The Displacement Current
Raise the charging rate to see the displacement current
The slider sets how fast the capacitor is charged. The length of the sky-blue arrows between the plates shows not the electric field itself but how fast it is growing, and the three gold rings are the magnetic field around the upper wire, the gap, and the lower wire. Raise the charging rate and all three rings grow alike; at a rate of 0 (at rest) they all vanish. Why a magnetic field circles a gap that no charge crosses is the question of this lesson.
One loop, two answers
Ampère's law says the field added up once around a loop (its circulation) is proportional to the current through the loop. ∮B·dl = μ₀I_enc (EM-14). Wrap an Ampère loop around the wire leading into a capacitor, pick a surface bounded by that loop, and count the current through it. Pick a flat surface that the wire pierces and a current I passes through; pick a surface that bulges out through the gap between the plates and the wire never pierces it, so the current is 0. Same loop, different surface, different answer: Ampère's law, always right in magnetostatics, is missing something in a circuit that is charging. That any surface on the same loop must give the same answer is the condition we met with curl and Stokes' theorem.
Maxwell's addition, the displacement current
Between the plates, charge piles up and the field E grows. Maxwell found that ε₀ times the rate of change of electric flux, ε₀ dΦ_E/dt, exactly fills in the current missing from the gap, and he called it the displacement current I_d = ε₀ dΦ_E/dt. The bulging surface catches none of the wire current but catches the same amount of displacement current instead, so now every surface gives μ₀I. That is also why the middle ring in the picture is drawn larger than the plates: only a loop running outside the plate edges encloses all the electric flux in the gap and gets the same field as the loops around the wire. A loop smaller than the plates encloses only part of the flux, and its field is weaker by that much. The corrected Ampère's law is ∮B·dl = μ₀(I_c + I_d).
The mirror of Faraday
Faraday's law said a changing magnetic field comes with an electric field. ∇ × E = -∂B/∂t (EM-20). Maxwell's term is its mirror image: a changing electric field comes with a magnetic field. ∇ × B = μ₀J + μ₀ε₀ ∂E/∂t. The two equations pair up in form but differ in sign: Faraday's is negative and Maxwell's is positive. Because the signs are opposite, combining the two equations gives the equation of a wave in which the two fields trade back and forth; with the same sign there would be no oscillation, only fields that blow up or die away in one direction. One changing field gives rise to the other, and that one gives rise to the first again: this is where the loop closes.
Opening the door to electromagnetic waves
With this term, even in empty space with no charges and no wires (J = 0), a changing electric field makes a magnetic field, and that field, as it changes, makes an electric field again (Faraday). The two fields carrying each other on through vacuum are an electromagnetic wave, and its speed, c = 1/√(μ₀ε₀), is set by the two constants alone. When Maxwell worked out this value, it matched the measured speed of light, and the conclusion that light is an electromagnetic wave came from there. The same term explains why current seems to flow across a capacitor in an AC circuit. The name "displacement" comes from Maxwell picturing a medium in the vacuum being pushed aside; today the term is understood with no such medium, as the change of the electric field itself standing in the same place as a current. When a dielectric fills the gap, though, part of the displacement current is a polarization current: bound charges in the dielectric really do shift a little.
Back to the first screen
On the first screen, at a charging rate of 0 there was no magnetic field, and the faster the charging, the stronger the field around the plate gap that no charge crosses. The electric field growing in the gap is worth exactly a current I_d = ε₀ dΦ_E/dt, equal to the wire current, and makes the same magnetic field that current would. That is why the loop around the wire and the loop around the gap had the same field. ∇ × B = μ₀J + μ₀ε₀ ∂E/∂t. A changing electric field gives rise to magnetism.
All four equations are now in place: Gauss's law (∇·E = ρ/ε₀), Gauss's law for magnetism (∇·B = 0), Faraday's law (∇ × E = -∂B/∂t), and Ampère's law completed by the displacement current (∇ × B = μ₀J + μ₀ε₀ ∂E/∂t). The next lesson (EM-22) gathers the four into Maxwell's equations and looks at how electromagnetic waves are hidden inside them.