​Two equilateral-triangular prisms P1 and P2 are kept with their sides parallel to each other, ​

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 \(\theta\) 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 \(\theta=\sin ^{-1}\left[\sqrt{\frac{3}{2}} \sin \left(\frac{\pi}{\beta}\right)\right]\), where the value of \(\beta\) is ______.

Answer 1

Correct answer: 12

At surface BC

\(\sqrt{\frac{3}{2}} \sin r_2=\sqrt{3} \sin 30\)

\(\sqrt{\frac{3}{2}} \sin r_2=\frac{\sqrt{3}}{2}\)

\(\sin _2=\frac{1}{\sqrt{2}}\)

\(\mathrm{r}_2=45^{\circ}\)

\(\mathrm{r}_1=60^{\circ}-45^{\circ}=15^{\circ}\)

At surface AB

\(1 \sin \theta=\sqrt{\frac{3}{2}} \sin 15^{\circ}\)

\(\theta=\sin ^{-1}\left[\sqrt{\frac{3}{2}} \sin \frac{\pi}{12}\right]\)

\(\beta=12\)