question_answer

A ray of light from denser medium strikes a rarer medium at angle of incidence \[i\]. The reflected and refracted rays make an angle of \[90{}^\circ \] with each other. The angle of reflection and refraction are r and \[r'.\] The critical angle is

A) \[\text{si}{{\text{n}}^{-\text{1}}}\text{ }\left( \text{tan r} \right)\]       

B)         \[{{\sin }^{-1}}(\tan i)\]  

C) \[\text{si}{{\text{n}}^{-1}}\text{ }\left( \text{tan r}' \right)\]     

D)        \[{{\sin }^{-1}}(\tan (i+r)\]

Correct Answer: B

Solution :

 From figure,\[i=r\]and\[r=90-r\]                 \[\mu =\frac{\sin r'}{\sin r}=\frac{\sin (90-r)}{\sin r}\]                    \[=\frac{\cos r}{\sin r}=\frac{1}{\tan r}\] We know,           \[\mu =\frac{1}{\sin C}\] \[\therefore \]  \[\frac{1}{\sin C}=\frac{1}{\tan r}\] \[\Rightarrow \]               \[\sin C=\tan r\] \[\therefore \]  \[C={{\sin }^{-1}}(\tan r)={{\sin }^{-1}}(\tan i)\]