22.10

Concepts & Calculations

In this chapter we have seen that there are three ways to create an induced emf in a coil: by changing the magnitude of a magnetic field, by changing the direction of the field relative to the coil, and by changing the area of the coil. We have already explored the first two methods, and Example 15 now illustrates the third method. In the process, it provides a review of Faraday’s law of electromagnetic induction.

| Concepts & Calculations Example 15 | | | The Emf Produced by a Changing Area

A circular coil of radius 0.11 m contains a single turn and is located in a constant magnetic field of magnitude 0.27 T. The magnetic field is parallel to the normal to the plane of the coil. The radius increases to 0.30 m in a time of 0.080 s. (a) Determine the emf induced in the coil, (b) The coil has a resistance of . Find the induced current.
Concept Questions and Answers Why is there an emf induced in the coil?

> Answer According to Faraday’s law of electromagnetic induction, Equation 22.3, an emf is induced whenever the magnetic flux through the coil changes. The magnetic flux, as expressed by Equation 22.2, depends on the area of the coil, which, in turn, depends on the radius. If the radius changes, the area changes, causing the flux to change and an induced emf to appear.

Does the magnitude of the induced emf depend on whether the area is increasing or decreasing?

> Answer No. The magnitude of the induced emf depends only on the magnitude of the rate at which the flux changes. It does not matter whether the flux is increasing or decreasing.

What determines the amount of current induced in the coil?

> Answer According to Ohm’s law (Equation 20.2), the current is equal to the emf induced in the coil divided by the resistance of the coil. Therefore, the larger the induced emf and the smaller the resistance, the larger will be the induced current.

If the coil is cut so it is no longer one continuous piece, are there an induced emf and induced current?

> Answer An induced emf is generated in the coil regardless of whether the coil is whole or cut. However, an induced current exists only if the coil is continuous. Cutting the coil is like opening a switch. There is no longer a continuous path for the current to follow, so the current stops.

Solution

a. | The average induced emf is given by Faraday’s law of electromagnetic induction.

| (22.3)

where and are the final and initial fluxes, respectively. The definition of magnetic flux is , according to Equation 22.2, where because the field is parallel to the normal. The area of a circle is . With these expressions for and A, Faraday’s law becomes

b. | The induced current I is equal to the induced emf divided by the resistance R of the coil:

| (20.2)

One of the most important applications of Faraday’s law of electromagnetic induction is the electric generator, because it is the source of virtually all the electrical energy that we use. A generator produces an emf as a rotating coil changes its orientation relative to a fixed magnetic field. Example 16 reviews the characteristics of this type of induced emf.

| Concepts & Calculations Example 16 | | | The Emf Produced by a Generator

The graph in Figure 22-31 shows the emf produced by a generator as a function of time t. The coil of the generator has an area of A = 0.15 m2 and consists of N = 10 turns. The coil rotates in a magnetic field of magnitude 0.27 T. (a) Determine the period of the motion. (b) What is the angular frequency of the rotating coil? (c) Find the value of the emf when , where T denotes the period of the coil motion. (d) What is the emf when t = 0.025 s?

Figure 22-31 A plot of the emf produced by a generator as a function of time.

Concept Questions and Answers Can the period of the rotating coil be determined from the graph?

> Answer Yes. The period is the time for the coil to rotate through one revolution, or cycle. During this time, the emf is positive for one half of the cycle and negative for the other half of the cycle.

The emf produced by a generator depends on its angular frequency. How is the angular frequency of the rotating coil related to the period?

> Answer According to Equation 10.6, the angular frequency is related to the period T by .

Starting at t = 0 s, how much time is required for the generator to produce its peak emf? Express the answer in terms of the period T of the motion (e.g., ).

> Answer An examination of the graph shows that the emf reaches its maximum or peak value one-quarter of the way through a cycle. Since the time to complete one cycle is the period T, the time to reach the peak emf is .

How often does the polarity of the emf change in one cycle?

> Answer The graph shows that in every cycle the emf has an interval of positive polarity followed by an interval of negative polarity. In other words, the polarity changes from positive to negative and then from negative to positive during each cycle. Thus, the polarity changes twice during each cycle.

Solution

a. | The period is the time for the generator to make one complete cycle. From the graph it can be seen that this time is .
b. | The angular frequency is related to the period by , so

c. | When , the emf has reached its peak value. According to Equation 22.4, the peak emf is

d. | The emf produced by the generator as a function of time is given by Equation 22.4 as

Problem solving insight In evaluating a calculator must be set to the radian mode (not the degree mode) because is expressed in units of rad/s.

At the end of the problem set for this chapter, you will find homework problems that contain both conceptual and quantitative parts. These problems are grouped under the heading Concepts & Calculations.

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