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Obtaining the interaction energy between two uniformly charged hemispherical surfaces in compact analytical form seems to be an impossible task to achieve under arbitrary conditions. However, we show in this work that one can obtain the interaction energy between two identical hemispherical surfaces with uniform surface charge density for the special condition of them touching each other along the ‘equator’. The mathematical solution method that we apply is remarkable in that the bulk of the treatment is analytic with the only drawback of having to rely on knowledge of certain special functions and their properties.

A common problem in electrostatics is determining the electrostatic potential due to a uniformly charged straight wire. The solution of this problem illustrates well the types of calculations that one must perform in order to obtain the electrostatic potential or field of a given continuous charge distribution. In this work, we reconsider and solve the problem of a uniformly charged straight wire via a new method that is different from the popular direct integration approach found in the majority of physics textbooks. The outcomes of the two methods are compared and the results suggest several interesting mathematical formulas involving special functions.

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We explain a general mathematical method that allows one to calculate the electrostatic potential created by a uniformly charged rectangular plate with arbitrary length and width at an arbitrary point in space. Exact analytical results for the electrostatic potential due to a uniformly charged finite rectangular plate are shown for special cases in order to illustrate the implementation of the formalism. Results of this nature are very important to various problems in physical sciences, applied mathematics and potential theory.