Electromagnetism / Electrodynamics I · Maxwell equations and conservation laws
A changing sheet charge requires surface-current divergence
Problem
A fixed sheet at $z=0$ carries surface charge $\sigma_s(x,t)$ and tangential current $\mathbf K_s(x,t)$, with no bulk current crossing the sheet. For any planar region $D$, $$\frac{d}{dt}\int_D\sigma_s\,dA=-\oint_{\partial D}\mathbf K_s\cdot\hat{\mathbf m}\,d\ell,$$ where $\hat{\mathbf m}$ is the outward in-plane normal. (a) Use the planar divergence theorem to derive the local surface continuity equation. (b) For $\sigma_s=A\cos(kx)\cos(\omega t)$, construct one periodic current of the form $K_x(x,t)\hat{\mathbf x}$. (c) Show that adding a spatially uniform $K_0(t)\hat{\mathbf y}$ changes neither the charge evolution nor the units.
Hint
The local sign is $\partial_t\sigma_s+\boldsymbol\nabla_\parallel\cdot\mathbf K_s=0$.
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