question_answer

One of the refracting surface of a prism of refractive index \[\sqrt{2}\] is silvered. The angle of the prism is equal to the critical angle of a medium of refractive index 2. A ray of light incident in the silvered surface passes through the prism and retraces its path after reflection at the silvered face. Then the angle of incidence on the un silvered surface is

A) \[{{0}^{o}}\]

B) \[{{30}^{o}}\]

C) \[{{45}^{o}}\]

D) \[{{60}^{o}}\]

Correct Answer: C

Solution :

[c] \[C={{\sin }^{-1}}\left( \frac{1}{\mu } \right)={{\sin }^{-1}}\left( \frac{1}{2} \right)={{30}^{o}}\]

\[\therefore \]Angle of the prism \[A={{30}^{o}}\] the ray trace it is path if it is incident normally as shown in fig.

Now,    \[{{r}_{2}}=0\]

\[A={{r}_{1}}+{{r}_{2}}={{30}^{o}}\]

Now,    \[\mu =\frac{\sin {{i}_{l}}}{\sin {{r}_{l}}}\]

or         \[\sqrt{2}=\frac{\sin {{i}_{l}}}{\sin {{30}^{o}}}\]

\[\therefore \]    \[\sin {{i}_{l}}=\sqrt{2}\sin {{30}^{o}}=\frac{1}{\sqrt{2}}\]

or         \[\sin {{i}_{l}}=\sin {{45}^{o}}\]

or         \[{{i}_{l}}={{45}^{o}}\]