question_answer

Let the refractive index of a denser medium with respect to a rarer medium be \[{{n}_{12}}\]and its c critical angle be \[{{\theta }_{C}}.\]At an angle of incidence A when light is travelling from denser medium to rarer medium, a part of the light is reflected and the rest is refracted and the angle between reflected and refracted rays is \[90{}^\circ \]. Angle A given by -    [JEE Online 08-04-2017]

A) \[{{\tan }^{-1}}(\sin {{\theta }_{C}})\]                    

B) \[\frac{1}{{{\tan }^{-1}}(\sin {{\theta }_{C}})}\]

C) \[{{\cos }^{-1}}(\sin {{\theta }_{C}})\]                    

D) \[\frac{1}{{{\cos }^{-1}}(\sin {{\theta }_{C}})}\]

Correct Answer: A

Solution :

\[\mu =\frac{{{\mu }_{R}}}{{{\mu }_{D}}}=\frac{\sin {{i}_{c}}}{\sin {{90}^{o}}}\] \[\frac{{{\mu }_{R}}}{{{\mu }_{D}}}=\sin {{i}_{i}}\] \[\mu =\frac{{{\mu }_{R}}}{{{\mu }_{D}}}=\frac{\operatorname{sinA}}{\operatorname{sinr}}\] \[=\frac{\sin A}{\sin (90-A)}=\frac{\operatorname{sinA}}{\operatorname{sinA}}\] \[\frac{{{\mu }_{R}}}{{{\mu }_{D}}}=\tan A\] \[\tan A=\sin {{\theta }_{C}}\] \[A={{\tan }^{-1}}(sin{{\theta }_{C}})\]