Displacement Current

The paradox that broke Ampère's law

Ampère's original 1820s law says the line integral of the magnetic field around any closed loop equals μ₀ times the current threading the loop. Wire carrying 1 A produces a B field that circles it; place an Amperian loop perpendicular to the wire and the integral ∮B·dl = μ₀ × 1 A. So far so good.

Now put a parallel-plate capacitor in the wire and start charging it. Pick an Amperian loop on the wire side of the capacitor: it encloses the wire's current, so ∮B·dl = μ₀I as before. Now slide the same loop's capping surface sideways so it bulges between the plates instead of cutting through the wire. The loop is unchanged; only the surface bounded by it has moved. Suddenly no current passes through the surface — the wire is on the other side of the cap — and Ampère's law as written gives ∮B·dl = 0.

Two perfectly valid capping surfaces, sharing the same boundary loop, give two different answers for the same integral. This is a mathematical contradiction. Stokes's theorem says the line integral of any field around a closed loop is fixed by the field's curl over any surface bounded by that loop — the answer cannot depend on which surface you choose. Maxwell saw this in 1861 and realized something had to fill in for the missing current between the plates.

Maxwell's fix: ε₀ ∂E/∂t

As the capacitor charges, the electric field between its plates rises in proportion to the accumulated charge. For ideal parallel plates with charge Q and area A, the field is E = Q/(ε₀A). Differentiate in time: ∂E/∂t = (1/(ε₀A)) dQ/dt = I/(ε₀A). Multiply through by ε₀A and you get I = ε₀A ∂E/∂t. The changing electric flux ε₀ ∂E/∂t through the gap exactly equals the current I in the wire.

Maxwell's modified Ampère's law is

∇×B = μ₀J + μ₀ε₀ ∂E/∂t

The new term μ₀ε₀ ∂E/∂t is what Maxwell called the displacement current density. With this addition, both Amperian surfaces give the same answer: cap-through-wire encloses real current I; cap-between-plates encloses displacement current ε₀A ∂E/∂t = I. Stokes's theorem is restored, charge conservation follows automatically, and electromagnetic waves become an inescapable consequence.

The Ampère-Maxwell law, term by term

Term	Physical meaning	Units	Dominant when	Example
μ₀J	Magnetic field from real charge flow	T/m	DC and low frequency	Solenoid, transmission line
μ₀ε₀ ∂E/∂t	Magnetic field from changing E	T/m	RF, capacitor gaps, free space	Charging capacitor, radio wave
Both terms	Inductor + capacitor in series	—	Always present	LC resonant circuit at MHz
∂D/∂t (in matter)	ε₀ ∂E/∂t + ∂P/∂t (vacuum + polarization)	A/m²	Dielectrics, high-ε_r media	Water at microwave frequency
Conservation	∇·J + ∂ρ/∂t = 0 (continuity)	A/m³	Always — implied by ∇·B = 0	Capacitor charge buildup
Wave production	Coupled with Faraday's −∂B/∂t	—	Vacuum, no charges	Light, radio, X-rays

The same equation describes a 60 Hz transformer (the J term dominates) and a 100 GHz waveguide (the ε₀ ∂E/∂t term dominates). It is the same physics — Maxwell's contribution is that he wrote it so both regimes are special cases of one law.

Worked example: 1 μF capacitor at the moment of switch-on

Consider a 1 μF parallel-plate capacitor with circular plates of radius r = 1 cm and gap d = 1 mm. It charges from 0 V through a 1 kΩ resistor up to 5 V; the time constant is τ = RC = 1 ms and the instantaneous current at t = 0 is I₀ = V/R = 5 mA. What is the displacement-current density between the plates at that instant?

The displacement current must equal the wire current by symmetry: I_D = ε₀A ∂E/∂t = I₀ = 5 mA. The plate area is A = πr² = π(0.01)² = 3.14 × 10⁻⁴ m², so the displacement-current density is J_D = I₀/A = 5 × 10⁻³ / 3.14 × 10⁻⁴ ≈ 16 A/m². That is a real, calculable number — and it sources a magnetic field around the symmetry axis. By Ampère's law applied to a circle of radius r within the gap, B(r) = μ₀ I_D r / (2 A) = (4π × 10⁻⁷)(5 × 10⁻³)(0.01) / (2 × 3.14 × 10⁻⁴) ≈ 100 nT at the edge of the plate.

Rowland's 1876 rotating-disk experiment measured exactly this kind of field: a charged disk spun fast enough to make the surface charge into a real current produced a measurable B field. Modern repetitions with high-Q LC tank circuits at MHz routinely verify the displacement-current contribution to within 1 % of theory.

Symmetry and the equations of light

With the displacement term in place, Faraday's law (∇×E = −∂B/∂t) and Ampère-Maxwell (∇×B = μ₀J + μ₀ε₀ ∂E/∂t) are mirror images of each other. A changing magnetic field sources circulating electric field; a changing electric field sources circulating magnetic field. This duality is what allows self-sustaining oscillation: an E that varies in time pumps B; the changing B pumps E; the wave propagates without any source.

Apply the curl operator to Faraday's law in vacuum (no charges, no currents):

∇×(∇×E) = −∂(∇×B)/∂t = −μ₀ε₀ ∂²E/∂t²

Using the vector identity ∇×(∇×E) = ∇(∇·E) − ∇²E = −∇²E (since ∇·E = 0 in vacuum), this becomes the wave equation ∇²E = μ₀ε₀ ∂²E/∂t² with phase velocity c = 1/√(μ₀ε₀) = 2.998 × 10⁸ m/s. The same calculation for B gives the same wave equation. Light is the simultaneous wave solution of these two equations — and it exists only because Maxwell's 1861 ε₀ ∂E/∂t coupled the two.

How charge conservation falls out automatically

Take the divergence of Ampère-Maxwell's law:

∇·(∇×B) = 0 = μ₀∇·J + μ₀ε₀ ∂(∇·E)/∂t

The left side is zero by the vector identity ∇·(∇×anything) = 0. On the right, substitute Gauss's law ∇·E = ρ/ε₀ to get

∇·J + ∂ρ/∂t = 0

This is the continuity equation — charge cannot vanish, only flow. Without the displacement term, the divergence of Ampère's law would imply ∇·J = 0 always, forbidding any capacitor from charging or any battery from supplying current to a circuit endpoint. Maxwell's addition makes charge conservation a theorem rather than an extra postulate.

Where the displacement current shows up in real engineering

• RF capacitors in LC tank circuits. A 100 pF capacitor in a 10 MHz tank carries ω·CV = 2π·10⁷·10⁻¹⁰·10 V = 6.3 mA of displacement current — same units, same magnetic-field signature as conduction current. Modern PCB simulators (Sonnet, ADS, HFSS) model both terms together; ignoring the displacement-current contribution above 100 MHz gives wildly wrong S-parameters for any geometry with electrodes close together.
• Touchscreen capacitive sensing. Every modern smartphone screen senses finger contact via the displacement current. A driver electrode generates a 100 kHz AC signal; the receiver picks up the ε₀ ∂E/∂t crossing the gap. When a finger lands, it shunts displacement current to ground, and the receiver records the change. Apple's Multi-Touch and Samsung's Synaptics chips both depend on Maxwell's term.
• Particle-accelerator RF cavities. The Large Hadron Collider's 400 MHz superconducting cavities deliver 5 MV/m gradient by storing energy in oscillating E and B fields that satisfy the full Ampère-Maxwell relation. The displacement current carries energy from cavity wall to beam axis; without it the cavity would behave as a static gap with no acceleration.
• Wireless power transfer (Qi, A4WP). 6.78 MHz resonant wireless chargers couple energy through near-field magnetic AND electric coupling — the displacement-current term enables the latter. Industrial systems above 10 kW use a hybrid approach that is impossible to design without keeping both terms.
• Atmospheric electricity and lightning return strokes. Right before a lightning strike, a 100 kV/m field builds up between cloud and ground in roughly 1 ms. The displacement current density ε₀ ∂E/∂t ≈ 10⁻⁶ A/m² × cloud-base area gives a global "Maxwell current" of ~2000 A continuously circulating through the atmosphere — measured by Carnegie and Wilson balloon campaigns of the 1920s and still tracked by the Schumann-resonance monitoring network.

Variants and extensions

• Polarization current in dielectrics. Inside a dielectric the displacement vector D = ε₀E + P contains a polarization term P that itself shifts real bound charges. Its time derivative ∂P/∂t is a real conduction current of bound charge, and together with ε₀ ∂E/∂t it makes up the full ∂D/∂t. In high-ε_r media most of "displacement current" is actually bound-charge motion.
• Magnetic-monopole symmetry. Hypothesizing magnetic charges introduces a "magnetic current" J_m = ρ_m v_m. The symmetric Ampère-Maxwell pair becomes −∇×E = μ₀J_m + ∂B/∂t and ∇×B = μ₀J + μ₀ε₀ ∂E/∂t. Maxwell's equations gain perfect E ↔ B duality.
• Quantum electrodynamics. Promote A^μ to a quantized field; the photon propagator includes both transverse and longitudinal modes. The displacement current is the classical analog of virtual photon exchange — the same physics, now obeyed by operator-valued fields.
• Computational electromagnetics (FDTD). Yee's 1966 finite-difference algorithm explicitly time-steps E and B in interleaved half-steps, with the displacement-current term appearing as the dE/dt update. Modern GPU FDTD simulates 10⁹ Yee cells per second.
• Cosmological electromagnetism. Maxwell's equations on a curved spacetime use covariant derivatives. The displacement current is preserved, and EM waves redshift with the cosmological scale factor — the CMB photons we see today are Maxwell-equation solutions that have been ∂E/∂t-coupling for 13.8 billion years.

Common pitfalls

• Thinking displacement current is a real current of charge. It is not. No electrons cross the gap of a vacuum capacitor. The name is a 19th-century mechanical metaphor. The physical quantity is just ε₀ times the time derivative of E.
• Forgetting that ∂E/∂t is a vector. The displacement-current density has the same direction as the changing E field, and the magnetic field it sources circles that direction via the right-hand rule. In a charging capacitor with E pointing from + plate to − plate, B circles in the same sense as it does around the connecting wire.
• Mixing free and bound currents in dielectrics. In a medium ∇×H = J_free + ∂D/∂t. If you write ∇×B/μ₀ = J_total + ε₀ ∂E/∂t and confuse the two, factors of μ_r and ε_r appear in the wrong places. Pick one form and stick with it.
• Neglecting the displacement current at low frequencies and then forgetting at high. At 60 Hz the displacement term is usually negligible compared to J — but in a high-voltage capacitor or an underwater cable at GHz it dominates. The transition happens when ωε ≈ σ.
• Confusing the displacement current density (J_D = ε₀ ∂E/∂t) with the displacement field (D = ε₀E + P). The field D is static; its time derivative ∂D/∂t is what behaves like a current density. They have different units and play different roles in the equations.