无限长双曲柱面带电导体的场分布

FIELD DISTRIBUTION OF INFINITELY LONG CONDUCTOR WITH HYPERBOLIC CYLINDERS

静电场问题最终归结于求解拉普拉斯方程,但由于具体问题的边界条件的复杂性,给求解带来了困难。本文引进椭圆柱面坐标,推导出了无限长带电体为用理想导体薄片制成的双曲柱面场分布的解析式。在此基础上计算出了面电荷密度、构成电容器时的电容量、几种特殊情况的场分布,通过与无限大的平板电容器情形比较验证了其正确性。本文可以给大家提供一点启发:根据不同的边界面,选择合适的坐标系,容易求得拉普拉斯方程解,甚至貌似不可求解的问题,通过坐标系变换也能求得解析式子。

The electrostatic field problem is ultimately attributed to solving the Laplace equation,but the complexity of the boundary conditions of the specific problem brings difficulties to the solution.In this paper,the elliptical cylinder coordinates are introduced to calculate the analytical formula of the hyperbolic cylindrical field distribution of the infinitely long conductor that is made of ideal conductor sheets.On this basis,the surface electric charge density,the capacitance of the form of the capacitor,and the field distribution of several special conditions are calculated.The correctness is verified by comparison with the infinite flat capacitor.This paper can give some inspirations.It is easy to find the solution of Laplace equation by choosing the appropriate coordinate system according to different boundary surfaces.Some seemingly unsolvable problems can also be solved by coordinate system transformation.