SPACE CHARGE BETWEEN PARALLEL PLANE ELECTRODES 77 



The curve a is obtained by making o-+ a maximum with respect to a 

 while holding (p constant. This gives 



( . \a= ^rain. = ^(1 + ip^^''Y\ (30) 



^^^ 1 cr = (1 + V'^/2)3/2, (31) 



Here ^ and a are coordinates of a point on the a curve and a = ^min. 

 is the parameter value for the potential distribution curve tangent to 

 curve a at that point. The o-+ curves give type C solutions for values 

 before the tangent point and give C overlap solutions beyond this 

 point. 



All the curves described in this section are readily transformed to 

 current voltage plots by the relationships 7 = Z7 = a^. 



The transit times for the various curves are found from equations 

 (7) and (9) using the value — (aFi)^'^ for the constant. Integrating 

 and measuring time from the first plane, they are 



TD = + (<^'/2 - «l/')'/2 - (1 - «l/2)l/2. (32) 



Te_ = - (<^l/2 _ ^1/2)1/2 + (1 _ c.l/2)l/2. ^2>Z) 



Te+ = + {ip"^ - a'ly + (1 - a'^y". (34) 



Integration Constant is Positive — Type D 



Type D solutions include those given by equations (23) and (24). 



Other solutions are obtained by giving the integration constant of 

 equation (7) a positive value, say + {^Viy^. Integrating the equa- 

 tion and measuring distances from the first plane in units of ^o we 

 obtain the two possibilities: 



a = -\- i,p''^ - 2,81/2) V.^'/- ^ ^1/2 _ (1 _ 2/31/2) Vl + ;Si^ (35) 

 which applies for tp > 1, and 



a = - (cp'i^ - 2/31/2) VTM^T^ + (1 - 2/31/2) Vr+~^, (36) 

 which applies for ^ < 1. Corresponding transit times are: 



r = -\- {<p"^ + /31/2)l/2 - (1 + ^1/2)1/2. (37) 



r = - (^1/2 + ^1/2)1/2 + (1 + /51/2)./2. (3g) 



Integration Constant is Positive — Type A 



The potential distribution curves of the A type are identical in 

 form with those of the D type, where ^ < 1, which result from the 



