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USAPhO 2022, Полуфинал — Задача 5. Question B2

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Всички материали от USAPhO 2022

USA Physics Olympiad Exam — Part A / Part B · 25 т.

Внимание: Бележка към темата

Both original question and worked-key packets have been read in full. They refer to a separate answer-sheet packet, including a blank plotting grid, which is not present in the retained question/key sources or the matching catalogue entry. All native question data and the actual key graph are retained. Printed variable/sign/wording defects and grading allowances remain source faithful and are documented.

Important Instructions for the Exam Supervisor — DO NOT DISTRIBUTE THIS PAGE

This examination has two parts. Part A has three questions and lasts for 90 minutes. Part B also has three questions and lasts for 90 minutes.
For each student, print out one copy of the exam and one copy of the answer sheets. Print everything single-sided, and do not staple anything. Divide the exam into the instructions (pages 2–3), Part A questions (pages 4–7), and Part B questions (pages 8–10).
Begin by giving students the instructions and all of the answer sheets. Let the students read the instructions and fill out their information on the answer sheets. They can keep the instructions for both parts of the exam. Also give students blank sheets of paper to use as scratch paper throughout the exam.
Students may bring calculators, but they may not use symbolic math, programming, or graphing features of these calculators. Calculators may not be shared, and their memory must be cleared of data and programs. Cell phones or other electronics may not be used during the exam or while the exam papers are present. Students may not use books or other references.
To start the exam, give students the Part A questions, and allow 90 minutes to complete Part A. Do not give students Part B during this time, even if they finish with time remaining. At the end of the 90 minutes, collect the Part A questions and answer sheets.
Then give students a 5 to 10 minute break. Then give them the Part B questions, and allow 90 minutes to complete Part B. Do not let students go back to Part A.
At the end of the exam, collect everything, including the questions, the instructions, the answer sheets, and the scratch paper. Students may not keep the exam questions, but they may be returned to the students after April 19th, 2022.
After the exam, sort each student’s answer sheets by page number. Scan every answer sheet, including blank ones.

We acknowledge the following people for their contributions to this year’s exam (in alphabetical order):

Tengiz Bibilashvilli, Abijith Krishnan, Brian Skinner, and Kevin Zhou.

Instructions for the Student

You should receive these instructions, the table on the next page, answer sheets, and blank paper for scratch work. Read this page carefully before the exam begins.
You may use a calculator, but its memory must be cleared of data and programs, and you may not use symbolic math, programming, or graphing features. Calculators may not be shared. Cell phones or other electronics may not be used during the exam or while the exam papers are present. You may not use books or other outside references.
When the exam begins, your proctor will give you the questions for Part A. You will have 90 minutes to complete three problems. Each question is worth 25 points, but they are not necessarily of the same difficulty. If you finish all of the questions, you may check your work, but you may not look at Part B during this time.
After 90 minutes, your proctor will collect the questions and answer sheets for Part A. You may then take a short break.
Then you will work on Part B. You have 90 minutes to complete three problems. Each question is worth 25 points. Do not look at Part A during this time. When the exam ends, you must return all papers to the proctor, including the exam questions.
Do not discuss the questions of this exam, or their solutions, until after April 19th, 2022. Violations of this rule may result in disqualification. Below are instructions for writing your solutions.
This year, all of your solutions will be written on official answer sheets. Nothing outside these answer sheets will be graded. Before the exam begins, write your name, student AAPT number, and proctor AAPT number as directed on the answer sheets.
There are several answer sheets per problem. If you run out of space for a problem, you may use the extra answer sheets, which are at the end of the answer sheet packet. To ensure this work is graded, you must indicate, at the bottom of your last answer sheet for that problem, that you are using these extra answer sheets.
Only write within the frame of each answer sheet. To simplify grading, we recommend drawing a box around your final answer for each subpart. You should organize your work linearly and briefly explain your reasoning, which will help you earn partial credit. You may use either pencil or pen, but sure to write clearly so your work will be legible after scanning.
You may have to fit a line to data, which you can plot on the answer sheet with grid lines. You may use a ruler, pencil, pen, or piece of paper as a straightedge.

Reference table of possibly useful information

$g=9.8\ \mathrm{N/kg}$	$G=6.67\times10^{-11}\ \mathrm{N\cdot m^2/kg^2}$
$k=1/4\pi\epsilon_0=8.99\times10^9\ \mathrm{N\cdot m^2/C^2}$	$k_m=\mu_0/4\pi=10^{-7}\ \mathrm{T\cdot m/A}$
$c=3.00\times10^8\ \mathrm{m/s}$	$k_B=1.38\times10^{-23}\ \mathrm{J/K}$
$N_A=6.02\times10^{23}\ (\mathrm{mol})^{-1}$	$R=N_Ak_B=8.31\ \mathrm{J/(mol\cdot K)}$
$\sigma=5.67\times10^{-8}\ \mathrm{J/(s\cdot m^2\cdot K^4)}$	$e=1.602\times10^{-19}\ \mathrm C$
$1\ \mathrm{eV}=1.602\times10^{-19}\ \mathrm J$	$h=6.63\times10^{-34}\ \mathrm{J\cdot s}=4.14\times10^{-15}\ \mathrm{eV\cdot s}$
$m_e=9.109\times10^{-31}\ \mathrm{kg}=0.511\ \mathrm{MeV}/c^2$	$(1+x)^n\approx1+nx$ for $\lvert x\rvert\ll1$
$\sin\theta\approx\theta-\theta^3/6$ for $\lvert\theta\rvert\ll1$	$\cos\theta\approx1-\theta^2/2$ for $\lvert\theta\rvert\ll1$

You may use this sheet for both parts of the exam.

End of Instructions for the Student

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN

Important Instructions for the Exam Supervisor — Key Packet Pagination

For each student, print out one copy of the exam and one copy of the answer sheets. Print everything single-sided, and do not staple anything. Divide the exam into the instructions (pages 2–3), Part A questions (pages 4–13), and Part B questions (pages 14–22).

Условие

Broken Vase

A uniformly charged ring of radius $d$ has total charge $Q$ and is fixed in place. A point charge $-q$ of mass $m$ is placed at its center. Both $Q$ and $q$ are positive. As a result, if the point charge is given a small velocity along the ring’s axis of symmetry, it will oscillate about the ring’s center.

a. Find the period $T$ of the oscillations.

For the rest of the problem, we will consider this situation in a reference frame where the ring is moving along its axis of symmetry with a constant speed $v$ , which may be comparable to the speed of light $c$ . In this frame, the ring and point charge have charge $Q'$ and $q'$ , and the point charge still oscillates about the ring’s center.

b. What is the period $T'$ of oscillation of the point charge in this frame?

c. When the charge is a small distance $\Delta x$ from the center of the ring, find the restoring force in terms of $q'$ , $Q'$ , $v$ , $d$ , $\Delta x$ , and fundamental constants.

d. Suppose the restoring force has the form $F=-k\,\Delta x$ . Find the period of the resulting oscillations in terms of $k$ , $m$ , $v$ , and fundamental constants.

e. Suppose the electric charge transforms between reference frames as $Q'=\gamma^nQ$ and $q'=\gamma^nq$ . By combining your answers to parts (c) and (d), and comparing to part (b), find the value of $n$ .

To solve this problem, you will need the following results from relativity:

The Lorentz factor is defined as $\gamma=1/\sqrt{1-v^2/c^2}$ .
The momentum of a particle is $\mathbf p=\gamma m\mathbf v$ , and $m$ is the same in all frames.
The electromagnetic force on a charge $q$ is $\mathbf F=d\mathbf p/dt=q(\mathbf E+\mathbf v\times\mathbf B)$ .
The electric field of a charge $q$ at the origin with constant velocity $\mathbf v$ is radial, with magnitude

E=\frac q{4\pi\epsilon_0r^2}\frac{1-v^2/c^2}{(1-(v^2/c^2)\sin^2\theta)^{3/2}}

where $\theta$ is the angle of $\mathbf r$ to $\mathbf v$ .

Решение

a.

Suppose the charge is displaced by a small distance $\Delta x$ . The horizontal component of the electric field of the ring, at the charge, is

E_x\approx E\frac{\Delta x}d=\frac Q{4\pi\epsilon_0d^3}\Delta x.

The restoring force is linear in $\Delta x$ , so it acts like a spring with spring constant

k=\frac{Qq}{4\pi\epsilon_0d^3}.

Thus, the period of oscillation is

T=2\pi\sqrt{\frac mk}=\sqrt{\frac{16\pi^3\epsilon_0md^3}{Qq}}.

b.

By directly applying time dilation to the answer of part (a),

T'=\gamma T=\frac1{\sqrt{1-v^2/c^2}}\sqrt{\frac{16\pi^3\epsilon_0md^3}{Qq}}.

c.

The position of the charge, relative to each charge in the ring, is at $\theta\approx90^\circ$ . Thus, using the provided information, we have

E_x\approx\frac{Q'}{4\pi\epsilon_0d^2}\frac{1-v^2/c^2}{(1-v^2/c^2)^{3/2}}\sin\theta.

Thus, the restoring force is

F=-\frac{Q'q'}{4\pi\epsilon_0d^3}\frac1{\sqrt{1-v^2/c^2}}\Delta x.

d.

By the definition of force given in the information below, we have

F=\frac{dp}{dt}=m\frac d{dt}\left(\frac v{\sqrt{1-v^2/c^2}}\right).

Carrying out the derivative and simplifying gives

F=\frac m{(1-v^2/c^2)^{3/2}}\frac{dv}{dt}=-k\Delta x

For small oscillations, the velocity is always approximately equal to the original velocity $v$ . Thus, the motion is still simple harmonic, except that the mass is effectively

m_{\mathrm{eff}}=\frac m{(1-v^2/c^2)^{3/2}}.

Thus, we conclude

T'=2\pi\sqrt{\frac{m_{\mathrm{eff}}}k}=2\pi\sqrt{\frac mk\frac1{(1-v^2/c^2)^{3/2}}}.

e.

The final result of part (c) tells us that

k=\frac{Q'q'}{4\pi\epsilon_0d^3}\frac1{\sqrt{1-v^2/c^2}}.

Plugging this into the answer to part (d) gives

T'=\frac1{\sqrt{1-v^2/c^2}}\sqrt{\frac{16\pi^3\epsilon_0md^3}{Q'q'}}.

This is precisely the same as the answer to part (b) if $Q'q'=Qq$ , or in other words,

n=0.

That is, the electric charge is Lorentz invariant.

Оригинал в Архива: 2022-USAPhO-Exam.pdf · официални решения: 2022-USAPhO-solutions.pdf