Eddy Currents and Magnetic Braking

A wire loop is a convenience, not a requirement. Written as ∮E⋅dl = −dΦ/dt, Faraday's law says a changing flux produces an electric field whose lines close on themselves, and that field exists in whatever material sits in the region. Put metal there and the local form of Ohm's law, J = σE, converts it into current. Because no wire fixes the path, the current runs on every closed route the conductor offers, weighted towards the low-resistance ones — hence the name. Take a disc of radius R in a uniform field B(t) normal to its face. Symmetry makes the E lines circles about the axis, so a circle of radius r encloses flux πr²B and gives 2πrE = πr²|dB/dt|, or E = (r/2)|dB/dt|. Current density therefore grows linearly outward: the rim carries the strongest loops and the centre almost none. Lenz's law sets the sense — the loops' own field opposes the change that made them.

Current in a resistive medium dissipates at p = J⋅E = J²/σ watts per cubic metre, which is I²R written locally. For the disc with B = B₀ sin ωt, E(r) peaks at rωB₀/2, so the time-averaged rate is σr²ω²B₀²/8; integrating over shells 2πr t dr from 0 to R gives P = πσω²B₀²tR⁴/16. Numbers make the scalings concrete. A copper disc (σ = 5.96 × 10⁷ S m⁻¹) of radius 40 mm and thickness 0.50 mm in a 50 mT, 50 Hz field absorbs 3.7 W. Halve the radius and it falls to 0.23 W, a factor of 16, because P ∝ R⁴. Double the frequency and it quadruples. The dependence on σ looks backwards at first: the better the conductor, the more it heats, because the same emf drives proportionally more current. That holds only while the disc's own field barely opposes the applied one, which for a disc means μ₀σωR t / 4 ≪ 1 — about 0.12 here — and not merely that the metal is thinner than the skin depth δ = √(2/(μ₀σω)), 9.2 mm for copper at 50 Hz. Scale the radius up at fixed thickness and the screening grows while δ does not move. Thicker than that and the current crowds into a surface layer, and the loss instead falls roughly as 1/√σ.

When the heat is waste, attack the loop area. A transformer or motor core carries B along the iron, so slicing it into thin sheets laid parallel to B leaves the magnetic path untouched while confining every eddy loop to one sheet. Inside a sheet of thickness d the induced field is E = x dB/dt with x measured from the mid-plane; averaging x² across the thickness gives p = σω²B₀²d²/24 = π²σf²B₀²d²/6 per unit volume. The dependence is d², so N sheets in place of one slab of the same total thickness cut the loss by N², not by N. Grain-oriented silicon steel, σ ≈ 2.0 × 10⁶ S m⁻¹, at 1.5 T and 50 Hz with d = 0.35 mm gives about 2.3 kW m⁻³, near 0.30 W kg⁻¹. The varnish between sheets does not make the steel more resistive — the steel is unchanged — it only stops loops crossing from sheet to sheet. The rule needs d below the skin depth, about 0.80 mm for that steel at 50 Hz with μ(r) ≈ 4000; above a few kilohertz designers move to ferrites and powder cores, where σ itself is tiny.

Move a conductor through a non-uniform field, or sweep the field past the conductor, and the same currents appear — but now they sit in B, so the metal feels F = ∫ J × B dV, directed against the relative motion by Lenz's law. At low speed J ∝ v, so the drag is linear: F = −bv, with b ≈ Cσwa²B² for a sheet of thickness w and a field region of size a, C a geometry factor of order 0.1 to 1. Linear drag behaves nothing like Coulomb friction. Newton's second law gives m dv/dt = −bv, so v = v₀ exp(−t/τ) with τ = m/b: the speed decays exponentially and never reaches zero. The distance is finite, v₀τ, but the time is not, which is why a lorry's eddy-current retarder or a roller-coaster fin brake always sits alongside a friction brake that can hold at rest. Read the other way, the same linearity is a gift. A copper vane in a magnet gap hands an oscillator a clean −bv term with no stiction and no wear, and instrument designers choose b = 2√(km) to bring a balance or a galvanometer needle to rest in the shortest time without overshoot.

Magnetic forces do no work on charges: qv × B is always perpendicular to v. The field therefore cannot be the source of the heat that appears in a braked disc; it is a broker. Whatever agent changes the flux — the hand pushing the plate, gravity pulling the magnet, the primary winding driving the core — does work against the induced emf, and that work reappears as J²/σ dissipation in the conductor. The bookkeeping is exact: for a conductor dragged at steady speed, Fv = ∫(J²/σ) dV, with nothing stored once the transient has passed. That equality fixes the design limits. A 1200 kg vehicle at 30 m s⁻¹ carries 540 kJ of kinetic energy; dumped into a 10 kg copper rotor with c = 385 J kg⁻¹ K⁻¹, that is a 140 K temperature rise, so eddy brakes are rated by thermal capacity and cooling rather than by peak force. It also explains their fade: σ drops as the metal warms, and b drops with it.

The linear law assumes the induced currents' own field is negligible beside the applied one. The test is the magnetic Reynolds number R(m) = μ₀σvL. For copper μ₀σ = 74.9 s m⁻², so a magnet drifting at 0.05 m s⁻¹ past a 20 mm feature has R(m) = 0.075 — safely linear. A vehicle at 100 m s⁻¹ over aluminium (μ₀σ = 44.0 s m⁻²) with L = 0.3 m has R(m) ≈ 1.3 × 10³, a different regime: the induced currents largely exclude the field from the conductor, much as a mirror image would. There the drag rises with v, peaks near R(m) ≈ 1, then falls as 1/v, while the repulsive lift climbs to a plateau, so the lift-to-drag ratio grows roughly as R(m) and an electrodynamic suspension gets more efficient the faster it runs. The static case still fails: a magnet cannot hover over ordinary copper, because with no motion there is no dΦ/dt and the loops die away in milliseconds. Levitation needs relative motion, an AC drive whose induced current lags by nearly half a cycle, or a superconductor.