Environmental Engineering Reference
In-Depth Information
where
N
e
is the number of electrons per unit beam length,
I
is the electron cur-
rent, and
v
e
is the electron velocity in the beam. On the basis of the equation
E
D
d
'
/
dx
for the voltage difference between the beam and metal tube, this
gives
2
I
e
ln
0
a
,
' D
v
e
whereweassume
a
and ignore the potential variation inside the electron
current. Hence, when the electron beam enters a space inside the tube through
a grid or when it intersects a surface with the electric potential of the walls, each
electron loses energy
e
0
'
. Let the initial energy of electrons in the beam be
E
,sothe
energy becomes
E
e
'
after entry into the tube. This corresponds to the electron
velocity
s
2(
E
e
'
)
D
v
e
m
e
or to the electron current
'
p
(2
e
/
m
e
)(
E
e
'
)
e
'
v
e
I
D
D
.
2ln(
/
a
)
2ln(
/
a
)
From this we find the maximum current that is possible under given conditions,
s
2
3
m
e
E
3/2
ln(
1
3
I
B
D
,
(1.32)
/
a
)
which is called the Bursian current [35] and corresponds to the beam voltage with
respect to the walls:
2
E
3
e
'
D
.
max
The formula for the maximum electric current has the form of the three-halves
power law and may be written in the form
E
3/2
ln(
I
B
D
I
0
,
(1.33)
/
a
)
and if the electron energy is expressed in electronvolts, we have [34]
I
0
A.
In this case of propagation of the electron beam through a vacuum, the electric
potential brakes the electron beam because of the noncompensated charge in the
electron beam [34, 36, 37], which is similar to the previous case of electron ther-
moemission and propagation between plane electrodes. One can expect that neu-
tralization of the beam by introduction of the ion component in the beam region
will remove the current limit. Nevertheless, in the latter case of beam propagation
inside a c
yl
indrical tube the limiting current is conserved, but its value increases
D
12.7
μ
and is 3
p
3
I
B
[34, 37-39].
