22.8

Mutual Inductance and Self-Inductance

| Mutual Inductance

We have seen that an emf can be induced in a coil by keeping the coil stationary and moving a magnet nearby, or by moving the coil near a stationary magnet. Figure 22-26 illustrates another important method of inducing an emf. Here, two coils of wire are placed close to each other, the primary coil and the secondary coil. The primary coil is the one connected to an ac generator, which sends an alternating current Ip through it. The secondary coil is not attached to a generator, although a voltmeter is connected across it to register any induced emf.

Figure 22-26 An alternating current Ip in the primary coil creates an alternating magnetic field. This changing field induces an emf in the secondary coil.

The current-carrying primary coil is an electromagnet and creates a magnetic field in the surrounding region. If the two coils are close to each other, a significant fraction of this magnetic field penetrates the secondary coil and produces a magnetic flux. The flux is changing, since the current in the primary coil and its associated magnetic field are changing. Because of the change in flux, an emf is induced in the secondary coil.
The effect in which a changing current in one circuit induces an emf in another circuit is called mutual induction. According to Faraday’s law of electromagnetic induction, the average emf induced in the secondary coil is proportional to the change in flux passing through it. However, is produced by the change in current in the primary coil. Therefore, it is convenient to recast Faraday’s law into a form that relates to . To see how this recasting is accomplished, note that the net magnetic flux passing through the secondary coil is , where Ns is the number of loops in the secondary coil and is the flux through one loop (assumed to be the same for all loops). The net flux is proportional to the magnetic field, which, in turn, is proportional to the current Ip in the primary. Thus, we can write . This proportionality can be converted into an equation in the usual manner by introducing a proportionality constant M, known as the mutual inductance:

| (22.6)

Substituting this equation into Faraday’s law, we find that

| (22.7)

Writing Faraday’s law in this manner makes it clear that the average emf induced in the secondary coil is due to the change in the current in the primary coil.
Equation 22.7 shows that the measurement unit for the mutual inductance M is V·s/A, which is called a henry (H) in honor of Joseph Henry: 1 V·s/A = 1 H. The mutual inductance depends on the geometry of the coils and the nature of any ferromagnetic core material that is present. Although M can be calculated for some highly symmetrical arrangements, it is usually measured experimentally. In most situations, values of M are less than 1 H and are often on the order of millihenries (1 mH = 1 × 10−3 H) or microhenries .
A new technique that shows promise for the treatment of psychiatric disorders such as depression is based on mutual induction. This technique is called transcranial magnetic stimulation (TMS) and is a type of indirect and gentler electric shock therapy. In traditional electric shock therapy, electric current is delivered directly through the skull and penetrates the brain, disrupting its electrical circuitry and in the process alleviating the symptoms of the psychiatric disorder. The treatment is not gentle and requires an anesthetic, because relatively large electric currents must be used to penetrate the skull. In contrast, TMS produces its electric current by using a time-varying magnetic field. A primary coil is held over the part of the brain to be treated (see Figure 22-27), and a time-varying current is applied to this coil. The arrangement is analogous to that in Figure 22-26, except that the brain and the electrically conductive pathways within it take the place of the secondary coil. The magnetic field produced by the primary coil penetrates the brain and, since the field is changing in time, it induces an emf in the brain. This induced emf causes an electric current to flow in the conductive brain tissue, with therapeutic results similar to those of conventional electric shock treatment. The current delivered to the brain, however, is much smaller than that in the conventional treatment, so that patients receive TMS treatments without anesthetic and without severe after-effects such as headaches and memory loss. TMS remains in the experimental stage, however, and the optimal protocol for applying the technique has not yet been determined.

Figure 22-27 In the technique of transcranial magnetic stimulation (TMS), a time-varying electric current is applied to a primary coil, which is held over a region of the brain, as this photograph illustrates. The time varying magnetic field produced by the coil penetrates the brain and creates an induced emf within it. This induced emf leads to an induced current that disrupts the electric circuits of the brain, thereby relieving some of the symptoms of psychiatric disorders such as depression. (Courtesy Mark S. George, Xingbao Li, and Chris Molnar, Medical University of South Carolina)

The physics of transcranial magnetic stimulation (TMS).

| Self-Inductance

In all the examples of induced emfs presented so far, the magnetic field has been produced by an external source, such as a permanent magnet or an electromagnet. However, the magnetic field need not arise from an external source. An emf can be induced in a current-carrying coil by a change in the magnetic field that the current itself produces. For instance, Figure 22-28 shows a coil connected to an ac generator. The alternating current creates an alternating magnetic field that, in turn, creates a changing flux through the coil. The change in flux induces an emf in the coil, in accord with Faraday’s law. The effect in which a changing current in a circuit induces an emf in the same circuit is referred to as self-induction.

Figure 22-28 The alternating current in the coil generates an alternating magnetic field that induces an emf in the coil.

When dealing with self-induction, as with mutual induction, it is customary to recast Faraday’s law into a form in which the induced emf is proportional to the change in current in the coil rather than to the change in flux. If is the magnetic flux that passes through one turn of the coil, then is the net flux through a coil of N turns. Since is proportional to the magnetic field, and the magnetic field is proportional to the current I, it follows that . By inserting a constant L, called the self-inductance, or simply the inductance, of the coil, we can convert this proportionality into Equation 22.8:

| (22.8)

Faraday’s law of induction now gives the average induced emf as

| (22.9)

Like mutual inductance, L is measured in henries. The magnitude of L depends on the geometry of the coil and on the core material. By wrapping the coil around a ferromagnetic (iron) core, the magnetic flux—and therefore the inductance—can be increased substantially relative to that for an air core. Because of their self-inductance, coils are known as inductors and are widely used in electronics. Inductors come in all sizes, typically in the range between millihenries and microhenries. Example 13 shows how to determine the emf induced in a solenoid from knowledge of its self-inductance and the rate at which the current in the solenoid changes.

A N A L Y Z I N G M U L T I P L E - C O N C E P T P R O B L E M S

| Example 13 | | | The Emf Induced in a Long Solenoid

A long solenoid of length 8.0 × 10−2 m and cross-sectional area 5.0 × 10−5 m2 contains 6500 turns per meter of length. Determine the emf induced in the solenoid when the current in the solenoid changes from 0 to 1.5 A during the time interval from 0 to 0.20 s.
Reasoning According to the relation , the emf E induced in the solenoid depends on the self-inductance L of the solenoid and the rate at which the solenoidal current changes. The variables and are known, and we can evaluate the self-inductance by using the definition, , where N is the number of turns in the coil, is the magnetic flux that passes through one turn, and I is the current. The magnetic flux is given by (Equation 22.2), and the magnetic field inside a solenoid has the value of (Equation 21.7), where n is the number of turns per unit length. These relations will enable us to evaluate the emf induced in the solenoid.
Knowns and Unknowns The data for this problem are listed below:

Description | Symbol | Value
Length of solenoid |  | 8.0 × l0−2m
Cross-sectional area of solenoid | A | 5.0 × 10−5 m2
Number of turns per unit length | n | 6500 turns/meter
Change in current |  | 1.5 A
Change in time |  | 0.20 s
Unknown Variable |  |
Emf induced in solenoid | E | ?

Modeling the Problem

| Step 1 Emf Due to Self-Induction
| The emf E produced by self-induction is given at the right by Faraday’s law of induction in the form expressed in Equation 22.9, where L is the self-inductance of the solenoid, is the change in current, and is the change in time. Both and are known (see the data table), and L will be evaluated in the next step.

| (22.9)

| Step 2 Self-Inductance
| The self-inductance L is defined by (Equation 22.8), where N is the number of turns, is the magnetic flux that passes through one turn, and I is the current in the coil. The number of turns is equal to the number n of turns per meter of length times the length , of the solenoid, or . Substituting this expression for N into gives

This result for the self-inductance can be substituted into Equation 22.9, as indicated at the right. Note from the data table that n and , are known. The magnetic flux and current I are not known and will be dealt with in Step 3.

| Step 3 Magnetic Flux
| According to the discussion in Section 22.3, the magnetic flux is given by (Equation 22.2), where B is the magnitude of the magnetic field that penetrates the surface of area A, and is the angle between the magnetic field and the normal to the surface. In the case of a solenoid, the interior magnetic field is directed perpendicular to the plane of the loops (see Section 21.7), so . Thus, the magnetic flux is . According to the discussion in Section 21.7, the magnitude of the magnetic field inside a long solenoid is given by (Equation 21.7). Substituting this expression for B into the equation for the flux yields

All the variables on the right side of this equation, except the current, are known. We now substitute this expression into Equation 22.10, as indicated in the right column. Note that the current I appears in both the numerator and denominator in the expression for the self-inductance L, so I will be algebraically eliminated from the final result.
Solution Algebraically combining the results of the three steps, we have

The self-inductance of the solenoid is , and the emf induced in the solenoid is

Related Homework: Problems 48, 52

| The Energy Stored in an Inductor

An inductor, like a capacitor, can store energy. This stored energy arises because a generator does work to establish a current in an inductor. Suppose an inductor is connected to a generator whose terminal voltage can be varied continuously from zero to some final value. As the voltage is increased, the current I in the circuit rises continuously from zero to its final value. While the current is rising, an induced emf appears across the inductor. Conforming to Lenz’s law, the polarity of the induced emf E is opposite to that of the generator voltage, so as to oppose the increase in the current. Thus, the generator must do work to push the charges through the inductor against this induced emf. The increment of work done by the generator in moving a small amount of charge through the inductor is , according to Equation 19.4. Since is the current I, the work done is

In this expression represents the work done by the generator to increase the current in the inductor by an amount . To determine the total work W done while the current is changed from zero to its final value, all the small increments must be added together. This summation is left as an exercise (see Problem 72). The result is , where I represents the final current in the inductor. This work is stored as energy in the inductor, so that

| (22.10)

It is possible to regard the energy in an inductor as being stored in its magnetic field. For the special case of a long solenoid, it is found in Example 13 that the self-inductance is , where n is the number of turns per unit length, A is the cross-sectional area, and is the length of the solenoid. As a result, the energy stored in a long solenoid is

Since at the interior of a long solenoid (Equation 21.7), this energy can be expressed as

The term is the volume inside the solenoid where the magnetic field exists, so the energy per unit volume, or energy density, is

| (22.11)

Although this result was obtained for the special case of a long solenoid, it is quite general and is valid for any point where a magnetic field exists in air or vacuum or in a nonmagnetic material. Thus, energy can be stored in a magnetic field, just as it can in an electric field.

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.