Formula of Self Inductance: L = NΦ/I, Unit Henry, Examples

The formula of self inductance is $L = \frac{N\Phi}{I}$ : the total flux linkage of a coil, divided by the current that creates it. A coil with a large L sets up a lot of flux for each ampere, and so it fights hard against any change in that current.

The same quantity appears in the induced emf rule $\varepsilon = -L\frac{dI}{dt}$ , which is how many exam questions define it. For a long solenoid you can also calculate it from the shape alone. This page explains all three forms, with the henry as the unit, and links them to other electricity and magnetism formulas.

Quick facts

Defining formula
$L = \frac{N\Phi}{I}$
Emf form
$\varepsilon = -L\frac{dI}{dt}$
Long solenoid
$L = \frac{\mu_0 N^2 A}{l}$
SI unit
henry (H) = Wb/A = V s/A
Dimensional formula
[M¹ L² T⁻² I⁻²]
Type
Scalar, always positive
Stored energy
$U = \frac{1}{2}LI^2$

Reading the formula of inductance term by term

Symbols used for self inductance
Symbol	Meaning	SI unit
$L$	Self inductance (also called the coefficient of self induction)	H
$N$	Number of turns in the coil	no unit
$\Phi$	Flux through a single turn	Wb
$N\Phi$	Flux linkage of the whole coil	Wb
$I$	Current flowing in the same coil	A
$\frac{dI}{dt}$	Rate at which that current changes	A/s

Flux linkage grows in step with the current, so their ratio is a fixed number for a given coil. That number depends on the turns, the size, and the core material, not on how much current flows. An iron core raises L a great deal because it multiplies the flux. Note that this L is inductance and must not be confused with the length l of a solenoid.

The emf definition and Lenz's law

When the current in a coil changes, its own flux changes, and Faraday's law induces an emf in the same coil:

$\varepsilon = -\frac{d(N\Phi)}{dt} = -L\frac{dI}{dt}$

The minus sign is Lenz's law at work. The induced emf opposes the change, so it pushes back when current rises and tries to keep it flowing when current falls. This gives a second definition: a coil has a self inductance of 1 H if a current change of 1 A every second induces 1 V across it. Because of this back emf, an inductor is often called the electrical version of inertia. The electromotive force formula page covers emf in general.

Practice problems using the formula of self inductance

Example 1: from flux linkage

A coil of 200 turns carries 2.0 A. The flux through each turn is $5.0 \times 10^{-4}$ Wb. Find L.

$L = \frac{N\Phi}{I} = \frac{200 \times 5.0 \times 10^{-4}}{2.0} = 0.050\ \text{H}$

That is 50 mH.

Example 2: a solenoid with area in square centimetres

An air cored solenoid has 500 turns, length 25 cm and cross section 4.0 cm². Find its self inductance.

Convert: l = 0.25 m, A = 4.0 cm² = $4.0 \times 10^{-4}$ m².

$L = \frac{\mu_0 N^2 A}{l} = \frac{(4\pi \times 10^{-7})(500)^2(4.0 \times 10^{-4})}{0.25} \approx 5.0 \times 10^{-4}\ \text{H}$

So L is about 0.50 mH.

Example 3: back emf when a switch opens

The current in a 0.20 H coil drops from 3.0 A to zero within 0.040 s. What average emf appears across it?

$|\varepsilon| = L\frac{\Delta I}{\Delta t} = 0.20 \times \frac{3.0}{0.040} = 15\ \text{V}$

The emf acts to keep the current flowing, which is why switches on large coils can spark.

Self inductance of a long solenoid

Inside a long solenoid the field is $B = \mu_0 n I$ , with $n = N/l$ turns per metre. The flux through one turn is $BA$ , so

$L = \frac{N \cdot \mu_0 \frac{N}{l} I \cdot A}{I} = \frac{\mu_0 N^2 A}{l}$

The current cancels, as it should. Doubling the turns makes L four times larger, because N enters twice: once in the field and once in the linkage. With a core of relative permeability $\mu_r$ , replace $\mu_0$ with $\mu_0\mu_r$ . For the field inside, see the magnetic field formula, and for flux itself the magnetic flux formula.

Two coils linked by flux need mutual inductance M as well; see the mutual inductance formula page for that case.

Henry, energy and the dimensional formula

From $L = \varepsilon / (dI/dt)$ , one henry is one volt second per ampere. Writing the volt in base units, [M L² T⁻³ I⁻¹], and multiplying by [T] over [I] gives [M¹ L² T⁻² I⁻²]. Each line of that working is laid out under dimensional formula of self inductance.

A current carrying inductor stores energy in its magnetic field: $U = \frac{1}{2}LI^2$ . For the coil in Example 1, $U = \frac{1}{2}(0.050)(2.0)^2 = 0.10$ J. A quick check: H times A² is (V s/A)(A²) = V A s = W s = J, so the units agree. In AC circuits the same L gives an inductive reactance $X_L = \omega L$ .

Common slips with inductance

Dropping N. Φ in the formula is the flux through one turn. Without N you get the wrong flux linkage.
Forgetting to square N in the solenoid formula.
Leaving area in cm². Remember that one square centimetre equals 10⁻⁴ m².
Thinking L depends on current. For an air core it does not. A bigger current gives bigger flux, and the ratio stays fixed.
Ignoring the minus sign in words. When asked for the direction, say the emf opposes the change in current.

Frequently asked questions

What is the formula of self inductance?

Self inductance is L = NΦ / I, the flux linkage per unit current. It also appears as ε = −L dI/dt, and for a long solenoid L = μ₀N²A / l.

What is the formula of inductance for a coil with a core?

Multiply the air value by the relative permeability of the core: L = μ₀μᵣN²A / l. Soft iron cores can raise the inductance hundreds of times.

Which SI unit measures self inductance?

The henry, symbol H. A henry is a weber of linkage for every ampere, the same as a volt second per ampere. Small coils are often rated in millihenry (mH) or microhenry (μH).

Is self inductance a scalar or a vector?

Self inductance is scalar: no direction, just a positive size fixed by the coil. The minus sign in ε = −L dI/dt belongs to the emf, not to L.

What is the difference between self inductance and mutual inductance?

Self inductance links a coil to changes in its own current. Mutual inductance links one coil to changes of current in a nearby coil. Both are measured in henry.