Question

A point source $S$ is placed at the bottom of a $12mm$ high transparent block of diamond (refractive index = $2.4$ ). The block is immersed in an optically rarer liquid as shown in the figure. It is found that the light emerging from the block to the liquid forms a circular bright spot of diameter $18mm$ on the top of the block. If the refractive index of the liquid is $\frac{x}{25}$ , what is the of $x$ ?

Answer 1

Answer: 36

As the light is passing from optically denser medium to rarer medium, let us assume $\theta$ is the critical angle.
From Pythagoras theorem,
$SB=\sqrt{S{O}^2+O{B}^2}=\sqrt{12^2+9^2}=15mm$
$\therefore sin\theta =\frac{OB}{SB}=\frac{9}{15}=\frac{3}{5}$ $sin\theta =\frac{{\mu }_2}{{\mu }_1}=\frac{\mu }{2.4}$
$\therefore \frac{3}{5}=\frac{\mu }{2.4}$ $\Rightarrow \mu =\frac{36}{25}$