22.5

Lenz’s Law

An induced emf drives current around a circuit just as the emf of a battery does. With a battery, conventional current is directed out of the positive terminal, through the attached device, and into the negative terminal. The same is true for an induced emf, although the locations of the positive and negative terminals are generally not as obvious. Therefore, a method is needed for determining the polarity or algebraic sign of the induced emf, so the terminals can be identified. As we discuss this method, it will be helpful to keep in mind that the net magnetic field penetrating a coil of wire results from two contributions. One is the original magnetic field that produces the changing flux that leads to the induced emf. The other arises because of the induced current, which, like any current, creates its own magnetic field. The field created by the induced current is called the induced magnetic field.
To determine the polarity of the induced emf, we will use a method based on a discovery made by the Russian physicist Heinrich Lenz (1804–1865). This discovery is known as Lenz’s law.

| Lenz’s Law
| The induced emf resulting from a changing magnetic flux has a polarity that leads to an induced current whose direction is such that the induced magnetic field opposes the original flux change.

Lenz’s law is best illustrated with examples. Each will be worked out according to the following reasoning strategy:

| Reasoning Strategy
| Determining the Polarity of the Induced Emf

1. | Determine whether the magnetic flux that penetrates a coil is increasing or decreasing.
2. | Find what the direction of the induced magnetic field must be so that it can oppose the change in flux by adding to or subtracting from the original field.
3. | Having found the direction of the induced magnetic field, use RHR-2 (see Section 21.7) to determine the direction of the induced current. Then the polarity of the induced emf can be assigned because conventional current is directed out of the positive terminal, through the external circuit, and into the negative terminal.

| Conceptual Example 8 | | | The Emf Produced by a Moving Magnet

Figure 22-15a shows a permanent magnet approaching a loop of wire. The external circuit attached to the loop consists of the resistance R, which could be the resistance of the filament in a light bulb, for instance. Find the direction of the induced current and the polarity of the induced emf.

Figure 22-15 (a) As the magnet moves to the right, the magnetic flux through the loop increases. The external circuit attached to the loop has a resistance R. (b) The polarity of the induced emf is indicated by the + and − symbols.

Reasoning and Solution We apply Lenz’s law, the essence of which is that the change in magnetic flux must be opposed by the induced magnetic field. The magnetic flux through the loop is increasing, since the magnitude of the magnetic field at the loop is increasing as the magnet approaches. To oppose the increase in the flux, the direction of the induced magnetic field must be opposite to the field of the bar magnet. Since the field of the bar magnet passes through the loop from left to right in part a of the drawing, the induced field must pass through the loop from right to left, as in part b. To create such an induced field, the induced current must be directed counterclockwise around the loop, when viewed from the side nearest the magnet. (See the application of RHR-2 in the drawing.) The loop behaves as a source of emf, just like a battery. Since conventional current is directed into the external circuit from the positive terminal, point A in Figure 22-15b must be the positive terminal, and point B must be the negative terminal.

In Conceptual Example 8 the direction of the induced magnetic field is opposite to the direction of the external field of the bar magnet. Problem solving insight: However, the induced field is not always opposite to the external field, because Lenz’s law requires only that it must oppose the change in the flux that generates the emf. Conceptual Example 9 illustrates this point.

| Conceptual Example 9 | | | The Emf Produced by a Moving Copper Ring

In Figure 22-16 there is a constant magnetic field in a rectangular region of space. This field is directed perpendicularly into the page. Outside this region there is no magnetic field. A copper ring slides through the region, from position 1 to position 5. For each of the five positions, determine whether an induced current exists in the ring and, if so, find its direction.

Figure 22-16 A copper ring passes through a rectangular region where a constant magnetic field is directed into the page. The picture shows that a current is induced in the ring at locations 2 and 4.

Reasoning and Solution Lenz’s law requires that the induced magnetic field must oppose the change in flux. Sometimes this means that the induced field will be opposite to the external field, as in Example 8, but not always. We will see that the induced field must sometimes have the same direction as the external field in order to oppose the flux change.
Position 1: No flux passes through the ring because the magnetic field is zero outside the rectangular region. Consequently, there is no change in flux and no induced emf or current.
Position 2: As the ring moves into the field region, the flux increases and there is an induced emf and current. To determine the direction of the current, we require that the induced field point out of the plane of the paper, opposite to the external field. Only then can the induced field oppose the increase in the flux, in accord with Lenz’s law. To produce such an induced field, RHR-2 indicates that the direction of the induced current must be counterclockwise, as the drawing shows.
Position 3: Even though a flux passes through the moving ring, there is no induced emf or current because the flux remains constant within the rectangular region. To induce an emf, it is not sufficient just to have a flux; the flux must change as time passes.
Position 4: As the ring leaves the field region, the flux decreases, and the induced field must oppose this change. Since the change is a decrease in flux, the induced field must point in the same direction as the external field—namely, into the paper. With this orientation, the induced field will increase the net magnetic field through the ring and thereby increase the flux. To produce an induced field pointing into the paper, RHR-2 indicates that the induced current must be clockwise, opposite to what it is in position 2. By comparing the results for positions 2 and 4, it should be clear that the induced magnetic field is not always opposite to the external field.
Position 5: As in position 1, there is no induced current, since the magnetic field is everywhere zero.
Related Homework: Problem 31

Lenz’s law should not be thought of as an independent law, because it is a consequence of the law of conservation of energy. The connection between energy conservation and induced emf has already been discussed in Section 22.2 for the specific case of motional emf. However, the connection is valid for any type of induced emf. In fact, the polarity of the induced emf, as specified by Lenz’s law, ensures that energy is conserved.

Check Your Understanding 3

A circular loop of wire is lying flat on a horizontal table, and you are looking down at it. A magnetic field has a constant direction that is perpendicular to the table, and an induced current flows clockwise in the loop. Is the field directed toward you or away from you, and is its magnitude increasing or decreasing? (There are two possible answers, and they are given at the end of the book.)
Background: Lenz’s law states that the induced current has a direction such that the induced magnetic field opposes the changing flux that gives rise to the current.
For similar questions (including calculational counterparts), consult Self-Assessment Test 22.1, which is described next.

Test your understanding of the material in Sections 22.1, 22.2, 22.3, 22.4 and 22.5 :

· | Induced EMF and Induced Current
· | Motional EMF
· | Magnetic Flux
· | Faraday’s Law of Electromagnetic Induction
· | Lenz’s Law

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