Question 31: Ray Optics

Question

Two equilateral-triangular prisms $\mathrm{P}_1$ and $\mathrm{P}_2$ are kept with their sides parallel to each other, in vacuum, as shown in the figure. A light ray enters prism $\mathrm{P}_1$ at an angle of incidence $	heta$ such that the outgoing ray undergoes minimum deviation in prism $\mathrm{P}_2$. If the respective refractive indices of $\mathrm{P}_1$ and $\mathrm{P}_2$ are $\sqrt{\frac{3}{2}}$ and $\sqrt{3}$, then $	heta=\sin ^{-1}\left[\sqrt{\frac{3}{2}} \sin \left(\frac{\pi}{\beta}ight)ight]$, where the value of $\beta$ is ____.

Accepted Answer

At surface BC
$$\begin{aligned}
& \sqrt{\frac{3}{2}} \sin r_2=\sqrt{3} \sin 30 \
& \sqrt{\frac{3}{2}} \sin r_2=\frac{\sqrt{3}}{2} \
& \operatorname{sinr}_2=\frac{1}{\sqrt{2}} \
& r_2=45^{\circ} \
& r_1=60^{\circ}-45^{\circ}=15^{\circ}
\end{aligned}$$
At surface $$A B$$
$$\begin{gathered}
1 \sin 	heta=\sqrt{\frac{3}{2}} \sin 15^{\circ} \
	heta=\sin ^{-1}\left[\sqrt{\frac{3}{2}} \sin \frac{\pi}{12}ight] \
\beta=12
\end{gathered}$$

Ray Optics

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