Q.Consider a metal ring kept (supported by a cardboard) on top of a fixed solenoid carrying a current $I$. The centre of the ring coincides with the axis of the solenoid. If the current in the solenoid is switched off, what will happen to the ring?

Mutual Inductance: From Intuition to Definition

Imagine you have two separate coils of wire placed near each other. You connect one coil to a battery — current starts flowing through it. Now, something strange happens in the other coil, which isn't connected to anything: a voltage appears across its ends. That voltage can even light a small bulb for an instant.

This is mutual inductance in action. One circuit "feels" the changing current in another circuit, even though they are not physically connected.

The Core Intuition

The key idea is changing magnetic fields. When current flows through a coil, it creates a magnetic field around it. If that current changes (increases or decreases), the magnetic field also changes. This changing field reaches the second coil. And a changing magnetic field, by Faraday's law, induces an electromotive force (emf) in any nearby conductor.

So mutual inductance is simply: how effectively a change in current in one coil induces a voltage in another coil.

The Precise Definition

Let's formalise this. Consider two coils: coil 1 and coil 2. Let $I_1$ be the current in coil 1. This current produces a magnetic flux $\Phi_{21}$ through coil 2 (the flux from coil 1 that passes through coil 2).

The mutual inductance $M$ (also written $M_{21}$) is defined as the constant of proportionality between the current $I_1$ and the flux it produces in coil 2:

Similarly, if current $I_2$ flows in coil 2, it produces a flux $\Phi_{12}$ through coil 1:

Now, by Faraday's law, the induced emf in coil 2 due to a changing current in coil 1 is:

And the induced emf in coil 1 due to a changing current in coil 2 is:

The negative sign is Lenz's law — the induced emf opposes the change that produced it.

Units

The SI unit of mutual inductance is the henry (H), named after Joseph Henry. From the definition:

One henry means that a current change of 1 ampere per second induces an emf of 1 volt in the other coil.

What Determines Mutual Inductance?

$M$ depends on:

• Geometry: size, shape, number of turns of both coils
• Relative position: how close they are and how they are oriented
• Core material: if a magnetic material (like iron) is present, $M$ increases dramatically

For two coaxial solenoids of length $l$, with $N_1$ and $N_2$ turns, and cross-sectional area $A$, the mutual inductance is: …