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The refractive index of the material of a prism is $\sqrt{2}$ and the angle of the prism is $30^\circ$. One of the two refracting surfaces of the prism is made a mirror inwards, by silver coating. A beam of monochromatic light entering the prism from the other face will retrace its path (after reflection from the silvered surface) if its angle of incidence on the prism is
Explanation
Normal incidence on the silvered face makes $r = A = 30^\circ$; then Snell's law gives $i = 45^\circ$.
Detailed Solution
For retracing its path, the light ray should be normally incident on the silvered face.
So the angle of refraction at the first face is $r = 30^\circ$.
Applying Snell's law at M: $\frac{\sin i}{\sin 30^\circ} = \frac{\sqrt{2}}{1}$
$\sin i = \sqrt{2}\times\frac{1}{2} = \frac{1}{\sqrt{2}}$, i.e. $i = 45^\circ$
So the angle of refraction at the first face is $r = 30^\circ$.
Applying Snell's law at M: $\frac{\sin i}{\sin 30^\circ} = \frac{\sqrt{2}}{1}$
$\sin i = \sqrt{2}\times\frac{1}{2} = \frac{1}{\sqrt{2}}$, i.e. $i = 45^\circ$