SPACE CHARGE BETWEEN PARALLEL PLANE ELECTRODES 77 



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

 while holding (p constant. This gives 



. . J « = <Pmin. = ^(1 + <p"')-\ (30) 



^^^ 1 ,r = (1 + <p'''y'\ (31) 



Here <p and a are coordinates of a point on the a curve and a = <pmin. 

 is the parameter value for the potential distribution curve tangent to 

 curve a at that point. The a+ 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 = Zy = 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 



T^ = + (<^l/2 _ ^1/2)1/2 - (1 - al/2)l/2. (32) 



Tc- = - (^1/2 - a^/2)i/2 + (1 - ai/2)i/2. (33) 



r,+ = + (^1/2 - ai/2)i/2 + (1 - ai/2)i/2. (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 + (/SFO^'^. Integrating the equa- 

 tion and measuring distances from the first plane in units of 5o we 

 obtain the two possibilities: 



a = + (^1/2 - 2/31/2) V^l/2 _^^l/2 _ (1 _ 2)31/2) Vl + ^1/2^ (35) 



which applies for ^ > 1, and 



a = - (^1/2 _ 2/31/2) V^i^2+7i72 + (1 - 2)81/2) Vr+^, (35) 

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



r = + (^1/2 + /31/2)l/2 - (1 + /31/2)l/2. (37) 



X = - (^1/2 + /31/2)l/2 -I- (1 + /31/2)'/2, (38) 



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 (p < 1, which result from the 



