Mutual Inductance

Relates current change in one coil to induced emf in another coil. Theory, formulas, worked examples and interactive graphs.

Conceptual context

Mutual inductance appears when the magnetic field created by one coil links another coil. Physical proximity is not enough: there must be common flux effectively crossing the receiving circuit. Two coils may be near each other and still couple weakly if their axes, cores, or field paths do not favor linkage.

This leaf explains how a variation of $I_1$ produces an emf $\varepsilon_2$ in another coil, what role $M$ plays, and why coefficient $k$ separates coil size from coupling quality. It is needed for transformers, inductive sensors, ignition coils, and contactless energy transfer.

[!NOTE] Mutual inductance does not describe an isolated coil. It describes a magnetic relation between two coils.

🟢 Essential level

A primary coil carrying current creates a magnetic field. If part of that field crosses a secondary coil, flux linkage $\lambda_{21}$ appears. While primary current stays steady, that flux may exist without producing sustained emf. The new phenomenon begins when $I_1$ changes.

The secondary coil responds to the flux change with an emf $\varepsilon_2$. That emf does not require the secondary to carry a closed current; it can be measured even with an open circuit. The important point is change of magnetic linkage, not electrical contact between coils.

A simple way to see it is to imagine a transmitting coil and a receiving coil. The transmitter decides how the field changes; the receiver responds only to the part of that field that actually crosses it. If that shared part is small, induction is small too.

[!WARNING] The basic error is looking only at proximity. The right question is how much flux from one coil is linked with the other.

🔵 Formal level

Basic formulas

In the linear regime, primary current and secondary flux linkage keep a proportional relation. The operational definition of mutual inductance is:

Definition of mutual inductance

The same relation can be read as mutual flux produced by a primary current:

Linear mutual flux linkage

Both expressions describe the same linear coupling from two complementary angles.

Real student questions

Does the secondary need to be connected for emf to exist? No. $\varepsilon_2$ may exist with an open circuit. What is missing then is sustained secondary current.

Why is $M$ not simply $L_1$ or $L_2$? Because $L_1$ and $L_2$ describe each coil with itself. $M$ describes flux shared between two coils.

Can there be mutual flux without induced voltage? Yes. If $I_1$ is steady, $\lambda_{21}$ may remain constant and produce no average emf.

What does small $k$ mean? It means much of the flux created by the primary does not link effectively with the secondary.

Final synthesis

Mutual inductance measures how one coil changes electrically because of flux created by another. $M$ summarizes coupling, $\varepsilon_2$ expresses induced response, and $k$ separates self-size from quality of magnetic linkage.