$A$ thin oil layer floats on water. $A$ ray of light making an angle of incidence of $40^o$ shines on the oil layer. The angle of refraction of the light ray in water is......$^o$ $({\mu _{oil}} = 1.45, {\mu _{water}} = 1.33)$

The refractive index of air with respect to vacuum is . . . . . . .

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.

When a light ray enters from medium $i$ to medium $j$,if $_i\mu_j$ represents the refractive index,what is the product of $_2\mu_1 \times _3\mu_2 \times _4\mu_3$?

If the velocity of light in the first medium is $v_1$ and in the second medium is $v_2$,then the refractive index of the second medium with respect to the first medium is given by:

$A$ ray of light passes from vacuum into a medium of refractive index $n$. If the angle of incidence is twice the angle of refraction,then the angle of incidence in terms of refractive index is