Q.Using the example of a parallel-plate capacitor being charged, explain qualitatively how a magnetic field can exist at a point between the plates even though no conduction current flows there, and write the formula for the displacement current in terms of the rate of change of electric flux.

Setup. A parallel-plate capacitor is being charged by a steady current $I$ flowing in the connecting wire. Consider the region between the plates -- no wire, and no conduction current, exists there.

Why a magnetic field exists there anyway. As charge accumulates on the plates, the electric field $E$ between them (and hence the electric flux $\Phi_E=EA$ through a surface placed in the gap) grows continuously with time. Maxwell's insight was that this changing flux is, for the purpose of producing $\vec{B}$, exactly equivalent to a current -- the displacement current

and it can be shown that $I_d$ works out to be numerically EQUAL to the conduction current $I$ charging the capacitor, at every instant. So the Ampère-Maxwell law, $\oint\vec{B}\cdot d\vec{l}=\mu_0(I_c+I_d)$, gives $\mu_0 I$ whichever surface is chosen -- $\mu_0 I_c=\mu_0 I$ for the flat surface pierced by the wire, or $\mu_0 I_d=\mu_0 I$ for the bulged surface through the gap -- and a genuine magnetic field, of the same strength either way, is found circling the gap region exactly as it circles the connecting wire.

Track the changing electric flux between the plates and apply Maxwell's displacement-current formula to it.

• Saying there is NO magnetic field between the plates because there is no conduction current there -- this is precisely the error the displacement current concept corrects.
• Forgetting that $I_d$ must come out numerically equal to $I$ for the law to be consistent between the two surfaces.