Review Problems EXAM#2 (Chapters 3-5)
PHY-361 Spring 2007
1.
A 0.200-pm photon scatters from a free electron that is initially at rest. For what
photon scattering angle will be the kinetic energy of the recoiling electron equal the
energy of the scattered photon?
Solution:
The kinetic energy of the recoiling electron is:
)
1
(
2
±
²
²
=
³
c
m
KE
where m is
the rest mass of the electron and
2
2
1
1
c
v
±
=
²
.
The initial energy of the photon (and of the electron at rest) is
0
±
h
(and
2
c
m
±
respectively). The energy of the photon and of the electron after the scattering is
±
h
and
±
²
²
2
c
m
respectively. The conservation of the energy states that:
±
²
²
³
³
+
³
=
³
+
2
2
0
c
m
h
c
m
h
, or in terms of the photon wavelengths:
±
²
²
³
³
+
=
³
+
2
2
0
/
/
c
m
hc
c
m
hc
(1)
Now one has to use the Compton formula for the wavelength of the scattered photon:
))
cos(
1
(
0
±
²
²
³
´
+
=
where
)
(
427
.
2
pm
mc
h
=
=
±
and
±
is the scattering angle of the
photon.
Since the problem says that the kinetic energy of the recoiling electron has to be
equal with the energy of the scattered photon we can write the electron’s kinetic energy:
±
²
³
/
)
1
(
2
hc
h
c
m
KE
=
=
´
µ
µ
=
(2)
From equation (1) the kinetic energy of the recoiling electron is:
±
±
²
/
/
)
1
(
0
2
hc
hc
c
m
KE
³
=
³
´
´
=
(3)
Using equation (2) and (3) we can write:
±
±
±
/
/
/
0
hc
hc
hc
²
=
0
/
/
2
±
±
hc
hc
=
,
±
±
=
0
2
or
)
cos
1
(
2
0
0
±
²
²
³
´
+
=
.
Therefore
±
=
²
/
cos
1
0
³
´
or
9176
.
0
/
1
cos
0
=
±
²
=
³
´
or
o
a
42
.
23
)
9176
.
0
cos(
=
=
±
2.
A particle of rest mass
m
0
and kinetic energy
2m
0
c
2
strikes and sticks to a stationary
particle of rest mass
2m
0
. Find the rest mass
M
0
of the composite particle.
Solution: We have to use the principle of conservation of energy: the initial energy of the
two particles has to be equal with their final energy after their collision.
Their initial energy is the sum:
2
0
2
0
2
1
2
)
(
c
m
c
m
KE
E
E
+
+
=
+
(1)
since particle 2 (mass
2m
0
c
2
) is at rest.
The final energy is the total energy of the composite particle:
2
2
2
0
/
1
/
c
v
c
M
±
where
v
is the velocity of the composite particle.

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