A hemispherical vessel is completely filled with a liquid of refractive index $\mu$. A small coin is kept at the lowest point $(\mathrm{O})$ of the vessel as shown in figure. The minimum value of the refractive index of the liquid so that a person can see the coin from point E (at the level of the vessel) is

• A

  $\sqrt{3}$
• B

  $\frac{\sqrt{3}}{2}$
• C

  $\frac{3}{2}$
• Correct answer
  D

  $\sqrt{2}$

Solution

For the rays from coin to reach the point $E$, the refracted rays must grazing the surface, i.e. they must be incident at critical angle $\theta_c$ inside the liquid. $\mu=\frac{1}{\sin \theta_c}$ $\mu$ is minimum when $\theta_c$ is maximum. Maximum value of $\theta_c=45^{\circ}$ $\Rightarrow \mu$ has a minimum value of $\sqrt{2}$.

Hence, the correct answer is option: 'D'.