Distribution of potential in cylindrical thermionic valves

Abstract

This paper presents a derivation of the steady-state potential distribution in a space bounded by coaxial cylinders, between which a homogeneous electron current flows. The problem is not formally soluble, but the solution is represented by twenty-four particular solutions covering the whole range of values of the variables likely to occur in practice. The potential equation, derived through the equation of motion of an electron, is reduced to a non-dimensional second-order form. The complete solution of this is obtainable by evaluating a singly infinite set of solutions, as the independent variable has an arbitrary zero. The physical meaning of this form of solution is expressed in similarity theorems. The behaviour of the solution in limiting cases, and the approximate graphical solution of the first-order form, suggested the procedure adopted in solving the second-order equation, which was suitable for integration by means of the differential analyser. In some solutions, a singularity in the potential equation was avoided by numerical integration of the equation of motion, to give initial conditions for the mechanical integration away from the singularity. The solutions are tabulated. Some applications of the results to thermionic valve problems are illustrated by numerical examples.