A light ray enters through a right angled prism at point $P$ with the angle of incidence $30°$ as shown in figure. It travels through the prism parallel to its base $BC$ and emerges along the face $AC$. The refractive index of the prism is:
Answer: (D) $\dfrac{\sqrt{5}}{2}$
The prism is right-angled at $A$, so $\angle B + \angle C = 90°$.
At face $AB$: the ray inside is parallel to $BC$, so it makes angle $B$ with face $AB$. The angle of refraction is $r_1 = 90° - B$. Snell's law:
$$\sin 30° = \mu\sin(90° - B) = \mu\cos B \;\Rightarrow\; \mu\cos B = \frac{1}{2}$$
At face $AC$: the ray (parallel to $BC$) makes angle $C = 90° - B$ with face $AC$, so the angle of incidence is $B$. The ray emerges along the face, so $B$ is the critical angle:
$$\mu\sin B = 1$$
Squaring and adding:
$$\mu^2 = 1 + \frac{1}{4} = \frac{5}{4} \;\Rightarrow\; \mu = \frac{\sqrt{5}}{2}$$
Solution by Sreeraj P, M.Sc Physics