Summary

The study of the response of divergence-free electric fields near corners and edges, resembling singularities that accumulate charges, is significant in modern engineering technology. A sharp point can mathematically be modelled with respect to the tip of the one sheet of a double cone. Here, we investigate the behaviour of the generated harmonic potential function close to the apex of a single-sheeted two-hyperboloid with elliptic cross-section, whose asymptote is the corresponding elliptic double cone with one sheet present. Hence, the electrostatic potential problem, involving a single sheet of a two-hyperboloid, is developed using the theory of ellipsoidal-hyperboloidal harmonics, wherein the particular consideration enforces as solution in terms of generalised Lamé functions of non-integer order. A numerical method to determine these functions is outlined and tested. We demonstrate our technique to the solution of a classical boundary value problem in electrostatics, referring to a metallic and charged single-sheeted elliptic two-hyperboloid and its double-cone limit. Semi-analytical expressions for the related fields are derived, all cases being accompanied by the necessary numerical implementation.

中文翻译：

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椭圆形截面的单双曲面和双锥的静电势

概括

对角和边缘附近的无散度电场的响应进行研究（类似​​于累积电荷的奇异点），在现代工程技术中具有重要意义。可以在数学上相对于双圆锥的一张纸的尖端对尖点建模。在这里，我们研究生成的谐波势函数的行为，该函数接近于具有椭圆形横截面的单片双曲线体的顶点，其渐近线是相应的椭圆形双锥，其中存在一张椭圆形。因此，使用椭球-双曲面谐波理论开发了涉及单张双曲面的静电势问题，其中就非整数阶广义Lamé函数而言，必须特别考虑作为解决方案。概述并测试了确定这些功能的数值方法。我们展示了一种解决静电学中经典边值问题的方法的技术，该技术涉及金属和带电单片椭圆双双曲面及其双锥极限。推导了相关领域的半解析表达式，所有情况都伴随有必要的数值实现。