How Solenoid Inductance Is Calculated
A solenoid is a coil of wire wound in tightly spaced turns, typically around a cylindrical form. When a current I flows through it, the coil produces a nearly uniform magnetic field inside its core. For an ideal solenoid — one whose length is much greater than its diameter, with turns wound evenly and closely together — Ampère's law gives an interior field of B = μ₀μrnI, where n = N/l is the number of turns per unit length. Multiplying by the cross-sectional area A gives the flux through one turn, and multiplying by the total number of turns N gives the flux linkage. Dividing that flux linkage by the current defines the self-inductance: L = μ₀μrN²A / l, where A = π(d/2)² is the coil's cross-sectional area from its diameter d.
Deriving the formula from Ampère's and Faraday's laws
Starting from Ampère's law for a long solenoid, the interior magnetic field is B = μ₀μrNI/l. The flux through a single turn of area A is Φ = BA, and because all N turns are linked by essentially the same flux, the total flux linkage is λ = NΦ = μ₀μrN²AI/l. Since inductance is defined by λ = LI, the I terms cancel and you're left with L = μ₀μrN²A/l — an inductance that depends only on geometry and core material, not on the current itself.
Turns, geometry, and core material
Three factors control the result: the number of turns N (inductance scales with N², so doubling the turns quadruples L), the coil's cross-sectional area A and length l, and the core's relative permeability μr. An air or vacuum core has μr ≈ 1; inserting a ferromagnetic core (iron, ferrite, permalloy) can raise μr into the hundreds or thousands, multiplying the inductance by that same factor — which is why transformer and inductor cores use magnetic materials rather than air.
When the long-solenoid approximation breaks down
This formula assumes the field lines close entirely inside a long, thin coil, which holds well when the length is at least about four times the diameter. For short, fat coils (length comparable to or smaller than the diameter), fringing fields at the ends become significant and the ideal formula overestimates inductance — sometimes by 10% or more. Precision work on short coils applies Nagaoka's coefficient, a correction factor derived from elliptic integrals that accounts for the coil's finite length-to-diameter ratio.