One face of the glass prism is silver polished. $A$ light ray falls at an angle of $45^{\circ}$ on the other face. After reflection it is subsequently reflected from the silvered face and then retraces its path. The refracting angle of prism is $30^{\circ}$. The refractive index of the prism is

• A

  $\frac{3}{2}$
• B

  $\sqrt{2}$
• C

  $\frac{\sqrt{3}}{2}$
• Correct answer
  D

  $\sqrt{3}$

Solution

$\angle r=90^{\circ}-\left[180^{\circ}-\left(90^{\circ}+30^{\circ}\right)\right]$
$\begin{aligned} & =90^{\circ}-\left[180^{\circ}-120^{\circ}\right] \\ & =90^{\circ}-60^{\circ} \\ & =30^{\circ} \\ \angle i & =45^{\circ} \\ \mu \quad & =\frac{\sin i}{\sin r} \mu=\frac{\sin 45^{\circ}}{\sin 30^{\circ}} \\ \mu & =\frac{\frac{1}{\sqrt{2}}}{\frac{1}{2}}=\sqrt{2}\end{aligned}$

Hence, the correct answer is option: 'D'.