The Complex Formulation of the Equations for Two-Dimensional Space-Charge Flow†

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Abstract
The equations satisfied by an electron for congruent space-charge flow, in the presence of a- magnetic field are quoted. It is shown that if the space-charge is constant along the lines of flow, and the flow is two-dimensional, these space-charge-flow equations may be written in a very simple form using a complex variable formulation. The equations are solved for a constant magnetic field normal to the flow, and flow along conic sections is obtained. The equations are also solved in the presence of space-charge but the absence of magnetic fields, and exact solutions for a periodic flow, similar to periodic focussing, are obtained. Lastly, it is shown that if the flow is along the level lines of a harmonic function, the complex variable formulation may also profitably be used. A solution is obtained in which the flow is along equiangularspirals, and which may be obtained from a space-charge-limited cathode.

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