$A$ needle is lying at the bottom of a water tank of height $12 \,cm$. The apparent depth of the needle measured by a microscope is $9 \,cm$. If the water is replaced by a liquid of refractive index of $1.5$ of same height, the distance through which the microscope has to be moved to focus the needle again is (in $\,cm$)

The absolute refractive indices of glass and water are $\frac{3}{2}$ and $\frac{4}{3}$,respectively. The ratio of the velocity of light in glass to that in water will be:

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.

$A$ light beam is traveling from Region $I$ to Region $IV$ (Refer Figure). The refractive indices in Regions $I, II, III$ and $IV$ are $n_0, \frac{n_0}{2}, \frac{n_0}{6}$ and $\frac{n_0}{8}$,respectively. The angle of incidence $\theta$ for which the beam just misses entering Region $IV$ is:

The phase difference between the incident wave and the reflected wave is $180^o$ when a light ray:

$A$ light wave travels through three transparent materials of equal thickness. Rank in order,from the highest to lowest,the indices of refraction $n_1, n_2$ and $n_3$.