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 $\frac{1}{3}rd$ 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

Answer: 3.00

Let $O$ be the centre of the sphere. The incident ray $PQ$ 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)
$=120^{\circ }-2r$
Deviation suffered by reflected ray $={\delta }_R$
$=60^{\circ }-r+180^{\circ }-2r+60^{\circ }-r$
(clockwise) (clockwise) (clockwise)
$=300^{\circ }-4r$
Given that, ${\delta }_T=\frac{1}{3}{\delta }_R$
$120^{\circ }-2r=\frac{1}{3}\left(300^{\circ }-4r\right)$
$120^{\circ }-2r=100^{\circ }-\frac{4r}{3}$
$20^{\circ }=\frac{2r}{3}$
$\Rightarrow r=\frac{60^{\circ }}{2}=30^{\circ }$
Now for refraction at point $Q$,
$1\times \sin 60^{\circ }=\mu \sin 30^{\circ }$
$\frac{\sqrt{3}}{2}=\frac{\mu }{2}\Rightarrow \mu =\sqrt{3}$