ECE 329
Homework 6
Due: Tue, Oct 7, 2008, 5PM
1.
a) Prove the identity
H
· ∇ ×
E

E
· ∇ ×
H
=
∇·
(
E
×
H
)
for arbitrary vector Felds
E
and
H
by expanding both sides of the identity under the assumption
that Cartesian components of
E
and
H
are di±erentiable.
b) Another important identity is
∇×∇×
F
=
∇
(
∇·
F
)
∇
2
F
,
where
∇
2
F
on right is known as
Laplacian
of
F
and it is deFned in Cartesian coordinates via
∇
2
F
≡
(
∂
2
∂x
2
+
∂
2
∂y
2
+
∂
2
∂z
2
)(ˆ
xF
x
+ˆ
yF
y
+ˆ
zF
z
)
.
²or the special case of
F
=
x
2
ˆ
x
+
z
ˆ
y
, show that
∇×∇×
F
and
∇
(
∇ ·
F
)
∇
2
F
are equal in
consistency with the general identity quoted above.
2. Consider an inFnite surface current
J
s
=

ˆ
xJ
so
cos(
ωt
)
³owing on
z
=0
surface, where
J
so
>
0
is the
realvalued amplitude of the surface current measured in A/m units. It is found that
J
s
injects Feld
energy into propagating electromagnetic waves away from
z
=0
plane at an average rate of 1 W/m
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 Spring '08
 FRANKE
 Electron, Polarization, Electric charge

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