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

===== The bulk EM Lagrangian is the usual =====

LEM=−14μ0FμνFμν,Fμν=∂μAν−∂νAμ.\mathcal{L}_{\rm EM}=-\frac{1}{4\mu_0}F^{\mu\nu}F_{\mu\nu},
\qquad F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu.LEM=−4μ01FμνFμν,Fμν=∂μAν−∂νAμ.
Varying the total action w.r.t. AμA_\muAμ gives Maxwell’s equations in the bulk with a surface current contribution from the shell:

∂νFνμ(x)  =  μ0 Jshellμ(t,θ,φ)  δ(r−r0)\partial_\nu F^{\nu\mu}(x) \;=\; \mu_0\, J^\mu_{\rm shell}(t,\theta,\varphi)\; \delta(r-r_0)∂νFνμ(x)=μ0Jshellμ(t,θ,φ)δ(r−r0)
where the shell current 4-vector (localized at r0r_0r0) is

  Jshellμ(t,θ,φ)  =  ℏ  iq  [ ϕ∗ Dμϕ  −  ϕ (Dμϕ)∗ ]∣r=r0  \boxed{\;
J^\mu_{\rm shell}(t,\theta,\varphi)
\;=\; \hbar \; i q \; \big[\,\phi^''\,D^\mu\phi \;-\; \phi\, (D^\mu\phi)^''\,\big]\Big|_{r=r_0}
\;}Jshellμ(t,θ,φ)=ℏiq[ϕ∗Dμϕ−ϕ(Dμϕ)∗]r=r0
Interpretation: JshellμJ^\mu_{\rm shell}Jshellμ is nonzero only on the sphere; its timelike component is the surface charge density and tangential components give the surface current density.

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