Question

A monochromatic light is incident lfom air on a refracting surface of a prism of angle 75${}^{\circ }$ and refractive index $n_0=\sqrt{3}.$ The other refracting surface of the prism is coated by a thin film of material of refractive index n as shown in figure. The light suffers total internal reflection at the coated prism surface for an incidence angle of $\theta \le 60^{\circ }$. The value of $n^2$ is ______.

Answer 1

Answer: 1.50

At $\theta =60^{\circ }$ ray incidents at critical angle at second surface
So,
$sin\theta =\sqrt{3}sin{r}_1$
$\frac{\sqrt{3}}{2}=\sqrt{3}sin{r}_1$
$r_1=30^{\circ }$
$r_2=45^{\circ }=C$
$\sqrt{3}sin45^{\circ }=nsin90^{\circ }$
$n=\sqrt{\frac{3}{2}}\Rightarrow n^2=\frac{3}{2}$