Displacement field

The electric displacement field D is defined as D = ε₀E + P, where E is the electric field and P is the polarization density. Its great virtue is that its divergence picks up only the free charge density: ∇·D = ρ_f. The bound charges that polarization carries inside a dielectric are entirely absorbed into the definition of D, so when you write Gauss's law for the free charges alone — the deliberately placed charges on capacitor plates, the conduction charges on a metal — D is what you should use.

Without D, applying Gauss's law to a parallel-plate capacitor filled with dielectric requires you to track three charge layers: the free charge on the plates, the bound charge on the dielectric's outer faces (which partially cancels the field), and any free charge that might have leaked into the bulk. With D, you wrap a Gaussian surface around the plate, count only the free surface charge density σ_f sitting on it, and out pops D = σ_f directly — no bookkeeping over induced bound layers. The price of this convenience is that D is not the field that exerts force on a test charge; only E is. Knowing D in a region tells you what free charges are around but not, by itself, how strongly things will be pushed; for that you need to combine D with the material relation D = εE to recover E.

For linear isotropic media D = εE, where ε is the permittivity of the material, and the calculation collapses to a straightforward arithmetic problem. For anisotropic crystals ε becomes a tensor and D and E need not be parallel. For nonlinear or hysteretic materials (ferroelectrics) the relationship is more complicated and history-dependent. But the underlying logic is always the same: D is the surrogate that lets you reason about the free-charge structure of a problem without the dielectric polluting your Gauss integrals.

James Clerk Maxwell introduced the displacement field — and the closely related "displacement current" ∂D/∂t that completes Ampère's law — in his 1865 paper A Dynamical Theory of the Electromagnetic Field. The word "displacement" reflected the mid-Victorian picture of the dielectric as a sea of mechanical molecules that physically displace under an applied field; the modern interpretation in terms of polarization came later, but the variable and its name stuck.