## Description

Problem 1. A cylindrical conductor with radius a and resistivity ρ carries a constant current I.

(a) What are the magnitude and direction of the electric field vector E and the magnetic

field vector B at a point just outside the wire and at a distance a from the axis of

the cylinder?

(b) What are the magnitude and direction of the Poynting vector S at the same point?

(c) Find the rate of flow of energy into the volume occupied by a length l of the conductor.

Hint. What is the physical meaning of the Poynting vector integrated over a closed

surface?

(d) How does your result compare to the rate of generation of thermal energy in the same

volume? Why does the Poynting vector point in the direction found in (b)?

(2 + 2 + 1/2 + 1/2 + 1 marks)

Problem 2. Light traveling downward is incident on a horizontal film of thickness d as shown in the

figure below. The incident ray splits into two rays, A and B. Ray A reflects from the top

of the film. Ray B reflects form the bottom of the film and then refracts back into the

material that is above the film. If the film has parallel faces, show that rays A and B end

up parallel to each other.

(3 marks)

Problem 3. An inside corner of a cube is lined with mirrors to make a corner reflector. A ray of light

is reflected successively from each of three mutually perpendicular mirrors. Show that its

final direction is always exactly opposite to its initial direction.

(4 marks)

Problem 4. We want to rotate the direction of polarization of a beam of linearly polarized light by 90o

using one ore more polarizing sheets.

(a) What is the minimum number of sheets required?

(b) What is the minimum number of sheets required if the transmitted intensity is to be

more than 60% of the original density.

(1 + 3 marks)

Problem 5. For interference of waves from two coherent sources show that the nodal lines and the

antinodal lines are families of hyperbolas.

(2 marks)

Problem 6. Consider a two-slit interference experiment in which the two slits are of different widths.

As measured on a distant screen, the amplitude of the wave from the first slit is E, while

the amplitude of the wave from the second slit is 2E.

(a) Show that the intensity at any point in the interference pattern is

I = Imax

5

9

+

4

9

cos φ

,

where φ is the phase difference between the two waves as measured at a particular

point of the screen and Imax is the maximum intensity in the pattern.

(b) Graph I versus φ. What is the minimum value of intensity, and for which values of φ

does it occur?

(3 + 1 marks)

Problem 7. Consider a two-slit interference pattern, for which the intensity distribution is given by

formula I = Imax cos2

(πdy/λl) (for the meaning of the symbols, see the lecture). Let θm

be the angular position of the m-th bright fringe, where the intensity is Imax. Assume that

θm is small, so that sin θm ≈ θm. Let θ

+

m and θ

−

m be the two angles on either side of θm

for which I =

1

2

Imax. The quantity ∆θm = |θ

+

m − θ

−

m| is the half-width if the m-th fringe.

Calculate ∆θm. How does it depend on m?

(4 marks)

Problem 8. A compact disk (CD) is read from the bottom by a semiconductor laser with wavelength 790

nm passing through a plastic substrate of refractive index 1.8. When the beam encounters

a pit, part of the beam is reflected from the pit and part from the flat region between

the pits (see the figure), so these two beams interfere with each other. What must the

minimum pit depth be so that the part of the beam reflected from a pit cancels the part

of the beam reflected from the flat region? (It is this cancellation that allows the player

to recognize the beginning and the end of a pit.)

(4 marks)