Self-inductance

Switch a current I into any closed loop and the loop finds itself threaded by the field that current is making. Biot–Savart gives every element of that field as a fixed geometric factor times I, so doubling the current doubles B everywhere at once, doubles the flux Φ_B = ∫B⋅dA through one turn, and doubles the flux linkage Λ = NΦ_B — the flux counted once for each of the N turns it passes through. What survives the doubling is the ratio: L = Λ/I = NΦ_B/I is the same number at 0.10 A and at 10 A. That ratio is the self-inductance, and its unit is the henry, with 1 H = 1 Wb A⁻¹ = 1 V s A⁻¹ = 1 Ω s. Fix the sense of positive flux to match the sense of positive current by the right-hand rule and L comes out positive for every circuit, because a current's own flux always links it the same way round.

Faraday's law does not care whose current made the flux. Apply ε = −dΛ/dt to a coil's own linkage and substitute Λ = LI. If the coil is rigid, so that L is a constant, the product rule leaves one term: ε = −L dI/dt. Hold the current steady and the emf is exactly zero no matter how large that current is; change it and the emf is set by the rate alone. The minus sign is Lenz's law written for a single circuit: the induced emf acts against the change in the current, never against the current itself. In the passive sign convention, with the terminal voltage measured from the end the current enters, this appears as v = L dI/dt: positive while the current rises, so the coil draws energy from the source, and negative while it falls, so the coil returns it. If the geometry is allowed to move, L becomes a function of time and ε = −L dI/dt − I dL/dt; everything in this lesson assumes the first term alone.

The recipe never changes. Assume a current I in the circuit; find B from Ampère's law or Biot–Savart; integrate B over the surface bounded by one turn to get Φ_B; multiply by N and divide by I. The current cancels in the last step, and that cancellation is the proof that L is pure geometry. Run it on a long solenoid of N turns, length ℓ and cross-section A. Ampère's law on a rectangular loop straddling the winding gives a uniform axial interior field B = μ₀NI/ℓ, with almost nothing outside. The flux through one turn is Φ_B = BA = μ₀NIA/ℓ, the linkage is N times that, and L = μ₀N²A/ℓ. For N = 500 turns over ℓ = 0.25 m with A = 4.0 cm², L = (4π × 10⁻⁷)(500)²(4.0 × 10⁻⁴)/0.25 = 5.0 × 10⁻⁴ H. The N² is the signature of self-inductance: N turns make N times the field, and that field then links N turns.

A toroid shows what to do when B varies across the cross-section. Ampère's law on a circle of radius r inside the windings gives B = μ₀NI/(2πr), so the field is strongest against the inner wall. Take the cross-section to be a rectangle of height h running from inner radius a to outer radius b, and slice it into strips of width dr at radius r. Each strip carries dΦ_B = B h dr, so the flux through one turn is Φ_B = (μ₀NIh/2π) ∫dr/r from a to b = (μ₀NIh/2π) ln(b/a). Multiply by N and divide by I to get L = μ₀N²h ln(b/a)/(2π). This result is exact for an ideal winding, because a toroid's flux never leaves the ring and there are no ends to leak from. The logarithm is what a 1/r field always leaves behind, and the same integral returns for a coaxial cable, where the flux sits in the gap between the conductors and the circuit has only one turn.

Numbers make the minus sign concrete. Take L = 0.50 mH carrying 3.0 A and let the current fall to zero in 2.0 ms: dI/dt = −1500 A s⁻¹, so ε = −L dI/dt = +0.75 V, an emf pointing the way the current was already flowing and trying to keep it alive. Now open a mechanical switch, so the same 3.0 A collapses in 2.0 μs. The rate is a thousand times larger and so is the emf: 750 V, across a coil that a 1.5 V cell was happily driving. That voltage appears across the separating contacts, and it is why they arc, why relay coils are fitted with a flyback diode, and why an inductor's terminals deserve respect after the supply is switched off. Nothing about the coil changed — only dI/dt did.

Two assumptions carry everything above: that B is proportional to I, and that the geometry holds still. Fill a 1.5 mH air solenoid with a ferromagnetic core and, while the material stays linear, the flux rises by a factor of its relative permeability — perhaps 2000, taking L to roughly 3 H for no extra copper. But iron's B against H curve bends. Once the domains are aligned, extra current adds almost no extra flux and Λ(I) flattens; beyond that knee L = Λ/I is no longer a constant, and designers quote the differential inductance dΛ/dI instead, which collapses towards the air-core value in saturation. Air and vacuum have no such limit, and their L holds up to the current that melts the wire. A second limit appears at high frequency: current crowds towards the conductor surfaces and the turn-to-turn capacitance of the winding starts carrying its own current, so a real coil stops behaving like a pure inductance well before Λ = LI itself fails.