Displacement current

Displacement current is the term J_d = ε₀ ∂E/∂t that Maxwell added to Ampère's law in 1861–1865 to patch a subtle inconsistency. The original Ampère form, ∇×B = μ₀J, requires ∇·J = 0 (by taking the divergence of both sides, since ∇·(∇×B) = 0). But the continuity equation ∂ρ/∂t + ∇·J = 0 means ∇·J is generally non-zero whenever charge density changes in time. The easiest example is a charging capacitor: current flows into one plate, charge builds up, but in the gap between the plates no actual current flows. Ampère's original law applied to a loop encircling the wire gives μ₀I; applied to a loop in the gap gives 0. The same loop, just shifted in space, would produce contradictory predictions for the magnetic field.

Maxwell's fix was to add a term ε₀ ∂E/∂t on the right-hand side: ∇×B = μ₀(J + ε₀ ∂E/∂t). The modified equation is consistent with charge conservation (take the divergence, use Gauss's law ∇·E = ρ/ε₀, and the continuity equation follows automatically). Physically, the extra term says that a changing electric field produces a magnetic field just as a current of charges does. In the capacitor gap, E is ramping up as the plates charge, and ε₀ ∂E/∂t replaces the missing conduction current exactly — the magnetic field loops around the gap the same way it loops around the wire.

The consequence was enormous. With the displacement-current term included, the coupled Maxwell equations in free space (ρ = 0, J = 0) permit self-sustaining wave solutions: a changing E creates a changing B via displacement current, the changing B creates a changing E via Faraday's law, and the pattern propagates through empty space. The speed comes out to c = 1/√(ε₀μ₀) ≈ 3×10⁸ m/s, which Maxwell immediately recognised as the speed of light. His 1865 paper concludes that light is an electromagnetic wave — one of the great deductive reveals in the history of science, and all of it hinges on the innocuous-looking extra term he inserted to keep Ampère's law self-consistent.

Maxwell introduced the displacement-current term in On Physical Lines of Force (1861–1862) and used it in A Dynamical Theory of the Electromagnetic Field (1865) to derive the wave equation. Hertz confirmed the prediction experimentally in 1887–1888, generating and detecting EM waves in the lab. The name "displacement current" is a historical artefact of Maxwell's original mechanical model (he pictured a kind of elastic displacement of an ether medium); the term has nothing to do with ordinary current but the name stuck.