ECE 329
Homework 6
Due: Tue, March 3, 2009, 5PM
1. Verify that vector identity
H
·
∇ ×
E

E
·
∇ ×
H
=
∇
·
(
E
×
H
)
holds for
E
= 2ˆ
xe

α
z
and
H
= 4ˆ
ye

α
z
by expanding both sides of the identity. Treat
α
as a real
constant.
2. Consider an infinite surface current density
J
s
=

ˆ
xJ
so
cos(
ω
t
)
flowing on
z
= 0
surface, where
J
so
>
0
is realvalued amplitude of the monochromatic surface
current measured in A/m units. It is found that
J
s
injects field energy into propagating transverse
electromagnetic (TEM) waves away from the
z
= 0
plane at an average rate of 1 W/m
2
— that is,
the magnitude of the average Poynting vector
¯
P
is
1
2
W/m
2
for the waves excited by the surface
current.
a) Denoting the TEM waves excited by
J
s
(above and below the
z
= 0
plane) as
E
= ˆ
xE
o
cos(
ω
t
∓
β
z
)
V/m and
H
=
±
ˆ
yH
o
cos(
ω
t
∓
β
z
)
A/m, where wavenumber
β
=
ω
c
, determine the numerical
values of wave amplitudes
E
o
and
H
o
in V/m and A/m units (assuming wave propagation in
free space).
Hint:
equate the expression for the magnitude of
¯
P
for each wave to
1
2
W/m
2
.
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 Spring '08
 FRANKE
 Electromagnet, Electric charge, Fundamental physics concepts, JSO, current density Js, surface current Js

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