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

=== Start from the standard fine-structure constant ===

α  =  e24πε0 ℏ c\boxed{\displaystyle \alpha \;=\; \frac{e^2}{4\pi\varepsilon_0 \,\hbar \, c}}α=4πε0ℏce2
(we can use ℏ=h/2π\hbar=h/2\piℏ=h/2π whenever you prefer that explicit factor).

If the total charge eee is distributed uniformly on the spherical surface radius RRR,

σ  =  e4πR2.\sigma \;=\; \frac{e}{4\pi R^2}.σ=4πR2e.
So

e2  =  (4πR2)2σ2  =  16π2σ2R4.e^2 \;=\; \big(4\pi R^2\big)^2\sigma^2 \;=\; 16\pi^2 \sigma^2 R^4.e2=(4πR2)2σ2=16π2σ2R4.
Substitute into α\alphaα:

α  =  16π2σ2R44πε0ℏc  =  4π σ2R4ε0ℏc.\alpha \;=\; \frac{16\pi^2 \sigma^2 R^4}{4\pi\varepsilon_0 \hbar c}
          \;=\; \frac{4\pi\, \sigma^2 R^4}{\varepsilon_0 \hbar c}.α=4πε0ℏc16π2σ2R4=ε0ℏc4πσ2R4.
(Using ℏ\hbarℏ makes the geometry of 2π2\pi2π transparent; if you prefer h/2πh/2\pih/2π just substitute.)

Therefore

α  =  4π σ2R4ε0ℏc⟺σ  =  α ε0 ℏ c4π R4\boxed{\displaystyle
\alpha \;=\; \frac{4\pi\, \sigma^2 R^4}{\varepsilon_0 \hbar c}
}
\qquad\Longleftrightarrow\qquad
\boxed{\displaystyle
\sigma \;=\; \sqrt{\frac{\alpha\,\varepsilon_0\,\hbar\,c}{4\pi\,R^4}}
}α=ε0ℏc4πσ2R4⟺σ=4πR4αε0ℏc
Interpretation:
* this equation expresses the electromagnetic coupling α\alphaα directly in terms of the surface geometric variables σ\sigmaσ and RRR, and universal constants ε0,ℏ,c\varepsilon_0,\hbar,cε0,ℏ,c.
* if the surface radius RRR is fixed by the boundary variational problem (see below), then σ\sigmaσ must take the value above to give the measured α\alphaα; conversely if σ\sigmaσ is determined microscopically the observed α\alphaα fixes an effective geometric radius.

Two known equivalent forms are especially useful:

α=reλC,withre=e24πε0mec2,λC=ℏmec.\alpha = \frac{r_e}{\lambda_C},\quad
\text{with}\quad r_e=\frac{e^2}{4\pi\varepsilon_0 m_e c^2},\quad
\lambda_C=\frac{\hbar}{m_e c}.α=λCre,withre=4πε0mec2e2,λC=mecℏ.
This shows α\alphaα is the ratio of two characteristic lengths — classically interpretable electrostatic radius vs. quantum action length — which is exactly the geometric viewpoint we want.