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

=== Varying the bulk action plus the coupling gives the Maxwell equations with both bulk and surface sources: ===

∂νFνμ(x)  =  1ε0[Jbulkμ(x)+  δ(r−r0) jsurfμ(t,Ω)],\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],∂νFνμ(x)=ε01[Jbulkμ(x)+δ(r−r0)jsurfμ(t,Ω)],
where δ(r−r0)\delta(r-r_0)δ(r−r0) localizes the surface current at the shell radius r0r_0r0, and JbulkJ_{\rm bulk}Jbulk denotes any bulk 4-current.

This single compact relation is the mathematical statement of “Jμ\muAJ^\muA_\muJμ\muA on the boundary sources the field.” It leads directly to the standard boundary jump conditions below.