KAHIBARO
Table of Contents
Coupled Circuits
Mutual inductance describes how a changing current in one circuit can produce an electromotive force in a nearby circuit. This happens because the current in the first circuit creates a magnetic field, and if that magnetic field passes through the second circuit, a change in the field changes the magnetic flux through it. By Faraday's law, that changing flux induces an emf.
Mutual inductance is therefore a measure of how strongly two circuits are magnetically linked. If the magnetic field produced by one circuit passes effectively through the other, the mutual inductance is large. If only a small part of the field links the second circuit, the mutual inductance is small.
Basic Idea
Consider two coils placed near each other. Let coil 1 carry a current $I_1$. This current creates a magnetic field, and part of that field passes through coil 2. The magnetic flux through coil 2 due to current in coil 1 is proportional to $I_1$, as long as the material and geometry remain fixed and the system behaves linearly.
We write
$$
\Phi_{21} \propto I_1
$$
where $\Phi_{21}$ means the magnetic flux through coil 2 caused by current in coil 1. If coil 2 has $N_2$ turns, we often use the flux linkage $N_2 \Phi_{21}$. The constant of proportionality is the mutual inductance $M$:
$$
N_2 \Phi_{21} = M I_1
$$
Similarly, if current $I_2$ flows in coil 2, it produces flux through coil 1:
$$
N_1 \Phi_{12} = M I_2
$$
A very important result is that the same mutual inductance $M$ appears in both directions.
For two coupled coils in a linear medium,
$$
N_2 \Phi_{21} = M I_1, \qquad N_1 \Phi_{12} = M I_2
$$
The mutual inductance $M$ measures how effectively current in one circuit creates magnetic flux in the other.
Induced EMF from Mutual Inductance
If the current in coil 1 changes with time, then the flux through coil 2 changes with time. This induces an emf in coil 2:
$$
\mathcal{E}_2 = -N_2 \frac{d\Phi_{21}}{dt}
$$
Using $N_2 \Phi_{21} = M I_1$, we get
$$
\mathcal{E}_2 = -M \frac{dI_1}{dt}
$$
In the same way, a changing current in coil 2 induces an emf in coil 1:
$$
\mathcal{E}_1 = -M \frac{dI_2}{dt}
$$
The negative sign comes from Lenz's law. The induced emf always acts to oppose the change that produced it.
The induced emf due to mutual inductance is
$$
\mathcal{E}_2 = -M \frac{dI_1}{dt}, \qquad \mathcal{E}_1 = -M \frac{dI_2}{dt}
$$
A mutual emf appears only when the current in the other circuit changes with time.
Meaning of the Symbol $M$
The quantity $M$ is called the mutual inductance. Its SI unit is the henry, $\mathrm{H}$, the same unit used for self-inductance.
From
$$
\mathcal{E} = -M \frac{dI}{dt}
$$
we see that
$$
1 \ \mathrm{H} = 1 \ \frac{\mathrm{V \cdot s}}{\mathrm{A}}
$$
A mutual inductance of $1 \ \mathrm{H}$ means that a current change of $1 \ \mathrm{A/s}$ in one circuit induces an emf of $1 \ \mathrm{V}$ in the other circuit.
What Affects Mutual Inductance
Mutual inductance depends on the geometry and arrangement of the two circuits. It becomes larger when the magnetic coupling between them becomes stronger.
Important factors include the number of turns in the coils, their size, their shape, the distance between them, their relative orientation, and the magnetic properties of any core material between or inside them.
If two coils are far apart, little magnetic flux from one passes through the other, so $M$ is small. If they are close together and wound on a common iron core, much more flux links them, so $M$ is much larger.
Mutual Inductance of Two Solenoids
A common example is two long solenoids wound on the same core. Suppose coil 1 has $N_1$ turns, length $\ell$, and carries current $I_1$. The magnetic field inside a long solenoid is approximately
$$
B_1 = \mu \frac{N_1}{\ell} I_1
$$
If coil 2 surrounds the same cross-sectional area $A$, then the flux through each turn of coil 2 due to coil 1 is
$$
\Phi_{21} = B_1 A = \mu \frac{N_1}{\ell} I_1 A
$$
Multiplying by $N_2$ gives
$$
N_2 \Phi_{21} = \mu \frac{N_1 N_2 A}{\ell} I_1
$$
So the mutual inductance is
$$
M = \mu \frac{N_1 N_2 A}{\ell}
$$
This formula is valid for closely coupled long solenoids with shared area and length.
For two long, closely coupled solenoids on the same core,
$$
M = \mu \frac{N_1 N_2 A}{\ell}
$$
where $\mu$ is the permeability of the core material, $A$ is the cross-sectional area, and $\ell$ is the length.
Symmetry of Mutual Inductance
An important physical fact is that the mutual inductance is the same whichever coil is considered the source:
$$
M_{21} = M_{12} = M
$$
This means that current in coil 1 producing flux in coil 2 gives the same mutual inductance value as current in coil 2 producing flux in coil 1. The fluxes themselves need not be numerically identical, because the coils may have different numbers of turns, but the proportionality constant relating flux linkage to current is the same.
Coupling Strength
Not all of the magnetic flux created by one coil necessarily passes through the other. Because of this, engineers often describe the magnetic link with a coupling coefficient $k$, where
$$
0 \le k \le 1
$$
If $k = 1$, the coupling is perfect, meaning all relevant flux from one coil links the other. If $k$ is much less than 1, the coupling is weak.
The relation between mutual inductance and the self-inductances $L_1$ and $L_2$ is
$$
M = k \sqrt{L_1 L_2}
$$
This is useful when the two self-inductances are known.
The coupling coefficient satisfies
$$
M = k \sqrt{L_1 L_2}, \qquad 0 \le k \le 1
$$
Stronger magnetic coupling means larger $k$ and larger $M$.
Direction and Sign
The magnitude of mutual inductance is positive, but when writing circuit equations, the sign of the induced voltage depends on coil orientation. The winding direction of the coils matters. If the coils are wound so that their magnetic effects reinforce each other in a chosen direction, one sign convention is used. If they oppose each other, the sign changes.
At an introductory level, the key idea is simple: always use Lenz's law to determine the direction of the induced emf. The induced effect opposes the change in current that caused it.
Example
Suppose two coils have mutual inductance
$$
M = 0.20 \ \mathrm{H}
$$
and the current in coil 1 changes at the rate
$$
\frac{dI_1}{dt} = 3.0 \ \mathrm{A/s}
$$
Then the induced emf in coil 2 is
$$
\mathcal{E}_2 = -M \frac{dI_1}{dt}
$$
so
$$
\mathcal{E}_2 = -(0.20)(3.0) = -0.60 \ \mathrm{V}
$$
The magnitude of the induced emf is $0.60 \ \mathrm{V}$. The negative sign shows that its direction opposes the change in current.
Mutual Inductance and Transformers
Mutual inductance is the basic physical principle behind transformers. In a transformer, an alternating current in one coil creates a changing magnetic flux in an iron core, and this changing flux induces a voltage in the second coil. The detailed operation of transformers belongs to a later chapter, but the essential connection is that transformers work because of mutual inductance.
Visualizing Two Coupled Coils
Summary Table
| Quantity | Meaning | Formula |
|---|---|---|
| Mutual inductance | Magnetic coupling between two circuits | $N_2 \Phi_{21} = M I_1$ |
| Induced emf in coil 2 | emf caused by changing current in coil 1 | $\mathcal{E}_2 = -M \dfrac{dI_1}{dt}$ |
| Induced emf in coil 1 | emf caused by changing current in coil 2 | $\mathcal{E}_1 = -M \dfrac{dI_2}{dt}$ |
| Unit of $M$ | henry | $1\ \mathrm{H} = 1\ \mathrm{V \cdot s/A}$ |
| Coupling relation | relation to self-inductances | $M = k\sqrt{L_1L_2}$ |
Key Point
Mutual inductance is the property by which a changing current in one circuit induces an emf in another circuit through a changing magnetic flux. It is strongest when the two circuits are arranged so that much of the magnetic field from one links the other.
Core idea of mutual inductance:
A changing current in one circuit creates a changing magnetic flux through a second circuit, which induces an emf in the second circuit:
$$
\mathcal{E} = -M \frac{dI}{dt}
$$
This is the foundation of magnetic coupling between circuits.