\(F={m}_{p}\cfrac{c^2}{r}={m}_{p}r.\omega^2={m}_{p}r.(2\pi .f)^2\)
this is also the force driving a photon to light speed \(c\) and the drag force such that the photon is at terminal velocity. For a given medium \(F\) is a constant.
\(\omega^{ 2 }=\cfrac { F }{ { m }_{ p } } \cfrac { 1 }{ r }\)
\(\log(\omega)=\cfrac { 1 }{ 2 } \log(\cfrac { F }{ { m }_{ p } } )-\cfrac { 1 }{ 2 } \log(r)\)
A plot of \(log(\omega)\) vs \(\log(r)\) can then provide a value for \(\cfrac{F}{{m}_{p}}\)
Since \(F\) is characteristic of a particular medium, we generate a number of y-intercepts \({y}_{i}\)for different mediums \({n}_{i}\). We know that,
\(F\propto {c}^{2}\) and in a medium of refractive index \(n_i\), \({v}^2_{s}=\cfrac{{c}^{2}}{n^2_i}\)
as such,
\({F}_{i}=\cfrac{F_o}{{n}_{i}}\), where \(F_o\) is the driving force in vacuum.
\({y}_{i} =\cfrac { 1 }{ 2 } \log(\cfrac { F_o }{ { m }_{ p } } ) - \cfrac { 1 }{ 2 }log{({n}_{i})}\)
A plot of \({y}_{i}\) vs \(log{({n}_{i})}\) will provide a value for \(\cfrac{F_o}{{m}_{p}}\).
From the Post "Wave Front and Wave Back" , after total internal reflection,
\({r}_{c2} = \cfrac{{r}_{c1}}{cos(\theta)}.\sqrt{(\cfrac{{n}_{1}}{{n}_{2}})^2.sin^2{(\theta)}-1}\)
\({r}_{c2} = {r}_{c1}\) when
\(cos{(\theta)}=((\cfrac{n_1}{n_2})^2-1)^\cfrac{1}{2} ({(\cfrac{n_1} {n_2})^2+1})^{-\cfrac{1}{2}}\)
A diagram of total internal reflection at this angle is shown,
As the ray enters the second medium \({n}_{2}\) photons speed up tracing a longer path. The intrusion into \({n}_{2}\) is given by \(\cfrac{2r.cos{(\theta)}}{tan{(\theta)}}\) which can be measured. Given different frequency \(f\), \(r\) is different at the same angle of incident \(\theta\). Hopefully the first plot can be obtained. Repeating the measurements using different mediums \({n}_{i}\), the second plot may be obtained. And so, \(\cfrac{F_o}{{m}_{p}}\) the photon acceleration under the driving force in free space can be approximated.
