Fourier transform method for the electrostatic self-energy of a solid sphere with uniform volume charge density

Abstract

The problem of a solid sphere with uniform volume charge density is encountered in virtually all undergraduate calculus-based physics textbooks dealing with the topic of electromagnetism. This example illustrates well the use of Gauss’s law and from there one can easily derive all the quantities of interest such as electrostatic field, potential, self-energy, and so on. Undergraduate physics majors are also well aware of the theory of Fourier transforms from having taken mathematics courses. Nevertheless, despite its great utility, the Fourier transform method is rarely mentioned as a powerful tool to solve physics problems at this level. To address this shortcoming, in this work we propose a possible scenario which may allow an instructor to introduce this powerful method to a proper undergraduate audience without any major pedagogical drawback. The case study that we choose is that of a solid sphere with uniform volume charge density. Specifically, we show the calculation of its electrostatic self-energy by using Fourier transform techniques. The main idea of this work is to draw reader’s attention to the versatility of the approach that can, in principle, be applied to other more geometrically complicated bodies where Gauss’s law does not lead to simple solutions. Concurrently, this work also provides instructional approaches that intertwine content-specific and pedagogical viewpoints that can be useful to all undergraduate students and teachers who wish to enhance their command of the subject.

Interaction energy between two identical hemispherical surfaces with uniform surface charge density

“…Now let us assume that some charge is contained in a spherically symmetric body in such a way that its density at a given point does not depend on direction. For instance, assume that a spherical surface is uniformly charged with constant surface charge density or a solid sphere contains charge with constant volume charge density [1]. This is the scenario of a charge distribution with spherical symmetry.…”