$A$ tank is filled with water to a height of $12.5 \, cm$. The apparent depth of a needle lying at the bottom of the tank is measured by a microscope to be $9.4 \, cm$. If water is replaced by a liquid of refractive index $1.63$ up to the same height,what will be the apparent depth of the needle in $cm$?

The optical properties of a medium are governed by the relative permittivity $(\epsilon_r)$ and relative permeability $(\mu_r)$. The refractive index is defined as $n = \sqrt{\epsilon_r \mu_r}$. For ordinary material $\epsilon_r > 0$ and $\mu_r > 0$ and the positive sign is taken for the square root. In $1964$,a Russian scientist $V$. Veselago postulated the existence of material with $\epsilon_r < 0$ and $\mu_r < 0$. Since then,such 'metamaterials' have been produced in the laboratories and their optical properties studied. For such materials $n = -\sqrt{\epsilon_r \mu_r}$. As light enters a medium of such refractive index,the phases travel away from the direction of propagation.
$(i)$ According to the description above,show that if rays of light enter such a medium from air (refractive index $= 1$) at an angle $\theta_i$ in the $2^{nd}$ quadrant,then the refracted beam is in the $3^{rd}$ quadrant.
$(ii)$ Prove that Snell's law holds for such a medium.

$A$ bird is flying at $12 \ m$ height above the water surface and a fish is swimming $16 \ m$ below the water surface $(\mu_{\text{water}} = 4/3)$. Find the distance of the bird with respect to the fish as seen by the fish. (in $m$)

Immiscible transparent liquids $A, B, C, D$ and $E$ are placed in a rectangular glass container,forming layers based on their densities. The refractive indices of the liquids are given in the table below. The container is illuminated from the side,and a small piece of glass with a refractive index of $1.61$ is gently dropped into the liquid layers. In which liquid will the glass piece not be visible as it descends?
| Liquid | Refractive Index |
| :--- | :--- |
| $A$ | $1.51$ |
| $B$ | $1.53$ |
| $C$ | $1.61$ |
| $D$ | $1.52$ |
| $E$ | $1.65$ |

An initially parallel cylindrical beam travels in a medium of refractive index $\mu(I) = \mu_0 + \mu_2I$,where $\mu_0$ and $\mu_2$ are positive constants and $I$ is the intensity of the light beam. The intensity of the beam is decreasing with increasing radius. The speed of light in the medium is

The difference of speed of light in the two media $A$ and $B$ $(v_{A}-v_{B})$ is $2.6 \times 10^{7} \, m/s$. If the refractive index of medium $B$ is $1.47$,then the ratio of refractive index of medium $B$ to medium $A$ is: (Given: speed of light in vacuum $c = 3 \times 10^{8} \, m/s$)