Length contraction of current

Length contraction of current is the application of special-relativistic length contraction to the lattice of stationary positive ions in a current-carrying wire, viewed from the rest frame of the drift electrons. In the lab frame, the lattice ions are at rest with line density n₀ (positive charge per unit length, n₀·e where e is the elementary charge), and the drifting electrons have line density n₀ as well (negative, equal magnitude) — the wire is net-neutral. Boost into the electrons' rest frame and the situation inverts: the cyan electrons sit still, but the magenta lattice now drifts at −v_drift relative to the boosted observer. The lattice's spacing shrinks by 1/γ, so its measured line density in the boosted frame becomes γ·n₀.

Symmetrically, the cyan electrons in the lab frame had line density n₀ but were moving at +v_drift; their rest-frame density is therefore n₀/γ (lower, by inverse contraction). In the electron rest frame they sit still at this lower density. The asymmetry — γ·n₀ for the lattice, n₀/γ for the electrons — produces a net positive charge density per unit length of e·n₀·(γ − 1/γ). This is the electric field source in the boosted frame that, evaluated as a Coulomb attraction between two parallel current-carrying wires, exactly reproduces the magnetic force F = μ₀I²/(2πd) of the lab frame after accounting for the transverse-force factor of 1/γ between frames. The "click" moment is visible in §11.4's two-panel scene: as the slider moves β past 0.3, the magenta lattice in the right panel visibly bunches together while the cyan electrons stay at their original spacing, and the + density readout climbs from zero to a finite value. Same physical force, different observer; magnetic attraction is the relativistic consequence of electrostatic attraction.