329
5.6. Space charge limited emission
is expressed as U
� (0) = 0. From before we also have U (0) = 0,
in which case, the integration constant is zero. Taking the square
root of both sides, we obtain
√
U
−1/4 dU = 2 α dx.
(5.125)
Integrating the left side between the limits U = 0 and U = U a ,
and integrating the right side between the limits x = 0 and x = s,
4
√
U
3/4
a
= 2 α s.
(5.126)
3
Squaring both sides, substituting for α, and rearranging factors,
we obtain the desired expression for the magnitude of the current
density j as
U
3/2
2 e
4 f 0
a
j =
2
.
(5.127)
9
m s
This is known as the Child–Langmuir equation for space charge
limited emission.