Assumes no initial electron
velocity. 
Assumes no ions (ions generally
increase electron current 100% (halve impedance), but are a small fraction
of total current) 
Length is generally cathode length
plus twice the AK gap to account for spreading of the electron source. 
Can also be expressed: 

where, I is in kA, l and r_{A }in
cm, and V in Volts. (note: mc^{2}/e=0.511MV) 
The first 14 power series
coefficients are as follows: 

0 
0.0 

1 
0.40 

2 
+0.09166667 

3 
0.01424242 

4 
+0.001679275 

5 
0.0001612219 

6 
+0.00001293486 

7 
8.87693E7 

8 
+5.46192E8 

9 
2.94843E9 

10 
+1.36026E10 

11 
7.1101E12 

12 
+2.6644E13 

13 
+1.2526E15 
(from Currents Limited by Space
Charge Between Coaxial Cylinders by Irving Langmuir and Katharine B.
Blodgett, Physical Review II Vol.22, October 1923) 