Formula of Mutual Inductance: M = N₂Φ₂/I₁, Henry, Examples

The formula of mutual inductance is $M = \frac{N_2 \Phi_2}{I_1}$ : the total flux linked with a second coil, divided by the current in the first coil that produces it. When that current changes, the second coil gets an induced emf $\varepsilon_2 = -M\frac{dI_1}{dt}$ . The SI unit of M is the henry (H).

Mutual inductance describes how strongly two circuits are magnetically coupled. It depends only on the coils, their turns, sizes, spacing and the core between them, not on the current. Transformers, wireless phone chargers and induction cooktops all depend on it. The single coil version, where a coil acts on itself, is the self inductance formula. Both belong to electromagnetic induction in the electricity and magnetism formulas section.

Quick facts

Definition
$M = \frac{N_2 \Phi_2}{I_1}$
Induced emf
$\varepsilon_2 = -M\frac{dI_1}{dt}$
Coaxial solenoids
$M = \frac{\mu_0 N_1 N_2 A}{l}$
SI unit
henry (H) = Wb/A = V s/A
Dimensional formula
[M¹ L² T⁻² I⁻²]
Coupling
$M = k\sqrt{L_1 L_2}$ , with 0 ≤ k ≤ 1
Symmetry
$M_{12} = M_{21}$

Explaining the formula of mutual inductance

Symbols for two coupled coils
Symbol	Meaning	Unit
$M$	Mutual inductance	H
$I_1$	Current in the primary (first) coil	A
$N_2$	Turns on the secondary (second) coil	no unit
$\Phi_2$	Flux through each secondary turn, due to $I_1$	Wb
$\frac{dI_1}{dt}$	Rate of change of primary current	A/s

The product $N_2\Phi_2$ is the flux linkage of the secondary. Doubling the primary current doubles that linkage, so their ratio M stays fixed. The flux idea is developed on the magnetic flux formula page. The minus sign in the emf form is Lenz's law: the secondary current flows so as to oppose the change in flux.

Deriving M for two coaxial solenoids

Wind a secondary of $N_2$ turns tightly over a long primary solenoid of length l, area A and $N_1$ turns. The primary field inside is

$B_1 = \mu_0 \frac{N_1}{l} I_1$

Each secondary turn encloses $\Phi_2 = B_1 A$ , so the linkage is $N_2 \mu_0 \frac{N_1}{l} I_1 A$ . Divide by $I_1$ :

$M = \frac{\mu_0 N_1 N_2 A}{l}$

With an iron core of relative permeability $\mu_r$ , multiply by $\mu_r$ . The Class 12 syllabus asks you to know this derivation. The field of a solenoid is covered on the magnetic field formula page.

Mutual inductance examples

Example 1: from flux and current

A current of 1.5 A in a primary coil sets up a flux of 3.0 × 10⁻⁵ Wb through each of the 200 turns of a nearby coil. Find M.

$M = \frac{N_2\Phi_2}{I_1} = \frac{200 \times 3.0 \times 10^{-5}}{1.5} = \frac{6.0 \times 10^{-3}}{1.5} = 4.0 \times 10^{-3}\ \text{H} = 4.0\ \text{mH}$

Example 2: induced emf, with a time in ms

For the same pair, the primary current falls from 2.0 A to zero in 5.0 ms. Find the average emf induced in the secondary.

Convert: Δt = 5.0 × 10⁻³ s, so $\frac{\Delta I_1}{\Delta t} = \frac{2.0}{5.0 \times 10^{-3}} = 400\ \text{A/s}$ .

$|\varepsilon_2| = M\frac{\Delta I_1}{\Delta t} = (4.0 \times 10^{-3})(400) = 1.6\ \text{V}$

Example 3: coaxial solenoids

A primary of 1000 turns, 50 cm long and 10 cm² in cross section, carries a 400 turn secondary wound over it. Find M.

Convert: l = 0.50 m, A = 1.0 × 10⁻³ m².

$M = \frac{(4\pi \times 10^{-7})(1000)(400)(1.0 \times 10^{-3})}{0.50} = 1.0 \times 10^{-3}\ \text{H}$

Factors that change mutual inductance

Turns: M grows with $N_1 N_2$ .
Area: a larger shared cross section links more flux.
Distance and angle: moving the coils apart, or turning their axes at right angles, cuts the shared flux and lowers M.
Core: soft iron raises M by the factor $\mu_r$ .

The coupling coefficient k ties M to the two self inductances: $M = k\sqrt{L_1 L_2}$ . Perfect coupling, with every field line of one coil passing through the other, gives k = 1. Loosely placed coils have k much less than 1.

Unit and dimensions

From $\varepsilon = M\,dI/dt$ , one henry is the mutual inductance that gives 1 V in the secondary when the primary current changes by 1 A each second. Hence 1 H = 1 V s/A = 1 Wb/A.

$[M] = \frac{[\text{emf}][T]}{[I]} = [M L^2 T^{-2} I^{-2}]$

Self inductance has exactly the same bracket, worked through on its own dimensions page. Do not mix up the symbol M for mutual inductance with the M for mass in the bracket.

Exam traps

Using N₁ in the definition. The turns that count are on the coil where the flux is linked, the secondary.
Expecting emf from steady current. A constant primary current induces nothing; only a change does.
Area in cm². 10 cm² is 1.0 × 10⁻³ m².
Length in cm. Put l in metres in the solenoid formula.

M has the same value whichever coil you call the primary:

M_{12} = M_{21}

Frequently asked questions

What is the formula of mutual inductance?

M = N₂Φ₂ / I₁, the flux linkage of the secondary per ampere in the primary. The emf it produces is ε₂ = −M dI₁/dt.

Which unit measures mutual inductance?

The henry, H. A pair of coils has M = 1 H when 1 A in the primary links 1 Wb with the secondary, so 1 H = 1 Wb/A = 1 V s/A.

What is the mutual inductance of two coaxial solenoids?

For a secondary wound over a long primary, M = μ₀N₁N₂A / l, multiplied by μᵣ if there is a magnetic core.

What is the difference between self and mutual inductance?

Self inductance is a coil's emf in response to its own changing current. Mutual inductance is the emf in one coil caused by a changing current in another coil.

Does mutual inductance depend on current?

No. It depends on the geometry of the coils and the core material. Changing the current changes the flux, but the ratio M stays the same.