Solenoid Inductance Calculator — L = µ₀µᵣN²A/l
Calculate the self-inductance of a solenoid coil using the long-solenoid formula L = µ₀µᵣN²A/l. Enter the number of turns, coil length and diameter, and the relative permeability of the core. Optionally enter the current to also compute stored magnetic energy and flux linkage.
cm
cm
A
Calculated from L = µ₀ × µᵣ × N² × A / l (long solenoid approximation)
- 1
Coil radius (m)
0.03 ÷ 2 = 0.015 - 2
Cross-sectional area (m²)
π × 0.015² = 0.00070686 - 3
Inductance (µH)
µ₀ × 1 × 200² × 0.00070686 ÷ 0.15 × 10⁶ = 236.871µ₀ = 4π×10⁻⁷ H/m is the permeability of free space.
How does this calculator work?
L (H) = µ₀ × µᵣ × N² × π(d/2)² / l. With µ₀ = 4π × 10⁻⁷ H/m, a 200-turn air-core coil 15 cm long and 3 cm in diameter gives about 71 µH. Doubling turns quadruples L; inserting a µᵣ = 100 ferrite core multiplies it by 100. Stored energy = ½LI² joules. The formula is the long-solenoid approximation (accurate when l/d > 5).
Formula
How this is calculated
A solenoid is a tightly wound helical coil that produces a nearly uniform magnetic field inside. Its self-inductance — the ratio of magnetic flux linkage to current — is given by L = µ₀ × µᵣ × N² × A / l, where µ₀ = 4π × 10⁻⁷ H/m is the permeability of free space, µᵣ is the relative permeability of the core material, N is the total number of turns, A is the cross-sectional area (π r²), and l is the physical length of the coil.
The formula is the ideal long-solenoid approximation and is accurate when the length is much greater than the diameter (aspect ratio l/d > ~5). For shorter or wider coils, fringe effects at the ends increase actual inductance above the formula; the Nagaoka coefficient correction brings it within 1–2% for any aspect ratio, but for most practical coil design purposes the long-solenoid formula is sufficient.
Core material is the most powerful design variable: an iron core with µᵣ = 1 000 increases inductance by a factor of 1 000 over an air-core coil with the same geometry. Practical ferrite cores span µᵣ ≈ 10–3 000 depending on composition and frequency; consult the core datasheet. Inductance scales with N² — doubling the turns quadruples inductance — and linearly with cross-sectional area but inversely with length. Stored magnetic energy ½LI² and flux linkage NΦ = LI are also computed if you enter the operating current.
Frequently asked questions
The formula L = µ₀µᵣN²A/l assumes the coil is much longer than its diameter, so the magnetic field is uniform inside and negligible outside. When l/d < 2–3 the fringing at the ends introduces significant error. The Nagaoka coefficient (a tabulated correction factor between 0 and 1) can restore accuracy, but for l/d > 5 the error is less than 1%.
Inserting a ferromagnetic core multiplies inductance by the relative permeability µᵣ of the material. Air-core coils have µᵣ = 1. Ferrite cores used in RF inductors are typically µᵣ = 10–3 000; silicon steel used in power transformers can reach µᵣ = 1 000–10 000 at low flux densities. At high current the core saturates, µᵣ drops sharply, and inductance collapses — the saturation current must stay below the core's rated value.
The output is given in microhenries (µH), millihenries (mH), and henries (H) simultaneously, since practical values vary enormously — a small RF choke might be 1–100 µH, while a power-line filter inductor could be 1–100 mH. Stored energy is in microjoules (µJ) and flux linkage in microweber-turns (µWb).
Also known as
TG we-Calculate Editorial Team. (2026). Solenoid Inductance Calculator — L = µ₀µᵣN²A/l [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/solenoid-inductance-calculator
TG we-Calculate Editorial Team. "Solenoid Inductance Calculator — L = µ₀µᵣN²A/l." TG we-Calculate. 2026. https://we-calculate.com/calculator/solenoid-inductance-calculator.
TG we-Calculate Editorial Team, "Solenoid Inductance Calculator — L = µ₀µᵣN²A/l," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/solenoid-inductance-calculator
@misc{wecalculate_solenoid_inductance_calculator, title = {Solenoid Inductance Calculator — L = µ₀µᵣN²A/l}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/solenoid-inductance-calculator}}, year = {2026}, note = {TG we-Calculate} }
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