# Problem Set 1

## Prof. Smoot's Physics 139 class

## Due Week January 25, 1999

1. (1.2 Mould) Light from a distant star approaches the earth at an angle theta.
The light enters the open end of a terrestrial telescope that is inclined
at an angle theta-prime.

(a) Show that:

tan(theta-prime) = sin(theta) /[b + cos(theta)]

where b = v/c, and v is the velocity of the earth relative to
the fixed stars.
Find also the relativisitic formula taking into account length contraction.

(b) Find the expresion for tan(theta-prime), when v/c << cos(theta)

(c) Suppose b = v/c = 0.5.
How would the appearance of the canopy of stars be distorted?
Check for theta = w = 0, 45, 90, 135, and 180 degrees finding theta-prime
in each case.
What value of theta gives theta-prime = 90^{o}.

2. (1.4 Mould) Let the leg of the Michelson-Morley apparatus along the x -axis
be equal
to L_{1}, and the leg along the y axis be equal to L_{2}.
The earth's velocity relative to the ether is v.
Show that the fringe shift for a 90^{o} rotation is:
n = b^{2} (L_{1} + L_{2}) / lambda where b = v /c.
A 90^{o} rotation is the difference in difference in phase shifts
for the interferometer aligned with motion and rotated 90^{o}.

3. (1.5 Mould) Michelson and Morley did not actually use a length L = 11 m
in their apparatus.
Both lengths ( L_{1} and L_{2}) were folded back on themselves
a number of times with mirrors.
Show that if the length L_{1} is folded in two,
with both parts along the x -axis,
the time for light to go back and forth over the complete path is still equal
to:
t = 2 L_{1} / [c (1 - b^{2})] ~ 2 L_{1} (1 + b^{2})/c.

4. (1.6 Mould) Show that FitzGerald's contraction hypothesis explains the
result of the Michelson-Morley experiment (limit your proof to the case
L_{1} = L_{2}).
What happens if L_{1} is not equal to L_{2}?
Does FitzGerald's contraction still explain the experimental result?

5. ( 2-3 French) In a modern Fizeau experiment (see French for a description
of the original experiment), a laser is in one arm of a triangular arrangement
of mirrors (see figure).
A slab of glass (n =1.5) 1 cm thick is inserted in another arm of the system.
The laser light can travel in a closed path in either direction.
When the slab is moved with a speed of 1 cm/sec in the direction indicated.
What is now the optical path difference?
(This difference which can be converted to an audible beat note if the samples
of the counter-circulating beams are suitably combined in a detector.)
Note: this differs importantly from the original Fizeau arrangement in
that the boundaries of the medium change
position while the light passes through.
Your calculation must take account of this fact.