Homework 4 Answers, 95.658 Spring 2011, Electromagnetic Theory II
Dr. Christopher S. Baird, UMass Lowell
Problem 1
Consider a hollow rectangular waveguide consisting of perfect conducting walls enclosing free space,
with its axis in the
z
dimension, its width in the
x
dimension, and its height in the
y
dimension. Let us
assume that the waveguide's width is so much greater than its height, that all the fields depend only on
y
and
z
. Additionally, assume that whatever source is creating the fields inside the waveguide is only
able to provide waves polarized such that the transverse component of the electric field vector points in
the positive
y
direction. Assume harmonic time dependence at a single frequency. Assume we know the
axial (
z
) components of the fields. Starting with Maxwell's equations, apply all of these assumptions
and derive all of the other components of the fields in terms of the known axial components. Do not go
so far as to apply boundary conditions.
Solution
Assume harmonic time dependence. The wave is free along the waveguide's axis (the parallel direction)
and depends only on
z
and
y
:
E
x , y , z ,t
=
E
y
e
ik z
−
i
t
and
B
x , y , z ,t
=
B
y
e
i k z
−
i
t
We must be careful and remember that
k
is unspecified at this point and is not the overall wave number
of the whole wave.
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 Spring '11
 Staff
 Mass, Work, Magnetic Field, harmonic time dependence

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