Environmental Engineering Reference
In-Depth Information
dw
x
dt
C
w
x
τ
D
dw
y
dt
C
w
y
τ
D
ω
a
x
e
i
ω
t
C
ω
H
w
y
,
H
w
x
,
dw
z
dt
C
w
z
τ
D
a
z
e
i
ω
t
,
(4.131)
where the Larmor frequency is
ω
D
eH
/(
m
e
c
)and
a
x
D
eE
x
/
m
e
,
a
z
D
H
eE
z
/
m
e
. This set of equations has the following steady-state solution that is inde-
pendent of the initial conditions:
)
a
x
e
i
ω
t
2
a
x
e
i
ω
t
τ
(1
C
i
ωτ
ω
τ
H
w
x
D
C
ω
τ
,
w
y
D
C
ω
τ
,
2
H
2
H
1
ω
2
2
C
2
i
ωτ
1
ω
2
2
C
2
i
ωτ
a
z
e
i
ω
t
D
τ
w
z
.
(4.132)
1
C
i
ωτ
Expressions (4.132) for the electron drift velocity allow one to determine the con-
ductivity tensor
Σ
α
that is introduced in terms of a generalized Ohm's law that
establishes the connection between components the current density vector
i
α
and
the electric field strength vector
E
in the form
i
α
D
Σ
α
E
,
where
Σ
α
is the conductivity tensor. This equation is valid at low electric field
strengths, so the electron drift velocity is small compared with electron thermal
velocity. Because the electron current density is
i
α
D
eN
e
w
e
,
we have the general expression for the conductivity tensor of an ionized gas:
0
@
1
A
1
C
i
ωτ
ω
τ
0
H
C
(
ω
2
)
τ
2
C
2
i
ωτ
C
(
ω
2
)
τ
2
C
2
i
ωτ
2
H
2
H
1
ω
1
ω
ω
τ
1
C
i
ωτ
Σ
α
D
Σ
0
.
(4.133)
H
0
C
(
ω
2
H
ω
2
)
τ
2
C
2
i
ωτ
C
(
ω
2
H
ω
2
)
τ
2
C
2
i
ωτ
1
1
1
0
0
1
C
i
ωτ
N
e
e
2
where
/
m
e
is the plasma conductivity in a constant electric field for the
tau approximation in accordance with (3.42). This expression may be used for the
analysis of the plasma interaction with external fields.
Σ
D
τ
0
4.5.2
Hall Effect
In the absence of a magnetic field, the plasma conductivity is a scalar quantity.
When a magnetic field is applied to a weakly ionized gas, the conductivity acquires
a tensor character. This means that an electric current can occur in directions in
which the electric field component is zero. If we consider the case where a con-
stant electric field strength is perpendicular to a magnetic field, then (4.133) yields
(
ω
D
0)
1
ω
τ
H
Σ
D
Σ
D
Σ
,
Σ
D
Σ
D
Σ
.
(4.134)
xx
yy
0
yx
xy
0
2
H
2
H
1
C
ω
τ
2
1
C
ω
τ
2
