$A$ 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.

Write the $SI$ unit of ${\epsilon _0}\left( {\frac{{d{\Phi _E}}}{{dt}}} \right)$.

$A$ quantity $X$ is given by $\varepsilon_0 L \frac{\Delta V}{\Delta t}$,where $\varepsilon_0$ is the permittivity of free space,$L$ is the length,$\Delta V$ is a potential difference,and $\Delta t$ is a time interval. The dimension of $X$ is the same as that of

Derive the expression for the displacement current term in the modified Ampere's circuital law. Define displacement current and state its $SI$ unit.

$A$ long straight cable of length $l$ is placed symmetrically along $z-$ axis and has radius $a (a << l)$. The cable consists of a thin wire and a co-axial conducting tube. An alternating current $I(t) = I_0 \sin(2\pi \nu t)$ flows down the central thin wire and returns along the co-axial conducting tube. The induced electric field at a distance $s$ from the wire inside the cable is $\vec{E}(s,t) = \mu_0 I_0 \nu \cos(2\pi \nu t) \ln(s/a) \hat{k}$.
$(i)$ Calculate the displacement current density inside the cable.
$(ii)$ Integrate the displacement current density across the cross-section of the cable to find the total displacement current $I_d$.
$(iii)$ Compare the conduction current $I_0$ with the displacement current $I_d$.

Sea water at frequency $v = 4 \times 10^8\, Hz$ has permittivity $\epsilon \approx 80\epsilon_0$,permeability $\mu = \mu_0$ and resistivity $\rho = 0.25\,\Omega m$. Imagine a parallel plate capacitor immersed in sea water and driven by an alternating voltage source $V(t) = V_0 \sin(2\pi vt)$. What fraction of the conduction current density is the displacement current density?