镜像法求解平面上球形突起电场方法探究

Research on using mirror image method to solve the electric field of spherical projection on the plane

镜像法可以用来求解高对称性系统中的静电场.本文讨论这样一种模型:在无穷大的接地导体平面上有一个球形的突起,点电荷位于系统的对称轴上,与导体面没有接触.通过对球的半径、球心与平面距离、电荷到球顶的最短距离三者关系的讨论,推导出了三种收敛的情形.前两种情况下,无论点电荷处在何种位置,总能得到稳定的解,并且结合MATLAB模拟,给出了空间的电场电势分布.第三种情况无法用镜像法求解,体现了镜像法的局限性.

The mirror image method can be used to solve the electrostatic field in the system with high symmetry.This paper discusses a model in which there is a circular projection on the plane of an infinite grounded conductor,and the point charge is located on the axis of symmetry of the system and has no contact with the surface of the conductor.Through the discussion on the relations among the radius of the circle,the distance between the center of the circle and the plane,and the shortest distance between the charge and the dome,three convergent models are summarized.In the first two cases,no matter where the point charge is,a stable solution can always be obtained,and the spatial electric field distribution is given by combining with MATLAB simulation.The latter case cannot be solved by the mirror image method,which reflects the limitations of the mirror image method.