Environmental Engineering Reference
In-Depth Information
Figure 5.9
Two types of magnetohydrodynamic waves: (a) magnetic sound; (b) Alfvén waves.
parameters, and therefore the plasma behavior establishes the dispersion relation
for the electromagnetic wave. We employ Maxwell's equations [38, 39] as they were
represented by Heaviside [40] for the electromagnetic wave:
1
c
@
H
4
c
j
1
c
@
E
curl
E
D
, l
H
D
.
(5.67)
@
t
@
t
Here
E
and
H
are the electric and magnetic fields in the electromagnetic wave,
j
is
the density of the electron current produced by the action of the electromagnetic
wave, and
c
is the light velocity. Applying the curl operator to the first equation
in (5.67) and the operator
t
) to the second equation, and then eliminating
the magnetic field from the resulting equations, we obtain
(1/
c
)(
@
/
@
2
4
c
2
@
j
c
2
@
1
E
r
div
E
Δ
E
C
D
0.
@
t
@
t
2
We assume the plasma to be quasineutral, so div
E
D
0. The electric current is
due to motion of the electrons, so
j
D
eN
0
w
,where
N
0
is the average number
density of electrons and
w
is the electron velocity due to the action of the electro-
magnetic field. The equation of motion for the electrons is
m
e
d
w
/
dt
D
e
E
,which
leads to the relation
@
j
@
e
2
N
0
m
e
E
.
eN
0
d
dt
D
D
t
Hence, we obtain
Δ
E
ω
2
p
c
2
E
C
2
c
2
@
1
E
D
0
(5.68)
@
t
2
for the electric field of the electromagnetic wave, where
p
is the plasma frequency.
Writing the electric field strength in the form (5.48) and substituting it into the
above equation, we obtain the dispersion relation for the electromagnetic wave [41]:
ω
2
2
p
c
2
k
2
.
ω
D
ω
C
(5.69)
If the plasma density is low (
N
e
!
!
0), the dispersion relation agrees with
that for an electromagnetic wave propagating in a vacuum,
0,
ω
p
ω
D
kc
. Equation (5.69)
