ECE 201A
Homework 8 solution
1.
(a)
Maxwell’s equations in a uniform, source free and isotropic dielectric medium are
0
E
j
H
ωμ
∇×
= −
G
G
H
j
E
ωε
∇×
=
G
G
0
H
•
∇
=
G
0
E
∇ •
=
G
N
(
)
(
)
2
0
0
0
E
E
E
j
H
j
j
E
ωμ
ωμ
ωε
⎛
⎞
∇×∇×
= ∇ ∇ •
− ∇
= −
∇×
= −
⎜
⎟
⎝
⎠
G
G
G
G
G
2
2
0
0
E
E
ω μ ε
∇
+
=
G
G
(b)
ˆ
T
z
a
z
∂
∇ = ∇
+
∂
, where
ˆ
ˆ
T
x
y
a
a
x
y
∂
∂
∇
=
+
∂
∂
.
Hence
2
2
2
2
T
z
∂
∇
= ∇ •∇ = ∇
+
∂
.
If
(
)
,
j
z
E
E x y e
β
−
=
G
G
,
(
)
(
)
2
2
2
2
2
2
2
2
2
,
,
T
T
T
j
z
j
z
E
E x y e
E
E x y
e
E
E
z
z
β
β
β
−
−
⎛
⎞
∂
∂
∇
=
∇
+
= ∇
+
= ∇
−
⎜
⎟
∂
∂
⎝
⎠
G
G
G
G
G
G
Substituting this result in to the wave equation
2
2
2
2
2
0
0
0
T
E
E
E
E
E
ω μ ε
β
ω μ ε
∇
+
= ∇
−
+
=
G
G
G
G
G
, or
(
)
2
2
2
0
0
T
E
E
ω μ ε
β
∇
+
−
=
G
G
.
(c)
For a TEM wave
T
E
E
=
G
G
and
T
H
H
=
G
G
, i.e., fields have only
x
and
y
components.
Then
0
E
j
H
ωμ
∇×
= −
G
G
becomes
0
ˆ
T
z
T
T
a
E
j
H
z
ωμ
∂
⎛
⎞
∇
+
×
= −
⎜
⎟
∂
⎝
⎠
G
G
0
z directed
x and y directed
x and y directed
ˆ
T
T
T
z
T
E
E
a
j
H
z
ωμ
∂
∇
×
+
×
= −
∂
G
G
G
±²³²´
±²³²´
±²³²´
Hence for the equality to hold the z component of this vector equation should vanish.
Therefore
z directed
0
T
T
E
∇
×
=
G
±²³²´
.
Hence
T
E
G
can be taken as the gradient of a scalar function since
(
)
0
T
T
φ
∇
× ∇
≡
.
Then

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