We solve the electrostatic boundary problems of a dielectric or conducting hemispheroid (half-spheroid) under arbitrary excitation. The solutions are obtained by expanding the potentials as series of spheroidal harmonics, and integrating over the boundary to obtain matrix equations which can be used to solve for the coefficients. The solutions are used to derive the capacity, polarizability and spatial fields. We simplify the results to that for a hemisphere, which for specific excitation fields agrees with the literature. We make a link to the T-matrix method, and present analytic expressions for the T-matrix and auxiliary Q and P matrices in the electrostatic limit. We show that the standard T-matrix approach of the extended boundary condition method (EBCM) cannot be used for this geometry, and that the P and Q-matrices do not match the EBCM form.

中文翻译：

---

电介质半球体的静电边界问题和 T 矩阵


我们求解了任意激发下电介质或导电半球体（半球体）的静电边界问题。通过将电位扩展为一系列球谐函数，并在边界上积分以获得可用于求解系数的矩阵方程来获得解。这些解用于推导容量、极化率和空间场。我们将结果简化为半球的结果，对于特定的激发场，这与文献一致。我们链接到 T 矩阵方法，并提供了静电极限中 T 矩阵和辅助 Q 和 P 矩阵的解析表达式。我们表明，扩展边界条件法 （EBCM） 的标准 T 矩阵方法不能用于这种几何结构，并且 P 和 Q 矩阵与 EBCM 形式不匹配。