Environmental Engineering Reference
In-Depth Information
shows that electromagnetic waves do not propagate in a plasma if their frequencies
are lower than the plasma frequ
ency
ω
p
. A characteristic damping distance for
such waves is of the order of
c
/
q
ω
p
ω
2
according to dispersion relation (5.69).
5.2.6
The Faraday Effect in a Plasma
The Faraday effect manifests itself as a rotation of the polarization vector of an
electromagnetic wave propagating in a medium in an external magnetic field. The
interaction of an electromagnetic wave with the medium results in electric currents
being induced in this medium by the electromagnetic wave, and these currents act
on the propagation of the electromagnetic wave. If this medium is subjected to
a magnetic field, different interactions occur for waves with left-handed as com-
pared with right-handed circular polarization. Hence, electromagnetic waves with
different circular polarizations propagate with different velocities, and propagation
of electromagnetic waves with plane polarization is accompanied by rotation of
the polarization vector of the electromagnetic wave. This effect was discovered by
Faraday in 1845 [42, 43] and is known by his name. The Faraday effect has been
observed in different media [44-48], and the specific characteristic of a plasma is
the interaction between an electromagnetic wave with a gas of free electrons. The
Faraday effect in a plasma will be analyzed below.
We consider an electromagnetic wave in a plasma propagating along the
z
-axis
while being subjected to an external magnetic field. The wave and the constant
magnetic field
H
are in the same direction. We treat a frequency regime such that
we can ignore ion currents compared with electron currents. Hence, one can ig-
nore motion of the ions. The electron velocity under the action of the field is given
by (4.132). The electric field strengths of the electromagnetic wave corresponding
to right-handed (subscript
C
)andleft-handed(subscript
) circular polarization
are given by
iE
y
)
e
i
ω
t
iE
y
)
e
i
ω
t
E
C
D
(
E
x
C
,
E
D
(
E
x
.
Using the criterion
1 for a plasma without collisions, we obtain on the
basis of expressions (4.132) for the electron drift velocities the following current
densities:
ωτ
i
2
p
E
C
iN
e
e
2
E
C
m
e
(
ω
j
C
D
eN
e
(
w
x
C
iw
y
)
D
D
,
ω
C
ω
H
)
4
π
(
ω
C
ω
H
)
2
p
E
i
ω
j
D
eN
e
(
w
x
iw
y
)
D
.
4
π
(
ω
ω
H
)
If we employ in (5.68) the harmonic dependence (5.48) on time and spatial coordi-
nates of wave parameters, we have
4
π
ω
j
c
2
i
ω
E
c
2
2
k
2
E
D
0 .
(5.70)
