Environmental Engineering Reference
In-Depth Information
In considering this plasma in an external magnetic field, we employ the usual
geometry, where the electric field is directed along the
x
-axis and the magnetic field
is directed along the
z
-axis. Then the electron drift velocity
w
in crossed electric and
magnetic fields is given by (4.136), and in the case when the rate of electron-atom
collisions is independent of the electron velocity
const, it is convenient to
reduce the indicated formulas to the following vector form:
ν
D
e
E
ν
e
E
ω
H
w
D
2
)
C
,
(5.140)
2
H
2
H
m
e
(
ω
C
ν
m
e
(
ω
C
ν
2
)
where the vector
e
H
/(
m
e
c
) is parallel to the magnetic field.
The physical framework for this problem is such that electrodes collect the elec-
tron current in the
x
direction; hence, there is no electric current along the
y
-axis.
The electron drift velocity along the
x
-axis creates the electric field along the
y
-axis,
where the strength of this electric field is
ω
D
H
m
e
e
w
m
e
e
ω
E
D
w
,
(5.141)
H
as follows from (5.140).
A perturbation in the electron number density causes a perturbation of the elec-
tric field strength and the electron drift velocity. We can determine these perturba-
tions from (5.138), which is now
dW
e
dt
dW
e
dt
eEwN
e
eE
0
wN
e
D
C
,
(5.142)
T
where (
dW
e
/
dt
)
T
3
N
e
T
e
(
m
e
/
M
) is established by energy changes in electron-
atom collisions, and was evaluated above.
The relationship between perturbations
w
0
and
N
e
can be derived from the
continuity equation (4.1) for electrons. This can be used in the steady-state form
div(
N
e
w
)
D
0 since the instability develops slowly. Writing the dependence on
r
in
the form exp(
i
D
k
r
), where
is the wave vector, we obtain
k
N
e
(
w
0
k
)
N
e
(
w
k
)
C
D
0 .
(5.143)
We can now apply (5.140) to eliminate the perturbed drift velocity of the electrons
from this expression. First, we find the direction of
w
0
. Since the perturbation de-
velops slowly, we can write
E
0
D
r
'
0
,where
'
0
is the perturbation of the electric
k
'
0
,thatis,
the vectors
E
0
and
k
are either parallel or antiparallel. Then, using the relation-
ship (5.140) between vectors
w
0
and
E
0
,weobtain
'
0
exp(
i
k
r
)andobtain
E
0
D
i
potential. As before, we assume
.
k
k
ω
H
w
0
D
const
ν
Multiplying vector
w
0
by itself, we evaluate the constant in the above expression,
and obtain
w
0
ν
k
(
k
ω
H
)
w
0
D˙
k
q
.
2
H
ω
C
ν
2