From the post "Wrong, Wrong, Wrong" dated 25 May 2015,
\(F=\int{F_{\rho}}dx\)
where \(F_{ \rho }=i\sqrt { 2{ mc^{ 2 } } } \, G.tanh\left( \cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } (x-x_{ z }) \right)\)
\(F=2{ mc^{ 2 } }.ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } (x-x_{ z }))\)
within the boundary of \(\psi\) where \(0\le x \le a_{\psi}\). For, \(x\ge a_{\psi}\), we use Gaussian flux,
\(F=F_{\small{G}}\cfrac{1}{4\pi r^2}\)
at \(r=a_{\psi}\)
\(F_{\small{G}}\cfrac{1}{4\pi a^2_{\psi}}=\cfrac{q}{4\pi\varepsilon_o a^2_{\psi}}=2{ mc^{ 2 } }.ln(cosh(\pi))\)
for a point mass \(x_z=0\) (in retrospect \(x_z\) was not necessary, \(x_z\) was to prevent the function \(F_{\rho}\) from blowing up at \(x=0\)).
\(F_{ { G } }=8mc^{ 2 }\pi a^{ 2 }_{ \psi }.ln(cosh(\pi ))\)
where \(ln(cosh(\pi))=2.4503\).
Also,
\(\cfrac{q}{\varepsilon_o}=4\pi a^2_{\psi}m.2c^2ln(cosh(\pi))\)
where \(m\) is a point mass/inertia in the respective field. Which suggests,
\(\varepsilon_o=\cfrac{1}{2c^2ln(cosh(\pi))}=\cfrac{1}{2*299792458^2*ln(cosh(\pi))}\)
\(\varepsilon_o=2.2704e-18\)
when
\(q=4\pi a^2_{\psi}m\)
That \(q\) is the distribution of \(m\) on the surface of a sphere of radius \(a_{\psi}\). Which seems to solve the problem of extending a point mass of mass density \(m\) to a mass of finite extent \(a_{\psi}\).
But, the quoted value of \(\varepsilon_o\) is \(8.8542e-12\) from the definition,
\(\varepsilon_o=\cfrac{1}{\mu_oc^2}\)
where \(\mu_o=4\pi \times 10^{-7}\)
we have \(4\pi\approx12.5664\) vs \(2ln(cosh(\pi))\approx4.9006\) and a scaling factor of \(10^{-7}\).
The derived \(\varepsilon_o\) and defined value do not match.
