Displacement current

The displacement current density is

${\mathbf{J}}_d=\frac{\partial \mathbf{D}}{\partial t}{\text{(A/m}}^2\text{)}.$

It has units of current density and behaves like a current source for $\mathbf{H}$, but it does not involve any actual motion of charges. It is Maxwell’s modification to Ampère’s law, extending the law from magnetostatics to fully dynamic problems:

$\nabla \times \mathbf{H}=\mathbf{J}+\frac{\partial \mathbf{D}}{\partial t}.$

The $\mathbf{J}$ on the right is the conduction current (actual moving charge); the ${\partial }_t\mathbf{D}$ term is the displacement current. Both have units A/m², and they appear on equal footing as sources of curling $\mathbf{H}$.

Why it’s necessary: the capacitor paradox

Consider a capacitor being charged: a real conduction current $I$ flows in the connecting wires.

Two imaginary surfaces $S_1$ (cutting through the wire, capturing the conduction current $I$) and $S_2$ (passing between the capacitor plates, where no conduction current flows).

Apply Ampère’s circuital law $\oint \mathbf{H}\cdot d\mathbf{l}=I_{\text{enc}}$ to the loop $C$ around the wire:

• For surface $S_1$: $I_{\text{enc}}=I$ — full conduction current.
• For surface $S_2$ (between the plates, no wire passes through): $I_{\text{enc}}=0$ — no conduction current.

But both surfaces have the same boundary $C$. Ampère’s law would give two different answers for the same line integral. Contradiction.

Maxwell’s fix: add the ${\partial }_t\mathbf{D}$ term. Between the plates, $\mathbf{D}$ changes in time as charge accumulates, and ${\int }_{S_2}{\partial }_t\mathbf{D}\cdot d\mathbf{s}$ exactly equals $I$. The paradox dissolves — total enclosed “current” (conduction + displacement) is the same for either surface.

Why “displacement”

The name is historical (Maxwell’s term) and a bit misleading. There’s no actual displacement of charge in vacuum across the capacitor gap. What’s happening is that the electric flux $\mathbf{D}$ between the plates is changing in time as more charge accumulates on the plates. This time-varying $\mathbf{D}$ behaves as if it were a current — it sources $\mathbf{H}$ in the same way conduction current does.

In a vacuum or perfect dielectric, displacement current is the only “current” possible — no charges are present to conduct.

Quantitative example: capacitor

Parallel-plate capacitor, plate area $A$, separation $d$, dielectric $ϵ$. Connected to AC source $v(t)=V_0\cos \omega t$.

Electric field between plates: $E=v(t)/d=(V_0/d)\cos \omega t$, so $D=ϵE$. Displacement current density:

$J_d=\frac{\partial D}{\partial t}=-ϵ\frac{V_0\omega }{d}\sin \omega t.$

Total displacement current through the capacitor:

$I_d=J_{d}A=-\frac{ϵA{V}_0\omega }{d}\sin \omega t=-C{V}_0\omega \sin \omega t.$

Conduction current in the external wire is $i(t)=Cdv/dt=-C{V}_0\omega \sin \omega t$. The two are equal — current is continuous through the capacitor as far as $\mathbf{H}$ is concerned, even though no charge actually crosses the gap.

Charge continuity

Displacement current is what makes total current conserved. Taking the divergence of Ampère’s law:

$\nabla \cdot (\nabla \times \mathbf{H})=0=\nabla \cdot \mathbf{J}+\nabla \cdot \frac{\partial \mathbf{D}}{\partial t}=\nabla \cdot \mathbf{J}+\frac{\partial }{\partial t}(\nabla \cdot \mathbf{D})=\nabla \cdot \mathbf{J}+\frac{\partial {\rho }_v}{\partial t},$

using Gauss’s law on the last term. This gives the Charge continuity equation:

$\nabla \cdot \mathbf{J}=-\frac{\partial {\rho }_v}{\partial t},$

which says net outflow of conduction current equals the rate of charge decrease in a region. Charge conservation is guaranteed by Maxwell’s equations only because the displacement current term is present.

Why it predicts EM waves

The displacement current is what closes the feedback loop between $\mathbf{E}$ and $\mathbf{B}$:

• Faraday: changing $\mathbf{B}$ → curling $\mathbf{E}$.
• Maxwell-Ampère (with displacement current): changing $\mathbf{E}$ (via ${\partial }_t\mathbf{D}$) → curling $\mathbf{H}$ → curling $\mathbf{B}$.

Together: a perturbation in one field generates the other, which generates the first, propagating outward as an electromagnetic wave. Without the displacement current term, this self-sustaining wave would not exist in Maxwell’s framework.

The propagation speed falls out: $c=1/\sqrt{{\mu }_0{ϵ}_0}$. Maxwell predicted this in 1865 and immediately identified it with the experimentally measured speed of light, unifying optics with electromagnetism.