Environmental Engineering Reference
In-Depth Information
equation for an electric field in a plasma, which has the form
div
E
D
Δ
' D
4
π
e
(
N
i
N
e
).
(1.4)
Here
E
Dr'
is the potential of the electric field,
N
e
and
N
i
are the number densities of electrons and ions, respectively, and ions
are assumed to be singly charged. The effect of an electric field is to cause a re-
distribution of charged particles. According to the Boltzmann formula, the ion and
electron number densities are given by
is the electric field strength,
'
N
0
exp
T
,
N
e
N
0
exp
e
T
,
e
N
i
D
D
(1.5)
where
N
0
is the average number density of charged particles in a plasma and
T
is
the plasma temperature. Substitution of (1.5) into the Poisson equation (1.4) gives
N
0
e
sinh
e
T
.
Δ
' D
8
π
Assuming that
e
'
T
, we can transform this equation to the form
r
D
,
Δ
' D
(1.6)
where
N
e
e
2
1/2
T
r
D
D
(1.7)
8
π
is the so-called Debye-Hückel radius [19].
The solution of (1.6) describes an exponential decrease with distance from the
plasma boundary. For example, if an external electric field penetrates a flat bound-
ary of a uniform plasma, the solution of (1.6) has the form
E
D
E
0
exp
,
x
r
D
(1.8)
where
x
is the distance from the plasma boundary in the normal direction. If the
electron and ion temperatures are different, then (1.5) has the form
N
0
exp
,
N
e
N
0
exp
e
,
e
T
i
T
e
N
i
D
D
and we will obtain the same results as above, except that the Debye-Hückel radius
takes the more general form
4
N
0
e
2
1
T
e
1/2
1
T
i
r
D
D
π
C
.
(1.9)
Now let us calculate the field from a test charge placed in a plasma. In this case
the equation for the potential due to the charge has the form
d
2
dr
2
(
r
1
r
r
D
,
Δ
'
'
)
D
