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PHYSICS 232
Final Examination, May 8 , 1997, 9:00am-12:00pm
Instructor: P. Q. Hung

10 points for each problem. READ the problems carefully. SHOW CLEARLY YOUR WORK. DO NOT JUST write the answers down. ANSWERS WITHOUT EXPLANATION WILL BE GIVEN NO CREDIT.

Useful formulas :

$F = k \frac{q_{1} q_{2}}{r^2}$,$\vec{E} = k \frac{q}{r^2} \hat{r}$,$V= k \frac{q}{r}$,$V_B - V_A = -\int_{A}^{B} \vec{E} \cdot d\vec{s}$,$\Delta V = -\vec{E}.\vec{d}$

$\Phi = \oint \vec{E} \cdot d\vec{A} = 4 \pi k Q$,$C= \frac{Q}{V}$,$E = \frac{\sigma}{\epsilon_0}$

$C = \epsilon_0 \frac{A}{d}$,$I = \frac{dQ}{dt}$,R= V/I

$\vec{F} = q \vec{v} \times \vec{B}$,$\vec{F} = I \vec{l} \times \vec{B}$

$B= \mu_{0} I / 2\pi r$,$d\vec{B} = \frac{\mu_{0}}{4\pi} \frac{I d\vec{l} \times \vec{r}}{r^3}$,$B= \mu_{0} I /2 R$

$\oint \vec{B}.d\vec{s} = \mu_{0} I$,$\Phi_{m} = \int \vec{B}.d\vec{A}$,${\cal{E}} = \oint \vec{E}.d\vec{s} = -\frac{d\Phi_m}{dt}$

$v= \sqrt{T/\mu}$,$v= \lambda/\tau = f \lambda$, $k= 2\pi/\lambda$, $\omega = 2\pi f$,$f^{\prime} = f(\frac{v \pm v_{0}}{v \mp v_{s}})$

$sin a + sin b = 2 sin(\frac{a+b}{2}) cos(\frac{a-b}{2})$,$cos a + cos b = 2 cos(\frac{a+b}{2}) cos(\frac{a-b}{2})$

$S_{av} = \frac{E^{2}_{max}}{2\mu_{0} c}$,$S_{av} = \frac{P_{av}}{4\pi r^2}$

$\Delta t = \Delta t_{0} / \sqrt{1- v^2/c^2}$,$L = L_{0} \sqrt{1-v^2/c^2}$

$hf = h f^{\prime} + K$,$K_{max} = h f -\phi$,$\Delta \lambda = \frac{h}{mc} (1- cos \theta)$, $\lambda = \frac{h}{p}$,$E= h f = h c/\lambda$

$U = \frac{1}{2} Q V = \frac{1}{2} C V^2$,

$k = 8.99 \times 10^9 N\cdot m^2/C^2$,$\epsilon_0 = 1/4\pi k = 8.85 \times 10^{-12} C^2/N\cdot m^2 = 8.85 \times 10^{-12} F/m$,1 F = 1 C/V

$e= 1.6 \times 10^{-19} C$,1 T= 1 N/C(m/s), $\mu_{0} = 4\pi \times 10^{-7} H/m$,1 H/m = 1 T.m/A

1 V = 1 T.m²/s, $\mu_{0}= 4\pi \times 10^{-7} N/A^2$,1 W = 1 J/s, 1 V/m = 1 N/A.s= 1 N/C

$\frac{h}{mc} = 2.4 \times 10^{-12} m$,$hc = 1.24 \times 10^{-6} eV.m$,$1 eV = 1.6 \times 10^{-19} J$,$h=6.63\times 10^{-34} J.s$,$c= 3 \times 10^8 m/s$

1) An electron is fired directly toward the ecnter of a large metal plate that has excess negative charge with surface charge density $2.0 \times 10^{-6} C/m^2$. If the initial kinetic energy of the electron is 100 eV and if the electron is to stop (owing to the electrostatic repulsion from the plate) just as it reaches the plate, how far from the plate must it be fired?

2) A certain substance has a dielectric constant of 2.8 and a dielectric strength of $1.8 \times 10^{7}$ V/m. A dielectric strength is the maximum value of the electric field that a dielectric can tolerate without breakdown. If it is used as the dielectric material in a parallel-plate capacitor, what minimum area should the plates of the capacitor have to obtain a capacitance of $7.0 \times 10^{-2} \mu F$ and to ensure that the capacitor will be able to withstand a potential difference of 4000 V?

3) An electron that is moving through a uniform magnetic field has a velocity $\vec{v} = (40 km/s) \hat{i} + (35 km/s) \hat{j}$when it experiences a force $\vec{F} = -(4.2 \times 10^{-15} N) \hat{i} +(4.8 \times 10^{-15} N) \hat{j}$ due to the magnetic field. If B_x = 0, calculate the magnetic field $\vec{B}$ (the full vector and not just the magnitude).

4) Four long copper wires are parallel to each other, their cross sections forming the corners of a square with sides a as shown below. Each wire carries an identical current I out of the page as shown in the figure. What is the force per unit length (magnitude and direction) on any one wire?

5) A small circular loop of area 2.00 cm² is placed in the plane of, and concentric with, a large circular loop of radius 1.00 m. The current in the large loop is changed UNIFORMLY from 200A to -200A (a change in direction) in a time of 1.00 s, beginning at t=0 s. (a) What is the magnetic field at the center of the small circular loop at t=0, t=0.5 s, and t=1.00 s? 4 pts. (b) What emf is induced in the small loop at t=0.5 s? (Since the inner loop is small, we shall assume that the magnetic field $\vec{B}$due to the outer loop is uniform over the area of the small loop.) 6 pts.

6) A stretched string has a mass per unit length of 5.0 g/cm and a tension of 10 N. A sinusoidal wave on this string has an amplitude of 0.12 mm and a frequency of 100 Hz and is traveling toward decreasing x. Write an equation for this wave.

7) Two waves are propagating on the same very long string. A generator at one end of the string creates a wave given by $y = (6.0 cm) cos(\frac{\pi}{2} ((2.0 m^{-1}) x + (8.0 s^{-1}) t))$and one at the other end creates the wave $y = (6.0 cm) cos(\frac{\pi}{2} ((2.0 m^{-1}) x - (8.0 s^{-1}) t))$ . (a) Calculate the frequency, wavelength, and speed of each wave. 2 pts (b) Find the points on the string at which there is no motion (the nodes). 4 pts. (c) At which points is the motion of the string a maximum (the antinodes)? 4 pts.

8) An airplane flying at a distance of 10 km from a radio transmitter receives a signal of intensity $10 \mu W/m^2$. Calculate (a) the maximum amplitude of the electric field at the airplane due to this signal (4 pts); (b) the maximum amplitude of the magnetic field at the airplane (3 pts); (c) the total power of the transmitter, assuming the transmitter to radiate uniformly in all directions (3 pts).

9) A space traveler takes off from Earth and moves at speed 0.99 c toward the star Vega, which is 26 light-years distant ( 1 light-year = $9.46 \times 10^{15}$ m). How much time will have elapsed by Earth clocks (a) when the traveler reaches Vega (2 pts) and (b) when Earth observers receive words from the traveler that she has arrived (2 pts)? How much older will the traveler be (ACCORDING TO HER) when she reaches Vega than when she was when she started the trip (6 pts)?

10) Consider a collision between an X-ray photon of initial energy 50.0 keV and an electron at rest, in which the photon is scattered backward and the electron is knocked forward. (a) What is the energy of the back-scattered photon (7 pts)? (b) What is the kinetic energy of the electron (3 pts)? Express your answer in eV.

BONUS PROBLEM: 3 pts

A toy rocket moves at a speed of 242 m/s directly toward a stationary pole (through stationary air) while emitting sound waves at frequency f = 1250 Hz. (a) What frequency $f^{\prime}$ is sensed by a detector that is attached to the pole (1.5 pts)? (b) Some of the sound reaching the pole reflects back to the rocket, which has an onboard detector. What frequency $f^{\prime\prime}$does it detect (1.5 pts)?