Question

A small coin is resting on the bottom of a beaker filled with a liquid. A ray of light from the coin travels upto the surface of the liquid and moves along its surface (see figure).

How fast is the light travelling in the liquid ?

• A. $1.8\times 10^{8}m/s$
• B. $2.4\times 10^{8}m/s$
• C. $3.0\times 10^{8}m/s$
• D. $1.2\times 10^{8}m/s$

Answer 1

Answer: (A)

Critical angle is the angle of incidence in denser medium for which the angle of refraction in rarer medium is $90^{\circ }$.
As shown in figure, a light ray from the coin will not emerge out of liquid, if $i>C$.

Therefore, minimum radius $R$ corresponds to $i=C$. In $\Delta SAB$,
$\frac{R}{h}=tanC$
or $R=htanC$
or $R=\frac{h}{\sqrt{{\mu }^2-1}}$

Given, $R=3cm$, $h=4cm$
Hence, $\frac{3}{4}=\frac{1}{\sqrt{{\mu }^2-1}}$
or ${\mu }^2=\frac{25}{9}$ or $\mu =\frac{5}{3}$
But $\mu =\frac{c}{v}$ or $v=\frac{c}{\mu }$
$=\frac{3\times 10^8}{5/3}$
$=1.8\times 10^{8}m/s$