22.4

Faraday’s Law of Electromagnetic Induction

Two scientists are given credit for the discovery of electromagnetic induction: the Englishman Michael Faraday (1791–1867) and the American Joseph Henry (1797–1878). Although Henry was the first to observe electromagnetic induction, Faraday investigated it in more detail and published his findings first. Consequently, the law that describes the phenomenon bears his name.
CONCEPTS AT A GLANCE Faraday discovered that whenever there is a change in flux through a loop of wire, an emf is induced in the loop. In this context, the word “change” refers to a change as time passes. A flux that is constant in time creates no emf. The Concepts-at-a-Glance chart in Figure 22-12, emphasizes that Faraday’s law of electromagnetic induction is expressed by bringing together the idea of magnetic flux and the time interval during which it changes. In fact, Faraday found that the magnitude of the induced emf is equal to the time rate of change of the magnetic flux. This is consistent with the relation we obtained in Section 22.3 for the specific case of motional emf: .

Figure 22-12 CONCEPTS AT A GLANCE Faraday’s law of electromagnetic induction specifies the emf created by a change in the magnetic flux that occurs as time passes. The change in the flux and the time interval over which it occurs both appear in the statement of the law. This “mom-and-pop” hydroelectric generating station utilizes falling water to turn a turbine, which generates electricity. Faraday’s law of electromagnetic induction plays a vital role in this process. (Chris Knapton/Photo Researchers, Inc.)

Often the magnetic flux passes through a coil of wire containing more than one loop (or turn). If the coil consists of N loops, and if the same flux passes through each loop, it is found experimentally that the total induced emf is N times that induced in a single loop. An analogous situation occurs in a flashlight when two 1.5-V batteries are stacked in series on top of one another to give a total emf of 3.0 volts. For the general case of N loops, the total induced emf is described by Faraday’s law of electromagnetic induction in the following manner:

| Faraday’s Law of Electromagnetic Induction
| The average emf E induced in a coil of N loops is

| (22.3)

where is the change in magnetic flux through one loop and is the time interval during which the change occurs. The term is the average time rate of change of the flux that passes through one loop.
SI Unit of Induced Emf: volt (V)

Faraday’s law states that an emf is generated if the magnetic flux changes for any reason. Since the flux is given by Equation 22.2 as , it depends on three factors, B, A, and , any of which may change. Example 5 considers a change in B.

| Example 5 | | | The Emf Induced by a Changing Magnetic Field

A coil of wire consists of 20 turns, or loops, each of which has an area of 1.5 × 10−3 m2. A magnetic field is perpendicular to the surface of each loop at all times, so that . At time t0 = 0 s, the magnitude of the field at the location of the coil is B0 = 0.050 T. At a later time t = 0.10 s, the magnitude of the field at the coil has increased to B = 0.060 T. (a) Find the average emf induced in the coil during this time. (b) What would be the value of the average induced emf if the magnitude of the magnetic field decreased from 0.060 T to 0.050 T in 0.10 s?
Reasoning To find the induced emf, we use Faraday’s law of electromagnetic induction (Equation 22.3), combining it with the definition of magnetic flux from Equation 22.2. We note that only the magnitude of the magnetic field changes in time. All other factors remain constant.
Solution

a. | Since , the induced emf is

Problem solving insight The change in any quantity is the final value minus the initial value: e.g., the change in flux is and the change in time is .

b. | The calculation here is similar to that in part (a), except the initial and final values of B are interchanged. This interchange reverses the sign of the emf, so . Because the algebraic sign or polarity of the emf is reversed, the direction of the induced current would be opposite to that in part (a).

The next example demonstrates that an emf can be created when a coil is rotated in a magnetic field.

| Example 6 | | | The Emf Induced in a Rotating Coil

A flat coil of wire has an area of 0.020 m2 and consists of 50 turns. At t0 = 0 s the coil is oriented so the normal to its surface is parallel to a constant magnetic field of magnitude 0.18 T. The coil is then rotated through an angle of in a time of 0.10 s (see Figure 22-10). (a) Determine the average induced emf. (b) What would be the induced emf if the coil were returned to its initial orientation in the same time of 0.10 s?

Need more practice?

Interactive LearningWare22.2
A 75-turn conducting coil has an area of 8.5 × 10−3 m2, and the normal to the coil is parallel to a magnetic field . The coil has a resistance of . At what rate (in T/s) must the magnitude of change for an induced current of 7.0 mA to exist in the coil?
Related Homework: Problem 21

Reasoning As in Example 5 we can determine the induced emf by using Faraday’s law of electromagnetic induction, along with the definition of magnetic flux. In the present case, however, only (the angle between the normal to the surface of the coil and the magnetic field) changes in time. All other factors remain constant.
Solution

a. | Faraday’s law yields

b. | When the coil is rotated back to its initial orientation in a time of 0.10 s, the initial and final values of are interchanged. As a result, the induced emf has the same magnitude, but opposite polarity, so .

One application of Faraday’s law that is found in the home is a safety device called a ground fault interrupter. This device protects against electrical shock from an appliance, such as a clothes dryer. It plugs directly into a wall socket, as in Figure 22-13 or, in new home construction, replaces the socket entirely. The interrupter consists of a circuit breaker that can be triggered to stop the current to the dryer, depending on whether an induced voltage appears across a sensing coil. This coil is wrapped around an iron ring, through which the current-carrying wires pass. In the drawing, the current going to the dryer is shown in red, and the returning current is shown in green. Each of the currents creates a magnetic field that encircles the corresponding wire, according to RHR-2 (see Section 21.7). However, the field lines have opposite directions since the currents have opposite directions. As the drawing shows, the iron ring guides the field lines through the sensing coil. Since the current is ac, the fields from the red and green current are changing, but the red and green field lines always have opposite directions and the opposing fields cancel at all times. As a result, the net flux through the coil remains zero, and no induced emf appears in the coil. Thus, when the dryer operates normally, the circuit breaker is not triggered and does not shut down the current. The picture changes if the dryer malfunctions, as when a wire inside the unit breaks and accidentally contacts the metal case. When someone touches the case, some of the current begins to pass through the person’s body and into the ground, returning to the ac generator without using the return wire that passes through the ground fault interrupter. Under this condition, the net magnetic field through the sensing coil is no longer zero and changes with time, since the current is ac. The changing flux causes an induced voltage to appear in the sensing coil, which triggers the circuit breaker to stop the current. Ground fault interrupters work very fast (in less than a millisecond) and turn off the current before it reaches a dangerous level.

Figure 22-13 The clothes dryer is connected to the wall socket through a ground fault interrupter. The dryer is operating normally.

The physics of a ground fault interrupter.

Conceptual Example 7 discusses another application of electromagnetic induction—namely, how a stove can cook food without getting hot.

| Conceptual Example 7 | | | An Induction Stove

Figure 22-14 shows two pots of water that were placed on an induction stove at the same time. There are two interesting features in this drawing. First, the stove itself is cool to the touch. Second, the water in the ferromagnetic metal pot is boiling while that in the glass pot is not. How can such a “cool” stove boil water, and why isn’t the water in the glass pot boiling?

Figure 22-14 The water in the ferromagnetic metal pot is boiling. Yet the water in the glass pot is not boiling, and the stove top is cool to the touch. The stove operates in this way by using electromagnetic induction.

Reasoning and Solution The key to this puzzle is related to the fact that one pot is made from a ferromagnetic metal and one from glass. We know that metals are good conductors, while glass is an insulator. Perhaps the stove causes electricity to flow directly in the metal pot. This is exactly what happens. The stove is called an induction stove because it operates by using electromagnetic induction. Just beneath the cooking surface is a metal coil that carries an ac current (frequency about 25 kHz). This current produces an alternating magnetic field that extends outward to the location of the metal pot. As the changing field crosses the pot’s bottom surface, an emf is induced in it. Because the pot is metallic, an induced current is generated by the induced emf. The metal has a finite resistance to the induced current, however, and heats up as energy is dissipated in this resistance. The fact that the metal is ferromagnetic is important. Ferromagnetic materials contain magnetic domains (see Section 21.9), and the boundaries between them move extremely rapidly in response to the external magnetic field, thus enhancing the induction effect. A normal aluminum cooking pot, in contrast, is not ferromagnetic, so this enhancement is absent and such cookware is not used with induction stoves. An emf is also induced in the glass pot and the cooking surface of the stove. However, these materials are insulators, so very little induced current exists within them. Thus, they do not heat up very much and remain cool to the touch.

The physics of an induction stove.

Related Homework: Conceptual Question 8

Check Your Understanding 2

A coil is placed in a magnetic field, and the normal to the plane of the coil remains parallel to the field. Which one of the following options causes the average emf induced in the coil to be as large as possible? (a) The magnitude of the field is small, and its rate of change is large. (b) The magnitude of the field is large, and its rate of change is small. (c) The magnitude of the field is large, and it does not change. (The answer is given at the end of the book.)
Background: The average emf E induced in the coil is given by Faraday’s law of electromagnetic induction as , where is the magnetic flux and f is time. The magnetic flux depends on the magnetic field strength B, the area A of the coil, and the angle between the field and the normal to the coil, according to .
For similar questions (including calculational counterparts), consult Self-Assessment Test 22.1, which is described at the end of Section 22.5.

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