# MUTUAL INDUCTANCE

MUTUAL INDUCTANCE

In Section 29-5 we considered the magnetic interaction between two wires carrying steady currents; the current in one wire causes a magnetic, which exerts a force on the current in the second wire. But an additional  interaction arises between two circuits when there is a changing current in one of the circuits. Consider two neighboring coils of wire, as m Fig. 31-1. A current flowing in coil 1 produces a magnetic field Band nee a magnetic flux through coil 2. If the current in coil 1 changes, the through coil 2 changes as well; according to Faraday’s law, this an emf in coil 2. In this way, a change in the current in one can induce a current in a second circuit. Let’s analyze the situation shown in Fig. 31-1 in more detail. We will use lower case letters to represent quantities that vary with time; for example, a time-varying current is i, often with a subscript to identify the circuit. In Fig. 31 a current il in coil 1 sets up a magnetic field (as indicated by the blue lines), and some of these field lines pass through coil 2. We denote the magnetic flux through each turn of coil 2, caused by the current in coil I, as (If the flux is different through different turns of the coil, then CIIB2 denotes the average flux.) The magnetic field is proportional to ii’ so CIIB 2 is also proportional to it” When changes, We could represent the proportionality of  i, in the form <IJB2 = (co instead it is more convenient to include the number of turns N2′ in the Introducing a proportionality constant M21′ called the mutual inductance coils, we write If the coils are in vacuum, the flux each turn of coil 2 is directly the current . Then the mutual inductance M2′ is a constant that depends geometry of the two coils (the size, shape, number of turns, and orientation and the separation between the coils). If a magnetic material is present, M21 on the magnetic properties of the material. If the material has nonlinear mag erties, that is, if the relative permeability Km (defined in Section 29-9) is and magnetization is not proportional to magnetic field, then <lJB2 is no long proportional to il. In that case the mutual inductance also depends on the this discussion we will assume that any magnetic material present has co that flux is directly proportional to current and M21 depends on geometry on; We can repeat our discussion for the opposite case in which a changing coil 2 causes a changing flux <IJ8′ and an emf s, in coil 1. We might expect responding constant MI2 would be different from M21 because in general the are not identical and the flux through them is not the same. It turns out, MI2 is always equal to M2I’ even when the two coils are not symmetric. We value simply the mutual inductance, denoted by the symbol M with characterizes completely the induced-emf interaction of two coils. Then The negative signs in Eq. (31-4) are a reflection of Lenz’s law. The first that a change in current in coil 1 causes a change in flux through coil 2, in coil 2 that opposes the flux change; in the second equation the roles of are interchanged. the Joseph Henry (1797-1878), one of the discoverers of electromagnetic undue From Eq. (31-5), one is equal to one per ampere. Other equivalent obtained by using Eq. (31-4), are one volt-second per ampere or one ohm-second: 1H = 1 Wb/A = 1V ·s/A = 1 inductance can be a nuisance in electric circuits, since variations in current circuit can induce unwanted emf’s in other, nearby circuits. To minimize these , multiple-circuit systems must be designed so that the value of M is as small ase; for example, two coils would be placed far apart or oriented with their planes mutual inductance also has many useful applications. A transformer, used sparing-current circuits to raise or lower voltages, is fundamentally no different the two coils shown in Fig. 31-1. A time-varying alternating current in one coil of former produces an alternating emf in the other coil; the value of M, which on the geometry of the coils, determines the amplitude of the induced emf in and hence the amplitude of the output voltage. (We’ll describe transin more ‘detail in Chapter 32 after we’ve discussed alternating current in greater in figure 