Calculation of Axially Symmetric Fields

A practical method is developed for calculating the distribution of scalar fields in axially symmetric systems. To obtain a complete solution a knowledge of the potential distribution along a cylinder of constant radius is required; in many cases the distribution along the cylinder can be estimated with sufficient accuracy by an inspection of the electrode configuration. The solution is first obtained in the form of an integral. This is evaluated for the class of boundary potential functions of the form $$V_n\text{(1, z)=}\begin{array}{cc}{z}^n,& \text{z>0}\\ \text{0,}& \text{z<0.}\end{array}$$ In this expression, V_n(1, z) is the potential along a cylinder of unit radius, z is measured along the axis of the cylinder, and n is a positive integer (zero included). Numerical values are given in tabular form for the cases n=0, 1, 2 and 3. It is then shown how the solution for the above boundary potential functions can be used to obtain the solution for an arbitrary distribution of potential along a cylinder of constant radius. In particular, the potential distribution of a simple electron lens consisting of two coaxial cylindrical electrodes of equal diameters is expressed in terms of the tabulated functions. The calculated equipotential lines for one such case are compared graphically with a plot made in an electrolytic plotting tank. The comparison between the calculated and experimental results is quite favorable.