Question

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

Answer 1

Let $d$ be the diameter of the disc.

The spot will be invisible if the incident rays from the dot to the surface at $d/2$ exceeds the critical angle.

Given, angle of the incidence $=i$

critical angle for this liquid, $i_c={\sin }^{-1}\left(\frac{1}{\mu }\right)$

$\Rightarrow \sin i_c=\frac{1}{\mu }$

from figure, $\frac{d/2}{h}=\tan i$

$\Rightarrow \frac{d}{2}=h\tan i=h{\left[\sqrt{{\mu }^2-1}\right]}^{-1}$

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

$\text{Final answer}:\frac{2h}{\sqrt{{\mu }^2-1}}$