Chapter 14. Inductance
14.1 Mutual Inductance
Learning Objectives
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 (Figure 14.2). Coils 1 and 2 have [latex]{N}_{1}[/latex] and [latex]{N}_{2}[/latex] turns and carry currents [latex]{I}_{1}[/latex] and [latex]{I}_{2},[/latex] respectively. The flux through a single turn of coil 2 produced by the magnetic field of the current in coil 1 is [latex]{\text{Φ}}_{21},[/latex] whereas the flux through a single turn of coil 1 due to the magnetic field of [latex]{I}_{2}[/latex] is [latex]{\text{Φ}}_{12}.[/latex]
The mutual inductance [latex]{M}_{21}[/latex] of coil 2 with respect to coil 1 is the ratio of the flux through the [latex]{N}_{2}[/latex] turns of coil 2 produced by the magnetic field of the current in coil 1, divided by that current, that is,
Similarly, the mutual inductance of coil 1 with respect to coil 2 is
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 [latex]1\phantom{\rule{0.2em}{0ex}}\text{H}=1\phantom{\rule{0.2em}{0ex}}\text{V}·\text{s/A}[/latex]. From Equation 14.1 and Equation 14.2, we can show that [latex]{M}_{21}={M}_{12},[/latex] so we usually drop the subscripts associated with mutual inductance and write
The emf developed in either coil is found by combining Faraday’s law and the definition of mutual inductance. Since [latex]{N}_{2}{\text{Φ}}_{21}[/latex] is the total flux through coil 2 due to [latex]{I}_{1}[/latex], we obtain
where we have used the fact that M is a time-independent constant because the geometry is time-independent. Similarly, we have
In Equation 14.5, 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 cu