L. G. Raub 



r dr\ dr J 



4Trne, (3) 



where r is the radius of the equipotential surface having the poten- 

 tial V. 



If now we substitute the current per unit length of cathode, 



1 1 = 2Trrj 



for y in (2) we obtain an expression for the difference of potential 

 between the cathode and a point in the gas between the source of 

 ions, To, and the cathode glow, namely, 



dV 

 /i = — 2irrneU-T- • (4) 



or 



Equation (3) is expressed in electrostatic units; in practical 

 units it becomes 



r dr\ dr J 



'■Trne. (5) 



Solving for SV/Sr by use of (4) and (5) and making use of 

 the fact that if r^ro, 8F/8r = o, we have 



-= i.34Xioe(^-j ["^r-J • (6) 



This equation requires that /^ be negative, as it should be, for 

 the direction of flow of the positive electricity is in the direction 

 of decreasing gradient. However, for convenience we wish to use 

 /^ as a positive quantity, and (6) becomes 



dV 



Tr = '-^^ X ' 





Solving (7) for V and making use of the fact that when r^^ro, 

 F=Fo, we have 



(8) 



Fo-F=i.34XioM — I h-o log ^iro~-f-l 



where V is the potential at any radius r between the source of ions 

 r„ and the polarization region at the cathode. 



