74 BELL SYSTEM TECHNICAL JOURNAL 



This implies that all of the electrons at any plane have the same speed, 

 a condition which must be borne in mind when the present analysis is 

 applied. 



Integration of equation (6) yields 



^^^i(^- + const.) (7) 



The choice of zero for this constant leads to type B distributions. A 

 negative constant gives type C distributions, and some of the type D 

 distributions while a positive value gives type A and type D distribu- 

 tions. These will be considered in detail in the sections which follow. 

 Transit time solutions are obtained by writing the energy equation 

 for an electron in a conservative field which in practical units is 



dt \ mc / c 



Solving for / and introducing numerical values 



t= { Y ^~"' ^^ = 1-68 X 10-8 (v-^'Hx (9) 



= 1.68 X 10-« r (^)~'^~'^'^^- 



Specialization of this equation for the various types is carried out 

 below. 



Integration Constant Zero — Type B 



If the constant in equation (7) is set equal to zero, the next integra- 

 tion gives Child's equation, applicable to the type B distribution when 

 the correct values of current are used, corresponding to conditions 

 before and after the potential zero. Before the potential zero the 

 total current, i.e., the arithmetic sum of injected and reflected currents, 

 is (2 - Z)I so that 



(2-Z)/ = ^, (10) 



X 



where V is the potential x centimeters before the zero. Hence meas- 

 uring distance from the first plane in units of ^o and expressing po- 

 tential in units of Vi, 



1 <P"' ^1^ 



"- (2 - z)i/2 (2 - zy^' ^ ^ 



A 



