SPACE CHARGE BETWEEN PARALLEL PLANE ELECTRODES 75 

 Similar analysis for the region beyond the potential zero where 



(,2 Y3I2 



Z7 = ^, (12) 



yields 



"+ ~ {2 - zyi^^ z'l^' ^^ 



Introducing the identity 7 = o-^ in equation (13) gives the relationship 



Z \l/2 



Zy 



1- Z 



^«/^j\ 



+ ^'^' . (14) 



Some of the limiting curves associated with the B solutions are 

 closely related, as is implied by the use of a common letter. Curve c 

 in Fig. 2 corresponds to maximum values of ^p for fixed values of <t 

 and hence of 7 = a^. By inspection this is seen to be the same condi- 

 ion as is implied by curve c in Fig. 9. To find these curves we make 

 if) a maximum in (13) with respect to Z holding o- constant and obtain: 



fZ = 2<v,i/V(l + ^i/2), (15) 



^^ [a = {I + ^1/2)3/22-1/2. (16) 



From these the various other forms of curve c are readily found by the 

 relationships y = a^ and Z7 = Za"^. The curve b of Fig. 2 corre- 

 sponds to Z = 1 and gives in equation (13) 



(b) C7 = 1 + <p'i\ (17) 



The minimum transmitted current for fixed (p, curve / in Figs. 8 and 9, 

 is seen to correspond to 7 ^ co ; for this the virtual cathode recedes to 

 the first plane and 



(f) Z7 = ^'"- (18) 



Introducing dV/dx obtained from equation (7) with the correct 

 current and a zero constant into equation (9) and integrating gives 



cyi/4 

 ^ = (2 - Z) 1/2/1/2 + ^°"^^-' ^^^) 



which in units of /o measured from the first plane with potentials in 

 units of Vi gives 



'- '"' (20) 



(2-Z) 



1/2 



