question_answer

The refractive index of the material of a prism is \[\sqrt{2}\] and its refracting angle is \[{{30}^{\text{o}}}\]. One of the refracting surfaces of the prism is made a mirror inwards. A beam of monochromatic light entering the prism from the other face will retrace its path after reflection from the mirrored surface if its angle of incidence on the prism is:                                                                                                                                          

A) \[{{45}^{\text{o}}}\]   

B) \[{{60}^{\text{o}}}\]                              

C) \[{{0}^{\text{o}}}\]        

D) \[{{30}^{\text{o}}}\]

Correct Answer: A

Solution :

      According to the given condition, the beam of light will retrace its path after reflection from BC.  So \[\angle CPQ={{90}^{0}}\] Thus, angle of refraction at surface AC \[\angle PQN=\angle r={{90}^{0}}-{{60}^{0}}={{30}^{0}}\] By Snell's law \[\mu =\frac{\sin i}{\sin r}\] \[\Rightarrow \]       \[\sqrt{2}=\frac{\sin i}{\sin {{30}^{0}}}\] \[\therefore \]          \[\sqrt{2}\times \sin {{30}^{0}}=\sin i\] \[\Rightarrow \]       \[\sqrt{2}\times \frac{1}{2}=\sin i\] \[\Rightarrow \]       \[\sin i=\frac{1}{\sqrt{2}}=\sin {{45}^{0}}\] \[i={{45}^{0}}\]