--- 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 noưïng 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 AÁœ = —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. 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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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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. 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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 oŸ 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, Ø„ = 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