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USAPhO 2022, Полуфинал — Задача 6. Question B3

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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

Time Crystal

The kinetic energy $E$ , momentum $p$ , and velocity $v$ of a particle moving in one dimension satisfy

F=\frac{dp}{dt}=\frac{dE}{dx},\qquad v=\frac{dE}{dp}

where $F$ is the external force. For a free particle, the momentum and energy are related by $E=p^2/2m$ . However, when an electron moves inside a metal, its interactions with the crystal lattice of positively charged ions lead to a different relationship between momentum and energy. All of the above identities still apply, but now suppose that

E(p)=V(1-\cos(pb))

where $V$ and $b$ are constants that depend on the metal. This result is inherently quantum mechanical in origin, and as we will see, it leads to some rather strange behavior.

a.

First, we investigate the motion of the electron in general.

i. Find the velocity as a function of $p$ .

ii. The effective mass $m_*$ of the electron is defined so that it satisfies $F=m_*a$ . Find the effective mass as a function of $p$ .

b.

Now suppose a metal rod of infinite length, aligned with the $x$ -axis, contains conducting electrons of charge $-e$ , initially with zero momentum. At time $t=0$ , an electric field $\mathbf E=E_0\hat{\mathbf x}$ is turned on, and is experienced by every electron in the rod. Ignore the interactions of the electrons with each other.

i. For an electron that starts at $x=0$ at time $t=0$ , find its position $x(t)$ .

ii. If the number of conducting electrons per unit volume is $n$ , and the cross-sectional area of the rod is $A$ , find the average current in the rod over a long time.

iii. Now suppose that every time $\tau$ , each electron suffers a collision with the crystal lattice, causing its momentum to reset to zero. In the limit of frequent collisions, $eE_0b\tau\ll1$ , find the average current in the rod over a long time.

iv. If $\tau$ can be freely adjusted, estimate the maximum possible average current in the rod, up to a dimensionless constant.

Решение

a.

i. Using the provided identity,

v=\frac{dE}{dp}=Vb\sin(pb).

ii. By definition, we have

\frac{dp}{dt}=m_*\frac{dv}{dt}

so by the chain rule, we conclude

m_*=\left(\frac{dv}{dp}\right)^{-1}=\frac1{Vb^2}\frac1{\cos(pb)}.

Note that at some points, this mass is negative! Applying a forward force can cause an electron to accelerate backwards, which leads to the strange result we’ll find in part b.ii. (This doesn’t contradict conservation of momentum, because the forward momentum ends up imparted to the crystal lattice, causing the solid as a whole to recoil.)

b.

i. The momentum is $p=Ft=-eE_0t$ , so using the result of part a.i,

v(t)=-Vb\sin(eE_0tb).

Integrating both sides gives

x(t)=\frac V{eE_0}(\cos(eE_0tb)-1).

ii. The electrons just oscillate, with zero average velocity. Therefore, the average current they supply is zero! This strange phenomenon is called a Bloch oscillation. It doesn’t occur in ordinary metals because of the frequent collisions between electrons and ions, considered in the parts below.

iii. For small times, the velocity is approximately

v(t)\approx-(eE_0Vb^2)t

where we applied the small angle approximation to the result of part b.i. Thus, if collisions happen every time $\tau$ , the average velocity is

\bar v=-\frac12(eE_0Vb^2)\tau.

Therefore, the average current is

I=-nAe\bar v=\frac12nAe^2E_0Vb^2\tau.

The plus sign makes sense, since it means current flows parallel to the applied field.

iv. If $\tau$ is small, then more frequent collisions slow down the current, as we showed in part b.iii. But if $\tau$ is very large, then the current cancels itself out due to Bloch oscillations, as we showed in part b.ii. Therefore, the highest possible current occurs when the electrons have time to accelerate to a substantial fraction of their maximum possible speed $Vb$ , but collide before their effective mass goes negative. This occurs when

eE_0\tau b\sim1

and gives an average current

I\sim nAeVb.

Some students tried to answer this part using dimensional analysis, but that won’t work, since there are 5 variables and only 4 independent dimensions (mass, length, time and charge). Incidentally, you can find the dimensionless coefficient by solving a simple equation numerically, though this wasn’t required.

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