Environmental Engineering Reference
In-Depth Information
surface, that is,
0. An electron charge creates an electric field that slows
the electrons. We can analyze this relationship. The electric field strength
E
'
(0)
D
D
d
'
/
dx
satisfies the Poisson equation
i
r
m
e
2
e
dE
dx
D
4
π
eN
e
(
x
)
D
4
π
.
'
Multiplication of this equation by
E
/
dx
provides an integrating factor that
makes a simple integration possible. We obtain
D
d
'
i
r
m
e
'
2
e
E
2
E
0
D
C
16
π
,
(1.30)
where
E
0
E
(0).
We need to establish the boundary condition on the cathode. We consider the
regime where the current density of the beam is small compared with the electron
current density of thermoemission. This means that most of the emitted electrons
return to the metallic surface, and the external electric field does not significant-
ly alter the equilibrium between the emitted electrons and the surface. Then the
boundary condition on the cathode is the same as in the absence of the external
electric field, so
E
(0)
D
0. Formula (1.30) leads to the distribution of the electric
potential in the gap, given by
D
i
r
m
e
2
e
9
2/3
x
4/3
.
'
(
x
)
D
π
This can be inverted to obtain the connection between the electron current density
and the parameters of the gap [27-30]:
r
3/2
0
L
2
2
9
2
m
e
'
e
i
D
.
(1.31)
π
This dependence is known as the three-halves power law. This describes the be-
havior of a nonneutral plasma that is formed in the space between two plane elec-
trodes with different voltages. This voltage difference in this case is determined
by the charge that is created by charged atomic particles [29-31] and has a univer-
sal character; in particular, in a magnetron discharge [32, 33], where electrons are
magnetized and hence reproduction of a charge in a magnetron discharge results
from ion flux to the cathode with energy of hundreds of electronvolts that creates
secondary electrons. Along with this, metal atoms are sputtered, which determines
the applications of this gas discharge.
We consider one more example of transport of the flux of charged particles
through a vacuum: an electron beam of radius
a
that is fixed by a longitudinal
magnetic field inside a cylindrical metal tube of a radius
0
[34]. According to the
Gauss theorem, the electric field strength
E
at a distance
from the beam center
outside the beam surface is
4
π
N
e
2
I
E
D
D
v
e
,
π
e
2
