THE UNIVERSITY OF BRITISH COLUMBIA

Physics 108 FIRST MIDTERM - 6 February 2004

SOLUTIONS

Jess H. Brewer

time: 50 min

1.

"QUICKIES"   [60 marks - 10 each]

(a)

In the homework problem on "Stat Ec", a "system" of N shares of a stock worth $\varepsilon$ each has n shares bought, resulting in a net investment of $U = n \varepsilon$ in that stock. (Both $\varepsilon$ and U are measured in monetary units.) If we define the entropy of such a system as the log of the number of different ways n shares could be sold, and the economic temperature of the system in the usual way as the inverse of the rate of change of the entropy with respect to U (i.e. the slope of the curve of entropy as a function of U), then

i.

What fraction of the shares will be sold at infinite economic temperature? ¹

ii.

For what U is the stock "hottest"? ²

iii.

Describe the economic temperature of the stock at that U. ³

(b)

For an ideal gas of N particles in thermal equilibrium, the mean internal energy depends on . . .   [encircle all correct answers] ⁴
(i) temperature (ii) pressure (iii) volume

(c)

To prevent nitrogen bubbles forming in the blood ("the bends"), divers in high pressure chambers often breathe "heliox", a mixture of helium gas and oxygen gas. In a room filled with such a mixture, what is the ratio of the average speed of the ⁴He atoms to that of the ¹⁶O₂ molecules? ⁵

(d)

In a hydrogen atom, the electrostatic force between the proton and electron is $2.3 \times 10^{39}$ times greater than the gravitational force. If we can adjust the distance between the two particles, at what separation will the electrostatic and gravitational forces between them be equal? Explain. ⁶

(e)

An electric dipole, consisting of a 0.01 C positive electric charge 1 cm away from an equal-magnitude negative charge, is located at the centre of a sphere of radius 1 m. What is the average value of the dipole's electric field normal to the sphere's surface? Explain. ⁷

(f)

A finite conducting slab of thickness   d   and area   $A \gg d^2$   has a charge   Q   uniformly distributed over its surface. If the electric field $\vec{\mbox{\boldmath$E$\unboldmath }}$ is measured along an axis normal to the surface and passing near the centre of the slab, match up all the left and right side phrases that make up true sentences: ⁸
$\parbox{2.5in}{\raggedright ~\\ The electric field inside the slab . . . . . . \\ The electric field a distance $\sqrt{A}$\space from the surface of the slab}$ $\parbox{2.5in}{\raggedright cannot be determined from the informatio . . . . . . gnitude $E = Q/\epsilon_0 A$ . \\ ~\\ has a magnitude $E = Q/2\epsilon_0 A$ .}$

2.

Buckyball Settling   [20 marks] Two large meteors made of pure carbon collide just outside the Earth's atmosphere and produce a huge number of "buckyballs" (C₆₀ molecules) that then fall gently into the atmosphere and settle toward the ground. After the buckyballs have had plenty of time to reach thermal equilibrium, assuming perfectly still air at 300 K, at what altitude above sea level will the concentration of buckyballs (per cubic meter of air) be exactly 1/e of their concentration at sea level? (Here e is the base of the natural logarithm.) One buckyball has a mass of $1.1956 \times 10^{-24}$ kg. Assume that the C₆₀ molecules are chemically inert. ⁹

3.

Nonuniform Sphere of Charge   [20 marks] A sphere of radius R has a net charge Q distributed isotropically (but not uniformly) throughout its interior with a charge density $\rho$ (charge per unit volume) that depends linearly on the radius r < R from the centre of the sphere:   $\rho(r) \propto r$,   with $\rho = 0$ at r = 0.

(a)

[5 marks] What is the electric field $\vec{\mbox{\boldmath$E$\unboldmath }}$ outside the sphere (r > R)? ¹⁰

(b)

[15 marks] What is the electric field $\vec{\mbox{\boldmath$E$\unboldmath }}$ inside the sphere ($r \le R$)? ¹¹