Two methods for solving electrostatic problems with azimuthal symmetry

Abstract The study of electrostatic phenomena is the gateway to the physics described by Classical Electrodynamics. In this paper, we discuss in detail two methods based on the Uniqueness Theorem for solving electrostatic problems with azimuthal symmetry. The first one is the electrostatic potential extension from the axis of symmetry to an arbitrary point. The other consists in the mutual mapping between two potentials through an inversion transformation. We have prepared a list of six examples for which we calculate, completely or partially, the electrostatic potentials for different charge distributions using both methods. The electric field lines are analyzed and presented graphically in all cases.

摘要 静电现象研究是经典电动力学（Classical Electrodynamics）所阐释的物理学领域的入门内容。本文详细探讨了两种基于唯一性定理（Uniqueness Theorem）求解具有方位角对称性的静电问题的方法：第一种方法是将静电电势从对称轴延伸至任意点；第二种方法则是通过反演变换（inversion transformation）实现两种电势间的相互映射。我们整理了六个示例案例，借助这两种方法，针对不同电荷分布完整或部分地求解其静电电势，并针对所有案例均开展了电场线分析与图形化展示。