Science 1 Physics Midterm - 1 March 2001
 
S O L U T I O N S
 
(see Footnotes)
 

1.

QUICKIES   [60 marks - 10 each]

(a)

Encircle all the true completions of the sentence: The temperature T of an isolated system ¹

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.

(b)

A target is placed in a uniform magnetic field $\hbox{\boldmath$\vec{B}$\unboldmath } = B \hat{z}$. You want to hit it with a beam of positive particles of charge q and momentum p, using a ``beam weapon'' that you can place wherever you want, as long as it is not closer to the target than a distance R > 2p/qB. Along which direction(s) can you ``aim'' the beam and hit the target? ²

$+\hat{x}$ $-\hat{x}$ $+\hat{y}$ $-\hat{y}$ $+\hat{z}$ $-\hat{z}$

(c)

Explain in words (no mathematical symbols or equations) why the flux^* of any conserved quantity emitted isotropically at a constant rate from some source must always drop off as the inverse square of the distance from the centre of the source.
(^*flux: the flow of ``stuff'' per unit time per unit area normal to the flow) ³

(d)

A cube 1 m on a side completely surrounds an electric dipole consisting of a positive electric charge of 1 C located 10 cm away from an equal-magnitude negative charge. What is the average value, taken over the whole surface of the cube, of the electric field normal to the surface? Explain your answer. ⁴

(e)

The capacitor in the circuit shown is initially charged to $Q = C {\cal E}$ where ${\cal E}$ is the voltage on the battery. Describe in detail what happens when the switch is closed. ⁵

(f)

The initially uncharged capacitor (C = 1 F) is in series with a battery whose voltage is ${\cal E} = 10$ V and a resistor ( $R = 100 \; \Omega$). Describe in detail what happens when the switch is closed. ⁶

2.

Two-State System   [20 marks] A simple organism has only two possible microstates: ``asleep'' with zero energy or ``awake'' with energy $\varepsilon$. It is in thermal equilibrium with its environment at temperature $\tau \equiv k_{\scriptscriptstyle\rm B}T$.

(a)

For what $\tau$ is the organism 1/e times as likely to be awake as to be asleep? ( $e = 2.718\cdots$) ⁷

(b)

For what $\tau$ does it have an equal probability of being asleep or awake? ⁸

(c)

Sketch The probability of being awake as a function of temperature $\tau$ for a fixed microstate energy $\varepsilon$. ⁹

(d)

Sketch The probability of being awake as a function of the microstate energy $\varepsilon$ for a given temperature $\tau$. ¹⁰

(Include axis labels and vertical scales on your sketches.)

3.

Ampère's Law   [20 marks] A net current of I=1 A flows down a long, empty, thin-walled metal pipe of radius R = 10 cm. If the current density is uniformly distributed over the conducting shell, what is the direction of the vector magnetic field $\vec{B}$ at any point? (Indicate on the sketch and/or in words.) ¹¹

Find expressions for the magnetic field strength B as a function of r, the distance from the central axis. Then plot B(r) from r=0 to r=3R, labelling your axes clearly (including the vertical scale in Tesla). ¹²