Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

=== Take a thin “pillbox” (Gaussian thin slab) across Σ\SigmaΣ: normal coordinate nnn through the shell, thickness 2ε2\varepsilon2ε, and integrate the inhomogeneous Maxwell equation ∂νFνμ=Jbulkμ(x)+jsurfμ(y) δ(n)\partial_\nu F^{\nu\mu} = J^\mu_{\rm bulk}(x) + j^\mu_{\rm surf}(y)\,\delta(n)∂νFνμ=Jbulkμ(x)+jsurfμ(y)δ(n) across n∈[−ε,ε]n\in[-\varepsilon,\varepsilon]n∈[−ε,ε]. The delta captures the boundary surface current. ===

Integrating across the thin shell gives

lim⁡ε→0∫−ε+εdn  ∂νFνμ  =  nν[Fνμ]−+  =  jsurfμ,\lim_{\varepsilon\to 0}\int_{-\varepsilon}^{+\varepsilon} dn\; \partial_\nu F^{\nu\mu}
 \;=\;  n_\nu\bigl[F^{\nu\mu}\bigr]_{-}^{+}
 \;=\; j^\mu_{\rm surf},ε→0lim∫−ε+εdn∂νFνμ=nν[Fνμ]−+=jsurfμ,
where [F]−+≡F∣+−F∣−[F]_{-}^{+}\equiv F|_{+}-F|_{-}[F]−+≡F∣+−F∣− denotes the jump across the surface.

Specializing to familiar components (SI-style, re-inserting ε0,μ0\varepsilon_0,\mu_0ε0,μ0 if desired):
* Normal electric field (Gauss law)

n⋅(Eout−Ein)  =  σsurf/ε0,where σsurf=jsurf0.n\cdot\big(\mathbf{E}_{\rm out}-\mathbf{E}_{\rm in}\big)
\;=\; \sigma_{\rm surf}/\varepsilon_0,
\quad\text{where }\sigma_{\rm surf}=j^0_{\rm surf}.n⋅(Eout−Ein)=σsurf/ε0,where σsurf=jsurf0.
* Tangential magnetic: (Ampère boundary)

n×(Hout−Hin)  =  Ksurf,n\times\big(\mathbf{H}_{\rm out}-\mathbf{H}_{\rm in}\big)
\;=\; \mathbf{K}_{\rm surf},n×(Hout−Hin)=Ksurf,
where Ksurf\mathbf{K}_{\rm surf}Ksurf is the surface current (spatial part of jaj^aja).
* Tangential electric is continuous (assuming no delta in Faraday law),

n×(Eout−Ein)=0,n\times(\mathbf{E}_{\rm out}-\mathbf{E}_{\rm in})=0,n×(Eout−Ein)=0,
and normal BBB is continuous if no magnetic monopoles.

These are the standard Maxwell jump conditions — but we obtained them from the variational principle with a boundary action. That shows the electron-sphere acts as a physically active membrane that sources/absorbs field flux.