634 BELL SYSTEM TECHNICAL JOURNAL 



Part I — General Analysis 



In line with most previous electronic papers consideration is here 

 directed to the behavior of electrons between two parallel planes of 

 practically infinite extent. Differing, however, from earlier works, 

 neither plane is to be regarded as constituting a thermionic emitter or 

 cathode in the general sense. For certain special applications con- 

 ditions may be chosen so that one of the planes coincides with, and 

 assumes the properties of a zero potential cathode, but in developing 

 the general relations this idea is strictly avoided. It will therefore be 

 premised as a starting point that the velocity and the acceleration of 

 the electrons at one of these two planes are given as initial conditions. 

 It will be found that this generalization completely avoids certain 

 ambiguities which were discussed in the February 1935 issue of the 

 Proceedings of the Institute of Radio Engineers.* It also allows appli- 

 cation of the results to be made to a much wider range of devices, 

 including a fairly rigorous treatment of the negative grid triode. 



In the following analysis, and differing from previous references, a 

 change in the units employed has been made so that all quantities are 

 expressed immediately in the practical system of engineering units 

 (amperes, volts, ohms, coulombs, etc.) instead of in the electrostatic and 

 electromagnetic systems. This change has been found to be of great 

 advantage in the use of the equations, since it obviates all necessity for 

 the continuous and irritating transformation of units that accompanies 

 the electrostatic and electromagnetic systems. 



The analysis starts with the expression for the total current density 



/ = pw + e— , (1) 



where I is current density, amperes per cm.^, 

 p is charge density, coulombs per cm.'', 

 u is electron, or charge velocity, cm. per sec, 

 e ispermittivity of a vacuum, which is 1/36x101^ = 8.85 X 10"^^ 



farads/cm., 

 E is the electric intensity, volts per cm. 



The equation of motion of an electron is 



eE = kma, (2) 



where e is electronic charge, coulombs, 

 m is electronic mass, grams, 



