Question

A rectangular block is composed of three different glass prisms (with refractive indices ,$\mu_{1} ,\mu_{2}$ and $\mu_{3}$ ) as shown in the figure below. A ray of light incident normal to the left face emerges normal to the right face. Then, the refractive indices are related by

• A. $\mu^{2}_{1}+,\mu^{2}_{2}=2\mu^{2}_{3}$
• B. $\mu^{2}_{1}+,\mu^{2}_{2}=\mu^{2}_{3}$
• C. $\mu^{2}_{1}+,\mu^{2}_{3}=2\mu^{2}_{2}$
• D. $\mu^{2}_{2}+,\mu^{2}_{3}=2\mu^{2}_{1}$

Answer 1

Answer: (C)

Ray diagram of ray through the composition of prisms will be

By Snell’s law on surface $AB$ and $AC$, we have
$\mu_{1} \sin 45^{\circ}=\mu_{2} \sin \theta \ldots\left(i\right)$
and $\mu_{3} \sin 45^{\circ}=\mu_{2} \cos \theta \cdots\left(ii\right)$
As $\alpha -\theta=45^{\circ}$ from figure
Squaring and adding Eqs $\left(i\right)$ and $\left(ii\right),$
we get $\frac{\mu_{1}^{2}}{2}+\frac{\mu_{3}^{2}}{2}=\mu_{2}^{2}$