Question

A jar of height h is filled with a transparent liquid of refractive index $\mu$. At the center 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 center, the dot is invisible

Answer 1

Ray $OA$ is incident at $A$ with critical angle $i_C$ and its angle of refraction will be ${90}^o$. For other ray if incident angle is more than $i_c$ it will get total internal reflection from the surface so will not come out from liquid
So disc of diameter $d$ is required to stop the rays from $O$ out of liquid
$\tan i_c\frac{d/2}{h}or\tan i_c=\frac{d}{2h}\Rightarrow d=2h\tan i_c$
$\sin i_c=\frac{1}{\mu }=\frac{P}{H}$ (From figure)
$B^2=H^2+P^2$
$B=\sqrt{{\mu }^2-1}$
$\therefore \tan i=\frac{P}{B}=\frac{1}{\sqrt{{\mu }^2-1}}$
$\therefore d=2h\times \frac{1}{\sqrt{{\mu }^2-1}}$