Question

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. $si{n}^{-1}\left(\frac{1}{2}\right)$
• B. $si{n}^{-1}\left(\frac{\sqrt{3}}{2}\right)$
• C. $si{n}^{-1}\left(\frac{2}{\sqrt{3}}\right)$
• D. $si{n}^{-1}\left(\frac{1}{\sqrt{3}}\right)$

Answer 1

The correct option is D

$si{n}^{-1}\left(\frac{1}{\sqrt{3}}\right)$

Applying snell's law at the first surface

$\frac{sin\theta }{sinr}=\frac{2}{\sqrt{3}}...\left(1\right)$
at second surface
sin (90 - r) = $\frac{\sqrt{3}}{2}...\left(2\right)$
$\Rightarrow cosr=\frac{\sqrt{3}}{2}$
$\Rightarrow sinr=\frac{1}{2}$
$\Rightarrow 2sin\theta =\frac{2}{\sqrt{3}}$ from (1)
$\Rightarrow sin\theta =\frac{1}{\sqrt{3}}$
$\Rightarrow \theta =si{n}^{-1}\left(\frac{1}{\sqrt{3}}\right)$