Chapter 5
Cylindrical Cavities and Waveguides
We shall consider an electromagnetic field propagating inside a hollow (in the present case
cylindrical) conductor.
There are no sources inside the conductor, but we shall assume the
material is isotropic with electric permittivity
, and magnetic permeability,
.
The speed of
the propagating wave is 1
/
The direction of propagation will be along the cylindrical axis
which is the
z
̂
direction
We shall assume that
E
(
r
,
t
)
=
E
(
r
)
e
−
i
t
and
B
(
r
,
t
)
=
B
(
r
)
e
−
i
t
.
Maxwell’s equations give:
[∇
2
−
∂
2
∂
t
2
]
E
(
r
.
t
) =
0
[∇
2
−
∂
2
∂
t
2
]
B
(
r
.
t
) =
0
[∇
2
+
2
]
B
(
r
) =
0
[∇
2
+
2
]
E
(
r
) =
0
5.1
5.2
5.3
5.4
Since the wave is propagating along the
z
̂
direction we shall further assume that:
∇ ×
E
=
i
B
(
r
)
;
∇ ×
B
= −
i
E
(
r
)
.
5.5
5.6
Since the wave is propagating along the
z
̂
direction we shall further assume that:
E
(
r
) =
E
(
x
,
y
)
e
±
ikz
B
(
r
) =
B
(
x
,
y
)
e
±
ikz
5.7a
5.7b
Thus Eq. (5.3 and 5.4) become
[∇
t
2
+
2
−
k
2
]
E
(
x
,
y
) =
0
[∇
t
2
+
2
−
k
2
]
B
(
x
,
y
) =
0
5.8b
5.8a
where

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