Environmental Engineering Reference
In-Depth Information
In the case the electromagnetic wave propagating along the direction of the mag-
netic field on the basis of the above expressions for the density of the electron
currents, we obtain the dispersion relations for the electromagnetic waves with dif-
ferent circular polarizations in the form
2
p
ω
C
C
ω
ω
C
ω
2
p
ω
ω
ω
ω
k
2
C
c
2
2
C
D
0,
k
2
c
2
2
D
C
H
ω
C
H
ω
0 ,
(5.71)
where subscripts
refer to right-handed and to left-handed circular polar-
izations, respectively. On the basis of these dispersion relations, we can analyze the
propagation of an electromagnetic wave of frequency
C
and
ω
in a plasma in an exter-
nal magnetic field. At
z
0 we take the wave to be polarized along the direction
of the
x
-axis, so
E
D
i
E
exp(
D
t
), and we introduce the unit vectors
i
and
j
along
the
x
-axis and the
y
-axis, respectively. The electric field of this electromagnetic wave
in the plasma is
i
ω
2
(
E
C
C
2
i
(
E
C
E
D
i
E
x
C
j
E
y
D
E
)
C
E
).
We use the boundary condition
E
0
e
i
(
k
C
z
i
ω
t
)
E
0
e
i
(
k
z
i
ω
t
)
E
C
D
,
E
D
.
Introducing
k
D
(
k
C
C
k
)/2 and
k
D
k
C
k
,weobtaintheresult
Δ
E
0
e
i
(
kz
ω
t
)
.
i
cos
Δ
kz
2
C
j
sin
Δ
kz
2
E
D
(5.72)
From dispersion relations (5.71) it follows that
"
1
#
,
"
1
#
.
2
p
2
p
ω
ω
2
c
2
2
c
2
C
D
ω
D
ω
k
2
k
2
(5.73)
ω
C
ω
ω
ω
ω
(
H
)
ω
(
H
)
If we assume the inequalities
Δ
k
k
and
ω
ω
, relations (5.73) yield
H
2
p
ω
D
ω
H
c
Δ
k
2
.
(5.74)
ω
This result establishes the rotation of the polarization vector during propagation
of the electromagnetic wave in a plasma. The angle
for the rotation of the polar-
ization is proportional to the distance
z
of propagation. This is a general property
of the Faraday effect. In the limiting case
'
ω
ω
p
and
ω
ω
H
,wehave
p
ω
@'
@
D
Δ
2
D
ω
k
H
.
z
c
2
ω
2
For a numerical example we note that for maximum laboratory magnetic fields
H
H
/
c
is about 10 cm
1
, so for these plasma conditions
the Faraday effect is detectable for propagation distances of the order of 1 cm.
10
4
Gthefirstfactor
ω
