Forming electrodes for bipolar beams shaped as strips and solid or hollow cylinders

The configuration of forming electrodes for strip and cylindrical bipolar beams is calculated. The method applied for strip beams is based on the exact parametric equations of equipotential surfaces. An analytic extension of the equation for a planar diode and a solution to Laplace’s equation are used in the case of cylindrical beams. The solution to Laplace’s equation is represented in an integral form. The results obtained are compared to approximate solutions to the corresponding problems. The decompositions describing the explicit equations of the zero equipotential surfaces in axially symmetric and plane systems are analyzed. The behavior of these decompositions makes it possible to extend the results to 3D problems for beams having more complicated shapes.