68 BELL SYSTEM TECHNICAL JOURNAL 



The solution of (20) is 



Fo= -^'W^ -^^V/^ (22) 



2e Ze 



which is the well-known classical relation between the potential, the 

 current, and the position between two parallel planes where complete 

 space charge exists. The complete space-charge condition is postu- 

 lated by the boundary conditions selected for Uq and the implications 

 involved are discussed by I. Langmuir and Karl T. Compton.^ 



The alternating-current component of the potential is obtained by 

 integration of (21) as follows: 



- ^ Fi = I- {u,dx + U,U, +m, (23) 



m ot J 



whence, from (18), and in complex notation 



y^= - — ?^2 (^ + ^^)(cos ^1 + i sin ^i)[(^ sin ^ -f cos k) 

 e yp^ 



-f ^T^cos ^ - sin I)] 



2ma^^\\( ^ 2..\ ./2 2 . ..\1 



[(^ sin ^ + cos ^) + i{^ cos ^ - sin ^)] 



- cos ^ - i{^ + W - sin ^) 



+ constant. (24) 



With the attainment of (24), the fundamental relation between the 

 alternating-current component /i and the alternating-current poten- 

 tial Vi in the idealized parallel plate diode has been secured. In a 

 more general sense the equation is applicable between any two fictitious 

 parallel planes where one is located at an origin where the boundary 

 conditions for Uq are satisfied; namely, that the direct-current com- 

 ponents of the velocity and acceleration are zero, and the value of the 

 alternating-current velocity at a point, .ri, corresponding to the transit 

 angle, ^i, is given by M sin pt -\- N cos pt, or by M -f iN in complex 

 notation. 



Equation (24) contains an additive constant which always appears 

 in potential calculations. This constant disappears when the potential 

 difference is computed. For instance, suppose the potential difference 

 between planes where ^ has the values ^ and ^', respectively, is desired. 



" I. Langmuir and Karl T. Compton, "Electrical Discharges in Gases" — Part II, 

 Rev. Mod. Phys., Vol. 3, p. 191; April (1931). 



i 



