Ncert Exercises & Solutions

A jar of height h is filled with a transparent liquid of refractive index figure μ. 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.

Hint: The dot will be invisible if all the refracted rays are obstructed by the disc.

Step 1: Find the critical angle of incidence.

Let d be the diameter of the disc. The spot shall be invisible if the incident rays from the dot at O to the surface at the d/2 are at the critical angle.
Let i be the angle of incidence.
Using the relationship between refractive index and critical Angle,
$\sin i = \frac{1}{μ}$

Step 2: Find the diameter of the disc.
Using geometry and trigonometry,
Now,                                             $\frac{d / 2}{h} = \tan i$
$\Rightarrow$                                                 $\frac{d}{2} = h \tan i = h {[\sqrt{μ^{2} - 1}]}^{- 1}$
$∴$                                                 $d = \frac{2 h}{\sqrt{μ^{2} - 1}}$
This is the required expression of d.

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