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

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 :

[c] : According to Snell?s law\[\sin \theta =\mu \sin {{r}_{1}}\] or\[{{r}_{1}}={{\sin }^{-1}}\left( \frac{\sin \theta }{\mu } \right)\] Now, \[A={{r}_{1}}+{{r}_{2}}\] \[\therefore \]\[{{r}_{2}}=A-{{r}_{1}}\] \[=A-{{\sin }^{-1}}\left( \frac{\sin \theta }{\mu } \right)\]              ?(i) For the ray to get transmitted through the face \[AC,{{r}_{2}}\] must be less than critical angle, i,e.,\[{{r}_{2}}<{{\sin }^{-1}}\left( \frac{1}{\mu } \right)\]or\[A-{{\sin }^{-1}}\left( \frac{\sin \theta }{\mu } \right)<{{\sin }^{-1}}\left( \frac{1}{\mu } \right)\] (using (i)) \[\Rightarrow \]\[\theta >{{\sin }^{-1}}\left[ \mu \sin \left( A-{{\sin }^{-1}}\left( \frac{1}{\mu } \right) \right) \right]\]