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

=== Poynting theorem in differential form: ===

∂uem∂t+∇⋅S=−J⋅E,\frac{\partial u_{\rm em}}{\partial t} + \nabla\cdot\mathbf{S} = -\mathbf{J}\cdot\mathbf{E},∂t∂uem+∇⋅S=−J⋅E,
with uem=12(ε0E2+B2/μ0)u_{\rm em}=\tfrac12(\varepsilon_0 E^2 + B^2/\mu_0)uem=21(ε0E2+B2/μ0) and S=E×B/μ0\mathbf{S}=\mathbf{E}\times\mathbf{B}/\mu_0S=E×B/μ0.

Integrate this over a small 3-volume that hugs the surface: the net electromagnetic energy change inside the volume equals the negative of the work done on charges (bulk + surface) plus the net Poynting flux through the bounding surface. The surface term −Jsurf⋅E-\mathbf{J}_{\rm surf}\cdot\mathbf{E}−Jsurf⋅E is exactly the energy transfer into the surface degrees of freedom (occupation changes). That gives a clean “bookkeeping” identity: photons → electromagnetic field energy change → (Jsurf⋅E)(\mathbf{J}_{\rm surf}\cdot\mathbf{E})(Jsurf⋅E) → changes of surface internal energy / quantum state. This is what I meant earlier by “bookkeeping of energy flows”.

Link to Third Law: as temperature T→0T\to0T→0 the thermal/available excitations vanish, so the surface mode occupation freezes. In QAT language the absorption/emission events (which form the arrow of time locally) become exceedingly rare; the EM degrees of freedom lock into their minimal energy configuration — consistent qualitatively with the third law’s statement that dynamics freeze at absolute zero (and consistent with your Faraday-lines picture: field lines ‘lock’ when no excitations remain).