Ampere's law, Maxwell's modification

Ampere's law

The line integral of the magnetic field B round any closed path equals μ₀ times the conduction current passing through the area bounded by the path:

∮B·dl = μ₀ I

It gives B for symmetric current distributions (long wire, solenoid, toroid).

Maxwell's modification

Ampere's law fails for a charging capacitor. Take a circular loop around the wire. If the surface bounded by the loop cuts the wire, ∮B·dl = μ₀I. If the surface is a bulge passing between the plates, no conduction current crosses it and the law gives zero. The same loop cannot give two answers.

  wire ---->  ||   ||  ----> wire
              ||   ||
   S1: flat surface cuts the wire     -> I enclosed
   S2: bulged surface between plates  -> I = 0 (?)

Maxwell added a term: a changing electric flux acts like a current, the displacement current I_d = ε₀ dΦ_E/dt.

∮B·dl = μ₀(I + I_d) = μ₀I + μ₀ε₀ dΦ_E/dt

Displacement current

It is not a flow of charge. It is the current-like term due to a time-varying electric field. A changing E produces a magnetic field, just as a changing B produces an electric field (Faraday). This leads to electromagnetic waves.

Proof that I_d equals the conduction current

Let the capacitor plate area be A and the charge on the plates q. The field between plates is E = q/(ε₀A).

1. Electric flux through the plate area: Φ_E = EA = q/ε₀.
2. I_d = ε₀ dΦ_E/dt = ε₀ (1/ε₀) dq/dt = dq/dt.
3. The conduction current in the wire is I = dq/dt.

So I_d = I. The current is continuous through the capacitor, and both surfaces S1 and S2 now give the same ∮B·dl = μ₀I.