# 14.1 Mutual inductance

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By the end of this section, you will be able to:
• Correlate two nearby circuits that carry time-varying currents with the emf induced in each circuit
• Describe examples in which mutual inductance may or may not be desirable

Inductance is the property of a device that tells us how effectively it induces an emf in another device. In other words, it is a physical quantity that expresses the effectiveness of a given device.

When two circuits carrying time-varying currents are close to one another, the magnetic flux through each circuit varies because of the changing current I in the other circuit. Consequently, an emf is induced in each circuit by the changing current in the other. This type of emf is therefore called a mutually induced emf , and the phenomenon that occurs is known as mutual inductance ( M ) . As an example, let’s consider two tightly wound coils ( [link] ). Coils 1 and 2 have ${N}_{1}$ and ${N}_{2}$ turns and carry currents ${I}_{1}$ and ${I}_{2},$ respectively. The flux through a single turn of coil 2 produced by the magnetic field of the current in coil 1 is ${\text{Φ}}_{21},$ whereas the flux through a single turn of coil 1 due to the magnetic field of ${I}_{2}$ is ${\text{Φ}}_{12}.$

The mutual inductance ${M}_{21}$ of coil 2 with respect to coil 1 is the ratio of the flux through the ${N}_{2}$ turns of coil 2 produced by the magnetic field of the current in coil 1, divided by that current, that is,

${M}_{21}=\frac{{N}_{2}{\text{Φ}}_{21}}{{I}_{1}}.$

Similarly, the mutual inductance of coil 1 with respect to coil 2 is

${M}_{12}=\frac{{N}_{1}{\text{Φ}}_{12}}{{I}_{2}}.$

Like capacitance, mutual inductance is a geometric quantity. It depends on the shapes and relative positions of the two coils, and it is independent of the currents in the coils. The SI unit for mutual inductance M is called the henry (H)    in honor of Joseph Henry (1799–1878), an American scientist who discovered induced emf independently of Faraday. Thus, we have $1\phantom{\rule{0.2em}{0ex}}\text{H}=1\phantom{\rule{0.2em}{0ex}}\text{V}·\text{s/A}$ . From [link] and [link] , we can show that ${M}_{21}={M}_{12},$ so we usually drop the subscripts associated with mutual inductance and write

$M=\frac{{N}_{2}{\text{Φ}}_{21}}{{I}_{1}}=\frac{{N}_{1}{\text{Φ}}_{12}}{{I}_{2}}.$

The emf developed in either coil is found by combining Faraday’s law    and the definition of mutual inductance. Since ${N}_{2}{\text{Φ}}_{21}$ is the total flux through coil 2 due to ${I}_{1}$ , we obtain

${\epsilon }_{2}=-\frac{d}{dt}\left({N}_{2}{\text{Φ}}_{21}\right)=-\frac{d}{dt}\left(M{I}_{1}\right)=\text{−}M\frac{d{I}_{1}}{dt}$

where we have used the fact that M is a time-independent constant because the geometry is time-independent. Similarly, we have

${\epsilon }_{1}=\text{−}M\frac{d{I}_{2}}{dt}.$

In [link] , we can see the significance of the earlier description of mutual inductance ( M ) as a geometric quantity. The value of M neatly encapsulates the physical properties of circuit elements and allows us to separate the physical layout of the circuit from the dynamic quantities, such as the emf and the current. [link] defines the mutual inductance in terms of properties in the circuit, whereas the previous definition of mutual inductance in [link] is defined in terms of the magnetic flux experienced, regardless of circuit elements. You should be careful when using [link] and [link] because ${\epsilon }_{1}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{\epsilon }_{2}$ do not necessarily represent the total emfs in the respective coils. Each coil can also have an emf induced in it because of its self-inductance (self-inductance will be discussed in more detail in a later section).

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