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

TL;DR

This work reexamines the integral form of Gauss' law for arbitrarily moving charges within and outside a moving Gaussian surface that can expand, contract, and deform. It derives the time-dependent flux through the surface using the convective derivative in conjunction with Maxwell's equations, showing that deformation does not influence the flux while expansion/contraction affects it through the enclosed charge, yielding the evolution equation $\frac{\text{d}}{\text{d}t} \oint_{s(t)} \vec{E}\cdot d\vec{s}(t) = \frac{I^{(s)}_{\text{in}}(t)}{\epsilon_0}$ and its integrated form $\oint_{s(t)} \vec{E}\cdot d\vec{s}(t) = q_{\text{in}}(t)/\epsilon_0$. The results reinforce that Gauss' law retains its structure in non-static electrodynamics without requiring relativistic corrections and offer a clear, pedagogical framework for understanding flux evolution in moving geometries, illustrated by simple expanding/contracting surface examples.

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 $\frac{\text{d}}{\text{d}\text{t}}\oint_{s(t)}\vec{E}\cdot\text{d}\vec{s}(t)=\frac{I^{(s)}_{\text{in}}(t)}{ε_0}$ 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.

Figures (2)

• Figure 1: A schematic diagram for $i=1,2,3,4,5$, and $6$ with $n=6$. The (point) charges $q_2$, $q_3$, $q_4$, and $q_5$ are moving with velocities $\vec{v}_2$, $\vec{v}_3$, $\vec{v}_4$, and $\vec{v}_5$, respectively, inside the moving and deforming Gaussian surface $s(t)$ at time $t$. The surface becomes $s(t+\delta t)$ at time $t+\delta t$. While the charge $q_1$ is about to leave $s(t)$ with velocity $\vec{v}_1$, the charge $q_6$ which although is at rest, is about to come inside $s(t)$ due to motion of the surface, say, with the velocity $\vec{v}'$ at the point $\vec{r}'$ at time $t$.
• Figure 2: An example with a cylindrical shaped Gaussian surface expanding along the flat surfaces.