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

At $\theta = 60^{\circ}$ ray incidents at critical angle at second surface
So,
$sin\theta = \sqrt{3}\,sinr_{1}$
$\frac{\sqrt{3}}{2} = \sqrt{3}\,sin\,r_{1}$
$r_{1} = 30^{\circ}$
$r_{2} = 45^{\circ} = C$
$\sqrt{3}\,sin\,45^{\circ} = n \, sin \, 90^{\circ} $
$n = \sqrt{\frac{3}{2}}\Rightarrow n^{2} = \frac{3}{2}$