Съдържание

Условие Отговори Решение

USAPhO 1997, Полуфинал — Задача 3. A3

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Всички материали от USAPhO 1997

United States Physics Team — 1997 Exam 2

Указания

Entia non multiplicanda sunt praeter necessitatem

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN

Please show all work, and do not hesitate to use explanatory words in addition to equations. Partial credit will be awarded.
Begin each problem on a fresh sheet of paper. In the upper right-hand corner of each page, write your name, problem number, and the page number/total number of pages for this problem (e.g., "A2, 3/4" for "Problem A2, page 3 of four").
You may use a non-programmable calculator, but may not use any tables, books, notes, or collections of formulas.
Work Part A first. You have ninety 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.

Possibly useful information:

Gravitational field on Earth’s surface: $g = 9.8\ \mathrm{N/kg}$

Gravitational constant: $G = 6.67\times10^{-11}\ \mathrm{N\text{-}m^2 / kg^2}$

Coulomb constant: $k = 1 / 4\pi\epsilon_o = 8.99\times10^{9}\ \mathrm{N\text{-}m^2 / C^2}$

Speed of light in vacuum: $c = 3.00\times10^{8}\ \mathrm{m/s}$

Permeability of empty space: $\mu_o$ , where $\mu_o\epsilon_o = 1/c^2$

Ideal gas constant $R = 8.31\ \mathrm{J / mol\text{-}K}$

Boltzmann’s constant: $k_B = 1.38\times10^{-23}\ \mathrm{J/K}$

Avagadro’s number: $N_A = 6.02\times10^{23}$

1 electron volt = $1\ \mathrm{eV} = 1.60\times10^{-19}\ \mathrm{J}$

Planck’s constant: $h = 6.63\times10^{-34}\ \mathrm{J\text{-}s} = 4.14\times10^{-15}\ \mathrm{eV\text{-}s}$

$(1 - x)^{-1} = 1 + x + x^2 + x^3 + \ldots$ for $|x| < 1$ $\qquad$ $\cos\theta \approx 1 - \theta^2/2$ for $\theta \ll 1$

$(1 + x)^n = 1 + nx + (1/2!)n(n-1)x^2 + \ldots$ for $|x| < 1$ $\qquad$ $\sin\theta \approx \theta$ for $\theta \ll 1$

Условие

The circuit shown consists of:

a 10 volt battery,
a 1.0 µF capacitor,
a 4.0 µF capacitor,
a 50 Ω resistor,
three 25 Ω resistors, and four switches, labeled $S_o$ , $S_1$ , $S_2$ , $S_3$ . All switches are initially open. In all cases assume that the circuit is completely insulated from its surroundings.

Circuit: a top rail feeds three vertical branches through resistors 25Ω, 25Ω and 50Ω; the first branch contains the 4.0 µF capacitor and switch S2, the second the 1.0 µF capacitor and switch S1, the third the 10 V battery and switch S0; a fourth branch on the left with a 25Ω resistor and switch S3 connects the node between the first 25Ω resistor and the 4.0 µF capacitor to the bottom rail.

(a) Switches $S_o$ , $S_1$ , $S_2$ are closed. Switch $S_3$ remains open. After a very long time, what is the charge on each capacitor? [5 т.]

(b) Switch $S_3$ is also closed, so that all four switches are closed. After a very long time, what is the charge on each capacitor? [5 т.]

(c) Switches $S_o$ and $S_3$ are opened simultaneously. Switches $S_1$ and $S_2$ are left closed. After a very long time, what is the charge on each capacitor? [5 т.]

(d) Switches $S_2$ and $S_3$ are opened. Switches $S_o$ and $S_1$ are closed. After a very long time has passed you insert into the 1.0 µF capacitor a slab of dielectric material. After the dielectric is inserted, it totally fills the region between the plates. The material's dielectric constant is 3.0. What is the total work done in inserting the dielectric (that it, what is the net work done by you and the battery)? [5 т.]