USAPhO 2018, Полуфинал — Задача 5. Question B2

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

Внимание: Бележка към темата

The supervisor and student security dates differ in the original packet. The printed diode key retains its positive negative-voltage exponential and positive discharge derivative despite surrounding text/graphs. The radiation-pressure key retains inconsistent intensity/energy-density terminology and its classical electromagnetic definitions. Exact original sources and detailed defect evidence accompany the transcription.

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.

  • The first page that follows is a cover sheet. Examinees may keep the cover sheet for both parts of the exam.

  • The parts are then identified by the center header on each page. Examinees are only allowed to do one part at a time, and may not work on other parts, even if they have time remaining.

  • Allow 90 minutes to complete Part A. Do not let students look at Part B. Collect the answers to Part A before allowing the examinee to begin Part B. Examinees are allowed a 10 to 15 minutes break between parts A and B.

  • Allow 90 minutes to complete Part B. Do not let students go back to Part A.

  • Ideally the test supervisor will divide the question paper into 3 parts: the cover sheet (page 2), Part A (pages 3-5), Part B (pages 7-9), and several answer sheets for one of the questions in Part A (pages 11-13). Examinees should be provided parts A and B individually, although they may keep the cover sheet. The answer sheets should be printed single sided!

  • The supervisor must collect all examination questions, including the cover sheet, at the end of the exam, as well as any scratch paper used by the examinees. Examinees may not take the exam questions. The examination questions may be returned to the students after April 21, 2018.

  • Examinees are allowed calculators, but they may not use symbolic math, programming, or graphic features of these calculators. Calculators may not be shared and their memory must be cleared of data and programs. Cell phones, PDA’s or cameras may not be used during the exam or while the exam papers are present. Examinees may not use any tables, books, or collections of formulas.

Copyright ©2018 American Association of Physics Teachers.

USA Physics Olympiad Exam — Instructions

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN - Work Part A first. You have 90 minutes to complete all three problems. Each question is worth 25 points. Do not look at Part B during this time. - After you have completed Part A you may take a break. - Then work 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 all your work. Partial credit will be given. Do not write on the back of any page. Do not write anything that you wish graded on the question sheets. - Start each question on a new sheet of paper. Put your AAPT ID number, your proctor’s AAPT ID number, the question number and the page number/total pages for this problem, in the upper right hand corner of each page. For example,

Student AAPT ID # Proctor AAPT ID # A1 - 1/3

  • 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, PDA’s 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. - Questions with the same point value are not necessarily of the same difficulty. - In order to maintain exam security, do not communicate any information about the questions (or their answers/solutions) on this contest until after April 13, 2018.
Possibly Useful Information. You may use this sheet for both parts of the exam.
gg9.8 N/kg9.8\ \mathrm{N/kg}
k=1/(4πε0)k=1/(4\pi\varepsilon_0)8.99×109 N⋅m2/C28.99\times10^9\ \mathrm{N\cdot m^2/C^2}
cc3.00×108 m/s3.00\times10^8\ \mathrm{m/s}
NAN_A6.02×1023 (mol)−16.02\times10^{23}\ (\mathrm{mol})^{-1}
σ\sigma5.67×10−8 J/(s⋅m2⋅K4)5.67\times10^{-8}\ \mathrm{J/(s\cdot m^2\cdot K^4)}
1 eV1\ \mathrm{eV}1.602×10−19 J1.602\times10^{-19}\ \mathrm J
mem_e9.109×10−31 kg=0.511 MeV/c29.109\times10^{-31}\ \mathrm{kg}=0.511\ \mathrm{MeV}/c^2
sin⁡θ\sin\theta≈θ−16θ3\approx\theta-\frac16\theta^3 for ∣θ∣≪1\vert \theta\vert \ll1
GG6.67×10−11 N⋅m2/kg26.67\times10^{-11}\ \mathrm{N\cdot m^2/kg^2}
km=μ0/(4π)k_m=\mu_0/(4\pi)10−7 T⋅m/A10^{-7}\ \mathrm{T\cdot m/A}
kBk_B1.38×10−23 J/K1.38\times10^{-23}\ \mathrm{J/K}
R=NAkBR=N_Ak_B8.31 J/(mol⋅K)8.31\ \mathrm{J/(mol\cdot K)}
ee1.602×10−19 C1.602\times10^{-19}\ \mathrm C
hh6.63×10−34 J⋅s=4.14×10−15 eV⋅s6.63\times10^{-34}\ \mathrm{J\cdot s}=4.14\times10^{-15}\ \mathrm{eV\cdot s}
(1+x)n(1+x)^n≈1+nx\approx1+nx for ∣x∣≪1\vert x\vert \ll1
cos⁡θ\cos\theta≈1−12θ2\approx1-\frac12\theta^2 for ∣θ∣≪1\vert \theta\vert \ll1
Important Instructions for the Exam Supervisor — DO NOT DISTRIBUTE THIS PAGE

Ideally the test supervisor will divide the question paper into 3 parts: the cover sheet (page 2), Part A (pages 3-10), Part B (pages 12-18), and several answer sheets for one of the questions in Part A (pages 20-22). Examinees should be provided parts A and B individually, although they may keep the cover sheet. The answer sheets should be printed single sided!

Условие

In this problem, use a particle-like model of photons: they propagate in straight lines and obey the law of reflection, but are subject to the quantum uncertainty principle. You may use small-angle approximations throughout the problem.

A photon with wavelength λ\lambda has traveled from a distant star to a telescope mirror, which has a circular cross-section with radius RR and a focal length f≫Rf\gg R. The path of the photon is nearly aligned to the axis of the mirror, but has some slight uncertainty Δθ\Delta\theta. The photon reflects off the mirror and travels to a detector, where it is absorbed by a particular pixel on a charge-coupled device (CCD).

Suppose the telescope mirror is manufactured so that photons coming in parallel to each other are focused to the same pixel on the CCD, regardless of where they hit the mirror. Then all small cross-sectional areas of the mirror are equally likely to include the point of reflection for a photon.

a. Find the standard deviation Δr\Delta r of the distribution for rr, the distance from the center of the telescope mirror to the point of reflection of the photon.

b. Use the uncertainty principle, ΔrΔpr≥ℏ/2\Delta r\Delta p_r\ge\hbar/2, to place a bound on how accurately we can know the angle of the photon from the axis of the telescope. Give your answer in terms of RR and λ\lambda. If you were unable to solve part a, you may also give your answer in terms of Δr\Delta r.

c. Suppose we want to build a telescope that can tell with high probability whether a photon it detected from Alpha Centauri A came the left half or right half of the star. Approximately how large would a telescope have to be to achieve this? Alpha Centauri A is approximately 4×1016 m4\times10^{16}\ \mathrm m from Earth and has a radius approximately 7×108 m7\times10^8\ \mathrm m. Assume visible light with λ=500 nm\lambda=500\ \mathrm{nm}.

Решение

Оригинал в Архива: USAPhO-2018.pdf · официални решения: USAPhO-2018-Solutions.pdf