Revisiting the integral form of Gauss' law for a generic case of electrodynamics with arbitrarily moving Gaussian surface

TL;DR

This paper re-examines Gauss' law for dynamic, deforming Gaussian surfaces in electrodynamics, deriving a new evolution equation for the flux integral and clarifying its dependence on surface expansion but not deformation.

Contribution

It introduces a generalized evolution equation for the flux integral over moving and deforming surfaces in electrodynamics, extending classical Gauss' law to dynamic geometries.

Findings

Derived a time-dependent flux evolution equation for moving surfaces.

Showed flux depends on surface expansion but not on deformation.

Clarified the conditions under which Gauss' law applies to dynamic surfaces.

Abstract

We have re-examined the integral form of Gauss' law for arbitrarily moving charges inside and outside an arbitrarily expanding (or contracting) and deforming Gaussian surface. We have explicitly calculated the time-dependent Gauss' flux integral for such a generic non-static case with the Maxwell equations under consideration. We have obtained an evolution equation $\frac{\text{d}}{\text{d}\text{t}}\oint_{s(t)}\vec{E}\cdot\text{d}\vec{s}(t)=\frac{I^{(s)}_{\text{in}}(t)}{\epsilon_0}$ for the time-dependence of the flux-integral. We have pedagogically demonstrated that while the flux integral is dependent on the expansion/contraction of the surface, it is independent of its deformation.