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USAPhO 2019, Полуфинал — Задача 4. Question B1

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Всички материали от USAPhO 2019

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

Внимание: Бележка към темата

Original question scan and matching revised key have been read in full. Printed sign/variable/wording defects in A1, A2, A3 and B1 are retained and documented; key alternatives and all answer sheets are preserved. The scanned B2(c) typo remains source faithful, with revised-key wording difference explicitly recorded.

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

This examination consists of two parts. Part A has three questions and is allowed 90 minutes. Part B also has three questions and is allowed 90 minutes.
Divide the exam paper into 4 parts: the instructions (pages 2–3), Part A (pages 4–6), Part B (pages 8–10), and answer sheets for one of the questions in Part A (pages 12–13). The exam should be printed single-sided to facilitate dividing the test and scanning the answer sheets.
Provide students with the instructions for the competition (pages 2–3). Students can keep the pages for both parts of the exam, as they contain a reference list of physical constants.
Provide students with blank sheets of paper as scratch paper. Students are not allowed to bring their own papers.
Then provide students with Part A and the associated answer sheets, 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 solutions to Part A along with the answer sheets and questions.
Students are allowed a 10 to 15 minute break between Parts A and B. Then allow 90 minutes to complete Part B. Do not let students go back to Part A.
At the end of the exam, the supervisor must collect all papers, including the questions, the instructions, and any scratch paper used by the students. Students may not take the exam questions. The examination questions may be returned to the students after April 8th, 2019.
Students are allowed calculators, but they may not use symbolic math, programming, or graphical features of these calculators. Calculators may not be shared and their memory must be cleared of data and programs. Cell phones, PDAs, or cameras may not be used during the exam or while the exam papers are present. Students may not use any tables, books, or collections of formulas.

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

JiaJia Dong, Abijith Krishnan, Brian Skinner, and Kevin Zhou.

Instructions for the Student

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN

At the beginning of the exam, you shall be provided with the instruction sheets, blank papers (both for your answers and scratch work), and the exam packet.
Work on Part A first. You have 90 minutes to complete three problems. Each question is worth 25 points, but they are not necessarily of the same difficulty. Do not look at Part B during this time.
After you have completed Part A you may take a break. You may consider checking your answers to Part A with the remaining time as you will not be allowed to return to Part A once you start Part B.
Then 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.
Show your work and reasoning. Partial credit will be given if you make your reasoning clear. Do not write on the back of any page. Further guidance is given on the next page.
A hand-held calculator may be used. Its memory must be cleared of data and programs. You may use only the basic functions found on a simple scientific calculator. Calculators may not be shared. Cell phones, PDAs or cameras may not be used during the exam or while the exam papers are present. You may not use any tables, books, or collections of formulas.
In order to maintain exam security, do not communicate any information about the questions of this exam, or their solutions until after April 8th, 2019.

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

$g$	$9.8\ \mathrm{N/kg}$
$k=1/(4\pi\varepsilon_0)$	$8.99\times10^9\ \mathrm{N\cdot m^2/C^2}$
$c$	$3.00\times10^8\ \mathrm{m/s}$
$N_A$	$6.02\times10^{23}\ (\mathrm{mol})^{-1}$
$\sigma$	$5.67\times10^{-8}\ \mathrm{J/(s\cdot m^2\cdot K^4)}$
$1\ \mathrm{eV}$	$1.602\times10^{-19}\ \mathrm J$
$m_e$	$9.109\times10^{-31}\ \mathrm{kg}=0.511\ \mathrm{MeV}/c^2$
$\sin\theta$	$\approx\theta-\frac16\theta^3$ for $\vert \theta\vert \ll1$
$G$	$6.67\times10^{-11}\ \mathrm{N\cdot m^2/kg^2}$
$k_m=\mu_0/(4\pi)$	$10^{-7}\ \mathrm{T\cdot m/A}$
$k_B$	$1.38\times10^{-23}\ \mathrm{J/K}$
$R=N_Ak_B$	$8.31\ \mathrm{J/(mol\cdot K)}$
$e$	$1.602\times10^{-19}\ \mathrm C$
$h$	$6.63\times10^{-34}\ \mathrm{J\cdot s}=4.14\times10^{-15}\ \mathrm{eV\cdot s}$
$(1+x)^n$	$\approx1+nx$ for $\vert x\vert \ll1$
$\cos\theta$	$\approx1-\frac12\theta^2$ for $\vert \theta\vert \ll1$

Instructions for the Student — Further Guidance

Following is some further guidance on formatting your solutions. Start each question on a new sheet of paper. Put your AAPT ID number, your proctor’s AAPT ID number, the problem number and the page number and total number of pages for this problem, in the upper right hand corner of each page. As an example, the second page of your solution to B3 might look as follows.

student AAPT #29485 proctor AAPT #1038 B3: 2/2

Remember to also write the AAPT ID numbers on the provided answer sheets. Write single- sided to facilitate scanning. You may use either pencil or pen, but in either case, make sure to write sufficiently clearly so your work will be legible after scanning. To preserve anonymity of grading, do not write your name on any sheet.

End of Instructions for the Student

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN

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

Divide the exam paper into 4 parts: the instructions (pages 2–3), Part A (pages 4–12), Part B (pages 14–23), and answer sheets for one of the questions in Part A (pages 25–26). The exam should be printed single-sided to facilitate dividing the test and scanning the answer sheets.

Условие

Part B

Strain in the Membrane[^2]

Entire original neuron membrane illustration: outer/inner surface labels, both charged layers, d double arrow and epsilon/+/- markings.

The wall of a neuron is made from an elastic membrane, which resists compression in the same way as a spring. It has an effective spring constant $k$ and an equilibrium thickness $d_0$ . Assume that the membrane has a very large area $A$ and negligible curvature.

The neuron has “ion pumps” that can move ions across the membrane. In the resulting charged state, positive and negative ionic charge is arranged uniformly along the outer and inner surfaces of the membrane, respectively. The permittivity of the membrane is $\epsilon$ .

[^2]: This question inspired by Partenskii and Jordan, Physical Review E 80, 011112 (2009).

a. Suppose that, after some amount of work is done by the ion pumps, the charges on the outer and inner surfaces are $Q$ and $-Q$ , respectively. What is the thickness $d$ of the membrane?

b. Derive an expression for the voltage difference $V$ between the outer and inner surfaces of the membrane in terms of $Q$ and the other parameters given.

c. Suppose that the ion pumps are first turned on in the uncharged state, and the membrane is charged very slowly (quasistatically). The pumps will only turn off when the voltage difference across the membrane becomes larger than a particular value $V_{\mathrm{th}}$ . How large must the spring constant $k$ be so that the ion pumps turn off before the membrane collapses?

d.

How much work is done by the ion pumps in each of the following situations? Express your answers in terms of $k$ and $d_0$ .

i. $k$ is infinitesimally larger than the value derived in part (c).

ii. $k$ is infinitesimally smaller than the value derived in part (c).

Assume in each case that the membrane thickness $d$ cannot become negative.

Решение

a.

One charge layer by itself creates an electric field $E_1=Q/(2\epsilon A)$ in each direction. So the force between the two sides of the membrane $F_E=QE_1=Q^2/(2\epsilon A)$ .

This electric force is balanced by the spring force $F_s=kx$ , where $x=d_0-d$ . Equating these two forces and solving for $d$ gives gives

d=d_0-\frac{Q^2}{2\epsilon Ak}.

b.

The electric field inside the membrane (as produced by both the left and right plates) is $E=Q/(\epsilon_0\kappa A)$ . So the voltage between them is

V=Ed=\frac Q{\epsilon A}d.

Inserting the expression for $Q$ from part (a) gives

V=\frac Q{\epsilon A}\left(d_0-\frac{Q^2}{2\epsilon Ak}\right).

This equation implies that as the charge $Q$ is increased, the voltage first increases and then decreases again.

c.

The voltage $V$ first increases and then decreases as a function of $Q$ , which implies that there is a maximum voltage $V_{\max}$ to which the membrane can be charged. This voltage can be found by taking the derivative $dV/dQ$ and setting it equal to zero. This procedure gives

V_{\max}=\sqrt{\frac{kd_0^3}{\epsilon A}}\left(\frac23\right)^{3/2}.

The corresponding charge at the maximum voltage is given by

Q_{V\max}^2=\frac23\epsilon Akd_0.

For the ion pumps to turn off, we must have $V_{\max}>V_{\mathrm{th}}$ . Otherwise the pumps will continue to move charge across the membrane until it collapses. Setting $V_{\max}>V_{\mathrm{th}}$ and solving for $k$ gives

k>\left(\frac32\right)^3\frac{V_{\mathrm{th}}^2\epsilon A}{d_0^3}.

d.

i. If $k$ is larger than the value in part (c), then the threshold voltage $V_{\mathrm{th}}>V_{\max}$ , and the ion pumps turn off before the membrane thickness $d$ reaches zero. The work $W$ done by the ion pumps is equal to the potential energy of the system relative to the uncharged state (with $Q=0$ and $d=0$ ). That is, $W=\frac12kx^2+Q^2/(2C)$ , where $C=\epsilon A/d$ is the capacitance of the membrane. Writing this equation in terms of $Q$ gives $W=Q^2d_0/(2\epsilon A)-Q^4/(8\epsilon^2A^2k)$ .

$k$ being infinitesimally larger than the critical value means that the ion pumps turn off just as the voltage maximum $V_{\mathrm{th}}$ is reached. At this point the charge $Q$ approaches $Q_{V\max}$ , derived in the previous answer. Inserting the value into the expression for $W$ gives

W=\frac5{18}kd_0^2.

ii. If $k$ is smaller than the value in part (c), then the threshold voltage is larger than the stopping voltage $V_{\mathrm{th}}$ , and the ion pumps continue to work until the membrane collapses to $d=0$ . At this point there is no electrostatic energy in the membrane (the membrane capacitance is infinite), and

W=\frac12kd_0^2.

Оригинал в Архива: 2019_USAPhO.pdf · официални решения: 2019_USAPHO_revised-solutions.pdf