We report an exact result for the electrostatic potential energy stored in a hemispherical surface with uniform surface charge density. This system lacks the spherical symmetry of a uniformly charged spherical surface or its solid sphere counterpart. Consequently, a hemispherical surface with all its charge uniformly spread on its surface represents an interesting case study in electrostatics. In this work we combine the fact that the body has axial symmetry with clever mathematical tools to obtain an outcome that is remarkably simple. The final exact expression obtained for the electrostatic potential energy stored in a hemispherical surface with uniform surface charge density answers a long-standing question. Though the bulk of the treatment is analytical, the accuracy of the findings can also be ascertained by using symbolic computation software or numerical recipes. Hence, the present analytical results can be compared to ones obtained with various powerful numerical methods which nowadays are routinely used to check the correctness of analytical and semi-analytical calculations.

我们报告了存储在半球形表面且具有均匀表面电荷密度的静电势能的精确结果。该系统缺乏均匀带电球面或其固体球体对应物的球对称性。因此，一个半球形表面，其所有电荷均匀地分布在其表面上，代表了一个有趣的静电学案例研究。在这项工作中，我们结合了身体具有轴向对称性这一事实和巧妙的数学工具，从而获得了非常简单的结果。对于存储在具有均匀表面电荷密度的半球形表面中的静电势能所获得的最终精确表达式，回答了一个长期存在的问题。尽管大部分处理都是分析性的，结果的准确性也可以通过使用符号计算软件或数字配方来确定。因此，可以将当前的分析结果与使用各种强大的数值方法获得的结果进行比较，而这些方法如今已常规用于检查分析和半分析计算的正确性。