Question

A transparent solid cylindrical rod has a refractive index of $2/\sqrt{3}$. It is surrounded by air. A light ray is incident at the midpoint 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. ${sin}^{-1}\left(\frac{1}{2}\right)$

Answer 1

The correct option is C ${sin}^{-1}\left(\frac{1}{\sqrt{3}}\right)$

$\begin{array}{c}\text{At point}A\text{:}\\ n_1\sin \theta =n_2\sin r\Rightarrow \sin \theta =\frac{2}{\sqrt{3}}\sin r\end{array}$

$\sin r=\frac{\sqrt{3}\sin \theta }{2}-\left(i\right)$

$n_2\sin ({90}^{\circ }-r)=n_1\sin {90}^{\circ }\Rightarrow \frac{2}{\sqrt{3}}\cos r=1$

$\cos r=\frac{\sqrt{3}}{2}-\left(ii\right)$

$\text{from}\left(i\right)\&\left(ii\right):$

$\begin{array}{cc}{\sin }^{2}r+{\cos }^{2}r=1& \\ \frac{3}{4}{\sin }^2\theta +\frac{3}{4}=1& \Rightarrow \sin \theta =\sqrt{\frac{1}{3}}\\ & \Rightarrow \theta ={\sin }^{-1}\sqrt{\frac{1}{3}}\end{array}$