Physics 122 Final Exam - 20 April 1995

1.

``QUICKIES''   [16 marks - 2 each]

(a)

Show with sketches how to combine

i.

two identical capacitors to make an equivalent capacitance half as big;

ii.

two identical inductances to make an equivalent inductance half as big.

(b)

What is the direction of any electric field $\vec{E}$ just outside the surface of a conductor, and why?

(c)

Explain briefly why static magnetic fields cannot change the energy of a charged particle, no matter how much they may alter its direction of motion.

(d)

The idea that the ``circulation" of magnetic field ``lines" around an open surface is proportional to the net electric current flowing through that surface corresponds to which of the following Laws of electromagnetism?  [encircle one]

(i) Gauss' Law (ii) Faraday's Law (iii) Ampere's Law

(e)

Visible light is reflected from the surface of a still pool of water (index of refraction n_w = 1.33). If the surface of the water were coated with a film of plastic (index of refraction n_p = 1.44) only 10 nm thick, how would the intensity of the reflected light change, if at all? [A qualitative answer and explanation will suffice.]

(f)

What is the diffraction-limited angular resolution of a telescope with a circular aperture of 1.0 m for light of wavelength 500 nm?

(g)

A multiple-slit interference pattern is shown on a screen. Each of the secondary intensity maxima between the principal maxima has a full width (from the minimum on one side of the little bump to the minimum on the other side) of 6 mm. What is the distance on the screen from the central maximum to the first minimum on one side?

(h)

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

2.

Gauss' Law and Flat Earth   [8 marks] If $\vec{g}$ is the acceleration of a small test mass due to the gravitational attraction of a large accumulation of other mass, we can write GAUSS' LAW FOR GRAVITY in a form analogous to that for electrostatics, with the replacements $\vec{E} \to \vec{g}$ and $Q_{\rm encl}$ (charge within closed surface) $\to M_{\rm encl}$ (mass within closed surface).

(a)

[2 marks] Write down the general form of GAUSS' LAW FOR GRAVITY, including the correct constants and defining any terms not explained above.

(b)

[3 marks] Using the above LAW, derive a formula for the acceleration of gravity outside an infinite flat slab of mass containing $\sigma_m$ mass per unit surface area.

(c)

[3 marks] If a technologically advanced civilization were able to construct such a flat slab - not truly infinite in area, of course, but large enough that ``edge effects" could be ignored near the middle of the slab - out of a material with a uniform mass density $\rho_m$ per unit volume equal to the mean $\rho_m$ of the Earth, how thick must the slab be in order to produce the same surface gravity as we feel at the surface of the Earth? (Express your answer in units of R_E, the Earth's radius.)

3.

AC Circuits   [8 marks]

The circuit shown is driven by an AC power supply generating $V(t) = V_\circ \sin \omega t$, where $V_\circ = 150$ Volts and $\omega = 2\pi \times 60$ Hz. This voltage is applied to a resistance R, a capacitor C and an inductance L, connected in series.

(a)

[2 marks] If L=0, C=0 and $R= 10.0 \; \Omega$, what are the values of the maximum current i_m and the average current $\bar{\imath}$ in the circuit?

(b)

[3 marks] If $R= 10.0 \; \Omega$ and $C= 6.00 \; \mu$F, what value of the inductance L will give the largest possible amplitude of current oscillations in the circuit?

(c)

[3 marks] With the values of R, C and L given (or calculated) in the preceding part, what is the average power dissipated in the circuit?

4.

Electromagnetic Waves   [8 marks] A plane electromagnetic wave (in which the $\vec{E}$ and $\vec{B}$ fields vary only in the direction of propagation) has its electric field in the x direction and its magnetic field in the z direction.

(a)

[2 marks] Based only on the above information, what can you say about the direction of propagation of the wave?

(b)

[3 marks] If at one position in space $E_x = E_\circ \sin \omega t$ and $B_z = B_\circ \sin \omega t$, with $E_\circ = 40.0$ V/m and $B_\circ = 1.334 \times 10^{-7}$ T, what is the total energy U contained in one cubic kilometer of such a plane wave? (Assume that the wavelength is very small compared to a kilometer.)

(c)

[3 marks] The speed of light c is given in terms of the permittivity of free space $\epsilon_\circ$ and the permeability of free space $\mu_\circ$ as

$$c = {1 \over \sqrt{\mu_\circ \epsilon_\circ}} .$$

Prove that the right hand side of this equation has units of velocity.

5.

Diffraction Grating   [10 marks] The interference pattern shown is observed on a screen 1.0 m away from a small, flat, linear grating which is uniformly illuminated by laser light with a wavelength of 500 nm. The grating and the screen are parallel to each other and perpendicular to the direction of the laser beam, which is aimed at the x=0 position on the screen.

Make a drawing of the entire grating, showing all relevant dimensions.

6.

States of Mind and the Heat of Thought   [10 marks] Consider the following grossly over-simplified model: A mind is a system ${\cal S}$ capable of N distinct thoughts, of which only $n \le N$ are actually held in memory at any given time. Any given thought is either in memory or not; there is no middle ground. Furthermore, each thought takes the same amount of ``mental energy" $\varepsilon$, so that a mind with a total available mental energy U will always have exactly $n = U/\varepsilon$ thoughts in memory. A specific set of n thoughts can be considered one ``fully specified state of mind" for ${\cal S}$. We shall make the further (rather insulting) assumption that every possible fully specified state of mind with n thoughts is a priori equally likely.

(a)

[2 marks] How many different fully specified states with n thoughts could occupy a mind which has ``mental room" for N thoughts?

(b)

[2 marks] What is the entropy $\sigma$ of a mind which has the capacity for 12 thoughts but currently contains only 4?

(c)

[2 marks] Explain how to define a mental temperature $\tau$ for a given mind, assuming only that you know how the number of possible fully specified states of that mind depends on its total mental energy. [The specific form of said dependence need not be the one you gave above.]

(d)

[2 marks] If ${\cal S}_1$, which has very little mental energy U₁ but a large mental temperature $\tau_1$, is free to exchange thoughts with ${\cal S}_2$, which has enormous mental energy U₂ but a small mental temperature $\tau_2$, whose mental energy is most likely to increase?  Explain. (Assume both are isolated mentally from the rest of the world.)

(e)

[2 marks] Now suppose that one particular mind is in ``mental equilibrium" with the UBC intellectual community, which we shall assume has a ``mental temperature" of $\tau = 1$ mJ. If it takes that mind an extra mental energy of $\varepsilon = 4$ mJ to have a given thought (in addition to whatever other thoughts it might be having), what is the probability of that specific thought being present in that mind at any given time?