170 BELL SYSTEM TECHNICAL JOURNAL 



The field at the potential minimum is zero. The charge which has flowed 

 in behind the electron at the time tis—tlo. Hence, from Gauss's theorem 

 the potential gradient is 



dV/dx = /o t/e (43) 



where e is the dielectric constant of vacuum. We have for the acceleration 



* = ^ ^.. (44) 



me 



If at the time / = (at the potential minimum), x = 0, i; = ^o 



*=-^/^ + *o (45) 



m 2e 



x = -^^t' + XoL (46) 



Now the voltage V between the potential minimum and any point x must 

 be such that 



^2 _ ^2 ^ 2 ^ ^^ (47^ 



m 



r. e \m 2e/ 2e 



2^v (48) 



m 



At any fixed point x^ if we vary xq by a small amount dxa , we find by dif- 

 ferentiating (46) 



dt_ __ t 



dxo fe lo ^2 , .y (49) 



\m2e' +^; 



€ \m2e I 2e 



From (48) 



(50) 



Using (49) 



dV,=^ -^^t'dxo. (51) 



2e 



It now remains to evaluate /. For most cases, the thermal velocities at 

 the p>otential minimum are so small compared with the velocities in most 

 of the region between the minimum and the anode that we can take the 

 value of / for xo = 0. Then, from (45) and (47) 



