Question

The refractive index of the material of a prism is 2 and its refracting angle is 30º. 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 form the mirrored surface if its angle of incidence on the prism is

• A. 60º
• B. 0º
• C. 30º
• D. 45º

Answer 1

The Correct Option is D

To solve the given problem, we need to understand the behavior of light in a prism where one surface is mirrored. Let's break down the problem step-by-step:

1. The given refractive index of the prism, \mu, is 2, and the refracting angle A is 30º.
2. We need to determine the angle of incidence such that the light beam retraces its path after reflection from the mirrored surface.
3. For the light to retrace its path, the angle between the incident ray and the returning ray must be zero after the reflection.
4. Use Snell's Law at the first surface: n_1 \sin i = n_2 \sin r. Here n_1 = 1 (air), \mu = 2 (prism's index), i is angle of incidence, and r is angle of refraction.
   \sin i = 2 \sin r
5. The prism angle A = 30^\circ and r_1 + r_2 = A, where r_1 is angle of refraction on entry and r_2 is that before reflection.
6. Since the light retraces its path, r_2 = r_1,
   2r_1 = A \Rightarrow r_1 = 15^\circ
7. Substituting r_1 into Snell's Law:
   \sin i = 2 \sin 15^\circ
8. Calculate \sin 15^\circ = \frac{\sqrt{6} - \sqrt{2}}{4}:
   \sin i = 2 \times \frac{\sqrt{6} - \sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{2}
9. Converting to angle i gives us approximately 45^\circ.

Therefore, the correct angle of incidence for the light to retrace its path is 45º.