Съдържание

Условие Отговори Решение

USAPhO 2002, Полуфинал — Задача 1. A1

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Всички материали от USAPhO 2002

2002 Semi-Final Exam

Указания

INSTRUCTIONS

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. It may not be used in graphing mode. Calculators may not be shared. You may not use any tables, books, or collections of formulas. You may use a ruler or straight edge.
Questions with the same point value are not necessarily of the same difficulty.
Good luck!

Possibly Useful Information

Gravitational field at the Earth’s surface	$g = 9.8\ \mathrm{N/kg}$
Newton’s gravitational constant	$G = 6.67 \times 10^{-11}\ \mathrm{N\cdot m^2/kg^2}$
Coulomb’s constant	$k = 1/4\pi\varepsilon_o = 9.0 \times 10^{9}\ \mathrm{N\cdot m^2/C^2}$
Biot-Savart constant	$k_m = \mu_o/4\pi = 10^{-7}\ \mathrm{T\cdot m/A}$
Speed of light in a vacuum	$c = 3.0 \times 10^{8}\ \mathrm{m/s}$
Boltzmann’s constant	$k_B = 1.38 \times 10^{-23}\ \mathrm{J/K}$
Avogadro’s number	$N_A = 6.02 \times 10^{23}\ \mathrm{(mol)^{-1}}$
Ideal gas constant	$R = N_Ak_B = 8.31\ \mathrm{J/(mol\cdot K)}$
Stefan-Boltzmann constant	$\sigma = 5.67 \times 10^{-8}\ \mathrm{J/(s\cdot m^2\cdot K^4)}$
Elementary charge	$e = 1.6 \times 10^{-19}\ \mathrm{C}$
1 electron volt	$1\ \mathrm{eV} = 1.6 \times 10^{-19}\ \mathrm{J}$
Planck’s constant	$h = 6.63 \times 10^{-34}\ \mathrm{J\cdot s} = 4.14 \times 10^{-15}\ \mathrm{eV\cdot s}$
Electron mass	$m = 9.1 \times 10^{-31}\ \mathrm{kg} = 0.51\mathrm{MeV}/c^2$
Binomial expansion	$(1 + x)^n \approx 1 + nx$ for $\lvert x\rvert \ll 1$

Условие

A wire of length $l$ , radius $r_1$ , and resistivity $\rho$ is tightly wound in a single layer into the shape of a solenoid with circular cross section of radius $r_2$ . Assume it is an ideal solenoid. A DC voltage $V$ is placed across the ends of the solenoid.

a. What is the magnitude of the magnetic field inside the solenoid? [10 т.]

b. What is the self inductance of the solenoid? [10 т.]

c. If the DC voltage source is replaced with an AC source with rms voltage $\mathrm{V_{rms}}$ and frequency $f$ , what is the rms current $I_{\mathrm{rms}}$ through the solenoid? [5 т.]

Отговори

Решение

a. The magnitude of the magnetic field inside an ideal solenoid is $B = \mu_o nI$

where the number of turns per unit length $n$ is the inverse of the length per turn – the diameter $d$ of the wire.

n = \frac{1}{d} = \frac{1}{2r_1}.

The total resistance of the wire used to construct the solenoid is

R = \rho\frac{l}{A_1} = \rho\frac{l}{\pi r_1^{\,2}}.

The current

I = \frac{V}{R} = \frac{V\pi r_1^{\,2}}{\rho l}.

Combining to find

B = \mu_o\left(\frac{1}{2r_1}\right)\left(\frac{V\pi r_1^{\,2}}{\rho l}\right) = \frac{\mu_o \pi V r_1}{2\rho l}

b. The self inductance $L$ can be found from

LI = N\Phi \qquad \text{(A1-1)}

where $N$ is the total number of turns – the length of the wire divided by the circumference of one turn

N = \frac{l}{2\pi r_2}

and $\Phi$ is the flux through one turn

\Phi = B\pi r_2^{\,2} = \mu_o nI\pi r_2^{\,2} = \frac{\mu_o \pi r_2^{\,2}}{2r_1}I

combining with (A1-1)

L = \frac{N\Phi}{I} = \left(\frac{l}{2\pi r_2}\right)\left(\frac{\mu_o \pi r_2^{\,2}}{2r_1}\right) = \frac{\mu_o r_2 l}{4r_1}

c. The inductive impedance is $Z_L = \omega L$ . The total impedance of the circuit is $Z = \sqrt{R^2 + (\omega L)^2}$

I_{rms} = \frac{V_{rms}}{\sqrt{R^2 + (\omega L)^2}} = \frac{V_{rms}}{\sqrt{\left(\frac{\rho l}{\pi r_1^{\,2}}\right)^2 + \left(\frac{2\pi f\mu_o r_2 l}{4r_1}\right)^2}} = \frac{2r_1V_{rms}}{l\sqrt{\left(\frac{2\rho}{\pi r_1}\right)^2 + \left(\pi f\mu_o r_2\right)^2}}

Автори

The 2002 Semi-Final Examination questions were written by the coaches of the United States Physics Team. The coaches are: Academic Director, Mary Mogge – Professor of Physics at California State Polytechnic University, Pomona, CA; Senior Coach, Leaf Turner – Physicist in the Theoretical Division of Los Alamos National Laboratory, Los Alamos, NM; Warren Turner – Physics Teacher at Massachusetts Academy of Math and Science, Worcester, MA; Robert Shurtz – Physics Teacher at the Hawken School, Gates Mills, OH; Boris Zbarsky –undergraduate student at MIT, Cambridge, MA; Andrew Lin – undergraduate student at Yale University, New Haven, CT. Questions A2, A4, B1, and B2 were suggested by Leaf Turner and Questions A1 and A3 by Mary Mogge.

Оригинал в Архива: Semifinal 2002.pdf · официални решения: Semifinal 2002 Solutions.pdf