EAD 234B: E&M
Homework #1
Due Thursday, April 8, 2010
(1)
The electric and magnetic fields at t = 0 are
( )
( )
0
ˆ
ˆ
,
E
f
z x
B
B f
z y
=
=
G
G
with
( )
(
)
2
exp
f
z
az
=
−
a bellshape function.
(a) Find the electromagnetic energy and momentum density and their ratio.
(b) Show that at t > 0 there are two wavepackets travelling in the directions
ˆ
z
and
ˆ
z
−
.
(c) At what ratio
0
0
/
E
B
α
=
is there is just one wavepacket? How much of the initial energy of
the field is carried in the
ˆ
z
and
ˆ
z
−
directions?
(2)
Derive the dispersion relation for the ordinary mode
(
)
0
0
,
E
B
k
B
⊥
&
±
±
±
in a cold plasma
(KT
e
= 0) where the effects of collisions are accounted for through a constant collision frequency
v
.
You
may assume the ions are sufficiently massive that they are fixed and thus only need treat the electron
motion. Assuming real frequency
ω
, show that the index of refraction
n
=
c/v
ph
and the wave number are
now complex.
Find expressions for the attenuation length
δ
= [Im(
k
)]
1
in the three limits (a) low
frequency
v<
ω
<
ω
p
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 Spring '10
 NCL
 Frequency, Wavelength, Fundamental physics concepts, Wave propagation

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