Name:

PHYSICS 232
Midterm 2, April 3 , 1998, 5:30pm-6:45pm
Instructor: P. Q. Hung

25 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 (not necessarily applied to all problems below):

$F = k \frac{q_{1} q_{2}}{r^2}$

$\vec{E} = k \frac{q}{r^2} \hat{r}$

$\Phi = \oint \vec{E} \cdot d\vec{A} = 4 \pi k Q$

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

$\vec{F} = I \vec{l} \times \vec{B}$

$d\vec{B} = k_m \frac{I d\vec{s} \times \hat{r}}{r^2}$

$\oint \vec{B} \cdot d\vec{s} = \mu_{0} I$

$\Phi_m = \int \vec{B} \cdot d\vec{A}$

${\cal E} = \oint \vec{E} \cdot d\vec{s} = -\frac{d\Phi_m}{dt}$

$B= \frac{\mu_0 I \Phi}{4 \pi R}$, $\Phi$: angle subtended by arc of circle of radius R

$f^{'} = f_{0}\frac{v \pm v_{o}}{v \mp v_{s}}$, v_o = speed of observer and v_s= speed of source.

$k_m = 10^{-7} T \cdot m/A$

$\omega = 2\pi f$

$v=\omega r$

$\mu_{0} = 4 \pi k_m = 4 \pi \times 10^{-7} T \cdot m/A$

$k = 8.99 \times 10^9 N\cdot m^2/C^2$

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

$k = 8.99 \times 10^9 N\cdot m^2/C^2$

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

$e= 1.6 \times 10^{-19} C$

1) Consider the circuits shown in Fig.1a and Fig.1b. The curved segments are semi-circular arcs with radii indicated in the figures. Find the magnitude and direction of the magnetic field at point P for Fig.1a (12.5 pts) and for Fig.1b (12.5 pts).

2) EQUAL but OPPOSITE currents I travel in the inner and outer wires of a coaxial cable. The radius of the inner wire is R₁. The inside radius of the outer wire is R₂ and its outside radius is R₃. As a function of the distance from the central axis, find the magnetic field (a) inside the inner wire (10 pts); (b) in the region between the two wires (7 pts); (c) outside the outer wire (8 pts). Assume that the current is uniformly distributed in the two wires.

3) 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 200 A to -200 A in a time 1.00 s, beginning at t=0 s, as shown in the figure. (a) What is the magnetic field at the center of the small circular loop due to the current in the large loop at t=0, t=0.5 s, and t=1.00 s (12.5 pts)? (b) What emf is induced in the small loop at t=0.5 s (12.5 pts)? (Since the inner loop is small, assume the field $\vec{B}$ due to the outer loop is uniform over the area of the smaller loop.)

4) A whistle of frequency f₀= 512 Hz moves in a circle of radius 1 m. The frequency of revolution (for the circular motion) is 3.0 rev/s. What are (a) the lowest (12.5 pts) and (b) the highest frequencies (12.5 pts) heard by a listener a long distance away at rest with respect to the center of the circle? Take the speed of sound to be v = 330 m/s.