Solenoid Inductance

Equivalent forms

L = \mu_0 n^2 A \ell

B = \mu_0 n I

Inductance reduces to pure geometry times a constant of nature — N²A/ℓ scaled by μ₀.

Symbol	Name	SI unit	Dimension	Range
$L$	Inductanceoutput Solenoid self-inductance	H	M·L²·T⁻²·I⁻²	1e-9 – 1
$N$	Turns Number of wire turns	(dimensionless)	1	1 – 5000
$A$	Cross-section Coil cross-sectional area	m²	L²	0.00001 – 0.01
$\ell$	Length Solenoid length	m	L	0.01 – 1
$\mu_0$	Permeability Permeability of free space	H/m	M·L·T⁻²·I⁻²	0.00000125663706212 – 0.00000125663706212

Unit systems

Symbol	SI
$L$	H
$N$	(dimensionless)
$A$	m²
$\ell$	m
$\mu_0$	H/m

Where it holds

Accurate for long, tightly wound solenoids (length ≫ diameter) where the interior field is uniform and end fringing is negligible. Short or loosely wound coils need Nagaoka's correction factor.

Dimensional analysis

$\mu _{0} N^{2} A$ /ℓ $= (H/m)\cdot (m^{2}/m) = H \checkmark$

Discovery

Joseph Henry · 1832

Henry's powerful multi-turn electromagnets and coils led him to the self-inductance of a solenoid; the geometry-only formula falls straight out of Ampère's law for the interior field.

Try this

Wind more turns and a coil's inductance doesn't just grow — it grows with the SQUARE of the turns. Why squared?

A 500-turn air-core solenoid is 20 cm long with a 2 cm² cross-section. Find its inductance.

Research status: stable

Real-world applications

Designing chokes and RF coils
Relay and contactor coils
Wireless-charging transmit coils
Inductive sensors

Common misconceptions

Inductance scales linearly with turns — it scales with $N^{2}$
Adding more wire always helps — stretching the same turns over a longer ℓ actually lowers L

Experimental verification

Measure L of solenoids with varying N (fixed geometry) on an LCR meter; L plots as a clean parabola in N, and inserting a ferrite core multiplies it

by \mu _r

.

Derivation

Inside a long solenoid Ampère's law gives

B = \mu _{0}(N

/ℓ)I.

Flux per turn

\Phi = BA

, total linkage

\Lambda = N\Phi = \mu _{0}N^{2}A I

/ℓ.

Since

\Lambda = LI

,

L = \mu _{0}N^{2}A

/ℓ — the

N^{2}

comes from N turns each linking N-fold flux.

Limiting cases

ℓ → large (very long coil)⟶

L \to

small per turnStretching a coil thins the field; the long-solenoid formula is the ideal limit where end effects vanish.

iron core inserted⟶

L \to \mu _r \times

largerReplacing

\mu _{0}

with

\mu _{0}\mu _r

boosts L by the relative permeability, often hundreds-fold.

What if…

What if you double the turns?

Inductance quadruples — the defining $N^{2}$ behavior of a coil.

What if you slide in an iron core?

L jumps by the core's relative permeability $\mu _r$ (often $\times 100$ –1000), which is why transformers and chokes use cores.

1

500-turn air solenoid

Given ·

N:: 500
A:: 0.0002
\ell:: 0.2
\mu 0:: 0.00000125663706212

Find · L

Steps

$N^{2} = 250000$ ; $N^{2}A = 250000\times 2\times 10^{-4} = 50$
$\mu _{0}N^{2}A = 1.257\times 10^{-6} \times 50 = 6.28\times 10^{-5}$
$L = 6.28\times 10^{-5}/0.2 \approx 3.14\times 10^{-4} H \approx 0.31\,\mathrm{mH}$

Result · About 0.31 mH.

Self-Inductance→field solenoidenergy in inductormagnetic flux