USAPhO 2007, Полуфинал — Задача 4. A4

AAPT AIP United States Physics Team 2007 — 2007 Semi-Final Exam · 25 т.

Указания

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN

  • Work Part A first. You have 90 minutes to complete all four problems.

  • After you have completed Part A, you may take a break.

  • Then work Part B. You have 90 minutes to complete both problems.

  • Show all your work. Partial credit will be given.

  • Start each question on a new sheet of paper. Be sure to put your name in the upper right-hand corner of each page, along with the question number and the page number/total pages for this problem. For example,

    Doe, Jamie

    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.

  • Do not discuss the contents of this exam with anyone until after March 27th.

  • Good luck!

Possibly Useful Information - (Use for both part A and for part B)
Gravitational field at the Earth’s surfaceg=9.8 N/kgg = 9.8\ \mathrm{N/kg}
Newton’s gravitational constantG=6.67×10−11 N⋅m2/kg2G = 6.67 \times 10^{-11}\ \mathrm{N\cdot m^2/kg^2}
Coulomb’s constantk=1/4πεo=8.99×109 N⋅m2/C2k = 1/4\pi\varepsilon_o = 8.99 \times 10^{9}\ \mathrm{N\cdot m^2/C^2}
Biot-Savart constantkm=μo/4π=10−7 T⋅m/Ak_m = \mu_o/4\pi = 10^{-7}\ \mathrm{T\cdot m/A}
Speed of light in a vacuumc=3.00×108 m/sc = 3.00 \times 10^{8}\ \mathrm{m/s}
Boltzmann’s constantkB=1.38×10−23 J/Kk_B = 1.38 \times 10^{-23}\ \mathrm{J/K}
Avogadro’s numberNA=6.02×1023 (mol)−1N_A = 6.02 \times 10^{23}\ \mathrm{(mol)^{-1}}
Ideal gas constantR=NAkB=8.31 J/(mol⋅K)R = N_A k_B = 8.31\ \mathrm{J/(mol\cdot K)}
Stefan-Boltzmann constantσ=5.67×10−8 J/(s⋅m2⋅K4)\sigma = 5.67 \times 10^{-8}\ \mathrm{J/(s\cdot m^2\cdot K^4)}
Elementary chargee=1.602×10−19 Ce = 1.602 \times 10^{-19}\ \mathrm{C}
1 electron volt1 eV=1.602×10−19 J1\ \mathrm{eV} = 1.602 \times 10^{-19}\ \mathrm{J}
Planck’s constanth=6.63×10−34 J⋅s=4.14×10−15 eV⋅sh = 6.63 \times 10^{-34}\ \mathrm{J\cdot s} = 4.14 \times 10^{-15}\ \mathrm{eV\cdot s}
Electron massm=9.109×10−31 kg=0.511 MeV/c2m = 9.109 \times 10^{-31}\ \mathrm{kg} = 0.511\ \mathrm{MeV/c^2}
Binomial expansion(1+x)n≈1+nx(1 + x)^n \approx 1 + nx for ∣x∣≪1\lvert x\rvert \ll 1
Small angle approximationssin⁡θ≈θ\sin\theta \approx \theta; cos⁡θ≈1−12θ2\cos\theta \approx 1 - \frac{1}{2}\theta^2

Условие

A nonlinear circuit element can be made out of a parallel plate capacitor and small balls, each of mass mm, that can move between the plates. The balls collide inelastically with the plates, dissipate all kinetic energy as thermal energy, and immediately release the charge they are carrying to the plate. Almost instantaneously, the balls then pick up a small charge of magnitude qq from the plate; the balls are then repelled directly toward the other plate under electrostatic forces only. Another collision happens, kinetic energy is dissipated, the balls give up the charge, collect a new charge, and the cycle repeats. There are n0n_0 balls per unit surface area of the plate. The capacitor has a capacitance CC. The separation dd between the plates is much larger than the radius rr of the balls. A battery is connected to the plates in order to maintain a constant potential difference VV. Neglect edge effects and assume that magnetic forces and gravitational forces may be ignored.

a. Determine the time it takes for one ball to travel between the plates in terms of any or all of the following variables: mm, qq, dd, and VV. [5 т.]

b. Calculate the kinetic energy dissipated as thermal energy when one ball collides inelastically with a plate surface in terms of any or all of the following variables: mm, qq, dd, and VV. [5 т.]

c. Derive an expression for the current between the plates in terms of the permittivity of free space, ε0\varepsilon_0, and any or all of the following variables: mm, qq, n0n_0, CC, and VV. [5 т.]

d. Derive an expression for the effective resistance of the device in terms of ε0\varepsilon_0, and any or all of the following variables: mm, qq, n0n_0, CC, and VV. [5 т.]

e. Calculate the rate at which the kinetic energy of the balls is converted into thermal energy in terms of ε0\varepsilon_0, and any or all of the following variables: mm, qq, n0n_0, CC, and VV. [5 т.]

Отговори

Решение

Оригинал в Архива: olympiad_2007_semi-final.pdf · официални решения: olympiad_2007_semi-final_soln.pdf