SPACE CHARGE BETWEEN PARALLEL PLANE ELECTRODES 75 

 Similar analysis for the region beyond the potential zero where 



yields 



z/=^^ (12) 



1 , <p'^' f... 



""+ ~ (2 - Z)i/2 "t" Zi/2 • ^^^^ 



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



1/2 



Zy = 



2 - Z 



+ f 



,3/4 



(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 a- 

 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 

 <p a maximum in (13) with respect to Z holding a constant and obtain: 



.. rz = 2s.i/V(i + «p^'^), (15) 



^ ^ \ tr = (1 + <p^iy^2-"\ (16) 



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

 relationships 7 = 0-^ and Zy = ZcP-. The curve h of Fig. 2 corre- 

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



(b) 0- = 1 + ^^>\ (17) 



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

 is seen to correspond to 7 -^ 00 ; for this the virtual cathode recedes to 

 the first plane and 



(f) • Zy = <p^'\ (18) 



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

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



^ = (2 - zyu'i^ + ^°"^^-' ^^^^ 



which in units of to measured from the first plane with potentials in 

 units of Vi gives 



1 - ^'" ,.0^ 



