Solenoid Inductance Calculator — Henries from Turns, Length and Diameter

Background.

The inductance of a solenoid follows from its geometry alone: L = μ₀μᵣN²A/l, where N is the number of turns, A the cross-sectional area, l the winding length and μ₀ the permeability of free space. Enter turns, length, diameter and a core permeability and this page returns the inductance in microhenries and millihenries, along with the winding density and the length-to-diameter ratio.

Two scaling laws are worth carrying away. Inductance goes as the square of the turn count, so doubling the turns quadruples it — turns are by far the most effective lever. And it goes as the square of the diameter too, through the area, while it falls only in inverse proportion to the length. A short fat coil therefore has much more inductance than a long thin one with the same wire on it.

The important caveat is that this equation describes an ideal solenoid — one assumed to be infinitely long, with a uniform axial field inside and no field outside. A real coil leaks flux at both ends, and its true inductance is lower than the formula gives. The shorter the coil relative to its diameter, the larger the shortfall, which is why this page reports the length-to-diameter ratio next to the answer and says which side of the useful range you are on. No end-correction factor is applied here and none is quoted: the Nagaoka correction table could not be retrieved from a primary source for this page, and an invented correction would be worse than an honest upper bound. Treat the figure as a design starting point and measure the wound coil.

The permeability constant is versioned rather than assumed. μ₀ is taken as 1.256 637 061 27 × 10⁻⁶ N·A⁻², NIST's CODATA 2022 value, which since the 2019 redefinition of the SI is a measured quantity rather than exactly 4π × 10⁻⁷ — though it remains equal to that to within about two parts in ten billion, which is nowhere near any tolerance a wound coil has. Relative permeability is deliberately an input rather than a built-in table, because it belongs to a specific core at a specific flux density, frequency and temperature, not to a material in general.

What is solenoid inductance calculator?

A solenoid is a coil of wire wound in a helix, and its self-inductance is the ratio of its flux linkage to the current producing it, L = NΦ/I. For a long solenoid the internal field is very nearly uniform and equal to B = μ₀nI, where n = N/l is the turns per unit length. The flux through one turn is then BA, the flux linkage is N times that, and dividing by the current leaves L = μ₀N²A/l — OpenStax University Physics prints this as Equation 14.13. Adding a magnetic core multiplies the field, and hence the inductance, by the core's relative permeability μᵣ. The result depends only on geometry and material: nothing about the wire gauge, the current, or the frequency enters it. That is both the equation's strength and its limit. It is exact for an infinitely long, closely wound, single-layer coil and progressively optimistic for shorter ones, because a finite coil's field spreads and weakens near the ends so the real flux linkage is less than the ideal calculation assumes. For coils much longer than they are wide the error is small; for a squat coil it is substantial, and a correction factor derived from the coil's proportions is needed.

How to use this calculator.

1. 01Count the turns in the winding and enter them as N. Inductance goes as N², so this is the number to get right.
2. 02Measure the winding length along the coil's axis — the extent of the turns, not the length of the former or of the wire.
3. 03Enter the mean diameter of the winding. The page converts it to area with A = πd²/4.
4. 04Set the relative permeability: 1 for air or a plastic former, or the datasheet figure for a magnetic core at your working flux density.
5. 05Read the inductance, then check the length-to-diameter ratio beside it. At 10 or above the ideal formula is a fair model; well below that, treat the answer as an upper bound.
6. 06Measure the finished coil if the value matters. No geometric formula accounts for wire thickness, winding pitch, multiple layers or self-capacitance.

The formula.

The derivation runs through the field. Inside a long solenoid the magnetic field is essentially uniform and axial, with magnitude B = μ₀nI where n = N/l is the turns per metre. The flux through a single turn is Φ = BA, and the total flux linkage is NΦ. Self-inductance is flux linkage per unit current, L = NΦ/I, and substituting gives L = N(μ₀nI)A/I = μ₀N²A/l once n is written as N/l. A magnetic core multiplies B, and therefore L, by its relative permeability.

Working the default: 100 turns, 50 mm long, 20 mm in diameter, air core. Convert to metres first — the equation is only valid in SI base units. The radius is 0.01 m, so A = π × 0.01² = 3.14159265 × 10⁻⁴ m², which is 314.159265359 mm². The winding density is n = 100/0.05 = 2000 turns per metre. Then L = 1.25663706127 × 10⁻⁶ × 1 × 100² × 3.14159265 × 10⁻⁴ ÷ 0.05 = 7.89568352 × 10⁻⁵ H, which is 78.9568351983 µH or 0.0789568352 mH.

That value can be checked by a genuinely different route rather than by rearranging the same expression. Put 1 ampere through the coil: B = μ₀nI = 1.25663706127 × 10⁻⁶ × 2000 × 1 = 2.51327 × 10⁻³ T. The flux per turn is Φ = BA = 2.51327 × 10⁻³ × 3.14159 × 10⁻⁴ = 7.8957 × 10⁻⁷ Wb. Flux linkage is NΦ = 7.8957 × 10⁻⁵ Wb, and dividing by the 1 A gives 78.957 µH — the same answer, arrived at through the field rather than through the compact formula.

The scaling laws follow directly from where each symbol sits. N is squared, so doubling the turns to 200 gives 315.83 µH, four times as much. The area is proportional to d², so doubling the diameter to 40 mm also quadruples the inductance. Length is in the denominator, so doubling it to 100 mm halves the result. And μᵣ is a plain multiplier: a core of μᵣ = 125 turns 78.96 µH into 9869.6 µH, or 9.87 mH.

The honest limitation is the ideal-solenoid assumption. The derivation used B = μ₀nI, which holds strictly only for an infinitely long coil where the field is uniform throughout and zero outside. A finite coil's field weakens and spreads near the ends, so its actual flux linkage — and therefore its inductance — is lower than this calculation gives. The error grows as the coil gets shorter relative to its diameter, which is why the page prints the length-to-diameter ratio and flags anything under 10. It does not apply a numeric correction, and it deliberately quotes no percentage, because the correction factor could not be sourced from a primary reference for this page and a fabricated one would be worse than a clearly labelled upper bound.

Also outside the model: wire diameter and insulation thickness, winding pitch, multiple layers, the difference between mean and inner diameter, inter-winding capacitance, skin and proximity effects, and any frequency dependence at all. Real inductor datasheets quote an A_L value per core precisely because the geometric formula stops being adequate as soon as a core is involved.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax (Rice University), University Physics Volume 2, §14.2 'Self-Inductance and Inductors'. Primary source for the equation implemented here, Equation 14.13: 'L_solenoid = μ₀N²A/l', where μ₀ is the permeability of free space, N the number of turns, A the cross-sectional area and l the length. Also the source of the definition of self-inductance as flux linkage per unit current and of the henry as one volt-second per ampere. Note that this section contains no worked example applying the geometric formula, so the worked example on this page is computed here and cross-checked by the independent flux route. Retrieved 2026-07-29.
2. [2]NIST, CODATA Internationally Recommended 2022 Values of the Fundamental Physical Constants — vacuum magnetic permeability μ₀. Source of the constant used here: value '1.256 637 061 27 x 10⁻⁶ N A⁻²', standard uncertainty '0.000 000 000 20 x 10⁻⁶ N A⁻²', relative standard uncertainty '1.6 x 10⁻¹⁰'. Since the 2019 revision of the SI, μ₀ is a measured quantity rather than exactly 4π × 10⁻⁷. Retrieved 2026-07-29.
3. [3]Engineering LibreTexts / James M. Fiore, Introduction to Circuit Analysis, §6.2.2 'Inductance and Inductors'. Independent second authority for the same relation, written as 'L = μ(AN²)/l, where L is inductance in henries, μ is core material permeability, A is cross-sectional area, N is number of coils/turns, and l is coil length' — which makes explicit that the μ in the compact form is the absolute permeability μ₀μᵣ. Retrieved 2026-07-29.