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Условие Отговори Решение

USAPhO 2024, Полуфинал — Задача 1. Question A1 · Ping Pong

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Всички материали от USAPhO 2024

AAPT AIP UNITED STATES PHYSICS TEAM 2024 — USA Physics Olympiad Exam · 25 т.

Important Instructions for the Exam Supervisor

DO NOT DISTRIBUTE THIS PAGE

This examination has two parts. Each part 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). Page 11 is a graph page, in case students wish to use it as an additional page for problem B3.
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, collect signed Honesty Policy, 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. Give them the Honor Code Certification and collect signed Codes. Students may not keep the exam questions.
After the exam, sort each student’s answer sheets by page number. Scan every answer sheet, including blank ones. Everybody may discuss the questions after April 3rd. Until April 19th, hold on to all of the answer sheets in the event that your scans are lost or illegible.

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

Tengiz Bibilashvili, Kellan Colburn, Natalie LeBaron, Rishab Parthasarathy, Elena Yudovina, and Kevin Zhou.

Instructions for the Student

You should receive these instructions, the reference 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 April 4. Violations of this rule may result in disqualification.

Below are instructions for writing your solutions.

All of your solutions must be written on the 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.

Fundamental Constants

$g = 9.8\ \mathrm{N/kg}$ $\qquad$ $G = 6.67 \times 10^{-11}\ \mathrm{N \cdot m^2/kg^2}$

$k = 1/4\pi\epsilon_0 = 8.99 \times 10^{9}\ \mathrm{N \cdot m^2/C^2}$ $\qquad$ $k_{\mathrm{m}} = \mu_0/4\pi = 10^{-7}\ \mathrm{T \cdot m/A}$

$c = 3.00 \times 10^{8}\ \mathrm{m/s}$ $\qquad$ $k_{\mathrm{B}} = 1.38 \times 10^{-23}\ \mathrm{J/K}$

$N_{\mathrm{A}} = 6.02 \times 10^{23}\ (\mathrm{mol})^{-1}$ $\qquad$ $R = N_{\mathrm{A}} k_{\mathrm{B}} = 8.31\ \mathrm{J/(mol \cdot K)}$

$\sigma = 5.67 \times 10^{-8}\ \mathrm{J/(s \cdot m^2 \cdot K^4)}$ $\qquad$ $e = 1.602 \times 10^{-19}\ \mathrm{C}$

$1\ \mathrm{eV} = 1.602 \times 10^{-19}\ \mathrm{J}$ $\qquad$ $h = 6.63 \times 10^{-34}\ \mathrm{J \cdot s} = 4.14 \times 10^{-15}\ \mathrm{eV \cdot s}$

$m_e = 9.109 \times 10^{-31}\ \mathrm{kg} = 0.511\ \mathrm{MeV/c^2}$

Useful Approximations

(1+x)^n \approx 1 + nx + n(n-1)x^2/2 \text{ for } |nx| \ll 1

e^x \approx 1 + x + x^2/2 + x^3/6 \text{ for } |x| \ll 1

\sin\theta \approx \theta - \theta^3/6 \text{ for } |\theta| \ll 1

\cos\theta \approx 1 - \theta^2/2 \text{ for } |\theta| \ll 1

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

Условие

A thin wire of negligible resistance and total length $D$ is wound to form a thin cylindrical solenoid of length $\ell \ll D$ . A conducting sphere of radius $R \ll \ell$ is attached to each end of the solenoid. Initially there is no current in the wire, and the spheres have charges $Q$ and $-Q$ . Give all your answers in terms of $R$ , $\ell$ , $D$ , and the speed of light $c = 1/\sqrt{\mu_0\epsilon_0}$ .

a. Assume this system can be modeled as an LC circuit. What is the angular frequency of its oscillations?

This system loses energy because it emits electromagnetic radiation. Consider an electric dipole consisting of charges $\pm q_0\cos(\omega t)$ separated by distance $d$ , whose dipole moment oscillates with amplitude $p_0 = q_0 d$ . If $d$ is much smaller than the wavelength $\lambda$ of the radiation produced, then it can be shown that the power radiated is roughly (i.e. up to an order-one dimensionless factor)

P \sim \frac{\omega^4 p_0^2}{\epsilon_0 c^3}.

For the rest of the problem, your answers only need to be similarly rough estimates.

b. For this setup, the above formula applies if $D \gg D_0$ . Find a rough estimate for $D_0$ .

c. Assuming $D \gg D_0$ , estimate the number of oscillations that occurs until half the energy is lost.

Отговори

Решение

a. The capacitance of this system is

C = \frac{Q}{\Delta V} = \frac{Q}{Q/(2\pi\epsilon_0 R)} \approx 2\pi\epsilon_0 R

since the potential on the spheres is approximately $\pm Q/4\pi\epsilon_0 R$ . Its inductance can be found by considering the magnetic field energy,

\frac{1}{2}LI^2 = \frac{1}{2\mu_0}\int B^2\,dV.

If the radius of the solenoid is $a$ , then $D = 2\pi a n\ell$ , where $n$ is the number of turns per length. Then we have

\int B^2\,dV = (\mu_0 n I)^2 \pi a^2 \ell = \frac{\mu_0 I^2 D^2}{4\pi\ell}

which implies an inductance

L = \frac{\mu_0 D^2}{4\pi\ell}.

The angular frequency of LC oscillations is

\omega = \frac{1}{\sqrt{LC}} = \frac{c}{D}\sqrt{\frac{2\ell}{R}}.

We weren’t told the values of $a$ and $n$ , but the dependence on them simply dropped out.

b. The radiation produced has angular frequency $\omega$ , so wavelength

\lambda \sim \frac{c}{\omega} \sim D\sqrt{R/\ell}.

The Larmor formula works when $\ell \ll \lambda$ , which corresponds to $D \gg D_0 \sim \sqrt{\ell^3/R}$ .

c. The total stored energy is of order

E \sim \frac{Q^2}{C} \sim \frac{Q^2}{\epsilon_0 R}.

Thus, the typical number $N$ of cycles for the energy to decay is approximately the inverse of the fraction of the energy that is radiated away in each cycle, so

N \sim \frac{E}{P/\omega} \sim \frac{Q^2/\epsilon_0 R}{\omega^3 (Q\ell)^2/\epsilon_0 c^3} \sim \frac{(c/\omega)^3}{R\ell^2} \sim \sqrt{\frac{RD^6}{\ell^7}}.

This is roughly the quality factor of the LC circuit. Note that we implicitly assumed above that many oscillations occur before half the energy is lost. This assumption made sense because we know $N \gg \sqrt{RD_0^6/\ell^7} \sim \ell/R \gg 1$ .

Part (a) of this problem was inspired by problem 19.16 of Zangwill’s Modern Electrodynamics. Compared to that problem, we replaced a straight connecting wire with a solenoid, which makes the problem a bit easier, and allows the LC circuit description to work for a broader range of parameters.

Оригинал в Архива: USAPhO_Exam_24.pdf · официални решения: 2024-USAPhO-Exam_solutions.pdf