Continuing from the previous post "Photons Like Bubbles", if we invoke Hertz Law,
\(F=\cfrac{4}{3}E^{*}a_\psi^{1/2}d^{3/2}\)
and assume that the medium presents a constant resistance force to all photons,
\(A=a_\psi^{1/2}d^{3/2}\)
where \(A=\cfrac{3}{4E^{*}}F\) is a constant, we have
There is a new derivation in a later post "Success Is In The Way You Handle The Question".
\(n=\cfrac{a_\psi}{a_\psi-d}\)
as \(n\) is the refractive index of the medium which is also the ratio of the normal components of \(\psi\) just before and just after passing into the medium as derived from the previous post.
\(a_\psi(1-\cfrac{1}{n})=d\)
Substitute the above in to the expression for \(A\),
\(A=a_\psi^{2}(1-\cfrac{1}{n})^{3/2}\)
\(\cfrac { 1 }{ n } =1-\cfrac { A^{ 2/3 } }{ a_{ \psi }^{ 4/3 } } \)
\(n=\frac { a_{ \psi }^{ 4/3 } }{ a_{ \psi }^{ 4/3 }-B} \) and \(\lambda=2\pi a_\psi\)
where \(B=A^{ 2/3 } \) is a constant.
This is not Sellmeier equation. Furthermore, from Hertz Law,
\(\cfrac{1}{E^{*}}=\cfrac{1-\nu_p^2}{E_p}+\cfrac{1-\nu_m^2}{E_m}\)
where \(E_p\) and \(E_m\) are the elastic modulus of the photon and the medium respectively; \(\nu_p\) and \(\nu_m\) are the Poisson's ratios; it is not immediately clear in this analogy what exactly they are. We could have started instead with,
\(F=\cfrac{4}{3}E^{*}a_\psi^{1/2}d^{m}\)
where \(m\) captures the interaction between photons and the medium. The rest being geometry for the time being.
