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)  \[ta{{n}^{-1}}\left( \sin i \right)\]                            

D)  \[cot\left( \tan i \right)\]

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