The Displacement Current

The displacement current is simply a changing electric field acting as if it were a current: it makes a magnetic field around itself even though no charge is moving. The name is a leftover. Maxwell imagined empty space filled with a medium whose charges get slightly displaced whenever the field changes; that medium turned out not to exist, but the term and its name stuck. It's what completes Ampère's law, and two lessons from now it's what makes light possible. The formula $I_d = \varepsilon_0\,d\Phi_E/dt$ is just three pieces: $\varepsilon_0$ (the same constant as in Coulomb's law), $\Phi_E$ (how much electric field passes through a surface) and $d/dt$ (how fast that's changing). It looks like a strange new idea, but it's really one small term that keeps the old law from contradicting itself. Bear with it.

The puzzle. Charge a capacitor with a steady current $I$ and put a circular loop $C$ around the wire leading to it. Ampère's law says $\oint_C\mathbf{B}\cdot d\boldsymbol{\ell} = \mu_0 I_{\text{enc}}$, where $I_{\text{enc}}$ is the current through any surface whose edge is $C$. Stretch a flat disk across the loop: the wire pierces it, so $I_{\text{enc}} = I$. Now blow the same surface out into a bag that slips between the plates without touching the wire: nothing crosses it, so $I_{\text{enc}} = 0$. Same loop, same field, two answers. Something has to give.

Where the math breaks. Stokes' theorem turns Ampère's law into $\nabla\times\mathbf{B} = \mu_0\mathbf{J}$. The divergence of a curl is always zero, so this forces $\nabla\cdot\mathbf{J} = 0$: every region loses exactly as much current as it gains. But charge piles up on a capacitor plate, and conservation of charge says $\nabla\cdot\mathbf{J} = -\partial\rho/\partial t$, which isn't zero there. Ampère's law only holds for steady currents that never pile charge up.

Maxwell's repair. Gauss's law, $\rho = \varepsilon_0\nabla\cdot\mathbf{E}$, ties the piling charge to the field: $\partial\rho/\partial t = \nabla\cdot(\varepsilon_0\,\partial\mathbf{E}/\partial t)$. So

everywhere, always. Notice what that means: this combination never piles up anywhere, so unlike $\mathbf{J}$ alone, it's allowed to be a curl. Maxwell made it the source of $\mathbf{B}$: $\nabla\times\mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0\,\partial\mathbf{E}/\partial t$. The new term is the displacement current density, and its flux through a surface is $I_d = \varepsilon_0\,d\Phi_E/dt$.

It fixes the capacitor exactly. Between the plates $E = \sigma/\varepsilon_0 = Q/(\varepsilon_0A)$ (conductors), so the flux through the bag is $\Phi_E = EA = Q/\varepsilon_0$ and

The flat disk catches the conduction current $I$ and the bag catches the displacement current $I$, so both give $\oint\mathbf{B}\cdot d\boldsymbol{\ell} = \mu_0 I$. Current flows into one plate, continues across the gap as a changing field and flows out of the other.

The field in the gap. For circular plates of radius $R$, $E$ is uniform, so a circle of radius $r < R$ encloses only the fraction $r^2/R^2$ of the displacement current:

$B$ grows linearly out to the rim, then falls off as $\mu_0 I/(2\pi r)$ beyond it, exactly like the field around the wire (Biot–Savart). No charge crosses the gap, yet the magnetic field is there.

Direction. $I_d$ points the way $\mathbf{E}$ is growing. While the capacitor charges, that's from the + plate to the − plate, along $\mathbf{E}$. While it discharges, $\mathbf{E}$ shrinks and $I_d$ points from − to +. Either way it continues the current in the wires.

Why nobody noticed sooner. In copper at 1 GHz the displacement current is about $10^{-9}$ of the conduction current, and a capacitor charging at 2 A makes only microtesla fields in its gap. Experiments with steady currents could never see it. In empty space, though, it's the only current there is. Together with Faraday's law it lets a changing $\mathbf{E}$ make $\mathbf{B}$ and a changing $\mathbf{B}$ make $\mathbf{E}$, which, two lessons from now, is light.

A common mistake. Because it's called a current and measured in amperes, it's tempting to think some charge sneaks across the gap. None does. In vacuum nothing flows at all; the displacement current is the changing field itself. That's also why the "current through a capacitor" in your AC circuits is entirely displacement current. One smaller slip: for a loop smaller than the plates, use only the enclosed part of $I_d$, not all of it.

Every coupling and bypass capacitor on a circuit board passes AC this way: current flows in the leads and continues across the insulating gap as a changing field. Capacitive touchscreens sense a finger by the extra displacement current it draws from their electrodes, and capacitive couplers send power and data across air gaps in sealed equipment. In empty space it's the only current there is, and pairing it with Faraday's law is what makes radio and light possible.