Time dependent electromagnetic fields and 4-dimensional Stokes' theorem

Abstract

Stokes' theorem is central to many aspects of physics -- electromagnetism, the Aharonov-Bohm effect, and Wilson loops to name a few. However, the pedagogical examples and research work almost exclusively focus on situations where the fields are time-independent so that one need only deal with purely spatial line integrals ({\it e.g.} $\oint \mathbf{A}\cdot d\mathbf{x}$) and purely spatial area integrals ({\it e.g.} $\int (\nabla \times \mathbf{A})\cdot d\mathbf{a}=\int \mathbf{B}\cdot d\mathbf{a}$). Here we address this gap by giving some explicit examples of how Stokes' theorem plays out with time-dependent fields in a full 4-dimensional spacetime context. We also discuss some unusual features of Stokes' theorem with time-dependent fields related to gauge transformations and non-simply connected topology.

25 pages ReVTeX, 5 eps figures, 3 appendices. To be published American Journal of Physics

References in corpus (1)

• Revisiting the Marton, Simpson, and Suddeth experimental confirmation of the Aharonov-Bohm effect