Mutual Inductance: From Intuition to Definition
Imagine you have two separate coils of wire placed close to each other, but not electrically connected. You connect the first coil to a battery through a switch. The moment you close the switch, current begins to flow in coil 1. That current doesn't just stay in coil 1 — it creates a magnetic field around it. Some of that magnetic field spreads out and passes through the loops of coil 2.
Now, here is the key physical fact: a changing magnetic field through a coil induces an emf (electromotive force) in that coil. So when you close the switch, the current in coil 1 rises from zero to some steady value. During that brief rise, the magnetic field through coil 2 is also changing — from zero to some steady value. That changing field induces a temporary emf in coil 2. If coil 2 is part of a closed circuit, that induced emf will drive a current for as long as the field is changing.
The same thing happens in reverse: if you change the current in coil 2, it induces an emf in coil 1. This mutual influence — the ability of a changing current in one circuit to induce an emf in a nearby circuit — is called mutual inductance.
Mutual inductance is a geometric property. It depends only on the shapes, sizes, number of turns, and relative positions of the two coils, and on the magnetic properties of the medium between them. It does not depend on how much current is actually flowing.
The Precise Definition
Let coil 1 carry a current I1. This current produces a magnetic flux Φ21 through each turn of coil 2. The total flux linkage through coil 2 is N2Φ21, where N2 is the number of turns in coil 2.
For a given geometry, the flux through coil 2 is directly proportional to the current in coil 1:
N2Φ21=M21I1
The constant of proportionality M21 is called the mutual inductance of the pair of coils. The subscript 21 means "flux in coil 2 due to current in coil 1."
By Faraday's law, the induced emf in coil 2 is the negative rate of change of flux linkage:
E2=−dtd(N2Φ21)=−M21dtdI1
The negative sign indicates Lenz's law — the induced emf opposes the change that produces it.
E2=−MdtdI1
If you reverse the roles — change current in coil 2 and measure the induced emf in coil 1 — you get:
E1=−M12dtdI2
A remarkable fact (which follows from energy conservation) is that the two mutual inductances are equal:
M12=M21=M
So we simply call it M, the mutual inductance of the pair.
Units and Typical Values
The SI unit of mutual inductance is the henry (H). From the defining equation:
1 H=1 AV⋅s
One henry means that a current changing at the rate of 1 ampere per second induces an emf of 1 volt in the other coil. In practice, typical values are much smaller — millihenries (mH) for transformers and microhenries (μH) for circuits on a circuit board.
Mutual inductance is not a property of either coil alone. It is a property of the pair of coils and their relative arrangement. If you move the coils farther apart, M decreases. If you align them differently, M changes. If you remove one coil, the concept of mutual inductance becomes meaningless.
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