Pound To Rescue Permittivity

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.