Environmental Engineering Reference
In-Depth Information
In the limit case
1, the total current is directed perpendicular to both the
electric and the magnetic fields. In this case the plasma conductivity and electric
current do not depend on the collision time because the change of the direction of
electron motion is determined by the electron rotation in a magnetic field. In this
case Ohm's law has the form
ω
τ
H
ecN
e
E
x
i
y
D
H
.
We consider the case
M
/
m
e
,where
M
is the atom mass when the vari-
ation of the electron energy is small during the rotation period in the magnetic
field. If the transverse electric current does not reach the plasma boundary, this re-
sults in a separation of charges that in turn creates an electric field that slows and
eventually stops the electrons. This gives rise to an electric current in the direction
perpendicular to the electric and magnetic fields that is the Hall effect [108].
We now consider the kinetic description of the Hall effect. By analogy with the
case when an electron is moving in a gas in an electric field and the electron dis-
tribution function has the form (3.10), let us represent the electron distribution
function as
ω
τ
H
f
(
v
)
D
f
0
(
v
)
C
v
x
f
1
(
v
)
C
v
y
f
2
(
v
).
The electric and magnetic fields are taken to be constant and mutually perpendic-
ular, with the electric field along the
x
-axis and the magnetic field along the
z
-axis.
By analogy with the case of the constant electric field, when the electron drift veloc-
ity is given by (3.26), for the electron distribution functions we now have
a
df
0
d
a
ω
df
0
d
v
H
f
1
D
,
f
2
D
,
(4.135)
v
v
2
H
)
2
H
)
(
ν
2
C
ω
v
(
ν
2
C
ω
v
where
a
N
a
v
σ
ea
istherateofelectroncollisionswithatoms.
These equations lead to the following expressions for the components of the elec-
tron drift velocity:
D
eE
/
m
e
,and
ν
D
1
v
,
w
y
1
v
.
2
3
eE
3
m
e
d
d
ν
v
eE
3
m
e
d
d
ω
v
H
w
x
D
D
2
2
H
2
2
H
ν
3
C
ω
ν
2
C
ω
v
v
(4.136)
In the limit
, the first of these expressions transforms into (3.26).
In the case when the rate of electron-atom collisions
ω
ν
H
is independent
of the electron velocity or for a hydrodynamic description of electron motion in
crossed electric and magnetic field we have in the limit
ν
D
1/
τ
ω
τ
1
H
eE
x
m
e
c
E
x
w
y
ω
w
y
D
H
D
H
,
w
x
D
.
(4.137)
ω
τ
H
τ
As is seen, in the limit
1 an electron (or a charged particle) is moving
in crossed fields in the direction that is perpendicular to the field directions, and
ω
H
