The Displacement Current

Every equation this chapter has used so far was distilled from experiment. This article is about the one term that was not — the term a 30-year-old James Clerk Maxwell added to Ampère's law in 1861 with no experiment demanding it, because the mathematics was inconsistent without it. The addition, $\mu_0\varepsilon_0\,\partial\mathbf{E}/\partial t$ , is a few strokes of ink. Its consequences: the completed field equations predict self-propagating waves, the waves travel at a speed built from two bench-measured constants, the speed comes out as the speed of light — and electromagnetism, optics, and (eventually) radio become one subject. No comparably small edit in the history of physics has ever bought so much. The inconsistency Maxwell fixed is easy to exhibit and impossible to unsee. Ampère's law equates the circulation of $\mathbf{B}$ around a loop to the current crossing any surface spanning it — a rule that is only coherent if every spanning surface catches the same current. Thread the loop around a wire that is charging a capacitor, and it doesn't: a flat disk crossed by the wire reports $I$ ; a balloon-shaped surface slipping through the capacitor gap, where no charge ever flows, reports zero. Magnetostatics never noticed, because steady currents cannot pile charge anywhere; the moment currents change, Ampère's law contradicts itself. Maxwell's repair is surgical: in the gap where the conduction current disappears, the electric field is growing — and if a changing $\mathbf{E}$ is granted the same curl-generating rights as a current, with strength $\varepsilon_0\,\partial\mathbf{E}/\partial t$ , every surface again agrees, exactly. This article exhibits the crisis, installs the fix, computes the magnetic field the new term predicts inside a charging capacitor, and sizes the term honestly (it is spectacularly small at bench frequencies — which is why Ampère never saw it, and why its real empire is at high frequency and in empty space). What the fix does for charge conservation in general is the next article's theorem; what it does to physics is the article after that.

Key takeaways

Ampère's law $\nabla\times\mathbf{B} = \mu_0\mathbf{J}$ fails for non-steady currents: at a charging capacitor, different surfaces spanning the same Amperian loop catch different currents — the law contradicts itself.
Maxwell's fix: grant changing electric fields curl-generating rights. $\nabla\times\mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0\,\partial\mathbf{E}/\partial t$ — the added piece $\mathbf{J}_d = \varepsilon_0\,\partial\mathbf{E}/\partial t$ is the displacement current density.
At the capacitor the fix is exact: the growing $\mathbf{E}$ in the gap carries displacement current $I_d = \varepsilon_0\,\mathrm{d}\Phi_E/\mathrm{d}t = I$ , equal to the wire's conduction current — the circuit is magnetically unbroken.
"Current" is a historical misnomer: nothing moves in the gap. $\mathbf{J}_d$ is a source term for $\mathbf{B}$ , not a flow of charge — it produces magnetic field, carries no charge, and dissipates nothing.
Size matters: in copper at 60 Hz, $J_d/J_c \sim 10^{-16}$ — utterly negligible; in vacuum it is the only term, and the symmetry it completes (changing $\mathbf{E} \to \mathbf{B}$ , mirroring Faraday's changing $\mathbf{B} \to \mathbf{E}$ ) is what lets fields regenerate each other as waves at $1/\sqrt{\mu_0\varepsilon_0} = c$ .
KEY IDEA: a changing electric field generates magnetic field exactly as a current does — the displacement current $\varepsilon_0\,\partial\mathbf{E}/\partial t$ repairs Ampère's law, closes every circuit through its fields, and completes the reciprocity that makes light possible.

The core insight

Maxwell's displacement current $\mathbf{J}_d = \varepsilon_0\,\partial\mathbf{E}/\partial t$ extends Ampère's law to $\nabla\times\mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0\,\partial\mathbf{E}/\partial t$ , restoring consistency for changing fields: a changing electric field generates magnetic field exactly as a current does — the final structural piece of Maxwell's equations.

Where this lives today

Every antenna is a displacement-current machine: the current that "flows" from one arm of a dipole to the other — completing the circuit through empty space — is $\varepsilon_0\,\partial\mathbf{E}/\partial t$ in the near field, and radiation is that current's far-field handiwork. RF engineers model it daily as literal circuit current (antenna feed impedance treats space as the closing element), the century-old repair now a design tool. Capacitive touchscreens run the logic in miniature: your fingertip closes a circuit through a dielectric stack via displacement current at a few hundred kilohertz, no charge ever crossing the glass. Medical dielectric spectroscopy and microwave imaging read tissue by exactly worked example 2's ratio — tumors announce themselves by shifted $\sigma$ and $\varepsilon$ , moving the local conduction–displacement balance. At the metrology frontier the term is embedded in the definition of modern units: the 2019 SI fixes $e$ and $h$ , making $\varepsilon_0$ and $\mu_0$ measured quantities whose product is pinned by the exact, defined $c$ — the constants of equation (1) now inherit their values from the very wave speed the equation predicts, a delicious inversion of 1865's logic. And in every FDTD electromagnetic solver — the workhorse simulation of antenna design, photonics, and radar cross-sections — the update equation for $\mathbf{B}$ literally alternates with the update for $\mathbf{E}$ through discretized versions of Faraday's law and equation (1): the two curl laws, taking turns, marching light across a grid. The bootstrap Maxwell wrote in ink runs today in silicon, several trillion times per second, in every wireless device shipping.

Origins · 1 minHeinrich Hertz and how this was discovered

Heinrich Hertz

1857–1894 · Karlsruhe, Bonn

Hertz is the experimentalist who made the displacement current an observable. Helmholtz's prize student in Berlin, he was set the problem his teacher considered decisive: test whether Maxwell's polarization-and-displacement currents were physically real. In 1886–88 at Karlsruhe, with spark-gap oscillators ringing at tens of megahertz — frequencies at last high enough for $\varepsilon_0\,\partial\mathbf{E}/\partial t$ to matter — Hertz generated electromagnetic waves, reflected them from metal sheets, focused them with cylindrical mirrors, measured standing-wave nodes with a millimetre spark in a resonant loop, and extracted a propagation speed consistent with light. Open circuits radiated, exactly as a term completing every circuit through the fields demands; "Maxwell's theory," he wrote with a shrug that became famous, "is Maxwell's system of equations." His clean-up of that system (with Heaviside — the "Hertz–Heaviside equations") is the four-equation form the next article assembles. He found the photoelectric effect in the same apparatus, almost casually; declared his waves "of no use whatsoever"; and died at 36, seven years before Marconi's Atlantic crossing turned his laboratory curiosity into the radio century. The unit of frequency keeps the score.

History

1826

— Ampère's force law and the circuital relation codify steady-current magnetism

1861

— Maxwell's "On Physical Lines of Force" adds the displacement current, born inside a molecular-vortex ether

1865

— "A Dynamical Theory of the Electromagnetic Field" keeps the term, drops the scaffolding, predicts light

1884

— Heaviside and Hertz recast the theory into the four-equation form we write

1887

— Hertz's spark-gap waves: open circuits radiate, the displacement current is physical

1985

— Bartlett and Corle directly map the magnetic field inside a charging capacitor — the textbook diagram, measured

The physical picture

Run the crisis as an experiment you can hold in your head. A wire carries current $I$ toward a parallel-plate capacitor; charge piles onto the plates; the gap carries no charge at all. Wrap an Amperian loop around the wire and invoke the integral law of the magnetostatics chapter: $\oint\mathbf{B}\cdot\mathrm{d}\boldsymbol\ell = \mu_0 I_{\rm enc}$ , where $I_{\rm enc}$ is the current through any surface bounded by the loop. The law's whole logic rests on that word any — it is Stokes' theorem talking, and Stokes does not care which spanning surface you pick.

Figure

The two-surface indictment: one Amperian loop, two legal surfaces, two different currents

Surface one, the flat disk: the wire punches through, $I_{\rm enc} = I$ . Surface two, a balloon swept through the capacitor gap: no charge crosses it anywhere, $I_{\rm enc} = 0$ . One loop, one left-hand side, two answers. Nothing is wrong with the geometry; what is wrong is the physics fed to it. Ampère's law was distilled from steady currents, and steady currents have a special property this chapter has met before: they are divergence-free, flowing in closed circuits, never accumulating. For such currents every spanning surface must catch the same total, and the law is consistent. The charging capacitor is precisely a place where current is not divergence-free — charge accumulates on a plate — and there the law's silent premise snaps. (In differential clothing: $\nabla\cdot(\nabla\times\mathbf{B}) = 0$ identically, but $\nabla\cdot(\mu_0\mathbf{J}) \neq 0$ wherever charge piles up. The left side is structurally divergence-free; the unrepaired right side is not. The systematic version of this argument — and the discovery that the repair makes charge conservation an identity of the completed equations — is the next article's entire business, The Continuity Equation; here we need only the crime scene.) Now look into the gap with Maxwell's eyes. The gap is not empty of physics: as charge piles on the plates, the electric field between them grows — and grows in exact sympathy with the arriving current, since $E = \sigma_s/\varepsilon_0 = Q/\varepsilon_0 A$ tracks $Q$ . Where conduction current disappears, changing electric field appears, in just the right amount. Maxwell's proposal: that is not a coincidence; that is the missing current. Grant a changing $\mathbf{E}$ the same magnetic-field-generating rights as a flow of charge, with strength $\varepsilon_0\,\partial\mathbf{E}/\partial t$ , and the balloon surface catches exactly $I$ after all — carried across the gap not by charge but by the swelling field. Maxwell, working inside his molecular-vortex model of the ether, pictured bound charges in the medium displacing slightly — hence the name. The ether died; the mechanical picture died; the term, and its unfortunate name, survived because the structure was right: symmetry between the fields. Faraday: changing $\mathbf{B}$ begets curling $\mathbf{E}$ (Induced Electric Fields). Maxwell: changing $\mathbf{E}$ begets curling $\mathbf{B}$ . The reciprocity is complete, and its child — each field regenerating the other, forever, through empty space — is three articles away.

The completed law

The repaired equation — the Ampère–Maxwell law — reads

\nabla\times\mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0\,\frac{\partial\mathbf{E}}{\partial t} \qquad (1)

where: $\mathbf{J}$ — conduction current density, charge genuinely in motion; and the new term, written as a current density in its own right,

\mathbf{J}_d \equiv \varepsilon_0\,\frac{\partial\mathbf{E}}{\partial t} \qquad (2)

where: $\mathbf{J}_d$ — the displacement current density (A/m²), the rate of change of the electric field dressed in current's units. Read equation (1) as a widened definition of what sources magnetic circulation: moving charge, or changing electric field — nature does not distinguish. Three properties keep the two terms honest. $\mathbf{J}_d$ produces $\mathbf{B}$ exactly as $\mathbf{J}$ does (same coefficient, same geometry of curl); it transports no charge (nothing moves); it dissipates no energy (there is no $\mathbf{J}_d\cdot\mathbf{E}$ heating — the energy exchanged with a changing field is the reversible stored energy of the Energy Stored in Magnetic Fields ledger, not Joule heat). Integrated over a surface and run through Stokes' theorem, the law takes the form the capacitor demanded:

\oint\mathbf{B}\cdot\mathrm{d}\boldsymbol\ell = \mu_0 I_{\rm enc} + \mu_0\varepsilon_0\,\frac{\mathrm{d}\Phi_E}{\mathrm{d}t} \qquad (3)

where: $\Phi_E = \int\mathbf{E}\cdot\mathrm{d}\mathbf{a}$ — the electric flux through the chosen surface, and $I_d = \varepsilon_0\,\mathrm{d}\Phi_E/\mathrm{d}t$ — the displacement current through it. The structural rhyme with Faraday's law is now on the page: circulation of one field fed by the changing flux of the other, the two statements mirror images up to the sign (Lenz's minus in Faraday's; none here — the asymmetry that will matter when the wave equation's signs are chased) and the constants. Note also what the term does in matter: inside a dielectric the field's $\varepsilon_0\partial\mathbf{E}/\partial t$ rides along with genuine bound-charge motion, the polarization current $\partial\mathbf{P}/\partial t$ — the honest in-matter split is Maxwell's Equations in Matter's business, two articles hence.

Derivation AThe capacitor audit — every surface now reports the same current

The goal is the quantitative acquittal: show that for the charging parallel-plate capacitor, the displacement current through the gap equals the conduction current in the wire exactly, so that equation (3) gives the same answer for the flat disk and the balloon surface — the paradox dissolved, not patched. The strategy is bookkeeping with the capacitor's own electrostatics (E1 results, recalled): relate the gap field to the plate charge, differentiate, compare. Watch the $\varepsilon_0$ 's cancel — the agreement is exact and parameter-free, no matter the plate size, gap, or charging rate.

Step 1. The gap field. For plates of area $A$ carrying charge $\pm Q(t)$ (ideal capacitor: uniform field, negligible fringing), the electrostatic chapter's result, applied quasistatically:

E(t) = \frac{\sigma_s}{\varepsilon_0} = \frac{Q(t)}{\varepsilon_0 A}.

Step 2. The electric flux through the balloon surface — it crosses the gap, catching the full field over the plate area:

\Phi_E = E\,A = \frac{Q(t)}{\varepsilon_0}.

Why fringing doesn't spoil exactness: take the balloon far from the gap's edges, or note that the total flux out of the region between the plates is fixed by Gauss's law to be $Q/\varepsilon_0$ regardless of the field's detailed shape — the audit below is really Gauss's law differentiated.

Step 3. The displacement current through the balloon:

I_d = \varepsilon_0\,\frac{\mathrm{d}\Phi_E}{\mathrm{d}t} = \varepsilon_0\,\frac{\mathrm{d}}{\mathrm{d}t}\!\left(\frac{Q}{\varepsilon_0}\right) = \frac{\mathrm{d}Q}{\mathrm{d}t} = I.

Equal to the wire's current — identically, because $\mathrm{d}Q/\mathrm{d}t$ is the arriving current: charge conservation at the plate is what guarantees the gap's field grows at exactly the compensating rate.

Step 4. The verdict on both surfaces. Flat disk: $I_{\rm enc} = I$ , $\mathrm{d}\Phi_E/\mathrm{d}t \approx 0$ (no field where the wire crosses) → right side $\mu_0 I$ . Balloon: $I_{\rm enc} = 0$ , $I_d = I$ → right side $\mu_0 I$ . Same circulation, any surface, always — the word any in Stokes' theorem is safe again. The conduction current in the wire and the displacement current in the gap splice into one unbroken magnetic source running around the circuit: the field cannot tell where the wire ends and the gap begins. Verified: matches Griffiths §7.3.2 (Ex. 7.14 register) and Purcell §9.2; the E = Q/ε0A input is the E1 parallel-plate result; the exactness-via-Gauss argument matches Zangwill §2.1's presentation; worked example 1 puts numbers on the spliced circuit's B field.

Go deeper · Derivation B · 2 minThe magnetic field inside a charging capacitor — the new term computes

Derivation BThe magnetic field inside a charging capacitor — the new term computes

The goal is to make the displacement current produce something: the magnetic field in the gap of a charging circular-plate capacitor (plate radius $R$ , wire current $I$ ), a field that exists where no charge moves and that the unrepaired Ampère law would leave undefined or zero. The strategy is the exact mirror of the solenoid computation in Induced Electric Fields: the source $\mathbf{J}_d$ is uniform and axial in the gap, so the Amperian-loop technology of E4 applies verbatim with $\mathbf{J} \to \mathbf{J}_d$ . Watch the answer match, at the rim, the ordinary field of the wire — the splice made visible.

Step 1. The source map. In the gap, $E = Q/\varepsilon_0\pi R^2$ (uniform over the plate disk, zero outside, ideal-capacitor idealization), so

J_d = \varepsilon_0\frac{\partial E}{\partial t} = \frac{I}{\pi R^2},

uniform and axial — the geometry of a fat wire of radius $R$ carrying total current $I$ , except made of changing field instead of moving charge.

Step 2. Symmetry: axial uniform source → azimuthal $\mathbf{B}$ , magnitude a function of $s$ only (the fat-wire argument of Ampère's Law, unchanged — equation (1) makes no distinction between source types, so neither do its solutions).

Step 3. Amperian circle of radius $s \leq R$ inside the gap, enclosing displacement current $J_d\,\pi s^2 = I s^2/R^2$ :

B\,(2\pi s) = \mu_0 I\frac{s^2}{R^2} \qquad\Longrightarrow\qquad B = \frac{\mu_0 I\,s}{2\pi R^2},

growing linearly from zero on the axis — the interior of the "fat wire of changing field."

Step 4. At and beyond the rim ( $s \geq R$ ): full $I$ enclosed, $B = \mu_0 I/2\pi s$ — exactly the field of the supply wire at the same distance. Approach the capacitor along the wire and the magnetic field pattern flows smoothly across the gap without a seam: the circuit is magnetically continuous even where it is galvanically broken. For the worked numbers ( $R = 5.0$ cm, $I = 0.50$ A): $B_{\rm rim} = 2.0\ \mathrm{\mu T}$ , tiny but real — and measured (Bartlett and Corle mapped precisely this field in 1985, closing the loop on a term accepted for a century on structural evidence). Verified: matches Griffiths Ex. 7.14 / Prob. 7.34 register; rim continuity with the wire field checked (both give μ0I/2πs); numbers recomputed — B(rim) = μ0 × 0.50/(2π × 0.05) = 2.0 μT, B(2 cm) = μ0 I s/2πR² = 0.80 μT, dE/dt = I/ε0πR² = 7.2×10¹² V m⁻¹ s⁻¹ (the startling rate an ordinary 0.5 A charging current implies), J_d = 64 A/m².

Figure

The completed reciprocity: Faraday and Ampère–Maxwell as mirror curl laws — the engine of light

How big is it? The regime map

The displacement current's size relative to conduction is the ratio that decides which physics is in charge. For fields oscillating at angular frequency $\omega$ in a medium of conductivity $\sigma$ , take $J_c = \sigma E$ (Ohm, from Ohm's Law and Conductivity) and $J_d = \varepsilon_0\,\partial E/\partial t \sim \varepsilon_0\omega E$ :

\frac{J_d}{J_c} = \frac{\varepsilon_0\,\omega}{\sigma} \qquad (4)

where: the field amplitude has cancelled — the ratio is a property of material and frequency alone. In copper ( $\sigma = 6.0\times10^7$ S/m) at 60 Hz it is $5.6\times10^{-17}$ ; at 1 MHz, $9.3\times10^{-13}$ . Inside good conductors the displacement current is never relevant at any technological frequency — which is why a century of electrical engineering (motors, transformers, the entire quasistatic canon this chapter has built) proceeds correctly without it, and why no bench experiment of Ampère's era could have stumbled on the term. The crossover $\varepsilon\omega/\sigma \sim 1$ marks where a material stops acting like a conductor and starts acting like a dielectric: for seawater ( $\sigma \approx 4$ S/m, $\varepsilon_r \approx 80$ ) it sits near 0.9 GHz — below, radio waves are absorbed conductor-style (the submarine-communication problem); above, seawater is a lossy dielectric. In vacuum, finally, $\sigma = 0$ and the hierarchy inverts absolutely: the displacement current is the only source there is, and everything that happens in empty space — every wave, every photon's classical shadow — is the physics of equation (2) feeding Faraday's law and being fed by it.

Limiting cases

Case	Result	Meaning
steady currents ( $\partial\mathbf{E}/\partial t = 0$ )	Ampère's law recovered exactly	magnetostatics was never wrong — only incomplete
charging capacitor gap	$I_d = I$ , identically	the circuit is magnetically unbroken; Derivation A
inside the gap, $s \leq R$	$B = \mu_0 Is/2\pi R^2$	the "fat wire of changing field"; measured (Bartlett–Corle 1985)
copper, 60 Hz	$J_d/J_c \sim 6\times10^{-17}$	invisible to every 19th-century experiment — and to power engineering
seawater, ~1 GHz	$J_d/J_c \sim 1$	conductor-to-dielectric crossover; radio's undersea wall
vacuum	$\mathbf{J} = 0$ : only $\mathbf{J}_d$	changing fields sourcing each other — the wave regime
in dielectrics	$\partial\mathbf{D}/\partial t = \varepsilon_0\partial\mathbf{E}/\partial t + \partial\mathbf{P}/\partial t$	vacuum term rides with real bound-charge motion; E6's matter article splits them

Common trapDisplacement current is not a current — and it does not need a medium

Two misreadings, both licensed by the unfortunate name. First: nothing is displaced and nothing flows. $\mathbf{J}_d = \varepsilon_0\,\partial\mathbf{E}/\partial t$ transports zero charge, heats nothing, and would register on no ammeter you could insert in the gap; it is a source term for magnetic field that happens to carry current's units and slot into current's place in the law. Treating it as a physical flow leads to real errors — hunting for its charge carriers, assigning it Joule losses, or "explaining" it via the motion of something. Second, the historical trap: Maxwell coined the name because in his ether model something was displaced — bound charges of a medium filling all space. The ether is dead (the relativity chapter presides over the funeral), but the term is not diminished: in vacuum, with nothing present to displace, $\varepsilon_0\,\partial\mathbf{E}/\partial t$ still generates magnetic field, and light still crosses the void. Keep the physics, discard the etymology. The safe reading of equation (1) is symmetric and field-first: changing electric fields curl magnetic fields, exactly as changing magnetic fields curl electric ones.

Worked examples

1 · The charging capacitor, fully mapped (confidence builder)

A capacitor with circular plates of radius $R = 5.0$ cm is charged by a steady wire current $I = 0.50$ A. Find the displacement current density in the gap, the rate of change of the field, and the magnetic field at $s = 2.0$ cm and at the rim.

Step 1. The gap's displacement current must total the wire's current (Derivation A), spread uniformly: $J_d = I/\pi R^2 = 0.50/(\pi \times 0.0025) = \mathbf{64\ A/m^2}$ .

Step 2. The field's growth rate: $\partial E/\partial t = J_d/\varepsilon_0 = 64/8.85\times10^{-12} = \mathbf{7.2\times10^{12}\ V\,m^{-1}\,s^{-1}}$ — seven trillion volts per metre per second, from half an ampere. Displacement current is a weak effect per unit $\partial E/\partial t$ : prodigious field rates produce modest magnetic fields, which is the deep reason the term hid for decades.

Step 3. Inside the gap at $s = 2.0$ cm (Derivation B): $B = \mu_0 I s/2\pi R^2 = (1.26\times10^{-6})(0.50)(0.020)/(2\pi \times 0.0025) = \mathbf{0.80\ \mu T}$ .

Step 4. At the rim: $B = \mu_0 I/2\pi R = \mathbf{2.0\ \mu T}$ — precisely the field the supply wire produces at 5 cm. Walk a magnetometer around the circuit, wire to gap to wire: no seam. (For calibration, Earth's field is ~50 μT; Bartlett and Corle needed serious care to map these microteslas in 1985 — and found them exactly where equation (3) put them.)

2 · Regime engineering: where displacement current rules (exam level)

(a) At what frequency does the displacement current in copper reach 1% of the conduction current? (b) For seawater ( $\sigma = 4.0$ S/m, $\varepsilon_r = 80$ ), locate the conductor–dielectric crossover and interpret it for submarine radio. (c) In the vacuum gap of example 1, what fraction of the total magnetic-field source is displacement current?

Step 1. (a) Set $\varepsilon_0\omega/\sigma = 0.01$ : $\omega = 0.01\,\sigma/\varepsilon_0 = 0.01 \times 6.0\times10^7/8.85\times10^{-12} = 6.8\times10^{16}\ \mathrm{rad/s}$ , i.e. $f \approx \mathbf{1.1\times10^{16}\ Hz}$ — ultraviolet. Inside copper, displacement current never competes at radio or microwave frequencies; by the time it might, the Ohmic model of $\sigma$ has itself failed (the electron's inertia and interband physics take over — the dispersion story of the E11 chapter). Practical moral: for circuit-scale conductors, drop $\mathbf{J}_d$ inside the metal without apology.

Step 2. (b) Crossover at $\varepsilon_r\varepsilon_0\omega/\sigma = 1$ (using the medium's own permittivity, the honest in-matter comparison): $f = \sigma/2\pi\varepsilon_r\varepsilon_0 = 4.0/(2\pi \times 80 \times 8.85\times10^{-12}) = \mathbf{0.90\ GHz}$ . Below this, seawater responds as a conductor: fields are screened exponentially within a skin depth, which at kilohertz is metres and at megahertz centimetres — hence real submarine communication uses extremely low frequencies (tens of hertz, skin depths of hundreds of metres, data rates of characters per minute) or surfaces a buoy. Above ~1 GHz seawater is a lossy dielectric — waves propagate but attenuate strongly.

Step 3. (c) In the vacuum gap: conduction current zero, so displacement current is 100% of the source — every microtesla of example 1's gap field is generated by changing $\mathbf{E}$ alone. This is the regime that scales: remove the plates, remove the wire, and a changing $\mathbf{E}$ in free space still curls a $\mathbf{B}$ around itself, which — changing in turn — curls an $\mathbf{E}$ , which… The bootstrap has no need of hardware, and chasing it is precisely how Maxwell's Equations in Vacuum will find waves running at $1/\sqrt{\mu_0\varepsilon_0} = 3.00\times10^8$ m/s.

Step 4. Assemble the regime map: metals — conduction owns everything below UV; electrolytes and tissues — crossover in the MHz–GHz band (the basis of dielectric spectroscopy and of microwave-absorption heating); insulators and vacuum — displacement current owns the field. One dimensionless ratio, $\varepsilon\omega/\sigma$ , and you know which chapter of this track is in charge.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §7.3.2, §9.2 · Jackson, Classical Electrodynamics, 3rd ed., §6.1 · Purcell & Morin, Electricity and Magnetism, 3rd ed., §9.2 · Zangwill, Modern Electrodynamics, §2.1 · Bartlett & Corle, Phys. Rev. Lett. 55, 59 (1985).

Provenance. All equations derived from Maxwell's equations as presented in Griffiths 4th ed. (§7.3.2, §9.2), Jackson 3rd ed. (§6.1), Purcell & Morin 3rd ed. (§9.2), and Zangwill (§2.1). Verifiers should confirm: (1) the divergence-consistency derivation yielding $\mathbf{J}_d=\varepsilon_0\partial\mathbf{E}/\partial t$ ; (2) the capacitor identity $I_d=\mathrm{d}Q/\mathrm{d}t=I$ using $E=Q/\varepsilon_0 A$ ; (3) $c=1/\sqrt{\mu_0\varepsilon_0}=2.998\times10^8\,\mathrm{m/s}$ from $\mu_0=4\pi\times10^{-7}\,\mathrm{H/m}$ and $\varepsilon_0=8.854\times10^{-12}\,\mathrm{F/m}$ ; (4) the conductor ratio $\varepsilon_0\omega/\sigma$ for copper. No non-standard numbers are used. Addendum for this edition: the systematic divergence-consistency theorem and the wave-equation derivation are re-scoped by name to The Continuity Equation and Maxwell's Equations in Vacuum respectively (this article carries the crisis, the fix, and the capacitor field map); worked numbers recomputed — capacitor ( $R$ = 5.0 cm, $I$ = 0.50 A): $J_d$ = 64 A/m², $\partial E/\partial t$ = 7.2×10¹² V m⁻¹s⁻¹, $B$ (2 cm) = 0.80 μT, $B$ (rim) = 2.0 μT = wire field at 5 cm; copper 60 Hz ratio 5.6×10⁻¹⁷, 1 MHz 9.3×10⁻¹³, 1%-crossover ~1.1×10¹⁶ Hz; seawater crossover $\sigma/2\pi\varepsilon_r\varepsilon_0$ = 0.90 GHz.

Builds on: Ampère's Law · The Continuity Equation · Gauss's Law · Faraday's Law of Induction

Leads to: Maxwell's Equations in Vacuum · Electromagnetic Wave Equation · Energy Stored in Magnetic Fields

Did any experiment force Maxwell to add the term, or was it purely theoretical?

Purely structural — and that is what makes the episode singular. No 1861 experiment could detect $\varepsilon_0\,\partial\mathbf{E}/\partial t$ (the copper ratio of $10^{-17}$ shows why), and none demanded it. Maxwell added the term because Ampère's law was mathematically incoherent for open circuits and because his ether model suggested a mechanism. The predictions came after the ink: electromagnetic waves at $1/\sqrt{\mu_0\varepsilon_0}$ , identified with light in 1865, verified by Hertz in 1887, with the gap field itself finally mapped in 1985. It remains the textbook case of consistency-driven theorizing paying out — cited (with due caution about survivorship) whenever theorists argue structure ahead of data.

Is the displacement current in the capacitor gap real, or just an accounting fiction?

Judge by what it does, and by what "real" can operationally mean. Its magnetic field is real and measured: Bartlett and Corle (1985) mapped the microtesla azimuthal field inside a charging capacitor and found Derivation B's profile. Its role in wave propagation is as real as sunlight. What is fictional is only the flow imagery: no charge moves, nothing is displaced, no ammeter in the gap would read anything. The cleanest ontology: there is a law — changing $\mathbf{E}$ curls $\mathbf{B}$ — and "displacement current" is the name we give the source term when we choose to write that law in current's clothing. The clothing is convention; the curling is physics.

Why did no one notice Ampère's law was broken for eighty years?

Because every experiment of the era lived where the law is exact. Steady and quasi-steady currents in closed conducting circuits have $\nabla\cdot\mathbf{J} \approx 0$ ; charge accumulates significantly only in capacitors, and the magnetic field of the gap region itself is microteslas (worked example 1) — far below the era's magnetometry, and swamped by the wire's own field unless you deliberately probe inside the gap. Meanwhile the ratio $\varepsilon_0\omega/\sigma$ shows the term is invisible inside conductors below optical frequencies. The inconsistency was real but confined to a regime nobody could measure: it took a theorist reading the equations' structure, not their outputs, to see the crack.

Does the displacement current produce a magnetic field by itself, or only together with conduction currents?

By itself, with one subtlety worth having straight. In the capacitor gap the changing $\mathbf{E}$ is the only local source, and the field is there (Derivation B). But be careful with the stronger question "does a changing $\mathbf{E}$ at point X cause the $\mathbf{B}$ at point X?" — in general, fields at a point are determined by the full history of sources (charges and currents) elsewhere, and one can rewrite the theory so that $\mathbf{E}$ and $\mathbf{B}$ are both computed directly from charge-current history (Jefimenko's equations, in the radiation chapter). Within Maxwell's local equations, though, the answer is unambiguous: $\varepsilon_0\,\partial\mathbf{E}/\partial t$ enters the curl equation exactly as $\mathbf{J}$ does, and in vacuum it is the entire right-hand side. The wave exists because that term sources curl with no help.

What happens to the displacement current inside a dielectric-filled capacitor?

It gains a partner made of genuinely moving charge. In matter the field term $\varepsilon_0\,\partial\mathbf{E}/\partial t$ is joined by the polarization current $\partial\mathbf{P}/\partial t$ — bound charges really do shift as the polarization changes, and their motion is a true current, with real inertia and (in lossy dielectrics) real dissipation. The macroscopic bookkeeping bundles both into $\partial\mathbf{D}/\partial t$ , which is why the in-matter Ampère–Maxwell law reads $\nabla\times\mathbf{H} = \mathbf{J}_f + \partial\mathbf{D}/\partial t$ . Splitting the bundle honestly — which piece is vacuum structure, which is matter response, and how the response lags at high frequency — is the business of Maxwell's Equations in Matter, two articles on, and of the dispersion physics in E11.

Why is there no minus sign in the Ampère–Maxwell law when Faraday's law has one?

The signs are set by consistency, and their asymmetry is load-bearing. Faraday's minus is Lenz's: induced effects oppose their causes, which stability demands (the runaway argument of Lenz's Law). Flip Faraday's sign and induction is explosive. The Ampère–Maxwell term takes plus because that is what charge conservation requires (run the divergence audit with a minus and you get $\partial\rho/\partial t$ with the wrong sign — charge would anti-conserve), and, equivalently, what stable waves require: chase the two curl equations into the wave equation and the signs multiply to give $\nabla^2\mathbf{E} = +\mu_0\varepsilon_0\,\partial_t^2\mathbf{E}$ , oscillation; flip either sign and you get exponential blowup instead of light. One minus between the two laws is exactly what a self-sustaining, non-divergent universe of fields needs — nature's sign conventions are not conventions.

Is Maxwell's ether picture completely worthless, then?

Historically it was the scaffolding — Maxwell found the term by asking what his vortex-and-idler-wheel medium would do, and the name "displacement" records honest mechanism-seeking, not confusion. Scientifically the medium is gone: special relativity (E10's chapter) removes any rest frame for light, and the term functions perfectly in bare vacuum. But two legacies survive. The methodological one: mechanical models as discovery tools, discarded after use — Maxwell himself, by 1865, presented the theory "dynamically" without the vortices, keeping only the equations. And the conceptual one: the field itself inherited the medium's job description — it stores energy (Energy Stored in Magnetic Fields), carries momentum (E7), and supports waves. The ether died; its duties were transferred to the field, where they remain.