University of California, Berkeley
Physics 110B Spring 2004 (Strovink)
EXAMINATION 1
Directions: Do all three problems, which have unequal weight. This is a closed-book closed-note
exam except for Griffiths, Pedrotti, a copy of anything posted on the course web site, and anything
in your own original handwriting (not Xeroxed). Calculators are not needed, but you may use one if
you wish. Laptops and palmtops should be turned off. Use a bluebook. Do not use scratch paper –
otherwise you risk losing part credit. Show all your work. Cross out rather than erase any work that
you wish the grader to ignore. Justify what you do. Express your answer in terms of the quantities
specified in the problem. Box or circle your answer.
Problem 1. (35 points)
A particle traveling with respect to the lab frame
with velocity βcˆx1has a physical property repre-
sented by the contravariant Lorentz four-vector
hµ. It is known that pµhµ= 0, where pµis a
component of the particle’s four-momentum (ex-
pressed in covariant form). (As usual, repeated
indices are summed.)
Denote by hµthe components of has viewed in
the rest frame of the particle. Based on the in-
formation given above, some of the hµcould be
nonzero.
(a.) (15 points)
Can you tell whether his timelike? Explain.
(b.) (20 points)
As a function of βand of those components of
hwhich could be nonzero, calculate all four
components of hin the lab frame.
Problem 2. (30 points)
In a pair annihilation experiment, a positron
(mass m) with total energy E=γmc2hits an
electron (same mass, but opposite charge) at
rest. The two particles annihilate, producing
two photons. If one of the photons emerges at
angle θrelative to the incident positron direc-
tion, show that its energy is given by
mc2
=1−γ−1
γ+1cos θ.
Problem 3. (35 points)
y
S
β0c
source
y
x, x
S
θ
observer
Consider Lorentz frames Sand S, with spatial
origins coincident at t=t= 0. As usual, frame
Smoves in the ˆx=ˆxdirection with velocity
β0crelative to frame S. A wave is emitted by
a source that is at rest with respect to S.As
seen by an observer who is at rest at the origin
of the lab frame S, the wave shown in the fig-
ure travels with phase velocity βphcat an angle
θwith respect to the ˆxdirection (θ=0ifdi-
rectly approaching, θ=πif directly receding).
However, as seen by an observer who is at rest
with respect to the frame S, show that the wave
makes a different angle θwith respect to the ˆx
direction, where
tan θ=sin θ
γ0(cos θ−β0βph).