Mutual inductance
The coupling between two separated coils: M = Φ₁₂/I₂, the flux that coil 2's current produces through coil 1 per unit coil-2 current. The operating principle of every transformer.
Definition
Mutual inductance M quantifies the magnetic coupling between two separate coils. Run a current I₂ through coil 2, and some fraction of the magnetic flux it produces links through coil 1; the amount of that linked flux per unit I₂ is the mutual inductance, M = Φ₁₂/I₂. A changing current in coil 2 then induces an EMF in coil 1: V₁ = −M dI₂/dt. Reciprocity (a consequence of the underlying symmetry of Maxwell's equations) requires that M₁₂ = M₂₁ — the mutual inductance is the same number whether you call coil 2 the driver and coil 1 the target, or vice versa.
The magnitude of M depends on geometry, separation, and the presence of any ferromagnetic or shielding material between the two coils. For two ideal coaxial solenoids of the same length, perfectly coupled with a ferromagnetic core, M approaches its maximum value, the geometric mean M = √(L₁ L₂). In the general case this is written M = k √(L₁L₂), where 0 ≤ k ≤ 1 is the coupling coefficient, a dimensionless measure of how completely the two coils share flux. Tightly wound transformers on a single toroidal core achieve k ≈ 0.99; loosely coupled air-core coils in a radio oscillator might have k ≈ 0.1.
Mutual inductance is the operating principle of the transformer. Primary current in coil 1 creates flux, which threads through coil 2; changing primary current induces EMF in coil 2. For an ideal transformer with N₁ and N₂ turns, the EMFs are in the ratio V₁/V₂ = N₁/N₂, and the currents in the inverse ratio I₁/I₂ = N₂/N₁ (from conservation of power, ignoring small losses). Transformers step voltage up for efficient long-distance transmission and down for household delivery; they isolate circuits galvanically; they match impedances between amplifier stages; they inject signals into RF chains. In the 21st century, resonant mutual-inductance coupling powers phone charging pads, implanted medical devices, and contactless access cards. The underlying physics is the same as Henry's 1832 Albany Academy demonstration.