Displacement current

We know a changing magnetic field makes an electric field (Faraday). Maxwell asked: does a changing electric field make a magnetic field? It must.

The clue is a charging capacitor. Current flows in the wires, so there is a magnetic field around them. Between the plates there is only empty space and no current, yet a magnetic field is found there too. Something must be standing in for the current.

What is there is a changing electric field: as charge piles up on the plates, E between them grows. Maxwell called the effect of this changing field the displacement current:

i_d = ε₀ dΦ_E / dt, where Φ_E is the electric flux.

Between the plates of a charging capacitor, i_d is exactly equal to the conduction current i_c in the wires. So the 'current' is continuous all round the circuit: conduction current in the wires, displacement current across the gap.

Example: plates of area 0.01 m², with E between them growing at 10¹² V/m each second: i_d = 8.85 × 10⁻¹² × 0.01 × 10¹² ≈ 89 mA.

Maxwell added it to Ampere's law, giving the Ampere–Maxwell law:

∮ B · dl = µ₀(i_c + i_d)

This makes electricity and magnetism symmetric: a changing B makes E, and a changing E makes B. Each can keep making the other, and that is what lets a wave of fields travel through empty space (next lesson).

Note that the displacement current is not a flow of charges. It is a changing field that has the same magnetic effect as a current.