关键词: 格林函数, 电像法, 边界条件

Abstract: Electrostatic problems in spherical domains are classical problems in electrodynamics and are widely applied in engineering and technology. The method of images, pioneered by Kelvin, has become a central tool for such problems. This paper systematically studies Green’s functions for spherical boundaries. For three types of boundary conditions—Dirichlet, Neumann, and Robin—this paper derives the analytical solutions of Green’s functions both inside and outside the sphere, along with their corresponding image source configurations. For Neumann and Robin boundaries, the research starts from the freespace Green’s function. Legendre polynomial expansions are employed to introduce boundary correction terms. Closedform solutions are obtained through integration. The results show that under these two boundary conditions, the image sources on the Neumann line appear as a uniform and a powerlaw distribution of rays or line segments, respectively.

Key words: Green’s function, method of images, boundary conditions