question_answer

Monochromatic light is incident on a glass prism of angle A. If the refractive index of the material of the prism is \[\mu ,\] a ray, incident at an angle \[\theta ,\] on the face AB would get transmitted through the face AC of the prism provided. [JEE Main Solved Paper-2015 ]

A) \[\theta >{{\cos }^{-1}}\left[ \mu \sin \left( A+{{\sin }^{-1}}\left( \frac{1}{\mu } \right) \right) \right]\]

B) \[\theta <{{\cos }^{-1}}\left[ \mu \sin \left( A+{{\sin }^{-1}}\left( \frac{1}{\mu } \right) \right) \right]\]

C) \[\theta >{{\sin }^{-1}}\left[ \mu \sin \left( A-{{\sin }^{-1}}\left( \frac{1}{\mu } \right) \right) \right]\]

D) \[\theta <{{\sin }^{-1}}\left[ \mu \sin \left( A-{{\sin }^{-1}}\left( \frac{1}{\mu } \right) \right) \right]\]

Correct Answer: C

Solution :

                By snell?s law at 1st refracting surface is \[1.\sin \theta =\mu .\sin {{r}_{1}}\] \[\sin {{r}_{1}}=\frac{\sin \theta }{\mu }\] \[{{r}_{1}}<{{\theta }_{1}},\] ray to transmitted through surface AC, \[{{r}_{2}}<{{\theta }_{c}}\]\[{{r}_{2}}<{{\sin }^{-1}}\left( \frac{1}{\mu } \right)\] \[A-{{r}_{1}}<{{\sin }^{-1}}\left( \frac{1}{\mu } \right)\] \[A-{{\sin }^{-1}}\left( \frac{\sin \theta }{\mu } \right)<{{\sin }^{-1}}\left( \frac{1}{\mu } \right)\] \[A-{{\sin }^{-1}}\left( \frac{1}{\mu } \right)<{{\sin }^{-1}}\left( \frac{\sin \theta }{\mu } \right)\] \[\sin \left( A-{{\sin }^{-1}}\left( \frac{1}{\mu } \right) \right)<\frac{\sin \theta }{\mu }\] \[\mu \sin \left( A-{{\sin }^{-1}}\left( \frac{1}{\mu } \right) \right)<\sin \theta \] \[\theta >{{\sin }^{-1}}\left( \mu \sin \left( A-{{\sin }^{-1}}\left( \frac{1}{\mu } \right) \right) \right)\]