On the Theory of Space Charge Between Parallel Plane Electrodes

The problem of the potential distribution, current, and electron transit time resulting from the perpendicular injection of electrons into the space between parallel planes is considered. The electrons are assumed to be injected uniformly with velocities corresponding to the potential of the plane through which they are injected. Consideration of all possible solutions of the basic equation shows that four general types of potential distribution are possible. Curves are given which enable the easy calculation of transmitted current and transit time and show the complete potential distribution for any concrete example. The case for current injected through both planes is also considered. The complete mathematical treatment is given in the appendix. C P A C E charge has been studied extensively since the publication of ^ the first papers on the subject by Child 1 in 1911 and Langmuir 2 in 1913. These papers in common with many which followed 3,4,5> 6 - 7 dealt for the most part with potential distributions which occur when electrons are injected into a region with relatively small initial velocities. The problem of space charge between parallel planes when the electrons possess arbitrary initial velocities, although contained implicitly in some of this early work, was first considered in detail by Gill 8 in 1925. Gill appears to have clearly understood the phenomenon but did not publish a complete analysis. Other workers, beginning with Tonks 9 in 1927, have considered various aspects of the problem,10, 11 and recently Plato, Kleen, and Rothe 12,13 published an extensive analysis.