$A$ parallel plate capacitor with circular plates of radius $1\, m$ has a capacitance of $1 \;nF$. At $t=0,$ it is connected for charging in series with a resistor $R=1 \;M \Omega$ across a $2 \;V$ battery (Figure). Calculate the magnetic field at a point $P$,halfway between the centre and the periphery of the plates,after $t=10^{-3}\; s$.
(The charge on the capacitor at time $t$ is $q(t)=C V[1-\exp (-t / \tau)],$ where the time constant $\tau$ is equal to $C R .$ )

Give the equation that relates the speed of light $c$,the permeability of free space ${\mu _0}$,and the permittivity of free space ${\epsilon _0}$.

$A$ 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 $I_d$.

Imagine a world where free magnetic charges exist. In this world,a circuit is made with a $U$-shape wire and a rod free to slide on it. $A$ current carried by free magnetic charges can flow in the circuit. When the circuit is placed in a uniform electric field $E$ perpendicular to the plane of the circuit and the rod is pulled to the right with a constant speed $v$,the magnetic $emf$ in the circuit and the direction of the corresponding current,arising because of changing electric flux will be ($l$ is the length of the rod and $c$ is the speed of light).

What was used to detect the radio waves?

$A$ 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.