# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## Induced Emf and Magnetic Flux

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate the flux of a uniform magnetic field through a loop of arbitrary orientation.
2. 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 . 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.
An experiment easily performed and often done in physics labs is illustrated in . 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.
The method of inducing an emf used in most electric generators is shown in . 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).
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
where  is the magnetic field strength over an area , at an angle

    with the perpendicular to the area as shown in . Any change in magnetic flux  This process is defined to be electromagnetic induction. Units of magnetic flux  are . As seen in , ⊥, 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.
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 ). This is also true for the bar magnet and coil shown in . 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.

### Test Prep for AP Courses

### Section Summary

1. The crucial quantity in induction is magnetic flux , defined to be , where  is the magnetic field strength over an area  at an angle  with the perpendicular to the area.
2. Units of magnetic flux  are .
3. Any change in magnetic flux  induces an emf—the process is defined to be electromagnetic induction.

### Conceptual Questions

### Problems & Exercises


# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## Faraday’s Law of Induction: Lenz’s Law

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate emf, current, and magnetic fields using Faraday’s Law.
2. 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
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 . 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 .)
For practice, apply these steps to the situations shown in  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 (). 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.
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.
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.

### Section Summary

1. Faraday’s law of induction states that the emfinduced by a change in magnetic flux is
    when flux changes by

2. If emf is induced in a coil,

   is its number of turns.
3. The minus sign means that the emf creates a current  and magnetic field  that oppose the change in flux  —this opposition is known as Lenz’s law.

### Conceptual Questions

### Problems & Exercises


# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## Motional Emf

### Learning Objectives

By the end of this section, you will be able to:
1. 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 . 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.
To find the magnitude of emf induced along the moving rod, we use Faraday’s law of induction without the sign:
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.
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 (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 , 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 , 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.  indicates feasibility in principle.

### Section Summary

1. An emf induced by motion relative to a magnetic field

 is called a motional emf and is given by

where  is the length of the object moving at speed  relative to the field.

### Conceptual Questions

### Problems & Exercises


# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## Eddy Currents and Magnetic Damping

### Learning Objectives

By the end of this section, you will be able to:
1. Explain the magnitude and direction of an induced eddy current, and the effect this will have on the object it is induced in.
2. 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 that current as an eddy current. Eddy currents can produce significant drag, called magnetic damping, on the motion involved. Consider the apparatus shown in , 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 (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?
 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.
When a slotted metal plate enters the field, as shown in , 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.

### 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 .) In most balances, magnetic damping is accomplished with a conducting disc that rotates in a fixed field.
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 .) This works for all metals, not just ferromagnetic ones. A magnet can separate out the ferromagnetic materials alone by acting on stationary trash.
Other major applications of eddy currents are in metal detectors and braking systems in trains and roller coasters. Portable metal detectors () 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.  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 .
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.

### Section Summary

1. Current loops induced in moving conductors are called eddy currents.
2. They can create significant drag, called magnetic damping.

### Conceptual Questions

### Problems & Exercises


# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## Electric Generators

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate the emf induced in a generator.
2. 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.
The emf calculated in  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 .
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 ). 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
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 . Here , so that , and
Noting that the area of the loop is , and allowing for  loops, we find that
is the emf induced in a generator coil of  turns and area  rotating at a constant angular velocity

 in a uniform magnetic field . This can also be expressed as
where
is the maximum (peak) emf. Note that the frequency of the oscillation is , and the period is .  shows a graph of emf as a function of time, and it now seems reasonable that AC voltage is sinusoidal.
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.
 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.
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.  shows a cutaway view of a steam turbine; steam moves over the blades connected to the shaft, which rotates the coil within the generator.
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.

### Test Prep for AP Courses

### Section Summary

1. An electric generator rotates a coil in a magnetic field, inducing an emfgiven as a function of time by

where  is the area of an -turn coil rotated at a constant angular velocity  in a uniform magnetic field .
2. The peak emf  of a generator is

### Conceptual Questions

### Problems & Exercises


# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## Back Emf

### Learning Objectives

By the end of this section, you will be able to:
1. 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 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 .)
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 . 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.

### Section Summary

1. Any rotating coil will have an induced emf—in motors, this is called back emf, since it opposes the emf input to the motor.

### Conceptual Questions

### Problems & Exercises


# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## Transformers

### Learning Objectives

By the end of this section, you will be able to:
1. Explain how a transformer works.
2. 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 ) 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 . 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 voltage at the user’s location.
The type of transformer considered in this text—see —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.
For the simple transformer shown in , 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
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
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:
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
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.
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 . This is not really a practical alternative, and AC is in common use wherever it is necessary to increase or decrease voltages.
Transformers have many applications in electrical safety systems, which are discussed in Electrical Safety: Systems and Devices.

### Test Prep for AP Courses

### Section Summary

1. Transformers use induction to transform voltages from one value to another.
2. For a transformer, the voltages across the primary and secondary coils are related by

where  and  are the voltages across primary and secondary coils having  and  turns.
3. The currents  and  in the primary and secondary coils are related by .
4. A step-up transformer increases voltage and decreases current, whereas a step-down transformer decreases voltage and increases current.

### Conceptual Questions

### Problems & Exercises


# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## Electrical Safety: Systems and Devices

### Learning Objectives

By the end of this section, you will be able to:
1. 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.
 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 , 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.
There are three connections to earth or ground (hereafter referred to as “earth/ground”) shown in . 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.  shows a more pictorial version of how the three-wire system is connected through a three-prong plug to an appliance.
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.
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 . 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.
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 . Current driven by the induced case emf is called a leakage current, although current does not necessarily pass from the resistor to the case.
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 . 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.
 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.
Another induction-based safety device is the isolation transformer, shown in . 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.
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.

### Test Prep for AP Courses

### Section Summary

1. Electrical safety systems and devices are employed to prevent thermal and shock hazards.
2. Circuit breakers and fuses interrupt excessive currents to prevent thermal hazards.
3. The three-wire system guards against thermal and shock hazards, utilizing live/hot, neutral, and earth/ground wires, and grounding the neutral wire and case of the appliance.
4. A ground fault interrupter (GFI) prevents shock by detecting the loss of current to unintentional paths.
5. An isolation transformer insulates the device being powered from the original source, also to prevent shock.
6. Many of these devices use induction to perform their basic function.

### Conceptual Questions

### Problems & Exercises


# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## Inductance

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate the inductance of an inductor.
2. Calculate the energy stored in an inductor.
3. 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 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 , where simple coils induce emfs in one another.
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
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  have a small  compared with the transformer coils in . 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
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 .)
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
where  is the self-inductance of the device. A device that exhibits significant self-inductance is called an inductor, and given the symbol in .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 .
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 .)
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  depends on how 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

   of a solenoid is fixed, the change in flux is

  .

To find

  , we note that the magnetic field of a solenoid is given by . (Here , where

   is the number of coils and

   is the solenoid’s length.) Only the current changes, so that . Substituting

   into  gives
This simplifies to
This is the self-inductance of a solenoid of cross-sectional area

   and length

  . Note that the inductance depends only on the physical characteristics of the solenoid, consistent with its definition.
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 .

### 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
This expression is similar to that for the energy stored in a capacitor.

### Section Summary

1. Inductance is the property of a device that tells how effectively it induces an emf  in another device.
2. Mutual inductance is the effect of two devices in inducing emfs in each other.
3. A change in current  in one induces an emf  in the second:

        where

   is defined to be the mutual inductance between the two devices, and the minus sign is due to Lenz’s law.
4. Symmetrically, a change in current  through the second device induces an emf  in the first:

        where

   is the same mutual inductance as in the reverse process.
5. Current changes in a device induce an emf in the device itself.
6. Self-inductance is the effect of the device inducing emf in itself.
7. The device is called an inductor, and the emf induced in it by a change in current through it is

where  is the self-inductance of the inductor, and  is the rate of change of current through it. The minus sign indicates that emf opposes the change in current, as required by Lenz’s law.
8. The unit of self- and mutual inductance is the henry (H), where .
9. The self-inductance  of an inductor is proportional to how much flux changes with current. For an -turn inductor,

10. The self-inductance of a solenoid is

where  is its number of turns in the solenoid,  is its cross-sectional area,  is its length, and  is the permeability of free space.
11. The energy stored in an inductor  is

### Conceptual Questions

### Problems & Exercises


# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## RL Circuits

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate the current in an RL circuit after a specified number of characteristic time steps.
2. Calculate the characteristic time of an RL circuit.
3. Sketch the current in an RL circuit over time.
We know that the current through an inductor  cannot be turned on or off instantaneously. The change in current changes flux, inducing an emf opposing the change (Lenz’s law). How long does the opposition last? Current will flow and can be turned off, but how long does it take?  shows a switching circuit that can be used to examine current through an inductor as a function of time.
When the switch is first moved to position 1 (at ), the current is zero and it eventually rises to , where

   is the total resistance of the circuit. The opposition of the inductor  is greatest at the beginning, because the amount of change is greatest. The opposition it poses is in the form of an induced emf, which decreases to zero as the current approaches its final value. The opposing emf is proportional to the amount of change left. This is the hallmark of an exponential behavior, and it can be shown with calculus that
is the current in an RL circuit when switched on (Note the similarity to the exponential behavior of the voltage on a charging capacitor). The initial current is zero and approaches  with a characteristic time constant

for an RL circuit, given by
where  has units of seconds, since

  .
 In the first period of time , the current rises from zero to , since . The current will go 0.632 of the remainder in the next time . A well-known property of the exponential is that the final value is never exactly reached, but 0.632 of the remainder to that value is achieved in every characteristic time . In just a few multiples of the time , the final value is very nearly achieved, as the graph in (b) illustrates.
The characteristic time  depends on only two factors, the inductance  and the resistance . The greater the inductance , the greater  is, which makes sense since a large inductance is very effective in opposing change. The smaller the resistance , the greater  is. Again this makes sense, since a small resistance means a large final current and a greater change to get there. In both cases—large  and small  —more energy is stored in the inductor and more time is required to get it in and out.
When the switch in (a) is moved to position 2 and cuts the battery out of the circuit, the current drops because of energy dissipation by the resistor. But this is also not instantaneous, since the inductor opposes the decrease in current by inducing an emf in the same direction as the battery that drove the current. Furthermore, there is a certain amount of energy, , stored in the inductor, and it is dissipated at a finite rate. As the current approaches zero, the rate of decrease slows, since the energy dissipation rate is . Once again the behavior is exponential, and

 is found to be
(See (c).) In the first period of time  after the switch is closed, the current falls to 0.368 of its initial value, since . In each successive time , the current falls to 0.368 of the preceding value, and in a few multiples of , the current becomes very close to zero, as seen in the graph in (c).
In summary, when the voltage applied to an inductor is changed, the current also changes, but the change in current lags the change in voltage in an RL circuit. In Reactance, Inductive and Capacitive, we explore how an RL circuit behaves when a sinusoidal AC voltage is applied.

### Section Summary

1. When a series connection of a resistor and an inductor—an RL circuit—is connected to a voltage source, the time variation of the current is

 where
  is the final current.
2. The characteristic time constant  is  , where

    is the inductance and

   is the resistance.
3. In the first time constant , the current rises from zero to , and 0.632 of the remainder in every subsequent time interval .
4. When the inductor is shorted through a resistor, current decreases as

Here  is the initial current.
5. Current falls to  in the first time interval , and 0.368 of the remainder toward zero in each subsequent time .

### Problem Exercises


# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## Reactance, Inductive and Capacitive

### Learning Objectives

By the end of this section, you will be able to:
1. Sketch voltage and current versus time in simple inductive, capacitive, and resistive circuits.
2. Calculate inductive and capacitive reactance.
3. Calculate current and/or voltage in simple inductive, capacitive, and resistive circuits.
Many circuits also contain capacitors and inductors, in addition to resistors and an AC voltage source. We have seen how capacitors and inductors respond to DC voltage when it is switched on and off. We will now explore how inductors and capacitors react to sinusoidal AC voltage.

### Inductors and Inductive Reactance

Suppose an inductor is connected directly to an AC voltage source, as shown in . It is reasonable to assume negligible resistance, since in practice we can make the resistance of an inductor so small that it has a negligible effect on the circuit. Also shown is a graph of voltage and current as functions of time.
The graph in (b) starts with voltage at a maximum. Note that the current starts at zero and rises to its peak after the voltage that drives it, just as was the case when DC voltage was switched on in the preceding section. When the voltage becomes negative at point a, the current begins to decrease; it becomes zero at point b, where voltage is its most negative. The current then becomes negative, again following the voltage. The voltage becomes positive at point c and begins to make the current less negative. At point d, the current goes through zero just as the voltage reaches its positive peak to start another cycle. This behavior is summarized as follows:
Current lags behind voltage, since inductors oppose change in current. Changing current induces a back emf . This is considered to be an effective resistance of the inductor to AC. The rms current  through an inductor  is given by a version of Ohm’s law:
where

   is the rms voltage across the inductor and  is defined to be
with  the frequency of the AC voltage source in hertz (An analysis of the circuit using Kirchhoff’s loop rule and calculus actually produces this expression).  is called the inductive reactance, because the inductor reacts to impede the current.  has units of ohms (, so that frequency times inductance has units of ), consistent with its role as an effective resistance. It makes sense that  is proportional to , since the greater the induction the greater its resistance to change. It is also reasonable that  is proportional to frequency , since greater frequency means greater change in current. That is,  is large for large frequencies (large , small ). The greater the change, the greater the opposition of an inductor.
Note that although the resistance in the circuit considered is negligible, the AC current is not extremely large because inductive reactance impedes its flow. With AC, there is no time for the current to become extremely large.

### Capacitors and Capacitive Reactance

Consider the capacitor connected directly to an AC voltage source as shown in . The resistance of a circuit like this can be made so small that it has a negligible effect compared with the capacitor, and so we can assume negligible resistance. Voltage across the capacitor and current are graphed as functions of time in the figure.
The graph in  starts with voltage across the capacitor at a maximum. The current is zero at this point, because the capacitor is fully charged and halts the flow. Then voltage drops and the current becomes negative as the capacitor discharges. At point a, the capacitor has fully discharged ( on it) and the voltage across it is zero. The current remains negative between points a and b, causing the voltage on the capacitor to reverse. This is complete at point b, where the current is zero and the voltage has its most negative value. The current becomes positive after point b, neutralizing the charge on the capacitor and bringing the voltage to zero at point c, which allows the current to reach its maximum. Between points c and d, the current drops to zero as the voltage rises to its peak, and the process starts to repeat. Throughout the cycle, the voltage follows what the current is doing by one-fourth of a cycle:
The capacitor is affecting the current, having the ability to stop it altogether when fully charged. Since an AC voltage is applied, there is an rms current, but it is limited by the capacitor. This is considered to be an effective resistance of the capacitor to AC, and so the rms current  in the circuit containing only a capacitor  is given by another version of Ohm’s law to be
where  is the rms voltage and  is defined (As with , this expression for  results from an analysis of the circuit using Kirchhoff’s rules and calculus) to be
where  is called the capacitive reactance, because the capacitor reacts to impede the current.  has units of ohms (verification left as an exercise for the reader).  is inversely proportional to the capacitance ; the larger the capacitor, the greater the charge it can store and the greater the current that can flow. It is also inversely proportional to the frequency ; the greater the frequency, the less time there is to fully charge the capacitor, and so it impedes current less.
Although a capacitor is basically an open circuit, there is an rms current in a circuit with an AC voltage applied to a capacitor. This is because the voltage is continually reversing, charging and discharging the capacitor. If the frequency goes to zero (DC),  tends to infinity, and the current is zero once the capacitor is charged. At very high frequencies, the capacitor’s reactance tends to zero—it has a negligible reactance and does not impede the current (it acts like a simple wire). Capacitors have the opposite effect on AC circuits that inductors have.

### Resistors in an AC Circuit

Just as a reminder, consider , which shows an AC voltage applied to a resistor and a graph of voltage and current versus time. The voltage and current are exactly in phase in a resistor. There is no frequency dependence to the behavior of plain resistance in a circuit:

### Section Summary

1. For inductors in AC circuits, we find that when a sinusoidal voltage is applied to an inductor, the voltage leads the current by one-fourth of a cycle, or by a

   phase angle.
2. The opposition of an inductor to a change in current is expressed as a type of AC resistance.
3. Ohm’s law for an inductor is

where  is the rms voltage across the inductor.
4.  is defined to be the inductive reactance, given by

with  the frequency of the AC voltage source in hertz.
5. Inductive reactance  has units of ohms and is greatest at high frequencies.
6. For capacitors, we find that when a sinusoidal voltage is applied to a capacitor, the voltage follows the current by one-fourth of a cycle, or by a

   phase angle.
7. Since a capacitor can stop current when fully charged, it limits current and offers another form of AC resistance; Ohm’s law for a capacitor is

where  is the rms voltage across the capacitor.
8.  is defined to be the capacitive reactance, given by

9.  has units of ohms and is greatest at low frequencies.

### Conceptual Questions

### Problems & Exercises


# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## RLC Series AC Circuits

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate the impedance, phase angle, resonant frequency, power, power factor, voltage, and/or current in a RLC series circuit.
2. Draw the circuit diagram for an RLC series circuit.
3. Explain the significance of the resonant frequency.

### Impedance

When alone in an AC circuit, inductors, capacitors, and resistors all impede current. How do they behave when all three occur together? Interestingly, their individual resistances in ohms do not simply add. Because inductors and capacitors behave in opposite ways, they partially to totally cancel each other’s effect.  shows an RLC series circuit with an AC voltage source, the behavior of which is the subject of this section. The crux of the analysis of an RLC circuit is the frequency dependence of  and , and the effect they have on the phase of voltage versus current (established in the preceding section). These give rise to the frequency dependence of the circuit, with important “resonance” features that are the basis of many applications, such as radio tuners.
The combined effect of resistance , inductive reactance , and capacitive reactance  is defined to be impedance, an AC analogue to resistance in a DC circuit. Current, voltage, and impedance in an RLC circuit are related by an AC version of Ohm’s law:
Here  is the peak current,  the peak source voltage, and

   is the impedance of the circuit. The units of impedance are ohms, and its effect on the circuit is as you might expect: the greater the impedance, the smaller the current. To get an expression for  in terms of

  , , and , we will now examine how the voltages across the various components are related to the source voltage. Those voltages are labeled , , and  in .
Conservation of charge requires current to be the same in each part of the circuit at all times, so that we can say the currents in , , and  are equal and in phase. But we know from the preceding section that the voltage across the inductor  leads the current by one-fourth of a cycle, the voltage across the capacitor  follows the current by one-fourth of a cycle, and the voltage across the resistor  is exactly in phase with the current.  shows these relationships in one graph, as well as showing the total voltage around the circuit , where all four voltages are the instantaneous values. According to Kirchhoff’s loop rule, the total voltage around the circuit

   is also the voltage of the source.
You can see from  that while  is in phase with the current,  leads by

  , and  follows by

  . Thus  and  are

   out of phase (crest to trough) and tend to cancel, although not completely unless they have the same magnitude. Since the peak voltages are not aligned (not in phase), the peak voltage  of the source does not equal the sum of the peak voltages across , , and . The actual relationship is
where , , and  are the peak voltages across , , and , respectively. Now, using Ohm’s law and definitions from Reactance, Inductive and Capacitive, we substitute  into the above, as well as , , and , yielding
 cancels to yield an expression for

  :
which is the impedance of an RLC series AC circuit. For circuits without a resistor, take

; for those without an inductor, take ; and for those without a capacitor, take .

### Resonance in RLC Series AC Circuits

How does an RLC circuit behave as a function of the frequency of the driving voltage source? Combining Ohm’s law, , and the expression for impedance

   from  gives
The reactances vary with frequency, with  large at high frequencies and  large at low frequencies, as we have seen in three previous examples. At some intermediate frequency , the reactances will be equal and cancel, giving  —this is a minimum value for impedance, and a maximum value for  results. We can get an expression for  by taking
Substituting the definitions of  and ,
Solving this expression for  yields
where  is the resonant frequency of an RLC series circuit. This is also the natural frequency at which the circuit would oscillate if not driven by the voltage source. At , the effects of the inductor and capacitor cancel, so that , and  is a maximum.
Resonance in AC circuits is analogous to mechanical resonance, where resonance is defined to be a forced oscillation—in this case, forced by the voltage source—at the natural frequency of the system. The receiver in a radio is an RLC circuit that oscillates best at its . A variable capacitor is often used to adjust  to receive a desired frequency and to reject others.  is a graph of current as a function of frequency, illustrating a resonant peak in  at . The two curves are for two different circuits, which differ only in the amount of resistance in them. The peak is lower and broader for the higher-resistance circuit. Thus the higher-resistance circuit does not resonate as strongly and would not be as selective in a radio receiver, for example.

### Power in RLC Series AC Circuits

If current varies with frequency in an RLC circuit, then the power delivered to it also varies with frequency. But the average power is not simply current times voltage, as it is in purely resistive circuits. As was seen in , voltage and current are out of phase in an RLC circuit. There is a phase angle  between the source voltage  and the current , which can be found from
For example, at the resonant frequency or in a purely resistive circuit , so that . This implies that  and that voltage and current are in phase, as expected for resistors. At other frequencies, average power is less than at resonance. This is both because voltage and current are out of phase and because  is lower. The fact that source voltage and current are out of phase affects the power delivered to the circuit. It can be shown that the average power is
Thus  is called the power factor, which can range from 0 to 1. Power factors near 1 are desirable when designing an efficient motor, for example. At the resonant frequency, .
Power delivered to an RLC series AC circuit is dissipated by the resistance alone. The inductor and capacitor have energy input and output but do not dissipate it out of the circuit. Rather they transfer energy back and forth to one another, with the resistor dissipating exactly what the voltage source puts into the circuit. This assumes no significant electromagnetic radiation from the inductor and capacitor, such as radio waves. Such radiation can happen and may even be desired, as we will see in the next chapter on electromagnetic radiation, but it can also be suppressed as is the case in this chapter. The circuit is analogous to the wheel of a car driven over a corrugated road as shown in . The regularly spaced bumps in the road are analogous to the voltage source, driving the wheel up and down. The shock absorber is analogous to the resistance damping and limiting the amplitude of the oscillation. Energy within the system goes back and forth between kinetic (analogous to maximum current, and energy stored in an inductor) and potential energy stored in the car spring (analogous to no current, and energy stored in the electric field of a capacitor). The amplitude of the wheels’ motion is a maximum if the bumps in the road are hit at the resonant frequency.
A pure LC circuit with negligible resistance oscillates at , the same resonant frequency as an RLC circuit. It can serve as a frequency standard or clock circuit—for example, in a digital wristwatch. With a very small resistance, only a very small energy input is necessary to maintain the oscillations. The circuit is analogous to a car with no shock absorbers. Once it starts oscillating, it continues at its natural frequency for some time.  shows the analogy between an LC circuit and a mass on a spring.

### Section Summary

1. The AC analogy to resistance is impedance

  , the combined effect of resistors, inductors, and capacitors, defined by the AC version of Ohm’s law:
    where  is the peak current and  is the peak source voltage.
2. Impedance has units of ohms and is given by .
3. The resonant frequency , at which , is

4. In an AC circuit, there is a phase angle  between source voltage  and the current , which can be found from

5.  for a purely resistive circuit or an RLC circuit at resonance.
6. The average power delivered to an RLC circuit is affected by the phase angle and is given by
     is called the power factor, which ranges from 0 to 1.

### Conceptual Questions

### Problems & Exercises


# Electromagnetic Waves

## Introduction to Electromagnetic Waves

The beauty of a coral reef, the warm radiance of sunshine, the sting of sunburn, the X-ray revealing a broken bone, even microwave popcorn—all are brought to us by electromagnetic waves. The list of the various types of electromagnetic waves, ranging from radio transmission waves to nuclear gamma-ray (-ray) emissions, is interesting in itself.
Even more intriguing is that all of these widely varied phenomena are different manifestations of the same thing—electromagnetic waves. (See .) What are electromagnetic waves? How are they created, and how do they travel? How can we understand and organize their widely varying properties? What is their relationship to electric and magnetic effects? These and other questions will be explored.

### Discovering a New Phenomenon

It is worth noting at the outset that the general phenomenon of electromagnetic waves was predicted by theory before it was realized that light is a form of electromagnetic wave. The prediction was made by James Clerk Maxwell in the mid-19th century when he formulated a single theory combining all the electric and magnetic effects known by scientists at that time. “Electromagnetic waves” was the name he gave to the phenomena his theory predicted.
Such a theoretical prediction followed by experimental verification is an indication of the power of science in general, and physics in particular. The underlying connections and unity of physics allow certain great minds to solve puzzles without having all the pieces. The prediction of electromagnetic waves is one of the most spectacular examples of this power. Certain others, such as the prediction of antimatter, will be discussed in later modules.


# Electromagnetic Waves

## Maxwell’s Equations: Electromagnetic Waves Predicted and Observed

### Learning Objectives

By the end of this section, you will be able to:
1. Restate Maxwell’s equations.
The Scotsman James Clerk Maxwell (1831–1879) is regarded as the greatest theoretical physicist of the 19th century. (See .) Although he died young, Maxwell not only formulated a complete electromagnetic theory, represented by Maxwell’s equations, he also developed the kinetic theory of gases and made significant contributions to the understanding of color vision and the nature of Saturn’s rings.
Maxwell brought together all the work that had been done by brilliant physicists such as Oersted, Coulomb, Gauss, and Faraday, and added his own insights to develop the overarching theory of electromagnetism. Maxwell’s equations are paraphrased here in words because their mathematical statement is beyond the level of this text. However, the equations illustrate how apparently simple mathematical statements can elegantly unite and express a multitude of concepts—why mathematics is the language of science.
Maxwell’s equations encompass the major laws of electricity and magnetism. What is not so apparent is the symmetry that Maxwell introduced in his mathematical framework. Especially important is his addition of the hypothesis that changing electric fields create magnetic fields. This is exactly analogous (and symmetric) to Faraday’s law of induction and had been suspected for some time, but fits beautifully into Maxwell’s equations.
Symmetry is apparent in nature in a wide range of situations. In contemporary research, symmetry plays a major part in the search for sub-atomic particles using massive multinational particle accelerators such as the new Large Hadron Collider at CERN.
Since changing electric fields create relatively weak magnetic fields, they could not be easily detected at the time of Maxwell’s hypothesis. Maxwell realized, however, that oscillating charges, like those in AC circuits, produce changing electric fields. He predicted that these changing fields would propagate from the source like waves generated on a lake by a jumping fish.
The waves predicted by Maxwell would consist of oscillating electric and magnetic fields—defined to be an electromagnetic wave (EM wave). Electromagnetic waves would be capable of exerting forces on charges great distances from their source, and they might thus be detectable. Maxwell calculated that electromagnetic waves would propagate at a speed given by the equation
When the values for  and  are entered into the equation for
 , we find that
which is the speed of light. In fact, Maxwell concluded that light is an electromagnetic wave having such wavelengths that it can be detected by the eye.
Other wavelengths should exist—it remained to be seen if they did. If so, Maxwell’s theory and remarkable predictions would be verified, the greatest triumph of physics since Newton. Experimental verification came within a few years, but not before Maxwell’s death.

### Hertz’s Observations

The German physicist Heinrich Hertz (1857–1894) was the first to generate and detect certain types of electromagnetic waves in the laboratory. Starting in 1887, he performed a series of experiments that not only confirmed the existence of electromagnetic waves, but also verified that they travel at the speed of light.
Hertz used an AC  (resistor-inductor-capacitor) circuit that resonates at a known frequency  and connected it to a loop of wire as shown in . High voltages induced across the gap in the loop produced sparks that were visible evidence of the current in the circuit and that helped generate electromagnetic waves.
Across the laboratory, Hertz had another loop attached to another  circuit, which could be tuned (as the dial on a radio) to the same resonant frequency as the first and could, thus, be made to receive electromagnetic waves. This loop also had a gap across which sparks were generated, giving solid evidence that electromagnetic waves had been received.
Hertz also studied the reflection, refraction, and interference patterns of the electromagnetic waves he generated, verifying their wave character. He was able to determine wavelength from the interference patterns, and knowing their frequency, he could calculate the propagation speed using the equation  (velocity—or speed—equals frequency times wavelength). Hertz was thus able to prove that electromagnetic waves travel at the speed of light. The SI unit for frequency, the hertz (), is named in his honor.

### Section Summary

1. Electromagnetic waves consist of oscillating electric and magnetic fields and propagate at the speed of light

  . They were predicted by Maxwell, who also showed that
        where

2. Maxwell’s prediction of electromagnetic waves resulted from his formulation of a complete and symmetric theory of electricity and magnetism, known as Maxwell’s equations.
3. These four equations are paraphrased in this text, rather than presented numerically, and encompass the major laws of electricity and magnetism. First is Gauss’s law for electricity, second is Gauss’s law for magnetism, third is Faraday’s law of induction, including Lenz’s law, and fourth is Ampere’s law in a symmetric formulation that adds another source of magnetism—changing electric fields.

### Problems & Exercises


# Electromagnetic Waves

## Production of Electromagnetic Waves

### Learning Objectives

By the end of this section, you will be able to:
1. Describe the electric and magnetic waves as they move out from a source, such as an AC generator.
2. Explain the mathematical relationship between the magnetic field strength and the electrical field strength.
3. Calculate the maximum strength of the magnetic field in an electromagnetic wave, given the maximum electric field strength.
We can get a good understanding of electromagnetic waves (EM) by considering how they are produced. Whenever a current varies, associated electric and magnetic fields vary, moving out from the source like waves. Perhaps the easiest situation to visualize is a varying current in a long straight wire, produced by an AC generator at its center, as illustrated in .

The electric field () shown surrounding the wire is produced by the charge distribution on the wire. Both the  and the charge distribution vary as the current changes. The changing field propagates outward at the speed of light.
There is an associated magnetic field () which propagates outward as well (see ). The electric and magnetic fields are closely related and propagate as an electromagnetic wave. This is what happens in broadcast antennae such as those in radio and TV stations.
Closer examination of the one complete cycle shown in  reveals the periodic nature of the generator-driven charges oscillating up and down in the antenna and the electric field produced. At time , there is the maximum separation of charge, with negative charges at the top and positive charges at the bottom, producing the maximum magnitude of the electric field (or -field) in the upward direction. One-fourth of a cycle later, there is no charge separation and the field next to the antenna is zero, while the maximum -field has moved away at speed .
As the process continues, the charge separation reverses and the field reaches its maximum downward value, returns to zero, and rises to its maximum upward value at the end of one complete cycle. The outgoing wave has an amplitude proportional to the maximum separation of charge. Its wavelength is proportional to the period of the oscillation and, hence, is smaller for short periods or high frequencies. (As usual, wavelength and frequency are inversely proportional.)

### Electric and Magnetic Waves: Moving Together

Following Ampere’s law, current in the antenna produces a magnetic field, as shown in . The relationship between  and  is shown at one instant in  (a). As the current varies, the magnetic field varies in magnitude and direction.

The magnetic field lines also propagate away from the antenna at the speed of light, forming the other part of the electromagnetic wave, as seen in  (b). The magnetic part of the wave has the same period and wavelength as the electric part, since they are both produced by the same movement and separation of charges in the antenna.
The electric and magnetic waves are shown together at one instant in time in . The electric and magnetic fields produced by a long straight wire antenna are exactly in phase. Note that they are perpendicular to one another and to the direction of propagation, making this a transverse wave.
Electromagnetic waves generally propagate out from a source in all directions, sometimes forming a complex radiation pattern. A linear antenna like this one will not radiate parallel to its length, for example. The wave is shown in one direction from the antenna in  to illustrate its basic characteristics.
Instead of the AC generator, the antenna can also be driven by an AC circuit. In fact, charges radiate whenever they are accelerated. But while a current in a circuit needs a complete path, an antenna has a varying charge distribution forming a standing wave, driven by the AC. The dimensions of the antenna are critical for determining the frequency of the radiated electromagnetic waves. This is a resonant phenomenon and when we tune radios or TV, we vary electrical properties to achieve appropriate resonant conditions in the antenna.

### Receiving Electromagnetic Waves

Electromagnetic waves carry energy away from their source, similar to a sound wave carrying energy away from a standing wave on a guitar string. An antenna for receiving EM signals works in reverse. And like antennas that produce EM waves, receiver antennas are specially designed to resonate at particular frequencies.
An incoming electromagnetic wave accelerates electrons in the antenna, setting up a standing wave. If the radio or TV is switched on, electrical components pick up and amplify the signal formed by the accelerating electrons. The signal is then converted to audio and/or video format. Sometimes big receiver dishes are used to focus the signal onto an antenna.
In fact, charges radiate whenever they are accelerated. When designing circuits, we often assume that energy does not quickly escape AC circuits, and mostly this is true. A broadcast antenna is specially designed to enhance the rate of electromagnetic radiation, and shielding is necessary to keep the radiation close to zero. Some familiar phenomena are based on the production of electromagnetic waves by varying currents. Your microwave oven, for example, sends electromagnetic waves, called microwaves, from a concealed antenna that has an oscillating current imposed on it.

### Relating -Field and -Field Strengths

There is a relationship between the - and -field strengths in an electromagnetic wave. This can be understood by again considering the antenna just described. The stronger the -field created by a separation of charge, the greater the current and, hence, the greater the -field created.
Since current is directly proportional to voltage (Ohm’s law) and voltage is directly proportional to -field strength, the two should be directly proportional. It can be shown that the magnitudes of the fields do have a constant ratio, equal to the speed of light. That is,
is the ratio of -field strength to -field strength in any electromagnetic wave. This is true at all times and at all locations in space. A simple and elegant result.
The result of this example is consistent with the statement made in the module Maxwell’s Equations: Electromagnetic Waves Predicted and Observed that changing electric fields create relatively weak magnetic fields. They can be detected in electromagnetic waves, however, by taking advantage of the phenomenon of resonance, as Hertz did. A system with the same natural frequency as the electromagnetic wave can be made to oscillate. All radio and TV receivers use this principle to pick up and then amplify weak electromagnetic waves, while rejecting all others not at their resonant frequency.

### Test Prep for AP Courses

### Section Summary

1. Electromagnetic waves are created by oscillating charges (which radiate whenever accelerated) and have the same frequency as the oscillation.
2. Since the electric and magnetic fields in most electromagnetic waves are perpendicular to the direction in which the wave moves, it is ordinarily a transverse wave.
3. The strengths of the electric and magnetic parts of the wave are related by
    which implies that the magnetic field

### Conceptual Questions

### Problems & Exercises