How this instrument works
Inductance measures how much magnetic flux a coil links for each ampere of current flowing through it. Inside a long solenoid, Ampère's law gives a uniform field B = μ₀nI, where n = N⁄l is the number of turns packed into each metre of length. Each of the N turns encloses that field over the cross-sectional area A, so the total flux linkage is N times B times A. Divide by the current and one factor of N cancels, but the n hidden inside B already carried another N⁄l — that double appearance is where the square comes from: L = μ₀N²A⁄l.
The shape of the formula rewards packing turns close together far more than it rewards adding coil length: because N is squared while l divides only once, a coil wound twice as densely on the same wire — half the length, same turn count — sees its inductance roughly double, while doubling the turns alone quadruples it. That asymmetry is why radio tuning coils and electromagnet windings are built as tight, many-turn spirals rather than long, sparse ones: turns are the expensive multiplier, length the cheap one.
The formula assumes an idealized long, thin solenoid — length several times the diameter — so the field stays uniform inside the winding and effectively vanishes outside it, an approximation that breaks down for short, fat coils where flux leaks out the open ends before completing its loop. Real short coils fall below this ideal value by a correction factor known as Nagaoka's coefficient, and slipping in a magnetic core multiplies the whole result by that material's relative permeability, since air and vacuum are what μ₀ alone describes.
- Enter Number of turns — the total winding count end to end, not turns per unit length.
- Enter Coil cross-sectional area — the area each turn encloses; for a round coil this is πr² using the coil's radius.
- Enter Coil length — the axial length the winding actually occupies, not the length of wire used to make it.
- Read Inductance, H — switch the unit menu to mH or µH for small air-core coils.
Worked example — 500-turn coil, 10 cm² area, 10 cm long
Wind 500 turns of wire onto a form with a 10 cm² cross-section (Area = 0.001 m²) over a 10 cm length (lCoil = 0.1 m) — a size typical of an air-core inductor for an antenna tuner. The formula gives L = (1.25663706212×10⁻⁶)(500²)(0.001)÷0.1. Squaring the turns first: 500² = 250,000. Multiply by μ₀: 1.25663706212×10⁻⁶ × 250,000 = 0.31415926553. Multiply by the area: × 0.001 = 0.00031415926553. Divide by the length: ÷ 0.1 = 0.0031415926553 H — about 3.14 mH.
A quick cross-check uses turns per metre instead of the raw turn count: n = N⁄l = 500⁄0.1 = 5000 turns per metre, and L = μ₀n²Al = (1.25663706212×10⁻⁶)(5000²)(0.001)(0.1), which multiplies out to the identical 0.0031415926553 H. Both routes land on 3.1416 mH, and that is not a coincidence — the digits after the decimal point trace π, since μ₀ itself is defined from 4π×10⁻⁷ H/m.
Questions
Why is inductance proportional to the square of the number of turns?
Because turns matter twice over. Doubling N doubles the magnetic field each turn contributes, and it also doubles the number of turns that field links, so the flux linkage — and therefore L — scales as N². A 500-turn coil has 25 times the inductance of an otherwise identical 100-turn coil, not 5 times.
Does this formula work for a coil of any shape?
Only for a long, thin solenoid, one whose length is several times its diameter, so the internal field stays roughly uniform and little leaks out the ends. Short, fat coils need a correction factor called Nagaoka's coefficient, always below 1, because the ideal formula assumes flux stays confined that a stubby winding lets escape.
What happens if the coil is wound around an iron or ferrite core?
Multiply the result by the core material's relative permeability, μr. Air and vacuum have μr = 1, which is what this calculator assumes; a ferrite rod can carry μr in the hundreds, so the same 500-turn, 10 cm coil could jump from 3.14 mH into the henry range once a core is inserted.
Why is Coil cross-sectional area the loop area, not the wire's area?
Because A here is the area each turn of wire encloses — for a circular coil, πr² using the winding's radius, not the copper conductor's radius. Enter the wire's own cross-section by mistake and the result understates the inductance by several orders of magnitude, since the winding radius normally dwarfs the wire diameter.
How does this differ from a toroid or a single straight wire?
A toroid's field stays entirely inside its core, so its formula swaps the length-over-area ratio for the core's mean radius and cross-section. A single straight wire has a small self-inductance from its own field, found with a logarithmic formula involving its length and radius — a different geometry from a wound coil entirely.
Can this formula size a radio-frequency tuning coil?
Yes — it is the standard starting point for air-core RF coils, though at high frequencies skin effect and inter-turn capacitance shift the real inductance a little from this DC value. Most builders use the formula to pick a turn count, then trim the physical coil to reach the target frequency.