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

=== Start with the bulk continuity equation ∂μJtotμ=0\partial_\mu J^\mu_{\rm tot}=0∂μJtotμ=0, where the total current includes bulk + surface delta. Integrate over the pillbox volume around the surface and shrink to the surface to obtain the surface continuity (bookkeeping law for charge on the shell): ===

  ∂tσsurf  +  ∇∥⋅Ksurf  =  − n⋅Jbulk∣Σ.  \boxed{\;
\partial_t \sigma_{\rm surf} \;+\; \nabla_{\parallel}\cdot \mathbf{K}_{\rm surf}
\;=\; -\, n\cdot \mathbf{J}_{\rm bulk}\big|_{\Sigma}.
\;}∂tσsurf+∇∥⋅Ksurf=−n⋅JbulkΣ.
Interpretation:
* The time change of the surface charge density σsurf\sigma_{\rm surf}σsurf plus the divergence of the surface (tangential) current equals minus the flux of bulk current into the surface. That right-hand side is precisely absorption/emission: photons (and EM momentum/energy) coming from the bulk and coupling to the boundary change the surface occupation (the electron sphere). This is the explicit “bookkeeping of flows” you asked about: Noether current (next) tells you how the surface charge evolves and how it ties to bulk flux.