Displacement Current

The trouble with the old Ampère's law. Ampère's circuital law, as met in the chapter on Moving Charges and Magnetism, states that the line integral of the magnetic field $\vec{B}$ around any closed loop equals $\mu_0$ times the conduction current $I_c$ passing through ANY surface bounded by that loop:

This works perfectly for a steady current in an ordinary wire. But consider a parallel-plate capacitor being CHARGED by a wire carrying a current $I$. Draw an Amperian loop encircling the connecting wire, then consider two different surfaces bounded by this same loop: a flat disc that the wire pierces (through which the conduction current $I$ clearly passes), and a bulged, balloon-shaped surface that instead passes THROUGH THE GAP between the capacitor's plates (through which, since the plates are not touching and no charge physically crosses the gap, the conduction current is exactly ZERO). Ampère's law, applied to the same loop, must give the SAME answer whichever surface is chosen -- but the old law, as written, gives $\mu_0 I$ for one surface and $0$ for the other, a flat contradiction.

Maxwell's fix: the displacement current. Maxwell resolved this by noticing that, while no CHARGE crosses the gap between the plates, the ELECTRIC FIELD between the plates (and hence the electric flux $\Phi_E$ through the bulged surface) is changing continuously as the capacitor charges up. He proposed that a changing electric flux acts, for the purpose of producing a magnetic field, exactly as if it WERE a current -- a fictitious-sounding but physically real quantity he named the displacement current, $I_d$:

where $\epsilon_0$ is the permittivity of free space and $\Phi_E$ is the electric flux through the chosen surface. Including this term, the corrected, Ampère-MAXWELL law reads

and now gives the SAME answer for either surface: for the flat disc, $I_c=I$ and $I_d=0$; for the bulged surface through the gap, $I_c=0$ but $I_d$ works out to be exactly equal to $I$ (the rate at which charge is piling up on the plates equals the rate at which the electric flux between them is growing, times $\epsilon_0$) -- so both surfaces agree. …