用镜像法求解几类典型静电场问题的教学研究

Pedagogy research on image method to solve typical electrostatic field problems

研究经典电磁学中的各类静电和静磁问题一般需要在对应边界条件下求解Laplace方程或者Poisson方程,然而由于直接求解这些方程所用的数学技巧复杂,学生很难从纷繁的数学公式中提炼出清晰的物理图像.本文基于唯一性定理介绍了镜像法的基本概念、物理思想和应用对象,并结合具体的边界条件讨论了如何较为简单地求解几类典型的静电场问题.研究结果表明,恰当地运用镜像法可以降低学生解答某些静电学问题的难度,从而有助于达到较好的教学效果.

The traditional route of researching all electrostatic and magnetostatic problems is to directly solve the relevant Laplace or Poisson equations under their corresponding boundary conditions, however students often feel difficult to draw out clear physical pictures of the considered real cases from the intricate mathematic formulas due to the complexities of mathematical methods. With respect to the difficult students might encounter, the present paper aims to introduce the basic mathematical principles, physical ideas and application objects of image method, which essentially bases on the uniqueness theorem of partial differential equations, and illustrate how to use the image method to more easily solve several kinds of typical electrostatic field problems with specific boundary configurations. Our result shows that properly implementing image method can alleviate the hardships of understanding some electrostatic phenomena and consequently improve teaching effect.