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PHY 506
Classical Electrodynamics II
Spring 2011
Final Exam with solutions
Problem F.1
(to be graded of 200 points). Use the Born approximation to calculate the differential
crosssection of scattering of a plane wave normally incident on a very thin, round dielectric disk of
radius
R
. Analyze the result in the limits
kR
<< 1 and
kR
>> 1.
Solution
: This problem is just a particular case of Optional Problem O.10, with
q
z
L
<< 1, and
q
=
k
sin
,
where
is the scattering angle between vectors
k
and
k
0
. (Note that it is different from angle
between
vectors
k
and
E
!) Thus we can immediately write the differential crosssection:
2
1
2
2
4
2
4
2
/
)
sin
(
sin
sin
)
1
(
16
kR
kR
J
R
L
k
d
d
r
.
(*)
If the disk radius is small (
kR
<< 1), this result is reduced to Eq. (8.53), with
V
=
R
2
d
. Taking
into account that
3
8
sin
2
sin
0
3
4
2
d
d
,
the full crosssection, in this limit, is
2
4
2
4
)
1
(
6
r
R
L
k
.
For a large disk (
kR
>> 1), the last factor in Eq. (*) limits scattering to small forward and back
angles, with
sin
~ 1/
kR
<< 1, and sin
1, so that
may be calculated as
.
)
1
(
2
4
)
1
(
16
4
2
/
)
(
)
1
(
16
4
4
2
2
4
2
4
0
2
1
2
4
2
4
0
2
1
2
4
2
4
1
1
r
r
r
R
L
k
d
J
R
L
k
d
kR
kR
J
R
L
k
d
d
d
d
d
d
Thus,
is described by essentially the same formula as at
kR
<< 1, just with a different numerical
coefficient.
Problem F.2
(300 points). A relativistic particle with energy
E
and rest mass
m
collides with a similar
particle, initially at rest in the laboratory reference frame. Find:
(i) the velocity of the center of mass of the system, in the lab frame,
(ii) the total energy of the system, in the centerofmass frame, and
(iii) final velocities of both particles (in the lab frame), if they move along the same direction.
Solutions
:
(i) Before the collision, the total momentum
p
s
of the system in the lab frame equals that (
p
) of
the only moving particle:
2
/
1
2
2
s
)
(
/
mc
c
p
p
E
,
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C:\User\Teaching\506\S11\Fin'sol.doc
while the total energy is
2
s
mc
E
E
.
By definition, the center of mass of a system is an imaginary particle having the same total momentum
p
s
and
total mass
M
s
as the system. (In relativity, the latter is the dynamic mass
M
s
=
E
s
/c
2
.) Applying to
the c.o.m. the general formula discussed in class,
p
=
M
u
, i.e.
=
p
/
Mc
=
p
c
/
E
, we get
1
2
/
1
2
2
2
2
/
1
2
2
s
)
(
/
mc
mc
mc
c
mc
c
E
E
E
E
,
so that
2
/
1
2
2
2
/
1
2
s
2
1
1
mc
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This note was uploaded on 09/10/2011 for the course PHY 506 taught by Professor Stephens,p during the Spring '08 term at SUNY Stony Brook.
 Spring '08
 Stephens,P

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