question_answer

A transparent cube of side d, made of a material of refractive index \[{{\mu }_{2}},\]is immersed in a liquid of refractive index \[{{\mu }_{1}}({{\mu }_{1}}<{{\mu }_{2}}).\]A ray is incident on the face AB at an angle \[\theta \](shown in the figure). Total internal reflection takes place at point E on the face BC. The \[\theta \] must satisfy: [JEE Main 12-4-2019 Afternoon]

A) \[\theta <{{\sin }^{-1}}\frac{{{\mu }_{1}}}{{{\mu }_{2}}}\]

B) \[\theta <{{\sin }^{-1}}\sqrt{\frac{\mu _{2}^{2}}{\mu _{1}^{2}}-1}\]

C) \[\theta >{{\sin }^{-1}}\frac{{{\mu }_{1}}}{{{\mu }_{2}}}\]   

D)   \[\theta >{{\sin }^{-1}}\sqrt{\frac{\mu _{2}^{2}}{\mu _{1}^{2}}-1}\]

Correct Answer: B

Solution :

            \[\sin c=\frac{{{\mu }_{1}}}{{{\mu }_{2}}}\] \[{{\mu }_{1}}\sin \theta ={{\mu }_{2}}\sin ({{90}^{o}}-C)\] \[\sin \theta =\frac{{{\mu }_{2}}\sqrt{1-\frac{\mu _{1}^{2}}{\mu _{2}^{2}}}}{{{\mu }_{1}}}\] \[\theta <{{\sin }^{-1}}\sqrt{\frac{\mu _{2}^{2}-\mu _{1}^{2}}{\mu _{1}^{2}}}\] For TIR \[\theta <{{\sin }^{-1}}\sqrt{\frac{\mu _{2}^{2}}{\mu _{1}^{2}}-1}\]