Ph 214, Spring 2008
Exam # 2
Question
#1
Name:SOLUTIONS
1.
Short answer questions
[20 pts] Answer all three questions and try to limit yourself to the space provided.
Make minimal use of equations and explain clearly. If you need extra space, use page # 3 and clearly identify
which part you are answering.
(a)
[5
pts]
Spacetime
events
The
spacetime
coordinates
(as observed in the ground frame) for three pairs of events
(
A
1
, A
2)
,
(
B
1
, B
2) & (
C
1
, C
2) are shown in the figure at right.
Is it possible to find a frame moving with velocity
v
relative to
the ground in which one of these pairs of events is observed to
take place at the same location? If so, for which pair and what is
the velocity of this frame relative to the ground?
If two events are
timelike separated
[1 pt]
in frame
S
, it is pos
sible to find a frame
S
′
in which the events occur at the same
location but at different times.
Timelike intervals are defined
as those for which (
c
Δ
t
)
2
>
(Δ
x
)
2
−→
(Δ
s
)
2
>
0.
Comput
ing the intervals for our events, we obtain the following results:
(Δ
x, c
Δ
t
) = (
−
2
,
3)
,
(4
,
2) & (4
,
4) for pairs
A
,
B
and
C
, respec
tively. Therefore, only
pair A is timelike separated .
[2 pt]
These
two events occur at the same location if the
ct
′
axis of the new
frame passes through them. The slope of this axis in the
S
frame is
1
β
=
3
−
2
−→
the velocity of this frame
relative to the ground is
v
=
βc
=
−
2
3
c
.
[2 pt]
(b)
[5 pts]
Photon Decay
Is it possible for a single photon to decay into a pair of identical massive particles?
Explain.
No, because it is not possible to conserve both energy and momentum simultaneously in this process.
[2 pt]
E
M, v
M, v
θ
θ
(1)
(2)
(3)
We can see this by writing the 4momenta before and after the decay:
vector
P
1
=
parenleftbigg
E
c
,
E
c
,
0
,
0
parenrightbigg
,
vector
P
2
= (
γMc, γMv
cos
θ, γMv
sin
θ,
0) &
vector
P
3
= (
γMc, γMv
cos
θ,
−
γMv
sin
θ,
0). Conservation of energy implies
E
c
= 2
γMc
and conservation of xmomentum leads to
E
c
= 2
γMv
cos
θ
.
Substituting the first equation into the second gives us
2
γMc
= 2
γMv
cos
θ
−→
v
=
c
cos
θ
≥
c
which is not physically allowed for particles with nonzero rest mass .
[3 pt]
(c)
[10 pts]
Modified Michelson Interferometer
The upper figure
at right shows the standard configuration of a Michelson interferometer with arms of equal length
L
. An engineer
wants to use this device to measure the index of refraction
n
of an unknown liquid. The liquid completely fills
a transparent container of length
d
(
L > d
).
Using monochromatic light of wavelength
λ
, the engineer first
measures the fringe pattern with the standard configuration. The liquid is then inserted along one arm of the
interferometer (lower figure) and a new fringe pattern is measured.
1
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Ph 214, Spring 2008
Exam # 2
Question
#1
Name:SOLUTIONS
⊙
[5pts] What is the difference in light travel time along the arms for
the new configuration? (
Assume that the container is of the same index
as the liquid and of negligible thickness
)
We will refer to the upper arm as #1 and the arm with the liquid as #2.
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 Spring '08
 Faculty
 Special Relativity, pts, ground frame

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