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)) \[\Rightarrow \]\[\theta >{{\sin }^{-1}}\left[ \mu \sin \left( A-{{\sin }^{-1}}\left( \frac{1}{\mu } \right) \right) \right]\]You need to login to perform this action.
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