How does Ampere-Maxwell law explain the flow of current through a capacitor when it is being charged by a battery? Write the expression for the displacement current in terms of the rate of change of electric flux.

Explanation

Consider charging of a parallel plate capacitor by a time-varying current iw Let us find the magnetic field at a point P in a region outside the capacitor. For this we consider a plane circular loop of radius r whose plane is perpendicular to the direction of the current carrying wire and which is centred symmetrically with respect to the wire [Fig]. Using symmetry condition and applying Ampere.s circuital law, we have
`B(2pir)=mu_(0)i_((t))" ....(i)"`
However, if we consider a different surface, having the same boundary [either as shown in Fig. or Fir. and apply Ampere.s circuital law as before, we find
`B(2pir)=mu_(0)(0)=0" ....(ii)"`
It is because nocurrent passes through the surface of Figs. and 8.04. It causes a contradiction because we get a finite value of magnetic field B by doing calculation in one way and zero value of B by doing calculation in another way. To remove this contradiction Maxwell introduced the concept of displacement current

`i_(D)=in_(0)(dphi_(E))/(dt),` where `(dphi_(E))/(dt)` is the rate of change of electric flux between the plates of given capacitor.
Now, the Ampere - Maxwell.s circuital law is expressed as
`oint vecB.vecdl=mu_(0)[i_((t))+i_(D)]`
Thus, Ampere - Maxwell law successfully explains the flow the current through a capacitor when it is being charged (or discharged) by a battery.