March 16, 2008
PHY 712 – Problem Set # 15
Continue reading Chapter 6 of Jackson.
1. Suppose that an electromagnetic wave of pure frequency ωis traveling along the z-axis
of a wave guide having a square cross section with side dimension acomposed of a
medium having permittivity constant ²and permeability constant µ. It is known to
have the form:
E(r, t) = <½H0eikz−iωt (iµω)π
asin µπx
a¶ˆy¾
H(r, t) = <½H0eikz−iωt ·−ik π
asin µπx
a¶ˆx + cos µπx
a¶ˆz¸¾.
Here H0denotes a real amplitude, and the parameter kis assumed to be real and equal
to
k≡sω2−µπ
a¶2
.
Find the form of the time-averaged Poynting vector
hSiavg ≡1
2< {E(r, t)×H∗(r, t)}
for this electromagnetic wave.