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

=== Integrate Maxwell’s equations across a thin radial pillbox that straddles the spherical shell. Standard results: ===

(1) Normal electric field jump (Gauss)

n^ ⁣⋅ ⁣(Eout−Ein)  =  σ(θ,φ)ε0,\hat n\!\cdot\!(\mathbf{E}_{\rm out}-\mathbf{E}_{\rm in})
\;=\;
\frac{\sigma(\theta,\varphi)}{\varepsilon_0},n^⋅(Eout−Ein)=ε0σ(θ,φ),
where σ=\sigma=σ= surface charge density (the time-component of jsurfμj^\mu_{\rm surf}jsurfμ), and n^\hat nn^ is the outward radial normal.

(2) Tangential magnetic field jump (Ampère / boundary current)

n^×(Bout−Bin)  =  μ0 j∥(θ,φ),\hat n\times(\mathbf{B}_{\rm out}-\mathbf{B}_{\rm in})
\;=\;
\mu_0\,\mathbf{j}_{\parallel}(\theta,\varphi),n^×(Bout−Bin)=μ0j∥(θ,φ),
where j∥\mathbf{j}_{\parallel}j∥ is the surface current vector (spatial part of jsurfμj^\mu_{\rm surf}jsurfμ tangent to the sphere), and μ0ε0=1/c2\mu_0\varepsilon_0 = 1/c^2μ0ε0=1/c2.

These relations are the classical electromagnetic signature of a charged, current-carrying spherical shell — the exact way charge/current placed on a 2-D sphere create the fields in the 3-D bulk.

Note: For a perfectly symmetric uniform shell σ=\sigma=σ= const and j∥=0\mathbf{j}_{\parallel}=0j∥=0, Newton/Gauss shell theorem gives Ein=0, Eout=Q/(4πε0r02) n^\mathbf{E}_{\rm in}=0,\ \mathbf{E}_{\rm out}=Q/(4\pi\varepsilon_0 r_0^2)\,\hat nEin=0, Eout=Q/(4πε0r02)n^. But in QAT the symmetry breaks locally during absorption/emission events so the local jumps are nontrivial.