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. ${\mu }_1^2+,{\mu }_2^2=2{\mu }_3^2$
• B. ${\mu }_1^2+,{\mu }_2^2={\mu }_3^2$
• C. ${\mu }_1^2+,{\mu }_3^2=2{\mu }_2^2$
• D. ${\mu }_2^2+,{\mu }_3^2=2{\mu }_1^2$

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

${\mu }_1\sin 45^{\circ }={\mu }_2\sin \theta \dots \left(i\right)$

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 $\frac{{\mu }_1^2}{2}+\frac{{\mu }_3^2}{2}={\mu }_2^2$