THE UNIVERSITY OF BRITISH COLUMBIA

Physics 108 SECOND MIDTERM - 12 March 2004

SOLUTIONS

Jess H. Brewer

time: 50 min

1.

"QUICKIES"   [60 marks - 10 each]

(a)

An electric field of 100 V/m acts between two parallel capacitor plates separated by an air gap. Without altering the charge on either plate, we now completely fill the space between the plates with strontium titanate, whose dielectric constant is $\kappa = 310$. What is then the electric field between the plates? ¹

(b)

Two resistors have a net resistance of 16 $\Omega$ when connected in series and 3 $\Omega$ when connected in parallel. What are their individual resistances? ²

(c)

Six identical capacitors of capacitance C=1 F and six identical resistors of resistance $R=1 \; \Omega$ are arranged in a cube as shown. Before it is closed at t=0, there is a potential of 1 V across the switch S.

i.

[4 marks] What is the net charge stored in the circuit before the switch is closed? ³

ii.

[4 marks] How long will it take for the net charge on any capacitor to drop to 1/e of its initial value? [ $e \equiv 2.71828\cdots$] ⁴

iii.

[2 marks] Would there be an unique answer to the previous question if one of the capacitors had a different capacitance from the others? Explain. ⁵

(d)

Describe in detail what happens when the switch is closed at t=0 in each of the following circuits, where in each case C = 0.1 F, $R = 10 \; \Omega$ and ${\cal E} = 10$ V.

The capacitor is initially charged to Q = 1 C. ⁶

The capacitor is initially uncharged. ⁷

(e)

A velocity selector has a magnetic field $\vec{\mbox{\boldmath$B$\unboldmath }}$ perpendicular to an electric field of 30,000 V/m. Charged particles with velocity $v = 2.0 \times 10^5$ m/s move through the device undeflected. What is the strength of $\vec{\mbox{\boldmath$B$\unboldmath }}$ [in Tesla]? ⁸

(f)

A proton is trapped in a magnetic mirror system as shown. In its spiraling motion, the proton's total kinetic energy and the magnetic flux encircled by its orbit both remain constant. Explain why the total kinetic energy of the proton remains constant. ⁹

2.

Coaxial Cable   [20 marks] A net current of I=100 A flows down a long, thin central wire and back along a long, thin-walled cylindrical outer conductor of radius R=1 cm, as shown. The returning current density is uniformly distributed over the outer conducting shell.

(a)

[5 marks]   In what direction is the vector magnetic field $\vec{B}$ in various regions? (Indicate on the sketch and/or in words.) ¹⁰

(b)

[10 marks]   Find a complete expression for the magnetic field strength B as a function of r (the distance from the central axis), I and R. ¹¹

(c)

[5 marks]   Plot B(r) from r=0 to r=2R, labeling your axes clearly (including the vertical scale in Tesla and the horizontal scale in meters). ¹²

Many people solved a different problem, usually an adaptation of the homework problem in which the current is uniformly distributed over a solid cylindrical conductor. Others failed to recognize that the same current flows up the central wire in one direction and back down the outer shell in the opposite direction. Still others thought the outer shell was not for the full length of the wire. I suppose I should take greater care in the wording of the question, but it is important to ask if you are confused about the terms of reference. I may not answer all your questions, but I will be glad to resolve unintended ambiguities.

3.

Athletic Potential   [20 marks] A sprinter on the UBC track team runs at a speed of 7 m/s while holding a 6 m long pole vault pole horizontal at right angles to her direction of motion.¹³ The pole has a fine copper wire down the center from one end to the other. If the Earth's magnetic field at the location of the runner has a magnitude of $0.5 \times 10^{-4}$ T and makes an angle of $40^\circ$ with the horizontal, what is the potential difference [in volts] between one end of the pole and the other? ¹⁴