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

==== Start from the usual bulk Poynting theorem (unchanged): ====

∂tuem+∇ ⁣⋅S  =  −E ⁣⋅ ⁣Jbulk,\partial_t u_{\rm em} + \nabla\!\cdot \mathbf{S} \;=\; -\mathbf{E}\!\cdot\!\mathbf{J}_{\rm bulk},∂tuem+∇⋅S=−E⋅Jbulk,
where uem=ε02∣E∣2+12μ0∣B∣2u_{\rm em}=\frac{\varepsilon_0}{2}|\mathbf{E}|^2+\frac{1}{2\mu_0}|\mathbf{B}|^2uem=2ε0∣E∣2+2μ01∣B∣2 and S=1μ0E×B\mathbf{S}=\frac{1}{\mu_0}\mathbf{E}\times\mathbf{B}S=μ01E×B.

Integrate this equation over a thin volume VϵV_\epsilonVϵ that encloses the shell and let the radial thickness ϵ→0\epsilon\to0ϵ→0. Using the delta surface sources from (1), one obtains the surface energy balance (exact):

∂tusurf(t,Ω)  =  −n^⋅S∣r=r0  −  E∥ ⁣⋅j∥  +  Qint.(5)\partial_t u_{\rm surf}(t,\Omega)
\;=\;
-\hat n\cdot\mathbf{S}\Big|_{r=r_0}
\;-\;
\mathbf{E}_\parallel\!\cdot\mathbf{j}_\parallel
\;+\;
\mathcal{Q}_{\rm int}.
\tag{5}∂tusurf(t,Ω)=−n^⋅Sr=r0−E∥⋅j∥+Qint.(5)
Meaning of terms:
* ∂tusurf\partial_t u_{\rm surf}∂tusurf: rate of change of the internal energy density of the shell degrees of freedom (quantum excitations, electron energy levels, etc.).
* −n^⋅S∣r0-\hat n\cdot\mathbf{S}\big|_{r_0}−n^⋅Sr0: net Poynting flux entering the shell from the bulk (photon energy arriving).
* −E∥⋅j∥-\mathbf{E}_\parallel\cdot\mathbf{j}_\parallel−E∥⋅j∥: power extracted from the tangential fields by currents on the shell (work done on charges → changes in internal state).
* Qint\mathcal{Q}_{\rm int}Qint: internal dissipation or quantum transition work (e.g., radiative relaxation, non-radiative loss). In a pure conservative model Qint=0\mathcal{Q}_{\rm int}=0Qint=0 and all energy is accounted between the Poynting flux and stored energy.

Equation (5) is the balance-sheet entry of the ledger: incoming field energy (Poynting) is either stored as shell internal energy (raising occupation, changing states) or spent doing work on currents (which can then reradiate).

Absorption and Emission:
* Absorption event: n^⋅S>0\hat n\cdot\mathbf{S}>0n^⋅S>0 (Poynting into shell) raises usurfu_{\rm surf}usurf or does E∥⋅j∥\mathbf{E}_\parallel\cdot\mathbf{j}_\parallelE∥⋅j∥ work. Normal flux term in (2) is positive.
* Emission event: the shell reconfigures (via j∥\mathbf{j}_\parallelj∥) so that radiative fields carry energy out; n^⋅S<0\hat n\cdot\mathbf{S}<0n^⋅S<0 on average (outgoing Poynting).

All is exact — the EM ledger is balanced at every moment.