\(q|_{a_\psi}=2\dot{x}F_{\rho}|_{a_\psi}\)
and at a distance \(r\), \(r\gt a_{\psi}\) from the center of the field,
\(\cfrac{1}{4\pi r^2}\cfrac{q|_{a_\psi}}{\varepsilon_o}=\dot{x}F_{\rho}|_{r}\) defined this way
where \(\dot{x}=v\).
This was Coulomb's inverse square law, as a constant power is divided over the surface area of a sphere at radial distance, \(r\) from the center of the field.
So, with
\(\cfrac{1}{\varepsilon_o}=2c^2ln(cosh(\theta_{\psi}))\) and,
\(F_{\rho}|_{r}=\cfrac{F_{\rho}|_{a_\psi}}{4\pi r^2}.\cfrac{2}{\varepsilon_o}\)
here the factor \(\cfrac{2}{\varepsilon_o}\) behaves like surface area,
\({F_{\rho}|_{a_\psi}}.\cfrac{2}{\varepsilon_o}=\Psi_c\)
is the total flux emanating from a sphere of surface area of \(4\pi a_{\psi}\), at a distance \(a_{\psi}\) from the center of the field. So that, \(F_{\rho}|_{r}\) is simply,
\(F_{\rho}|_{r}=\cfrac{\Psi_c}{4\pi r^2}\)
And when \(\dot{x}=v=c\), we have,
\(\cfrac{1}{4\pi r^2}\cfrac{q|_{a_\psi}}{\varepsilon_o}=F_{\rho}|_{r}.c\)
This is different from Coulomb's
\(\cfrac{1}{4\pi r^2}\cfrac{q}{\varepsilon_o}=F\)
From Newton force,
\(F=ma=m\cfrac{dv}{dt}\)
An incremental work done by this force,
\(F.\Delta x=m\cfrac{dv}{dt}\Delta x\)
\(32\pi^4v^2_x+v^2_t=32\pi^4c^2\)
that originated from considering the different in speed limits in the time dimension and the space dimension and that energy is conserved across the two dimensions. At the speed limit, a particle passes over to the orthogonal dimension; in space, light speed brings a particle to the time dimension at \(v_t=0\). So, we have,
\(F.{ \Delta x }=\left\{ m(v_{ p })+\Delta m(v_{ p }) \right\} \cfrac { dv_{ p } }{ dt } { \Delta x_{ p } }\)
where \(m=m(v_p)\), the particle mass is a function of its velocity, and the particle is displaced by \(\Delta x_p\) instead. \(\Delta m(v_p)\) is due to a change \(v_p\), \(\Delta v_p\) with \(\Delta x_p\),
\(F.{ \Delta x }=m(v_{ p })\cfrac { dv_{ p } }{ dt } { \Delta x_{ p } }+\Delta m(v_{ p })\cfrac { dv_{ p } }{ dt } { \Delta x_{ p } }\)
\( F.{ \Delta x }=m(v_{ p })\cfrac { dv_{ p } }{ dt } { \Delta x_{ p } }+\cfrac { dm(v_{ p }) }{ dv_{ p } } \Delta v_{ p }.\cfrac { dv_{ p } }{ dt } { \Delta x_{ p } }\)
\( F.{ \Delta x }=m(v_{ p })\cfrac { dv_{ p } }{ dt } { \Delta x_{ p } }+\cfrac { dm(v_{ p }) }{ dt } \Delta v_{ p }.{ \Delta x_{ p } }\)
Over time \(\Delta t\),
\(F.{ \cfrac { \Delta x }{ \Delta t } }=m(v_{ p })\cfrac { dv_{ p } }{ dt } { \cfrac { \Delta x_{ p } }{ \Delta t } }+\cfrac { dm(v_{ p }) }{ dt } \cfrac { \Delta v_{ p } }{ \Delta t } .{ \Delta x_{ p } }\)
as \(\Delta t\rightarrow 0\)
\(\cfrac{\Delta x_p}{\Delta t}=v_p\), \(\cfrac{\Delta v_p}{\Delta t}=a\)
\(F.{ \cfrac { \Delta x }{ \Delta t } }=m(v_{ p })v_p\cfrac { dv_{ p } }{ dt } +\cfrac { dm(v_{ p }) }{ dt } a.{ \Delta x_{ p } }\)
since \(\Delta t\) is small and \(\Delta x\) is small, \(a\) is a constant, basically,
\( v=u+at\)
\( \Delta x=ut+\cfrac { 1 }{ 2 } at^{ 2 }\)
\( \Delta x=u\cfrac { (v-u) }{ a } +\cfrac { 1 }{ 2 } \cfrac { (v-u)^{ 2 } }{ a } \)
\( 2a\Delta x=2(uv-u^{ 2 })+(v-u)^{ 2 }=v^{ 2 }-u^{ 2 }\)
when \( u=0\), the particle is subjected to the field from rest,
\( 2a\Delta x=v^{ 2 }\)
\( a\Delta x=\cfrac { 1 }{ 2 } v^{ 2 }\)
\(F.{ v }=m(v_{ p }).v_{ p }\cfrac { dv_{ p } }{ dt } +\cfrac { 1 }{ 2 } v^{ 2 }_{ p }\cfrac { dm(v_{ p }) }{ dt } \)
\(F.{ v }=m(v_{ p })\cfrac{1}{2}\cfrac { dv_{ p }^2 }{ dt } +\cfrac { 1 }{ 2 } v^{ 2 }_{ p }\cfrac { dm(v_{ p }) }{ dt }=\cfrac { d }{ dt }\left\{\cfrac{1}{2}m(v_p).v_P^2\right\}=\cfrac { dKE }{ dt } \)
\(v\) is independent of \(v_p\), it is a parameter associated with the field only.
From previously,
\(c=n\sqrt { \cfrac {(32\pi ^{ 4 }-1) }{ 128\pi\theta_{\psi}ln(cosh(\theta_{\psi}))tanh(\theta_{\psi}) } }\)
\(n=\left(\cfrac{\theta_{\psi}}{0.7369}\right)^3\)
\(c\) is the flow of energy from the center of the field, irrespective of the velocity of a test charge in the field. \(c\) is a constant between all interacting particles with the same \(n\) number of constituent basic particle.
When \(v=c\), the idea of \(F\) changing \(KE\) directly and \(F.c\) changing \(KE\) directly differ by a constant.
\(F.c=\cfrac { dKE }{ dt }\)
\(F.c\) is decreasing with distance \(r\) from the center of the field because the fixed total amount of energy from the field passes through a bigger surface area \(4\pi r^2\) at \(r\).
...
This is still prelude to an expression for mass \(m\), but correct for \(F.c\) that change \(KE\) directly.
\(\cfrac{1}{4\pi r^2}\cfrac{q|_{a_\psi}}{\varepsilon_o}=F_{\rho}|_{r}.c\ne F_{newton}\)
Two things have changed, a field is quantifies by its power output \(F.c\) and the force in a field is different from the Newtonian force.
