id	question	solution	final_answer	context	image	modality	difficulty	is_multiple_answer	unit	answer_type	error	question_type	subfield	subject	language
0	"(f) Now suppose that we attach the point-like charge to a wall with a rod of length a. Any perturbation from the equilibrium will cause a perturbation of the polarization of the sphere. Prove that this equilibrium is stable and find the frequency of oscillation around it. The charge and rod have masses $m$ and $M$, respectively and assume that $L>R+a$.

<image_1>"	['Force is attractive. Therefore, any deviation from the equilibrium $(\\theta=0)$ induces an attractive torque towards $\\theta=0$.\n\nForce at the equilibrium would be the same as in part (e) with $L$ replaced by $L-a$. Since we are looking for small oscillations, we need to calculate the torque in the first order approximation of $\\theta$. Denote $\\beta$ as the angle between the line segment from the center of the sphere to the particle and from the particle to the wall. Then the equations of motion become\n\n$$\n\\begin{aligned}\n\\tau=-F(\\theta) a \\sin (\\beta) & \\approx-F(0) a \\frac{L}{L-a} \\theta \\\\\n& =I \\frac{d^{2} \\theta}{d t^{2}} \\\\\n& =\\left(m+\\frac{M}{3}\\right) a^{2} \\frac{d^{2} \\theta}{d t^{2}}\n\\end{aligned}\n$$\n\nThus it follows that\n\n$$\n\\omega=\\frac{q}{(L-a)^{2}-R^{2}} \\sqrt{\\frac{R L}{4 \\pi \\epsilon_{0} a\\left(m+\\frac{M}{3}\\right)}}\n$$\n\nIn the limit $a<<L$, we obtain the formula\n\n$$\n\\omega=\\frac{q}{L^{2}-R^{2}} \\sqrt{\\frac{R L}{4 \\pi \\epsilon_{0} a\\left(m+\\frac{M}{3}\\right)}}\n$$']		"5. Polarization and Oscillation 

In this problem, we will understand the polarization of metallic bodies and the method of images that simplifies the math in certain geometrical configurations.

Throughout the problem, suppose that metals are excellent conductors and they polarize significantly faster than the classical relaxation of the system.
Context question:
(a) Explain in words why there can't be a non-zero electric field in a metallic body, and why this leads to constant electric potential throughout the body.
Context answer:
开放性回答


Context question:
(b) Laplace's equation is a second order differential equation

$$
\nabla^{2} \phi=\frac{\partial^{2} \phi}{\partial x^{2}}+\frac{\partial^{2} \phi}{\partial y^{2}}+\frac{\partial^{2} \phi}{\partial z^{2}}=0
\tag{8}
$$

Solutions to this equation are called harmonic functions. One of the most important properties satisfied by these functions is the maximum principle. It states that a harmonic function attains extremes on the boundary.

Using this, prove the uniqueness theorem: Solution to Laplace's equation in a volume $V$ is uniquely determined if its solution on the boundary is specified. That is, if $\nabla^{2} \phi_{1}=0$, $\nabla^{2} \phi_{2}=0$ and $\phi_{1}=\phi_{2}$ on the boundary of $V$, then $\phi_{1}=\phi_{2}$ in $V$.

Hint: Consider $\phi=\phi_{1}-\phi_{2}$.
Context answer:
\boxed{证明题}


Context question:
(c) The uniqueness theorem allows us to use ""image"" charges in certain settings to describe the system. Consider one such example: There is a point-like charge $q$ at a distance $L$ from a metallic sphere of radius $R$ attached to the ground. As you argued in part (a), sphere will be polarized to make sure the electric potential is constant throughout its body. Since it is attached to the ground, the constant potential will be zero. Place an image charge inside the sphere to counter the non-uniform potential of the outer charge $q$ on the surface. Where should this charge be placed, and what is its value?
Context answer:
\boxed{$x=\frac{R^{2}}{L}$ , $q_{0}=-q \frac{R}{L}$}


Context question:
(d) Argue from the uniqueness theorem that the electric field created by this image charge outside the sphere will be the same as the field created by the complicated polarization of the sphere.
Context answer:
开放性回答


Context question:
(e) Find the force of attraction between the charge and the sphere.
Context answer:
\boxed{$F=\frac{1}{4 \pi \epsilon_{0}} \frac{q^{2} \frac{R}{L}}{\left(L-\frac{R^{2}}{L}\right)^{2}}$}
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proof	Electromagnetism	Physics	English
1	(c) Discuss the factors that limit the final speed of the rocket.	['The ratio of the mass of the rocket it self (the spaceship and the fuel container) to the mass of the fuel it can carry.\n\nThe fuel eject speed $u$.']		"4. The fundamental rocket equation

In this problem, we will investigate the acceleration of rockets.

Rocket repulsion

In empty space, an accelerating rocket must ""throw"" something backward to gain speed from repulsion. Assume there is zero gravity.

The rocket ejects fuel from its tail to propel itself forward. From the rocket's frame of reference, the fuel is ejected at a constant (relative) velocity $\mathbf{u}$. The rate of fuel ejection is $\mu=d m / d t<0$, and this rate is constant until the fuel runs out.
<image_1>

Figure 1: Rocket (and fuel inside) with mass $m$, ejecting fuel at rate $\mu=d m / d t$ with relative velocity $\mathbf{u}$.

(Note: the $m$ in $\mu=d m / d t$ denotes the mass of rocket plus fuel, not the mass of an empty rocket.)
Context question:
(a) Consider a small time interval $\Delta t$ in the rocket's frame. $\Delta t$ is small enough that the rocket's frame can be considered an inertial frame (i.e., the frame has no acceleration). The amount of fuel ejected in this time is $\Delta m=|\mu| \Delta t$. (See Figure 2.)
<image_2>

Figure 2: Rocket gaining speed $\Delta \mathbf{v}$ during interval $\Delta t$ by ejecting mass $\Delta m$ with relative velocity u.

Write down the momentum-conservation relation between $\mathbf{u}, \Delta \mathbf{v}, \Delta m$ and $m$ (the total mass of the rocket and the fuel inside at the moment). What is the acceleration $\mathbf{a}=d \mathbf{v} / d t$ (independent of reference frame) in terms of $\mathbf{u}, \mu$ and $m$ ?
Context answer:
\boxed{$\mathbf{u} \Delta m+m \Delta \mathbf{v}=0$ , $\mathbf{a}=-\frac{\mu}{m} \mathbf{u}$}


Context question:
(b) Suppose that the rocket is stationary at time $t=0$. The empty rocket has mass $m_{0}$ and the rocket full of fuel has mass $9 m_{0}$. The engine is turned on at time $t=0$, and fuel is ejected at the rate $\mu$ and relative velocity $\mathbf{u}$ described previously. What will be the speed of the rocket when it runs out of fuel, in terms of $u=|\mathbf{u}|$ ?



(Hint: useful integral formula: $\int_{a}^{b} \frac{1}{x} d x=\ln \left(\frac{b}{a}\right)$, and $\ln (10) \approx 2.302, \ln (9) \approx 2.197$.)
Context answer:
$u \ln 10$
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proof	Mechanics	Physics	English
2	c - Show that this so-called Bragg reflection yields the same conditions for the maxima as those that you found in $b$.	['c - In Bragg reflection, the path difference for constructive interference between neighbouring planes:\n\n$$\n\\Delta=2 \\cdot a \\cdot \\sin (\\phi) \\approx 2 \\cdot a \\cdot \\phi=n \\cdot \\lambda \\quad \\rightarrow \\quad \\frac{x}{L} \\approx 2 \\cdot \\phi \\approx \\frac{n \\cdot \\lambda}{a} \\quad \\rightarrow \\quad x \\approx \\frac{n \\cdot \\lambda \\cdot L}{a}\n$$\n\nHere $\\phi$ is the angle of diffraction.\n\nThis is the same condition for a maximum as in section $b$.']		"X-ray Diffraction from a crystal.

We wish to study X-ray diffraction by a cubic crystal lattice. To do this we start with the diffraction of a plane, monochromatic wave that falls perpendicularly on a 2-dimensional grid that consists of $\mathrm{N}_{1} \times \mathrm{N}_{2}$ slits with separations $\mathrm{d}_{1}$ and $\mathrm{d}_{2}$. The diffraction pattern is observed on a screen at a distance $L$ from the grid. The screen is parallel to the grid and $\mathrm{L}$ is much larger than $\mathrm{d}_{1}$ and $\mathrm{d}_{2}$.
Context question:
a - Determine the positions and widths of the principal maximum on the screen. The width is defined as the distance between the minima on either side of the maxima.
Context answer:
$(x_{n_{1},} y_{n_{2}})=(\frac{n_{1} \cdot \lambda \cdot L}{d_{1}}, \frac{n_{2} \cdot \lambda \cdot L}{d_{2}})$;
$2 \cdot \Delta x=2 \cdot \frac{\lambda \cdot L}{N_{1} \cdot d_{1}} ; \quad 2 \cdot \Delta y=2 \cdot \frac{\lambda \cdot L}{N_{2} \cdot d_{2}}$


Extra Supplementary Reading Materials:

We consider now a cubic crystal, with lattice spacing a and size $\mathrm{N}_{0} \cdot a \times \mathrm{N}_{0} \cdot a \times \mathrm{N}_{1} \cdot a \cdot \mathrm{N}_{1}$ is much smaller than $\mathrm{N}_{0}$. The crystal is placed in a parallel X-ray beam along the z-axis at an angle $\Theta$ (see Fig. 1). The diffraction pattern is again observed on a screen at a great distance from the crystal.

<image_1>

Figure 1 Diffraction of a parallel X-ray beam along the z-axis. The angle between the crystal and the $y$-axis is $\Theta$.
Context question:
b - Calculate the position and width of the maxima as a function of the angle $\Theta$ (for small $\Theta)$.
Context answer:
position is $(x_{n_{1}},y_{n_{2}},y_{n_{3}}^{\prime})=(\frac{n_{1} \lambda L}{a},\frac{n_{2} \lambda L}{a \cos (\theta)},\frac{n_{3} \lambda L}{a \sin (\theta)})$, and the width $\Delta x=2 \frac{\lambda L}{N_{0} a}$, $\Delta y=2 \frac{\lambda L}{N_{0} a \cos (\theta)}$, $\Delta y^{\prime}=2 \frac{\lambda L}{N_{1} a \sin (\theta)}$


Extra Supplementary Reading Materials:

The diffraction pattern can also be derived by means of Bragg's theory, in which it is assumed that the X-rays are reflected from atomic planes in the lattice. The diffraction pattern then arises from interference of these reflected rays with each other."	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proof	Modern Physics	Physics	English
3	$a_{2}$ - Discuss the stability of the equilibrium for each case.	['$\\mathrm{a}_{2}$ - The situation is stable if the moment $M=m_{2} \\cdot L \\cdot \\ddot{\\alpha} \\cdot L=m_{2} \\cdot L^{2} \\cdot \\ddot{\\alpha}$ changes sign in a manner opposed to that in which the sign of $\\alpha-\\alpha_{0}$ changes:\n\n$\\operatorname{sign}\\left(\\alpha-\\alpha_{0}\\right) \\quad-\\quad+\\quad-\\quad+\\quad-\\quad+\\quad-\\quad+\\quad-\\quad+$\n\n| $\\alpha$ | 0 | $\\pi / 2$ | $\\pi$ | $3 \\pi / 2$ |  | $2 \\pi$ |\n| :---: | :---: | :---: | :---: | :---: | :---: | :---: |\n| $\\operatorname{sign}(\\mathrm{M})$ |  | + | + | $-\\quad+$ | + |  |\n| $\\alpha$ | 0 | $\\pi / 2$ | $\\pi$ | $3 \\pi / 2$ |  | $2 \\pi$ |\n\nThe equilibrium about the angles 0 en $\\pi$ is thus stable, whereas that around $\\pi / 2$ and $3 \\pi / 2$ is unstable.']		"Electric experiments in the magnetosphere of the earth. 

In May 1991 the spaceship Atlantis will be placed in orbit around the earth. We shall assume that this orbit will be circular and that it lies in the earth's equatorial plane.

At some predetermined moment the spaceship will release a satellite $S$, which is attached to a conducting rod of length $\mathrm{L}$. We suppose that the rod is rigid, has negligible mass, and is covered by an electrical insulator. We also neglect all friction. Let $\alpha$ be the angle that the rod makes to the line between the Atlantis and the centre of the earth. (see Fig. 1).

S also lies in the equatorial plane.

Assume that the mass of the satellite is much smaller than that of the Atlantis, and that $\mathrm{L}$ is much smaller than the radius of the orbit.

<image_1>

Figure 1 The spaceship Atlantis (A) with a satellite (S) in an orbit around the earth. The orbit lies in the earth's equatorial plane.

The magnetic field (B) is perpendicular to the diagram and is directed towards the reader.
Context question:
$a_{1}$ - Deduce for which value(s) of $\alpha$ the configuration of the spaceship and satellite remain unchanged (with respect to the earth)? In other words, for which value(s) of $\alpha$ is $\alpha$ constant?
Context answer:
\boxed{$0,\pi,\frac{\pi}{2},\frac{3\pi}{2}$}
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']	Multimodal	Competition	False				Theorem proof	Electromagnetism	Physics	English
4	$\mathrm{b}$ - Express the period of the swinging in terms of the period of revolution of the system around the earth.	['$\\mathrm{b}$ - For small values of $\\alpha$ equation (2) becomes:\n\n$$\n\\ddot{\\alpha}+3 \\cdot \\Omega^{2} \\cdot \\alpha=0\n$$\n\nThis is the equation of a simple harmonic motion.\n\nThe square of the angular frequency is:\n\n$$\n\\omega^{2}=3 . \\Omega^{2}\n$$\n\nso:\n\n$$\n\\omega=\\Omega \\cdot \\sqrt{3} \\quad \\rightarrow \\quad T_{1}=\\frac{2 \\pi}{\\omega}=\\frac{1}{3} \\sqrt{3} \\cdot\\left(\\frac{2 \\pi}{\\Omega}\\right) \\approx 0,58 \\cdot T_{0}\n$$']		"Electric experiments in the magnetosphere of the earth. 

In May 1991 the spaceship Atlantis will be placed in orbit around the earth. We shall assume that this orbit will be circular and that it lies in the earth's equatorial plane.

At some predetermined moment the spaceship will release a satellite $S$, which is attached to a conducting rod of length $\mathrm{L}$. We suppose that the rod is rigid, has negligible mass, and is covered by an electrical insulator. We also neglect all friction. Let $\alpha$ be the angle that the rod makes to the line between the Atlantis and the centre of the earth. (see Fig. 1).

S also lies in the equatorial plane.

Assume that the mass of the satellite is much smaller than that of the Atlantis, and that $\mathrm{L}$ is much smaller than the radius of the orbit.

<image_1>

Figure 1 The spaceship Atlantis (A) with a satellite (S) in an orbit around the earth. The orbit lies in the earth's equatorial plane.

The magnetic field (B) is perpendicular to the diagram and is directed towards the reader.
Context question:
$a_{1}$ - Deduce for which value(s) of $\alpha$ the configuration of the spaceship and satellite remain unchanged (with respect to the earth)? In other words, for which value(s) of $\alpha$ is $\alpha$ constant?
Context answer:
\boxed{$0,\pi,\frac{\pi}{2},\frac{3\pi}{2}$}


Context question:
$a_{2}$ - Discuss the stability of the equilibrium for each case.
Context answer:
\boxed{证明题}


Extra Supplementary Reading Materials:

Suppose that, at a given moment, the rod deviates from the stable configuration by a small angle. The system will begin to swing like a pendulum."	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161J/49tf084cLxk8nYD3U8ZNDGXWLYapYKItbsjtlhzhZWH8BP93PegBWZ9ct1vbQG21ux4ZH6kdCSB1yAcUkjnU4Ytd0xfJ1G1G24t25O08lSB/EQBzSySNfxJr+jjF7FkXNufl83HDZ78YOKSWRSsfiXRQSrDN1AeNwz8zsOu4AcCgAmkZtniXSEJkUbb21HLMo5KgdA2eppZpBaSr4g0sedaTfLeQJyfTjsMEnNEsy6a6eIdMy+mzgNdwgfcXqXAH8XTiiRk0Kcahajfo12MTxrz5JI6gD1J5oAHZNIuRq1gRPo96czovIDHgMOwQYPFLuTQLoSKVm0K+O4nqsTHncT/dx2oKpoF3sYCfQb/5R3ELHAVFH908kmkjWPR7k6TfAT6PfHbA7HIDHqh7BQB1oAI1TQ5/sNwfN0S85hkfkRE8/MT6kjFJDEumSPoGpHfps4P2OY/8ALMdlLHncSTg0sUa2craBqh86zuMmyuH5JPXn0xkYpIUJ3+G9YfLKC1ldHkuoH3yegYE9KACKJcSeGdZJMbjFpP0JXPCg92AHWiONr6GTw/q5xeRfNbTr8okxyMd+ABmiKM6nC+g6o5i1G2G62uV+9t6K4P8AeI6ihFfWrZ7G7P2bWrE/I6dSP4cHvkDmgARZNZt20vUGEWtWZDRTAbRKw5DAf3ckZFBWXXrYKWFvr+nNuQsMBm6biB1U5PGaNz6/a+ZgW2v6edrqpwC3XYD3U8c0O0msW41TT1EWtWR2zQdBKw4KH/ZyetACuz63At/Zj7NrVjw8b9x0OQOuQDj+lJI/9pRR69paGK/t123Ns3J2nkqQP4iO9LLK17FH4g0f/j7iO25t2+UOBw2e/HOKSWZQI/E2jZMTjddwEYyufmcjruAHAoAJnJ2eJdHXJwFvbUcswHO0DoGBIyaWaQWMy6/pg86yuOLuBOTnpx2GOc0Syrpcqa9po8zS7gbruEf8s1PO8AdWJxxQ7JoNwL63HmaLdgeag58onjge5PNAA7R6RcjVLEifRr5t06DkKx4DA9AoweKUFNAusjE+hX5zk8iJm5LE/wBw+lG2PQLowyAT6FfHauOVhY4CoB02nk5zSRhNIuTo9/ifSL07beRuQGPVD2CgDrmgAiVNEnOnXP7zRbzmGR+fKJ55J9SeKIYl0528PaoS+nTgraTHqi9AhP8AeyTg0QxrayN4e1U+daz5NncPyWPXn0wSMUkMZkD+G9XcmRRusro8s6jgOT0DZzxQAsUSssvhnWclWA+yzjjcM/KqnruAHWkjjbUIX0DVjtvYfmt5l+USAcjHfgAZojQ6pDJoepsYdTtRutrheu3orgn+IgHIoRX1u2eyuj9l1uxOVdPTqMHuCBzQAKsutWzabfMItbsjuilHCysP4gP7vI4oIk162GCLfX9ObcpYY3HpuIHVTzgUZl16285FFvr+nnbIqnAZhyUB7qTjmh2fV7ddX05RHrFmds8B+USkcFG77ckkHFACuz61AmpWI+z6zYnEkT9xnnIHXIBx/SkklN+ieINKQxXsAxdWzcnb1Kkf3sd6WWRryOLxDowzcpxc27fKHHRie+QM4olmSPy/Euj5a3kG67gI6r1LkddwHagAmkIZPEukLuBAW9thyWA/hA6BgTyaWWQWE66/pg86wuMC6gTnB6bh2AHeklkXSZV13Tv3mlXABu4R/wAsged4A6tkjIodk0C5F3D+80S84lRefKY8DAHYknNACs6aLdDUrMifRb4lplHzBWJxuB6BcdqQFdAuguRLoV8dwJ5ETNyWJ/u9MClCx6Ddm2kAm0K+O1e6xMeAgHTb70kSppNy2i6hifSr04tZG5G48lD2CgAc5oAWJU0WZtNuyJNGux+4lbpESOhPqSeKbFCtgzeHdUy9hKCtnMTyq9Apbu2e9LDGIHbw9qx822my1ncPyWxzz2BBIxSQxGVX8N6vIfOjG6zu25Z1HAcnoGz2oAIo/MSXw1rOembW4HG5c/KqnruAHJoSNtShfQdWIXUICHgnX5fM28gjvgYGaI1OqwvompN9n1W0+a3nXrt6KwP94gHIoVX1y2a1uD9l1ywOQ6d+4AJ6ggDNAAol1q2On3jiHXbI7o5QMLIw/iA/u+1B8zX7ZWT/AEbX9PJIDdz03EDqp5xS5m121+0xoLfXrA7ZIwcKzDkpnupPeh2fVrdNa0xQmrWpKzQH5fNK8FG77ckkHFAA7PrMKapp4+z6xZfLNE4zkdCCB1JHSiWU3scfiHSkKXkAxdWp5O3GSuP73vRLI10kPiLRRmdeLm3b5Q46MT3yADiiaZIjH4l0fLW0oDXUGOqnkuR/exxigAmfY6eJdHXeGAS9t15LAfwjsGB6miWQadMNd00edp91gXcK889Nw7ALzmiV10eZdb0/95pNyAbuFf8AliDzvAHViSMih2Tw/ci5i/e6He8SIvPksemAOxJOaAFdo9Fuv7QtSJ9EvjulUDKox43ey47VAlnaeHb7/UQSaJfHcrmMHyXbkkt/dPYVOEj0K6NpMBNod6dqHqsTHgIB/dx3pIkXSrg6JqJE+l3mRazNzhjyUPYKBjBzQBDb2NloVy2mXVvEdKu1IgnKDMe7qrN6knikt9Ms9PaTw5qVtF9inUpaz7QDs/uFupb3qeKNYXbw5q58yGTLWdw/JbHOT6EEjH0pIojcrJ4c1eQi4jG60ujyzKOA+em7PagCh/YlpdyJ4f1hWhkgybWeA+UJVPCrxySAKzNM8E6ez3GlX1xepqUasYpPtDBZfcD0HGa6KMNq0Mmjakfs+rWfzQTL129FYE/xEDkUKr67am3nP2TXbAj5k79wAT1BAGaAObsPBNld2Nzpkl1ew61AhQM9y22Q/wB8L6UW/gyz1LSLiGznvLfWrcYKzXBbkH72PQ9hXSkza5ai7gQW+vWB2yRA4DsOdme6570O7arAmt6WoTVLY7J4D8vnY4KnvgZODQBh6Zc+Or3TpZItRsUltkw9u1vl1I6A+5xSW9947utEk1CHULJ5Yly9sLf51b+7161uSubhY/EeiqGmHFzbtwHXoxI9QAcUSzLbGPxJpOXtJQGu4cYwp5L4H8XtQBn6fqXiyKxTWbm8tL6zjGZre3h2yE9xn1Hetfw54xHiW6RI9LubeHaXE0mCpI4IHvz+hqKRl0aYazYfPpF0AbmJR/qs87gB1JJ5FXdMsJLHVw9m4fSrlTKBniF8cBR6EEmgDo6KKKAONjQKZPDWs5dHH+i3HQSDsg9wAM0scZ1BJPD+rn/TYvmtZ+nm7eQw9gcUkanUY5NA1UmLULYbredeMpnClW7tgc0Kj63A1jdn7PrViQUkXjfjkAHuDgZoAVFl1iBtLvpBDrVkd0ExH+sI6SAentQC+vW4T/j017T2LLu64zjeR3DAHAoHna5bl1Ag16wfawHyiRh2z3X3oZn1q3XVNPXydXtGKyxN8vmEYBB77euDQAjF9bt1v7MG11uw4libkgd1I6EsBxSyStqESa/pkRj1G3Gy6terEdTH6Zz3okka+ij8QaONt1FkXFu3y7h0YsOuQAcUksyoqeJdJBMRUNd2wHzBepO3ru9qAFmk3hPEmjDzGICXluoyXx/APQqScmiaQWUi+INJH2ixuSPtcCclskDf7BRnIomddNkGu6cN+nzgfa4U58vuSFH8WTzRIyaFcfbrf97ot6Qs0a8+U7cDaB0HJzQArSpo066tZsJ9FvSHmA5CE8mTntjtSYTw/cBgfM0G9O7PUW7NyWJ9CSMClxHoV55T7Z9Ev32rn5lRj29AvvSIi6PcHSb0ifSLwn7PK/IVjk4PYAcYoAEVNCn/ALOu/n0S8BEEp6Qk/wABPUliTg0RwpYSN4e1Qb9OuBstJc4wM8R56596IVW0lbw7qjeZayqWs7h/QcjJ6BgTxSRxeYH8N6s5Dqpa0uj94qOAc/3vagBYkJMnhvWm37h/olx90SDsg91AGaI0OoxyeH9WOL+H57afp523kOPYHGRSIj6nE+h6ofJ1G2GbeZeMr/DtbuSBzQqvrtu1ldH7PrVgw2uvG/HPB7qcDNACqsuswPpt44g1uyO6GYjG8jpJj09qQF9etx1tNe08kqG646biO4YA4pcy67b+bGBBr2nthl+6HYds9196RpG1mBNX05fK1W1JEsL/AC+Zt4O7vjrj1oAGZ9agXUrIG21qx+SaFuSB1KEdMkAYOaWWVr+JPEGlRFdQgG26tRyzDqY/TOe9EkhvIovEejgrcRgi5t2+Xcufm3L13ADikmnWEL4l0oM1uyg3luB823udo53e1ACzSAhfEmjDzdwC3lunJcDt7FSTmiaQWMi+INK/0jT7rBu4UGck8b+egAzkUkzLpco1vTx5mm3GPtUKc7O5IA7kk5pZGXQbn7ZABNol6QJUXny2bgbQOi8nNACvImi3C6naMJ9FvTvnA5EZPJkJ9MdqQBPD1yB/rNBvDnJ5FuW5JJ9CSMUoEeh3f2eXbPod+2FJO5Y2Pb0C+9JGi6RP/Y98fP0q8ybaZ/m2k84PYAZGKABETQ5zpd78+i3YIt5j0hzxsJ6ksScURwpZu/hzVcvYXA2Ws2ccHpHnrn3ohVbaVvDmrN5lvIpazuH647ZPQMCeKSKLz1fw5q7MsqqTaXR6legIb+97UALEhfzPDWtHczDNrcfdEg6gD3AAzRGjalE+gau23UYDvtp+nnbeQ49gSMikRX1SKTRdTPk6na4aCZeMjPy7W7kgc0oD67bNa3P7jXLBvlZeN5HOFPdTgA0ACrLrcB0+5cW+uWJLwykY3EcCTHcZ7UmX1233KDaa7p5OFbrjpux6MM4oPm65bCeACDXrBtrIflEjAdOeq570rSNq8Caxpo8rVLUlZoX+XzMcHPfHBxQAjM2swJqtipt9ZsflmhPJA6shHTJAHNLLI15GniLSYiL6EbLq0HLOByY/QHPekkk+1RxeJNGBWdBi5tm+Xcuedw67sDiiaYW23xJpQLWzqGvLcD5gvUnaOd3tQAs0oAXxJo6+ajYW7t15LY4wPTaetErrp8q69pf+kabdkG6hQZyTxv8AYAZyKJnTSZRrNgPN0u5A+0wpzs9SAO5JOaHK6DcfaYgJtDviBIq8+W7cDaB0XrmgBWkj0O4XUbZhNod4d0uORETyZCT2x2pAE8PXGw/vNCvOVY8iAtyST3yTxSgR6Hd/ZZ9s+iX77Y2Y7lRj2PYLjvSRRrpU39jXx87TbrJtpn+baepB7ADIxQAIi6JOdJvcvo91xbS9oc8CMnqSSTiiOFLZ38N6tl7G4Gy1mzjIJ4jB65x3ohQQO3hzVmLwOpayuG647Et2YE8Ckji+1JJ4c1ZmWeNc2t1/FtzgEMf4sdhQAscZl8zw5rLbpMZtbj7okHUAe4AGaI1fU4n0LVW2anbnfbz9PP28hwB2BIyKRFfVYX0fUv3Oq2nzQzLxuX+Eq3ckAZpR5mu2728/7jXdPbhl+UORzgHup4BoAFWXXLc2M7i212xO6GUjGSOBJjuPaky+u24kjBtNc0/Pyt19M4/2gDig+brlsLq2Ah12wbDI3yiRhxg55255zileRtVhTWtLHl6lbErPC/y78cEN39cUAIzvrECavYoYNYsfkntz1GeShHTJAHNLNK91GniPSEP2yIbLu0HLSAcmP0Bz3pJJRcJH4k0gFZFGLq3YYJXqSw67sDiiaZbQr4l0oF7WRQbyBR8wXqSF67s44oAWaQIV8R6QPOjkwt3bpyXxgcem05zRI66bKuvaYRPpl4Q11GnPJ48z/dABzSSumkTDV7EeZpVyALmFOfLJ4+VR3yTmnOF0C686LE2iXzYkUciJjwAo7L1zQANImhXK39ufN0O9O6XHSEnnzCfTHagBPD9x5L/vNDvPuOeRATycnvknigCPRLoWdxi40TUH2xO3zKjN2PYLgDBpI0XTJv7E1A+bpt0CbWaTnaeuDngYJGKABIho0zaNf/Po90CttIekXbyyepJyeaIokt3fw1q+XtJxttJumRniMHrkAZzRFH5LP4a1di0TKTZ3Lddo7lv72e1JHF9sSTw7qzlLmJc21z/EV6Ahv72M5AoAWOM3Ak8Oayd0w+a1uOnmY5GPoMZpI1k1SJ9D1RvL1W1O+3nPHmkciQAdgSOKEV9WhfSdRPk6raYMUq8bh22nvkAZoBfXrZ4Zh9n17T25I+UORzgHuvQGgAVZNct/skz/AGbXdPO6KRu56eZjuD6UZfXbcTwj7Jrmn8MjdQM88f7QBxQTJrlut7agQ65YnDxt8okYcYOeduTkcUryNqkMet6WBHqNvlZ4G+XeBwcjr0BxQAju2sQJrGnoYdWsxtuLbq2DyYz7n1omdriNPEWkKTcxjZd2o5aQL/yzx0BB70SyrNHH4m0hSGUf6XbkYJXqxI67gAABRLKtiy+I9LBezlAN5Ao5VepIXruyRkUALNJ5JTxFpC+dBLhbu3T+Lscem3nNErppk413TSJ9LvTuuUXkZ6eZ9AAciiZ10eb+1bMebpN1gXMSc+WT/dA9yc0ME0C4MiYm0K/bBUfMsbHgAdgpFAAXTQLkXkJ83Qr07pMDiAnJMhPoeOKFCaBcfZ5fn0O95jc8iAkdz3yTxSrs0W6+w3OJ9Fv2xC5+YIx5wewUADBpIo006ZtC1E+bp9zk2sz84PXBzwMEjH0oAEhXR5m0O/8An0i6BS2kJ4iB4EZPUk5PNEUawNJ4a1j5raYbbSboGBPEY75AGc0kUXlM/hvV2LRlc2VyfvFRxkt/eFEcf26OTw9qzGO7hXNtcjg7TwpVv72AcgUALGhulk8O6wczrhrW46ebjkY+mBmiNZNVifRdScR6vaHdBPjAlIH+sA9MnpSKr6xC+l358jV7PmKVONw6jae+QBmhTJr1sUkAt9e09sHb8odgM4B7rnGTQAASa9b/AGeU/Ztd087o3Ydeg3kdweeKCX1y3FzBm01zT+HRuoGecj/aAOKGMmt2631mBDrdi2Hib5fMI4we+3JNK8janDHrulDy7+DieBvl3gcHI68DOKAB5G1WBNb06Mw6tZjbc2x5fHUxntn3pJXaZU8R6OhM6AJeWo5aTH8A9CCetEs4kSPxNpCn7ubu3Iw23qSy9dwAGBRNKtg6eItLBeylUC7t1HKr1yFH8WSM0ALLILdl8Q6QPPt5vlu4E/iPAyPTbzmlkkj0uddb04ifSb077hE6Ak48z2GO1JKyaLONTtB5ukXeBcRJz5ZPGVA9yc0N5eg3eRtm0K/bBA+ZY2PGB6LjvQAFk8P3K3ER87Qr35mPaAtyXJ9DxxSoE0C4+yT/AD6Hef6qQ9IcjoT3yTxQipotz/Z13i40a+b9zI3zBGPJB7BQAMUkKLYTNoOpHzbG4BNrPJzjvyTwMEjH0oAEiXSZW0LURv0q4Gy1lJ4jHQR56k+9EcYjMnhrWfngmGLS46B8niMd8gDNJDEE3eGtWYldm6yum5YqOASf72e1ESG/jl8P6qxivYBm2uR12nhSrf3sA5FACxxm8WTw9rBzdId1rcdBLjkEfTAzQiy6tC+kai4j1i0JaCcjAlIH+sA9MnpSIj61A+m3x8jWLI5jkTjd3GD3yAM0qmXXrZtwEGvae21tvyh2HOAe6k96AEAfXrfyHb7Lr2n5aMt6Zx5mO4PPBoZn1y3W8th9k1zT+JI25IHcEf7QHFDF9at01GxAh1qzbDxt8vmEcYbuV5JFLJI2oRR69pI8u9g4uIG+XcOhLD2AOKAEeVtUhTXdOiMWqWo2XNqeXx1MZ7Z96JnMwXxJo6l5QPLvLZeWkx/B6Agk80SzK6ReJtIB2lQ13b4w23qSV67u2KJpF06QeINN/eWEygXkC8lB1yFH8WTzQAs0gtnXxDpC+faz8XcCfxHgbv8AgIzmleVNJnGs6ewn0e9IedV6KSeZPYY7U2Z10Sf+0LUebo15gTxpz5bHj5QO3JzTiI9Cu8fLPod+2OOVjY8Y9AuO9ACEp4fuBNGwl0C9O49xAW5Lk+hOOKVAmgXH2K4IfRLwfupD0hJ/hJ75J4pEVdGuf7Mu8T6Pen9xK3zBWPJB7BRgYoijWxlbQNTbzbOcFrSeTnHfkngEEjH0oAEhTS5W0DUsvpdyClrKT9wdo89Sfekjj2mTw1rWXikwLS46CQdkHuABmiGL7/hvVnPCbrK5P3io4Bz/AHs9qI0bUEk0DVWMeoW67re4XjKE4BU/3sDmgAjQ3ySeH9XObuM77afp52OQR7A4zSosusQtpWoOItZszugnIwJCOkgHp7Uio+twPYXh+z6zYkFJE43Y5AU9wcDNL++122LYEGvWD7WA+UOw7A9196AAeZr1uImP2XXtPbcm7rjpvx3DAHApGL63At9af6Lrdh8ssbckDuCPVgOKGZ9at01SwAi1e0YrJE3y+YRwQe+0ZOKWR2voo/EGkDZdxZFxbt8u4dGLDrkDOKACSVtSiTXtMiMep2w2XVr1cjqYz2znvSSyeYF8S6OvmSY2XlsvJkx/APQgk80SzKFj8TaQCUKhrq2x823qSVHO72omddNkXXtN+fTpwPtcC87B1yAP4iTzQAs0gs3XxBpK/aLO4IF3AnOc4G72CjOaVpE0eddXsWE2jXpDzKvRCeTJz2x2pJWXQ5vt9uPN0W8wJo158tjwNoHQcnNKRHoV2Y32z6Jftgc7lRienoF96AEOzw/cq6HzdBvTuPcQFuS5PoTjAq7pFrc6Tqv2GMNLpsqtNbv18r/ZJ75yTVJEXRrj+yr0ibSLwk28r/NtY5JB7ADjFXNGNzpmpDRpy0sGGmtZuuUHUMe2C2B7UAdLRRRQBxqh9dhNnOfs2u6f88b9Wx0D+nzAGgmTXYRcQ/6Lrun/ACuo5bHdfT5gKGZtbtkvLYCDW7BjuR/l3Y4JYd1POKJHbU4o9c0v91qNt8k8Ld1/i3Ad8A4oAVpH1aFdZ06Py9Xs/kuIByT3MX50kj/a0TxHo67rlR5d5Cv/AC1C9UOf7pJ+tEsvnCPxLpKkSKB9rtsZbb3BUfx0srraSDxDpQ320oC3lugyeOcAdmyeaACSTyWXxJow823m/wCPuFeN/OC+T02jJ96Hmj0uVdd05vM0m7/eXCjoueTL+VErro9x/a1kPP0q9IFzEnzbSeAQOwGTmnZj0O786MrPod83zY+cRse+emygBpKeHphdQ/NoV580i9BAzcl/X5iR9KAE8Oz/AGWf5tCvBtjY/dgLcBPX5snn2oiVNCuPsU/77Rb4lonb5hGx5OT2U8YFEKLpkjaFqH7zTrgf6LO3Oz2JPfJ4oAEiTSZm0LUVD6Vd/u7dzwFB6R0Rx+SzeG9ZPmwT/wDHpMeN/onH90Ac0QwLGJPDerFvJZT9kuM4IXsNx/jpI0N9DL4e1glbqMA29wvy7hn5Qp9cAZxQAscf2pW8Naw2bhP3lpOesm3kPj1Bx9aREk1WJ9G1CTy9XtPnt7g9WI6S4oVG1m3fStQPlatZkPFKvy78cgqe4zjNB83XLfy3Kwa/YMSvG0SMOhx12UAKvma7Abeb/Rdd0/5kbqcdA3p8wFBL67ALuAG213T/AJZFHLbepT0+YAUjM2twJfWf+j61YsQ8b/Luxwdw7qcHFEjtqMMWvaUPK1C3G24t27r/ABBgP4sDg0ADyPqkS61p0ezVbQeXcW45L45MX50skhuVTxLow3TL+7u4F/5abcAqc/3eeaJpd4TxLpKncF/0y2xlivUjaP480TOtnMPEGljzLScbby3QZ6eg7HJOaACSUQMviTRx5trPzeQr/GehfJ6bQD9aHmj0mVdcsG8zR7v57hR91SeTLRI66Nc/2rZgT6RekC4jXnYxwAQOygZyKUGPQrvzFK3Gh37c/wAYjY989AmKAA+X4dnE8Xz6FefMy9BAzc7vX5ifwpAE8PzmzuBu0K8+WJicLAW4CevJJ57URImhz/2fcfvtHvSWhkb5tjHk5J6DkYohVdNlOgaifM0+4U/ZJ252juCT/Fk8UACQx6ZK2g6kol0u7Hl2zt0H/TOiOPy2bw3rJ82KX/jznb+P0UY/ugCiKABX8NauW8tl/wBEuM4JXPADf36SNGv4ZNA1c7buHm3uF+UMv8IB9cAZxQAsaG8RvDesNm7T95aTH/lpt5Dgf7Jx9aREk1aJtIvpPJ1izPmW9wfvNjgSYoVW1q1fTL/EOr2R3RyJ8ocjkEeq5xkUp83XLfGVt9fsGOONodh0OOuygBFMmuwGCT/RNd0/lT1bHQN6fMAaGL65CL63X7PrunfJKg5YA8mP0+bApXZtbgTUbP8A0fWbEkPG/wAu7HB3D0IBxSSOdQji8QaUPKv7cFZ7du69WDAfxYHBNAA0kmpRJrelx7NTthsubcdZMcmLn3pZJPPCeI9GXfL/AKu7gX/lrt4KnPTbzSTS/c8SaSpOF/0y2Ay20ckBf7+aWZ1sJhr2mDzbO4G27t05x/ujsck5oAJJBbOviTRz5tncZN3Co+/28zJ6bQDxQ8sejTLrVi2/Rro77gD7sWeTJQ7Lot1/admPO0e+IE6L82xycAr6KOcilXy9Cugfln0O/Ycn5hGx7k9AmKAEPl+HZxMnzaFejcw6LCW53evzE0Kq6BM2n3I36HdjZC54WAtwI/Xkk/SiJE0Wc6dc/vtHvCTDI3zbGPJyT2yRj6UkMa6fI3h/UzvsJl/0Sdv4R6Mx/jyeKAFjhTT5W8P6ool0y8+S2c9ME8RflRHHgt4b1lvNjk/4852/jxzt4/ugD60RwB1k8NawzYK4tLnODt6ABv7+KEjbUYZdB1Y7L2EAwTr8oYdtp9QAM0AEatfI/h3VnzexfvLS4PHmbeQ4H+ySPrSKj61E2lXsnka1ZZe3nP3s9BKBQqvrds+n3uIdasmyki/KHYcgr3K9M9qD5uu2wKEW+v2DHbkbQ7DgEjqUzQAoMmvQeW3+i67p2CD1bHTPp8wFIWbXIBqFqv2fXNP+SWPqeeTH6fNgfSh2bWIE1Oy/caxZZEkT8bgOCWHoQDiiV/tyR+INKHl3sClbi3bqV6tuUfx4AweuDQASSPqMSa7pUZGpWw2XVsvWTA5iOfelkk80J4k0Yb3OVu4F48zbwQc9NvNJNLs2eJdJUldv+mWyjLbRyQF/v560ssi6dMNd0webYXGFuoE5x/ujsck5oASSRbR18R6QfNsbn5rqJeA3bzMn+6AfrSvKmiTLrFkd+i3Z8y4A6RZ5MtDFNFuTqNrifRr4jzkHzBHJwCvYIOc0q+XoV3hiJ9Dv2GCfmEbHuT0CYoAQiPw7OGHzaDedR/DAWH5/MTQsaaDMdMuhv0O8ysLE4EBb/ln68kn6URIuiznS7r97pF3kwStzsJ5OSfcjFJBGtizeHtTO+xlU/Y7hv4V9Cx/jyTjvQA5IkspH8O6qPN067+S2kbpyeIvwFJHHnd4b1g+YJPms52/jxz/47gfWiKASrJ4a1gt93/RbnODt6ABv7+PSiNG1KCTQ9VOzULfBgmX5d46jafbAzQAkatfxv4e1ZiL6D95aznrLt5EgHsfzpFR9ahbTLx/I1qxPmQTH73oJce9KofW7ZrC8xBrdi2UkHyiRhyCO5TJHtQRLr1sJIyLfX9PJwGG0Ow4BYdSmSaADMmvQYP8Aomu6eR7nGfy+YD8KGZtbhXUbRfs+t2HyTRdW55MeenzcUO7axAmq2H7jVrMkSxScbgODuH0zillf7dHH4h0lSl5EpW5tj1K9WDKP48AY74oASRnv401/SVI1CD5Lq2XrLj/ln+BpZJN2zxHow37vlvIV48zGAc56befrRNJ5W3xJpIJjZR9ttl5JXqQF7PkjNErrp041vTR52n3OFuoUGcHp8o7ck5oAHlWykHiTST5un3Xz3SL0PbzOemAKGkTQ5hqtmfM0S8+ecD7sWeTJ+JxQxj0W7N/a7bjRb9v3qD5gjdAV7BBg5pRs0G7EbYn0S/b5WPzCNjzyegTHSgA/d+HZ8H5tCvPygLD8/mJ/CkWNdCmbS7wb9FvDshY9Ic8CP8aIUTRpjpN4fM0m7BNvK3Plk8nJ+pGKSCJbNm8O6oxazkU/Y7g9VU8YLH+MknFACxxraSP4d1YedYXXy2sj9CeojGPQCiOPfv8ADesnzCfmtJz1kxk/+O8UkcInSTw3rBIdVza3A4JXphW/vgZoSNtUhfRNUPl6jbYaCZPl3gcgKe+ABmgAjR9Qjfw/qr41CD95bXB6y4HEn4GgK+twNp123ka3YfPDMevPAk9OeaB5muWxs7orDrli3yuBtEjDnI7lMkZoIk163WeIi217T26H5d7DjLDuhycds0AL+812DI/0XXdPPPc4z+XzAfhSM761AuqWaeRrdj+7nhHLY6mL05odn1aCPV9O/capaZE0T8bh0bcPoDilll+1pF4i0hSlzGo+1Wx6lTyQVH8ePWgBJGa+iTxBpKkX0P7u6t16ybeDHk+hOfelkkzs8SaMPMV/lu4F4EnQHOem3n60TSeQ6+JNJXdE6gXtsvJK4zgDs+SPeiV10uYa1p487Tbri5hTnBPA2jtgk5oAHlSxkXxFpJ83Tbs+ZconQknHm8+1DOmhTDUbU+Zod588qj7sJPPmevzHH0pWMeiXf2y32z6LqDfvFHzBGPAI7BMUibNCuhbykT6Hfk7Gb5hG55OT2T07UAKfL8OzbW+bQrzoT0gLdvX5ifwpFiTQ5m0e9+fRrv5LdjwIweBFRCi6RMdHvD5ml3WTbTNzszyck+5GKIYVtN/hvVCzWkikWdwTghegBY/x56UACRraSN4b1gebZXOVtJGP3u/l8f3QBRHH5obw5rLb3+9aTt/y0xyD/wAB4ojiFxHJ4a1glZFUfZrgfLlc/KFb+/gc45pEjbVYH0bUj5epWuGhmX5d2OV2nuBxmgAjR9RjbQdVkK6nb5e1uT96XA/1oH1oVX1yE2Fy32fXNPPmRSHr6B/T5uaP3uuWxtbkrDr1g3yuBtEjDnI7lM9aCX122W5g/wBH1ywblT8u9hwdw6lDzjtQAfPrsPmJ/omvafwwHJx6enzAfhQ0j6zEurWMfka1ZfJPADk+pi9OtEjNqkMes6d+51S0yJoX43Do24euAcUSTfaUj8R6SpWdAPtdqeu3qQVH8dABKzXsS+ItIX/S4/3d3br/AMtdvWMk+hJ+tLJJt2+JdGHmRyDF1CvAfsW/4CMn3pJn+zuPEmkqWhkUC9t15O3rwvZ8nnvSyuulXH9sWA87TLs4uYk52noCo7AZOaAB5o9PkXxBpjebpd4RJcoOnJ5l/KgsmgTi/tzu0O8+eRBwsBbkv68nHHalJj0O7NzAVn0W/b94B8wRjxkHoExSR7NBuRaykT6HfEmNj8wjc5LEnsvp2oAP3fh2fyZfm0K84Un7sBPRfX5iT9KFhj0aZtFv1D6Pdjy7djwI8niKiFU0mU6LfHzdMuQTbTNzszzgk98niiGAWwfw1qhY2zqfsdyTg7ewLH+PNABHGLZ38N6wPNtLgEWczH7/AP0zGP7oAojj88N4b1lt8oO+0mb/AJabeQ2P9k4+tJHGbqKTw3rB2zRgG2uB8u5c4UKf7wA5pUjbV4H0fUj5WqWhDRSp8u/HI2nqRwM0AJGj6lE2h6nJs1W2y9tcnrIR0lxQofXYDZXDfZtd0/8AeRP1bb0Enp8wFA83XLbyJysGvWDcN91ZGHcdylDFtct0u7cfZ9csG+ZH+XcRwSw7qecUABMmuwCeH/Rdd087WA5bHdfT5gKGkfV4l1iwj8rV7P5Li3HJI6mL86WR21SKPWtM/dalagrPC38S5+YMB3wDikmmE6x+JNJUiVFH2u2xltuMkFR/HQAsjG8RfEejruuk/d3cC8ebt6pz/dJNEknllfEujDzYZf8Aj7hX+PsWyem0Z+tJM4tZV8R6UC8EqhLy3Tngc4A7Nk8mllkXSLn+1rEedpV4QLmJPm2seAQOwGTmgAeZNMkTXtNPmaTd4e4UdFz1l/KglPD0y3kHzaHefNInRYGbkv6/McfSnEx6Hd+fFtuNEv2+YD5hGx756bKbEF0K4NnORNol8S0Tt8wjY8nJ7L0wKAFHl+HZ/s82G0K94Rj92AtwF9fmJP0pEiTSJm0PUFEmk3f7u3Y8BATxFRAqaVKdEv8AEmm3AzazNzs46Envk8URQCIP4a1VmMDKfslxnBC9huPV6ACKPyHfw3rB822nyLOY/wAXon/AQBzV3Rru5tb4aPqBLTxgvBKesyDgnHYDKiqUcZvIZfDusErcx4NvcL8u5c/KFP8AewOav6LfSTXJsNQRBqVqMeZtx5ieq98DjPvQB0VFFFAHFS3Meowx+ItJdBfQ/u7mENt83b1Qn0Uk/Wia7iPleJdGdHWQEXUGdokHRmPuozVFfg/oqKVXVdaUE5wLhB3/ANyhfg9oiJsXVdaVfQXCYx9NlAF+a9t7CSPxFpkqtp10PMuYs4GO8mP71JJcW+gzjUbWRG0W8AM8IPyw553gdcsTzVH/AIU7ofleX/amteXjGz7Qm36Y2UN8HtFaPyzqutFPQ3CY/wDQO1AF8z2/hy72NIr6HenaFzxAzcBQOuGJP0oVrfRrs6ReNHNo963lwbjxGe0YA/hqg3we0V1CvqussAcgNcIcf+OUrfB/RXIL6rrTYORm4Q4Pt8lAF2FobSZvDuqyrcWdyf8ARZnOSx67MDsoA5ogaGbf4Z1eZZXwZLOdzkuF53exU9PpVJvg/orMGOq60WHQm4QkeuPk4oPwf0UuJDqutFwMAm4TP57OKALsLJqMT6DqVyF1C2Xfa3ecuwA4kz03UiTLrlu9hdzJba3p4zHMrAsqnADA9MsBVP8A4U/ovmb/AO1da39N32hM49M7KB8INGVy41bWgxGCftK5/wDQKALvnpr9t5gmS113TTglWBOOuwHphhjNI9yNYtl1WxeODWLJtk8KsAJCP+WZP92qY+EGiqzMura2pblsXK8nsT8lC/CDRUzt1bW1ySTtuVGT7/JQBdkuo9Qhj8RaQ6C8iPl3UIbHm7TgqSf7pz9aJruEeV4m0aRGjkH+kwZwJB0Ln3UZqinwf0WNdq6rrSrnOBcJj/0CgfB7RFj2DVNZC8/KLhMc9eNlAF6W9t9MkTxDp0qtplyPMuogcADvJjruoee38PXAv7aRDol4AZoQeIS38QHX5i3NUf8AhTuieX5X9qaz5eMbPtCYx6fcoPwe0RkCNqmssoGADcJjHp9ztQBfM1v4duzDI6SaFenaBniBm4CAdcNk/ShTbaRdHRr50n0m9Plw7jwhzxHgfw1Qb4P6K6hX1XWWUdA1whx/45St8H9FcqW1XWmKkEE3CHB9vkoAuwtDbSt4c1aZbi0uTm0mc/e77cDsoxRDJFOH8M6xMssoBks7hzkuF53cdCDVI/B/RWZWbVdaLL90m4Q49cfJxSH4P6KXDnVda3gYDfaEz+ezjNAF6Jk1SGXQtRuVTUbVd9td5y2O0mf72aFlXXLd7K6lS11vTsMkqnJC/wALZ6fMPyql/wAKf0XzPM/tXWt/Td9oTOM+uykHwf0VXLjVdaDEYJ+0Jk/+OUAXjOmv2vnpIlrrmnHaxU8+pTPTDYGaRrkaxbLq+nvHFq9m2yeANgSkf8syT/DVJfg/oqszLqutAscsRcIM/X5KVfhBoqbtura2u45OLleT/wB8UAXZLuK9jj8R6O6/akOy5hzjzQvBBJ/u8/Wia7hTyvE2jSI0Mo/0q3zgSZ4Mh91GeKor8HtFRdqarrSqewuExn1+5QPg9ogj2DVdaCf3RcJj342UAXpr230qRNf0+VW0u5G+5iBwEHeTHXdSvPb+Hrj7ZBIh0S9wZIgeIS38QHXLE81QPwe0QxeUdU1ny8Y2faExj0+5Q3we0Vl2NqmslMdDcIf/AGTtQBf8238O3TW0rrJoV6dqgtxAW+URgdcNyfalja20q5bQ9QaO40q9JS3LnIBP/LP/AHRWe3we0V1CvqutMB0DXCHH/jlK3wf0V2Vm1XWmKnKk3CHafb5KAL0LQ28r+G9WmW4t5zm0mc8t328dNvGKSCSO4D+GtXnV51UvaXLHLOF53+gINUm+D+iswZtV1osucE3CEjPXHycUh+D+ilw51XWt6jAb7QmR7Z2d6AL0Tpq0EmiahcLFqdoN9vdg5bHQSA/3qBMuu272txMltrennKSKwJA6Bs9PmAGapf8ACn9F8wyf2rrW8jG77QmcemdlA+EGihy41bWwxwCftK8gdB9ygC4bhNftvtMUkdrrmmnY+1h9THn0PFD3K6tbLrOmvHHq1odk8AbAlI/5Zkn+GqY+D+iqzFdV1tS3LEXKjd9fkoX4QaKgIXVtbUHk4uV5P/fFAF2W7iu408SaO6eep23UOcebt4IP+7z9aJruCExeJtIlRrecbrq3zgODjMh91AxVFfg9oqKVTVdaCk9BcIB1/wBygfB7RFj8sarrQT+79oTHvxsoAvS3sGjSprtjKraVcjfcxZ4jHUyAddxpXnt/DtwLmKRG0S9/1kYPEJPcDr8xPNUP+FPaIYvLOqaz5eMbftCYx6fcob4PaI6bH1XWWXjg3CHp0/g7UAX/ADLfw9eNaTOkuh3pKoC3EBbgRgeh5NEb22mXLaDqLx3Gl3xKW5dgcE/8s+P4RVBvg/orgB9V1pgOgNwhxzx/B2pW+EGiuVLatrbFTkZuUOPp8nFAF6JoYpH8N6tMtxDL81pO55bvjjpt4x9KbBJFch/Der3CNcIPMtLosCzgciT0BBJqmfg/orMGOq60WXOCbhCRkc/wcUH4QaKXDnVtb3gYDfaUyPx2UAXI2TV4ZNFv7hYdVsgXt7oHLqM4EgPTdSiZdet2t5pktdb0/lXVgSBnjB6fNgZql/wp/RQ5f+1da3kYLfaEz9M7KF+EGiqxYarrYY4yftK846fwUAXTcLr1r9rhZLbXNObZIqtxnqYwfQ8UPcjVbdNc01kj1S1bZcQBsCUgf6sk/wAPeqS/B/REZmXVdaUtyxFwg3H1PyUL8H9FQEJqutLk5OLhOT6/coAuy3cdzHH4k0d085fluYM4Em3hs5/u8/Wia7gtzH4m0iVGtLgbrqHOA+esp77lAxVFfg9oqoVXVdaC+guEA6/7lA+D2iCPyxqushP7v2hMe/GygC/Ld2+izprdjKraTdDfcxZ4jHUyAdSSaHnt/D1x56SI2iXo+dAeISe4HX5s81QPwe0Qx+WdV1ny8AbftCY/9Aob4PaI6hW1XWWXjg3CHp0/g7UAXw9v4fu2sZ3SXRL3CRgtxCTwIwPQjmiN7bTblvD+pOlxpt6dlsztnJPWM+wGMVQb4P6K4AfVdaYD7u64Q4/8c7UN8H9FcqW1XWmKnKk3CHHHb5OKAL0LQxu3hvVplnik+a0mc5LY5/DacflSQOl0j+HNXuFa6iBktLonLOB/y09M81TPwf0VmVjqutFlzgm4QkZ6/wAHFH/Cn9FLh/7V1reBgN9oTI/HZQBcjdNZhfSL6cQ6tYjzLe5BywGcCQHpk0CZdetzFJMlrrmnEYdWBIHsenzYFUh8H9FEhkGra0HIALfaEyQOgzspR8INFVmZdW1sM2MkXK846fwUAXDcrrtr9theO21vTyElRWGCepjz/dNLLcLqkEevaW6R6nakpcQhsecVHMZJ/hGSapL8INFQsV1bW1LHLEXKjcfU/JSJ8H9EQEJqutKCcnFwnJ9fud6AL013FMkXiTRnUyfduYAdolxwc/7oz9aWa8t7VovEukyq1ncjfdRZwGBPMhH94YAqgvwe0RE2LqutBT1UXCYP/jlA+D2iCPyv7U1ny8Y2/aEx+WygC9LdwaJMusWcqto90N9xEDxFnkygdSScUrz2/h25MqyK2h3uAyA8QlhxgdfmyfpVA/B7RDH5Z1XWSnHym4TH/oFDfB7RXUK+qayy+jXCH/2SgC+Ht9Bu2024eOXRL0+XEpPEJPAjA/u4oieDT7hvD+pyJcadeHbau5zk/wDPM47AAVQb4P6K+A+q60wByAbhDz6/c7UN8H9FdlZ9V1piv3SbhDjjt8nFAF+BodzeGtXlWdHy1pO5yWxz7Y2nGPpTYXjvUk8O6tcqbyFS9rdlgWcD/lp6bqpn4P6KXDnVdaLjoxuEyPX+DjNH/CoNF8wSf2tre8DG77SmcfXZQBcSSPWoH0q+nSDWLEb4LgMCwGcCQH1bml85dft2R5Vtdc04j5lbJAz0z0+bAqkPg/owkLjVtbDEAFvtKZOOnOygfB/RVYsNV1oMcZP2hOcdM/JQBcNz/blqL+3ZLbW7AhJo1bgnGTHk/wAJpZLlNSgTX9LdE1K3Pl3MIbHmlR/qyfQZJqkvwf0VCxXVdaUscti4Qbj6n5KRPg9oiAhdV1pQTnAuE6+v3O9AF6W7icR+JNGdCxytzBnaJMcEn/dGaWW9t7RovEmlSq1jcLvuYs4BBPMhHdqoL8HtERCi6rrSoRjaLhMf+gUD4PaII/L/ALU1ny8Y2faExj0xsoAvS3MGhzrq1nKraNdgPcQ54izyZAOuScZpWnt/Dtzu8xW0K+ABUHiAt0AHX5sn6VQPwe0Vk8s6rrRTGNpuEx/6BQ/we0R12tqussvXBuEOD6/coAvq9toV2dLuXjl0a9by4VLDERPAjA/u4pIpILCd/D2qSx3FhdnbayOwOT/zzwOygCqT/CDRZAA+ra22DkZuUOD/AN8UN8INGdlZtW1pivQm5U445x8nFAF6FoWZvDWrypMGBktJ3OS+Oc+204x9KSFkvo38Papcr9tgXfa3ZOXcD/lpnpuqkfg/orOHOq60XHRjcJkD0+5xmj/hT+i79/8Aautb8EbvtCZx6Z2UAXElTW4W0u8nW31nTxvinByyr0Dg9MtS+cmvWxzMlrruncEqwJA/u56YYCqQ+D+ihy41XWg5AG77QmeOn8FA+EGihmYatrYZj8xFyvPHGfkoAuNc/wBtWy6jaMlvrVgQk0atwx6mPJ/hNLLcx6lBH4h0l0W/h/d3MIOPO2j7hJ7AkmqS/B/RUJK6rrQ3HLYuEGTjqfkpF+D2ioCF1XWlBJOBcIOvX+DvQBemu4mEXiXRnR94xdQE7RIBwWPuozSzXtvYvH4j0uVWsLoeZcxZwCCeZMf3qoL8HtEVNi6rrSp/dFwmP/QKP+FPaIIvK/tTWfLxjb9oTGPTGygC9JcW+g3A1O1lVtGvADPDn5Yc87wOuWPWlae38OXODIr6FenbjPEDN0UDr8xJ+lUG+D2itHsOq60V9DcJj/0DtQ/we0V1CvqussAcgG4Q45/3KAL6tbaLdtpN26TaPet5cIY8RntGMfw0kLw2Uz+HtVlS4sbon7JK7A5PXZj0UAc1Sb4P6K+N+q6020gjNwhwf++KG+EGjMys2ra2WU8E3KHHrj5OKALsDQy7/DWryrMSDJaXDnJfbzu9ipxj6UkLx6hE+gandKL+3Xfa3ZYF2A6SZ/vVTPwf0UuHOq60XHRvtCZ/9A4o/wCFQaLv3/2tre/GN32lM49M7KALiyprcD6ddzJba1p43RzKwLBTwHB6fMPyrS0TULfVLlWuIo11WzBhlAOSFPPH1wDWCPhBowcuNW1rcerfaVz/AOgVq+HPh9pvhnVH1C0vb+aV42jZbiRWU5xzgKORjH40AdbRRRQB/9k=']	Multimodal	Competition	False				Theorem proof	Electromagnetism	Physics	English
5	"a) Consider a plane-parallel transparent plate, where the refractive index, $n$, varies with distance, $z$, from the lower surface (see figure). Show that $n_{A} \sin \alpha=n_{B} \sin \beta$. The notation is that of the figure.

<image_1>"	['a) \n\n<img_4495>\n\n\nFrom the figure we get\n\n$n_{A} \\sin \\alpha=n_{1} \\sin \\alpha_{1}=n_{2} \\sin \\alpha_{2}=\\ldots=n_{B} \\sin \\beta$']			['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proof	Optics	Physics	English
6	"b) Assume that you are standing in a large flat desert. At some distance you see what appears to be a water surface. When you approach the ""water"" is seems to move away such that the distance to the ""water"" is always constant. Explain the phenomenon."	['b) The phenomenon is due to total reflexion in a warm layer of air when $\\beta=90^{\\circ}$. This gives\n\n\n\n$$\nn_{A} \\sin \\alpha=n_{B}\n$$']		"

Context question:
a) Consider a plane-parallel transparent plate, where the refractive index, $n$, varies with distance, $z$, from the lower surface (see figure). Show that $n_{A} \sin \alpha=n_{B} \sin \beta$. The notation is that of the figure.

<image_1>
Context answer:
\boxed{证明题}
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proof	Optics	Physics	English
7	"C.3 Is the equation of a thin optical lens

$$
\frac{1}{b}+\frac{1}{c}=\frac{1}{f}
$$

fulfilled for the electrostatic lens? Show it by explicitly calculating $1 / b+1 / c$."	['From the previous answer we obtain:\n\n$$\n\\frac{1}{b}+\\frac{1}{c}=-\\frac{e q \\beta d}{E}\n$$\n\nComparing with the answer of C. 1 we immediately obtain\n\n$$\n\\frac{1}{b}+\\frac{1}{c}=\\frac{1}{f}\n$$\n\ni.e. the equation of a thin optical lens is valid for an electrostatic lens as well.']		"Electrostatic lens

Consider a uniformly charged metallic ring of radius $R$ and total charge $q$. The ring is a hollow toroid of thickness $2 a \ll R$. This thickness can be neglected in parts A, B, C, and E. The $x y$ plane coincides with the plane of the ring, while the $z$-axis is perpendicular to it, as shown in Figure 1. In parts A and B you might need to use the formula (Taylor expansion)

$$
(1+x)^{\varepsilon} \approx 1+\varepsilon x+\frac{1}{2} \varepsilon(\varepsilon-1) x^{2}, \text { when }|x| \ll 1 \text {. }
$$

<image_1>

Figure 1. A charged ring of radius $R$.

Part A. Electrostatic potential on the axis of the ring (1 point)
Context question:
A.1 Calculate the electrostatic potential $\Phi(z)$ along the axis of the ring at a $z$ distance from its center (point A in Figure 1).
Context answer:
\boxed{$\Phi(z)=\frac{q}{4 \pi \varepsilon_{0}} \frac{1}{\sqrt{R^{2}+z^{2}}}$.}


Context question:
A.2 Calculate the electrostatic potential $\Phi(z)$ to the lowest non-zero power of $z$, assuming $z \ll R$.
Context answer:
\boxed{$\Phi(z) \approx \frac{q}{4 \pi \varepsilon_{0} R}(1-\frac{z^{2}}{2 R^{2}})$}


Context question:
A.3 An electron (mass $m$ and charge $-e$ ) is placed at point A (Figure 1, $z \ll R$ ). What is the force acting on the electron? Looking at the expression of the force, determine the sign of $q$ so that the resulting motion would correspond to oscillations. The moving electron does not influence the charge distribution on the ring.
Context answer:
$F(z)=-\frac{q e}{4 \pi \varepsilon_{0} R^{3}} z$, q>0


Context question:
A.4 What is the angular frequency $\omega$ of such harmonic oscillations?
Context answer:
\boxed{$\omega=\sqrt{\frac{q e}{4 \pi m \varepsilon_{0} R^{3}}}$}


Extra Supplementary Reading Materials:

Part B. Electrostatic potential in the plane of the ring

In this part of the problem you will have to analyze the potential $\Phi(r)$ in the plane of the ring $(z=0)$ for $r \ll R$ (point $\mathrm{B}$ in Figure 1). To the lowest non-zero power of $r$ the electrostatic potential is given by $\Phi(r) \approx q\left(\alpha+\beta r^{2}\right)$.
Context question:
B.1 Find the expression for $\beta$. You might need to use the Taylor expansion formula given above.
Context answer:
\boxed{$\beta=\frac{1}{16 \pi \varepsilon_{0} R^{3}}$}


Context question:
B.2 An electron is placed at point $\mathrm{B}$ (Figure $1, r \ll R$ ). What is the force acting on the electron? Looking at the expression of the force, determine the sign of $q$ so that the resulting motion would correspond to harmonic oscillations. The moving electron does not influence the charge distribution on the ring.
Context answer:
$F(r)=+\frac{q e}{8 \pi \varepsilon_{0} R^{3}} r . \quad q<0$.


Extra Supplementary Reading Materials:

Part C. The focal length of the idealized electrostatic lens: instantaneous charging

One wants to build a device to focus electrons-an electrostatic lens. Let us consider the following construction. The ring is situated perpendicularly to the $z$-axis, as shown in Figure 2. We have a source that produces on-demand packets of non-relativistic electrons. Kinetic energy of these electrons is $E=m v^{2} / 2$ ( $v$ is velocity) and they leave the source at precisely controlled moments. The system is programmed so that the ring is charge-neutral most of the time, but its charge becomes $q$ when electrons are closer than a distance $d / 2(d \ll R)$ from the plane of the ring (shaded region in Figure 2, called ""active region""). In part $\mathrm{C}$ assume that charging and de-charging processes are instantaneous and the electric field ""fills the space"" instantaneously as well. One can neglect magnetic fields and assume that the velocity of electrons in the $z$-direction is constant. Moving electrons do not perturb the charge distribution on the ring.

<image_2>

Figure 2. A model of an electrostatic lens.
Context question:
C.1 Determine the focal length $f$ of this lens. Assume that $f \gg d$. Express your answer in terms of the constant $\beta$ from question B. 1 and other known quantities. Assume that before reaching the ""active region"" the electron packet is parallel to the $z$-axis and $r \ll R$. The sign of $q$ is such so that the lens is focusing.
Context answer:
\boxed{$f=-\frac{E}{e q d \beta}$}


Extra Supplementary Reading Materials:

In reality the electron source is placed on the $z$-axis at a distance $b>f$ from the center of the ring. Consider that electrons are no longer parallel to the $z$-axis before reaching the ""active region"", but are emitted from a point source at a range of different angles $\gamma \ll 1$ rad to the $z$-axis. Electrons are focused in a point situated at a distance $c$ from the center of the ring.
Context question:
C.2 Find $c$. Express your answer in terms of the constant $\beta$ from question B.1 and other known quantities.
Context answer:
\boxed{$c=-\frac{1}{\frac{e q \beta d}{E}+\frac{1}{b}}$.}
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proof	Electromagnetism	Physics	English
8	D.2 The resulting potential energy has a sixfold rotational symmetry axis, i.e., the potential distribution is invariant with respect to a rotation by a multiple of $60^{\circ}$ around the origin. Provide a simple argument to prove that this is indeed the case.	"[""Argument: Observe that rotation by $60^{\\circ}$ maps the three vectors $\\vec{b}_{1,2,3}$ into the relabelled triplet of $-\\vec{b}$ 's.""]"		"Particles and Waves

Wave-particle duality, which states that each particle can be described as a wave and vice versa, is one of the central concepts of quantum mechanics. In this problem, we will rely on this notion and just a few other basic assumptions to explore a selection of quantum phenomena covering the two distinct types of particles of the microworld-fermions and bosons.

 Part A. Quantum particle in a box

Consider a particle of mass $m$ moving in a one-dimensional potential well, where its potential energy $V(x)$ is given by

$$
V(x)= \begin{cases}0, & 0 \leq x \leq L \\ \infty, & x<0 \text { or } x>L\end{cases}
\tag{1}
$$

While classical particle can move in such a potential having any kinetic energy, for quantum particle only some specific positive discrete energy levels are allowed. In any such allowed state, the particle can be described as a standing de Broglie wave with nodes at the walls.
Context question:
A.1 Determine the minimal possible energy $E_{\min }$ of the quantum particle in the well. Express your answer in terms of $m, L$, and the Planck's constant $h$.
Context answer:
\boxed{$E_{\min }=\frac{h^{2}}{8 m L^{2}}$}


Extra Supplementary Reading Materials:

The particle's state with minimal possible energy is called the ground state, and all the rest allowed states are called excited states. Let us sort all the possible energy values in the increasing order and denote them as $E_{n}$, starting from $E_{1}$ for the ground state.
Context question:
A.2 Find the general expression for the energy $E_{n}$ (here $n=1,2,3, \ldots$ ).
Context answer:
\boxed{$E_{n}=\frac{h^{2} n^{2}}{8 m L^{2}}$}


Context question:
A.3 Particle can undergo instantaneous transition from one state to another only by emitting or absorbing a photon of the corresponding energy difference. Find the wavelength $\lambda_{21}$ of the photon emitted during the transition of the particle from the first excited state $\left(E_{2}\right)$ to the ground state $\left(E_{1}\right)$.
Context answer:
\boxed{$\lambda_{21}=\frac{8 m c L^{2}}{3 h}$}


Extra Supplementary Reading Materials:

Part C. Bose-Einstein condensation

This part is not directly related to Parts A and B. Here, we will study the collective behaviour of bosonic particles. Bosons do not respect the Pauli exclusion principle, and-at low temperatures or high densitiesexperience a dramatic phenomenon known as the Bose-Einstein condensation (BEC). This is a phase transition to an intriguing collective quantum state: a large number of identical particles 'condense' into a single quantum state and start behaving as a single wave. The transition is typically reached by cooling a fixed number of particles below the critical temperature. In principle, it can also be induced by keeping the temperature fixed and driving the particle density past its critical value.

We begin by exploring the relation between the temperature and the particle density at the transition. As it turns out, estimates of their critical values can be deduced from a simple observation: Bose-Einstein condensation takes place when the de Broglie wavelength corresponding to the mean square speed of the particles is equal to the characteristic distance between the particles in a gas.
Context question:
C.1 Given a non-interacting gas of ${ }^{87} \mathrm{Rb}$ atoms in thermal equilibrium, write the expressions for their typical linear momentum $p$ and the typical de Broglie wavelength $\lambda_{\mathrm{dB}}$ as a function of atom's mass $m$, temperature $T$ and physical constants.
Context answer:
\boxed{$p=\sqrt{3 m k_{\mathrm{B}} T}$ , $\lambda_{\mathrm{dB}}=\frac{h}{\sqrt{3 m k_{\mathrm{B}} T}}$}


Context question:
C.2 Calculate the typical distance between the particles in a gas, $\ell$, as a function of particle density $n$. Hence deduce the critical temperature $T_{c}$ as a function of atom's mass, their density and physical constants.
Context answer:
\boxed{$\ell=n^{-1 / 3}$ , $T_{c}=\frac{h^{2} n^{2 / 3}}{3 m k_{\mathrm{B}}}$}


Extra Supplementary Reading Materials:

To realize BEC in the lab, the experimentalists have to cool gases to temperatures as low as $T_{c}=100 \mathrm{nK}$.
Context question:
C.3 What is the particle density of the Rb gas $n_{c}$ if the transition takes place at such= a temperature? For the sake of comparison, calculate also the 'ordinary' particle density $n_{0}$ of an ideal gas at the standard temperature and pressure (STP), i.e. $T_{0}=300 \mathrm{~K}$ and $p_{0}=10^{5} \mathrm{~Pa}$. How many times is the 'ordinary' gas denser? You may assume that the mass of the atoms is equal to 87 atomic mass units $\left(m_{\mathrm{amu}}\right)$.
Context answer:
Expression: $n_{c}=\frac{\left(3 \cdot 87 m_{\mathrm{amu}} k_{\mathrm{B}} T_{c}\right)^{3 / 2}}{h^{3}}$.

Numerical value: $n_{c} \approx 1.59 \cdot 10^{18} \mathrm{~m}^{-3}$.

Expression: $n_{0}=p /\left(k_{\mathrm{B}} T\right)$.

Numerical value: $n_{0} / n_{c} \approx 1.5 \cdot 10^{7}$.


Extra Supplementary Reading Materials:

Part D. Three-beam optical lattices

The first Bose-Einstein condensates were produced back in 1995, and since then the experimental work has branched out in diverse directions. In this part, you will investigate one particularly fruitful idea to load the condensate into spatially periodic potentials created by interfering a number of coherent laser beams. Due to the periodic nature of the resulting interference patterns, they are referred to as optical lattices. The potential energy $V(\vec{r})$ of an atom moving in an optical lattice is proportional to the local intensity of the light, and in your calculations you may assume that

$$
V(\vec{r})=-\alpha\left\langle|\vec{E}(\vec{r}, t)|^{2}\right\rangle
\tag{2}
$$

Here, $\alpha$ is a positive constant, and the angle brackets indicate time-averaging which eliminates the rapidly oscillating terms. The electric field produced by the $i$-th laser is described by

$$
\vec{E}_{i}=E_{0, i} \vec{\varepsilon}_{i} \cos \left(\vec{k}_{i} \cdot \vec{r}-\omega t\right)
\tag{3}
$$

with the amplitude $E_{0, i}$, the wave vector $\vec{k}_{i}$, and the unit-length polarization vector $\vec{\varepsilon}_{i}$.

<image_1>

(a)

<image_2>

(b)

<image_3>

(c)

Figure 2. (a) Three-beam optical lattice: three plane waves with wave vectors $\vec{k}_{1,2,3}$ intersect and interfere in the area indicated by the grey circle. (b) Symmetries of a regular hexagon: solid and dashed lines show two sets of symmetry axes. (c) Saddle point: a point on a surface where the slopes in orthogonal directions are all zero, but which is not a local extremum of the plotted function. Travelling along the trajectory marked by the full line one encounters an apparent minimum. Additional analysis of the perpendicular direction (dashed line) is needed to distinguish a true minimum from a saddle point (shown).

Your task is to study triangular optical lattices that are produced by interfering three coherent laser beams of equal intensity. A typical setup is shown in Fig. 2a. Here, all three beams are polarized in the $z$ direction, propagate in the $x y$ plane and intersect at equal angles of $120^{\circ}$. Choose the direction of the $x$ axis parallel to the wave vector $\vec{k}_{1}$.
Context question:
D.1 Using Eqs. 2 and 3 obtain the expression for the potential energy $V(\vec{r})$ as a function of $\vec{r}=(x, y)$ in the plane of the beams.

Hint: the result can be neatly expressed as a constant term plus a sum of three cosine functions of arguments $\vec{b}_{i} \cdot \vec{r}$. Please write your result in this form and identify the vectors $\vec{b}_{i}$.
Context answer:
\boxed{$V(\vec{r})=-\alpha E_{0}^{2}\left(\frac{3}{2}+\sum_{j=1}^{3} \cos \vec{b}_{j} \cdot \vec{r}\right)$,$\vec{b}_{1}=\vec{k}_{2}-\vec{k}_{3}, \quad \vec{b}_{2}=\vec{k}_{3}-\vec{k}_{1}, \quad \vec{b}_{3}=\vec{k}_{1}-\vec{k}_{2}$}
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proof	Modern Physics	Physics	English
9	"(i) Show that the two waves observed at an angle $\theta$ to a normal to the slits, have a resultant amplitude A which can be obtained by adding two vectors, each having magnitude $a$, and each with an associated direction determined by the phase of the light wave.

Verify geometrically, from the vector diagram, that

$$
A=2 a \cos \theta
$$

where

$$
\beta=\frac{\pi}{\lambda} d \sin \theta
$$"	['(i) Vector Diagram\n<img_4350>\n\nIf the phase of the light from the first slit is zero, the phase from second slit is\n\n$$\n\\phi=\\frac{2 \\pi}{\\lambda} d \\sin \\theta\n$$\n\nAdding the two waves with phase difference $\\phi$ where $\\xi=2 \\pi\\left(f t-\\frac{x}{\\lambda}\\right)$,\n\n$$\n\\begin{aligned}\n& a \\cos (\\xi+\\phi)+a \\cos (\\xi)=2 a \\cos (\\phi / 2)(\\xi+\\phi / 2) \\\\\n& a \\cos (\\xi+\\phi)+a \\cos (\\xi)=2 a \\cos \\beta\\{\\cos (\\xi+\\beta)\\}\n\\end{aligned}\n$$\n\nThis is a wave of amplitude $A=2 a \\cos \\beta$ and phase $\\beta$. From vector diagram, in isosceles triangle $\\mathrm{OPQ}$,\n\n$$\n\\beta=\\frac{1}{2} \\phi=\\frac{\\pi}{\\lambda} d \\sin \\theta \\quad(N B \\quad \\phi=2 \\beta)\n$$\n\nand\n\n$$\nA=2 a \\cos \\beta\n$$\n\nThus the sum of the two waves can be obtained by the addition of two vectors of amplitude a and angular directions 0 and $\\phi$.']		"Figure 1.1

<image_1>

A plane monochromatic light wave, wavelength $\lambda$ and frequency $f$, is incident normally on two identical narrow slits, separated by a distance $d$, as indicated in Figure 1.1. The light wave emerging at each slit is given, at a distance $x$ in a direction $\theta$ at time $t$, by

$$
y=a \cos [2 \pi(f t-x / \lambda)]
$$

where the amplitude $a$ is the same for both waves. (Assume $x$ is much larger than $d$ )."	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proof	Optics	Physics	English
10	"(ii) The double slit is replaced by a diffraction grating with $N$ equally spaced slits, adjacent slits being separated by a distance $d$. Use the vector method of adding amplitudes to show that the vector amplitudes, each of magnitude $a$, form a part of a regular polygon with vertices on a circle of radius $R$ given by

$$
R=\frac{a}{2 \sin \beta}
$$

Deduce that the resultant amplitude is

$$
\frac{a \sin N \beta}{\sin \beta}
$$



and obtain the resultant phase difference relative to that of the light from the slit at the edge of the grating."	"[""(ii) Each slit in diffraction grating produces a wave of amplitude a with phase $2 \\beta$ relative to previous slit wave. The vector diagram consists of a 'regular' polygon with sides of constant length $a$ and with constant angles between adjacent sides.\n\nLet $\\mathrm{O}$ be the centre of circumscribing circle passing through the vertices of the polygon. Then radial lines such as OS have length $\\mathrm{R}$ and bisect the internal angles of the polygon. Figure 1.2.\n\nFigure 1.2\n\n<img_4428>\n\n\n\n\n\n$$\n\\begin{aligned}\n& O\\hat{ S }T=O\\hat{ T }S=\\frac{1}{2}(180-\\phi) \\\\\n& \\text { and } T \\hat{O }S=\\phi\n\\end{aligned}\n$$\nIn the triangle $T O S$, for example\n$$\n\\begin{gathered}\na=2 R \\sin (\\phi / 2)=2 R \\sin \\beta \\text { as }(\\phi=2 \\beta) \\\\\n\\therefore R=\\frac{a}{2 \\sin \\beta} \\quad \\text { (1) }\n\\end{gathered}\n$$\n\nAs the polygon has $N$ faces then:\n\n$$\nT \\hat{O} Z=N(T \\hat{O} Z)=N \\phi=2 N \\beta\n$$\n\nTherefore in isosceles triangle TOZ, the amplitude of the resultant wave, TZ, is given by\n\n$$\n2 R \\sin N \\beta \\text {. }\n$$\n\nHence form (1) this amplitude is\n\n$$\n\\frac{a \\sin N \\beta}{\\sin \\beta}\n$$\n\nResultant phase is\n\n\n\n$$\n\\begin{aligned}\n& =Z \\hat{T }S \\\\\n& =O \\hat{T} S-O \\hat{T} Z \\\\\n& \\left(90-\\frac{\\phi}{2}\\right)-\\frac{1}{2}(180-N \\phi) \\\\\n& -\\frac{1}{2}(N-1) \\phi \\\\\n& =(N-1) \\beta\n\\end{aligned}\n$$""]"		"Figure 1.1

<image_1>

A plane monochromatic light wave, wavelength $\lambda$ and frequency $f$, is incident normally on two identical narrow slits, separated by a distance $d$, as indicated in Figure 1.1. The light wave emerging at each slit is given, at a distance $x$ in a direction $\theta$ at time $t$, by

$$
y=a \cos [2 \pi(f t-x / \lambda)]
$$

where the amplitude $a$ is the same for both waves. (Assume $x$ is much larger than $d$ ).
Context question:
(i) Show that the two waves observed at an angle $\theta$ to a normal to the slits, have a resultant amplitude A which can be obtained by adding two vectors, each having magnitude $a$, and each with an associated direction determined by the phase of the light wave.

Verify geometrically, from the vector diagram, that

$$
A=2 a \cos \theta
$$

where

$$
\beta=\frac{\pi}{\lambda} d \sin \theta
$$
Context answer:
\boxed{证明题}
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proof	Optics	Physics	English
11	"(v) Show that the number of principal maxima cannot exceed

$$
\left(\frac{2 d}{\lambda}+1\right)
$$"	['(v) Adjacent max. estimate $I_{l}$ :\n\n$$\n\\sin ^{2} N \\beta=1, \\quad \\beta=2 \\pi p \\mp \\frac{3 \\pi}{2 N} \\text { i.e } \\beta= \\pm \\frac{3 \\pi}{2 N}\n$$\n\n$$\n\\begin{gathered}\n{\\left[\\beta=\\pi p \\pm \\frac{\\pi}{2 N}\\right] \\text { does not give a maximum as can be observed from the graph. }} \\\\\n\\qquad I_{1}=a^{2} \\frac{1}{\\frac{3 \\pi^{2}}{2 n}}=\\frac{a^{2} N^{2}}{23} \\text { for } N \\gg>1\n\\end{gathered}\n$$\n\nAdjacent zero intensity occurs for $\\beta=\\pi \\rho \\pm \\frac{\\pi}{N}$ i.e. $\\delta= \\pm \\frac{\\pi}{N}$\n\nFor phase differences much greater than $\\delta, \\quad \\mathrm{I}=\\mathrm{a}^{2}\\left(\\frac{\\sin N \\beta}{\\sin \\beta}\\right)=a^{2}$']		"Figure 1.1

<image_1>

A plane monochromatic light wave, wavelength $\lambda$ and frequency $f$, is incident normally on two identical narrow slits, separated by a distance $d$, as indicated in Figure 1.1. The light wave emerging at each slit is given, at a distance $x$ in a direction $\theta$ at time $t$, by

$$
y=a \cos [2 \pi(f t-x / \lambda)]
$$

where the amplitude $a$ is the same for both waves. (Assume $x$ is much larger than $d$ ).
Context question:
(i) Show that the two waves observed at an angle $\theta$ to a normal to the slits, have a resultant amplitude A which can be obtained by adding two vectors, each having magnitude $a$, and each with an associated direction determined by the phase of the light wave.

Verify geometrically, from the vector diagram, that

$$
A=2 a \cos \theta
$$

where

$$
\beta=\frac{\pi}{\lambda} d \sin \theta
$$
Context answer:
\boxed{证明题}


Context question:
(ii) The double slit is replaced by a diffraction grating with $N$ equally spaced slits, adjacent slits being separated by a distance $d$. Use the vector method of adding amplitudes to show that the vector amplitudes, each of magnitude $a$, form a part of a regular polygon with vertices on a circle of radius $R$ given by

$$
R=\frac{a}{2 \sin \beta}
$$

Deduce that the resultant amplitude is

$$
\frac{a \sin N \beta}{\sin \beta}
$$



and obtain the resultant phase difference relative to that of the light from the slit at the edge of the grating.
Context answer:
\boxed{证明题}


Context question:
(iv) Determine the intensities of the principal intensity maxima.
Context answer:
\boxed{$N^{2} a^{2}$}
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proof	Optics	Physics	English
12	"(ii) Verify that the system has simple harmonic solutions of the form

$$
u_{n}=a_{n} \cos \omega t
$$

with accelerations, $\left(-\omega^{2} u_{n}\right)$ where $a_{n}(n=1,2,3)$ are constant amplitudes, and $\omega$, the angular frequency, can have 3 possible values,

$$
\omega_{o} \sqrt{3}, \omega_{o} \sqrt{3} \text { and } 0 \text {. where } \omega_{o}^{2}=\frac{k}{m} \text {. }
$$"	"[""(ii) Equation of motion of the n'th particle:\n\n$$\n\\begin{aligned}\n& m \\frac{d^{2} u_{n}}{d t^{2}}=k\\left(u_{1+n}-u_{n}\\right)+k\\left(u_{n-1}-u_{n}\\right) \\\\\n& \\frac{d^{2} u_{n}}{d t^{2}}=k\\left(u_{1+n}-u_{n}\\right)+\\omega_{o}^{2}\\left(u_{n-1}-u_{n}\\right)\n\\end{aligned}\n$$\n\nSubstituting $u_{n}(t)=u_{n}(0) \\sin \\left(2 n s \\frac{\\pi}{N}\\right) \\cos \\omega_{s} t$\n\n$$\n\\begin{aligned}\n& -\\omega_{s}^{2}\\left(\\sin \\left(2 n s \\frac{\\pi}{N}\\right)\\right)=\\omega_{o}^{2}\\left[\\sin \\left(2(n+1) s \\frac{\\pi}{N}\\right)-2 \\sin \\left(2 n s \\frac{\\pi}{N}\\right)+\\sin \\left(2(n-1) s \\frac{\\pi}{N}\\right)\\right] \\\\\n& -\\omega_{s}^{2}\\left(\\sin \\left(2 n s \\frac{\\pi}{N}\\right)\\right)=2 \\omega_{o}^{2}\\left[\\frac{1}{2} \\sin \\left(2(n+1) s \\frac{\\pi}{N}\\right)+\\sin \\left(2 n s \\frac{\\pi}{N}\\right)-\\frac{1}{2} \\sin \\left(2(n-1) s \\frac{\\pi}{N}\\right)\\right] \\\\\n& -\\omega_{s}^{2}\\left(\\sin \\left(2 n s \\frac{\\pi}{N}\\right)\\right)=2 \\omega_{o}^{2}\\left[\\sin \\left(2 n s \\frac{\\pi}{N}\\right) \\cos \\left(2 s \\frac{\\pi}{N}\\right)-\\sin \\left(2 n s \\frac{\\pi}{N}\\right)\\right] \\\\\n& \\therefore \\omega_{s}^{2}=2 \\omega_{o}^{2}\\left[1-\\cos \\left(2 s \\frac{\\pi}{N}\\right)\\right]: \\quad(s=1,2, \\ldots . . N)\n\\end{aligned}\n$$\n\nAs $2 \\sin ^{2} \\theta=1-\\cos 2 \\theta$\n\nThis gives\n\n$$\n\\omega_{s}=2 \\omega_{o} \\sin \\left(\\frac{s \\pi}{N}\\right) \\quad(s=1,2, \\ldots N)\n$$\n\n$\\omega_{s}$ can have values from 0 to $2 \\omega_{o}=2 \\sqrt{\\frac{k}{m}}$ when $N \\rightarrow \\infty$; corresponding to range $s=1$ to $\\frac{N}{2}$.""]"		"Three particles, each of mass $m$, are in equilibrium and joined by unstretched massless springs, each with Hooke's Law spring constant $k$. They are constrained to move in a circular path as indicated in Figure 3.1.

Figure 3.1

<image_1>
Context question:
(i) If each mass is displaced from equilibrium by small displacements $u_{1}, u_{2}$ and $u_{3}$ respectively, write down the equation of motion for each mass.
Context answer:
$$
\begin{aligned}
& m \frac{d^{2} u_{1}}{d t^{2}}=k\left(u_{2}-u_{1}\right)+k\left(u_{3}-u_{1}\right) \\
& m \frac{d^{2} u_{2}}{d t^{2}}=k\left(u_{3}-u_{2}\right)+k\left(u_{1}-u_{2}\right) \\
& m \frac{d^{2} u_{3}}{d t^{2}}=k\left(u_{1}-u_{3}\right)+k\left(u_{2}-u_{3}\right)
\end{aligned}
$$
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Pz968g1H4j+Nte+IF7YeAItP1PTdMByMAJcAqqku7uuQH3bShGevzCgD3GkJwOK888PfFnSZkXTfFTx6B4hgwl1aXQMcecAhlc8BWDAgE5HPUDcc/wAQ/EbXNdaTTvhnp6atIExPqR4igZlbCrv2qXGA2SSB0IOeADpvGvxD0TwPYiXUJfOunx5dlCy+cwO7DFSQQnynLdvc8VxVyfHfxX0wG1UeFvDVzBgmVhNPfIzdcDBVdqj5eNwY8srcdJ4T+F2maROmra251zxEzeY+o3eW2kbduxWJA27RhuWHYgYUd7sGO/THWgDkfB/w38NeChv0yzaS8ZSr3l0wklIyeAcALwcHaBkAZzjNddtGe/50oABzS0AIFAOe9LRRQAUVi+JvEtv4W0mXUry1u5reJdzm2jD7BkDJyR6iuf1P4paVpnhu11l9O1OZLiBLkwwwhmgic4RpDu2ruyMDOT6daAO6pGOFJAya47XPiTpGieH7bVzb3t2lxapdiK1h3PHC4G15MkBByByeTnGcHGpeazpl74EuNbubeSfSptMa7lh2gu8Ji3lcZxkrkYz+NAG4GzwRg+9Opq47Dr+tOoA5P4myJD8N9clkhjmRLfc0UhYLIAw+U7SDg9OCD7iuI/Z4YS+F9ZnUeTHJqTbbZA/lxfIpwpYn1xyScBck5r0fxlok3iPwdqukW8qRT3VuyRvJnaG6jOOcZHPB+hrnPhN4I1HwJ4bu9O1Oe2muJrxpwbaR2QIURQMMowcqe3QjmgDv6KKKACkb7p4zS0EZGDQB8ueCPEuleFPjDruoeLxHHcma6iNxbRGSG3nMhLlRguAcMqkc4bB4JI7vStZj+NXidF+zJbeFtCljuXtLiMNLeTsHCbxyqoMNlcnPQ7tw2ei6v4D8La5qDahqeh2dzdsAGmdPmYDpnHXjjnsMdK8uufhJ4y0XxPql54H1yx0rTrs/LB50qkDHQrsYZBLYOcgHIxnFAHbfEDxPdW89t4O0AZ8Sa1CyW7mTYlrFzulZgQwIUOVxnlT1xhrNnFoXwm+HiGcJHa2kamd4U+e6nIAJAJyWZugJwBgZCrx5f4f8SSfCjXtYl8fWF5ea1fyfur+32zPLEo5wzlSEJCkAemCF2iu40iKbXbseOfGM8FloapDc6PY3N2vlWpIwJpflCmQ5UoSW2+YR1AwAYfhT4N2mrS3XifxrELi81XN4dOi8yBLZ5GLkMdwbcM42ngfMDu4I5/wj4/8ACvw68beI9Gt/7SbQp7tUt+RIltIrMkjcMSyHghhliqKCCRmvQtW8Uaz4vaHTPAhMVrLJtu/ELoDDAgAJWIH7787fY8erJqTf8Ij8KfDEl2trDY2qKsZaOMGa6cBiqk9XY/NyTxkngUAbmgeJdK8T6auo6PdC6tGZkEmxkyQeRhgD6dq1Q3SvB/D/AIe+Mdw2papZ3+maB/aV49xLZzQoMsQAWCiN9uemSdxK5OScnf8AAPj+7tdY8QaJ458UaNJdadLGsV15iQLIxDCRASEDbCoH3QQS3UYoA9bopoY5wadQAUUUUAFFFFAAeBXl/wAaJ5LvQ9I8LW7AXGuahFBj0jVgS34NsNeoHpXiHi3xNGnxklv2tLi+g8LaXJMIIVJBmdckscYQBXGSehTAycAgHqKeIvDOliLTTrelW7W4ECwPdxqybflC4JyCMYxW4rh1DKQVIyCDwRXK+ELOx1O0tPFciR3OpalbRyNcsgzGpGREn91VyRjOSRkkmurHWgBaKKKACiiigAooooAKKKRjgEgZ9hQAtFcdpXjmTU/H174V/saWB7OATzXDzqwAIUqAFzydw7+tdjQAUUUUAFFFFABRRRQAUUUh4FAC1heLfFWn+DtBk1fUkmeBHRNkChnZmOMAEge/XoK1bu5a2sp7gW8tw0UTOIYQC8hAztXJAJPQcjrXzdrviP4j/FPwx5MHhqP+yGnEqTWsLAvtJAXc7kMAT/COo7YIoAoeLfE3hz4mfFDw9Nm/tdOcQ2Vz58KBh+8YggiQjad+CT90ZOGxivTj8P5/htfyeI/BEM97GwZb7Sp2Ls0BdWIt9uCZFCsBvLEg9zwdbRdK8HeOtCjjl8M2+l6hZoIpbL7OILnT2bcwAwFKglmdTgA5JwDkCLw/rHiLwnrseg+ONSt762v32aTqccYUyPuC+TIABhiNrDOR94b2OMAFu40XwX8XvDkGoNEkxkRQt1CVW5tmxny2YZwRvOUOVyc4PBrsdL0mx0XTLfTdNtktrO3QJFEmcAfXqTnJJPJJJPWuC1z7X4C8aW2u6eiL4Z1WYR63GzBY7a4Zwq3WWb5S29QxAwQmWySCPSA2SR6cGgBVUL0paKKACiiigAoPSig9KAPLPjjeTy+F9O8OWYL3mt38dukeeWVWB/8AQvL/ADrL+K9uND+H+m+FrAebf61fRQvt+9MVwTj2BEagdhgdq7zU/CH9rePdH8ST337jS4ZEhsvJzmRwcvu3em3jH8I5pNW8HDWPHmjeI7i+zBpUbrDZGLO6Vs/Pu3cY+XjHVetAHF/EuzOjfD2x8N2hT+0tevILRyP4sY4H+yu1EA64x3r1K30uzi0eLSTbxyWUcAtvJlUMrRhdu0g8EEcYrmPF/gi78TeI/D+qW+sGxGktI4UQCQktjlc8A8dSDiuts7WKzto7eEMI4xtG5yzH3JPJJ5JJ5JoAsYooooACM01UC9M4xjFOooAKKKKACiiigAPIpu0DOOKdRQBjeJ/DOl+LNDm0rVYPNgk+ZGBw8T9nQ9mGfyJBBBIPzj44+G+qfD7SrIS61fal4YuLpTexW2YAj8c7CzLuZQQGIIBAz2z9TkZGKq3unWepWj2l/aw3dtJjfDcRrIj4ORlSMHkA/hQB5A37QvhO0soYNN0LU/LiCxpCY4oo40AwAuHOMAAAYHH0xXV6P4R1DVfEUXirxdcQXVzHEv8AZ2nRxEQWOdrFgH5MuQPmIBGP90I7xr8KtC8VaBDp9rBa6RPbupt7m2tI8ouWJjwMHYS7HaCPmOeeQfJfE9z488H3+i6Z4t8V3y6feyxr9u027IKQK/74H92JDIA6EMe3HzdFAPVvFPjeaTxBN4H8O2U91rk9qS9yjhIrDeMCR25Pyhg2Mc5UDJbjm7P4MeB/COgXWoeKbmTUEhQPLczyPBGgBOAiI2cnKjBZiSBjGcVs6R41+Hvh+0GieE3W5uGV5oLHT7eaZp5AvTdtILEJyWPAGSQOaNK8Kav4yuX1bx9Cos2Jax8PBsxWo6K8jKRvkxu6g43Hp91QDk/h/wDGTwjoGiW3h+5/tWG3tpp1hvJ4lkAhLu0e/Yd27aVXCqQCPTp7lb3EV1BFPBKksMqCRJI23K6kZBBHBBHevL/HPizQNO0O98D+GrC3vNanVraDSrO0DRwl8szkAbBt5Yjkg4JGMkHhrx9N4V8L6ZY+LPDGsaRBZW8dsb4Q+fbhURVVnZeULNwBgjJHPXAB6rRSBs9qWgAooooARvumvN/AXhHUY7HxTdeJbT7Pf+ILuQyxGRJCsJUhRlSR/G9ekkZGKQrnufzoA83+F2m+KtH0SDQdXsltLbTZ5VS4MiubmMklQoGdoBOdx5xgAcnb6TSY5paACiiigAooooAKKKKACmuypGzsQFUZJJwAKdXGfFTXf7B+HGsXCtiaaH7NDjrukO3j3AJP/AaAOO+H2sR22m+LviJfW880V9fOV2bA6W8YO3G9lz124BySBjNem6FrkuuW4uG0bUtPiZdy/bljRj7bQ5Zf+BAV5V4R/tPw54q8IeE9bggj099Lea2t1JIS7yzsznu4w2B0XeMcjJ9pXGcDoP8AP+frQA+iiigAooooAKKKKAA8Cmsx7Clb7p5x7+leUfHbVGt/BYgstdhsNQjnjne1F55U80B3RkKoOWG5gT2wjelACa3f+PNV+L8/hnS9bg0TTI7BbyKU2kc7SRjClsMD8xkJXGVAVc4/vV/Bg1n4TQDRfFLwS+Hbm4H2TVoZiUtZXziORWwUQ7d2QNqs5yTuJXmI/gprN3oNh4i0nxi9zqzW6XFsxWSMFWUMFSQtvXguQSozkZC8mvXNM8TWOr3o8Oa3afY9d+ziafT7mPcjgYy8b8pImenOeDkAqcAEHijwHZa5qkGv2Ez6d4jtNptr6N2w23OEkToyHcQehxxnHFc7p2s6N8Z/C9/4d1WF9M1q2YG4tGAMtu6NjzI9w6ZyrcAjcVPUExeI4fEXw31m68QeHdLt7zwo8AN5pEDeULZ1GDKigEIOhYqOQWLDgOK0/hd/iX4p8P6pqfgqTStJht55NQN3IIZriVhgIAhEhCuAwZtuQWOBwGAK+i+BvGni3wnJpviTxuz6YLuS1mtYrVJpJlgnKk+ewDgl42wSDxgnOSo9pH3qitbK1sbWO1s7eK3t4xiOKFAiIPQKOBUwAHSgBaKKKACiiigAooooAQKBQFA9aWigBNoNGKWigAooooAKKKKACiiigAooooAKKKKACiiigAPIrK1/w5pHifS207WrJLy1Lq+x2KkMOhVlIKnr0I4JHQmtWigD5+m8A+J/h14+vNf8HeHbfVNKEJaBZ5BJJApwXVASH3jDKpAYlXx8zEitvwV468T/ABUm1KwjNvomm28CC4ubMM11lycLG7NtQlVf5tpxjjkgj2TaBk96+a9Q8HfEL4UWWpeIdN8QW7WQfErqd7uruoV2jkUqGJK5wSRzyRzQB7lY6VoPgTQLq64giijMt9fzfPPcFcsXlfG52JZvxbCgcCvLfEVx4i+NsF5ZeFZ7ey8NWU4jkmu3eP8AtCQAHOFQnavBCnH3lY84C63gPwW/ivTrPxX4y1t/ERukE9vYu+62tmyQTsB2s2AAV2gKdwIY4I6O5+J2gx3f9j+GrefxBqSqBHa6Um6JBgBS8v3FjyyguCdueRxQB3QJzzmnA59K8W0j4S+L11xvFdz41Gn+ILne8/lWYuEUN/BlmAZQu3AIwMDH3Qa7HwHf+KDq/iHRvEl7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']	Multimodal	Competition	False				Theorem proof	Mechanics	Physics	English
13	a) Show that the radial deviation of the electrons from the injection radius is finite.	['3. Finiteness of $r_{1}$ and direction of the drift velocity\n\nIn the magnetic field the lines of force are circles with their centres on the symmetry axis (z-axis) of the toroid.\n\nIn accordance with the symmetry of the problem, polar coordinates $r$ and $\\varphi$ are introduced into the plane perpendicular to the z-axis (see figure below) and the occurring vector quantities (velocity, magnetic field $\\overrightarrow{\\mathrm{B}}$, Lorentz force) are divided into the corresponding components.\n\n<img_4485>\n\nSince the angle of aperture of the beam can be neglected examine a single electron injected tangentially into the toroid with velocity $\\mathrm{v}_{0}$ on radius $\\mathrm{R}$.\n\nIn a static magnetic field the kinetic energy is conserved, thus\n\n$$\n\\mathrm{E}=\\frac{\\mathrm{m}}{2}\\left(\\mathrm{v}_{\\mathrm{r}}^{2}+\\mathrm{v}_{\\varphi}^{2}+\\mathrm{v}_{\\mathrm{z}}^{2}\\right)=\\frac{\\mathrm{m}}{2} \\mathrm{v}_{0}^{2}\n\\tag{12}\n$$\n\nThe radial points of inversion of the electron are defined by the condition\n\n$$\n\\mathrm{v}_{\\mathrm{r}}=0\n$$\n\nUsing eq. (12) one obtains\n\n$$\n\\mathrm{v}_{0}^{2}=\\mathrm{v}_{\\varphi}^{2}+\\mathrm{v}_{\\mathrm{z}}^{2}\n\\tag{13}\n$$\n\nSuch an inversion point is obviously given by\n\n$$\n\\mathrm{r}=\\mathrm{R} \\cdot\\left(\\mathrm{v}_{\\varphi}=\\mathrm{v}_{0}, \\mathrm{v}_{\\mathrm{r}}=0, \\mathrm{v}_{\\mathrm{z}}=0\\right)\n$$\n\nTo find further inversion points and thus the maximum radial deviation of the electron the components of velocity $\\mathrm{v}_{\\varphi}$ and $\\mathrm{v}_{\\mathrm{Z}}$ in eq. (13) have to be expressed by the radius.\n\n$\\mathrm{V}_{\\varphi}$ will be determined by the law of conservation of angular momentum. The Lorentz force obviously has no component in the $\\varphi$ - direction (parallel to the magnetic field). Therefore it cannot produce a torque around the z-axis. From this follows that the $\\mathrm{z}$-component of the angular momentum is a constant, i.e. $\\mathrm{L}_{\\mathrm{z}}=\\mathrm{m} \\cdot \\mathrm{v}_{\\varphi} \\cdot \\mathrm{r}=\\mathrm{m} \\cdot \\mathrm{v}_{0} \\cdot \\mathrm{R}$ and therefore \n$$\n\\mathrm{v}_{\\varphi}=\\mathrm{v}_{0} \\cdot \\frac{\\mathrm{R}}{\\mathrm{r}}\n\\tag{14}\n$$\n\n$\\mathrm{v}_{\\mathrm{z}}$ will be determined from the equation of motion in the z-direction. The z-component of the Lorentz force is $\\mathrm{F}_{\\mathrm{z}}=-\\mathrm{e} \\cdot \\mathrm{B} \\cdot \\mathrm{v}_{\\mathrm{r}}$. Thus the acceleration in the $\\mathrm{z}$-direction is\n\n$$\n\\mathrm{a}_{\\mathrm{z}}=-\\frac{\\mathrm{e}}{\\mathrm{m}} \\cdot \\mathrm{B} \\cdot \\mathrm{v}_{\\mathrm{r}} .\n\\tag{15}\n$$\n\nThat means, since B is assumed to be constant, a change of $v_{z}$ is related to a change of $r$ as follows:\n\n$$\n\\Delta \\mathrm{v}_{\\mathrm{z}}=-\\frac{\\mathrm{e}}{\\mathrm{m}} \\cdot \\mathrm{B} \\cdot \\Delta \\mathrm{r}\n$$\n\nBecause of $\\Delta \\mathrm{r}=\\mathrm{r}-\\mathrm{R}$ and $\\Delta \\mathrm{v}_{\\mathrm{z}}=\\mathrm{v}_{\\mathrm{z}}$ one finds\n\n$$\n\\mathrm{v}_{\\mathrm{z}}=-\\frac{\\mathrm{e}}{\\mathrm{m}} \\cdot \\mathrm{B} \\cdot(\\mathrm{r}-\\mathrm{R})\n\\tag{16}\n$$\n\nUsing eq. (14) and eq. (15) one obtains for eq. (13)\n\n$$\n1=\\left(\\frac{\\mathrm{R}}{\\mathrm{r}}\\right)^{2}+\\mathrm{A}^{2} \\cdot\\left(\\frac{\\mathrm{r}}{\\mathrm{R}}-1\\right)^{2}\n\\tag{17}\n$$\n\nwhere $\\mathrm{A}=\\frac{\\mathrm{e}}{\\mathrm{m}} \\cdot \\mathrm{B} \\cdot \\frac{\\mathrm{R}}{\\mathrm{v}_{0}}$\n\nDiscussion of the curve of the right side of eq. (17) gives the qualitative result shown in the following diagram:\n\n<img_4532>\n\nHence $\\mathrm{r}_{1}$ is finite. Since $\\mathrm{R} \\leq \\mathrm{r} \\leq \\mathrm{r}_{1}$ eq. (16) yields $\\mathrm{v}_{\\mathrm{z}}<0$. Hence the drift is in the direction of the negative $z$-axis.']		"A beam of electrons emitted by a point source $\mathrm{P}$ enters the magnetic field $\vec{B}$ of a toroidal coil (toroid) in the direction of the lines of force. The angle of the aperture of the beam $2 \cdot \alpha_{0}$ is assumed to be small $\left(2 \cdot \alpha_{0}<<1\right)$. The injection of the electrons occurs on the mean radius $\mathrm{R}$ of the toroid with acceleration voltage $\mathrm{V}_{0}$.

Neglect any interaction between the electrons. The magnitude of $\vec{B}, B$, is assumed to be constant.

<image_1>


Data:

$\frac{\mathrm{e}}{\mathrm{m}}=1.76 \cdot 10^{11} \mathrm{C} \cdot \mathrm{kg}^{-1} ; \quad \mathrm{V}_{0}=3 \mathrm{kV} ; \quad \mathrm{R}=50 \mathrm{~mm}$
Context question:
1. To guide the electron in the toroidal field a homogeneous magnetic deflection field $\vec{B}_{1}$ is required. Calculate $\vec{B}_{1}$ for an electron moving on a circular orbit of radius $R$ in the torus.
Context answer:
\boxed{$1.48 \cdot 10^{-2}$}


Context question:
2. Determine the value of $\vec{B}$ which gives four focussing points separated by $\pi / 2$ as indicated in the diagram.

Note: When considering the electron paths you may disregard the curvature of the magnetic field.
Context answer:
\boxed{$0.37 \cdot 10^{-2}$}


Extra Supplementary Reading Materials:

3. The electron beam cannot stay in the toroid without a deflection field $\vec{B}_{1}$, but will leave it with a systematic motion (drift) perpendicular to the plane of the toroid.

Note: The angle of aperture of the electron beam can be neglected. Use the laws of conservation of energy and of angular momentum."	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proof	Electromagnetism	Physics	English
14	"A.6 For the purpose of this task only, we assume a weak thermal interaction between the tube and the gas. As a result, the standing sound wave remains almost unchanged, but the gas can exchange a small amount of heat with the tube. The heating due to viscosity can be neglected.

For each of the points in Figure 2 (A, C at the edges of the tube, B at the center) state whether the temperature of the tube at that point will increase, decrease or remain the same over a long time.

<image_2>

Figure 2"	['The movement of the gas parcels inside the tube conveys heat along its boundary. To determine the direction of the convection, we combining the result of Task A. 5 and the expression (1) for $u(x, t)$. We see that when $0<x<\\frac{L}{2}$, the gas is colder when the displacement $u(x, t)$ is positive. Likewise, when $\\frac{L}{2}<x<L$, the gas is colder when the displacement $u(x, t)$ is negative. Hence, heat flows into the gas near the point $B$, cooling it down, and out of the gas near the points $A$ and $C$, heating them up.']		"Thermoacoustic Engine

A thermoacoustic engine is a device that converts heat into acoustic power, or sound waves - a form of mechanical work. Like many other heat machines, it can be operated in reverse to become a refrigerator, using sound to pump heat from a cold to a hot reservoir. The high operating frequencies reduce heat conduction and eliminate the need for any working chamber confinement. Unlike many other engine types, the thermoacoustic engine has no moving parts except the working fluid itself.

The efficiencies of thermoacoustic machines are typically lower than other engine types, but they have advantages in set up and maintenance costs. This creates opportunities for renewable energy applications, such as solar-thermal power plants and utilization of waste heat. Our analysis will focus on the creation of acoustic energy within the system, ignoring the extraction or conversion for powering external devices.

Part A: Sound wave in a closed tube

Consider a thermally insulating tube of length $L$ and cross-sectional area $S$, whose axis lies along the $x$ direction. The two ends of the tube are located at $x=0$ and $x=L$. The tube is filled with an ideal gas and is sealed on both ends. At equilibrium, the gas has temperature $T_{0}$, pressure $p_{0}$ and mass density $\rho_{0}$. Assume that viscosity can be ignored and that the gas motion is only in the $x$ direction. The gas properties are uniform in the perpendicular $y$ and $z$ directions.

<image_1>

Figure 1
Context question:
A.1 When a standing sound wave forms, the gas elements oscillate in the $x$ direction with angular frequency $\omega$. The amplitude of the oscillations depends on each element's equilibrium position $x$ along the tube. The longitudinal displacement of each gas element from its equilibrium position $x$ is given by

$$
u(x, t)=a \sin (k x) \cos (\omega t)=u_{1}(x) \cos (\omega t)
\tag{1}
$$

(please note the $u$ here describes the displacement of a gas element)

where $a \ll L$ is a positive constant, $k=2 \pi / \lambda$ is the wavenumber and $\lambda$ is the wavelength. What is the maximum possible wavelength $\lambda_{\max }$ in this system?
Context answer:
\boxed{$\lambda_{\max }=2 L$}


Extra Supplementary Reading Materials:

We will assume throughout the question an oscillation mode of $\lambda=\lambda_{\max }$.

Now, consider a narrow parcel of gas, located at rest between $x$ and $x+\Delta x(\Delta x \ll L)$. As a result of the displacement wave of Task A.1, the parcel oscillates along the $x$ axis and undergoes a change in volume and other thermodynamic properties.

Throughout the following tasks assume all these changes to the thermodynamic properties to be small compared to the unperturbed values.
Context question:
A.2 The parcel volume $V(x, t)$ oscillates around the equilibrium value of $V_{0}=S \Delta x$ and has the form

$$
V(x, t)=V_{0}+V_{1}(x) \cos (\omega t) .
\tag{2}
$$

Obtain an expression for $V_{1}(x)$ in terms of $V_{0}, a, k$ and $x$.
Context answer:
\boxed{$V_{1}(x)=a k V_{0} \cos (k x)$}


Context question:
A.3 Assume that the total pressure of the gas, as a result of the sound wave, takes the approximate form

$$
p(x, t)=p_{0}-p_{1}(x) \cos (\omega t) .
\tag{3}
$$

Considering the forces acting on the parcel of gas, compute the amplitude $p_{1}(x)$ of the pressure oscillation to leading order, in terms of the position $x$, the equilibrium density $\rho_{0}$, the displacement amplitude $a$ and the wave parameters $k$ and $\omega$.
Context answer:
\boxed{$p_{1}(x)=a \frac{\omega^{2}}{k} \rho_{0} \cos (k x)$}


Extra Supplementary Reading Materials:

At acoustic frequencies, the thermal conductivity of the gas can be neglected. We will treat the expansion and contraction of gas parcels as purely adiabatic, satisfying the relation $p V^{\gamma}=$ const, where $\gamma$ is the adiabatic constant.
Context question:
A.4 Use the relation above and the results of the previous tasks to obtain an expression for the speed of sound waves $c=\omega / k$ in the tube, to first order. Express your answer in terms of $p_{0}, \rho_{0}$ and the adiabatic constant $\gamma$.
Context answer:
\boxed{$c=\sqrt{\frac{\gamma p_{0}}{\rho_{0}}}$}


Context question:
A.5 The change in the gas temperature due to the adiabatic expansion and contraction, as a result of the sound wave, takes the form:

$$
T(x, t)=T_{0}-T_{1}(x) \cos (\omega t)
\tag{4}
$$

Compute the amplitude $T_{1}(x)$ of the temperature oscillations in terms of $T_{0}, \gamma$, $a, k$ and $x$.
Context answer:
\boxed{$a k(\gamma-1) T_{0} \cos (k x)$}
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proof	Optics	Physics	English
15	(k)  Prove that the actual thermodynamic efficiency $\varepsilon$ for the winter Hadley circulation is always smaller than $\varepsilon_{i}$, showing all mathematical steps.	['$$\n\\begin{aligned}\n\\varepsilon & =\\frac{W_{\\text {net }}}{Q_{\\text {loss }}+W_{\\text {net }}} \\\\\n\\frac{1}{\\varepsilon}-1 & =\\frac{Q_{\\text {loss }}}{W_{\\text {net }}}=\\frac{R T_{C} \\ln \\left(\\frac{p_{D}}{p_{C}}\\right)}{R\\left(T_{H}-T_{C}\\right) \\ln \\left(\\frac{p_{E}}{p_{A}}\\right)} \\\\\n& =\\frac{T_{C} \\ln \\left(\\frac{p_{D}}{p_{B}} \\times \\frac{p_{B}}{p_{C}}\\right)}{\\left(T_{H}-T_{C}\\right) \\ln \\left(\\frac{p_{E}}{p_{A}}\\right)} \\\\\n& >\\frac{T_{C} \\ln \\left(\\frac{p_{D}}{p_{B}}\\right)}{\\left(T_{H}-T_{C}\\right) \\ln \\left(\\frac{p_{E}}{p_{A}}\\right)} \\quad \\text { as } \\frac{p_{B}}{p_{C}}>1 \\\\\n& =\\frac{T_{C}}{T_{H}-T_{C}} \\quad \\text { using equation (*) in part (h) } \\\\\n\\frac{1}{\\varepsilon} & >1+\\frac{T_{C}}{T_{H}-T_{C}}=\\frac{T_{H}}{T_{H}-T_{C}} \\\\\n\\varepsilon & <\\frac{T_{H}-T_{C}}{T_{H}}=\\varepsilon_{i}\n\\end{aligned}\n$$']		"The schematic below shows the Hadley circulation in the Earth's tropical atmosphere around the spring equinox. Air rises from the equator and moves poleward in both hemispheres before descending in the subtropics at latitudes $\pm \varphi_{d}$ (where positive and negative latitudes refer to the northern and southern hemisphere respectively). The angular momentum about the Earth's spin axis is conserved for the upper branches of the circulation (enclosed by the dashed oval). Note that the schematic is not drawn to scale.

<image_1>
Context question:
(a)  Assume that there is no wind velocity in the east-west direction around the point $\mathrm{X}$. What is the expression for the east-west wind velocity $u_{Y}$ at the points Y? Convention: positive velocities point from west to east.

(The angular velocity of the Earth about its spin axis is $\Omega$, the radius of the Earth is $a$, and the thickness of the atmosphere is much smaller than $a$.)
Context answer:
\boxed{$u_{Y}=\Omega a\left(\frac{1}{\cos \varphi_{d}}-\cos \varphi_{d}\right)$}


Context question:
(b)  Which of the following explains ultimately why angular momentum is not conserved along the lower branches of the Hadley circulation?

Tick the correct answer(s). There can be more than one correct answer.

(I) There is friction from the Earth's surface.

(II) There is turbulence in the lower atmosphere, where different layers of air are mixed

(III) The air is denser lower down and so inertia slows down the motion around the spin axis of the Earth.

(IV) The air is moist at the lower levels causing retardation to the wind velocity.
Context answer:
(I),(II)


Extra Supplementary Reading Materials:

Around the northern winter solstice, the rising branch of the Hadley circulation is located at the latitude $\varphi_{r}$ and the descending branches are located at $\varphi_{n}$ and $\varphi_{s}$ as shown in the schematic below. Refer to this diagram for parts (c), (d) and (e).

<image_2>
Context question:
(c)  Assume that there is no east-west wind velocity around the point $\mathrm{Z}$. Given that $\varphi_{r}=-8^{\circ}, \varphi_{n}=28^{\circ}$ and $\varphi_{s}=-20^{\circ}$, what are the east-west wind velocities $u_{P}, u_{Q}$ and $u_{R}$ respectively at the points $\mathrm{P}, \mathrm{Q}$ and $\mathrm{R}$ ?

(The radius of the Earth is a $=6370 \mathrm{~km}$.)

Hence, which hemisphere below has a stronger atmospheric jet stream?

(I) Winter Hemisphere

(II) Summer Hemisphere

(III) Both hemispheres have equally strong jet streams.
Context answer:
\boxed{(I)}


Context question:
(d)  The near-surface branch of the Hadley circulation blows southward across the equator. Mark by arrows on the figure below the direction of the eastwest component of the Coriolis force acting on the tropical air mass

(A) north of the equator;

(B) south of the equator.

<image_3>
Context answer:
<image_4>


Context question:
(e)  From your answer to part (d) and the fact that surface friction nearly balances the Coriolis forces in the east-west direction, sketch the near-surface wind pattern in the tropics near the equator during northern winter solstice.
Context answer:
<image_5>


Extra Supplementary Reading Materials:

Suppose the Hadley circulation can be simplified as a heat engine shown in the schematic below. Focusing on the Hadley circulation reaching into the winter hemisphere as shown below, the physical transformation of the air mass from A to B and from $\mathrm{D}$ to $\mathrm{E}$ are adiabatic, while that from $\mathrm{B}$ to $\mathrm{C}, \mathrm{C}$ to $\mathrm{D}$ and from $\mathrm{E}$ to $\mathrm{A}$ are isothermal. Air gains heat by contact with the Earth's surface and by condensation of water from the atmosphere, while air loses heat by radiation into space.

<image_6>
Context question:
(f)  Given that atmospheric pressure at a vertical level owes its origin to the weight of the air above that level, order the pressures $p_{A}, p_{B}, p_{C}, p_{D}, p_{E}$, respectively at the points $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}$ by a series of inequalities.

(Given that $p_{A}=1000 \mathrm{hPa}$ and $p_{D}=225 \mathrm{hPa}$. Note that $1 \mathrm{hPa}$ is $100 \mathrm{~Pa}$.)
Context answer:
$p_{E}>p_{A}>p_{D}>p_{B}>p_{C}$


Context question:
(g)  Let the temperature next to the surface and at the top of the atmosphere be $T_{H}$ and $T_{C}$ respectively. Given that the pressure difference between points $\mathrm{A}$ and $\mathrm{E}$ is $20 \mathrm{hPa}$, calculate $T_{C}$ for $T_{H}=300 \mathrm{~K}$.

Note that the ratio of molar gas constant $(R)$ to molar heat capacity at constant pressure $\left(c_{p}\right)$ for air, $\kappa$, is $2 / 7$.
Context answer:
\boxed{195}


Context question:
(h)  Calculate the pressure $p_{B}$.
Context answer:
\boxed{220}


Context question:
(i) For an air mass moving once around the winter Hadley circulation, using the molar gas constant, $R$, and the quantities defined above, obtain expressions for

(A)  the net work done per unit mole $W_{\text {net }}$ ignoring surface friction;
Context answer:
\boxed{$R\left(T_{H}-T_{C}\right) \ln \left(\frac{p_{E}}{p_{A}}\right)$}


Context question:
(B)  the heat loss per unit mole $Q_{\text {loss }}$ at the top of the atmosphere.
Context answer:
\boxed{$R T_{C} \ln \left(\frac{p_{D}}{p_{C}}\right)$}


Context question:
(j)  What is the value of the ideal thermodynamic efficiency $\varepsilon_{i}$ for the winter Hadley circulation?
Context answer:
\boxed{0.35}
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proof	Thermodynamics	Physics	English
16	2.  Now consider the case in part A. Plot the time ct versus the position $x$ of the particle. Draw the $x^{\prime}$ axis and $c t^{\prime}$ axis when $\frac{g t}{c}=1$ in the same graph using length scale $x\left(c^{2} / g\right)$ and $c t\left(c^{2} / g\right)$.	['The position of the particle is given by eq.(6).\n\n\n<img_4553>\n\nFigure 2: Minkowski Diagram']		"Global Positioning System (GPS) is a navigation technology which uses signal from satellites to determine the position of an object (for example an airplane). However, due to the satellites high speed movement in orbit, there should be a special relativistic correction, and due to their high altitude, there should be a general relativistic correction. Both corrections seem to be small but are very important for precise measurement of position. We will explore both corrections in this problem.

First we will investigate the special relativistic effect on an accelerated particle. We consider two types of frame, the first one is the rest frame (called $S$ or Earth's frame), where the particle is at rest initially. The other is the proper frame (called $S^{\prime}$ ), a frame that instantaneously moves together with the accelerated particle. Note that this is not an accelerated frame, it is a constant velocity frame that at a particular moment has the same velocity with the accelerated particle. At that short moment, the time rate experienced by the particle is the same as the proper frame's time rate. Of course this proper frame is only good for an infinitesimally short time, and then we need to define a new proper frame afterward. At the beginning we synchronize the particle's clock with the clock in the rest frame by setting them to zero, $t=\tau=0$ ( $t$ is the time in the rest frame, and $\tau$ is the time shown by particle's clock).

By applying equivalence principle, we can obtain general relativistic effects from special relavistic results which does not involve complicated metric tensor calculations. By combining the special and general relativistic effects, we can calculate the corrections needed for a GPS (global positioning system) satellite to provide accurate positioning.

Some mathematics formulas that might be useful

- $\sinh x=\frac{e^{x}-e^{-x}}{2}$
- $\cosh x=\frac{e^{x}+e^{-x}}{2}$
- $\tanh x=\frac{\sinh x}{\cosh x}$
- $1+\sinh ^{2} x=\cosh ^{2} x$
- $\sinh (x-y)=\sinh x \cosh y-\cosh x \sinh y$



- $\int \frac{d x}{\left(1-x^{2}\right)^{\frac{3}{2}}}=\frac{x}{\sqrt{1-x^{2}}}+C$
- $\int \frac{d x}{1-x^{2}}=\ln \sqrt{\frac{1+x}{1-x}}+C$


Part A. Single Accelerated Particle 

Consider a particle with a rest mass $m$ under a constant and uniform force field $F$ (defined in the rest frame) pointing in the positive $x$ direction. Initially $(t=\tau=0)$ the particle is at rest at the origin $(x=0)$.
Context question:
1.  When the velocity of the particle is $v$, calculate the acceleration of the particle, $a$ (with respect to the rest frame).
Context answer:
\boxed{$a=\frac{F}{\gamma^{3} m}$}


Context question:
2.  Calculate the velocity of the particle $\beta(t)=\frac{v(t)}{c}$ at time $t$ (in rest frame), in terms of $F, m, t$ and $c$.
Context answer:
\boxed{$\beta=\frac{\frac{F t}{m c}}{\sqrt{1+\left(\frac{F t}{m c}\right)^{2}}}$}


Context question:
3.  Calculate the position of the particle $x(t)$ at time $t$, in term of $F, m, t$ and $c$.
Context answer:
\boxed{$x=\frac{m c^{2}}{F}\left(\sqrt{1+\left(\frac{F t}{m c}\right)^{2}}-1\right)$}


Context question:
4.  Show that the proper acceleration of the particle, $a^{\prime} \equiv g=F / m$, is a constant. The proper acceleration is the acceleration of the particle measured in the instantaneous proper frame.
Context answer:
\boxed{证明题}


Context question:
5.  Calculate the velocity of the particle $\beta(\tau)$, when the time as experienced by the particle is $\tau$. Express the answer in $g, \tau$, and $c$.
Context answer:
\boxed{$\beta=\tanh \frac{g \tau}{c}$}


Context question:
6. ( $\mathbf{0 . 4} \mathbf{~ p t s )}$ Also calculate the time $t$ in the rest frame in terms of $g, \tau$, and $c$.
Context answer:
\boxed{$t=\frac{c}{g} \sinh \frac{g \tau}{c}$}


Extra Supplementary Reading Materials:

Part B. Flight Time 

The first part has not taken into account the flight time of the information to arrive to the observer. This part is the only part in the whole problem where the flight time is considered. The particle moves as in part A.
Context question:
1.  At a certain moment, the time experienced by the particle is $\tau$. What reading $t_{0}$ on a stationary clock located at $x=0$ will be observed by the particle? After a long period of time, does the observed reading $t_{0}$ approach a certain value? If so, what is the value?
Context answer:
\boxed{证明题}


Context question:
2.  Now consider the opposite point of view. If an observer at the initial point $(x=0)$ is observing the particle's clock when the observer's time is $t$, what is the reading of the particle's clock $\tau_{0}$ ? After a long period of time, will this reading approach a certain value? If so, what is the value?
Context answer:
\boxed{证明题}


Extra Supplementary Reading Materials:

Part C. Minkowski Diagram

In many occasion, it is very useful to illustrate relativistic events using a diagram, called as Minkowski Diagram. To make the diagram, we just need to use Lorentz transformation between the rest frame $S$ and the moving frame $S^{\prime}$ that move with velocity $v=\beta c$ with respect to the rest frame.

$$
\begin{aligned}
& x=\gamma\left(x^{\prime}+\beta c t^{\prime}\right), \\
& c t=\gamma\left(c t^{\prime}+\beta x^{\prime}\right), \\
& x^{\prime}=\gamma(x-\beta c t), \\
& c t^{\prime}=\gamma(c t-\beta x) . \\
& \gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
\end{aligned}
$$

<image_1>

Let's choose $x$ and $c t$ as the orthogonal axes. A point $\left(x^{\prime}, c t^{\prime}\right)=(1,0)$ in the moving frame $S^{\prime}$ has a coordinate $(x, c t)=(\gamma, \gamma \beta)$ in the rest frame $S$. The line connecting this point and the origin defines the $x^{\prime}$ axis. Another point $\left(x^{\prime}, c t^{\prime}\right)=(0,1)$ in the moving frame $S^{\prime}$ has a coordinate $(x, c t)=(\gamma \beta, \gamma)$ in the rest frame $S$. The line connecting this point and the origin defines the $c t^{\prime}$ axis. The angle between the $x$ and $\mathrm{x}^{\prime}$ axis is $\theta$, where $\tan \theta=\beta$. A unit length in the moving frame $S^{\prime}$ is equal to $\gamma \sqrt{1+\beta^{2}}=\sqrt{\frac{1+\beta^{2}}{1-\beta^{2}}}$ in the rest frame S.

To get a better understanding of Minkowski diagram, let us take a look at this example. Consider a stick of proper length $L$ in a moving frame S'. We would like to find the length of the stick in the rest frame S. Consider the figure below.



<image_2>

The stick is represented by the segment AC. The length $\mathrm{AC}$ is equal to $\sqrt{\frac{1+\beta^{2}}{1-\beta^{2}}} L$ in the $S$ frame. The stick length in the $S$ frame is represented by the line $A B$.

$$
\begin{aligned}
\mathrm{AB} & =\mathrm{AD}-\mathrm{BD} \\
& =A C \cos \theta-A C \sin \theta \tan \theta \\
& =L \sqrt{1-\beta^{2}}
\end{aligned}
$$
Context question:
1.  Using a Minkowski diagram, calculate the length of a stick with proper length $L$ in the rest frame, as measured in the moving frame.
Context answer:
\boxed{$\sqrt{1-\beta^{2}} L$}
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proof	Modern Physics	Physics	English
17	"1.  Let us consider an oven source of silver atoms, which has a small opening. The atoms stream out of the opening along $-y$ direction (see Figure below) and experience a spatial varying field $\boldsymbol{B}_{1}$. The field $\boldsymbol{B}_{1}$ has strong bias field component in the $z$ direction, where the atoms with different magnetic moment $\mu_{z}= \pm \gamma \hbar$ are split in the $z$ direction. At a distance $D$ from the oven source, a screen $S C_{1}$ is put to allow only spin up atoms to pass (blocking spin down atoms). Thus, at the instant after passing the screen, the atoms are prepared in spin up states. After the screen, the atoms enter a region of nonhomogenous field $\boldsymbol{B}_{2}$ where the atoms feel a force

$$
F_{x}=\mu_{x} C
$$

The field $\boldsymbol{B}_{2}$ has strong bias field component in the $x$ direction, where the atoms have magnetic moment $\mu_{x}= \pm \gamma \hbar$.



<image_1>

In order to determine $\mu_{x}$ by observing the splitting in $x$ direction, show that the following condition must be fulfilled:

$$
\frac{1}{\hbar}\left|\mu_{x}\right| \Delta x C t \gg 1
$$

where $t$ is the duration after leaving the screen $S C_{1}$ and $\Delta x$ is the opening width on $S C_{1}$."	['In the $x$ direction, the uncertainty in position due to the screen opening is $\\Delta x$. According to the uncertainty principle, the atom momentum uncertainty $\\Delta p_{x}$ is given by\n\n$$\n\\Delta p_{x} \\approx \\frac{\\hbar}{\\Delta x},\n$$\n\nwhich translates into an uncertainty in the $x$ velocity of the atom,\n\n$$\nv_{x} \\approx \\frac{\\hbar}{m \\Delta x}\n$$\n\nConsequently, during the time of flight $t$ of the atoms through the device, the uncertainty in the width of the beam will grow by an amount $\\delta x$ given by\n\n$$\n\\delta x=\\Delta v_{x} t \\approx \\frac{\\hbar}{m \\Delta x} t\n$$\n\n\n\nSo, the width of the beams is growing linearly in time. Meanwhile, the two beams are separating at a rate determined by the force $F_{x}$ and the separation between the beams after a time $t$ becomes\n\n$$\nd_{x}=2 \\times \\frac{1}{2} \\frac{F_{x}}{m} t^{2}=\\frac{1}{m}\\left|\\mu_{x}\\right| C t^{2}\n$$\n\nIn order to be able to distinguish which beam a particle belongs to, the separation of the two beams must be greater than the widths of the beams; otherwise the two beams will overlap and it will be impossible to know what the $x$ component of the atom spin is. Thus, the condition must be satisfied is\n\n$$\n\\begin{aligned}\nd_{x} & \\gg \\delta x \\\\\n\\frac{1}{m}\\left|\\mu_{x}\\right| C t^{2} & \\gg \\frac{\\hbar}{m \\Delta x} t, \\\\\n\\frac{1}{\\hbar}\\left|\\mu_{x}\\right| \\Delta x C t & \\gg 1 .\n\\end{aligned}\n\\tag{15}\n$$']		"All matters in the universe have fundamental properties called spin, besides their mass and charge. Spin is an intrinsic form of angular momentum carried by particles. Despite the fact that quantum mechanics is needed for a full treatment of spin, we can still study the physics of spin using the usual classical formalism. In this problem, we are investigating the influence of magnetic field on spin using its classical analogue.

The classical torque equation of spin is given by

$$
\boldsymbol{\tau}=\frac{d \boldsymbol{L}}{d t}=\boldsymbol{\mu} \times \boldsymbol{B}
$$

In this case, the angular momentum $\boldsymbol{L}$ represents the ""intrinsic"" spin of the particles, $\boldsymbol{\mu}$ is the magnetic moment of the particles, and $\boldsymbol{B}$ is magnetic field. The spin of a particle is associated with a magnetic moment via the equation

$$
\boldsymbol{\mu}=-\gamma \boldsymbol{L}
$$

where $\gamma$ is the gyromagnetic ratio.

In this problem, the term ""frequency"" means angular frequency (rad/s), which is a scalar quantity. All bold letters represent vectors; otherwise they represent scalars.

Part A. Larmor precession
Context question:
1.  Prove that the magnitude of magnetic moment $\mu$ is always constant under the influence of a magnetic field $\boldsymbol{B}$. For a special case of stationary (constant) magnetic field, also show that the angle between $\boldsymbol{\mu}$ and $\boldsymbol{B}$ is constant.

(Hint: You can use properties of vector products.)
Context answer:
\boxed{证明题}


Context question:
2.  A uniform magnetic field $\boldsymbol{B}$ exists and it makes an angle $\phi$ with a particle's magnetic moment $\boldsymbol{\mu}$. Due to the torque by the magnetic field, the magnetic moment $\boldsymbol{\mu}$ rotates around the field $\boldsymbol{B}$, which is also known as Larmor precession. Determine the Larmor precession frequency $\omega_{0}$ of the magnetic moment with respect to $\boldsymbol{B}=B_{0} \boldsymbol{k}$.
Context answer:
\boxed{$\omega_{0}=\gamma B_{0}$}


Extra Supplementary Reading Materials:

Part B. Rotating frame

In this section, we choose a rotating frame $S^{\prime}$ as our frame of reference. The rotating frame $S^{\prime}=\left(x^{\prime}, y^{\prime}, z^{\prime}\right)$ rotates with an angular velocity $\omega \boldsymbol{k}$ as seen by an observer in the laboratory frame $S=(x, y, z)$, where the axes $x^{\prime}, y^{\prime}, z^{\prime}$ intersect with $x, y, z$ at time $t=0$. Any vector $\boldsymbol{A}=A_{x} \boldsymbol{i}+A_{y} \boldsymbol{j}+A_{z} \boldsymbol{k}$ in a lab frame can be written as $\boldsymbol{A}=A_{x}{ }^{\prime} \boldsymbol{i}^{\prime}+A_{y}{ }^{\prime} \boldsymbol{j}^{\prime}+A_{z}{ }^{\prime} \boldsymbol{k}^{\prime}$ in the rotating frame $S^{\prime}$. The time derivative of the vector becomes

$$
\frac{d \boldsymbol{A}}{d t}=\left(\frac{d A_{x}{ }^{\prime}}{d t} \boldsymbol{i}^{\prime}+\frac{d A_{y}{ }^{\prime}}{d t} \boldsymbol{j}^{\prime}+\frac{d A_{z}{ }^{\prime}}{d t} \boldsymbol{k}^{\prime}\right)+\left(A_{x}{ }^{\prime} \frac{d \boldsymbol{i}^{\prime}}{d t}+A_{y}{ }^{\prime} \frac{d \boldsymbol{j}^{\prime}}{d t}+A_{z}{ }^{\prime} \frac{d \boldsymbol{k}^{\prime}}{d t}\right)
$$



$$
\left(\frac{d \boldsymbol{A}}{d t}\right)_{l a b}=\left(\frac{d \boldsymbol{A}}{d t}\right)_{r o t}+(\omega \mathbf{k} \times \boldsymbol{A})
$$

where $\left(\frac{d \boldsymbol{A}}{d t}\right)_{l a b}$ is the time derivative of vector $\boldsymbol{A}$ seen by an observer in the lab frame, and $\left(\frac{d A}{d t}\right)_{\text {rot }}$ is the time derivative seen by an observer in the rotating frame. For all the following problems in this part, the answers are referred to the rotating frame $S^{\prime}$.
Context question:
1.  Show that the time evolution of the magnetic moment follows the equation

$$
\left(\frac{d \boldsymbol{\mu}}{d t}\right)_{r o t}=-\gamma \boldsymbol{\mu} \times \boldsymbol{B}_{e f f}
$$

where $\boldsymbol{B}_{\text {eff }}=\boldsymbol{B}-\frac{\omega}{\gamma} \boldsymbol{k}^{\prime}$ is the effective magnetic field.
Context answer:
\boxed{证明题}


Context question:
2.  For $\boldsymbol{B}=B_{0} \boldsymbol{k}$, what is the new precession frequency $\Delta$ in terms of $\omega_{0}$ and $\omega$ ?
Context answer:
\boxed{$\Delta =\gamma B_{0}-\omega$}


Context question:
3.  Now, let us consider the case of a time-varying magnetic field. Besides a constant magnetic field, we also apply a rotating magnetic field $\boldsymbol{b}(t)=b(\cos \omega t \boldsymbol{i}+\sin \omega t \boldsymbol{j})$, so $\boldsymbol{B}=B_{0} \boldsymbol{k}+\boldsymbol{b}(t)$. Show that the new Larmor precession frequency of the magnetic moment is

$$
\Omega=\gamma \sqrt{\left(B_{0}-\frac{\omega}{\gamma}\right)^{2}+b^{2}}
$$
Context answer:
\boxed{证明题}


Context question:
4.  Instead of applying the field $\boldsymbol{b}(t)=b(\cos \omega t \boldsymbol{i}+\sin \omega t \boldsymbol{j})$, now we apply $\boldsymbol{b}(t)=b(\cos \omega t \boldsymbol{i}-\sin \omega t \boldsymbol{j})$, which rotates in the opposite direction and hence $\boldsymbol{B}=B_{0} \boldsymbol{k}+b(\cos \omega t \boldsymbol{i}-\sin \omega t \boldsymbol{j})$. What is the effective magnetic field $\boldsymbol{B}_{\text {eff }}$ for this case (in terms of the unit vectors $\boldsymbol{i}^{\prime}, \boldsymbol{j}^{\prime}, \boldsymbol{k}^{\prime}$ )? What is its time average, $\overline{\boldsymbol{B}_{\text {eff }}}$ (recall that $\overline{\cos 2 \pi t / T}=\overline{\sin 2 \pi t / T}=0$ )?
Context answer:
\boxed{$\mathbf{B}_{\mathrm{eff}}=\left(B_{0}-\frac{\omega}{\gamma}\right) \mathbf{k}^{\prime}+b\left(\cos 2 \omega t \mathbf{i}^{\prime}-\sin 2 \omega t \mathbf{j}^{\prime}\right)$ , $\overline{\mathbf{B}_{\mathrm{eff}}}=\left(B_{0}-\frac{\omega}{\gamma}\right) \mathbf{k}^{\prime}$}


Extra Supplementary Reading Materials:

Part C. Rabi oscillation  

For an ensemble of $N$ particles under the influence of a large magnetic field, the spin can have two quantum states: ""up"" and ""down"". Consequently, the total population of spin up $N_{\uparrow}$ and down $N_{\downarrow}$ obeys the equation

$$
N_{\uparrow}+N_{\downarrow}=N
$$

The difference of spin up population and spin down population yields the macroscopic magnetization along the $z$ axis:

$$
M=\left(N_{\uparrow}-N_{\downarrow}\right) \mu=N \mu_{z} .
$$

In a real experiment, two magnetic fields are usually applied, a large bias field $B_{0} \boldsymbol{k}$ and an oscillating field with amplitude $2 b$ perpendicular to the bias field $\left(b \ll B_{0}\right)$. Initially, only the large bias is applied, causing all the particles lie in the spin up states ( $\boldsymbol{\mu}$ is oriented in the $z$-direction at $t=0$ ). Then, the oscillating field is turned on, where its frequency $\omega$ is chosen to be in resonance with the Larmor precession frequency $\omega_{0}$, i.e. $\omega=\omega_{0}$. In other words, the total field after time $t=0$ is given by

$$
\boldsymbol{B}(t)=B_{0} \boldsymbol{k}+2 b \cos \omega_{0} t \boldsymbol{i} .
$$
Context question:
1.  In the rotating frame $S^{\prime}$, show that the effective field can be approximated by

$$
\boldsymbol{B}_{\text {eff }} \approx b \boldsymbol{i}^{\prime},
$$

which is commonly known as rotating wave approximation. What is the precession frequency $\Omega$ in frame $S^{\prime}$ ?
Context answer:
\boxed{$\Omega=\gamma b$}


Context question:
2.  Determine the angle $\alpha$ that $\boldsymbol{\mu}$ makes with $\boldsymbol{B}_{\text {eff }}$. Also, prove that the magnetization varies with time as

$$
M(t)=N \mu(\cos \Omega t) .
$$
Context answer:
\boxed{证明题}


Context question:
3.  Under the application of magnetic field described above, determine the fractional population of each spin up $P_{\uparrow}=N_{\uparrow} / N$ and spin down $P_{\downarrow}=N_{\downarrow} / N$ as a function of time. Plot $P_{\uparrow}(t)$ and $P_{\downarrow}(t)$ on the same graph vs. time $t$. The alternating spin up and spin down population as a function of time is called Rabi oscillation.
Context answer:
\boxed{$P_{\downarrow}=\sin ^{2} \frac{\Omega t}{2}$ , $P_{\uparrow}=\cos ^{2} \frac{\Omega t}{2}$}


Extra Supplementary Reading Materials:

Part D. Measurement incompatibility  

Spin is in fact a vector quantity; but due to its quantum properties, we cannot measure each of its components simultaneously (i.e. we can know both $|\boldsymbol{\mu}|$ and $\mu_{z}$ as in above problems; but not all $|\boldsymbol{\mu}|, \mu_{x}, \mu_{y}$, and $\mu_{z}$ simultaneously). In this problem, we will do a calculation based on the Heisenberg uncertainty principle (using the relation $\Delta p_{q} \Delta q \geq \hbar$ ) to show how these measurements are incompatible with each other."	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proof	Modern Physics	Physics	English
18	2.  The atoms are initially prepared in the spin up states right after leaving the screen, where $\mu_{z}=\gamma \hbar=\left|\mu_{x}\right|$. This means the atoms will precess at rates covering a range of values $\Delta \omega$ with respect to the $x$ component of $\boldsymbol{B}_{2}$, specifically $B_{2 x}=B_{0}+C x$. Prove that the spread in the precession angle $\Delta \omega t$ is so large and hence we cannot measure both $\mu_{x}$ and $\mu_{z}$ simultaneously. In other words, the measurement of $\mu_{x}$ destroys the information on $\mu_{z}$.	['As the atoms pass through the screen, the variation of magnetic field strength across the beam width experienced by the atoms is\n\n$$\n\\Delta B=\\Delta x \\frac{d B}{d x}=C \\Delta x\n$$\n\nThis means the atoms will precess at rates covering a range of values $\\Delta \\omega$ given by\n\n$$\n\\Delta \\omega=\\gamma \\Delta B=\\frac{\\mu_{z}}{\\hbar} \\Delta B=\\frac{\\left|\\mu_{x}\\right|}{\\hbar} C \\Delta x\n$$\n\nand, if previous condition in measuring $\\mu_{x}$ is satisfied,\n\n$$\n\\Delta \\omega t \\gg 1 .\n\\tag{16}\n$$\n\nIn other words, the spread in the angle $\\Delta \\omega t$ through which the magnetic moments precess is so large that the $z$ component of the spin is completely randomized or the measurement uncertainty is very large.']		"All matters in the universe have fundamental properties called spin, besides their mass and charge. Spin is an intrinsic form of angular momentum carried by particles. Despite the fact that quantum mechanics is needed for a full treatment of spin, we can still study the physics of spin using the usual classical formalism. In this problem, we are investigating the influence of magnetic field on spin using its classical analogue.

The classical torque equation of spin is given by

$$
\boldsymbol{\tau}=\frac{d \boldsymbol{L}}{d t}=\boldsymbol{\mu} \times \boldsymbol{B}
$$

In this case, the angular momentum $\boldsymbol{L}$ represents the ""intrinsic"" spin of the particles, $\boldsymbol{\mu}$ is the magnetic moment of the particles, and $\boldsymbol{B}$ is magnetic field. The spin of a particle is associated with a magnetic moment via the equation

$$
\boldsymbol{\mu}=-\gamma \boldsymbol{L}
$$

where $\gamma$ is the gyromagnetic ratio.

In this problem, the term ""frequency"" means angular frequency (rad/s), which is a scalar quantity. All bold letters represent vectors; otherwise they represent scalars.

Part A. Larmor precession
Context question:
1.  Prove that the magnitude of magnetic moment $\mu$ is always constant under the influence of a magnetic field $\boldsymbol{B}$. For a special case of stationary (constant) magnetic field, also show that the angle between $\boldsymbol{\mu}$ and $\boldsymbol{B}$ is constant.

(Hint: You can use properties of vector products.)
Context answer:
\boxed{证明题}


Context question:
2.  A uniform magnetic field $\boldsymbol{B}$ exists and it makes an angle $\phi$ with a particle's magnetic moment $\boldsymbol{\mu}$. Due to the torque by the magnetic field, the magnetic moment $\boldsymbol{\mu}$ rotates around the field $\boldsymbol{B}$, which is also known as Larmor precession. Determine the Larmor precession frequency $\omega_{0}$ of the magnetic moment with respect to $\boldsymbol{B}=B_{0} \boldsymbol{k}$.
Context answer:
\boxed{$\omega_{0}=\gamma B_{0}$}


Extra Supplementary Reading Materials:

Part B. Rotating frame

In this section, we choose a rotating frame $S^{\prime}$ as our frame of reference. The rotating frame $S^{\prime}=\left(x^{\prime}, y^{\prime}, z^{\prime}\right)$ rotates with an angular velocity $\omega \boldsymbol{k}$ as seen by an observer in the laboratory frame $S=(x, y, z)$, where the axes $x^{\prime}, y^{\prime}, z^{\prime}$ intersect with $x, y, z$ at time $t=0$. Any vector $\boldsymbol{A}=A_{x} \boldsymbol{i}+A_{y} \boldsymbol{j}+A_{z} \boldsymbol{k}$ in a lab frame can be written as $\boldsymbol{A}=A_{x}{ }^{\prime} \boldsymbol{i}^{\prime}+A_{y}{ }^{\prime} \boldsymbol{j}^{\prime}+A_{z}{ }^{\prime} \boldsymbol{k}^{\prime}$ in the rotating frame $S^{\prime}$. The time derivative of the vector becomes

$$
\frac{d \boldsymbol{A}}{d t}=\left(\frac{d A_{x}{ }^{\prime}}{d t} \boldsymbol{i}^{\prime}+\frac{d A_{y}{ }^{\prime}}{d t} \boldsymbol{j}^{\prime}+\frac{d A_{z}{ }^{\prime}}{d t} \boldsymbol{k}^{\prime}\right)+\left(A_{x}{ }^{\prime} \frac{d \boldsymbol{i}^{\prime}}{d t}+A_{y}{ }^{\prime} \frac{d \boldsymbol{j}^{\prime}}{d t}+A_{z}{ }^{\prime} \frac{d \boldsymbol{k}^{\prime}}{d t}\right)
$$



$$
\left(\frac{d \boldsymbol{A}}{d t}\right)_{l a b}=\left(\frac{d \boldsymbol{A}}{d t}\right)_{r o t}+(\omega \mathbf{k} \times \boldsymbol{A})
$$

where $\left(\frac{d \boldsymbol{A}}{d t}\right)_{l a b}$ is the time derivative of vector $\boldsymbol{A}$ seen by an observer in the lab frame, and $\left(\frac{d A}{d t}\right)_{\text {rot }}$ is the time derivative seen by an observer in the rotating frame. For all the following problems in this part, the answers are referred to the rotating frame $S^{\prime}$.
Context question:
1.  Show that the time evolution of the magnetic moment follows the equation

$$
\left(\frac{d \boldsymbol{\mu}}{d t}\right)_{r o t}=-\gamma \boldsymbol{\mu} \times \boldsymbol{B}_{e f f}
$$

where $\boldsymbol{B}_{\text {eff }}=\boldsymbol{B}-\frac{\omega}{\gamma} \boldsymbol{k}^{\prime}$ is the effective magnetic field.
Context answer:
\boxed{证明题}


Context question:
2.  For $\boldsymbol{B}=B_{0} \boldsymbol{k}$, what is the new precession frequency $\Delta$ in terms of $\omega_{0}$ and $\omega$ ?
Context answer:
\boxed{$\Delta =\gamma B_{0}-\omega$}


Context question:
3.  Now, let us consider the case of a time-varying magnetic field. Besides a constant magnetic field, we also apply a rotating magnetic field $\boldsymbol{b}(t)=b(\cos \omega t \boldsymbol{i}+\sin \omega t \boldsymbol{j})$, so $\boldsymbol{B}=B_{0} \boldsymbol{k}+\boldsymbol{b}(t)$. Show that the new Larmor precession frequency of the magnetic moment is

$$
\Omega=\gamma \sqrt{\left(B_{0}-\frac{\omega}{\gamma}\right)^{2}+b^{2}}
$$
Context answer:
\boxed{证明题}


Context question:
4.  Instead of applying the field $\boldsymbol{b}(t)=b(\cos \omega t \boldsymbol{i}+\sin \omega t \boldsymbol{j})$, now we apply $\boldsymbol{b}(t)=b(\cos \omega t \boldsymbol{i}-\sin \omega t \boldsymbol{j})$, which rotates in the opposite direction and hence $\boldsymbol{B}=B_{0} \boldsymbol{k}+b(\cos \omega t \boldsymbol{i}-\sin \omega t \boldsymbol{j})$. What is the effective magnetic field $\boldsymbol{B}_{\text {eff }}$ for this case (in terms of the unit vectors $\boldsymbol{i}^{\prime}, \boldsymbol{j}^{\prime}, \boldsymbol{k}^{\prime}$ )? What is its time average, $\overline{\boldsymbol{B}_{\text {eff }}}$ (recall that $\overline{\cos 2 \pi t / T}=\overline{\sin 2 \pi t / T}=0$ )?
Context answer:
\boxed{$\mathbf{B}_{\mathrm{eff}}=\left(B_{0}-\frac{\omega}{\gamma}\right) \mathbf{k}^{\prime}+b\left(\cos 2 \omega t \mathbf{i}^{\prime}-\sin 2 \omega t \mathbf{j}^{\prime}\right)$ , $\overline{\mathbf{B}_{\mathrm{eff}}}=\left(B_{0}-\frac{\omega}{\gamma}\right) \mathbf{k}^{\prime}$}


Extra Supplementary Reading Materials:

Part C. Rabi oscillation  

For an ensemble of $N$ particles under the influence of a large magnetic field, the spin can have two quantum states: ""up"" and ""down"". Consequently, the total population of spin up $N_{\uparrow}$ and down $N_{\downarrow}$ obeys the equation

$$
N_{\uparrow}+N_{\downarrow}=N
$$

The difference of spin up population and spin down population yields the macroscopic magnetization along the $z$ axis:

$$
M=\left(N_{\uparrow}-N_{\downarrow}\right) \mu=N \mu_{z} .
$$

In a real experiment, two magnetic fields are usually applied, a large bias field $B_{0} \boldsymbol{k}$ and an oscillating field with amplitude $2 b$ perpendicular to the bias field $\left(b \ll B_{0}\right)$. Initially, only the large bias is applied, causing all the particles lie in the spin up states ( $\boldsymbol{\mu}$ is oriented in the $z$-direction at $t=0$ ). Then, the oscillating field is turned on, where its frequency $\omega$ is chosen to be in resonance with the Larmor precession frequency $\omega_{0}$, i.e. $\omega=\omega_{0}$. In other words, the total field after time $t=0$ is given by

$$
\boldsymbol{B}(t)=B_{0} \boldsymbol{k}+2 b \cos \omega_{0} t \boldsymbol{i} .
$$
Context question:
1.  In the rotating frame $S^{\prime}$, show that the effective field can be approximated by

$$
\boldsymbol{B}_{\text {eff }} \approx b \boldsymbol{i}^{\prime},
$$

which is commonly known as rotating wave approximation. What is the precession frequency $\Omega$ in frame $S^{\prime}$ ?
Context answer:
\boxed{$\Omega=\gamma b$}


Context question:
2.  Determine the angle $\alpha$ that $\boldsymbol{\mu}$ makes with $\boldsymbol{B}_{\text {eff }}$. Also, prove that the magnetization varies with time as

$$
M(t)=N \mu(\cos \Omega t) .
$$
Context answer:
\boxed{证明题}


Context question:
3.  Under the application of magnetic field described above, determine the fractional population of each spin up $P_{\uparrow}=N_{\uparrow} / N$ and spin down $P_{\downarrow}=N_{\downarrow} / N$ as a function of time. Plot $P_{\uparrow}(t)$ and $P_{\downarrow}(t)$ on the same graph vs. time $t$. The alternating spin up and spin down population as a function of time is called Rabi oscillation.
Context answer:
\boxed{$P_{\downarrow}=\sin ^{2} \frac{\Omega t}{2}$ , $P_{\uparrow}=\cos ^{2} \frac{\Omega t}{2}$}


Extra Supplementary Reading Materials:

Part D. Measurement incompatibility  

Spin is in fact a vector quantity; but due to its quantum properties, we cannot measure each of its components simultaneously (i.e. we can know both $|\boldsymbol{\mu}|$ and $\mu_{z}$ as in above problems; but not all $|\boldsymbol{\mu}|, \mu_{x}, \mu_{y}$, and $\mu_{z}$ simultaneously). In this problem, we will do a calculation based on the Heisenberg uncertainty principle (using the relation $\Delta p_{q} \Delta q \geq \hbar$ ) to show how these measurements are incompatible with each other.
Context question:
1.  Let us consider an oven source of silver atoms, which has a small opening. The atoms stream out of the opening along $-y$ direction (see Figure below) and experience a spatial varying field $\boldsymbol{B}_{1}$. The field $\boldsymbol{B}_{1}$ has strong bias field component in the $z$ direction, where the atoms with different magnetic moment $\mu_{z}= \pm \gamma \hbar$ are split in the $z$ direction. At a distance $D$ from the oven source, a screen $S C_{1}$ is put to allow only spin up atoms to pass (blocking spin down atoms). Thus, at the instant after passing the screen, the atoms are prepared in spin up states. After the screen, the atoms enter a region of nonhomogenous field $\boldsymbol{B}_{2}$ where the atoms feel a force

$$
F_{x}=\mu_{x} C
$$

The field $\boldsymbol{B}_{2}$ has strong bias field component in the $x$ direction, where the atoms have magnetic moment $\mu_{x}= \pm \gamma \hbar$.



<image_1>

In order to determine $\mu_{x}$ by observing the splitting in $x$ direction, show that the following condition must be fulfilled:

$$
\frac{1}{\hbar}\left|\mu_{x}\right| \Delta x C t \gg 1
$$

where $t$ is the duration after leaving the screen $S C_{1}$ and $\Delta x$ is the opening width on $S C_{1}$.
Context answer:
\boxed{证明题}
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proof	Modern Physics	Physics	English