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Classical Electrodynamics
Physics 6561
Homework 4. Due: In class on Tuesday, 9/30
1. Angular momentum of EM ﬁeld
12.19 from Jackson:
Sourcefree EM ﬁelds exist in a localized region of space. Consider the various conserva
tion laws that are contained in the integral of
∂
α
M
αβγ
= 0 over all of space, where
M
is as
deﬁned in class
T
αβ
x
γ

T
αγ
x
β
.
(a) Show that when
β,γ
are both spatial indices conservation of the total ﬁeld angular
momentum follows.
(b) Show that when
β
= 0 the conservation law is
d
~
X
dt
=
c
2
~
P
EM
E
EM
,
where
~
X
is the center of mass of the electromagnetic ﬁeld deﬁned by
~
X
Z
ud
3
x
=
Z
~xud
3
x,
where
u
is the electromagnetic energy density and
~
P
EM
and
E
EM
are the total energy and
momentum of the ﬁelds.
2. Microscopic expression for magnetization
Problem 6.7 from Jackson
The microscopic current
~
j
(
~x,t
) is written as
~
j
(
~x,t
) =
X
j
q
j
~v
j
δ
(
~x

~x
j
(
t
))
.
Just as for the charge density, this current can be broken up into a “free” (conduction)
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 Fall '08
 LIPSON/POLLOCK

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