Flux linkage
The total flux threading a multi-turn coil, summed across all turns: λ = N Φ for N turns each enclosing flux Φ. The quantity Faraday's law naturally refers to for real coils rather than single loops.
Definition
Flux linkage, usually written λ, is the total magnetic flux threading through a coil, counted once for each turn the flux passes through. For a simple N-turn coil where every turn encloses the same flux Φ (a good approximation for tightly-wound solenoids), the flux linkage is just λ = N Φ. For a more complicated geometry — overlapping coils, coils with varying pitch, coils wound around non-uniform cores — λ is the proper sum ∑ Φ_k over all turns, and different turns may enclose different fluxes.
The significance of flux linkage is that Faraday's law, when applied to real coils rather than idealised single loops, refers to λ rather than to Φ: EMF = −dλ/dt. For the tight-winding case this is just EMF = −N dΦ/dt, which is why a 1000-turn coil produces 1000× the induced voltage of a single-turn coil in the same field. For the general case it is why engineers compute self-inductance as L = λ/I rather than as Φ/I — the factor of N is baked into λ from the start.
Flux linkage also gives a uniform way to talk about inductance. For a single air-core solenoid, L = μ₀ n² ℓ A = N × (Φ/I) = λ/I. For two mutually coupled coils, M₁₂ = λ₁₂/I₂, where λ₁₂ is the flux linkage through coil 1 due to the current in coil 2. Writing everything in terms of λ avoids the bookkeeping confusion of "flux per turn" vs "total flux" and scales cleanly from single-loop circuits (λ = Φ, N = 1) to complex multi-winding machines.