Question

An arrangement of plane mirrors and a spherical glass surface is shown in figure. $O$ is the centre of the spherical portion. The incident light ray from air makes an angle of ${60}^{\circ }$ with the normal and the normal makes an angle of ${60}^{\circ }$ with each plane mirrors. The value of refractive index for which a light ray retraces its path will be

• A. $\sqrt{6}$
• B. $\sqrt{5}$
• C. $\sqrt{2}$
• D. $\sqrt{3}$

Answer 1

The correct option is D $\sqrt{3}$

For the light ray to retrace its path, the light after refraction from the spherical glass surface must fall perpendicular to the inclined plane mirror.

From $△ABO$,
$\angle BAO={180}^{\circ }-{90}^{\circ }-{60}^{\circ }={30}^{\circ }$
So,
Angle of incidence, $i={60}^{\circ }$
Angle of refraction, $r={30}^{\circ }$
We know that, by Snell's law,
$\mu =\frac{\sin i}{\sin r}$
$[$when first medium is air$]$
$\Rightarrow \mu =\frac{\sin {60}^{\circ }}{\sin {30}^{\circ }}=\sqrt{3}$

Why this question? It uniquely combines the application of refraction at an interface and reversibility of light after reflection. Tips: Solve in $2$ steps. Begin from Snell's law at interface followed by reflection.