ELECTRON STREAMS IN A DIODE S31 



this general solution, f It involves a wave equation, and the results are in 

 exact agreement with the small-signal calculations for the motions of 

 electrons as individual particles. It is therefore believed that the general 

 solution reconciles the two lines of approach to the theory of electron 

 streams. 



With this solution available, the situation is comparable to that encount- 

 ered in two-dimensional potential theory; assignment of definite functions 

 to two arbitrary functions gives a solution for a particular problem in 

 electron streams, but it is then difficult to determine just what problem 

 has been solved. In the case of small signals the general solution does not 

 greatly shorten the calculations, and it probably should not be regarded as 

 a labor saving tool in comparison to any particular solution when the latter 

 is already known. It is more probable that the broader solution will serve 

 as a guide for general reasonings about electron streams, and as a guide to 

 approximations that can be used in particular problems. 



1. The Parallel Plane Diode 



The diode of this article is shown in Fig. 1. It is two parallel planes indi- 

 cated as (a) and (b), and separated by a distance /. The first plane (a) may 

 be a thermionic cathode that emits electrons, or it may be a grid through 

 which a stream of electrons is injected into the diode. The second plane (b) 

 may be a metallic plate that receives the electrons after they have traversed 

 the diode space, or it may be a grid that permits the electron stream to pass 

 out of the diode. The dimensions of the diode are assumed small compared 

 to the electromagnetic wave-length at any frequency involved, that is, 

 small compared to the velocity of Hght divided by the frequency; and the 

 separation of the planes is assumed small compared to their lateral dimen- 

 sions. Under these conditions the electric intensity is parallel to the x-axis, 

 and the electrons move in that direction only. 



The electron stream injected through the first plane may vary with time, 

 both in charge density and electron velocity; and the voltages at the two 

 planes may also vary with time. The total current flowing in the diode space 

 is then the sum of two components : a conduction current resulting from the 

 motion of electrons, and a displacement current arising from the time rate 

 of change of electric intensity. The displacement current can flow even when 

 there are no electrons in the diode space; it is then the familiar a-c. current, 

 flowing between two plates of a condenser. But, when electrons are present, 

 the two currents interact with each other and they both flow in a compli- 

 cated manner. 



t H. W. Konig also demonstrated the existence of a general solution; and he developed 

 the solution for the particular case of a sinusoidal current.^ 



