Question

A point source of light is placed at the bottom of the water lake. If the area of the illuminated circle on the surface is equal to 3 times the square of the depth of the lake. The refractive index of water is :

• A. $\sqrt{\frac{\pi }{3}+1}$
• B. $\sqrt{\frac{\pi }{3}}+1$
• C. $\frac{\pi }{3}+1$
• D. $\frac{\pi }{4}+1$

Answer 1

The correct option is A $\sqrt{\frac{\pi }{3}+1}$

Light of the source emerges out of water only when the incident angle at the water air boundary equals critical angle. $\left({\theta }_c\right)$

Let depth of lake $=H$

and radius of Illuminated circle $=R$

Given : $\pi R^2=3H^2$

$\Rightarrow H^2=\frac{\pi }{3}{R}^2$ ...(1)

From figure,

$\sin {\theta }_c=\frac{R}{\sqrt{R^2+H^2}}$

also we know, $\sin {\theta }_c=\left(\frac{1}{\mu }\right)$ $[\mu :$ Refractive index of water$]$

So we get, from above:-

$\Rightarrow \mu =\frac{\sqrt{R^2+H^2}}{R}=\frac{R\sqrt{1+\frac{\pi }{3}}}{R}$ [from (1)]

$\Rightarrow \mu =\sqrt{1+\frac{\pi }{3}}$