Physics 2- Chapter 23
INTRODUCTION TO ELECTROMAGNETIC INDUCTION, AC CIRCUITS AND ELECTRICAL TECHNOLOGIES
Nature’s displays of symmetry are beautiful and alluring. A butterfly’s wings exhibit an appealing symmetry in a complex system. (See Figure 23.2.) The laws of physics display symmetries at the most basic level—these symmetries are a source of wonder and imply deeper meaning. Since we place a high value on symmetry, we look for it when we explore nature. The remarkable thing is that we find it.
The hint of symmetry between electricity and magnetism found in the preceding chapter will be elaborated upon in this chapter. Specifically, we know that a current creates a magnetic field. If nature is symmetric here, then perhaps a magnetic field can create a current. The Hall effect is a voltage caused by a magnetic force. That voltage could drive a current. Historically, it was very shortly after Oersted discovered currents cause magnetic fields that other scientists asked the following question: Can magnetic fields cause currents? The answer was soon found by experiment to be yes. In 1831, some 12 years after Oersted’s discovery, the English scientist Michael Faraday (1791–1862) and the American scientist Joseph Henry (1797–1878) independently demonstrated that magnetic fields can produce currents. The basic process of generating emfs (electromotive force) and, hence, currents with magnetic fields is known as induction; this process is also called magnetic induction to distinguish it from charging by induction, which utilizes the Coulomb force.
Today, currents induced by magnetic fields are essential to our technological society. The ubiquitous generator—found in automobiles, on bicycles, in nuclear power plants, and so on—uses magnetism to generate current. Other devices that use magnetism to induce currents include pickup coils in electric guitars, transformers of every size, certain microphones, airport security gates, and damping mechanisms on sensitive chemical balances. Not so familiar perhaps, but important nevertheless, is that the behavior of AC circuits depends strongly on the effect of magnetic fields on currents.
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23.1 Induced Emf and Magnetic Flux
LEARNING OBJECTIVES
By the end of this section, you will be able to:
Calculate the flux of a uniform magnetic field through a loop of arbitrary orientation.
Describe methods to produce an electromotive force (emf) with a magnetic field or magnet and a loop of
wire.
The apparatus used by Faraday to demonstrate that magnetic fields can create currents is illustrated in Figure 23.3. When the switch is closed, a magnetic field is produced in the coil on the top part of the iron ring and transmitted to the coil on the bottom part of the ring. The galvanometer is used to detect any current induced in the coil on the bottom. It was found that each time the switch is closed, the galvanometer detects a current in one direction in the coil on the bottom. (You can also observe this in a physics lab.) Each time the switch is opened, the galvanometer detects a current in the opposite direction. Interestingly, if the switch remains closed or open for any length of time, there is no current through the galvanometer. Closing and opening the switch induces the current. It is the change in magnetic field that creates the current. More basic than the current that flows is the emf that causes it. The current is a result of an emf induced by a changing magnetic field, whether or not there is a path for current to flow.
23.1 • Induced Emf and Magnetic Flux 1005
FIGURE 23.3 Faraday’s apparatus for demonstrating that a magnetic field can produce a current. A change in the field produced by the top coil induces an emf and, hence, a current in the bottom coil. When the switch is opened and closed, the galvanometer registers currents in opposite directions. No current flows through the galvanometer when the switch remains closed or open.
An experiment easily performed and often done in physics labs is illustrated in Figure 23.4. An emf is induced in the coil when a bar magnet is pushed in and out of it. Emfs of opposite signs are produced by motion in opposite directions, and the emfs are also reversed by reversing poles. The same results are produced if the coil is moved rather than the magnet—it is the relative motion that is important. The faster the motion, the greater the emf, and there is no emf when the magnet is stationary relative to the coil.
FIGURE 23.4 Movement of a magnet relative to a coil produces emfs as shown. The same emfs are produced if the coil is moved relative to the magnet. The greater the speed, the greater the magnitude of the emf, and the emf is zero when there is no motion.
The method of inducing an emf used in most electric generators is shown in Figure 23.5. A coil is rotated in a magnetic field, producing an alternating current emf, which depends on rotation rate and other factors that will be explored in later sections. Note that the generator is remarkably similar in construction to a motor (another symmetry).
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FIGURE 23.5 Rotation of a coil in a magnetic field produces an emf. This is the basic construction of a generator, where work done to turn the coil is converted to electric energy. Note the generator is very similar in construction to a motor.
So we see that changing the magnitude or direction of a magnetic field produces an emf. Experiments revealed that there is a crucial quantity called the magnetic flux, , given by
23.1
where is the magnetic field strength over an area , at an angle with the perpendicular to the area as shown in
Figure 23.6. Any change in magnetic flux induces an emf. This process is defined to be electromagnetic induction. Units of magnetic flux are . As seen in Figure 23.6, ⊥, which is the component of perpendicular to the area . Thus magnetic flux is , the product of the area and the component of the magnetic field perpendicular to it.
FIGURE 23.6 Magnetic flux is related to the magnetic field and the area over which it exists. The flux is related to induction; any change in induces an emf.
All induction, including the examples given so far, arises from some change in magnetic flux . For example, Faraday changed and hence when opening and closing the switch in his apparatus (shown in Figure 23.3). This is also true for the bar magnet and coil shown in Figure 23.4. When rotating the coil of a generator, the angle and, hence, is changed. Just how great an emf and what direction it takes depend on the change in and how rapidly the change is made, as examined in the next section.
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23.2 Faraday’s Law of Induction: Lenz’s Law
LEARNING OBJECTIVES
By the end of this section, you will be able to:
Calculate emf, current, and magnetic fields using Faraday’s Law.
Explain the physical results of Lenz’s Law
Faraday’s and Lenz’s Law
Faraday’s experiments showed that the emf induced by a change in magnetic flux depends on only a few factors. First, emf is directly proportional to the change in flux . Second, emf is greatest when the change in time is smallest—that is, emf is inversely proportional to . Finally, if a coil has turns, an emf will be produced that is times greater than for a single coil, so that emf is directly proportional to . The equation for the emf induced by a change in magnetic flux is
23.2
This relationship is known as Faraday’s law of induction. The units for emf are volts, as is usual.
The minus sign in Faraday’s law of induction is very important. The minus means that the emf creates a current I and magnetic field B that oppose the change in flux —this is known as Lenz’s law. The direction (given by the minus sign) of the emfis so important that it is called Lenz’s law after the Russian Heinrich Lenz (1804–1865), who, like Faraday and Henry,independently investigated aspects of induction. Faraday was aware of the direction, but Lenz stated it so clearly that he is credited for its discovery. (See Figure 23.7.)
FIGURE 23.7 (a) When this bar magnet is thrust into the coil, the strength of the magnetic field increases in the coil. The current induced in the coil creates another field, in the opposite direction of the bar magnet’s to oppose the increase. This is one aspect of Lenz’s law—induction opposes any change in flux. (b) and (c) are two other situations. Verify for yourself that the direction of the induced
shown indeed opposes the change in flux and that the current direction shown is consistent with RHR-2.Problem-Solving Strategy for Lenz’s Law
To use Lenz’s law to determine the directions of the induced magnetic fields, currents, and emfs:
1. Make a sketch of the situation for use in visualizing and recording directions.
2. Determine the direction of the magnetic field B.
3. Determine whether the flux is increasing or decreasing.
4. Now determine the direction of the induced magnetic field B. It opposes the change in flux by adding or
23.2 • Faraday’s Law of Induction: Lenz’s Law 1007
1008 23 • Electromagnetic Induction, AC Circuits, and Electrical Technologies
subtracting from the original field.
Use RHR-2 to determine the direction of the induced current I that is responsible for the induced magnetic
field B.
The direction (or polarity) of the induced emf will now drive a current in this direction and can be
represented as current emerging from the positive terminal of the emf and returning to its negative terminal.
For practice, apply these steps to the situations shown in Figure 23.7 and to others that are part of the following text material.
Applications of Electromagnetic Induction
There are many applications of Faraday’s Law of induction, as we will explore in this chapter and others. At this juncture, let us mention several that have to do with data storage and magnetic fields. A very important application has to do with audio and video recording tapes. A plastic tape, coated with iron oxide, moves past a recording head. This recording head is basically a round iron ring about which is wrapped a coil of wire—an electromagnet (Figure 23.8). A signal in the form of a varying input current from a microphone or camera goes to the recording head. These signals (which are a function of the signal amplitude and frequency) produce varying magnetic fields at the recording head. As the tape moves past the recording head, the magnetic field orientations of the iron oxide molecules on the tape are changed thus recording the signal. In the playback mode, the magnetized tape is run past another head, similar in structure to the recording head. The different magnetic field orientations of the iron oxide molecules on the tape induces an emf in the coil of wire in the playback head. This signal then is sent to a loudspeaker or video player.
FIGURE 23.8 Recording and playback heads used with audio and video magnetic tapes. (credit: Steve Jurvetson)
Similar principles apply to computer hard drives, except at a much faster rate. Here recordings are on a coated, spinning disk. Read heads historically were made to work on the principle of induction. However, the input information is carried in digital rather than analog form – a series of 0’s or 1’s are written upon the spinning hard drive. Today, most hard drive readout devices do not work on the principle of induction, but use a technique known as giant magnetoresistance. (The discovery that weak changes in a magnetic field in a thin film of iron and chromium could bring about much larger changes in electrical resistance was one of the first large successes of nanotechnology.) Another application of induction is found on the magnetic stripe on the back of your personal credit card as used at the grocery store or the ATM machine. This works on the same principle as the audio or video tape mentioned in the last paragraph in which a head reads personal information from your card.
Another application of electromagnetic induction is when electrical signals need to be transmitted across a barrier. Consider the cochlear implant shown below. Sound is picked up by a microphone on the outside of the skull and is used to set up a varying magnetic field. A current is induced in a receiver secured in the bone beneath the skin and transmitted to electrodes in the inner ear. Electromagnetic induction can be used in other instances where electric signals need to be conveyed across various media.
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FIGURE 23.9 Electromagnetic induction used in transmitting electric currents across mediums. The device on the baby’s head induces an electrical current in a receiver secured in the bone beneath the skin. (credit: Bjorn Knetsch)
Another contemporary area of research in which electromagnetic induction is being successfully implemented (and with substantial potential) is transcranial magnetic simulation. A host of disorders, including depression and hallucinations can be traced to irregular localized electrical activity in the brain. In transcranial magnetic stimulation, a rapidly varying and very localized magnetic field is placed close to certain sites identified in the brain. Weak electric currents are induced in the identified sites and can result in recovery of electrical functioning in the brain tissue.
Sleep apnea (“the cessation of breath”) affects both adults and infants (especially premature babies and it may be a cause of sudden infant deaths [SID]). In such individuals, breath can stop repeatedly during their sleep. A cessation of more than 20 seconds can be very dangerous. Stroke, heart failure, and tiredness are just some of the possible consequences for a person having sleep apnea. The concern in infants is the stopping of breath for these longer times. One type of monitor to alert parents when a child is not breathing uses electromagnetic induction. A wire wrapped around the infant’s chest has an alternating current running through it. The expansion and contraction of the infant’s chest as the infant breathes changes the area through the coil. A pickup coil located nearby has an alternating current induced in it due to the changing magnetic field of the initial wire. If the child stops breathing, there will be a change in the induced current, and so a parent can be alerted.
EXAMPLE 23.1
Calculating Emf: How Great Is the Induced Emf?
Calculate the magnitude of the induced emf when the magnet in Figure 23.7(a) is thrust into the coil, given the
23.2 • Faraday’s Law of Induction: Lenz’s Law 1009
Making Connections: Conservation of Energy
Lenz’s law is a manifestation of the conservation of energy. The induced emf produces a current that opposes the change in flux, because a change in flux means a change in energy. Energy can enter or leave, but not instantaneously. Lenz’s law is a consequence. As the change begins, the law says induction opposes and, thus, slows the change. In fact, if the induced emf were in the same direction as the change in flux, there would be a positive feedback that would give us free energy from no apparent source—conservation of energy would be violated.
following information: the single loop coil has a radius of 6.00 cm and the average value of since the bar magnet’s field is complex) increases from 0.0500 T to 0.250 T in 0.100 s.
Strategy
To find the magnitude of emf, we use Faraday’s law of induction as stated by sign that indicates direction:
Solution
We are given that and , but we must determine the change in flux
(this is given,
, but without the minus
23.3
before we can find emf.
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Since the area of the loop is fixed, we see that Now , since it was given that
23.4
changes from 0.0500 to 0.250 T. The area of the loop is
. Thus, Entering the determined values into the expression for emf gives
Discussion
23.5
23.6
While this is an easily measured voltage, it is certainly not large enough for most practical applications. More loops in the coil, a stronger magnet, and faster movement make induction the practical source of voltages that it is.
PHET EXPLORATIONS
Faraday's Electromagnetic Lab
Play with a bar magnet and coils to learn about Faraday's law. Move a bar magnet near one or two coils to make a light bulb glow. View the magnetic field lines. A meter shows the direction and magnitude of the current. View the magnetic field lines or use a meter to show the direction and magnitude of the current. You can also play with electromagnets, generators and transformers!
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LEARNING OBJECTIVES
By the end of this section, you will be able to:
• Calculate emf, force, magnetic field, and work due to the motion of an object in a magnetic field.
As we have seen, any change in magnetic flux induces an emf opposing that change—a process known as induction. Motion is one of the major causes of induction. For example, a magnet moved toward a coil induces an emf, and a coil moved toward a magnet produces a similar emf. In this section, we concentrate on motion in a magnetic field that is stationary relative to the Earth, producing what is loosely called motional emf.
One situation where motional emf occurs is known as the Hall effect and has already been examined. Charges moving in a magnetic field experience the magnetic force , which moves opposite charges in opposite directions and produces an . We saw that the Hall effect has applications, including measurements of and . We will now see that the Hall effect is one aspect of the broader phenomenon of induction, and we will find that motional emf can be used as a power source.
Consider the situation shown in Figure 23.10. A rod is moved at a speed along a pair of conducting rails separated by a distance in a uniform magnetic field . The rails are stationary relative to and are connected to a stationary resistor . The resistor could be anything from a light bulb to a voltmeter. Consider the area enclosed by the moving rod, rails, and resistor. is perpendicular to this area, and the area is increasing as the rod moves. Thus the magnetic flux enclosed by the rails, rod, and resistor is increasing. When flux changes, an emf is induced according to Faraday’s law of induction.
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FIGURE 23.10 (a) A motional is induced between the rails when this rod moves to the right in the uniform magnetic field. The magnetic field is into the page, perpendicular to the moving rod and rails and, hence, to the area enclosed by them. (b) Lenz’s law gives the directions of the induced field and current, and the polarity of the induced emf. Since the flux is increasing, the induced field is in the opposite direction, or out of the page. RHR-2 gives the current direction shown, and the polarity of the rod will drive such a current. RHR-1 also indicates the same polarity for the rod. (Note that the script E symbol used in the equivalent circuit at the bottom of part (b) represents emf.)
To find the magnitude of emf induced along the moving rod, we use Faraday’s law of induction without the sign:
23.7
Here and below, “emf” implies the magnitude of the emf. In this equation, and the flux . We have and , since is perpendicular to . Now , since is uniform. Note that the area swept out by the rod is . Entering these quantities into the expression for emf yields
Finally, note that , the velocity of the rod. Entering this into the last expression shows that is the motional emf. This is the same expression given for the Hall effect previously.
23.8 23.9
23.3 • Motional Emf 1011
1012 23 • Electromagnetic Induction, AC Circuits, and Electrical Technologies
Making Connections: Unification of Forces
There are many connections between the electric force and the magnetic force. The fact that a moving electric field produces a magnetic field and, conversely, a moving magnetic field produces an electric field is part of why electric and magnetic forces are now considered to be different manifestations of the same force. This classic unification of electric and magnetic forces into what is called the electromagnetic force is the inspiration for contemporary efforts to unify other basic forces.
To find the direction of the induced field, the direction of the current, and the polarity of the induced emf, we apply Lenz’s law as explained in Faraday's Law of Induction: Lenz's Law. (See Figure 23.10(b).) Flux is increasing, since the area enclosed is increasing. Thus the induced field must oppose the existing one and be out of the page. And so the RHR-2 requires that I be counterclockwise, which in turn means the top of the rod is positive as shown.
Motional emf also occurs if the magnetic field moves and the rod (or other object) is stationary relative to the Earth (or some observer). We have seen an example of this in the situation where a moving magnet induces an emf in a stationary coil. It is the relative motion that is important. What is emerging in these observations is a connection between magnetic and electric fields. A moving magnetic field produces an electric field through its induced emf. We already have seen that a moving electric field produces a magnetic field—moving charge implies moving electric field and moving charge produces a magnetic field.
Motional emfs in the Earth’s weak magnetic field are not ordinarily very large, or we would notice voltage along metal rods, such as a screwdriver, during ordinary motions. For example, a simple calculation of the motional emf of a 1 m rod moving at 3.0 m/s perpendicular to the Earth’s field gives
. This small value is consistent with experience. There is a spectacular exception, however. In 1992 and 1996, attempts were made with the space shuttle to create large
motional emfs. The Tethered Satellite was to be let out on a 20 km length of wire as shown in Figure 23.11, to create a 5 kV emf by moving at orbital speed through the Earth’s field. This emf could be used to convert some of the shuttle’s kinetic and potential energy into electrical energy if a complete circuit could be made. To complete the circuit, the stationary ionosphere was to supply a return path for the current to flow. (The ionosphere is the rarefied and partially ionized atmosphere at orbital altitudes. It conducts because of the ionization. The ionosphere serves the same function as the stationary rails and connecting resistor in Figure 23.10, without which there would not be a complete circuit.) Drag on the current in the cable due to the magnetic force does the work that reduces the shuttle’s kinetic and potential energy and allows it to be converted to electrical energy. The tests were both unsuccessful. In the first, the cable hung up and could only be extended a couple of hundred meters; in the second, the cable broke when almost fully extended. Example 23.2 indicates feasibility in principle.
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EXAMPLE 23.2
Calculating the Large Motional Emf of an Object in Orbit
FIGURE 23.11 Motional emf as electrical power conversion for the space shuttle is the motivation for the Tethered Satellite experiment. A 5 kV emf was predicted to be induced in the 20 km long tether while moving at orbital speed in the Earth’s magnetic field. The circuit is completed by a return path through the stationary ionosphere.
Calculate the motional emf induced along a 20.0 km long conductor moving at an orbital speed of 7.80 km/s perpendicular to the Earth’s magnetic field.
Strategy
This is a straightforward application of the expression for motional emf— .
Solution
Entering the given values into gives
Discussion
23.10
The value obtained is greater than the 5 kV measured voltage for the shuttle experiment, since the actual orbital motion of the tether is not perpendicular to the Earth’s field. The 7.80 kV value is the maximum emf obtained when
and .
23.4 Eddy Currents and Magnetic Damping
LEARNING OBJECTIVES
By the end of this section, you will be able to:
Explain the magnitude and direction of an induced eddy current, and the effect this will have on the
object it is induced in.
Describe several applications of magnetic damping.
Eddy Currents and Magnetic Damping
As discussed in Motional Emf, motional emf is induced when a conductor moves in a magnetic field or when a magnetic field moves relative to a conductor. If motional emf can cause a current loop in the conductor, we refer to
23.4 • Eddy Currents and Magnetic Damping 1013
1014 23 • Electromagnetic Induction, AC Circuits, and Electrical Technologies
that current as an eddy current. Eddy currents can produce significant drag, called magnetic damping, on the motion involved. Consider the apparatus shown in Figure 23.12, which swings a pendulum bob between the poles of a strong magnet. (This is another favorite physics lab activity.) If the bob is metal, there is significant drag on the bob as it enters and leaves the field, quickly damping the motion. If, however, the bob is a slotted metal plate, as shown in Figure 23.12(b), there is a much smaller effect due to the magnet. There is no discernible effect on a bob made of an insulator. Why is there drag in both directions, and are there any uses for magnetic drag?
FIGURE 23.12 A common physics demonstration device for exploring eddy currents and magnetic damping. (a) The motion of a metal pendulum bob swinging between the poles of a magnet is quickly damped by the action of eddy currents. (b) There is little effect on the motion of a slotted metal bob, implying that eddy currents are made less effective. (c) There is also no magnetic damping on a nonconducting bob, since the eddy currents are extremely small.
Figure 23.13 shows what happens to the metal plate as it enters and leaves the magnetic field. In both cases, it experiences a force opposing its motion. As it enters from the left, flux increases, and so an eddy current is set up (Faraday’s law) in the counterclockwise direction (Lenz’s law), as shown. Only the right-hand side of the current loop is in the field, so that there is an unopposed force on it to the left (RHR-1). When the metal plate is completely inside the field, there is no eddy current if the field is uniform, since the flux remains constant in this region. But when the plate leaves the field on the right, flux decreases, causing an eddy current in the clockwise direction that, again, experiences a force to the left, further slowing the motion. A similar analysis of what happens when the plate swings from the right toward the left shows that its motion is also damped when entering and leaving the field.
FIGURE 23.13 A more detailed look at the conducting plate passing between the poles of a magnet. As it enters and leaves the field, the change in flux produces an eddy current. Magnetic force on the current loop opposes the motion. There is no current and no magnetic drag when the plate is completely inside the uniform field.
When a slotted metal plate enters the field, as shown in Figure 23.14, an emf is induced by the change in flux, but it is less effective because the slots limit the size of the current loops. Moreover, adjacent loops have currents in opposite directions, and their effects cancel. When an insulating material is used, the eddy current is extremely small, and so magnetic damping on insulators is negligible. If eddy currents are to be avoided in conductors, then they can be slotted or constructed of thin layers of conducting material separated by insulating sheets.
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23.4 • Eddy Currents and Magnetic Damping 1015
FIGURE 23.14 Eddy currents induced in a slotted metal plate entering a magnetic field form small loops, and the forces on them tend to cancel, thereby making magnetic drag almost zero.
Applications of Magnetic Damping
One use of magnetic damping is found in sensitive laboratory balances. To have maximum sensitivity and accuracy, the balance must be as friction-free as possible. But if it is friction-free, then it will oscillate for a very long time. Magnetic damping is a simple and ideal solution. With magnetic damping, drag is proportional to speed and becomes zero at zero velocity. Thus the oscillations are quickly damped, after which the damping force disappears, allowing the balance to be very sensitive. (See Figure 23.15.) In most balances, magnetic damping is accomplished with a conducting disc that rotates in a fixed field.
FIGURE 23.15 Magnetic damping of this sensitive balance slows its oscillations. Since Faraday’s law of induction gives the greatest effect for the most rapid change, damping is greatest for large oscillations and goes to zero as the motion stops.
Since eddy currents and magnetic damping occur only in conductors, recycling centers can use magnets to separate metals from other materials. Trash is dumped in batches down a ramp, beneath which lies a powerful magnet. Conductors in the trash are slowed by magnetic damping while nonmetals in the trash move on, separating from the metals. (See Figure 23.16.) This works for all metals, not just ferromagnetic ones. A magnet can separate out the ferromagnetic materials alone by acting on stationary trash.
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FIGURE 23.16 Metals can be separated from other trash by magnetic drag. Eddy currents and magnetic drag are created in the metals sent down this ramp by the powerful magnet beneath it. Nonmetals move on.
Other major applications of eddy currents are in metal detectors and braking systems in trains and roller coasters. Portable metal detectors (Figure 23.17) consist of a primary coil carrying an alternating current and a secondary coil in which a current is induced. An eddy current will be induced in a piece of metal close to the detector which will cause a change in the induced current within the secondary coil, leading to some sort of signal like a shrill noise. Braking using eddy currents is safer because factors such as rain do not affect the braking and the braking is smoother. However, eddy currents cannot bring the motion to a complete stop, since the force produced decreases with speed. Thus, speed can be reduced from say 20 m/s to 5 m/s, but another form of braking is needed to completely stop the vehicle. Generally, powerful rare earth magnets such as neodymium magnets are used in roller coasters. Figure 23.18 shows rows of magnets in such an application. The vehicle has metal fins (normally containing copper) which pass through the magnetic field slowing the vehicle down in much the same way as with the pendulum bob shown in Figure 23.12.
FIGURE 23.17 A soldier uses a metal detector to search for explosives and weapons. (credit: U.S. Army)
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23.5 • Electric Generators 1017
FIGURE 23.18 The rows of rare earth magnets (protruding horizontally) are used for magnetic braking in roller coasters. (credit: Stefan Scheer, Wikimedia Commons)
Induction cooktops have electromagnets under their surface. The magnetic field is varied rapidly producing eddy currents in the base of the pot, causing the pot and its contents to increase in temperature. Induction cooktops have high efficiencies and good response times but the base of the pot needs to be ferromagnetic, iron or steel for induction to work.
23.5 Electric Generators
LEARNING OBJECTIVES
By the end of this section, you will be able to:
Calculate the emf induced in a generator.
Calculate the peak emf which can be induced in a particular generator system.
Electric generators induce an emf by rotating a coil in a magnetic field, as briefly discussed in Induced Emf and Magnetic Flux. We will now explore generators in more detail. Consider the following example.
EXAMPLE 23.3
Calculating the Emf Induced in a Generator Coil
The generator coil shown in Figure 23.19 is rotated through one-fourth of a revolution (from to ) in 15.0 ms. The 200-turn circular coil has a 5.00 cm radius and is in a uniform 1.25 T magnetic field. What is the average emf induced?
1018 23 • Electromagnetic Induction, AC Circuits, and Electrical Technologies
FIGURE 23.19 When this generator coil is rotated through one-fourth of a revolution, the magnetic flux zero, inducing an emf.
Strategy
We use Faraday’s law of induction to find the average emf induced over a time :
changes from its maximum to
We know that and , and so we must determine the change in flux to find emf.
23.11
23.12 23.13
23.14
Solution
Since the area of the loop and the magnetic field strength are constant, we see that
Now, , since it was given that
The area of the loop is
Discussion
goes from to
. Thus
, and
. Entering this value gives
This is a practical average value, similar to the 120 V used in household power.
The emf calculated in Example 23.3 is the average over one-fourth of a revolution. What is the emf at any given instant? It varies with the angle between the magnetic field and a perpendicular to the coil. We can get an expression for emf as a function of time by considering the motional emf on a rotating rectangular coil of width and height in a uniform magnetic field, as illustrated in Figure 23.20.
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FIGURE 23.20 A generator with a single rectangular coil rotated at constant angular velocity in a uniform magnetic field produces an emf that varies sinusoidally in time. Note the generator is similar to a motor, except the shaft is rotated to produce a current rather than the other way around.
Charges in the wires of the loop experience the magnetic force, because they are moving in a magnetic field. Charges in the vertical wires experience forces parallel to the wire, causing currents. But those in the top and bottom segments feel a force perpendicular to the wire, which does not cause a current. We can thus find the induced emf by considering only the side wires. Motional emf is given to be , where the velocity v is perpendicular to the magnetic field . Here the velocity is at an angle with , so that its component perpendicular to is (see Figure 23.20). Thus in this case the emf induced on each side is , and they are in the same direction. The total emf around the loop is then
23.15
This expression is valid, but it does not give emf as a function of time. To find the time dependence of emf, we assume the coil rotates at a constant angular velocity . The angle is related to angular velocity by , so that
Now, linear velocity is related to angular velocity by Noting that the area of the loop is , and allowing for
is the emf induced in a generator coil of turns and area magnetic field . This can also be expressed as
where
23.16
23.17
23.18
in a uniform
23.19 23.20
is the maximum (peak) emf. Note that the frequency of the oscillation is , and the period is
. Figure 23.21 shows a graph of emf as a function of time, and it now seems reasonable that AC
voltage is sinusoidal.
. Here , so that , and loops, we find that
rotating at a constant angular velocity
23.5 • Electric Generators 1019
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FIGURE 23.21 The emf of a generator is sent to a light bulb with the system of rings and brushes shown. The graph gives the emf of the generator as a function of time. is the peak emf. The period is , where is the frequency. Note that the script E stands for emf.
The fact that the peak emf, , makes good sense. The greater the number of coils, the larger their area, and the stronger the field, the greater the output voltage. It is interesting that the faster the generator is spun (greater ), the greater the emf. This is noticeable on bicycle generators—at least the cheaper varieties. One of the authors as a juvenile found it amusing to ride his bicycle fast enough to burn out his lights, until he had to ride home lightless one dark night.
Figure 23.22 shows a scheme by which a generator can be made to produce pulsed DC. More elaborate arrangements of multiple coils and split rings can produce smoother DC, although electronic rather than mechanical means are usually used to make ripple-free DC.
FIGURE 23.22 Split rings, called commutators, produce a pulsed DC emf output in this configuration. EXAMPLE 23.4
Calculating the Maximum Emf of a Generator
Calculate the maximum emf, , of the generator that was the subject of Example 23.3.
Strategy
Once , the angular velocity, is determined, can be used to find . All other quantities are
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known.
Solution
Angular velocity is defined to be the change in angle per unit time:
One-fourth of a revolution is radians, and the time is 0.0150 s; thus,
23.21
23.22
104.7 rad/s is exactly 1000 rpm. We substitute this value for and the information from the previous example into , yielding
23.23
In real life, electric generators look a lot different than the figures in this section, but the principles are the same. The source of mechanical energy that turns the coil can be falling water (hydropower), steam produced by the burning of fossil fuels, or the kinetic energy of wind. Figure 23.23 shows a cutaway view of a steam turbine; steam moves over the blades connected to the shaft, which rotates the coil within the generator.
FIGURE 23.23 Steam turbine/generator. The steam produced by burning coal impacts the turbine blades, turning the shaft which is connected to the generator. (credit: Nabonaco, Wikimedia Commons)
Generators illustrated in this section look very much like the motors illustrated previously. This is not coincidental. In fact, a motor becomes a generator when its shaft rotates. Certain early automobiles used their starter motor as a generator. In Back Emf, we shall further explore the action of a motor as a generator.
23.6 Back Emf
LEARNING OBJECTIVES
By the end of this section, you will be able to:
• Explain what back emf is and how it is induced.
It has been noted that motors and generators are very similar. Generators convert mechanical energy into electrical energy, whereas motors convert electrical energy into mechanical energy. Furthermore, motors and generators have the same construction. When the coil of a motor is turned, magnetic flux changes, and an emf (consistent with Faraday’s law of induction) is induced. The motor thus acts as a generator whenever its coil rotates. This will happen
Discussion
The maximum emf is greater than the average emf of 131 V found in the previous example, as it should be.
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whether the shaft is turned by an external input, like a belt drive, or by the action of the motor itself. That is, when a motor is doing work and its shaft is turning, an emf is generated. Lenz’s law tells us the emf opposes any change, so that the input emf that powers the motor will be opposed by the motor’s self-generated emf, called the back emf of the motor. (See Figure 23.24.)
FIGURE 23.24 The coil of a DC motor is represented as a resistor in this schematic. The back emf is represented as a variable emf that opposes the one driving the motor. Back emf is zero when the motor is not turning, and it increases proportionally to the motor’s angular velocity.
Back emf is the generator output of a motor, and so it is proportional to the motor’s angular velocity . It is zero when the motor is first turned on, meaning that the coil receives the full driving voltage and the motor draws maximum current when it is on but not turning. As the motor turns faster and faster, the back emf grows, always opposing the driving emf, and reduces the voltage across the coil and the amount of current it draws. This effect is noticeable in a number of situations. When a vacuum cleaner, refrigerator, or washing machine is first turned on, lights in the same circuit dim briefly due to the drop produced in feeder lines by the large current drawn by the motor. When a motor first comes on, it draws more current than when it runs at its normal operating speed. When a mechanical load is placed on the motor, like an electric wheelchair going up a hill, the motor slows, the back emf drops, more current flows, and more work can be done. If the motor runs at too low a speed, the larger current can overheat it (via resistive power in the coil, ), perhaps even burning it out. On the other hand, if there is no mechanical load on the motor, it will increase its angular velocity until the back emf is nearly equal to the driving emf. Then the motor uses only enough energy to overcome friction.
Consider, for example, the motor coils represented in Figure 23.24. The coils have a equivalent resistance and are driven by a 48.0 V emf. Shortly after being turned on, they draw a current
and, thus, dissipate of energy as heat transfer. Under normal operating conditions for this motor, suppose the back emf is 40.0 V. Then at operating speed, the total
voltage across the coils is 8.0 V (48.0 V minus the 40.0 V back emf), and the current drawn is . Under normal load, then, the power dissipated is
. The latter will not cause a problem for this motor, whereas the former 5.76 kW would burn out the coils if sustained.
23.7 Transformers
LEARNING OBJECTIVES
By the end of this section, you will be able to:
Explain how a transformer works.
Calculate voltage, current, and/or number of turns given the other quantities.
Transformers do what their name implies—they transform voltages from one value to another (The term voltage is used rather than emf, because transformers have internal resistance). For example, many cell phones, laptops, video games, and power tools and small appliances have a transformer built into their plug-in unit (like that in Figure 23.25) that changes 120 V or 240 V AC into whatever voltage the device uses. Transformers are also used at several points in the power distribution systems, such as illustrated in Figure 23.26. Power is sent long distances at high voltages, because less current is required for a given amount of power, and this means less line loss, as was discussed previously. But high voltages pose greater hazards, so that transformers are employed to produce lower
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voltage at the user’s location.
FIGURE 23.25 The plug-in transformer has become increasingly familiar with the proliferation of electronic devices that operate on voltages other than common 120 V AC. Most are in the 3 to 12 V range. (credit: Shop Xtreme)
FIGURE 23.26 Transformers change voltages at several points in a power distribution system. Electric power is usually generated at greater than 10 kV, and transmitted long distances at voltages over 200 kV—sometimes as great as 700 kV—to limit energy losses. Local power distribution to neighborhoods or industries goes through a substation and is sent short distances at voltages ranging from 5 to 13 kV. This is reduced to 120, 240, or 480 V for safety at the individual user site.
The type of transformer considered in this text—see Figure 23.27—is based on Faraday’s law of induction and is very similar in construction to the apparatus Faraday used to demonstrate magnetic fields could cause currents. The two coils are called the primary and secondary coils. In normal use, the input voltage is placed on the primary, and the secondary produces the transformed output voltage. Not only does the iron core trap the magnetic field created by the primary coil, its magnetization increases the field strength. Since the input voltage is AC, a time-varying magnetic flux is sent to the secondary, inducing its AC output voltage.
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FIGURE 23.27 A typical construction of a simple transformer has two coils wound on a ferromagnetic core that is laminated to minimize eddy currents. The magnetic field created by the primary is mostly confined to and increased by the core, which transmits it to the secondary coil. Any change in current in the primary induces a current in the secondary.
For the simple transformer shown in Figure 23.27, the output voltage depends almost entirely on the input voltage and the ratio of the number of loops in the primary and secondary coils. Faraday’s law of induction for the secondary coil gives its induced output voltage to be
23.24
where is the number of loops in the secondary coil and / is the rate of change of magnetic flux. Note that the output voltage equals the induced emf ( ), provided coil resistance is small (a reasonable assumption for transformers). The cross-sectional area of the coils is the same on either side, as is the magnetic field strength, and so is the same on either side. The input primary voltage is also related to changing flux by
23.25
The reason for this is a little more subtle. Lenz’s law tells us that the primary coil opposes the change in flux caused by the input voltage , hence the minus sign (This is an example of self-inductance, a topic to be explored in some detail in later sections). Assuming negligible coil resistance, Kirchhoff’s loop rule tells us that the induced emf exactly equals the input voltage. Taking the ratio of these last two equations yields a useful relationship:
23.26
This is known as the transformer equation, and it simply states that the ratio of the secondary to primary voltages in a transformer equals the ratio of the number of loops in their coils.
The output voltage of a transformer can be less than, greater than, or equal to the input voltage, depending on the ratio of the number of loops in their coils. Some transformers even provide a variable output by allowing connection to be made at different points on the secondary coil. A step-up transformer is one that increases voltage, whereas a step-down transformer decreases voltage. Assuming, as we have, that resistance is negligible, the electrical power output of a transformer equals its input. This is nearly true in practice—transformer efficiency often exceeds 99%. Equating the power input and output,
Rearranging terms gives
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23.27
23.28
23.29
A portable x-ray unit has a step-up transformer, the 120 V input of which is transformed to the 100 kV output needed by the x-ray tube. The primary has 50 loops and draws a current of 10.00 A when in use. (a) What is the number of loops in the secondary? (b) Find the current output of the secondary.
Strategy and Solution for (a)
We solve for , the number of loops in the secondary, and enter the known values. This gives
23.30
Discussion for (a)
A large number of loops in the secondary (compared with the primary) is required to produce such a large voltage. This would be true for neon sign transformers and those supplying high voltage inside TVs and CRTs.
Strategy and Solution for (b)
We can similarly find the output current of the secondary by solving for and entering known values. This gives
23.31
Discussion for (b)
As expected, the current output is significantly less than the input. In certain spectacular demonstrations, very large voltages are used to produce long arcs, but they are relatively safe because the transformer output does not supply a large current. Note that the power input here is . This equals the power output , as we assumed in the derivation of the equations used.
The fact that transformers are based on Faraday’s law of induction makes it clear why we cannot use transformers to change DC voltages. If there is no change in primary voltage, there is no voltage induced in the secondary. One possibility is to connect DC to the primary coil through a switch. As the switch is opened and closed, the secondary produces a voltage like that in Figure 23.28. This is not really a practical alternative, and AC is in common use wherever it is necessary to increase or decrease voltages.
Combining this with , we find that
is the relationship between the output and input currents of a transformer. So if voltage increases, current decreases. Conversely, if voltage decreases, current increases.
EXAMPLE 23.5
Calculating Characteristics of a Step-Up Transformer
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FIGURE 23.28 Transformers do not work for pure DC voltage input, but if it is switched on and off as on the top graph, the output will look something like that on the bottom graph. This is not the sinusoidal AC most AC appliances need.
EXAMPLE 23.6
Calculating Characteristics of a Step-Down Transformer
A battery charger meant for a series connection of ten nickel-cadmium batteries (total emf of 12.5 V DC) needs to have a 15.0 V output to charge the batteries. It uses a step-down transformer with a 200-loop primary and a 120 V input. (a) How many loops should there be in the secondary coil? (b) If the charging current is 16.0 A, what is the input current?
Strategy and Solution for (a)
You would expect the secondary to have a small number of loops. Solving values gives
for and entering known
Strategy and Solution for (b)
The current input can be obtained by solving
Discussion
for and entering known values. This gives
23.32
23.33
The number of loops in the secondary is small, as expected for a step-down transformer. We also see that a small input current produces a larger output current in a step-down transformer. When transformers are used to operate large magnets, they sometimes have a small number of very heavy loops in the secondary. This allows the secondary to have low internal resistance and produce large currents. Note again that this solution is based on the assumption of 100% efficiency—or power out equals power in ( )—reasonable for good transformers. In this case the primary and secondary power is 240 W. (Verify this for yourself as a consistency check.) Note that the Ni- Cd batteries need to be charged from a DC power source (as would a 12 V battery). So the AC output of the
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secondary coil needs to be converted into DC. This is done using something called a rectifier, which uses devices called diodes that allow only a one-way flow of current.
Transformers have many applications in electrical safety systems, which are discussed in Electrical Safety: Systems and Devices.
PHET EXPLORATIONS
Generator
Generate electricity with a bar magnet! Discover the physics behind the phenomena by exploring magnets and how you can use them to make a bulb light.
Click to view content (https://openstax.org/l/28gen).
23.8 Electrical Safety: Systems and Devices
LEARNING OBJECTIVES
By the end of this section, you will be able to:
• Explain how various modern safety features in electric circuits work, with an emphasis on how induction
is employed.
Electricity has two hazards. A thermal hazard occurs when there is electrical overheating. A shock hazard occurs when electric current passes through a person. Both hazards have already been discussed. Here we will concentrate on systems and devices that prevent electrical hazards.
Figure 23.29 shows the schematic for a simple AC circuit with no safety features. This is not how power is distributed in practice. Modern household and industrial wiring requires the three-wire system, shown schematically in Figure 23.30, which has several safety features. First is the familiar circuit breaker (or fuse) to prevent thermal overload. Second, there is a protective case around the appliance, such as a toaster or refrigerator. The case’s safety feature is that it prevents a person from touching exposed wires and coming into electrical contact with the circuit, helping prevent shocks.
FIGURE 23.29 Schematic of a simple AC circuit with a voltage source and a single appliance represented by the resistance . There are no safety features in this circuit.
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FIGURE 23.30 The three-wire system connects the neutral wire to the earth at the voltage source and user location, forcing it to be at zero volts and supplying an alternative return path for the current through the earth. Also grounded to zero volts is the case of the appliance. A circuit breaker or fuse protects against thermal overload and is in series on the active (live/hot) wire. Note that wire insulation colors vary with region and it is essential to check locally to determine which color codes are in use (and even if they were followed in the particular installation).
There are three connections to earth or ground (hereafter referred to as “earth/ground”) shown in Figure 23.30. Recall that an earth/ground connection is a low-resistance path directly to the earth. The two earth/ground connections on the neutral wire force it to be at zero volts relative to the earth, giving the wire its name. This wire is therefore safe to touch even if its insulation, usually white, is missing. The neutral wire is the return path for the current to follow to complete the circuit. Furthermore, the two earth/ground connections supply an alternative path through the earth, a good conductor, to complete the circuit. The earth/ground connection closest to the power source could be at the generating plant, while the other is at the user’s location. The third earth/ground is to the case of the appliance, through the green earth/ground wire, forcing the case, too, to be at zero volts. The live or hot wire (hereafter referred to as “live/hot”) supplies voltage and current to operate the appliance. Figure 23.31 shows a more pictorial version of how the three-wire system is connected through a three-prong plug to an appliance.
FIGURE 23.31 The standard three-prong plug can only be inserted in one way, to assure proper function of the three-wire system.
A note on insulation color-coding: Insulating plastic is color-coded to identify live/hot, neutral and ground wires but these codes vary around the world. Live/hot wires may be brown, red, black, blue or grey. Neutral wire may be blue, black or white. Since the same color may be used for live/hot or neutral in different parts of the world, it is essential to determine the color code in your region. The only exception is the earth/ground wire which is often green but may be yellow or just bare wire. Striped coatings are sometimes used for the benefit of those who are colorblind.
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The three-wire system replaced the older two-wire system, which lacks an earth/ground wire. Under ordinary circumstances, insulation on the live/hot and neutral wires prevents the case from being directly in the circuit, so that the earth/ground wire may seem like double protection. Grounding the case solves more than one problem, however. The simplest problem is worn insulation on the live/hot wire that allows it to contact the case, as shown in Figure 23.32. Lacking an earth/ground connection (some people cut the third prong off the plug because they only have outdated two hole receptacles), a severe shock is possible. This is particularly dangerous in the kitchen, where a good connection to earth/ground is available through water on the floor or a water faucet. With the earth/ground connection intact, the circuit breaker will trip, forcing repair of the appliance. Why are some appliances still sold with two-prong plugs? These have nonconducting cases, such as power tools with impact resistant plastic cases, and are called doubly insulated. Modern two-prong plugs can be inserted into the asymmetric standard outlet in only one way, to ensure proper connection of live/hot and neutral wires.
FIGURE 23.32 Worn insulation allows the live/hot wire to come into direct contact with the metal case of this appliance. (a) The earth/ ground connection being broken, the person is severely shocked. The appliance may operate normally in this situation. (b) With a proper earth/ground, the circuit breaker trips, forcing repair of the appliance.
Electromagnetic induction causes a more subtle problem that is solved by grounding the case. The AC current in appliances can induce an emf on the case. If grounded, the case voltage is kept near zero, but if the case is not grounded, a shock can occur as pictured in Figure 23.33. Current driven by the induced case emf is called a leakage
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current, although current does not necessarily pass from the resistor to the case.
FIGURE 23.33 AC currents can induce an emf on the case of an appliance. The voltage can be large enough to cause a shock. If the case is grounded, the induced emf is kept near zero.
A ground fault interrupter (GFI) is a safety device found in updated kitchen and bathroom wiring that works based on electromagnetic induction. GFIs compare the currents in the live/hot and neutral wires. When live/hot and neutral currents are not equal, it is almost always because current in the neutral is less than in the live/hot wire. Then some of the current, again called a leakage current, is returning to the voltage source by a path other than through the neutral wire. It is assumed that this path presents a hazard, such as shown in Figure 23.34. GFIs are usually set to interrupt the circuit if the leakage current is greater than 5 mA, the accepted maximum harmless shock. Even if the leakage current goes safely to earth/ground through an intact earth/ground wire, the GFI will trip, forcing repair of the leakage.
FIGURE 23.34 A ground fault interrupter (GFI) compares the currents in the live/hot and neutral wires and will trip if their difference exceeds a safe value. The leakage current here follows a hazardous path that could have been prevented by an intact earth/ground wire.
Figure 23.35 shows how a GFI works. If the currents in the live/hot and neutral wires are equal, then they induce equal and opposite emfs in the coil. If not, then the circuit breaker will trip.
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FIGURE 23.35 A GFI compares currents by using both to induce an emf in the same coil. If the currents are equal, they will induce equal but opposite emfs.
Another induction-based safety device is the isolation transformer, shown in Figure 23.36. Most isolation transformers have equal input and output voltages. Their function is to put a large resistance between the original voltage source and the device being operated. This prevents a complete circuit between them, even in the circumstance shown. There is a complete circuit through the appliance. But there is not a complete circuit for current to flow through the person in the figure, who is touching only one of the transformer’s output wires, and neither output wire is grounded. The appliance is isolated from the original voltage source by the high resistance of the material between the transformer coils, hence the name isolation transformer. For current to flow through the person, it must pass through the high-resistance material between the coils, through the wire, the person, and back through the earth—a path with such a large resistance that the current is negligible.
FIGURE 23.36 An isolation transformer puts a large resistance between the original voltage source and the device, preventing a complete circuit between them.
The basics of electrical safety presented here help prevent many electrical hazards. Electrical safety can be pursued to greater depths. There are, for example, problems related to different earth/ground connections for appliances in close proximity. Many other examples are found in hospitals. Microshock-sensitive patients, for instance, require special protection. For these people, currents as low as 0.1 mA may cause ventricular fibrillation. The interested reader can use the material presented here as a basis for further study.
23.9 Inductance
LEARNING OBJECTIVES
By the end of this section, you will be able to:
Calculate the inductance of an inductor.
Calculate the energy stored in an inductor.
Calculate the emf generated in an inductor.
Inductors
Induction is the process in which an emf is induced by changing magnetic flux. Many examples have been discussed so far, some more effective than others. Transformers, for example, are designed to be particularly effective at
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inducing a desired voltage and current with very little loss of energy to other forms. Is there a useful physical quantity related to how “effective” a given device is? The answer is yes, and that physical quantity is called inductance.
Mutual inductance is the effect of Faraday’s law of induction for one device upon another, such as the primary coil in transmitting energy to the secondary in a transformer. See Figure 23.37, where simple coils induce emfs in one another.
FIGURE 23.37 These coils can induce emfs in one another like an inefficient transformer. Their mutual inductance M indicates the effectiveness of the coupling between them. Here a change in current in coil 1 is seen to induce an emf in coil 2. (Note that " induced" represents the induced emf in coil 2.)
In the many cases where the geometry of the devices is fixed, flux is changed by varying current. We therefore concentrate on the rate of change of current, , as the cause of induction. A change in the current in one device, coil 1 in the figure, induces an in the other. We express this in equation form as
23.34
where is defined to be the mutual inductance between the two devices. The minus sign is an expression of Lenz’s law. The larger the mutual inductance , the more effective the coupling. For example, the coils in Figure 23.37 have a small compared with the transformer coils in Figure 23.27. Units for are , which is named a henry (H), after Joseph Henry. That is, .
Nature is symmetric here. If we change the current in coil 2, we induce an in coil 1, which is given by
23.35
where is the same as for the reverse process. Transformers run backward with the same effectiveness, or mutual inductance .
A large mutual inductance may or may not be desirable. We want a transformer to have a large mutual inductance. But an appliance, such as an electric clothes dryer, can induce a dangerous emf on its case if the mutual inductance between its coils and the case is large. One way to reduce mutual inductance is to counterwind coils to cancel the magnetic field produced. (See Figure 23.38.)
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23.9 • Inductance 1033
FIGURE 23.38 The heating coils of an electric clothes dryer can be counter-wound so that their magnetic fields cancel one another, greatly reducing the mutual inductance with the case of the dryer.
Self-inductance, the effect of Faraday’s law of induction of a device on itself, also exists. When, for example, current through a coil is increased, the magnetic field and flux also increase, inducing a counter emf, as required by Lenz’s law. Conversely, if the current is decreased, an emf is induced that opposes the decrease. Most devices have a fixed geometry, and so the change in flux is due entirely to the change in current through the device. The induced emf is related to the physical geometry of the device and the rate of change of current. It is given by
23.36
where is the self-inductance of the device. A device that exhibits significant self-inductance is called an inductor, and given the symbol in Figure 23.39.
FIGURE 23.39
The minus sign is an expression of Lenz’s law, indicating that emf opposes the change in current. Units of self- inductance are henries (H) just as for mutual inductance. The larger the self-inductance of a device, the greater its opposition to any change in current through it. For example, a large coil with many turns and an iron core has a large
and will not allow current to change quickly. To avoid this effect, a small must be achieved, such as by counterwinding coils as in Figure 23.38.
A 1 H inductor is a large inductor. To illustrate this, consider a device with that has a 10 A current flowing through it. What happens if we try to shut off the current rapidly, perhaps in only 1.0 ms? An emf, given by
, will oppose the change. Thus an emf will be induced given by
. The positive sign means this large voltage is in the
same direction as the current, opposing its decrease. Such large emfs can cause arcs, damaging switching equipment, and so it may be necessary to change current more slowly.
There are uses for such a large induced voltage. Camera flashes use a battery, two inductors that function as a transformer, and a switching system or oscillator to induce large voltages. (Remember that we need a changing magnetic field, brought about by a changing current, to induce a voltage in another coil.) The oscillator system will do this many times as the battery voltage is boosted to over one thousand volts. (You may hear the high pitched whine from the transformer as the capacitor is being charged.) A capacitor stores the high voltage for later use in powering the flash. (See Figure 23.40.)
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FIGURE 23.40 Through rapid switching of an inductor, 1.5 V batteries can be used to induce emfs of several thousand volts. This voltage can be used to store charge in a capacitor for later use, such as in a camera flash attachment.
It is possible to calculate for an inductor given its geometry (size and shape) and knowing the magnetic field that it produces. This is difficult in most cases, because of the complexity of the field created. So in this text the inductance is usually a given quantity. One exception is the solenoid, because it has a very uniform field inside, a nearly zero field outside, and a simple shape. It is instructive to derive an equation for its inductance. We start by noting that the induced emf is given by Faraday’s law of induction as and, by the definition of self-inductance, as . Equating these yields
Solving for gives
This equation for the self-inductance of a device is always valid. It means that self-inductance effective the current is in creating flux; the more effective, the greater / is.
Let us use this last equation to find an expression for the inductance of a solenoid. Since the area
23.37
23.38
depends on how
fixed, the change in flux is
by . (Here
current changes, so that
This simplifies to
of a solenoid is . To find , we note that the magnetic field of a solenoid is given
, where
is the number of coils and . Substituting into
is the solenoid’s length.) Only the gives
23.39
23.40
. Note that the inductance depends
Calculate the self-inductance of a 10.0 cm long, 4.00 cm diameter solenoid that has 200 coils.
Strategy
This is a straightforward application of , since all quantities in the equation except are known.
Solution
Use the following expression for the self-inductance of a solenoid:
This is the self-inductance of a solenoid of cross-sectional area and length only on the physical characteristics of the solenoid, consistent with its definition.
EXAMPLE 23.7
Calculating the Self-inductance of a Moderate Size Solenoid
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The cross-sectional area in this example is
200, and the length is 0.100 m. We know the permeability of free space is these into the expression for gives
Discussion
This solenoid is moderate in size. Its inductance of nearly a millihenry is also considered moderate.
23.42
One common application of inductance is used in traffic lights that can tell when vehicles are waiting at the intersection. An electrical circuit with an inductor is placed in the road under the place a waiting car will stop over. The body of the car increases the inductance and the circuit changes sending a signal to the traffic lights to change colors. Similarly, metal detectors used for airport security employ the same technique. A coil or inductor in the metal detector frame acts as both a transmitter and a receiver. The pulsed signal in the transmitter coil induces a signal in the receiver. The self-inductance of the circuit is affected by any metal object in the path. Such detectors can be adjusted for sensitivity and also can indicate the approximate location of metal found on a person. See Figure 23.41.
FIGURE 23.41 The familiar security gate at an airport can not only detect metals but also indicate their approximate height above the floor. (credit: shankar s/Flickr)
Energy Stored in an Inductor
We know from Lenz’s law that inductances oppose changes in current. There is an alternative way to look at this opposition that is based on energy. Energy is stored in a magnetic field. It takes time to build up energy, and it also takes time to deplete energy; hence, there is an opposition to rapid change. In an inductor, the magnetic field is directly proportional to current and to the inductance of the device. It can be shown that the energy stored in an inductor is given by
23.9 • Inductance 1035
23.41
, is given to be
. Substituting
This expression is similar to that for the energy stored in a capacitor.
EXAMPLE 23.8
Calculating the Energy Stored in the Field of a Solenoid
23.43
How much energy is stored in the 0.632 mH inductor of the preceding example when a 30.0 A current flows through it?
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Strategy
The energy is given by the equation , and all quantities except are known.
Solution
Substituting the value for found in the previous example and the given current into gives
Discussion
23.44
This amount of energy is certainly enough to cause a spark if the current is suddenly switched off. It cannot be built up instantaneously unless the power input is infinite.