m_e={2R_{\infty}h\over c\alpha^2}
m_e{\alpha^2\over R_{\infty}}={2h\over c}
m_e{\alpha^2\over \pi R_{\infty}}={2h\over \pi c}
Let r_e={\alpha^2\over\pi R_{\infty}}, then:
m_er_e={2h\over\pi c}
Now, because for every action there is an equal and opposite reaction, for every force there is an equal and opposite force, for every torque, there is an equal and opposite torque, equate m_pr_p to m_er_e to balance torque/spin between proton and electron:
m_er_e={2h\over\pi c}=m_pr_p
\therefore {m_p\over m_e}={r_e\over r_p}={\alpha^2\over\pi r_pR_{\infty}}=1836.15267
Where:
m_pr_p={2h\over\pi c}=4\ell m_{\ell}
r_p=0.841235640294664\;fm
Compare to TOP_PCG calculated r_p:
r_p=0.841235640294664\;fm
r_p=0.841235640479985\;fm <~~ TOP_PCG
Compare to TOP_PCG calculated r_p:
r_p=0.841235640294664\;fm
r_p=0.841235640479985\;fm <~~ TOP_PCG
m_p= proton mass
m_e= electron mass
r_p= proton radius
r_e= effective torque arm radius for electron
\alpha= fine-structure constant
h= Planck constant
c= speed of light
R_{\infty}= Rydberg constant
\ell= Planck length
m_{\ell}= Planck mass
QED.
