Question

A ray is incident on a glass sphere as shown. The opposite surface of the sphere is partially silvered. If the net deviation of the ray transmitted at the partially silvered surface is $\frac{1}{3} r d$ of the net deviation suffered by the ray reflected at the partially silvered surface (after emerging out of the sphere), The refractive index of the sphere $\frac{a}{\sqrt{3}}$ then the value of $a$ is.

Answer 1

Let $O$ be the centre of the sphere. The incident ray $P Q$ travels along the path as shown in figure. The transmitted ray is $RW$ and reflected ray is $ST$.

Deviation suffered by transmitted ray $=\delta T$
$=60^{\circ}- r +60^{\circ}- r$
(clockwise) (clockwise)
$=120^{\circ}-2 r$
Deviation suffered by reflected ray $=\delta_{R}$
$=60^{\circ}- r +180^{\circ}-2 r +60^{\circ}- r$
(clockwise) (clockwise) (clockwise)
$=300^{\circ}-4 r$
Given that, $\delta_{ T }=\frac{1}{3} \delta_{ R }$
$120^{\circ}-2 r=\frac{1}{3}\left(300^{\circ}-4 r\right)$
$120^{\circ}-2 r=100^{\circ}-\frac{4 r}{3}$
$20^{\circ}=\frac{2 r}{3} $
$\Rightarrow r=\frac{60^{\circ}}{2}=30^{\circ}$
Now for refraction at point $Q$,
$1 \times \sin 60^{\circ}=\mu \sin 30^{\circ}$
$\frac{\sqrt{3}}{2}=\frac{\mu}{2} \Rightarrow \mu=\sqrt{3}$