Question

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: 

(1) \(\frac{\sqrt{5}}{4}\)

(2) \(\frac{\sqrt{5}}{2}\)

(3) \(\frac{\sqrt{3}}{4}\)

(4) \(\frac{\sqrt{3}}{2}\) 

Answer 1

Correct option is : (2) \(\frac{\sqrt{5}}{2}\) 

In prism, \(r_1 + c = A\)

\(r_1 = 90^\circ - c \ \ \ .....(1)\)

\(\sin c = \frac{1}{\mu} \Rightarrow \cos c = \frac{\sqrt{\mu^2-1}}{\mu}\)

\( \Rightarrow\) Apply Snell's law, on incidence surface

\(1. \sin 30^\circ= \mu \sin(r_1) \)

\(\Rightarrow 1 \times \frac 12 = \mu \times \sin (90^\circ - c)\)

\(\frac 12 = \mu \times \frac{\sqrt{\mu ^2 - 1}} \mu\)

On squaring

\(\frac 14 = \mu ^2-1\)

\(\Rightarrow \mu^2 = \frac54 \)

\(\Rightarrow \mu = \frac{\sqrt 5} 2\)