eq. (33.6) could be written +E= Eoefen, +But we don't need to be that fancy now. +Eor a sinusoidal #, as in Eq. (33.6), Ø/O is the same as j7, and 9E /9z +is —?k„E, and so on for the other components. You can see why it is very +convenient 6o use the form in Eq. (33.6) when working with diferential equations—— +diferentiations are replaced by multiplications. One further useful point: The +operation W = (0/9z,0/Øu,Ø/9z) gets replaced by the three multiplications +(—/k„,—¿ky,—ik„). But these three factors transform as the components of the +vector &, so the operator V gets replaced by multiplication with —¿È: +— ->i +V—› —¡k. (33.8) +This remains true for any W operation—whether it is the gradient, or the +divergence, or the curl. For instance, the z-component of V x # is +0y _ ØE„ +Ôz Ôy ` +Tf both „ and F„ vary as e”'*”, then we get +—‡k„ uy + thụ E„, +which is, you see, the z-component of —¿k x E. +So we have the very useful general fact that whenever you have to take the +gradient of a vector that varies as a wave in three dimensions (they are an +important part of physics), you can always take the derivations quickly and +almost without thinking by remembering that the operation V is equivalent to +multiplication by —¿k. +--- Trang 426 --- +For instance, the Earaday equation +VxE-=-— +becomes for a wave +—¿k x E — —iuHB. +This tells us that +5ö=——,, (33.9) +which corresponds to the result we found earlier for waves in free space—that +B,ïn a wave, is at right angles to # and to the wave direction. (In free space, +œ/È = c.) You can remember the sign in Eq. (33.9) from the fact that k is in the +đirection of Poynting's vector S = cạc?E x Ö. +TÍ you use the same rule with the other Maxwell equations, you get again the +results of the last chapter and, in particular, that +But since we know that, we won't do it again. +TÍ you want to entertain yourself, you can try the following terrifying problem +that was the ultimate test for graduate students back in 1890: solve Maxwells +equations for plane waves in an ønwsotropic crystal, that is, when the polariza- +tion ? is related to the electric ñeld by a tensor of polarizability. You should, +Of course, choose your axes along the principal axes of the tensor, so that the +relations are simplest (then y„ = œ„„, „ = ayl, and P, = œ„¿F,), but let +the waves have an arbitrary direction and polarization. You should be able +to fnd the relations between # and #Ö, and how k varies with direction and +wave polarization. 'Phen you will understand the optics oŸ an anisotropic crystal. +It would be best to starb with the simpler case of a birefringent crystal——like +calcite—for which two of the polarizabilities are equal (say, œạ = œ„), and see lf +you can understand why you see double when you look through such a crystal. +TÍ you can do that, then try the hardest case, in which all three œ's are different. +Then you will know whether you are up to the level of a graduate student of 1890. +In this chapter, however, we will consider only isotropic substances. +Ỷ "¬ "n cz ⁄ +h _E, tu ' "¬ ' +¬ - - ... * Er +s. G MS NG ẻ ký +R ]{“= x +Ta : ` Fig. 33-3. The propagation vectors k, kí, +¬ ¬¬ ... and k” for the incident, reflected, and trans- +: " mitted waves. +We know from experience that when a plane wave arrives at the boundary +between two different materials—say, air and glass, or water and oil—there is a +wave reflected and a wave transmitted. Suppose we assume no more than that and +see what we can work out. We choose our axes with the z-plane in the surface and +the z#-plane perpendicular to the incident wave surfaces, as shown in Fig. 33-3. +--- Trang 427 --- +The electric vector of the inecident wave can then be written as +E¡= EogcẴet=Er), (33.11) +Since & is perpendicular to the z-axis, +k-r= k„z + kuU. (33.12) +We write the refected wave as +E„ = Epel6 kim), (33.13) +so that its frequeney is œ', its wave number is k', and its amplitude is Eạ. (We +know, of course, that the frequeney is the same and the magnitude of kf is the +same as for the incident wave, but we are not going to assume even that. We +will let it come out oŸ the mathematical machinery.) Finally, we write for the +transmitted wave, ¬ +E,= Epel@ tk), (33.14) +W©e know that one of Maxwell's equations gives Pq. (33.9), so for each of the +waves we have +kx E, khxE k”xE +B,= ———, B„= ; " b.= „ —, (33.15) +Also, if we call the indexes of the two media ø and nạ, we have from aq. (33.10) +k” = kệ + kỹ = _ (33.16) +Since the refected wave is in the same material, then +2 _ Tị +kÝ= _¬ (33.17) +whereas for the transmitted wave, +„a T2 +k^ = _ (33.18) +‹ _ RYV +33-3 The boundary conditions ¬ "¬ ` +All we have done so far is to describe the three waves; our problem now is to ` nó t - +work out the parameters of the refected and transmitted waves in terms of those _ nh; - +of the incident wave. How can we do that? The three waves we have described ¬. +satisfy Maxwells equations in the uniform material, but Maxwell°s equations qui th +must also be satisfed a# the boundary bebween the two diferent materials. So ¬ E +we mus now look at what happens right at the boundary. We will ñnd that ¬. „ +Maxwell's equations demand that the three waves ft together in a certain way. ¬ +As an example of what we mean, the -component of the electric fñeld E/ must ` tÓ, : +be the same on both sides of the boundary. 'This is required by Faraday's law, ¬_— +0B ——. +VxE=--—_ 33.19 m ,a ề +5c: (33.19) +as we can see in the following way. Consider a little recbtangular loop I` which E F1g. Tp Ngài: concron Eya = +straddles the boundary, as shown in Fig. 33-4. Equation (33.19) says that the y+ is obtained from #, E - ds = 0. +line integral of E around l is equal to the rate of change of the fux of Ö through +the loop: +đEsas= củ, Bsndn +Now imagine that the rectangle is very narrow, so that the loop encloses an +inũnitesimal area. If Ö remains fnite (and there's no reason it should be inũnite +at the boundary!) the ñux through the area is zero. So the line integral of E +--- Trang 428 --- +must be zero. lf F„¡ and F„z are the components of the field on the two sides of +the boundary and ïf the length of the rectangle is , we have +đi — Jua[ = 0Ö +Đi = Eụa, (33.20) +as we have said. This gives us one relation among the fields of the three waves. +'The procedure of working out the consequences of Maxwell's equations at the +boundary is called “determining the boundary conditions.” Ordinarily, it is done +by ñnding as many equations like Eq. (33.20) as one can, by making arguments +about little rectangles like I'in Fig. 33-4, or by using little gaussian surfaces that +straddle the boundary. Although that is a perfectly good way of proceeding, it +gives the impression that the problem of dealing with a boundary is difÑferent for +every diferent physical problem. +For example, in a problem of heat ow across a boundary, how are the +temperatures on the ©wo sides related? Well, you could argue, for one thing, that +the heat fow £o the boundary from one side would have to equal the Ñow ad +from the other side. It is usually possible, and generally quite useful, to work +out the boundary conditions by making such physical arguments. There may be +times, however, when in working on some problem you have only some equations, +and you may not see right away what physical arguments to use. So although we +are at the moment interested only in an electromagnetic problem, where we cøn +make the physical arguments, we want to show you a method that can be used +for any problem——a general way of fñnding what happens at a boundary directly +from the diferential equations. +W© begin by writing all the Maxwell equations for a dielectric—and this time +we are very specifc and write out explicitly all the components: +V.E=-—-——— +9l„ 0E, 0E, 9P 0P 0P, +——+— + —_ Ì=-|—>+—+— 33.21 +ST tt) (tt) ) +VxE-=-— +9E, 0Ey öB„ +— —Ủ—___“ 33.22 +Øy Øz Øt ( 8) +g91„ ðØE, By +—- — ZẨ—__ 33.22b +Øz Øz lôI2 ) +ðy 0l„ 8B, +—_ “=—_ .“ 33.22 +Ôx — Ôụ ðt (33.22c) +V.B=0 +0B, 0B, 0B, +—=—= + = +. ===0 33.23 +Øz + ỡy + Øz ( ) +16Ø0P 6E +Vxb=_—-_—+¬+— +, €0 ðt + ðt +8B 3B 1.ØP 8E +2 Z 1U œ bã +—_——#]=—-—1+_-__“ 33.24 +_`- 2) cọ Øi " 9 ' 8) +3B 3B 1 0P, 8E +2 ba z U Ụ +—_ — “| =_—-."1+-." 33.24b +“l5 h dạ Ôi ` Ôf (3.24) +3B 3B 1 0P, ðØE +2 ụ % Z Z +—_— | =——_=— +-_Z 33.24 +`. mì sụ Ôt ` Ôt (33.24) +--- Trang 429 --- +NÑow these equations must all hold in region 1 (to the left of the boundary) +and in region 2 (to the right of the boundary). We have already written the +solutions in regions 1 and 2. Finally, they must also be satisled #w the boundary, +which we can call region 3. Although we usually think of the boundary as being +sharply discontinuous, in reality it is not. The physical properties change very +rapidly but not infñnitely fast. In any case, we can imagine that there is a very +rapid, but continuous, transition of the index between region I and 2, in a short +distance we can call region 3. Also, any fñeld quantity like ;, or #„, etc., will +make a similar kind of transition in region 3. In this region, the diferential +equations must still be satisfed, and it is by following the diferential equations +in this region that we can arrive at the needed “boundary conditions.” +For instance, suppose that we have a boundary between vacuum (region l1) P. +and glass (region 2). There is nothing to polarize in the vacuum, so ? = 0. Let's PB +say there is some polarization s in the glass. Between the vacuum and the glass : +there is a smooth, but rapid, transition. If we look at any component of , say „, (a) ' +it might vary as drawn in Eig. 33-5(a). Suppose now we take the first of our +cquations, Eq. (33.21). It involves derivatives of the components of with respect +to ø, , and z. The - and z-derivatives are not interesting; nothing spectacular is — +happening in those directions. But the z-derivative of „ will have some very large 1= ¬ - +values in region 3, because of the tremendous slope of „. The derivative ØPz„/9z REGION 1 ! REGION 3 ! REGION 2 +will have a sharp spike at the boundary, as shown in Eig. 33-5(b). IÝ we imagine ôP, +squashing the boundary to an even thinner layer, the spike would get much higher. " +Tí the boundary is really sharp for the waves we are interested in, the magnitude +of 8P„/Ø+ in region 3 will be much, mụuch greater than any contributions we ' +might have from the variation of in the wave away from the boundary——so ' +we ignore any variations other than those due to the boundary. 0), +Now how can Bd. (33.21) be satisfed if there is a whopping bịg spike on the ' +right-hand side? Only if there is an equally whopping big spike on the other side. ' +Something on the left-hand side must also be big. The only candidate is Ø⁄+„/Øz, ` +because the variations with and z are only those small efects in the wave we ' ' +Jusb mentioned. So —eco(Ø+/Ø+z) must be as drawn in Eig. 33-5(c)—just a copy ' +of 0P„/Øz. We have that ¬.—. +Ø„ 9Ð, 8x +TÍ we integrate this equation with respect to # across region 3, we conclude that (© Ị ị +co(E„a — E„ì) = —(f„s — mi). (33.25) | | +In other words, the jump in eog„ in going from region 1 to region 2 must be +cqual to the ]ump in — Tỳ. * +W© can rewrite Eq. (33.25) as +Fig. 33-5. The fields In the transition +co>a + Đ»ya = coE„¡ + Tàn, (33.26) region 3 between two different materials in +regions 1 and 2. +which says that the quantity (eo Z„ + P„) has equal values in region 2 and region 1. +People say: the quantity (eo„ + Ty) 1s continuous across the boundary. WWe +have, in this way, one of our boundary conditions. +Although we took as an illustration the case in which + was zero because +region l1 was a vacuum, it is clear that the same argument applies for any two +materials in the two regions, so Eq. (33.26) is true in general. +Let's now go through the rest of Maxwell's equations and see what each of +them tells us. We take next Eq. (33.22a). There are no #-derivatives, so it doesnE +tell us anything. (Remember that the fields #hemseloes do not get especially +large at the boundary; only the derivatives with respect to + can become so huge +that they dominate the equation.) Next, we look at Eq. (33.22b). Ahl There is +an #-derivativel We have Ø⁄;z/Øz on the left-hand side. Suppose it has a huge +derivative. But wait a moment†l 'There is nothing on the right-hand side to match +it with; therefore y cœnnot have any ]ump in going from region Ì to region 2. [HÝ +it dịd, there would be a spike on the left of Ðq. (33.22b) but none on the right, +--- Trang 430 --- +and the equation would be false.| 5o we have a new condition: +Đà = E„. (33.27) +By the same argument, Eq. (33.22c) gives +đua = đấu. (33.28) +Thịs last result is just what we gọt ín Eq. (33.20) by a line integral argument. +W©e go on to Bd. (33.23). The only term that could have a spike is 9ÖB„/9z. +But there's nothing on the right to match it, so we conclude that +Ba = Bại. (33.29) +On to the last of Maxwell's equationsl Equation (33.24a) gives nothing, +because there are no z-derivatives. Pquation (33.24b) has one, —c?ØB;/Øz, but +again, there is nothing to match it with. We get +B;a = Bại. (33.30) +'The last equation is quite similar, and gives +DĐụa = Bụi. (33.31) +Table 33-1 +The last three equations gives us that ạ = ị. We want to emphasize, _~ +however, that we get this result only when the materials on both sides of the Boundary conditions at the surface of a +boundary are nonmagnetic—or rather, when we can neglect any magnetic effects dielectric +ofthe materials. 'Phis can usually be done for most materials, except ferromagnetic Ea PO.=(eoE.+P +ones. (We will treat the magnetic properties of materials in some later chapters.) (oi + PỊ)z = (co; + P›); +Our program has netted us the six relations between the fñelds in region 1 Œị)y = (Ea)y +and those in region 2. We have put them all together in Table 33-1. We can now (E1)z = (Ea); +use them to match the waves in the two regions. We want to emphasize, however, B:=b› +that the idea we have just used will work in an physical situation in which you (The surface is in the gz-plane) +have diferential equations and you want a solution that crosses a sharp boundary +between bwo reglons where some property changes. For our presen% purposes, +we could have easily derived the same equations by using arguments about the +fuxes and circulations at the boundary. (You might see whether you can get the +same result that way.) But now you have seen a method that will work in case +you ever get stuck and don” see any easy argument about the physics of what is +happening at the boundary——you can just work with the equations. +33-4 The reflected and transmitted waves +Now we are ready to apply our boundary conditions to the waves we wrote +down in Section 33-2. We had: +E, = EoclGt-ResRuU), (33.32) +E, = Epclet-Rie-RiU), (33.33) +E, = Epcl6 1e Ei0), (33.34) +kx E;, +B,=———, (33.35) +We have one further bit of knowledge: # ¡is perpendicular to Its propagation +vector k for each wave. +--- Trang 431 --- +The results will depend on the direction of the #-vector (the “polarization”) of +the incoming wave. The analysis is much simplifed iŸ we treat separately the case +oŸ an incident wave with its E-vecbor paraliel to the “plane of incidence” (that is, +the z-plane) and the case of an incident wave with the E-vector perpendicular +to the plane of incidence. AÁ wave of any other polarization is just a linear ÓC huy xà ÂM +combination of two such waves. In other words, the reflected and transmitted ¬— ¬ +Intensities are diferent for diferent polarizations, and it is easiest to pick the " vẽ " ¬ _¬ k” +two simplest cases and treat them separately. `, ng „nh: cHỈ c +We© will carry through the analysis for an incoming wave polarized perpendic- ¬ >> +ular to the plane of ineidence and then just give you the result for the other. We nh TU ĐÔNG Tu Bì +are cheating a little by taking the simplest case, but the principle is the same for Số ga XIN NG +both. So we take that ; has only a z-component, and since all the -vectors _ TT Tư ở Z x +are in the same direction we can leave off the vector signs. _. ¬-.. +So long as both materials are isotropic, the induced oscillations of charges in xo PB Z. +the material will also be in the z-direction, and the #-feld of the transmitted UY N _— SURFACE +and radiated waves will have only z-components. So for all the waves, „ and lr, ẹ ¬ , và l Tân +and „ and ự are zero. The waves will have their E- and -vectors as drawn in ¬ ¬" +Eig. 33-6. (WS are cutting a corner here on our original plan of getting everything N VỀn CÔ Tờ, ế nạ +from the equations. This result would also come out of the boundary conditionsg, `? `” +but we can save a lot of algebra by using the physical argument. When you have Fig. 33-6. Polarization of the reflected +Some spare time, see If you can get the same result from the equations. Ït is clear and transmitted waves when the E-field of +that what we have said agrees with the equations; it is just that we have not the incident wave is perpendicular to the +shown that there are no øfÖer possibilities.) plane of incidence. +NÑow our boundary conditions, Ðqs. (33.26) through (33.31), give relations +bebween the components of E and #Ö in regions 1 and 2. For region 2 we have +only the transmitted wave, but ín region l1 we have #uo waves. Which one do we +use? 'Phe fields in region 1 are, of course, the superposition of the fields of the +ineident and reflected waves. (Since each satisfes Maxwell”s equations, so does +the sum.) 5o when we use the boundary conditions, we must use that +tì =E,+ E„, ba = bi, +and similarly for the 's. +Eor the polarization we are considering, Pqs. (33.26) and (33.28) give us no +new information; only Eq. (33.27) is useful. It says that +đa + b„ = n, +d‡ the boundaru, that 1s, for z —= 0. So we have that +Eoe'6et=Ru9) + Eje'te 1=Ryw) — Elee ki), (33.38) +which must be true for ai £ and for ai! . Suppose we look first at =0. Then +we have +Eoc?t + Ejelet — Eletst +'This equation says that two oscillating terms are equal to a third oscillation. That +can happen only ïf all the oscillations have the same frequency. (It is impossible +for three—or any number—of such terms with diferent frequencies to add to +zero for all times.) So +œ” =ư =ứ. (33.39) +As we knew all along, the frequencies of the refected and transmitted waves are +the same as that of the incident wave. +W© should really have saved ourselves some trouble by putting that in at the +beginning, but we wanted to show you that i9 can also be got out of the equations. +'When you are doïng a real problem, it is usually the best thing to put everything +you know into the works right at the start and save yourself a lot of trouble. +By deñnition, the magnitude oŸ K is given by k = n”ằœ2/c, so we have also +that k2 k2 k2 +—z=-s=_.s (33.40) +Họ Ị Tị +--- Trang 432 --- +Now look at Eq. (33.38) for £ =0. Using again the same kind of argument +we have Just made, but this time based on the fact that the equation must hold +for all values of , we get that +kụ = ku = kụ. (33.41) +Erom Eq. (33.40), k2 = kỶ, so +2 /2 — L2 2 +ký + ký = kệ + kụ. +Combining this with Eq. (38.41), we have that +k2 — k2 +or that k¿ = +k„. The positive sign makes no sense; that would not give a +reflectcd wave, but another zncident wave, and we said at the start that we were +solving the problem of only one incident wave. So we have +k„ = —kạ. (33.42) +The two equations (33.41) and (33.42) give us that the angle of reflection is equal +to the angle of incidence, as we expected. (See Fig. 33-3.) The refected wave is +Eụ„ = EAclGttRsa—Euu), (33.43) +For the transmitted wave we already have that +kụ =Rụ, +_=.. (83.44) +so we can solve these to fnd k.. We get +r2 ——. r2 k2 ——. nộ k2 k2 33 45 +Suppose for a moment that + and mạ are real numbers (that the imaginary +parts of the indexes are very smaill). Then all the &?s are also real numbers, and +from Pig. 33-3 we ñnd that +T =sinÚ,, mm =sinÚy;. (33.46) +Erom (33.44) we get that +nạ sin Ö¿ = mạ sìn Ở¿, (33.47) +which is Snells law of refraction—again, something we already knew. lf the +indexes are not real, the wave numbers are cormmplex, and we have to use Eq. (33.45). +[We could still defime the angles Ø; and 6Ø; by Eq. (33.46), and Snells law, +Ea. (33.47), would be true in general. But then the “angles” also are complex +numbers, thereby losing their simple geometrical interpretation as angles. Ït is +best then to describe the behavior of the waves by their complex k„ or kZ values.] +So far, we haven'”t found anything new. We have just had the simple-minded +delight of getting some obvious answers from a complicated mathematical ma- +chinery. Now we are ready to fnd the amplitudes of the waves which we have +not yet known. sing our results for the œ¿'s and k's, the exponential factors In +Eq. (33.38) can be cancelled, and we get +Eo + Eh = E1. (33.48) +Since both #2 and #ÿ are unknown, we need one more relationship. We must use +another of the boundary conditions. The equations for „ and l„ are no help, +because all the #7s have only a z-component. So we must use the conditions +on Ö. Let)s try Eq. (33.29): +Đạya = Đại. +--- Trang 433 --- +trom Eqs. (33.35) through (33.37), +ID kị Đ, k2 +Bụi = “——, B„„ = -T—, D„ạị = _h +Recalling that œj“ = œ/ = ø and kj = kỳ = kụ, we get that +Eo + Eh = E1. +But this is just Eq. (33.48) all over againl We°ve just wasted time getting +something we already knew. +W©e could try Bq. (33.30), Ö¿a = ¿¡, but there are no z-components oŸ BI +So there's only one equation left: Eq. (33.31), Đụa = Bựụi. Eor the three waves: +kự„ Eị kị, Eú, hạ Fạụ +Đụi = — . Đựụy = — dt 7 Đụi = — “7 (33.49) +Putting for ¿, F„, and ; the wave expression for œ = 0 (to be at the boundary), +the boundary condition is +kự N Lại h Là Là kử : HÀ ,” +" Eaef@t—=RvU) + = Eaet« t—kyU) — — Eer« thu). +Again all ¿s and k„'s are equal, so this reduces to ¬š y +k„Eo + kị E) = kị EJ'. (8350) 225,2, +.ẻ vàn - _ " _ .| k” +This gives us an equation for the #?s that is diferent from Eq. (33.48). With `: xi +the two, we can solve for #ö and #Z. Remembering that k¿ = —k„, we get ¬ ">> B, +Eh= —- họ, (33.51) "“....... +% bu . ' Tin Tài Š x +20... 33.52 AC ' 8... +0 k„+kƑ ) Kể /IẾN ¬ SURFACE +These, together with Eq. (33.45) or Eq. (33.46) for kƒ, give us what we wanted .2 " Š " " +to know. We will discuss the consequeneces of this result in the next section. ¬ —-—.. " +Tf we begin with a wave polarized with its E-vector paraiiel to the plane of nà hàng : „.m, ti nạ +ineidenece, will have both z- and -components, as shown in EFig. 33-7. The ¬ +algebra is straightforward but more complicated. (The work can be somewhat Fig. 33-7. Polarization of the waves when +reduced by expressing things in this case in terms of the zmaønefic fields, which the E-field of the incident wave is parallel +are all in the z-direction.) One ñnds that to the plane of incidence. +n$k„ — n‡kƑ +Eộl=-#——zlE 33.53 +lái = nộp + nôRộ | (33.53) +2minak„ +EÌ= -s—— =s- bài. 33.54 +KổI= án ng ni; LÊ (33.54) +Let's see whether our results agree with those we got earlier. Equation (33.3) +is the result we worked out in Chapter 33 of Volume I for the ratio of the intensity +of the reflected wave to the intensity of the incident wave. Then, however, we +were considering only reai indexes. For real indexes (and &”s), we can write +kự„ = kcosØ; = cm" cos Ö;, +ký = k” cosØ; = “2 cọs 6;. +Substituting in Eq. (33.51), we have +Eh _ Hà CO5 Ổi — na COS Đi (33.55) +đo — mị cOS; + na cOS +--- Trang 434 --- +which does not look the same as Eq. (33.3). It will, however, if we use Snell?s law +to get rid of the n0 s. Setting nạ = ?m sỉn Ø;/ sin Ø;, and multiplying the numerator +and denominator by sinØ;, we get +E§ — cosØ;sinØ; — sin 0; cos 0; +Eo — cosØ;sinØ; + sinØ;cosØ;' +The numerator and denominator are just the sines of —(Ø; — Ø;) and (0Ø; + Ø;); we +Tq sin(Ø; — 8 +đồ __ múi 6) (33.56) +To sin(Ø; + 6,) +Since #2 and Fo are in the same material, the intensities are proportional to +the squares of the electric fields, and we get the same result as before. 5imilarly, +Eq. (33.53) is the same as Eq. (33.4). +For waves which arrive at normal ineidence, Ø; = 0 and 6Ø; = 0. Equa- +tion (33.56) gives 0/0, which is not very useful. We can, however, go back to +Eq. (33.55), which gives +I DẠC — 2 +"“—=[ =0) -[ 1—”2), (33.57) +1, To ?Ị Ƒ- Tủa +'This result, naturally, applies for “either” polarization, since for normal incidence +there is no special “plane of incidence.” +33-5 Reflection from metals +W©e can now use our results to understand the interesting phenomenon of +refection from metals. Why ¡is it that metals are shiny? WSe saw in the last +chapter that metals have an index of refraction which, for some frequencies, has +a large imaginary part. Let”s see what we would get for the reflected intensity +when light shines from air (with œ = 1) onto a material with ø = —nr. Then +Eq. (33.55) gives (for normal incidence) +Tộ _ l+nr : +FWErr ¬àð ⁄z +GREEN ⁄ +For the mtensift of the reflected wave, we want the square of the absolute values —` 2 "2 +of Ej and Eạụ: ` ⁄ : Z“ +Tạ IE2|? lI+znr|? (33 58) )Z“” +—-.--.—-. — . & Z +1, |Eal |L—¡ni? & 2| GLASS PLATE +Or ⁄ w Z l +1. 1+n} Z I ì +TL 1n. 1. (33.59) DRIED RED INK +For a material with an index which is a pure imaginary number, there ¡is 100 per- : +cent refection! Fig. 33-8. A material which absorbs light +Metals do not refect 100 percent, but many do reflect visible light very well. strongly at the írequency œ also reflects +. . x. . light of that frequency. +In other words, the imaginary part of their indexes is very large. But we have +seen that a large Imaginary part of the index means a strong absorption. So +there is a general rule that if øngy material gets to be a øer good absorber at +any frequency, the waves are strongly relected at the surface and very little gets +inside to be absorbed. You can see this efect with strong dyes. Pure crystals of +the strongest dyes have a “metallic” shine. Probably you have noticed that at the +edge of a bottle of purple ink the dried dye will give a golden metallic reflection, +or that dried red ink will sometimes give a greenish metallic relection. Red ink +absorbs out the greens of fransmiited light, so 1f the ink is very concentrated, 1t +will exhibit a strong surface reƒfiection for the frequencies of green light. +You can easily show this efect by coating a glass plate with red ink and +letting it dry. HÝ you direct a beam of white light at the back of the plate, as +shown in Fig. 33-8, there will be a transmitted beam of red light and a reflected +beam of green light. +--- Trang 435 --- +- Vi TT ÔN on cv ¬ X |E¿| +¬- - A cự x 1/kị ~ Ào x +va : ` ./ " : | +Fig. 33-9. Total internal reflection. +33-6 Total internal reflection +Tf light goes from a material like glass, with a real index ø greater than 1, +toward, say, air, with an index n¿ equal to 1, Snells law says that +sỉn Ø;¿ = nsin Ø;. +'The angle Ø; of the transmitted wave becomes 90° when the incident angle Ø; is +cqual to the “critical angle” đc given by +m sỉn Ø„ = 1. (33.60) +'What happens for Ø; greater than the critical angle? You know that there is total +internal refection. But how does that come about? +Let's go back to Eq. (33.45) which gives the wave number k; for the trans- +mitted wave. We would have +2 —_ 2 +k„^ = n” kỹ. +Now ky = ksin 6; and & = ¿n/c, so : ` " "- ` : x _Ằ- vẽ +/2 „2 2_. 2 h ` F "` Ề l 1 - " +TỶ nsỉn Ø; is greater than one, k2 is negatiue and kƒ" is a pure imaginary, say +¿kr. "= — kế _ +You know by now what that meansl The “transmitted” wave (Eq. 33.34) will t1 NG " +have the form ¬.— ¬ ++, = ERetFrsel@t—~Ru0), CN tin Mi VỆ ¬- +The wave amplitude either grows or drops of exponentially with increasing z. " : ' Š ⁄, - Xây ˆ „ . ị +Clearly, what we want here is the negative sign. Then the ampljtude of the ¬.‹= ¬ +wave to the right of the boundary will go as shown in Eig. 33-9. Notice that kr "¬ ¬- +1s @/6—which is of the order 1/^Ao, the reciprocal of the free-space wavelength of 8 5 ñn* | =1 nạ = nh, +the light. When light is totally relected from the inside oŸ a glass-air surface, +there are felds in the air, but they extend beyond the surface only a distance of Fig. 33-10. lf there is a small gap, internal +the order of the wavelength of the light. reflection Is not “total;” a transmitted wave +W© can now see how to answer the following question: If a light wave in gÌass appears beyond the gap. +arrives at the surface at a large enough angle, it is refect©ed; if another piece +of glass is brought up to the surface (so that the “surface” in efect disappears) +the light is transmitted. Exactly when does this happen? Surely there must be +continuous change from total reflection to no refection! 'Phe answer, oŸ cOurse, +1s that if the air gap is so small that the exponential tail of the wave in the +air has an appreciable strength at the second piece of glass, it will shake the +electrons there and generate a new wave, as shown in EFig. 33-10. Some light +--- Trang 436 --- +TRANSMITTER DETECTOR DETECTOR +IIIESÌI†, II +'©| '®] +TRANSMITTER DETECTOR DETECTOR TRANSMITTER DETECTOR DETECTOR +Fig. 33-11. A demonstration of the penetration of internally reflected waves. +will be transmitted. (Clearly, our solution is incomplete; we should solve all the +cquations again for a thin layer of air between two regions of glass.) +This transmission efect can be observed with ordinary light only if the air +gap is very small (of the order of the wavelength of light, like 105 em), but +1b is easily demonstrated with three-centimeter waves. Then the exponentially +decreasing field extends several centimeters. Á microwave apparatus that shows +the effect is drawn in Eig. 33-11. Waves from a small three-centimeter transmitter +are directed at a 45° prism of paraffin. 'The index of refraction of paraflin for +these frequencies is 1.50, and therefore the critical angle is 41.5°. So the wave is +totally refected from the 45° face and is picked up by detector A, as indicated +in Fig. 33-11(a). IÝ a second paraffin prism is placed in contact with the first, as +shown in part (b) of the fñgure, the wave passes straight through and is picked +up at detector . lÝ a gap of a few centimeters is left bebween the two prisms, +as in part (c), there are both transmitted and refected waves. The electric fñeld +outside the 45° face of the prism in EFig. 33-11(a) can also be shown by bringing +detector Ö to within a few centimeters of the surface. +--- Trang 437 --- +Tĩ:o IWqgjrtoffsrrt @oŸ Wq6Éor' +34-1 Diamagnetism and paramagnetism +In this chapter we are goïing to talk about the magnetic properties of materials. 34-1 Diamagnetism and +'The material which has the most striking magnetic properties is, oŸ course, iron. paramagnetism +Similar magnetic properties are shared also by the elements nickel, cobalt, and—— 34-2 Magnetic moments and angular +at sufficiently low temperatures (below 16°Œ)——by gadolinium, as well as by a momentum +number of peculiar alloys. That kind of magnetism, called ƒerromagnetism, 1s 34-3 The precession ofatomic magnets +sufficiently striking and complicated that we will discuss it in a special chapter. R : +l . 34-4 Diamagnetism +However, all ordinary substances do show some magnetic efects, although very , +small ones—a thousand to a million times less than the efects in ferromagnetic 3á-š Larmor's theorem +materials. Here we are going to describe ordinary magnetism, that is to say, the 34-6 Classical physïcs gives neither +magnetism of substances other than the ferromagnetic ones. diamagnetism nor +This small magnetism is of two kinds. 5ome materials are øffracted toward paramagnetism +magnetic fields; others are repelled. Unlike the electrical efect in matter, which 34-7 Angular momentum ỉn quantum +always causes dielectrics to be attracted, there are two signs to the magnetic efect. mechanics +These two signs can be easily shown with the help of a strong electromagnet 34-8 The magnetic energy of atoms +which has one sharply pointed pole piece and one flat pole piece, as drawn in +Fig. 34-1. The magnetic feld is much stronger near the pointed pole than near +the fat pole. If a small piece of material ¡is fastened to a long string and suspended +between the poles, there will, in general, be a small force on it. This small force +can be seen by the slight displacement oŸ the hanging material when the magnet +is turned on. The few ferromagnetic materials are attracted very strongly toward Reuieu: Section 15-1, “The Íorces on +the pointed pole; all other materials feel only a very weak force. Some are weakly a current loop; energy 0Ÿ a +attracted to the pointed pole; and some are weakly repelled. dipole.” +STRING +~-T->__SMALL PIECE OF MATERIAL +2 Bei== +LINES ÒE B Fig. 34-1. A smaill cylinder of bismuth is +weakly repelled by the sharp pole; a piece of +7 POLES OE A STRONG 2 aluminum ¡s attracted. +ELECTROMAGNET +The effect is most easily seen with a small cylinder of bismuth, which is +repelled from the high-feld region. Substances which are repelled in this way +are called đzørmnagnetic. Bismuth is one of the strongest diamagnetic materials, +but even with it, the effect is still quite weak. Diamagnetism is aÌlways very +weak. lf a small piece of aluminum is suspended bebween the poles, there is +also a weak force, but #ouørd the pointed pole. Substances like aluminum are +called parœmagnetic. (In such an experiment, eddy-current forces arise when the +magnet is turned on and of, and these can give off strong impulses. You must be +careful to look for the net displacement after the hanging object settles down.) +--- Trang 438 --- +W© want now to describe briely the mechanisms of these two efects. First, in +many substances the atoms have no permanent magnetic moments, or rather, all +the magnets within each atom balance out so that the ne£ moment of the atom is +zoro. The electron spins and orbital motions all exactly balance out, so that any +particular atom has no average magnetic moment. In these circumstances, when +you turn on a magnetic feld little extra currents are generated inside the atom by +induction. According to Lenz”s law, these currents are in such a direction as to +oppose the increasing field. So the induced magnetie moments of the atoms are +directed oppos#e to the magnetic field. 'This is the mechanism of diamagnetism. +Then there are some substances for which the atoms do have a permanent +magnetie moment——in which the electron spins and orbits have a net circulating +current that is not zero. So besides the diamagnetic efect (which is always +present), there is also the possibility of lining up the individual atomic magnetic +mmoments. In this case, the moments try to line up i#h the magnetic fñeld (in +the way the permanent dipoles of a dielectric are lined up by the electric feld), +and the induced magnetism tends to enhance the magnetic fñeld. These are the +paramagnetic substances. Paramagnetism is generally fairly weak because the +lining-up forces are relatively small compared with the forces from the thermal +motions which try to derange the order. lt also follows that paramagnetism is +usually sensitive to the temperature. (The paramagnetism arising from the spins +of the electrons responsible for conduction in a metal constitutes an exception. +W©e will not be discussing this phenomenon here.) Eor ordinary paramagnetism, +the lower the temperature, the stronger the efect. There is more lining-up at low +temperatures when the deranging efects of the collisions are less. Diamagnetism, +on the other hand, is more or less independent of the temperature. Ïn any +substance with built-in magnetic moments there is a diamagnetic as well as a +paramagnetic efect, but the paramagnetic efect usually dominates. +In Chapter I1 we described a ƒerroelectric material, in which all the electric +dipoles get lined up by their own mutual electric ñelds. It is also possible to +imagine the magnetic analog of ferroelectricity, in which all the atomic moments +would line up and lock together. If you make calculations of how this should +happen, you will ñnd that because the magnetic forces are so much smaller than +the electric forces, thermal motions should knock out this alignment even at +temperatures as low as a few tenths of a degree Kelvin. 5o it would be impossible +a% room temperature to have any permanent lining up of the magnets. +Ôn the other hand, this is exactly what does happen ín iron——it does get lined +up. There is an efective force between the magnetic moments of the diferent +atoms oŸ iron which is mụuch, much greater than the đirect rmagnefic interaction. +lt is an indirect efect which can be explained only by quantum mechanics. Ït is +about ten thousand times stronger than the direct magnetie interaction, and is +what lines up the moments in ferromagnetic materials. We discuss this special +interaction in a later chapter. +Now that we have tried to give you a qualitative explanation of diamagnetism +and paramagnetism, we must correct ourselves and say that # ¡s no‡ possible to +understand the magnetic efects of materials in any honest way from the point +of view of classical physics. Such magnetic efects are a comjpletcl quantum- +mechanical phenomenon. Tt is, however, possible to make some phoney classical +arguments and to get some idea of what is going on. We might put it this way. +You can make some classical arguments and get guesses as to the behavior of the +material, but these arguments are not “legal” in any sense because i% is absolutely +essential that quantum mechanics be involved in every one of these magnetic +phenomena. Ôn the other hand, there are situations, such as in a plasma or a +region of space with many free electrons, where the electrons do obey the laws +Of classical mechanics. And in those circumstances, some of the theorems from +classical magnetism are worth while. Also, the classical arguments are oŸ some +value for historical reasons. The first few times that people were able to guess at +the meaning and behavior of magnetic materials, they used classical arguments. +Finally, as we have already illustrated, classical mechanics can give us some useful +guesses as to what might happen——=even though the really honest way to study +--- Trang 439 --- +this subject would be to learn quantum mechanics fñrst and then to understand +the magnetism in terms of quantum mechanics. +Ôn the other hand, we don't want to wait until we learn quantum mechanies +inside out to understand a simple thing like diamagnetism. We will have to +lean on the classical mechanics as kind of half showing what happens, realizing, +however, that the arguments are really not correct. We therefore make a series +of theorems about classical magnetism that will confuse you because they will +prove diferent things. Except for the last theorem, every one of them will be +wrong. Purthermore, they will all be wrong as a description of the physical world, +because quantum mechanics is left out. +34-2 Magnetic moments and angular momentum +The fñrst theorem we want to prove from classical mechanics is the following: +T an electron is moving in a circular orbit (for example, revolving around a J +nucleus under the inuence of a central force), there is a definite ratio between +the magnetic moment and the angular momentum. Let's call .ƑJ the angular u +momentum and /ø¿ the magnetic moment of the electron in the orbit. “The +magnitude of the angular momentum is the mass oŸ the electron tỉimes the +velocity times the radius. (See Eig. 34-2.) It is directed perpendicular to the +plane of the orbit. +J = mớr. (34.1) t +(This is, of course, a nonrelativistic formula, but 1ÿ 1s a good approximatlon for Eig. 34-2. For any circular orbit the mag- +atoms, because for the electrons involved 0/e is generally of the order of e^/he : : . +netic moment is q/2m times the angular +1/187, or about 1 percent.) momentum J. +The magnetic moment of the same orbit is the current times the area. (See +Section 14-5.) The current is the charge per unit time which passes any point on +the orbit, namely, the charge g times the frequency of rotation. The frequency is +the velocity divided by the cireumference of the orbit; so +1=4 2mr- +The area is rr?, so the magnetic moment is +". (34.2) +lt is also directed perpendicular to the plane of the orbit. 5o .J and are in the +same direction: +u= 5. 7 (orbit). (34.3) +Theïr ratio depends neither on the velocity nor on the radius. Eor any particle +moving in a circular orbit the magnetic moment is equal to g/2m times the +angular momentum. Eor an electron, the charge is negative—we can call it —qe; +So for an electron +u= =¬ J (electron orbit). (34.4) +'That's what we would expect classically and, miraculously enough, it is also +true quantum-mechanically. It's one of those things. However, if you keep going +with the classical physics, you find other places where it gives the wrong answers, +and it is a great game to try to remember which things are right and which things +are wrong. We might as well give you immediately what is true #n general in +quantum mechanics. First, Ðq. (34.4) is true for orb#tal rmmotion, bụt that”s not +the only magnetism that exists. The electron also has a spin rotation about its +own axis (something like the earth rotating on its axis), and as a result of that +spin it has both an angular momentum and a magnetic moment. But for reasons +that are purely quantum-mechanical—there is no classical explanation——the ratio +OŸ to .Ƒ for the electron spin is twice as large as it is for orbital motion of the +spinning electron: +u= _ Ở (electron spin). (34.5) +--- Trang 440 --- +In any atom there are, generally speaking, several electrons and some combi- +nation oŸ spin and orbit rotations which builds up a total angular momentum and +a total magnetic moment. Although there is no classical reason why it should be +SO, ÌE is a0ø/s frue in quantum mechanics that (for an isolated atom) the direc- +tion of the magnetic moment is exactly opposite to the direction of the angular +momentum. 'The ratio of the two is not necessarily either —qe/1m or —qe/2m, +but somewhere in between, because there is a mixture of the contributions from +the orbits and the spins. We can write +— ( dc ) +=-—g| >- |3. (34.6) +where ø is a factor which is characteristic of the state of the atom. Iỳ would be 1 +for a pure orbital moment, or 2 for a pure spin moment, or some other number +in bebween for a complicated system like an atom. 'This formula does not, of +course, tell us very much. lt says that the magnetic moment is øarailel to the +angular momentum, but can have any magnitude. “The form of Eq. (34.6) is +convenient, however, because g—called the “Landé g-factor”—is a dimensionless +constant whose magnitude is of the order of one. It is one of the jobs of quanbun +mechanies to predict the g-factor for any particular atomiec state. +You might also be interested in what happens in nuclei. In nuclei there are +protons and neutrons which may move around in some kind oŸ orbit and at the +same tỉme, like an electron, have an intrinsic spin. Again the magnetic moment +is parallel to the angular momentum. Ônly now the order of magnitude of the +ratlo of the bwo is what you would expect for a pro‡on goïing around ïn a cirele, +with zm in Eq. (34.3) equal to the profon rmass. Therefore it is usual to write for +nuclei +u= s(s2) J, (34.7) +where ?n, is the mass of the proton, and ø——called the nucleør g-factor—is a +number near one, to be determined for each nucleus. +Another important diference for a nucleus is that the sp7n magnetic moment +of the proton does øø have a g-factor of 2, as the electron does. For a proton, +g= 2-(2.79). Surprisingly enough, the nøewfron also has a spin magnetic moment, +and i0s magnetic moment relative to its angular momentum is 2 - (—1.91). The +neutron, in other words, is not exactly “neutral” in the magnetic sense. It is like +a little magnet, and it has the kind oŸ magnetic moment that a rotating megafiue +charge would have. +34-3 The precession of atomic magnets +One of the consequences of having the magnetic moment proportional to +the angular momentum is that an atomic magnet placed in a magnetic fñield +will precess. First we will argue classically. Suppose that we have the magnetic +mmoment # suspended freely in a uniform magnetic fñeld. It will feel a torque 7, +cequal to ø x Ö, which tries to bring it in line with the fñield direction. But the +atomic magnet is a gyroscope—it has the angular momentum Ƒ. 'Pherefore the +torque due to the magnetic field wïll not cause the magnet to line up. Instead, the +magnet will precess, as we saw when we analyzed a gyroscope in Chapter 20 of +Volume ÏI. The angular momentuun—=and with it the magnetic morment——precesses +about an axis parallel to the magnetic ñeld. We can fñnd the rate of precession +by the same method we used in Chapter 20 of the first volume. +Suppose that in a small time A£ the angular momentum changes from .Ƒ +to J, as drawn in Eig. 34-3, staying always at the same angle Ø with respect +to the direction of the magnetic field Ö. Lets call «„ the angular velocity of +the precession, so that in the time A£ the angle øƒ precession 1s œ At. Erom +the geometry of the fgure, we see that the change of angular momentum in the +time Af is +A/ = (7sin 0)(œy Af). +--- Trang 441 --- +So the rate of change of the angular momentum is +¬ œpởỞ sỉn Ø, (34.8) +which must be equal to the torque: += #uBsin0. 34.9 +r=uBsin (34.9) Hy, +The angular velocity of precession is then : +öœ =EB, (34.10) | +J %€Ì5 +Substituting u/J from Eq. (34.6), we see that for an atomie system | 6 +=Ø0——) 34.11 +tp g 2m, ) ( ) +the precession Írequency is proportional to . It is handy to remember that for Fig. 34-3. An object with angular momen- +an atom (or electron) tum J and a parallel magnetic moment +Đ placed in a magnetic field B precesses with +f›ạ= > = (1.4 megacycles/gauss)g, (34.12) the angular velocity ¿;. +and that for a nucleus +tạ= 2 = (0.76 kilocycles/gauss) g. (34.13) +(The formulas for atoms and nuclei are different only because of the different +conventions for ø for the two cases.) +According to the class¿cal theory, then, the electron orbits—and spins—in an +atom should precess in a magnetic fñeld. Is it also true quantum-mechanically? It +1s essentially true, but the meaning of the “precession” is diferent. In quantum +mmechanics one cannot talk about the đecon of the angular momentum in the +same sense as one does classically; nevertheless, there is a very close analogy——so +close that we continue to call it “precession.” We will discuss it later when we +talk about the quantum-mechanical point of view. +34-4 Diamagnetism +Next we want to look at điamagnetism from the classical point of view. lt +can be worked out in several ways, but one of the nice ways is the following. B +Suppose that we slowly turn on a magnetic ñeld in the vicinity of an atom. Äs L +the magnetic fñeld changes an elecirc field is generated by magnetic induction. ⁄Z ⁄⁄ ⁄ > +trom Earadayˆs law, the line integral of E around any closed path is the rate of ⁄2 Z ⁄⁄ Path F +change of the magnetic Ñux through the path. Suppose we pick a path I' which is ⁄ ⁄⁄ +a circle oŸ radius r concentric with the center of the atom, as shown in Eig. 34-4. ⁄⁄⁄⁄% ⁄ +The average tangential electric ñeld #⁄ around this path is given by 4 =< +H2mr = — (Bmr2), +Fig. 34-4. The induced electric forces on +and there is a circulating electric feld whose strength is the electrons in an atom. +t=-—-;—. +The induced electric field acting on an electron in the atom produces a +torque equal to —qe#Zr, which must equal the rate of change of the angular +momentum đJ/di: +dJ — qạr? dB +— = —_—- 34.14 +dt 2_ di ' ) +--- Trang 442 --- +Integrating with respect to time rom zero field, we ñnd that the change in angular +mmomentum due to turning on the fñeld is +AJ= “a— b. (34.15) +Thịs is the extra angular momentum from the twist given to the electrons as the +ñeld is turned on. +This added angular momentum makes an extra magnetic moment which, +because it is an orbifal motion, is just —qe/2n tìmes the angular momentum. +The induced diamagnetic moment is +An=-=¿°AJ=-*%—B. (34.16) +2m 4m, +The minus sign (as you can see is right by using Lenzs law) means that the +added moment is opposite to the magnetic feld. +We would like to write Eq. (34.16) a little diferently. The r2 which appears +is the radius from an axis through the atom parallel to Ö, so if B is along the +z-direction, it is #2 + 2. IÝ we consider spherically symmetric atoms (or average +over atoms with their natural axes in all directions) the average of #2 + 9Ÿ is 2/3 +of the average of the square of the true radial distance from the center øø#n£ of +the atom. It is therefore usually more convenient to write ðq. (34.16) as +An=— 2° (3,„vB. (34.17) +In any case, we have found an induced atomic moment proportional to the +magnetic ñeld # and opposing it. 'This is diamagnetism of matter. It is this +magnetic efect that is responsible for the small force on a piece of bismuth ín a +nonuniform magnetic ñeld. (You could compute the force by working out the +energy of the induced moments in the fñeld and seeing how the energy changes as +the material is moved into or out oŸ the high-field region.) +We are still left with the problem: What is the mean square radius, (?)av? +Classical mechanics cannot supply an answer. We must go back and start over +with quantum mechanics. In an atom we cannot really say where an electron is, +but only know the probability that it will be at some place. IỶ we interpret (z?)av to +mean the average of the square of the distance from the center for the probability +distribution, the diamagnetic moment given by quantum mechanics is just the +same as formula (34.17). 'This equation, of course, is the moment for one electron. +The total moment is given by the sum over all the electrons in the atom. The +surprising thing is that the classical argument and quantum mechanics give +the same answer, although, as we shall see, the classical argument that gives +Eq. (34.17) is not really valid in classical mechanics. +The same diamagnetic efect occurs even when an atom already has a per- +manent moment. 'Then the system will precess in the magnetic field. As the +whole atom precesses, it takes up an additional small angular velocity, and that +slow turning gives a small current which represents a correction to the magnetiec +moment. 'Phis is just the diamagnetic effect represented in another way. But we +donˆt really have to worry about that when we talk about paramagnetism. Tf +the diamagnetic efect is frst computed, as we have done here, we don't$ have +to worry about the fact that there is an extra little current from the precession. +That has already been included in the diamagnetic term. +34-5 Larmor?s theorem +We can already conclude something from our results so far. Eirst of all, in +the classical theory the moment was always proportional to J, with a given +constant of proportionality for a particular atom. 'Phere wasn”t any spin of the +electrons, and the constant of proportionality was always —qe/2m; that is to +say, in Ðq. (34.6) we should set g = 1. The ratio oŸ ø& to .J was independent +--- Trang 443 --- +of the internal motion of the electrons. Thus, according to the classical theory, +all systems of electrons would precess with £Öe sơme angular velocity. (Thịis is +no‡ true in quantum mechanics.) This result is related to a theorem in classical +mechanics that we would now like to prove. Suppose we have a group of electrons +which are all held together by attraction toward a central point—as the electrons +are attracted by a nucleus. The electrons will also be interacting with each other, +and can, in general, have complicated motions. Suppose you have solved for +the motions with øo magnetic ñeld and then want to know what the motions +would be ui#h a weak magnetic fñeld. 'Phe theorem says that the motion with a +weak magnetic fñeld is always one oŸ the no-field solutions with an added rotation, +about the axis of the field, with the angular velociby œ„ = qeB/2m. (This is the +same as œ„, if g = 1.) There are, of course, many possible motions. The point is +that for every motion without the magnetic field there is a corresponding motion +in the field, which is the original motion plus a uniform rotation. 'Phis ¡is called +Larmor”s theorem, and œr is called the Earmor ƒrequenc0. +We would like to show how the theorem can be proved, but we will let you +work out the details. 'Take, first, one electron in a central force field. “The force on +it is just F'{r), directed toward the center. IÝ we now turn on a uniform magnetic +fñeld, there is an additional force, gu x #Ö; so the total force is +F{r) + qu x Ö. (34.18) +Now let”s look at the same system from a coordinate system rotating with angular +velocity œ about an axis through the center of force and parallel to Ö. This is +no longer an inertial system, so we have to put in the proper pseudo forces—the +centrifugal and Coriolis forces we talked about in Chapter 19 of Volume I. We +found there that in a frame rotating with angular velocity œ, there is an apparent +tangenfial force proportional to ø„, the radial component of velocity: +Tỳ = —2muty. (34.19) +And there is an apparent radial force which is given by +F} = muŸr + 2m0, (34.20) +where œ is the tangential component of the velocity, measured 7n the rotating +frame. (The radial component ơ; for rotating and inertial frames is the same.) +Now for small enough angular velocities (that is, iŸ œ¿r' < 0¿), we can neglect +the first term (centrifugal) in Eq. (34.20) in comparison with the second (Coriolis). +Then Eqs. (34.19) and (34.20) can be written together as +F'= —2mưœ x 0. (34.21) +TÍ we now cormmb?ne a rotation and a magnetic field, we must add the force in +Eq. (34.21) to that in Eq. (34.18). The total force is +Ft(r) + gu x B + 2m0 x œ (34.22) +[we reverse the cross product and the sign of Eq. (34.21) to get the last term]. +Looking at our result, we see that if +2mœ = —q +the two terms on the right cancel, and in the moving frame the only force is F'(r). +The motion of the electron is just the same as with no magnetic field——and, of +course, no rotation. We have proved Larmor”s theorem for one electron. Since +the proof assumes a small œ, it also means that the theorem is true only for weak +magnetic fields. The only thing we could ask you to improve on is 0o take the +case of many electrons mutually interacting with each other, but all in the same +central fñeld, and prove the same theorem. So no matter how complex an atom +1s, If it has a central field the theorem ¡is true. But that”s the end of the classical +mmechanics, because it isn't true in fact that the motions precess in that way. The +precession frequency œ„ oŸ Eq. (34.11) is only equal to œr, iŸ øg happens to be +cqual to 1. +--- Trang 444 --- +34-6 Classical physics gives neither diamagnetism nor paramagnetism +Now we would like to demonstrate that according to classical mechanics there +can be no diamagnetism and no paramagnetism at all. It sounds crazy——frst, +we have proved that there are paramagnetism, diamagnetism, precessing orbits, +and so on, and now we are going to prove that it is all wrong. Yesl—We are +going to prove that j#ƒ you follow the class¿cal mechanics far enough, there are no +such magnetic efects—fhe alÙ cancel ouf. TÝ you start a classical argument in a +certain place and don't go far enough, you can get any answer you want. But the +only legitimate and correct proof shows that there is no magnetic efect whatever. +Tt is a consequence of classical mechaniecs that if you have any kind of system—— +a gas with electrons, protons, and whatever——kept in a box so that the whole thing +can t turn, there will be no magnetic efect. It is possible to have a magnetic efect +1ƒ you have an isolated system, like a star held together by itself, which can start +rotating when you put on the magnetic ñeld. But if you have a piece of material +that is held in place so that it can”t start spinning, then there will be no magnetic +efects. What we mean by holding down the spin is summarized this way: At a +given temperature we suppose that there is onl one s‡øte of thermal equilibrium. +"The theorem then says that if you turn on a magnetic fñeld and wait for the system +to get into thermal equilibrium, there will be no paramagnetism or diamagnetism—— +there will be no induced magnetic moment. Proof: According to statistical +mechanics, the probability that a system will have any given state of motion is +proportional to e—U/*” where U is the energy of that motion. NÑow what is the +energy of motion? For a particle moving in a constant magnetic fñeld, the energy 1s +the ordinary potential energy plus „2/2, with nothing additional for the magnetic +ñeld. [You know that the forces from electromagnetic fields are g( + x ), and +that the rate of work #'- ø is just gE - ø, which is not afected by the magnetic +fñeld.] So the energy of a system, whether it is in a magnetic field or not, is always +given by the kinetic energy plus the potential energy. 5ince the probability of any +motion depends only on the energy——that is, on the velocity and position——it is the +same whether or not there is a magnetic field. Eor #hermal equilibrium, therefore, +the magnetic fñeld has no efect. If we have one system in a box, and then have +another system in a second box, this time with a magnetic fñield, the probability +of any particular velocity at any point in the first box is the same as in the second. +Tf the fñrst box has no average circulating current (which it will not have ïf it is +in equilibrium with the stationary walls), there is no average magnetic moment. +Since in the second box all the motions are the same, there is no average magnetic +moment there either. Hence, if the temperature is kept constant and thermal +equilibrium is re-established after the field is turned on, there can be no magnetic +moment induced by the field—according to classical mechanics. We can only get +a satisfactory understanding of magnetic phenomena from quantum mechanics. +Unfortunately, we cannot assume that you have a thorough understanding +of quantum mechanics, so this is hardly the place to discuss the matter. Ôn the +other hand, we donˆt always have to learn something frst by learning the exact +rules and then by learning how they are applied in diferent cases. Almost every +subject that we have taken up in this course has been treated in a different way. +In the case of electricity, we wrote the Maxwell equations on “Page One” and then +deduced all the consequences. That”s one way. But we will no# now try to begin +a new “Page One,” writing the equations of quantum mechanics and deducing +everything tom them. We will just have to tell you some of the consequences +of quantum mechanics, before you learn where they come from. So here we go. +34-7 Angular momentum in quantum mechanics +We have already given you a relation between the magnetic moment and the +angular momentum. 'That”s pleasant. But what do the magnetic moment and +the angular momentum ?neøn in quantum mechanics? In quantum mechanics it +turns out to be best to defñne things like magnetic moments in terms of the other +concepts such as energy, in order to make sure that one knows what it means. +--- Trang 445 --- +Now, 1È is easy to defne a magnetic moment in terms of energy, because the +energy of a moment in a magnetic field is, in the classical theory, - 1. Therefore, +the following defnition has been taken in quantum mechanics: lf we calculate +the energy of a system in a magnetic field and we fñnd that it is proportional +to the field strength (for small ñeld), the coefficient is called the component of +magnetic moment in the direction of the ñeld. (We donˆt have to get so elegant +for our work now; we can still think of the magnetic moment in the ordinary, to +some extent classical, sense.) +Now we would like to discuss the idea of angular momentum in quantum +mechanies—or rather, the characteristics of what, in quantum mechanies, is +called angular momentum. You see, when you go to new kinds of laws, you +can't just assume that each word is going to mean exactly the same thing. You +may think, say, “Oh, I know what angular momentum is. It's that thing that +1s changed by a torque.” But what”s a torque? In quantum mechanics we have +to have new definitions of old quantities. It would, therefore, be legally best to +call it by some other name such as “quantangular momentum,” or something +like that, because it is the angular momentum as deñned in quantum mechanics. +But ïÝ we can fñnd a quantity in quantum mechanics which is identical to our old +idea. of angular momentum when the system becomes large enough, there is no +use in inventing an extra word. We might as well just call it angular momentum. +With that understanding, this odd thing that we are about to describe 7s angular +momentum. lt is the thing which in a large system we recognize as angular +mmomentum in classical mechanics. +First, we take a system in which angular momentum is conserved, such as an +abom all by itself in empty space. NÑow such a thing (like the earth spinning on +its axis) could, in the ordinary sense, be spinning around any axis one wished to +choose. Ảnd for a given spin, there could be many diferent “states,” all of the +same energy, each “state” corresponding to a particular direction of the axis of the +angular momentum. 5o in the classical theory, with a given angular momentum, +there is an infinite number of possible states, all oŸ the same energy. +lt turns out in quantum mechanics, however, that several strange things +happen. Eirst, the number of states in which such a system can ez¿sf is limited—— +there is only a ñnite number. If the system is small, the ñnite number is very +small, and if the system is large, the fñnite number gets very, very large. Second, +we canwnot describe a “state” by giving the dieclion ofits angular momentum, but +only by giving the componen‡ of the angular momentum along some direction—say +in the z-direction. Classically, an object with a given total angular momentum .Ƒ +could have, for its z-component, any value from + to —ư. But quantum- +mechanically, the z-component oŸ angular momentum can have only certain +discrete values. Any given system——a particular atom, or a nucleus, or anything—— +with a given energy, has a characteristic number 7, and its z-component of angular +mmomentum can only be one of the following set of values: +: (34.23) +-(—2)h +~( — 1)h +The largest z-component is 7 times ñ; the next smaller is one unit of Ö less, and so +on down to —7h. The number 7 is called “the spin of the system.” (Some people call +it the “total angular momentum quantum number”; but well call it the “spin.”) +You may be worried that what we are saying can only be true Íor some +“special” z-axis. But that is not so. Eor a system whose spin is 7, the component +Of angular momentum along øn axis can have only one of the values in (34.23). +--- Trang 446 --- +Although it is quite mysterious, we ask you just to accept it for the moment. We +will come back and discuss the point later. You may at least be pleased to hear +that the z-component goes from some number to minus the sørme number, so +that we at least don't have to decide which is the plus direction of the z-axis. +(Certainly, if we said that it went from -Ƒ7 to minus a diferent amount, that +would be infnitely mysterious, because we wouldn't have been able to defñne the +z-axis, pointing the other way.) +Now 1ƒ the z-component of angular momentum must go down by integers +from +7 to —7, then j must be an integer. Nol NÑot quite; twice j must be an +integer. lt is only the đjerence between +7 and —j7 that must be an integer. +So, in general, the spin 7 is either an integer or a half-integer, depending on +whether 27 is even or odd. 'Take, for instance, a nucleus like lithium, which has a +spin of three-halves, 7 = 3/2. Then the angular momentum around the z-axis, in +units of ñ, is one of the following: +'There are four possible states, each of the same energy, ¡f the nucleus is in empty +space with no external fields. If we have a system whose spin is two, then the +z-component of angular momentum has only the values, in units of ñ, +TÍ you count how many states there are for a given 7, there are (27-1) possibilities. +In other words, if you tell me the energy and also the spin 7, it turns out that +there are exactly (27 + 1) states with that energy, each siate corresponding to +one of the diferent possible values of the z-component of the angular momentum. +We would like to add one other fact. If you pick out any atom of known j +at random and measure the z-component of the angular momentum, then you +may get any one of the possible values, and each of the values is eguali likely. +AII of the states are in fact single states, and each is just as good as any other. +Each one has the same “weight” in the world. (We are assuming that nothing +has been done to sort out a special sample.) This fact has, incidentally, a simple +classical analog. IÝ you ask the same question classically: What ¡is the likelihood +of a particular z-eomponent of angular momentum If you take a random sample +Of systems, all with the same total angular momentum?——the answer is that all +values from the maximum to the minimum are equally likely. (You can easily +work that out.) The classical result corresponds to the equal probability of the +(27 + 1) possibilities in quantum mechanics. +trom what we have so far, we can get another interesting and somewhat +surprising conclusion. In certain classical caleculations the quantity that appears +in the final result is the sguare of the magnitude of the angular momentum .j—in +other words, .Ƒ -.Ƒ. It turns out that it is often possible to guess at the correct +quantum-mechanical formula by using the classical calculation and the following +simple rule: Replace J2 = .J -.J by 7(7 + 1)ðZ. Thịs rule is commonly used, +and usually gives the correct result, but nø£ always. We can give the following +argument to show why you might expect this rule to work. +'The scalar product .Ƒ - J can be written as +J-J=J2+ J2 +}. +Sinee it is a scalar, it should be the same for any orientation of the spin. Suppose +we pick samples of any given atomiec system at random and make measurements +--- Trang 447 --- +Of J2, or Jộ, or J2, the œuerage 0alue should be the same for each. (There is no +special distinction for any one of the directions.) Therefore, the average of .Ÿ - JƑ +is just equal to three tỉmes the average of any component squared, say of J2; +(J - J)av = 3(22)av. +But since .Ƒ - .Ƒ is the same for all orientations, its average is, of course, just its +constant value; we have +J-J—=3(J72)v. (34.24) +T we now say that we will use the same equation for quantum mechanics, +we can easily fnd (72)„v. We just have to take the sum of the (27 + 1) possible +values of J2, and divide by the total number; +x2 7—] 2 vu. ". 1 2 _—_2z\2 +27T+1 +Eor a system with a spin of 3/2, ¡it goes like this: +2)2+ (1/2)2+(-1/2)?+(-3/2)2 bì +V3), — G/2)9 + (1/98 + (~1/8)8 + (S8/2),y - 5 „z +W© conclude that +J-J=3(77)¿„ = 35h” = š(š + 1)hẺ. +We will leave it for you to show that Eq. (344.25), together with Eq. (34.24), gÌves +the general result +J-J=7(7 + 1)Ẻ. (34.26) +Although we would think classically that the largest possible value of the z- +component of .Ƒ is Just the magnitude of .Jj——namely, w.Ƒ - .j—quantum mechan- +ically the maximum of 7; is always a little less than that, because 7ñ is always +less than 4⁄7(7 + 1)5. The angular momentum is never “completely along the +z-direction.” +34-8 The magnetic energy of atoms +Now we want to talk again about the magnetic moment. We have said that +in quantum mechanics the magnetic moment of a particular atomie system can +be written in terms of the angular momentum by Eq. (34.6); +M... (34.27) +where —qe and ?nm are the charge and mass of the electron. +An atomic magnet placed in an external magnetic fñeld will have an extra +magnetic energy which depends on the component of its magnetic moment along +the ñeld direction. We know that +Day = —b- Ö. (34.28) +Choosing our z-axis along the direction of Ö, +Duag —= —H¿„Ð. (34.29) +Using Eq. (34.27), we have that +Duag = g| — |J;Đ. +c~*(ẩn) +Quantum mechanics says that J; can have only certain values: 7ñ, (7 — 1)h, +...„ —7hR. Therefore, the magnetic energy of an atomic system is not arbitrary; +1 can have only certain values. lIts maximum value, for instance, is +ức : +=—— |hjB. +--- Trang 448 --- +The quantity qeh/2m is usually given the name “the Bohr magneton” and Uạng +written up: +qch J,=+P +HB—= g—- +The possible values of the magnetic energy are +Dũnag — gup Tễ, 0 F5 +where J;/ñ takes on the possible values 7, (7 — 1), (j—2),..., (—7+ 1), —ÿ. +=~zP +In other words, the energy of an atomic system is changed when it is put in a +magnetic feld by an amount that is proportional to the field, and proportional +to Jy. We say that the energy of an atomic system is “split into 27 + 1 levels” by a J=— 5h +magnetic ñeld. Eor instance, an atom whose energy is Ứo outside a magnetic fñeld +and whose 7 is 3/2, will have four possible energies when placed in a field. We Fig. 34-5. The possible magnetic energies +can show these energies by an energy-level diapram like that drawn in Eig. 34-5. of an atomic system with a spin of 3/2 in a +Any particular atom can have only one of the four possible energies in any given magnetic filed B. +ñeld . That is what quantum mechanics says about the behavior of an atomie +system in a magnetic field. mac } Ta 1n +The simplest “atomie” system is a single electron. The spin of an electron 7" +is 1/2, so there are two possible states: jJy = ñ/2 and J; = —ñ/2. For an electron, +at rest (no orbital motion), the spin magnetic moment has a g-value of 2, so the +magnetic energy can be either +/u;7. The possible energies in a magnetic fñeld 0 +are shown in Fig. 34-6. Speaking loosely we say that the electron either has its 5 +spin “up” (along the field) or “down” (opposite the field). +For systems with higher spins, there are more states. We can think that the , +spin is “up” or “down” or cocked at some “angle” in between, depending on the $z=— 2ñ +value of Jz. +We will use these quantum mechanical results to discuss the magnetic prop- Fig. 34-6. The two possible energy states +erties oŸ materials in the next chapter. of an electron in a magnetic field B. +--- Trang 449 --- +XPqr-drrttergjre©ffsite đrracÏ WetgyreoffC Oseradrite© +35-1 Quantized magnetic states +In the last chapter we described how in quantum mechanics the angular 35-1 Quantized magnetic states +mmomentum of a thing does not have an arbitrary direction, but its component 35-2 The Stern-Gerlach experiment +along a given aXis can take on only certain equally spaced, discrete values. lt 35-3 The Rabi moleeular-beam +sô shocking and peculiar thing. You may think that perhaps we should not g0 method +into such things until your mỉnds 8T mOF€ advanced and ready to accept this 35-4 The paramagnetism of bulk +kind of an idea. Actually, your minds will never become more advanced——in the . +sense of being able to accept such a thing easily. There isn't any descriptive way materials . . +of making it intelligible that isnt so subtle and advanced in its own form that 3-5 Cooling by adiabatic +1È is more complicated than the thing you were trying to explain. The behavior demagnetization +of matter on a small scale—as we have remarked many times—is diferent from 3ã-6 Nuclear magnetic resonance +anything that you are used to and is very strange indeed. Âs we proceed with +classical physics, it is a good idea to try to get a growing acquaintance with the +behavior of things on a small scale, at first as a kind of experience without any +deep understanding. nderstanding of these matters comes very slowly, if at +all. Of course, one does get better able to know what is going 0o happen in a +quantum-mechanical situation—If that is what understanding means——but one +never øgets a comfortable feeling that these quantum-mechanical rules are “natural” Reuicu: Chapter 11, Inside Dielecirics +Of course they are, but they are not natural to our own experience at an ordinary +level. We should explain that the attitude that we are going to take with regard +to this rule about angular momentum ¡is quite diferent from many of the other +things we have talked about. We are not going to try to “explain” it, but we must +at least £ell you what happens; it would be dishonest to describe the magnetic +properties of materials without mentioning the fact that the classical description +of magnetism——of angular momentum and magnetic moments—is incorrect. +One of the most shocking and disturbing features about quantum mechanics +is that if you take the angular momentum along any particular axis you fnd that +1b is always an integer or halinteger times h. 'Phis is so no matter which axis +you take. "The subtleties involved in that curious fact—that you can take any +other axis and fnd that the component for it is also locked to the same set of +values—we will leave to a later chapter, when you will experience the delight of +seeing how this apparent paradox is ultimately resolved. +We will now just accept the fact that for every atomic system there is a +number 7, called the sø#n of the system——which must be an integer or a halfˆ +integer—and that the component of the angular momentum along any particular +axis will always have one of the following values between +7 and —7Ï: +J„ = one of : -ñ, (35.1) +We have also mentioned that every simple atomic system has a magnetic +moment which has the same direction as the angular momentum. 'PThis is true +not only for atoms and nuclei but also for the fundamental particles. Each +fundamental particle has its own characteristic value of 7 and its magnetic +--- Trang 450 --- +jJ=1⁄2 J=1 _xk +Jy= +M2 +Uo B Úo z=9 B +(a) (@) >> +J=3/2 v2 +Rúp Jz—= +M2 +Uo hp B +húp + = 35⁄2 +Fig. 35-1. An atomic system with spin / (c) +has (2/ + 1) possible energy values in a > +magnetic field B. The energy splitting Is 1> +proportional to for small fields. T2 +moment. (For some particles, both are zero.) What we mean by “the magnetic +mmoment” in this statement is that the energy of the system in a magnetic fñeld, +say in the z-direction, can be written as —/u; for small magnetic fñelds. We +must have the condition that the field should not be too great, otherwise i9 could +disturb the internal motions of the system and the energy would not be a measure +of the magnetic moment that was there before the field was turned on. But If +the field is sufficiently weak, the feld changes the energy by the amount +AU = -hxÖ, (35.2) +with the understanding that in this equation we are to replace uy by +Hz =s(s*) đz, (35.3) +where J; has one of the values in Eq. (35.1). +Suppose we take a system with a spin j = 3/2. Without a magnetic feld, +the system has four diferent possible states corresponding to the diferent values +of J„, all of which have exactly the same energy. But the moment we turn on +the magnetic fñeld, there is an additional energy of interaction which separates +these states into four slightly diferent energy levels. 'Phe energies of these levels +are given by a certain energy proportional to Ö, multiplied by ñ times 3/2, 1/2, +—1/2, and —3/2—the values of J;. The splitting of the energy levels for atomic +systems with spins of 1/2, 1, and 3/2 are shown in the diagrams of Eig. 35-1. +(Remember that for any arrangement of electrons the magnetic moment is always +directed opposite to the angular momentum.) +You will notice from the diagrams that the “center of gravity” of the energy +levels is the same with and without a magnetic feld. Also notice that the spacings +from one level to the next are always equal for a given particle in a given magnetic +fñeld. We are going to write the energy spacing, for a given magnetic fñeld Ö, +as ñœ���——which is just a defnition oŸ œ„. Using Eqs. (35.2) and (35.3), we have +hư —= g>— R.B +Sh— tóm +OF : +œơẹ =gz— Ö. (35.4) +>m 35-2 +--- Trang 451 --- +The quantity ø(g/2m) is just the ratio of the magnetic moment to the angular +momentum——it is a property of the particle. Pquation (35.4) is the same formula +that we got in Chapter 34 for the angular velocity of precession in a magnetic +ñeld, for a gyroscope whose angular momentum is .ƑJ and whose magnetic moment +1S Jứ. +_———_ ọ +|=<= | —=——_ +GLASS +PLATE +VACUUM +Fig. 35-2. The experiment of Stern and Gerlach. +35-2 The Stern-Gerlach experiment +The fact that the angular momentum is quantized is such a surprising thing +that we will talk a little bit about it historically. It was a shock om the moment +it was discovered (although it was expected theoretically). It was first observed +in an experiment done in 1922 by Stern and Gerlach. lf you wish, you can +consider the experiment of Stern-Gerlach as a direct justification for a belief in +the quantization of angular momentum. Stern and Gerlach devised an experiment +for measuring the magnetic moment oŸ individual silver atoms. They produced a +beam of silver atoms by evaporating silver in a hot oven and letting some of them +come out through a series of small holes. This beam was directed between the +pole tips of a special magnet, as shown in Fig. 35-2. Theïr idea was the following. +Tƒ the silver atom has a magnetic moment #ø, then in a magnetic field #Ö it has +an energy —/;, where z is the direction of the magnetic fñeld. In the classical +theory, ; would be equal to the magnetic moment times the cosine of the angle +between the moment and the magnetic field, so the extra energy in the field +would be +AU = —hùBcos0. (35.5) +OŸ course, as the atoms come out of the oven, their magnetic moments would +point in every possible direction, so there would be all values of Ø. Now ïf the +magnetic ñeld varies very rapidly with z—If there is a strong fñeld gradient—then +the magnetic energy will also vary with position, and there will be a force on the +magnetic moments whose direction wiïll depend on whether cosine ổ is positive +or negative. 'Phe atoms will be pulled up or down by a force proportional to the +derivative of the magnetic energy; from the principle of virtual work, +ý = TC = e0 SẺ, (35.6) +Stern and Gerlach made their magnet with a very sharp edge on one oŸ the pole +tips in order to produce a very rapid variation of the magnetic ñeld. The beam +OŸ silver atoms was directed right along this sharp edge, so that the atoms would +feel a vertical force in the inhomogeneous feld. A silver atom with its magnetic +mmoment directed horizontally would have no force on it and would go straight past +the magnet. Ân atom whose magnetic moment was exactly vertical would have a +force pulling it up toward the sharp edge of the magnet. An atom whose magnetic +tmmoment was pointed downward would feel a downward push. 'Thus, as they left the +--- Trang 452 --- +magnet, the atoms would be spread out according to their vertical components of +magnetic moment. In the classical theory all angles are possible, so that when the +silver atoms are collected by deposition on a glass plate, one should expect a smear +oÝ silver along a vertical line. 'Phe height of the line would be proportional to the +magnitude of the magnetic moment. The abject failure of classical ideas was com- +pletely revealed when Stern and Gerlach saw what actually happened. 'They found +on the glass plate two distinct spots. The silver atoms had formed two beams. +That a beam of atoms whose spins would apparently be randomly oriented +gets split up into ©wo separate beams is most miraculous. How does the magnetic +moment no that it is only allowed to take on certain components in the direction +of the magnetic field? Well, that was really the beginning of the discovery of the +quantization of angular momentum, and instead of trying to give you a theoretical +explanation, we will just say that you are stuck with the result of this experiment +Just as the physicists of that day had to accept the result when the experiment +was done. It is an ezperimenial ƒact that the energy of an atom in a magnetic +ñeld takes on a series of individual values. For each of these values the energy +1s proportional to the feld strength. So in a region where the field varies, the +prineciple of virtual work tells us that the possible magnetie force on the atoms +will have a set of separate values; the force is different for each state, so the beam +of atoms is split into a small number of separate beams. From a measurement of +the defection of the beams, one can fñnd the strength of the magnetic moment. +35-3 The Rabi molecular-beam method +W©e would now like to describe an improved apparatus for the measurement +of magnetic moments which was developed by I. I. Rabi and his collaborators. +In the 5tern-Gerlach experiment the defection of atoms is very small, and the +measurement of the magnetic moment is not very precise. Rabis technique +permits a fantastic precision in the measurement of the magnetic moments. The +method is based on the fact that the original energy of the atoms in a magnetic +fñeld ¡is split up into a fñnite number of energy levels. That the energy of an atom +in the magnetic fñeld can have only certain discrete energies is really not more +surprising than the fact that atoms ¿n general have only certain discrete energy +levels—something we mentioned often in Volume I. Why should the same thing +no‡ hold for atoms in a magnetic field? It does. But ít is the attempt to correlate +this with the idea of an oriented magnetic tmmnormmen£ that brings out some of the +strange Iimplications of quantum mechanics. +When an atom has two levels which difer in energy by the amount AU, it +can make a transition from the upper level to the lower level by emitting a light +quantum of frequency œ, where +hưu = AU. (35.7) +"The same thing can happen with atoms in a magnetic ñeld. Only then, the energy +diferences are so small that the frequency does not correspond to light, but to mĩ- +crowaves or to radiofrequencies. The transitions from the lower energy level to an +upper energy level of an atom can also take place with the absorption of light or, in +the case of atoms in a magnetic field, by the absorption of microwave energy. Thus +1ƒ we have an atom in a magnetic ñeld, we can cause transitions om one state to +another by applying an additional electromagnetic ñeld of the proper frequenecy. +In other words, if we have an atom in a strong magnetic fñeld and we “tickle” the +atom with a weak varying electromagnetic ñeld, there will be a certain probability +of knocking it to another level if the frequenecy is near to the œ in Eq. (35.7). For +an atom in a magnetic fñeld, this frequency is just what we have earlier called ¿ +and ïE is given in terms of the magnetic fñeld by Bq. (35.4). TỶ the atom is tickled +with the wrong frequenecy, the chance of causing a transition is very smaill. Thus +there is a sharp resonanee at œp in the probability of causing a transition. By +measuring the frequenecy of this resonanece in a known magnetic fñeld , we can +measure the quantity ø(g/2m)——and hence the g-factor—with great precision. +--- Trang 453 --- +Tt is interesting that one comes to the same coneclusion from a classical point B +of view. According to the classical picture, when we place a smalÌ gyroscope +with a magnetic moment / and an angular momentum .Ƒ in an external magnetic +field, the gyroscope will precess about an axis parallel to the magnetic field. +(See Eig. 35-3.) Suppose we ask: How can we change the angle of the classical +øyroscope with respect to the fñeld—namely, with respect to the z-axis? 'The / +magnetic field produces a torque around a hor?zontal axis. Such a torque you | +would think is £rw¿ng to line up the magnet with the feld, but it only causes 6pC 3 u +the precession. IÝ we want to change the angle of the gyroscope with respect to +the z-axis, we must exert a torque on it øbou£‡ the z-azis. lÝ we applÌy a torque +which goes in the same direction as the precession, the angle of the gyroscope : : +will change to give a smaller component of .Ƒ in the z-direction. In Eig. 35-3, the F1g. 35-3. The classical precession of an +. . . . atom with the magnetic moment and the +angle between .ƒ and the z-axis would increase. IÝ we try to hinder the precession, angular momentum /. +Jj moves toward the vertical. +For our precessing atom In a uniform magnetic feld, how can we apply the +kind of torque we want? "The answer is: with a weak magnetic fñeld from the B +side. You might at fñrst think that the direction of this magnetic fñield would have +to rotate with the precession of the magnetic moment, so that it was always at +right angles to the moment, as indicated by the field ' ín Eig. 35-4(a). Such ° +a fñeld works very well, but an øiernating horizontal field is almost as good. Tf ì +we have a small horizontal feld , which is always in the z-direction (plus or J +minus) and which oscillates with the frequeney œ„, then on each one-half cycle ụ +the torque on the magnetic moment reverses, so that it has a cumulative effect +which is almost as effective as a rotating magnetic fñeld. Classically, then, we +would expect the component of the magnetic moment along the z-direction to (a) g...^*‹ +change if we have a very weak oscillating magnetic ñeld at a frequency which is ^ › +exactly œ;. Classically, oŸ course, „; would change continuously, but in quantum B +mnechanics the z-component of the magnetie moment cannot adjust continuousÌy. +lt must jump suddenly from one value to another. We have made the comparison +between the consequences of classical mechanics and quantum mechanics to give +you some clue as to what might happen classically and how ï§ is related to what TẦNG +actually happens in quantum mechanics. You will notice, incidentally, that the J +expected resonant frequency is the same in both cases. +One additional remark: EFrom what we have said about quantum mechanics, , +there is no apparent reason why there couldn't also be transitions at the fre- +quency 2ư„. It happens that there isn't any analog of this in the classical case, ___ +and also it doesnt happen in the quantum theory either—at least not for the 6) B =bcosupt +particular method of inducing the transitions that we have described. With an Eig. 35-4. The angle of precession of an +oscillating horizontal magnetic fñield, the probability that a frequency 24 would atomic magnet can be changed by a hori- +cause a jump of two steps at once is zero. lt is only at the frequenecy œ„ that zontal magnetic field always at right angles +transitions, either upward or downward, are likely to occur. to #, as in (a), or by an oscillating field, as +Now we are ready to describe Rabïs method for measuring magnetic moments. ¡in (b). +We will consider here only the operation for atoms with a spin of 1/2. A diagram +of the apparatus is shown in Fig. 35-5. There is an oven which gives out a stream +of neutral atoms which passes down a line of three magnets. Magnet 1 is jus§ +⁄⁄⁄⁄⁄⁄ 2 ` +4= +ñ/2 Ị >⁄ l4 ` I 8B; +. ỒZ Ôz +2⁄42 4 ` m=¬ c b | DETECTOR +OVEN ————— 7 _— —_— — — 7 ————— +b ? Ì _~~z-_- a +: MAGNET 1 MAGNÉT `Neàn 3 SUT $ +⁄⁄⁄⁄2?⁄ ⁄⁄2⁄2 SNNN +SLIT $¡ +Fig. 35-5. The Rabi molecular-beam apparatus. +--- Trang 454 --- +like the one in Eig. 35-2, and has a feld with a strong field gradient—say, with +9B,„/Ôz positive. TÝ the atoms have a magnetie moment, they will be deflected +downward if J¿ = +ñ/2, or upward if J¿ = —ñ/2 (since for electrons # is directed +opposite to .J). IÝ we consider only those atoms which can get through the slit 51, +there are tEwo possible trajectories, as shown. Atoms with J; = +Ï/2 must go +along curve ø to get through the slit, and those with J; = —h/2 must go along +curve Ù. Atoms which start out from the oven along other paths will not get +through the slit. +Magnet 2 has a uniform field. There are no forces on the atoms in this region, +so they go straight through and enter magnet 3. Magnet 3 is just like magnet 1 +but with the field #muerted, so that ØB,„/Øz has the opposite sign. The atoms +with 7; = +ñ/2 (we say “with spin up”), that felt a downward push in magnet 1, +get an uørd push in magnet 3; they continue on the path ø and go through +slit 52 to a debector. The atoms with J¿ = —ñ/2 (“with spin down”) also have +opposite forces in magnets 1 and 3 and go along the path b, which also takes +them through slit 52 to the detector. +The detector may be made in various ways, depending on the atom being +measured. Eor example, for atoms of an alkali metal like sodium, the detector +can be a thin, hot tungsten wire connect©ed to a sensitive current meter. When +sodium atoms land on the wire, they are evaporated of as NaT ions, leaving an +electron behind. 'There is a current from the wire proportional to the number of +sodium atoms arriving per second. +In the gap of magnet 2 there is a set of coils that produces a small horizontal +magnetic ñeld Bí. The coils are driven with a current which oscillates at a variable +Írequency œ. 5o bebween the poles of magnet 2 there is a strong, constant, vertical +ñeld Bọ and a weak, oscillating, horizontal field Bĩ. +Suppose now that the frequency œ of the oscillating feld is set at œ„—the CURRENE +“precession” frequency of the atoms in the field . 'The alternating fñeld will +cause some of the atoms passing by to make transitions from one J;y to the +other. An atom whose spin was initially “up” (2; = +ñ/2) may be flipped “down” Ị +(7; = —h/2). Now this atom has the direction oŸ its magnetic moment reversed, +so 1E will feel a dounard force in magnet 3 and will move along the path 4a, V +shown in Fig. 35-5. It will no longer get through the slit ŠS+ to the detector. +Similarly, some of the atoms whose spins were initially down (J; = —ñ/2) will | +have theïir spins flipped up (2; = +ñ/2) as they pass through magnet 2. They +will then go along the path Ù and will not get to the detector. Ị +Tf the oseillating fñeld #/ has a frequeney appreciably diferent from œp, ÌÊ t——————#—z>——x~> +will not cause any spin fips, and the atoms will follow their undisturbed paths +to the debector. So you can see that the “precession” frequency œ„ of the atoms Fig. 35-6. The current of atoms in the +in the field Bọ can be found by varying the frequeney œ of the fñeld untila — P€am decreases when w = œp. +decrease is observed in the current of atoms arriving at the detector. ÄÁ decrease +in the current will occur when œ is “in resonance” with œ„. A plot of the detector +current as a function of œ might look like the one shown in Fig. 35-6. Knowing +œ„, we can obtain the ø-value of the atom. +Such atomic-beam or, as they are usually called, “molecular” beam resonance +experiments are a beautiful and delicate way of measuring the magnetic properties +of atomic objects. 'The resonance frequency œ„ can be determined with great +precision——in fact, with a greater precision than we can measure the magnetic +ñeld Bọ, which we must know to ñnd g. +35-4 The paramagnetism of bulk materials +W©e would like now to describe the phenomenon of the paramagnetism of bulk +materials. Suppose we have a substance whose atoms have permanent magnetic +mmoments, for example a crystal like copper sulfate. In the crystal there are copper +ions whose inner electron shells have a net angular momentum and a net magnetic +moment. 5o the copper ion is an object which has a permanent magnetic moment. +Let”s say just a word about which atoms have magnetic moments and which ones +don?t. Any atom, like sodium for instance, which has an odđ number of electrons, +--- Trang 455 --- +will have a magnetic moment. Sodium has one electron in its unflled shell. This +electron gives the atom a spin and a magnetic moment. Ordinarily, however, +when compounds are formed the extra electrons in the outside shell are coupled +together with other electrons whose spin directions are exactly opposite, so that +all the angular momenta and magnetic moments of the valence electrons usually +cancel out. That”s why, in general, molecules do not have a magnetic moment. +Of course if you have a gas of sodium atoms, there is no such cancellation.* Also, +1f you have what is called in chemistry a “free radical”—an object with an odd +number of valence electrons—then the bonds are not completely satisied, and +there is a net angular momentum. +In most bulk materials there is a net magnetic moment only if there are atoms +present whose 7nnwer electron shell is not filled. Then there can be a net angular +mmomentum and a magnetic moment. Such atoms are found in the “transition +element” part of the periodic table—for instance, chromium, manganese, iron, +nickel, cobalt, palladium, and platinum are elements of this kind. Also, all of the +rare earth elements have unfilled inner shells and permanent magnetic moments. +There are a couple of other strange things that also happen to have magnetic +mmoments, such as liquid oxygen, but we will leave it to the chemistry department +to explain the reason. +Now suppose that we have a box full of atoms or molecules with permanent +mmoments—say a gas, or a liquid, or a crystal. We would like to know what +happens IÝ we apply an external magnetic field. With øoø magnetic feld, the +atoms are kicked around by the thermal motions, and the moments wind up +pointing ¡in all directions. But when there is a magnetic field, it acts to line up +the little magnets; then there are more moments lying toward the fñeld than away +from it. The material is “magnetized.” +We defne the rmagnetizatiion IM of a material as the net magnetic moment +per unit volume, by which we mean the vector sum of all the atomic magnetic +moments in a unit volume. lf there are W atoms per unit volume and their +đuerage moment is (6)av then jM can be written as times the average atomic +mmoment: +M = N(h)av. (85.8) +The defnition of MỸ corresponds to the defnition oŸ the electric polarization +of Chapter 10. +The classical theory of paramagnetism is just like the theory of the dielectric +constant we showed you in Chapter 11. One assumes that each of the atoms has a +magnetic moment , which always has the same magnitude but which can point +in any direction. In a ñeld #Ö, the magnetic energy is —/- = —uB cosØ, where +6 is the angle between the moment and the fñeld. EHrom statistical mechanics, the +relative probability of having any angle is e—°"e'8Y/'T so angles near zero are +more likely than angles near z. Proceeding exactly as we did in Section 11-3, +we fnd that for small magnetic fñelds Mƒ is directed parallel to Ö and has the +magnitude : +ẢM = BỊ (35.9) +[5ee Eq. (11.20).] This approximate formula is correct only for „Ð/k7' much less +than one. +W© fnd that the induced magnetization—the magnetic moment per unit +volume——is proportional to the magnetic fñeld. This is the phenomenon of +paramagnetism. You will see that the efect is stronger at lower temperatures and +weaker at higher temperatures. When we put a field on a substance, it develops, +for small fñields, a magnetic moment proportional to the fñeld. 'Phe ratio of ă +to (for smaill fñelds) is called the magnetic suscept¿bilitg. +Now we want to look a% paramagnetism from the point of view of quantum +mechanics. We take first the case of an atom with a spin of 1/2. In the absence +öŸ a magnetic fñeld the atoms have a certain energy, but in a magnetic field there +* Ordinary Na vapor is mostly monatomic, although there are also some molecules of Naa. +--- Trang 456 --- +are two possible energies, one for each value of J;. For J; = +Ï/2, the energy is +changed by the magnetic feld by the amount +AU =+g[#“\.-.P. (35.10) +(The energy shiít AU is positive for an atom because the electron charge is +negative.) Eor /JJ; = —ñ/2, the energy is changed by the amount +AUa=-g[“\...P. (35.11) +To save writing, let”s set +=Ø| 2— ]'za; 35.12 +"M1 (35.12 +AU = +ụhịạB. (35.13) +'The meaning of to is clear: —/uo is the z-component of the magnetic moment in +the up-spin case, and -+ọ 1s the z-component of the magnetic moment in the +down-spin case. +Now statistical mechanics tells us that the probability that an atom is in one +state or another is proportional to +eT (Pnergy of state)/kT- +With no magnetic feld the two states have the same energy; so when there is +equilibrium in a magnetic field, the probabilities are proportional to +c-AU/T, (35.14) +'The number of atoms per unit volume with spin up 1s +Nụp = ae Ho 8/t. (35.15) +and the number with spin down 1s +Naoyn = ae†toB/RT, (35.16) +The constant ø is to be determined so that +Áp + Naown — N, (35.17) +the total number of atoms per unit volume. So we get that +ah. ..nnr mẽ (35.18) +'What we are interested in is the aueraøe magnetic moment along the z-axis. +The atoms with spin up will contribute a moment of —/o, and those with spin +down will have a moment of +uo; so the average moment is +Nụ -~ + w own + +(U)av = NhpCHo) ‡ NaovnCEHo), (35.19) +The magnetic moment per unit volume Ä⁄ is then V()a¿v. Using Eqs. (35.15), +(35.16), and (35.17), we get that +c+toB/KT — c—=HoB/KT +Thịs is the quantum-mechanical formula for ă for atoms with 7 = 1/2. Inciden- +tally, this formula can also be written somewhat more concisely in terms of the +hyperbolic tangent function: +M = Nhẹ tanh ———. 35.21 +Họ tan kT ( ) +--- Trang 457 --- +A plot of MỸ as a function of B is given in Fig. 35-7. When Ö gets very large, +the hyperbolic tangent approaches 1, and Mƒ approaches the limiting value No. +So at hiph fields, the magnetization sœ£urates. We can see why that is; at high +enough fields the moments are all lined up in the same direction. In other words, +they are all in the spin-down state, and each atom contributes the moment /o. +In most normal cases—say, for typical moments, room temperatures, and the M +fñelds one can normally get (like 10,000 gauss)—the ratio oÐ/K7 is about 0.002. N , +One must go to very low temperatures 0o see the saturation. For normal temper- ” ¬aï%ẶẶằẶằ..ư+n +atures, we can usually replace tanh z by ø, and write / +NuậB 7 += ——_. 35.22 +KT (5.22) ị +Just as we saw in the classical theory, ÁM is proportional to . In fact, the +formula is almost exactly the same, except that there seems to be a factor of 1/3 ò 1 3 3 1 +missing. But we still need to relate the o in our quantum formula to the that uoB/kT +appears in the classical result, Eq. (35.9). +In the classical formula, what appears is ” = - , the square of the vector Fig. 35-7. The variation of the paramag- +1nagnetic morment, Or netic magnetization with the magnetic field +q 2 strength B. +":) J- J. (35.23) +W© pointed out in the last chapter that you can very likely get the right answer +from a classical caleulation by replacing .J -.Ƒ by 7( + 1)ñ2. In our particular +example, we have j = 1/2, so +7(7j+ 1)h? = 3hể, +Substituting this for j - JƑ in Eq. (35.23), we get +— (_ q\ 3]? +or in terms of uọ, defned in Eq. (35.12), we get +U-= 3u. +Substituting this for 2 in the classical formula, Eq. (35.9), does indeed reproduce +the correct quantum formula, Eq. (35.22). +The quantum theory of paramagnetism is easily extended to atoms of any +spin 7. The low-feld magnetization is +70 +1) u$B +ME=Ng?———^~P—, 35.24 +g TT (35.24) +up = TU (35.25) +1s a combination of constants with the dimensions of a magnetic moment. Most +atoms have momentfs of roughly this size. It is called the Pohr rmmagneton. The +Spin magnetic moment of the electron 1s almost exactly one Bohr magneton. +35-5 Cooling by adiabatic demagnetization +There is a very interesting special application of paramagnetism. At very low +temperatures it is possible to line up the atomic magnets in a strong field. lt +is then possible to get down to eztremelu low temperatures by a process called +adiabatic demagnetization. We can take a paramagnetic salt (for example, one +containing a number oŸ rare-earth atoms like praseodymium-ammonium-nitrate), +and start by cooling it down with liquid helium to one or ÿwo degrees absolute In +a strong magnetic field. Then the factor Ð/kT' is larger than l—say more like +2 or 3. Most of the spins are lined up, and the magnetization is nearly saturated. +--- Trang 458 --- +Let's say, to make ï§ easy, that the field is very powerful and the temperature +is very low, so that nearly all the atoms are lined up. 'Phen you isolate the salt +thermally (say, by removing the liquid helium and leaving a good vacuum) and +turn of the magnetic ñeld. 'The temperature of the salt goes way down. +Now iƒ you were to turn of the fñeld sưuddemiu, the jiggling and shaking, +of the atoms in the crystal lattice would gradually knock all the spins out of +alignment. Some of them would be up and some down. But ïf there is no field +(and disregarding the interactions between the atomic magnets, which will make +only a slight error), it takes no energy to turn over the atomic magnets. They +could randomize their spins without any energy change and, therefore, without +any temperature change. +Suppose, however, that while the atomic magnets are being fipped over by +the thermal motion there is still some magnetic field present. Then it requires +some work to fÑip them over opposite to the fñield——fhe must do t0uork against the +ƒield. 'Phis takes energy from the thermal motions and lowers the temperature. +So I1f the strong magnetic field is not removed too rapidly, the temperature of +the salt wïll decrease—It is cooled by the demagnetization. FTom the quantum- +mmechanical view, when the fñeld is strong all the atoms are in the lowest state, +because the odds against any beïng in the upper state are impossibly big. But as +the fñeld is lowered, i% gets more and more likely that thermal Ñuctuations will +knock an atom into the upper state. When that happens, the atom absorbs the +energy AU = nọ. So ïf the field is turned of slowly, the magnetic transitions +can take energy out of the thermail vibrations of the crystal, cooling it of. It is +possible in this way to go from a temperature of a few degrees absolute down to +a temperature of a few thousandths of a degree. +'Would you like to make something even colder than that? It turns out that +Nature has provided a way. We have already mentioned that there are also +magnetie moments for the atomic nuclei. Our formulas for paramagnetism work +Just as well for nuclel, except that the moments of nuclel are roughly a thousœnd +times smailler. [They are of the order of magnitude of gh/2mp, where my is +the proton mass, so they are smaller by the ratio of the masses of the electron +and probon.| With such magnetic moments, even at a temperature oŸ 2°K, the +factor B/KT is only a few parts in a thousand. But iŸ we use the paramagnetic +demagnetization process to get down to a temperature of a few thousandths of +a degree, //k7' becomes a number near lI—at these low bemperatures we can +begin to saturate the nuclear moments. That is good luck, because we can then +use the adiabatic demagnetization of the ø%ecleør magnetism to reach still lower +temperatures. Thus it is possible to do two stages of magnetic cooling. Pirst we +use adiabatic demagnetization of paramagnetie ions to reach a few thousandths +of a degree. Then we use the cold paramagnetie salt to cool some material which +has a strong nuclear magnetism. Finally, when we remove the magnetic field +from this material, its temperature will go down to within a rmllionth of a degree +of absolute zero—if we have done everything very carefully. +35-6 Nuclear magnetic resonance +W©e have said that atomic paramagnetism is very small and that nuclear +mmagnetism is even a thousand times smaller. Yet it is relatively easy to observe +the nuclear magnetism by the phenomenon of “nuclear magnetic resonanee.” +Suppose we take a substance like water, in which all of the electron spins are +exactly balanced so that their net magnetic moment is zero. The molecules +will still have a very, very tỉny magnetic moment due to the nuclear magnetic +mmoment of the hydrogen nuclei. Suppose we put a small sample of water in a +magnetic feld Ö. Since the protons (of the hydrogen) have a spin of 1/2, they +will have two possible energy states. If the water is in thermal equilibrium, there +will be slightly more protons in the lower energy states—with their moments +directed parallel to the fñeld. "There is a small net magnetic moment per unit +volume. Since the proton moment is only about one-thousandth of an atomic +moment, the magnetization which goes as 2—=using Eq. (35.22)—is only about +--- Trang 459 --- +one-millionth as strong as typical atomic paramagnetism. (That's why we have +to pick a material with no atomic magnetism.) IÝ you work it out, the difference +between the number of protons with spin up and with spin down is onÌy one part +in 10, so the efect is indeed very small! It can still be observed, however, in the +following way. +uppose we surround the water sample with a small coil that produces a small +horizontal oscillating magnetic field. If this fñeld oscillates at the frequency œ;, +it will induce transitions between the two energy states—just as we described +for the Rabi experiment in Section 35-3. When a proton fips from an upper +energy state to a lower one, it will give up the energy „ which, as we have +seen, is equal to ñưư„. lf it flips from the lower energy state to the upper one, iÈ +will absorb the energy hp from the coïl. Since there are slightly more protons in +the lower state than in the upper one, there will be a net øabsorpfion oŸ energy +from the coïil. Although the efect is very small, the slight energy absorption can +be seen with a sensitive electronic amplifer. +dust as in the Rabi molecular-beam experiment, the energy absorption will +be seen only when the oscillating feld is in resonance, that is, when +0) = 0y = s(s;-) +Tt is often more convenlent to search for the resonance by varying while keeping +œ fñxed. 'he energy absorption will evidently appear when +B= #my œ. +AUXILIARY +A typical nuclear magnetic resonance apparatus is shown in Fig. 35-8. A 5“ cols +high-frequency oscillator drives a small coïl placed between the poles of a large Nq ⁄24 OSCILLATOR +electromagnet. 'IWwo small auxiliary coils around the pole tips are driven with ⁄Z +a 60-cycle current so that the magnetic field is “wobbled” about its average WATER—-đ@ mm +value by a very small amount. Äs an example, say that the main current of the ⁄ ° +magnet is set to give a field of 5000 gauss, and the auxiliary coils produce a ⁄ Rr] 2 SIGNAL +variation of +1 gauss about this value. If the oscillator is set at 21.2 megacycles +per second, it will then be at the proton resonance each time the field sweeps +through 5000 gauss [using Eq. (34.13) with øg = 5.58 for the proton]. 05C L05COPE +The circuit of the oscillator is arranged to give an additional output signal (3 +proportional to any chønge in the power being absorbed from the oscillator. 'This +signal is fed to the vertical defection amplifier of an oscilloscope. The horizontal 2v +sweep of the oscilloscope is triggered once during each cycle of the field-wobbling 60 ~. oH SWEEP +frequency. (More usually, the horizontal deflection is made to follow in proportion SOURCE ——>——oTRIGGER +to the wobbling feld.) . . +Before the water sample is placed inside the high-frequency coil, the power _-  nuclear magnetlC resonance +drawn from the oscillator is some value. (It doesn't change with the magnetic : +fñeld.) When a small bottle of water is placed in the coil, however, a signal appears +on the oscilloscope, as shown in the fgure. We see a picture of the power being +absorbed by the fipping over of the protonsl +In practice, it is dificult to know how to set the main magnet to exactly +5000 gauss. What one does is to adjust the main magnet current until the +resonance signal appears on the oscilloscope. lt turns out that this is now the +most convenient way to make an accurate measurement of the strength of a +magnetic field. Of course, at some time sømecone had to measure accurately the +magnetic ñeld and frequency to determine the g-value of the proton. But now +that this has been done, a proton resonance apparatus like that of the figure can +be used as a “proton resonance magnetometer.” +W©e should say a word about the shape of the signal. If we were to wobble +the magnetic fñeld very slowly, we would expect to see a normal resonance Curve. +The energy absorption would read a maximum when œ; arrived exactly at the +oscillator frequency. “There would be some absorption at nearby frequencies +because all the protons are not in exactly the same fñeld—and diferent fñields +mean slightly diferent resonant Írequencies. +--- Trang 460 --- +One might wonder, incidentally, whether at the resonance frequency we should +see any sipnal at all. Shouldn”t we expect the high-frequency field to equalize the +populations of the two states—so that there should be no signal except when the +water is frst put in? Not exactly, because although we are ?rng to equalize the +two populations, the thermal motions on their part are trying to keep the proper +ratios for the temperature 7". If we sit at the resonanece, the power being absorbed +by the nuclei is just what is being lost to the thermal motions. 'Phere is, however, +relatively little “thermal contact” between the proton magnetic moments and +the atomic motions. The protons are relatively isolated down in the center of the +electron distributions. So in pure water, the resonance signal is, in fact, usually +too small to be seen. 'Fo increase the absorption, it is necessary to increase the +“thermal contact” 'Phis is usually done by adding a little iron oxide to the water. +The iron atoms are like small magnets; as they jiggle around in their thermal +dance, they make tiny Jiggling magnetic fields at the protons. 'These varying +fields “couple” the proton magnets to the atomiec vibrations and tend to establish +thermal equilibrium. It ¡is through this “coupling” that protons in the higher +energy states can lose their energy so that they are again capable of absorbing +energy from the oscillator. +In practice the output signal of a nueclear resonance apparatus does not look +like a normal resonance curve. lt is usually a more complicated signal with +oscillations——like the one drawn in the figure. Such signal shapes appear because +of the changing fñields. The explanation should be given in terms of quantum +mechanies, but it can be shown that in such experiments the classical ideas of +precessing moments always give the correcÿ answer. Classically, we would say +that when we arrive at resonance we start driving a lot of the precessing nuclear +magnets synchronously. In so doing, we make them precess £ogether. 'These +nuclear magnets, all rotating together, will set up an induced emf in the oscillator +coil at the frequenecy œ„. But because the magnetic feld is increasing with time, +the precession frequenecy is increasing also, and the induced voltage is soon at a +frequency a little higher than the oscillator frequency. As the induced emf goes +alternately in phase and out of phase with the oscillator, the “absorbed” power +goes alternately positive and negative. So on the oscilloscope we see the beat +note between the proton frequency and the oscillator frequency. Because the +proton frequencies are not all identical (diferent protons are in slightly diferent +felds) and also possibly because of the disturbance from the iron oxide in the +water, the freely precessing momenfs soon get out of phase, and the beat signal +disappears. +These phenomena of magnetic resonance have been put to use in many ways +as tools for finding out new things about matter—especially in chemistry and +nuclear physics. It goes without saying that the numerical values of the magnetiec +mmoments of nuclei tell us sormething about theïr structure. In chemistry, much has +been learned from the structure (or shape) of the resonances. Because of magnetic +fields produced by nearby nuelei, the exact position of a nuclear resonance is +shifted somewhat, depending on the environment in which any particular nucleus +fnds itself. Measuring these shifts helps determine which atoms are near which +other ones and helps to elucidate the details of the structure of molecules. Equally +important is the electron spin resonance of free radicals. Although not present to +any very large extent in equilibrium, such radicals are often intermediate states +of chemical reactions. A measurement of an electron spin resonance is a delicate +test for the presence of free radicals and is often the key to understanding the +mnechanism of certain chemical reactions. +--- Trang 461 --- +# orr-'oIittrgjït©fÉfsrtt +36-1 Magnetization currents +In this chapter we will discuss some materials in which the net efect of 30-1 Magnetization currents +the magnetic moments in the material is much greater than in the case of 36-2 The field H +paramagnetism or diamagnetism. The phenomenon is called ƒerrormnagnetism. 36-3 The magnetization curve +In paramagnetic and diamagnetic materials the induced magnetic moments are 36-4 Iron-core inductances +usually so weak that we don't have to worry about the additional fñelds produced +by the magnetic moments. For ƒerrornagnetic materials, however, the magnetic 36-5 Electromagnets +moments induced by applied magnetic felds are quite enormous and have a 36-6 5pontaneous magnetization +great efect on the fields themselves. In fact, the induced moments are so strong +that they are often the dominant efect in producing the observed fields. So one +of the things we will have to worry about is the mathematical theory of large +induced magnetic moments. That is, of course, just a technical question. “The +real problem is, why are the magnetic moments so strong—how does it all work? +We will come to that question in a little while. Reuieu: Chapter 10, Dielectrics +Pinding the magnetic fields of ferromagnetic materials is something like the Chapter 17, The baus oƒ In- +problem of fñnding the electrostatic feld in the presence of dielectrics. You will duction +remember that we frst described the internal properties of a dielectric in terms +of a vector field ?, the dipole moment per unit volume. “hen we figured out +that the efects of this polarization are equivalent to a charge density Øpoị øÏven +by the divergence of P: +Øpoai =—VW -P. (36.1) +'The total charge in any situation can be written as the sum of this polarization +charge plus all other charges, whose density we writ©e* øother. Then the Maxwell +equation which relates the divergence of # to the charge density becomes +V.E— P _ PslTPothe. +ÿ.E— _v.ự + other. +W© can then pull out the polarization part of the charge and put it on the other +side of the equation, to get the new law +vs. (coE + P) = fØother- (36.2) +The new law says the divergence of the quantity (eo -+ PP) is equal to the density +of the other charges. +Pulling and ? together as in Eq. (36.2), of course, is useful only if we know +some relation between them. We have seen that the theory which relates the +induced electric dipole moment to the field was a relatively complicated business +and can really only be applied to certain simple situations, and even then as an +approximation. We would like to remind you of one of the approximate ideas +we used. To fnd the induced dipole moment of an atom inside a dielectric, it is +necessary to know the electric fñeld that acts on an individual atom. We made the +approximation——which 1s not too bad in many cases—that the fñeld on the atom +is the same as it would be at the center of the small hole which would be left if +we took out the atom (keeping the dipole moments of all the neighboring atoms +* Tf all of the “other” charges were on conductors, øØother would be the same as Our /Ø£yee Of +Chapter 10. +--- Trang 462 --- +the same). You will also remember that the electric field in a hole in a polarized +dielectric depends on the shape of the hole. We summarize our earlier results +in Eig. 36-1. For a thin, disc-shaped hole perpendieular to the polarization, the ⁄4 ⁄ Z⁄ p ⁄ ⁄ ⁄ +electric field in the hole is given by ⁄) hư: F ⁄ +P P + P ⁄ lế +hole dielectric €0 › ⁄⁄ +which we showed by using Gauss' law. Ôn the other hand, in a needle-shaped slot ⁄ +parallel to the polarization, we showed——by using the fact that the curl of is ⁄⁄⁄ +zero—that the electric fields inside and outside of the slot are the same. Pinally, ⁄ +we Íound that for a spherical hole the electric fñeld was one-third of the way +between the fñield of the slot and the field of the disc: ⁄ ⁄ +1P ⁄ é⁄ ⁄⁄ +Thole = aieleetric + 3a (spherical hole). (36.3) ⁄ Ei ⁄ +'This was the field we used in thinking about what happens to an atom inside a 1„ +polarized dielectric. Ị +Now we have to discuss the analog of all this for the case of magnetism. One ⁄ +simple, short-cut way of doïing this is to say the jMf, the magnetic moment per +unit volume, is just like , the electric dipole moment per unit volume, and ⁄ +that, therefore, the negative of the divergence oŸ jMf is equivalent to a “magnetic ⁄ +charge density” ø„—whatever that may mean. The trouble is, of course, that ⁄ ⁄ ⁄ ⁄ ⁄ +there isnˆt any such thing as a “magnetic charge” in the physical world. As we Eue = E-{ P/ấcu ⁄ +know, the divergence of is always zero. But that does not stop us from making ⁄ ⁄⁄ +an artifcial ønalog and writing ⁄ Ei ⁄ +V.M =_— pm, (36.4) ⁄1⁄ +where it is to be understood that ø„ is purely mathematical. Then we could Z2 +make a complete analogy with the electrostatic case and use all our old equations +from electrostatics. People have often done something like that. In fact, histori- +cally, people even believed that the analogy was right. They believed that the ⁄ +quantity ø„„ represented the density of “magnetic poles.” ' hese days, however, we +know that the magnetization of materials comes from circulating currents within Fig. 36-1. The electric field in a cavity +the atoms——either from the spinning electrons or om the motion of the elecbrons ịn 3 dielectric depends on the shape of the +in the atom. It is therefore nicer from a physical point oŸ view to describe things CaVIVy. +realistically in terms of the atomic currents, rather than in terms of a density +of some mythical “magnetic poles.” Incidentally, these currents are sometimes +called “Ampèrian” currents, because Ampère first suggested that the magnetism +of matter came from circulating atomic currenfs. +The actual microscopic current density in magnetized matfter is, OÝ cOUrse, +very complicated. Its value depends on where you look in the atom——it's large in +some places and small in others; it goes one way in one part of the atom and +the opposite way in another part (jusb as the microscopic electric field varies +enormously inside a dielectric). In many practical problems, however, we are +interested only in the fñelds outside of the matter or in the aueraøe magnetic fñeld +inside of the matter—where we mean an average taken over many, many atoms. +lt is only for such rmacroscoøpic problems that it is convenient to describe the +magnetic state of the matter in terms of M, the average dipole moment per unit +volume. What we want to show now is that the atomic currents of magnetized +matter can give rise to certain large-scale currents which are related to /M. +'What we are goïing to do, then, is to separate the current density j——which is +the real source of the magnetic fields——into various parts: one part to describe +the circulating currents of the atomie magnets, and the other parts to describe +what other currents there may be. It is usually most convenient to separate the +currents into three parts. In Chapter 32 we made a distinction between the +currents which flow freely on conductors and the ones which are due to the back +and forth motions oŸ the bound charges in dielectrics. In Section 32-2 we wrote +7 — 2pol + đother› +--- Trang 463 --- +where jpọị represented the currents Írom the motion of the bound charges in +dielectrics and 7e took care of all other currents. NÑow we want to go further. +WWe want to sebarat€ ?ø¿n¿ inbo one part, 7ma„, which describes the average +currents inside of magnetized materials, and an additional term which we can +call 7cona for whatever is left over. The last term will generally refer to currents +in conductors, but it may also include other currents—for example the currents +from charges moving freely through empty space. 5o we will write for the total +current density: +bì = đpoi T đmag T đcond- (36.5) +OŸÝ course it is this total current which belongs in the Maxwell equation for the +curl of Ö: - +cvxp-=J+°E, (36.6) +€0 lôI) +Now we have to relate the current 2ma„ 0o the magnetization vector jM. 5o +that you can see where we are going, we will tell you that the result is goïing to +be that +đmag —= V x1M. (36.7) +Tf we are given the magnetization vector ƒ everywhere in a magnetic material, +the circulation current density is given by the curl of M. Let's see iŸ we can , +understand why this is so. C@ŒX©) +First, let”s take the case of a cylindrical rod which has a uniform magnetization ®1.Ằ1⁄40 29) +parallel to is axis. Physically, we know that such a uniform magnetization really +means a uniform density of atomic circulating currents everywhere inside the C)È⁄{}%*ŒX {1Z-€£>z⁄<) +material. Suppose we try to imagine what the actual currents would look like in +a cross section of the material. We would expect to see currents something like cfCX⁄€{XỚ Œ) @) C3) +those shown in Eig. 36-2. Each atomic curren goes around and around ïn a little ⁄ +circle, with all the circulating currents going around in the same direction. Now XØ 2 2 €CtX⁄O +what is the efective current of such a thing? Well, in most of the bar there is no (1 (2 CÀ C3X⁄C) +efect at all, because right next to each current there is another current going y +in the opposite direction. If we imagine a small surface—but one still quite a L_ CŒ