Displacement current

The electric flux through the flat bottom S in the gap is ΦE = EA = Q/ε₀, using Gauss's law.

As the charge changes, dΦE/dt = (1/ε₀) dQ/dt = i/ε₀, so ε₀ dΦE/dt = i. This is the missing term.

The current carried by moving charges in conductors is the conduction current ic. The new term, id = ε₀ dΦE/dt, is the displacement current (Maxwell's displacement current), caused by a changing electric field.

The total current through a surface is i = ic + id. Outside the plates ic = i and id = 0; in the gap ic = 0 and id = i. So the total is the same for every surface on the loop and B at P comes out the same.

The generalised law is the Ampere-Maxwell law: ∮B·dl = μ₀ic + μ₀ε₀ dΦE/dt.

In every respect the displacement current acts like a conduction current; above all, it produces a magnetic field just as a real current would. The field measured at a point M between the plates equals the field just outside at P.

A steady field in a conducting wire does not change, so there id = 0. In the charging capacitor ic and id sit in different regions; in most real media both are present together, because no medium conducts or insulates perfectly.

There can be large regions of space with no conduction current at all but a displacement current from a changing E. A magnetic field exists there even though no current source is nearby.