On some nonlinear extensions to Maxwell's electrodynamics

Abstract

Soon after Maxwell's celebrated synthesis of electrodynamics, the discovery of the electron raised fundamental questions about the nature and structure of elementary particles. The picture of a small, rigid, charged body of finite radius faced the difficulty of introducing non-electromagnetic forces or stresses to prevent the charge from dispersing, an assumption that was increasingly regarded as unsatisfactory. The alternative description in terms of a point charge was no more satisfactory, because its electromagnetic self-energy diverges. From the early twentieth century onward, several authors therefore explored modifications of Maxwell's theory, with nonlinear electrodynamics emerging as a particularly natural and radical possibility. In this paper, we revisit at the undergraduate level some of the nonlinear theories proposed in this context and explore their ability to address the classical self-energy problem of charged particles.