\(\varepsilon_o=\cfrac{1}{2c^2ln(cosh(\pi))}\)
after applying the \(\sqrt{3}\) factor to account for 3D space,
\(\varepsilon_o=\cfrac{1}{2c^2ln(cosh(\pi))}*\sqrt{3}\)
\(\varepsilon_o=\cfrac{1}{2*299792458^2*ln(cosh(\pi))}*\sqrt{3}=3.9325e(-18)\)
Why was it not necessary to divide by the Durian constant \(A_D=\cfrac{4}{3}\pi(2c)^3=9.029022e26\)?
Because the elementary charge \(q_e\) has absorbed the constant,
\(q_e\rightarrow \cfrac{q}{A_D}\)
Given one particle, \(q=1\)
\(q_e=\cfrac{1}{9.029022e26}=1.107540e-27\)
since by definition,
\(\varepsilon_o=\cfrac{1}{\mu_oc^2}=\cfrac{1}{4\pi\times10^{-7}c^2}\)
If we adjust \(q_e\) by the factor,
\(\cfrac{2ln(cosh(\pi))}{4\pi\times10^{-7}*\sqrt{3}}\)
to return to the definition of \(\varepsilon_o\). ie,
\(\cfrac{1}{2c^2ln(cosh(\pi))}*\sqrt{3}\rightarrow\cfrac{1}{\mu_oc^2}=\cfrac{1}{4\pi\times10^{-7}c^2} \)
As both denominator and numerator of \(\cfrac{q}{\varepsilon_o}\)are multiplied by the same factor, we have,
\(q_{adj}=q_e*\cfrac{2ln(cosh(\pi))}{4\pi\times10^{-7}*\sqrt{3}}\)
\(q_{adj}=\cfrac{1}{9.029022e26}*\cfrac{2ln(cosh(\pi))}{4\pi\times10^{-7}*\sqrt{3}}=2.493676e-21\)
and we further account for the fact that \(77\) particles coalesce up to the limit \(tanh(\cfrac{G}{\sqrt{2mc^2}}a_{\psi\,\pi})=1\)
\(\pi\rightarrow 3.135009 \)
\(ln(cosh(\pi))\rightarrow ln(cosh(3.135009))\)
that is,
\(\varepsilon_o=\cfrac{1}{2c^2ln(cosh(3.135009))}\)
\(q_{adj}=\cfrac{1}{9.029022e26}*\cfrac{2ln(cosh(3.135009))}{4\pi\times10^{-7}*\sqrt{3}}*77=1.914991e-19\)
we compare this with the quoted value of \(e=1.602 176 565e-19\), we are about \(1.195\) times off.
If we do not adjust for the factor \(\sqrt{3}\),
\(q_{adj}=\cfrac{1}{9.029022e26}*\cfrac{2ln(cosh(3.135009))}{4\pi\times10^{-7}}*77=3.316861e-19\)
we are about twice off the quoted value,
\(\cfrac{q_{adj}}{2}=\cfrac{3.316861e-19}{2}=1.658431e-19\)
What happened? If the experiment to obtain \(e\) is with point charges then we may not have to factor in \(\sqrt{3}\) for charged bodies in 3D space, but this results in a calculated elementary charge of twice the quoted value.
When we do factor in \(\sqrt{3}\), the discrepancy seems to reflect the ratio between the Durian constant and Avogadro constant,
\(\cfrac{A_D}{A_v}=\cfrac{9.029022e26}{6.02214e26}=1.49930\)
there is however no reason to adjust the Durian constant here. Furthermore, swapping Durian for Avogadro increases the calculated charge, \(q_e\) that is already too high.
Otherwise, another constant hangs on my wall!
Note: Dividing by the Durian constant was necessary to account for entanglement.
