Environmental Engineering Reference
In-Depth Information
4.5.7
High-Conductivity Plasma in a Magnetic Field
A plasma with high conductivity will have electrons with velocities considerably
greater than the velocities of the ions. Then the electric current is created by elec-
trons and is given by
j
e
D
eN
e
w
e
,
where
w
e
is the drift velocity of the electrons and
N
e
is their number density. If the
motion occurs in a magnetic field, an additional electric field is produced in the
laboratory frame of axes, whose strength is
1
c
[
w
e
1
ecN
e
[
j
H
] .
E
0
D
H
]
D
(4.156)
This field acts on the electrons, giving rise to an additional force acting on the entire
plasma. The force per unit volume of the plasma is
1
c
[
j
H
].
e
E
0
N
e
D
If the plasma conductivity is sufficiently high, its response to the electric field
(4.156) will result in the movement of electrons. This movement will continue un-
til separation of the electrons and ions gives rise to an internal electric field in the
plasma,
1
c
[
w
e
E
D
H
] ,
(4.157)
which will compensate for the field (4.156). We insert (4.157) into the Maxwell
equation
(
@
H
/
@
t
)
D
c
curl
E
, which yields
@
H
@
t
D
curl[
w
e
H
] .
(4.158)
We shall analyze the variation of the magnetic field and the plasma motion when
the electric current is due to electrons and the plasma conductivity is high. We
transform (4.158) by writing curl(
w
e
H
)
D
w
e
div
H
C
(
H
r
)
w
e
(
w
e
r
)
H
H
div
w
e
. Using the Maxwell equation div
H
D
0, and taking the expression for
div
w
e
from the continuity equation for electrons, we obtain
@
H
@
N
e
@
N
e
dt
C
N
e
(
w
e
t
(
w
e
r
)
H
r
)
N
e
D
(
H
r
)
w
e
.
We divide this equation by
N
e
and find that
N
e
N
e
r
d
dt
D
w
e
,
(4.159)
