Mutual Inductance

Mutual inductance is the phenomenon by which a changing current in one electrical circuit induces an electromotive force (emf) in a neighbouring circuit, without any direct electrical connection between them. This coupling occurs through the magnetic field: when current in the first coil (the primary) changes, its surrounding magnetic flux changes, and if that changing flux threads through a second coil (the secondary), Faraday's law demands an induced emf in the secondary. The effect is the foundation of transformers, wireless charging pads, and many sensor technologies. Think of two people on a trampoline. When one person bounces, they create waves in the trampoline surface that make the other person move — even though they never touch directly. In mutual inductance, the magnetic field is the 'trampoline surface': disturbances created by a changing current in coil 1 travel through space and influence coil 2. The stronger the coupling (more shared flux), the larger the induced emf for the same rate of current change. Mutual inductance M is measured in henries (H), named after Joseph Henry. A mutual inductance of 1 H means that a current changing at 1 A/s in the primary coil will induce exactly 1 V in the secondary coil. In practice, M depends on the geometry of both coils, their separation, their relative orientation, and the magnetic permeability of any material between them. Understanding M is essential for designing efficient power transformers and for analysing interference between nearby circuits.

Key takeaways
  • Current Creates a Magnetic Field
  • Magnetic Flux Links the Secondary Coil
  • Faraday's Law Produces an Induced emf
  • Reciprocity: M₁₂ = M₂₁
  • Coupling Coefficient k

Current Creates a Magnetic Field

Any current-carrying conductor produces a magnetic field around it (Ampère's law). For a coil of wire, this field is concentrated and directed along the coil's axis, forming a solenoid-like dipole pattern. The field strength is proportional to the current I₁ flowing through the primary coil.

The magnetic field lines from the primary spread through space. Some of those field lines pass through the area enclosed by the secondary coil. The total magnetic flux threading the secondary is Φ₂₁ = M · I₁, where M captures how efficiently the geometry of the two coils shares flux. If the coils are coaxial and close together, a large fraction of flux links them; if they are far apart or perpendicular, almost none does.

Faraday's Law Produces an Induced emf

Faraday's law states that the emf in any loop equals the negative rate of change of magnetic flux through it: ε = −dΦ/dt. Since Φ₂₁ = M·I₁, differentiating with respect to time gives ε₂ = −M(dI₁/dt). The minus sign (Lenz's law) tells us the induced emf opposes the change that caused it — if the primary current is increasing, the secondary emf drives a current that would create a field opposing the growing primary flux.

Reciprocity: M₁₂ = M₂₁

A beautiful and non-obvious result proved using the Neumann formula is that mutual inductance is symmetric: the emf induced in coil 2 due to changing current in coil 1 is governed by the same M as the emf induced in coil 1 due to changing current in coil 2. You don't need two different constants — the geometry determines one single value M for the pair of coils, regardless of which one is 'primary' and which is 'secondary'.

Coupling Coefficient k

In real situations, not all flux from coil 1 reaches coil 2 — some flux 'leaks' into the surrounding space. The coupling coefficient k = M / √(L₁L₂) quantifies how well the coils are magnetically linked, where L₁ and L₂ are their self-inductances. k ranges from 0 (no coupling) to 1 (perfect coupling, all flux shared). Ideal transformers assume k = 1; real transformers achieve k ≈ 0.95–0.99 using iron cores to guide the flux.

The Key Equation

ε2=−MdI1dt\varepsilon_2 = -M \frac{dI_1}{dt}
  • \varepsilon_2 — Induced emf in secondary (V (volts)): The electromotive force appearing across the terminals of the secondary coil due to flux change.
  • M — Mutual inductance (H (henries)): A geometric constant describing how effectively the two coils share magnetic flux. Depends on shape, size, separation, and orientation of both coils.
  • \frac{dI_1}{dt} — Rate of change of primary current (A/s (amperes per second)): How rapidly the current in the primary coil is changing with time. A steady current produces no induced emf.

Worked Example

Problem. Two coils are arranged so that their mutual inductance is M = 0.040 H. The current in the primary coil increases linearly from 0 A to 6.0 A in 0.30 s. (a) What is the magnitude of the induced emf in the secondary coil during this interval? (b) If the secondary has 200 turns and behaves like a solenoid, what is the average induced voltage per turn?

∣ε2∣=M ΔI1Δt=0.040×20=0.80 V|\varepsilon_2| = M\,\frac{\Delta I_1}{\Delta t} = 0.040 \times 20 = 0.80\text{ V}

Answer. The induced emf in the secondary is 0.80 V; the average voltage per turn is 4.0 mV.

Figure
Diagram
Mutual Inductance Between Two CoilsPrimary Coil (N₁)Current I₁(t)AC sourceMagnetic flux ΦSecondary Coil (N₂)Induced ε₂Vε₂ = −M (dI₁/dt)   M = k√(L₁L₂)dI₁/dt ≠ 0
Origins · 1 minJoseph Henry and how this was discovered
JH
Joseph Henry
1797–1878 · American

Joseph Henry independently discovered electromagnetic induction at around the same time as Michael Faraday (1831), and discovered the phenomenon of self-inductance. He performed meticulous experiments showing that a changing current in one wire could induce a current in a completely separate wire — the direct discovery of mutual inductance. The unit of inductance, the henry (H), is named in his honour. Henry built one of the earliest practical electric motors and an early version of the electric telegraph. His failure to publish his results promptly meant Faraday received primary credit for electromagnetic induction — a painful lesson in the importance of timely scientific communication.

  • Self-Inductance — Self-inductance describes how a coil opposes changes in its own current via its own magnetic flux — the single-coil analogue of mutual inductance. Understanding L is essential before tackling M.
  • Faraday's Law of Electromagnetic Induction — Mutual inductance is a direct application of Faraday's law: ε = −dΦ/dt. Mastering Faraday's law gives you the theoretical backbone behind why any induced emf appears.
  • Transformers — Transformers are the most important practical application of mutual inductance, using tightly coupled coils (k ≈ 1) to step AC voltages up or down according to the turns ratio.
  • Lenz's Law — The negative sign in ε₂ = −M(dI₁/dt) is Lenz's law: the induced emf always opposes the change producing it. Lenz's law determines the direction of induced currents in the secondary.