question_answer

A transparent solid cylindrical rod has a refractive index of \[\frac{2}{\sqrt{3}}\]. It is surrounded by air. A light ray is incident at the mid-point of one end of the rod as shown in the figure.

The incident angle \[(\theta )\] for which the light ray grazes along the wall of the rod is:

A) \[{{\sin }^{-1}}\left( \frac{\sqrt{3}}{2} \right)\]

B) \[{{\sin }^{-1}}\left( \frac{2}{\sqrt{3}} \right)\]

C) \[{{\sin }^{-1}}\left( \frac{1}{\sqrt{3}} \right)\]

D) \[{{\sin }^{-1}}\left( \frac{1}{2} \right)\]

Correct Answer: C

Solution :

[c] \[\sin \theta =\frac{2}{\sqrt{3}}\,\,\sin r=\frac{2}{\sqrt{3}}\,\,\cos i\]                ....(i)

and      \[\frac{2}{\sqrt{3}}\,\,\sin i=\sin 90{}^\circ \Rightarrow i={{60}^{o}}\]            ....(ii)

From (i) and (ii)

\[\sin \theta =\frac{1}{\sqrt{3}}\]