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

==== Vary StotS_{\rm tot}Stot with respect to Aμ(x)A_\mu(x)Aμ(x). Because SintS_{\rm int}Sint is localized on the shell, the variation yields the standard Maxwell equations with a delta-supported surface contribution: ====

∂νFνμ(x)  =  1ε0[Jbulkμ(x)  +  δ(r−r0) jsurfμ(t,Ω)].(1)\partial_\nu F^{\nu\mu}(x)
\;=\;
\frac{1}{\varepsilon_0}\Big[J_{\rm bulk}^\mu(x)
\;+\; \delta(r-r_0)\, j^\mu_{\rm surf}(t,\Omega)\Big].
\tag{1}∂νFνμ(x)=ε01[Jbulkμ(x)+δ(r−r0)jsurfμ(t,Ω)].(1)
Interpretation:
* JbulkμJ_{\rm bulk}^\muJbulkμ is any bulk 4-current (free charges in bulk).
* δ(r−r0)jsurfμ\delta(r-r_0) j^\mu_{\rm surf}δ(r−r0)jsurfμ is a distribution (a delta layer) representing the surface charge/current localized on the spherical shell. Equation (1) is the unchanged Maxwell equation; the only novelty is that the source includes a thin-shell term.