Question

A container contains water up to a height of $20cm$ and there is a point source at the centre of the bottom of the container. A rubber ring of radius $r$ floats centrally on the water. The ceiling of the room is $2.0m$ above the water surface. (a) Find the radius of the shadow of the ring formed on the ceiling if $r=15cm$. (b) Find the maximum value $r$ for which the shadow of the ring is formed on the ceiling. Refrective index of water $=4/3$.

Answer 1

$\left(a\right)r=15cm$

$\because d=\frac{d}{\left(\frac{n_{center}}{n_{air}}\right)}$

$\Rightarrow d=\frac{3d}{4}$

$\Rightarrow FC=\frac{3}{4}AC=\frac{3}{4}\times 20=15cm$

Now, $\because FC=BC\Rightarrow \angle BFC={450}^o$

$\Rightarrow EF=ED$

$\Rightarrow ED=200+15$

$\Rightarrow EG=215cm$

So radius of shadow $=215cm$

$\left(b\right)$ we know that at the maximum values of the light ray will undergo total internal reflection.

From figure:-

$\sin C=\frac{I}{n_{water}}=\frac{3}{4}$

Area from $\Delta ABD$

$\sin C=\frac{BD}{\sqrt{B{D}^2+A{D}^2}}$

$\Rightarrow \frac{3}{4}=\frac{BD}{\sqrt{B{D}^2+(20{)}^2}}$

$\Rightarrow 9B{D}^2+3600=16B{D}^2$

$\Rightarrow 7B{D}^2=3600$

$\Rightarrow B{D}^2=514.28$

$\Rightarrow BD=\sqrt{514.28}=22.68cm$