91Ó°ÊÓ

The refractive index \(n\) of a medium is a function only of the distance \(r\) from a fixed point \(O\). Prove that the equation of a light ray, assumed to lie in a plane through \(O\), travelling in the medium satisfies (in plane polar coordinates) $$ \frac{1}{r^{2}}\left(\frac{d r}{d \phi}\right)^{2}=\frac{r^{2}}{a^{2}} \frac{n^{2}(r)}{n^{2}(a)}-1 $$ where \(a\) is the distance of the ray from \(O\) at the point at which \(d r / d \phi=0\). If \(n=\left[1+\left(\alpha^{2} / r^{2}\right)\right]^{1 / 2}\) and the ray starts and ends far from \(O\), find its deviation (the angle through which the ray is turned) if its minimum distance from \(O\) is \(a\).