THE UNIVERSITY OF BRITISH COLUMBIA

Physics 108 Final Exam - 23 April 2004

SOLUTIONS

Jess H. Brewer & Erich Vogt

time: 2${1\over2}$ hours

1.

"QUICKIES"   [60 marks - 6 each]

(a)

i.

On the drawing, sketch in the electric field vectors at points A and B. ¹

ii.

If you take the electrostatic potential to be zero infinitely far from the charges,
is the potential at point A positive, negative or zero? ²
(Underline your answer.)

iii.

What about the potential at point B? ³ (Positive, negative or zero?)

(b)

An electron is moving with a finite constant velocity ${\mbox{\boldmath$v$\unboldmath }}$ in a region where there is a uniform, constant electric field ${\mbox{\boldmath$E$\unboldmath }}$ as well as a uniform, constant magnetic field ${\mbox{\boldmath$B$\unboldmath }}$. Of the three vectors ${\mbox{\boldmath$v$\unboldmath }}$, ${\mbox{\boldmath$E$\unboldmath }}$ and ${\mbox{\boldmath$B$\unboldmath }}$, which pairs of vectors must be perpendicular? ⁴

i.

The electric field and the velocity.

ii.

The velocity and the magnetic field.

iii.

The electric field and the magnetic field.

iv.

All the vectors are mutually perpendicular.

v.

No two vectors are necessarily perpendicular to each other.

(c)

The charge Q on any shorted capacitor will exhibit thermal fluctuations. If a 1 F capacitor is in thermal equilibrium at a temperature of 300 K, what is its root mean square charge   $Q_{rms} = \sqrt{\langle Q^2 \rangle}$ ? ⁵

(d)

An infinitely long straight wire passes over another such wire at right angles without touching the other wire, thus forming a rather exotic capacitor. The resultant capacitance is [underline the correct answer] ⁶
(a) zero. (b) nonzero but finite. (c) infinite.

(e)

Suppose that you want to make a plastic camera lens with a nonreflective coating of a transparent material (e.g. diamond) whose index of refraction is higher than that of the lens. What is the optimum thickness of the coating?   Explain. ⁷

(f)

The temperature T of an isolated system

i.

increases as the energy U of the system increases.

ii.

increases as the entropy S of the system increases.

iii.

decreases as the entropy S of the system increases.

iv.

does not depend on the magnitude of either U or S.

v.

increases as the rate of change dS/dU increases.

(Indicate any true statements; explain your reasoning briefly to ensure at least part credit.) ⁸

(g)

A loop of limp wire lies in a heap on a frictionless horizontal surface. What happens when a current is passed through the loop? ⁹

i.

The wire spreads out into a circular loop.

ii.

The wire clumps up into an even tighter tangle.

iii.

Nothing happens.

Explain your reasoning briefly to ensure at least part credit.

(h)

A length of wire carrying a current can be made into a single circular loop or a two-turn circular coil of half the diameter. If B₁ is the resultant magnetic field strength at the centre of the single loop, what is the ratio of B₁ to B₂, the field at the centre of the double loop? ¹⁰ (i) 1. (ii) 2. (iii) 1/2. (iv) 4. (v) 1/4.
Explain your reasoning briefly to ensure at least part credit.

(i)

A uniform current density flows down a long, straight copper tube. Which of the following statements is true? ¹¹

i.

There is a magnetic field inside the tube, but none outside.

ii.

There is a magnetic field ouside the tube, but none inside.

iii.

There is a magnetic field both inside and outside the tube.

iv.

There is no magnetic field either inside or outside the tube.

Explain your reasoning briefly to ensure at least part credit.

(j)

Match up the names on the left and the principles on the right with the equations in the middle. (For each match, draw a connecting line.) ¹²
$\parbox{1.25in}{\raggedright% Lorentz Force \\ [0.75in] Gauss' Law . . . . . . ostatics \\ [0.75in] Amp{\\lq e}re's Law \\ [0.75in] Faraday's Law \\ [0.75in] }$$\parbox{2in}{% \begin{displaymath}\int\!\!\!\!\int_{\cal S}\!\!\!\!\ . . . . . . th}} \cdot d\vec{\mbox{\boldmath$A$\unboldmath}} \end{displaymath}~\\ [0.2in] }$ $\parbox{2.2in}{\raggedright% A charged particle generally follows a . . . . . . es an electric field. ~\\ [0.25in] Moving charges generate magnetic fields. }$

Definitions:	$\vec{\mbox{\boldmath$D$\unboldmath }} \equiv \epsilon \vec{\mbox{\boldmath$E$\unboldmath }}$	$\epsilon \equiv \kappa \epsilon_{_0}$	$\kappa$ = dielectric constant.
	$\vec{\mbox{\boldmath$B$\unboldmath }} \equiv \mu \vec{\mbox{\boldmath$H$\unboldmath }}$	$\mu = \mu_{_0} (1 + \chi_m)$	$\chi_m$ = magnetic susceptibility.

(In vacuum, $\kappa = 1$ and $\chi_m = 0$.)

2.

ELECTRON SPIN IN THERMAL EQUILIBRIUM   [10 marks] An electron in a magnetic field B = 1 T is in thermal equilibrium with a heat reservoir at temperature $\tau$. The electron's magnetic moment $\mu_{\scriptscriptstyle\rm B}$ points in the direction opposite to its spin.

(a)

[3 marks] For what $\tau$ is the electron spin sure to be "up" (along the magnetic field)? ¹³

(b)

[3 marks] For what $\tau$ is its spin sure to be "down" (opposite to the magnetic field)? ¹⁴

(c)

[4 marks] For what $\tau$ is the electron spin equally likely to be "up" or "down"? ¹⁵

3.

LCR CIRCUITS   [10 marks] You are given the following components from which to build a circuit:

• A switch, initially open.
• Two 1 F capacitors, both initially charged to capacity at 1 V.
• Two 2 $\Omega$ resistors.
• Two 1 H inductances.

(a)

[5 marks] Using any combination of these 7 components, draw a circuit that will oscillate at exactly 0.5 s^-1 after the switch is closed. ¹⁶

(b)

[5 marks] If all 7 components are connected in series, describe in quantitative detail what happens after the switch is closed at t=0. ¹⁷

4.

RAIL GUN IN REVERSE   [10 marks]

A vertical bar of length $\ell = 50$ cm is pulled to the right at a constant speed v = 4.20 m/s through a constant, uniform magnetic field B = 0.675 T normal to the plane of the circuit. Assuming that the resistance of the bar and rails is negligible compared to that of the resistor ( $R = 12 \; \Omega$), find the power dissipated in the resistor. ¹⁸

5.

GRATING   [10 marks]

A grating is uniformly illuminated with infrared light of wavelength $\lambda = 1 \; \mu$m incident normal to the plane of the grating, producing the interference pattern shown at left on a screen 1 m away. Make a detailed sketch of the grating itself, showing all dimensions. ¹⁹