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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
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Verified Answer
The correct answer is:
$\mu_{1}^{2}+\mu_{3}^{2}=2 \mu_{2}^{2}$
for surface $\mathrm{AB}$ :
$\mu_{1} \sin 45^{\circ}=\mu_{2} \sin \theta_{1}$ $\ldots$ (1)
for surface $\mathrm{AC}$ :
$\mu_{2} \sin \left(\omega-\theta_{1}\right)=\mu_{3} \sin 45$ $\mu_{3} \sin 45^{\circ}=\mu_{2} \cos \theta_{1}$ $\ldots$ (2) $\omega=90^{\circ}$
Squaring and adding equation (1) \& (2)
$\frac{\mu_{1}^{2}}{2}+\frac{\mu_{3}^{2}}{2}=\mu_{2}^{2} \Rightarrow \mu_{1}^{2}+\mu_{3}^{2}=2 \mu_{2}^{2}$
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