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
k q
.
2 H
ω
C ν
2
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