In a plane electromagnetic wave, the electric field oscillates sinusoidally at a frequency of $2.0 \times 10^{10}\; Hz$ and amplitude $48\; Vm ^{-1}$

$(a)$ What is the wavelength of the wave?

$(b)$ What is the amplitude of the oscillating magnetic field?

$(c)$ Show that the average energy density of the $E$ field equals the average energy density of the $B$ field. $\left[c=3 \times 10^{8} \;m s ^{-1} .\right]$

The optical properties of a medium are governed by the relative permitivity $({ \in _r})$ and relative permeability $(\mu _r)$. The refractive index is defined as $n = \sqrt {{ \in _r}{\mu _r}} $. For ordinary material ${ \in _r} > 0$ 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 $\in _r < 0$ and $u_r < 0$. Since then such 'metamaterials' have been produced in the laboratories and their optical properties studied. For such materials $n =  – \sqrt {{ \in _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 $ in $2^{nd}$ quadrant, then the refracted beam is in the $3^{rd}$ quadrant.

$(ii)$ Prove that Snell's law holds for such a medium. 

A plane electromagnetic wave in a non-magnetic dielectric medium is given by $\vec E\, = \,{\vec E_0}\,(4 \times {10^{ – 7}}\,x – 50t)$ with distance being in meter and time in seconds. The dielectric constant of the medium is

A linearly polarized electromagnetic wave in vacuum is $E=3.1 \cos \left[(1.8) z-\left(5.4 \times 10^{6}\right) {t}\right] \hat{\text { i }}\, {N} / {C}$ is incident normally on a perfectly reflecting wall at $z=a$. Choose the correct option

A plane electromagnetic wave of angular frequency $\omega$ propagates in a poorly conducting medium of conductivity $\sigma$ and relative permittivity $\varepsilon$. Find the ratio of conduction current density and displacement current density in the medium.