The solenoid inductance calculator above works out how much inductance a cylindrical coil has, from the four things that actually determine it: how many turns are on it, how long the winding is, how wide it is, and what is inside it. It also applies the correction that most inductance calculators leave out, which matters because the textbook formula assumes an infinitely long coil and no coil you will ever wind is one.
Arb Digital builds free physics calculators that state their assumptions instead of hiding them. This page shows the raw ideal-solenoid figure and the corrected one side by side, prints the correction factor itself, and explains the range over which that correction is reliable. If you are winding a coil for a tuned circuit, the difference between the two numbers is the difference between hitting your resonant frequency and missing it.
What This Solenoid Inductance Calculator Does
Inductance is the constant that relates the magnetic flux a coil links to the current flowing through it. A coil with a large inductance opposes changes in current strongly; one with a small inductance barely notices them. For a solenoid the inductance depends only on geometry and the core material, not on the current, which is why a single number characterises the coil.
The tool returns the inductance in whichever of nanohenries, microhenries, millihenries or henries suits the magnitude, and scales automatically so a small RF coil and a large mains choke both read sensibly. The grid gives the cross-sectional area, the length correction factor being applied, the turn density in turns per metre, and the inductive reactance at a frequency you choose — because in practice the reactance is what your circuit sees.
The correction factor is the item worth watching. It is 1 for a hypothetical infinite coil and falls towards zero as the coil gets shorter relative to its diameter. At the defaults it sits near 0.85, meaning the uncorrected formula would have overstated the inductance by about 18 per cent.
How to Use It
- Count the turns carefully. Inductance scales with N squared, so a miscount of five turns in a hundred moves the answer by about ten per cent.
- Measure the wound length, not the former. The length in the formula is the distance the winding actually occupies. A coil wound over half a former has half the length, not the full one.
- Use the mean diameter. Measure to the centre of the wire, which is the former diameter plus one wire diameter. On a fine winding the difference is negligible; on heavy wire it is not.
- Set the core permeability honestly. Leave it at 1 for air, plastic or a copper former. For ferrite, a partly inserted or ungapped-versus-gapped core makes the effective figure very different from the material's bulk value.
- Compare the two models. Switch between the corrected and ideal settings to see how much the finite length is costing you. On a coil longer than about five diameters the two agree closely.
The Formula: How Solenoid Inductance Is Calculated
For a long solenoid the inductance is L = μ0 μr N² A ÷ l, where μ0 is the vacuum permeability, μr is the relative permeability of the core, N is the number of turns, A is the cross-sectional area of the coil and l is its wound length. OpenStax University Physics Volume 2, section 14.2 on self-inductance and inductors, derives exactly this result by finding the field inside the solenoid, computing the flux through one turn and applying L = NΦ ÷ I.
The vacuum permeability is not the round 4π × 10−7 it used to be by definition. Since the 2019 redefinition of the SI base units it is a measured quantity, and NIST's CODATA value for the vacuum magnetic permeability gives 1.25663706127 × 10−6 N/A², which is what this tool uses. The difference from the old exact value is about two parts in ten billion, so it changes nothing practical, but it is the correct constant.
The finite-length correction multiplies that result by k = l ÷ (l + 0.9r), where r is the coil radius. This is Wheeler's classic approximation, and it is within about one per cent of the exact Nagaoka coefficient for coils longer than about 0.8 of their radius. Below that it degrades, and for a very flat pancake winding a different formula entirely is required.
Work the defaults through. A hundred turns over 50 mm on a 20 mm diameter gives an area of π × 0.01² = 3.1416 × 10−4 m². The ideal inductance is 1.2566 × 10−6 × 10,000 × 3.1416 × 10−4 ÷ 0.05 = 78.96 µH. The correction factor is 0.05 ÷ (0.05 + 0.9 × 0.01) = 0.8475, so the corrected inductance is 66.9 µH. At 1 MHz that is a reactance of 2π × 106 × 66.9 × 10−6 = 420 Ω.
Why the Ideal Formula Always Reads High
The derivation of the ideal formula assumes every field line that goes up the middle of the coil comes back outside it, far away, and that the field inside is uniform right to the ends. Neither is true for a real coil. Near the ends the field lines spread out and some of them leave through the side of the winding without linking every turn, so the flux linkage is less than the ideal calculation assumes. Less flux linkage means less inductance.
The shorter the coil relative to its diameter, the larger the fraction of it that is near an end, and the worse the overestimate gets. For a coil ten diameters long the error is a couple of per cent. For one a single diameter long it is over 30 per cent. Because the error is always in the same direction, an uncorrected calculation will consistently give you a coil that resonates high, which is a frustrating way to lose an afternoon.
What This Calculator Deliberately Does Not Model
It is a geometry calculation, and there are several real effects it leaves out on purpose. It does not model self-capacitance between adjacent turns, which is what gives every coil a self-resonant frequency above which it stops behaving as an inductor at all. It does not model skin effect or proximity effect, which raise the effective resistance at high frequency and lower the Q. It does not model core saturation, which makes a ferrite-cored inductance current-dependent in a way the formula assumes it is not.
It also assumes a single-layer winding. A multi-layer coil has a larger mean diameter for the outer layers and different mutual coupling between them, and its inductance is higher than this formula predicts for the same turn count. Treat a multi-layer result as an underestimate rather than an answer.
Core permeability deserves particular caution. A ferrite rod pushed partway into a coil does not multiply the inductance by the material's bulk μr; the effective permeability depends on the rod's length-to-diameter ratio and how much of the winding it occupies, and it is usually a small fraction of the bulk figure. OpenStax University Physics Volume 2, section 12.7 on magnetism in matter, covers why permeability behaves the way it does in real materials.
Where This Sits Next to the Other Magnetics Tools
Arb Digital publishes several tools around coils and magnetic fields, and they answer genuinely different questions. This page returns an inductance, a property of the coil that exists whether or not any current is flowing. The solenoid magnetic field calculator returns a flux density inside the coil, which depends on the current and vanishes when you switch it off. The magnetic field of a wire calculator covers the field around a single straight conductor, where the field falls off as 1/r rather than being roughly uniform.
Downstream, the inductor energy calculator turns an inductance and a current into stored energy, the LC resonant frequency calculator pairs it with a capacitance to find the tuned frequency, and the reactance calculator and RLC impedance calculator take it into full circuit analysis. If you need to move a value between henries and its submultiples, the inductance converter does that job.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using the ideal formula on a short coil — it overstates the inductance every time, and by more than 30 per cent when the coil is about as long as it is wide.
- Entering the former length instead of the wound length — the formula wants the length the turns actually occupy, and a half-filled former gives half the length.
- Using a core's bulk permeability for a partly inserted rod — the effective permeability of an open magnetic path is far lower than the material figure, often by a factor of ten or more.
- Ignoring self-resonance — above its self-resonant frequency a coil looks capacitive, and no inductance formula describes it usefully there.
- Applying it to a multi-layer winding — the single-layer formula underestimates a multi-layer coil, sometimes badly, because the outer layers sit at a larger diameter.
Related Free Tools From Arb Digital
Pair this with the solenoid magnetic field calculator for the field the same coil produces, and the magnetic field of a wire calculator for a straight conductor. Take the inductance forward with the LC resonant frequency calculator, the inductor energy calculator and the reactance calculator. For the wire itself, the wire resistance calculator gives the DC resistance that sets the coil's Q, and the inductance converter handles units. The full set is on the free online tools hub.
Frequently Asked Questions
Each turn contributes to the field, and each turn also links the field that all the others produce. Doubling the turns doubles the field and doubles the number of turns linking it, so the flux linkage rises by a factor of four. That is why adding turns is such a powerful lever compared with changing the diameter.
It multiplies the ideal inductance by l divided by (l plus 0.9r), correcting for the flux that escapes near the ends of a finite coil. Use it for any coil shorter than about five diameters, which is most practical coils. It is within roughly one per cent of the exact Nagaoka value for coils longer than about 0.8 of their radius.
Only through the geometry it forces. Thicker wire means a larger mean coil diameter for the same former and a longer winding for the same turn count, and both change the answer. The wire size does not enter the formula directly, but it does determine the resistance and therefore the coil's Q.
No. A toroid confines almost all its flux inside the core and has a completely different inductance formula based on the mean magnetic path length. Applying the solenoid formula to a toroid will not give a meaningful answer.
The usual causes are a shorter wound length than assumed, a spaced rather than close winding, or measurement at a frequency close to self-resonance. A measurement made near self-resonance reads high rather than low, so a low reading almost always points at the geometry instead.
It multiplies the inductance by the effective relative permeability, which for a closed core can be in the thousands. For an open rod inside a coil the effective figure is far lower than the material's bulk value, because the return path through air dominates the reluctance of the magnetic circuit.
Not for an air core. Inductance is pure geometry there, and the same at one milliamp as at ten amps. With a ferromagnetic core it does depend on current, because approaching saturation reduces the effective permeability and therefore the inductance.
This tool is provided for educational and preliminary design use. It models a single-layer solenoid and ignores self-capacitance, skin effect, proximity effect and core saturation, so treat its output as a starting point for a coil design rather than a final value.
