question_answer

If a ray of light in a denser medium enters into a rarer medium at an angle of incidence i, the angle of reflection and refraction are respectively r and r. If the reflected and refracted rays are at right angles to each other, the critical angle for the given pair of media is:

A)  \[{{\sin }^{-1}}\left( \tan r \right)\]

B)  \[{{\sin }^{-1}}\left( \tan r \right)\]

C)  \[{{\tan }^{-1}}\left( \sin i \right)\]

D)  cot (tan i)

Correct Answer: B

Solution :

Here  \[i=r\] \[r=90-r\] So, \[\mu =\frac{\sin r}{\sin r}=\frac{\sin (90-r)}{\sin r}\] \[\mu =\frac{\cos r}{\sin r}=\frac{1}{\tan r}\] but       \[\mu =\frac{1}{\sin C}\]s where C is the critical angle So,       \[\frac{1}{\sin C}=\frac{1}{\tan r}\] \[\Rightarrow \]        \[\sin C=\tan r\] or         \[C={{\sin }^{-1}}(\tan r)\]