In My Dreams

From Hertz Law,

\(F=\cfrac{4}{3}E^{*}a_\psi^{1/2}d^{3/2}\)

Work done in moving against this force is,

\(W=\int_0^d{F}\,d(d)=\cfrac{4}{3}E^{*}a_\psi^{1/2}\int_0^d{d^{3/2}}\,d(d)\)

\(W=\cfrac{8}{15}E^{*}a_\psi^{1/2}d^{5/2}\)

where Young's Modulus \(E^*\) is of two parts,

\(\cfrac{1}{E^{*}}=\cfrac{1-\nu_p^2}{E_p}+\cfrac{1-\nu_m^2}{E_m}\)

where \(E_p\) characterize \(\psi\) as it deforms and \(E_m\) characterize the medium as it deforms \(\psi\).  If we incident a ray of photons into another ray of photons, then

\(E_p=E_m\) and \(\nu_p=\nu_m\)

 \(E^{*}=\cfrac{1}{2(1-\nu_p^2)}E_p\)

Using the Planck's Relationship derived earlier,

\(h.f_d=\psi_o+W\)

\(h.f_d=\psi_o+\cfrac{4}{15(1-\nu_p^2)}E_pa_\psi^{1/2}d^{5/2}\)

\(h.f_d=2\pi a_\psi.mc+\cfrac{4}{15(1-\nu_p^2)}E_pa_\psi^{1/2}d^{5/2}\)

Alternatively,

\(h.f_d=2\pi a_\psi.mc+\cfrac{8}{15}E^*a_\psi^{1/2}d^{5/2}\)

As \(a_\psi\) is a constant from a monochromatic source, this is just a linear equation in \(d^{2.5}\).  \(f_d\) can be measured experimentally.  The problem is to set up an experiment to vary \(d\), the distortion in \(\psi\), systematically.

In the diagram above, a ray, \(a_\psi\) is aimed at a cone of light also from a source \(a_\psi\) at various incident angles \(\theta\).  \(f_d\), an increased frequency is measured (if there is actually an increase) correspondingly to each \(\theta\).  By further changing \(a_\psi\), we may be able to estimate \(m\) (from the set of y-intercepts) and \(E^*\) (from the set of gradient values) assuming that both do not change with \(a_\psi\).

But the first question is: Does a ray of light refract in a cone of light?

In my dreams.