50 BELL SYSTEM TECHNICAL JOURNAL 



calculations. In this country Salzberg and Haeff ^^ have presented a 

 solution, apparently quite complete, which has not as yet been pub- 

 lished. The present independently derived solution differs from other 

 published work in the manner of presentation and in an attempt at 

 completeness. The results are presented in the form of curves which 

 give potential distributions, electron transit times, and current-voltage 

 relations, directly in terms of units which have a simple physical 

 significance. 



Statement of the Problem 



In common with some of the earlier treatments, the present dis- 

 cussion will be restricted to a consideration of the steady state potential 

 distributions which can exist in an evacuated region between two 

 parallel planes at known potentials when electrons having normal 

 velocities corresponding to these potentials are injected into the region 

 through one or both planes. For definiteness the case in which all 

 of the current is injected through one plane will be considered first. 

 It will then be shown that the results may be applied to the more 

 general problem. It is supposed that the electrostatic potential is a 

 function of one rectilinear coordinate only and that all the electrons 

 move parallel to this coordinate and have the same total energy. 

 The assumption of equal energy electrons leads to indeterminancy at 

 planes where the potential and its gradient are both zero. At these 

 planes the existence of non-uniform electron velocities will be recog- 

 nized in so far as it provides a selective mechanism to resolve the 

 mathematical indeterminancy.^^ 



Units 



It has been found convenient to express voltages, currents, distances 

 and transit times in terms of some derived units which are related to 

 these quantities in certain ways by the simple Child's law equation 

 for space charge limited current and by the corresponding transit 

 time equation. The use of such derived units makes it possible to 

 present a limited number of curves which are then applicable to a 

 wide variety of conditions and may be used with a minimum amount 

 of computations. The physical significance of the units also simplifies 

 the interpretation of the results. 



In order to understand how these units are obtained consider the 

 hypothetical situation shown in Fig. 1 in which an electron current 

 from a space charge limited cathode is injected into the region to the 



" B. Salzberg and A. V. Haeff, Proc. I.R.E., 25, 546 (May 1937).— Abstract only. 

 This paper appears in full in the January 1938 issue of the R.C.A. Review. 



" This procedure has been justified by Langmuir. See Reviews of Modern Physics, . 

 vol. 3, pp. 237-244 (1931). 



