Question

A jar of height h is filled with a transparent liquid of refractive index μ. At the centre of the jar on the bottom surface is a dot. Find the minimum diameter of a disc, such that when it is placed on the top surface symmetrically about the centre, the dot is invisible

Answer 1

Let us consider that $i_C$ is critical angle.

ray of light incident at an angle greater than $i_C$ are totally reflected within water and consequently cannot emerge out of the water surface.

from Snell's law,

$\quad 1sin90^{\circ}=\mu sini_C$

$\implies sini_C=\frac{1}{\mu}$

$\implies tani_C=\frac{1}{\sqrt{\mu^2-1}}$.

see figure, $tani_C=\frac{d/2}{h}$

so, $\frac{d/2}{h}=\frac{1}{\sqrt{\mu^2-1}}$

$d=\frac{2h}{\sqrt{\mu^2-1}}$

hence, minimum diameter of the disc is $\boxed{\bf{d=\frac{2h}{\sqrt{\mu^2-1}}}}$