Axisymmetric Electric Field Calculation with Zonal Harmonic Expansion

The electric potential and field of an axially symmetric electric system can be computed by expansion of the central and remote zonal harmonics, using the Legendre polynomials. Garrett showed the usefulness of the zonal harmonic expansion for magnetic field calculations, and the similar radial series expansion has been widely used in electron optics. In this paper, we summarize our experience of using the zonal harmonic expansion for practically interesting axisymmetric electric field computations. This method provides very accurate potential and field values, and it is much faster than calculations with elliptic integrals. We present formulas for the central and remote expansions and for the coefficients of the zonal harmonics (source constants) in the case of general axisymmetric electrodes and dielectrics. We also discuss the general convergence properties of the zonal harmonic series (proof, rate of convergence, and connection with complex series). Practical considerations about the computation method are given at the end. In our appendix, one can find many useful formulas about properties of the Legendre polynomials, various derivatives of the zonal harmonic functions, and a simple numerical integration algorithm.