March 16, 2008
PHY 712 – Problem Set # 15
Continue reading Chapter 6 of Jackson.
1. Suppose that an electromagnetic wave of pure frequency ω is traveling along the z-axis
of a wave guide having a square cross section with side dimension a composed of a
medium having permittivity constant ² and permeability constant µ. It is known to
have the form:
¶ ¾
½
µ
π
πx
ikz−iωt
ŷ
E(r, t) = < H0 e
(iµω) sin
a
a
½
·
µ
¶
µ
¶ ¸¾
π
πx
πx
ikz−iωt
H(r, t) = < H0 e
−ik sin
x̂ + cos
ẑ .
a
a
a
Here H0 denotes a real amplitude, and the parameter k is assumed to be real and equal
to
s
µ ¶2
π
2
.
k≡ ω −
a
Find the form of the time-averaged Poynting vector
1
hSiavg ≡ < {E(r, t) × H∗ (r, t)}
2
for this electromagnetic wave.