EM-21 · Time-varying field

The Displacement Current

A changing electric field makes a magnetic field. While a capacitor charges, current flows in the wire but no charge crosses the gap between the plates, yet a magnetic field circles that gap. The changing electric field acts like a current. I_d = ε₀ dΦ_E/dt.

Raise the charging rate to see the displacement current

Charge a capacitor. As you raise the charging rate, a magnetic field circles not only the wire but also the gap between the plates, where no charge crosses, in exactly the same way. At rate zero (static) there is no field. Why does a magnetic field circle an empty gap?

Charging rate (rate of field change)50%

Drag to orbit. The slider sets the charging rate.

The conduction current and the displacement current

I_d = ε₀ dΦ_E/dt · the faster the field changes, the bigger the displacement current

∮B·dl = μ₀(I_c + I_d) · the field circling the gap

Conduction I_c 50% · Displacement I_d 50%

Charging

The contradiction at a charging capacitor

Ampère's law says the circulation of the magnetic field is proportional to the current threading the loop: ∮B·dl = μ₀I_enc (EM-14). But take an Amperian loop around the wire feeding a capacitor and cap it with a surface. A flat surface pierced by the wire passes a current I; a surface that bulges through the gap between the plates is pierced by no wire, so the current is zero. Same loop, different surfaces, different answers. Ampère's law, perfect in magnetostatics, breaks in the time-varying case.

ObserveI_d = ε₀ dΦ_E/dt

The displacement current is the rate of change of the electric flux.

Maxwell's addition, the displacement current

Between the plates, charge piles up and the electric field E grows. Maxwell saw that the changing electric flux ε₀ dΦ_E/dt exactly fills in the current missing from the gap. He called it the displacement current I_d = ε₀ dΦ_E/dt. The bulging surface catches no conduction current but catches the same amount of displacement current. Now both surfaces give μ₀I. Continuity of current is restored. The corrected Ampère's law is ∮B·dl = μ₀(I_c + I_d).

Choose∮B·dl = μ₀ ?

The corrected Ampère's law adds both currents.

Fill in틈에서 I_d = ?

Even in the gap between the plates the current is unbroken.

The mirror of Faraday

Faraday said a changing magnetic field makes an electric field: ∇ × E = -∂B/∂t (EM-20). Maxwell's term is its mirror image: a changing electric field makes a magnetic field, ∇ × B = μ₀J + μ₀ε₀ ∂E/∂t. Note that the signs of the two equations differ — minus for Faraday, plus for Maxwell. That asymmetry makes the two fields chase each other rather than cancel. Now the two curl equations are symmetric: a changing field begets the other, and that field begets the first again.

On your own∇ × B = μ₀J + ?

A changing electric field makes a magnetic field.

Opening the door to electromagnetic waves

Thanks to this term, even in empty space with no charge and no wire (J = 0) a changing electric field makes a magnetic field, and that changing magnetic field makes an electric field again (Faraday). The two sustain each other and become an electromagnetic wave running through the vacuum. Its speed is c = 1/√(μ₀ε₀), arising from those two constants alone. The value Maxwell computed matched the measured speed of light: light itself is an electromagnetic wave. The same term explains why, in AC circuits, current appears to flow across a capacitor.

Back to the first screen

On the first screen there was no magnetic field at rate zero, and the faster you charged, the more a field grew around the empty gap between the plates. No real charge crosses that gap, yet the changing electric field there acts as a current of exactly I_d = ε₀ dΦ_E/dt — the same size as the current in the wire. That is why the field around the loop encircling the wire and the loop encircling the gap were identical. ∇ × B = μ₀J + μ₀ε₀ ∂E/∂t. A changing electric field gives birth to magnetism.

The displacement current is the term, made by a changing electric flux, that produces a magnetic field like a current does. I_d = ε₀ dΦ_E/dt. Added to Ampère's law it gives ∮B·dl = μ₀(I_c + I_d), so the continuity of current holds even across a capacitor gap. The differential form is ∇ × B = μ₀J + μ₀ε₀ ∂E/∂t, the mirror of Faraday (∇ × E = -∂B/∂t). This term makes an electromagnetic wave through vacuum possible.

What comes next

Now all four equations are gathered. Gauss (∇·E = ρ/ε₀), magnetic Gauss (∇·B = 0), Faraday (∇ × E = -∂B/∂t), and Ampère completed by the displacement current (∇ × B = μ₀J + μ₀ε₀ ∂E/∂t). Next (EM-22) gathers these four in one place as Maxwell's equations and sees how an electromagnetic wave hides within them.