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

==== Use the Maxwell stress tensor to compute the net momentum flux F\mathbf{F}F on the shell: ====

F  =  ∮shell ⁣ ⁣TEM⋅n^ dA  =  1c∮ ⁣ ⁣S cos⁡θ dA\mathbf{F} \;=\; \oint_{\text{shell}} \!\! \mathbf{T}_{\rm EM}\cdot \hat{\mathbf{n}}\, dA
\;=\; \frac{1}{c}\oint \!\! \mathbf{S}\,\cos\theta\, dAF=∮shellTEM⋅n^dA=c1∮ScosθdA
with Poynting vector S=E×B/μ0 \mathbf{S} = \mathbf{E}\times\mathbf{B}/\mu_0S=E×B/μ0.

Two key QAT ingredients:
# Acceleration of the shell (or equivalently, changing its state in time) introduces a phase lag between incident and re-emitted fields (the action quantum h/2πh/2\pih/2π sets the minimal lag scale).
# That lag yields a radiation-reaction-like term—but now geometrically averaged on the 2-D shell—producing a force proportional to the acceleration:

Freact  =  − meff a,meff  =  1c2 (bound EM energy on the shell  +  phase-lag dissipation).\mathbf{F}_{\rm react} \;=\; -\,m_{\rm eff}\,\mathbf{a}, 
\qquad
m_{\rm eff} \;=\; \frac{1}{c^{2}}\,\Big(\text{bound EM energy on the shell} \;+\; \text{phase-lag dissipation}\Big).Freact=−meffa,meff=c21(bound EM energy on the shell+phase-lag dissipation).
Thus the effective inertial mass meffm_{\rm eff}meff is literally energy per c2c^{2}c2 stored+cycled by photon–electron exchanges on the spherical boundary (plus ordinary rest mass). This uses only standard EM momentum flux and the geometry of the shell.

Why the familiar Ek=12mv2E_k=\tfrac{1}{2}mv^{2}Ek=21mv2?
Expanding total energy E ⁣= ⁣γmeffc2E \!=\! \gamma m_{\rm eff} c^{2}E=γmeffc2 at small vvv gives Ek ⁣≈ ⁣12meffv2E_k\!\approx\!\tfrac12 m_{\rm eff} v^{2}Ek≈21meffv2. The ½ has a clean geometric role in QAT: radius = half the diameter sets the fundamental half-turn/half-spin action increment (visible as h/2πh/2\pih/2π in angular phases), mirroring how angular quantization on the shell works.