Question

The ray of light enters from a rarer to the denser medium. The angle of incidence is $i$ . Then the reflected and refracted rays are mutually perpendicular to each other. The critical angle for the pair of media is.
\left( A \right){\sin ^{ - 1}}\left( {\tan i} \right) \\\ \left( B \right){\tan ^{ - 1}}\left( {\sin i} \right) \\\ \left( C \right){\sin ^{ - 1}}\left( {\cot i} \right) \\\ \left( D \right){\cos ^{ - 1}}\left( {\tan i} \right) \\\

Answer 1

In order to solve this question, we are going to first take the angles of the incidence and the refraction, then, by considering the conditions given or that, we simply find the critical angle by finding the value of refractive index in terms of the angle of the incidence and further simplifying.
According to the Snell’s law,
$\dfrac{{\sin i}}{{\sin r}} = \mu$
If $c$ is the critical angle, then,
$\mu = \dfrac{1}{{\sin c}}$
$\dfrac{1}{{\tan i}} = \cot i$

Complete step by step solution:
It is given that the ray if light enters from a rarer to the denser medium.
The angle of incidence here is $i$ ,
Let the angle of refraction be $r$
Now, as it is given that the reflected and the refracted rays are parallel to each other, so, this implies that the total internal reflection is taking place,
Therefore, $i + r = \dfrac{\pi }{2}$
Now, according to the Snell’s law,
$\dfrac{{\sin i}}{{\sin r}} = \mu$
Now simplifying it further, we get,
\dfrac{{\sin i}}{{\sin \left( {\dfrac{\pi }{2} - i} \right)}} = \mu \\\ \Rightarrow \dfrac{{\sin i}}{{\cos i}} = \mu \\\ \Rightarrow \tan i = \mu \\\
Now, let $c$ be the critical angle, then,
$\mu = \dfrac{1}{{\sin c}}$
Then, using the relation $\tan i = \mu$ in above equation to find the value of refractive index, we get,
\tan i = \dfrac{1}{{\sin c}} \\\ \Rightarrow c = {\sin ^{ - 1}}\left( {\dfrac{1}{{\tan i}}} \right) \\\ \Rightarrow c = {\sin ^{ - 1}}\left( {\cot i} \right) \\\
Thus, option $\left( C \right){\sin ^{ - 1}}\left( {\cot i} \right)$ is the correct answer.

Note:
The critical angle is the angle at which the ray moving from the denser to the rarer medium is refracted such that the ray makes an angle of ${90^ \circ }$ with the interface. Above this angle, any ray travelling from the denser to the rarer medium is internally reflected completely and the phenomenon is known as total internal reflection.