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Sabtu, 21 November 2009


The study of magnetism is a science in itself. Electrical and magnetic phe- nomena interact; a detailed study of magnetism and electromagnetism could easily fill a book. Magnetism exists whenever electric charges move relative to other objects or relative to a frame of reference.


The Earth has a core made up largely of iron heated to the extent that some of it is liquid. As the Earth rotates, the iron flows in complex ways. This flow gives rise to a huge magnetic field, called the geomagnetic field, that surrounds the Earth.


The geomagnetic field has poles, as a bar magnet does. These poles are near, but not at, the geographic poles. The north geomagnetic pole is located
in the frozen island region of northern Canada. The south geomagnetic pole
is in the ocean near the coast of Antarctica. The geomagnetic axis is thus somewhat tilted relative to the axis on which the Earth rotates. Not only this, but the geomagnetic axis does not exactly run through the center of the Earth. It’s like an apple core that’s off center.

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PART 2 Electricity, Magnetism, and Electronics


Charged particles from the Sun, constantly streaming outward through the solar system, distort the geomagnetic field. This solar wind in effect
“blows” the field out of shape. On the side of the Earth facing the Sun, the field is compressed; on the side of the Earth opposite the Sun, the field is stretched out. This effect occurs with the magnetic fields around the other planets, too, notably Jupiter.
As the Earth rotates, the geomagnetic field does a complex twist-and- turn dance into space in the direction facing away from the Sun. At and near the Earth’s surface, the field is nearly symmetrical with respect to the geo- magnetic poles. As the distance from the Earth increases, the extent of geomagnetic-field distortion increases.


The presence of the Earth’s magnetic field was noticed in ancient times. Certain rocks, called lodestones, when hung by strings, always orient them- selves in a generally north-south direction. Long ago this was correctly attributed to the presence of a “force” in the air. It was some time before the reasons for this phenomenon were known, but the effect was put to use by seafarers and land explorers. Today, a magnetic compass can still be a valuable navigation aid, used by mariners, backpackers, and others who travel far from familiar landmarks. It can work when more sophisticated navigational devices fail.
The geomagnetic field and the magnetic field around a compass nee- dle interact so that a force is exerted on the little magnet inside the com- pass. This force works not only in a horizontal plane (parallel to the Earth’s surface) but vertically, too, in most locations. The vertical com- ponent is zero at the geomagnetic equator, a line running around the globe equidistant from both geomagnetic poles. As the geomagnetic lat- itude increases toward either the north or the south geomagnetic pole, the magnetic force pulls up and down on the compass needle more and more. The extent of this vertical component at any particular location is called the inclination of the geomagnetic field at that location. You have noticed this when you hold a compass. One end of the needle seems to insist on touching the compass face, whereas the other end tilts up toward the glass.

Magnetic Force

As children, most of us discovered that magnets “stick” to some metals. Iron, nickel, and alloys containing either or both of these elements are known as ferromagnetic materials. Magnets exert force on these metals. Magnets generally do not exert force on other metals unless those metals carry electric currents. Electrically insulating substances never attract mag- nets under normal conditions.


When a magnet is brought near a piece of ferromagnetic material, the atoms in the material become lined up so that the metal is temporarily mag- netized. This produces a magnetic force between the atoms of the ferro- magnetic substance and those in the magnet.
If a magnet is near another magnet, the force is even stronger than it is when the same magnet is near a ferromagnetic substance. In addition, the force can be either repulsive (the magnets repel, or push away from each other) or attractive (the magnets attract, or pull toward each other) depend- ing on the way the magnets are turned. The force gets stronger as the mag- nets are brought closer and closer together.
Some magnets are so strong that no human being can pull them apart if they get “stuck” together, and no person can bring them all the way together against their mutual repulsive force. This is especially true of electromag- nets, discussed later in this chapter. The tremendous forces available are of use in industry. A huge electromagnet can be used to carry heavy pieces of scrap iron or steel from place to place. Other electromagnets can provide sufficient repulsion to suspend one object above another. This is called magnetic levitation.


Whenever the atoms in a ferromagnetic material are aligned, a magnetic field exists. A magnetic field also can be caused by the motion of electric charge carriers either in a wire or in free space.
The magnetic field around a permanent magnet arises from the same cause as the field around a wire that carries an electric current. The responsible

Electricity, Magnetism, and Electronics

factor in either case is the motion of electrically charged particles. In a
wire, the electrons move along the conductor, being passed from atom to atom. In a permanent magnet, the movement of orbiting electrons occurs in such a manner that an “effective current” is produced by the way the elec- trons move within individual atoms.
Magnetic fields can be produced by the motion of charged particles through space. The Sun is constantly ejecting protons and helium nuclei. These particles carry a positive electric charge. Because of this, they pro- duce “effective currents” as they travel through space. These currents in turn generate magnetic fields. When these fields interact with the Earth’s geomagnetic field, the particles are forced to change direction, and they are accelerated toward the geomagnetic poles.
If there is an eruption on the Sun called a solar flare, the Sun ejects more charged particles than normal. When these arrive at the Earth’s geomag- netic poles, their magnetic fields, collectively working together, can disrupt the Earth’s geomagnetic field. Then there is a geomagnetic storm. Such an event causes changes in the Earth’s ionosphere, affecting long-distance radio communications at certain frequencies. If the fluctuations are intense enough, even wire communications and electrical power transmission can be interfered with. Microwave transmissions generally are immune to the effects of geomagnetic storms. Fiberoptic cable links and free-space laser communications are not affected. Aurora (northern or southern lights) are frequently observed at night during geomagnetic storms.


Physicists consider magnetic fields to be comprised of flux lines, or lines of flux. The intensity of the field is determined according to the number of flux lines passing through a certain cross section, such as a centimeter squared (cm2) or a meter squared (m2). The lines are not actual threads in space, but it is intuitively appealing to imagine them this way, and their presence can be shown by simple experimentation.
Have you seen the classical demonstration in which iron filings are placed on a sheet of paper, and then a magnet is placed underneath the paper? The filings arrange themselves in a pattern that shows, roughly, the
“shape” of the magnetic field in the vicinity of the magnet. A bar magnet has a field whose lines of flux have a characteristic pattern (Fig. 14-1).
Another experiment involves passing a current-carrying wire through the paper at a right angle. The iron filings become grouped along circles

Fig. 14-1. Magnetic flux around a bar magnet.

centered at the point where the wire passes through the paper. This shows
that the lines of flux are circular as viewed through any plane passing through the wire at a right angle. The flux circles are centered on the axis of the wire, or the axis along which the charge carriers move (Fig. 14-2).


A magnetic field has a direction, or orientation, at any point in space near a current-carrying wire or a permanent magnet. The flux lines run parallel to the direction of the field. A magnetic field is considered to begin, or originate
PART 2 Electricity, Magnetism, and Electronics

Fig. 14-2. Magnetic flux produced by charge carriers traveling in a straight line.

at a north pole and to end, or terminate, at a south pole. These poles are not
the same as the geomagnetic poles; in fact, they are precisely the opposite! The north geomagnetic pole is in reality a south pole because it attracts the north poles of magnetic compasses. Similarly, the south geomagnetic pole is
a north pole because it attracts the south poles of compasses. In the case of a permanent magnet, it is usually, but not always, apparent where the magnetic poles are located. With a current-carrying wire, the magnetic field goes around and around endlessly, like a dog chasing its own tail.
A charged electric particle, such as a proton, hovering in space, is an
electric monopole, and the electrical flux lines around it aren’t closed. A
CHAPTER 14 Magnetism 351

positive charge does not have to be mated with a negative charge. The elec-
trical flux lines around any stationary charged particle run outward in all directions for a theoretically infinite distance. However, a magnetic field is different. Under normal circumstances, all magnetic flux lines are closed loops. With permanent magnets, there is always a starting point (the north pole) and an ending point (the south pole). Around the current-carrying wire, the loops are circles. This can be seen plainly in experiments with iron filings on paper.


You might at first think that the magnetic field around a current-carrying wire is caused by a monopole or that there aren’t any poles at all because the concentric circles apparently don’t originate or terminate anywhere. However, think of any geometric plane containing the wire. A magnetic dipole, or pair of opposite magnetic poles, is formed by the lines of flux going halfway around on either side. There in effect are two such “mag- nets” stuck together. The north poles and the south poles are thus not points but rather faces of the plane backed right up against each other.
The lines of flux in the vicinity of a magnetic dipole always connect the two poles. Some flux lines are straight in a local sense, but in a larger sense they are always curves. The greatest magnetic field strength around a bar magnet is near the poles, where the flux lines converge. Around a current- carrying wire, the greatest field strength is near the wire.

Magnetic Field Strength

The overall magnitude of a magnetic field is measured in units called webers, symbolized Wb. A smaller unit, the maxwell (Mx), is sometimes used if a magnetic field is very weak. One weber is equivalent to 100 mil- lion maxwells. Thus 1 Wb 108 Mx, and 1 Mx 10 8 Wb.


If you have a permanent magnet or electromagnet, you might see its
“strength” expressed in terms of webers or maxwells. More often, though,

PART 2 Electricity, Magnetism, and Electronics

you’ll hear or read about units called teslas (T) or gauss (G). These units
are expressions of the concentration, or intensity, of the magnetic field within a certain cross section. The flux density, or number of “flux lines per unit cross-sectional area,” is a more useful expressions for magnetic effects than the overall quantity of magnetism. Flux density is customarily denoted
B in equations. A flux density of 1 tesla is equal to 1 weber per meter squared (1 Wb/m2). A flux density of 1 gauss is equal to 1 maxwell per cen- timeter squared (1 Mx/cm2). It turns out that the gauss is equivalent to exactly 0.0001 tesla. That is, 1 G 10 4 T, and 1 T 104 G. To convert from teslas to gauss (not gausses!), multiply by 104; to convert from gauss
to teslas, multiply by 10 4.
If you are confused by the distinctions between webers and teslas or between maxwells and gauss, think of a light bulb. Suppose that a lamp emits 20 W of visible-light power. If you enclose the bulb completely, then
20 W of visible light strike the interior walls of the chamber, no matter how large or small the chamber. However, this is not a very useful notion of the brightness of the light. You know that a single bulb gives plenty of light for
a small walk-in closet but is nowhere near adequate to illuminate a gymna- sium. The important consideration is the number of watts per unit area. When we say the bulb gives off a certain number of watts of visible light, it’s like saying a magnet has an overall magnetism of so many webers or maxwells. When we say that the bulb produces a certain number of watts per unit area, it’s analogous to saying that a magnetic field has a flux den- sity of so many teslas or gauss.


When working with electromagnets, another unit is employed. This is the ampere-turn (At). It is a unit of magnetomotive force. A wire bent into a cir- cle and carrying 1 A of current produces 1 At of magnetomotive force. If the wire is bent into a loop having 50 turns, and the current stays the same, the resulting magnetomotive force becomes 50 times as great, that is, 50 At.
If the current in the 50-turn loop is reduced to 1/50 A or 20 mA, the mag- netomotive force goes back down to 1 At.
A unit called the gilbert is sometimes used to express magnetomotive force. This unit is equal to about 1.256 At. To approximate ampere-turns when the number of gilberts is known, multiply by 1.256. To approxi- mate gilberts when the number of ampere-turns is known, multiply by
CHAPTER 14 Magnetism 353

In a straight wire carrying a steady direct current surrounded by air or by free space (a vacuum), the flux density is greatest near the wire and dimin- ishes with increasing distance from the wire. You ask, “Is there a formula that expresses flux density as a function of distance from the wire?” The answer is yes. Like all formulas in physics, it is perfectly accurate only under idealized circumstances.
Consider a wire that is perfectly thin, as well as perfectly straight. Suppose that it carries a current of I amperes. Let the flux density (in teslas) be denoted B. Consider a point P at a distance r (in meters) from the wire, as measured along the shortest possible route (that is, within a plane perpendicular to the wire). This is illustrated in Fig. 14-3. The following formula applies:
B 2 10 7 (I/r)

In this formula, the value 2 can be considered mathematically exact to any desired number of significant figures.
As long as the thickness of the wire is small compared with the distance
r from it, and as long as the wire is reasonably straight in the vicinity of the point P at which the flux density is measured, this formula is a good indi- cator of what happens in real life.

What is the flux density in teslas at a distance of 20 cm from a straight, thin
wire carrying 400 mA of direct current?

First, convert everything to units in the International System (SI). This means
that r 0.20 m and I 0.400 A. Knowing these values, plug them directly into the formula:
B 2 10 7 (I/r)
2.00 10 7 (0.400/0.20)
4.0 10 7 T

In the preceding scenario, what is the flux density Bgauss (in gauss) at point P?
To figure this out, we must convert from teslas to gauss. This means that we must multiply the answer from the preceding problem by 104:

7 4
3 G

PART 2 Electricity, Magnetism, and Electronics

Fig. 14-3. Flux density varies inversely with the distance from a wire carrying direct current.


Any electric current, or movement of charge carriers, produces a magnetic field. This field can become intense in a tightly coiled wire having many turns and carrying a large electric current. When a ferromagnetic rod, called a core, is placed inside the coil, the magnetic lines of flux are con- centrated in the core, and the field strength in and near the core becomes tremendous. This is the principle of an electromagnet (Fig. 14-4).
Electromagnets are almost always cylindrical in shape. Sometimes the cylinder is long and thin; in other cases it is short and fat. Whatever the
CHAPTER 14 Magnetism

Fig. 14-4. A simple electromagnet.

ratio of diameter to length for the core, however, the principle is always the
same: The flux produced by the current temporarily magnetizes the core.


You can build a dc electromagnet by taking a large iron or steel bolt (such as a stove bolt) and wrapping a couple of hundred turns of wire around it. These items are available in almost any hardware store. Be sure the bolt is made of ferromagnetic material. (If a permanent magnet “sticks” to the bolt, the bolt is ferromagnetic.) Ideally, the bolt should be at least 3 8 inch
in diameter and several inches long. You must use insulated or enameled wire, preferably made of solid, soft copper. “Bell wire” works well.
Be sure that all the wire turns go in the same direction. A large 6-V
“lantern battery” can provide plenty of dc to operate the electromagnet. Never leave the coil connected to the battery for more than a few seconds at
a time. And do not—repeat, do not—use an automotive battery for this exper- iment. The near-short-circuit produced by an electromagnet can cause the acid from such a battery to violently boil out, and this acid is dangerous stuff.

PART 2 Electricity, Magnetism, and Electronics

Direct-current electromagnets have defined north and south poles, just
like permanent magnets. The main difference is that an electromagnet can get much stronger than any permanent magnet. You should see evidence of this if you do the preceding experiment with a large enough bolt and enough turns of wire. Another difference between an electromagnet and a permanent magnet is the fact that in an electromagnet, the magnetic field exists only as long as the coil carries current. When the power source is removed, the magnetic field collapses. In some cases, a small amount of residual magnetism remains in the core, but this is much weaker than the magnetism generated when current flows in the coil.


You might get the idea that the electromagnet can be made far stronger if, rather than using a lantern battery for the current source, you plug the wires into a wall outlet. In theory, this is true. In practice, you’ll blow the fuse or circuit breaker. Do not try this. The electrical circuits in some buildings are not adequately protected, and a short circuit can create a fire hazard. Also, you can get a lethal shock from the 117-V utility mains. (Do this experi- ment in your mind, and leave it at that.)
Some electromagnets use 60-Hz ac. These magnets “stick” to ferromag- netic objects. The polarity of the magnetic field reverses every time the direction of the current reverses; there are 120 fluctuations, or 60 complete north-to-south-to-north polarity changes, every second (Fig. 14-5). If a per- manent magnet is brought near either “pole” of an ac electromagnet of the same strength, there is no net force resulting from the ac electromagnetism because there is an equal amount of attractive and repulsive force between the alternating magnetic field and the steady external field. However, there
is an attractive force between the core material and the nearby magnet pro- duced independently of the alternating magnetic field resulting from the ac
in the coil.

Suppose that the frequency of the ac applied to an electromagnet is 600 Hz
instead of 60 Hz. What will happen to the interaction between the alternating magnetic field and a nearby permanent magnet of the same strength?

Assuming that no change occurs in the behavior of the core material, the
situation will be the same as is the case at 60 Hz or at any other ac frequency.


Fig. 14-5. Polarity change in an ac electromagnet.

Magnetic Materials

Some substances cause magnetic lines of flux to bunch closer together than they are in the air; other materials cause the lines of flux to spread farther apart. The first kind of material is ferromagnetic. Substances of this type are, as we have discussed already, “magnetizable.” The other kind of mate- rial is called diamagnetic. Wax, dry wood, bismuth, and silver are examples of substances that decrease magnetic flux density. No diamagnetic material reduces the strength of a magnetic field by anywhere near the factor that ferromagnetic substances can increase it.
The magnetic characteristics of a substance or medium can be quantified in two important but independent ways: permeability and retentivity.


Permeability, symbolized by the lowercase Greek mu ( ), is measured on a scale relative to a vacuum, or free space. A perfect vacuum is assigned, by

PART 2 Electricity, Magnetism, and Electronics

convention, a permeability figure of exactly 1. If current is forced through a
wire loop or coil in air, then the flux density in and around the coil is about the same as it would be in a vacuum. Therefore, the permeability of pure air
is about equal to 1. If you place an iron core in the coil, the flux density increases by a factor ranging from a few dozen to several thousand times, depending on the purity of the iron. The permeability of iron can be as low as about 60 (impure) to as high as about 8,000 (highly refined).
If you use special metallic alloys called permalloys as the core material
in electromagnets, you can increase the flux density, and therefore the local strength of the field, by as much as 1 million (106) times. Such substances thus have permeability as great as 106.
If, for some reason, you feel compelled to make an electromagnet that
is as weak as possible, you can use dry wood or wax for the core material. Usually, however, diamagnetic substances are used to keep magnetic objects apart while minimizing the interaction between them.


Certain ferromagnetic materials stay magnetized better than others. When
a substance such as iron is subjected to a magnetic field as intense as it can handle, say, by enclosing it in a wire coil carrying a high current, there will be some residual magnetism left when the current stops flowing in the coil. Retentivity, also sometimes called remanence, is a measure of how well a substance can “memorize” a magnetic field imposed on it and thereby become a permanent magnet.
Retentivity is expressed as a percentage. If the maximum possible flux density in a material is x teslas or gauss and then goes down to y teslas or gauss when the current is removed, the retentivity Br of that material is given by the following formula:

Br 100y/x
What is meant by maximum possible flux density in the foregoing defi- nition? This is an astute question. In the real world, if you make an elec- tromagnet with a core material, there is a limit to the flux density that can be generated in that core. As the current in the coil increases, the flux den- sity inside the core goes up in proportion—for awhile. Beyond a certain point, however, the flux density levels off, and further increases in current do not produce any further increase in the flux density. This condition is called core saturation. When we determine retentivity for a material, we
CHAPTER 14 Magnetism 359

are referring to the ratio of the flux density when it is saturated and the flux
density when there is no magnetomotive force acting on it.
As an example, suppose that a metal rod can be magnetized to 135 G when it is enclosed by a coil carrying an electric current. Imagine that this is the maximum possible flux density that the rod can be forced to have. For any substance, there is always such a maximum; further increasing the current in the wire will not make the rod any more magnetic. Now suppose that the cur- rent is shut off and that 19 G remain in the rod. Then the retentivity Br is
Br 100 19/135 100 0.14 14 percent
Certain ferromagnetic substances have good retentivity and are excellent for making permanent magnets. Other ferromagnetic materials have poor retentivity. They can work well as the cores of electromagnets, but they do not make good permanent magnets. Sometimes it is desirable to have a sub- stance with good ferromagnetic properties but poor retentivity. This is the case when you want to have an electromagnet that will operate from dc so that it maintains a constant polarity but that will lose its magnetism when the current is shut off.
If a ferromagnetic substance has poor retentivity, it’s easy to make it work as the core for an ac electromagnet because the polarity is easy to switch. However, if the retentivity is high, the material is “magnetically sluggish” and has trouble following the current reversals in the coil. This sort of stuff doesn’t function well as the core of an ac electromagnet.

Suppose that a metal rod is surrounded by a coil and that the magnetic flux
density can be made as great as 0.500 T; further increases in current cause no further increase in the flux density inside the core. Then the current is removed; the flux density drops to 500 G. What is the retentivity of this core material?

First, convert both flux density figures to the same units. Remember that 1 T
104 G. Thus the flux density is 0.500 104 5,000 G with the current and
500 G without the current. “Plugging in” these numbers gives us this:

Br 100 500/5,000 100 0.100 10.0 percent


Any ferromagnetic material, or substance whose atoms can be aligned per- manently, can be made into a permanent magnet. These are the magnets

PART 2 Electricity, Magnetism, and Electronics

you played with as a child (and maybe still play with when you use them
to stick notes to your refrigerator door). Some alloys can be made into stronger permanent magnets than others.
One alloy that is especially suited to making strong permanent magnets
is known by the trade name Alnico. This word derives from the chemical symbols of the metals that comprise it: aluminum (Al), nickel (Ni), and cobalt (Co). Other elements are sometimes added, including copper and titanium. However, any piece of iron or steel can be magnetized to some extent. Many technicians use screwdrivers that are slightly magnetized so that they can hold onto screws when installing or removing them from hard-to-reach places.
Permanent magnets are best made from materials with high retentivity. They are made by using the material as the core of an electromagnet for an extended period of time. If you want to magnetize a screwdriver a lit- tle bit so that it will hold onto screws, stroke the shaft of the screwdriver with the end of a bar magnet several dozen times. However, take note: Once you have magnetized a tool, it is practically impossible to completely demagnetize it.


Suppose that you have a long coil of wire, commonly known as a solenoid,
with n turns and whose length in meters is s. Suppose that this coil carries
a direct current of I amperes and has a core whose permeability is . The flux density B in teslas inside the core, assuming that it is not in a state of saturation, can be found using this formula:
B 4p 10 7 ( nI/s)

A good approximation is
B 1.2566 10 6 ( nI/s)

Consider a dc electromagnet that carries a certain current. It measures 20 cm
long and has 100 turns of wire. The flux density in the core, which is known not to be in a state of saturation, is 20 G. The permeability of the core mate- rial is 100. What is the current in the wire?

As always, start by making sure that all units are correct for the formula that
will be used. The length s is 20 cm, that is, 0.20 m. The flux density B is 20
G, which is 0.0020 T. Rearrange the preceding formula so it solves for I:
CHAPTER 14 Magnetism 361

6 ( nI/s)
6 ( n/s)
I 1 1.2566 10 6 ( n/sB)
I 7.9580 10
(sB/ n)

This is an exercise, but it is straightforward. Derivations such as this are subject to the constraint that we not divide by any quantity that can attain a value of zero in a practical situation. (This is not a problem here. We aren’t concerned with scenarios involving zero current, zero turns of wire, permeability of zero, or coils having zero length.) Let’s “plug in the numbers”:
I 7.9580 105 (0.20 0.0020)/(100 100)
7.9580 105 4.0 10 8

0.031832 A 31.832 mA

This must be rounded off to 32 mA because we are only entitled to claim two significant figures.

Magnetic Machines

A solenoid, having a movable ferromagnetic core, can do various things. Electrical relays, bell ringers, electric “hammers,” and other mechanical devices make use of the principle of the solenoid. More sophisticated elec- tromagnets, sometimes in conjunction with permanent magnets, can be used to build motors, meters, generators, and other devices.


Figure 14-6 is a simplified diagram of a bell ringer. Its solenoid is an elec- tromagnet. The core has a hollow region in the center, along its axis, through which a steel rod passes. The coil has many turns of wire, so the electromagnet is powerful if a substantial current passes through the coil. When there is no current flowing in the coil, the rod is held down by the force of gravity. When a pulse of current passes through the coil, the rod is pulled forcibly upward. The magnetic force “wants” the ends of the rod,

PART 2 Electricity, Magnetism, and Electronics

Steel plate(ringer)

Fig. 14-6. A bell ringer using a solenoid.

which is the same length as the core, to be aligned with the ends of the core.
However, the pulse is brief, and the upward momentum is such that the rod passes all the way through the core and strikes the ringer plate. Then the steel rod falls back down again to its resting position, allowing the plate to reverberate. Some office telephones are equipped with ringers that produce this noise rather than conventional ringing, buzzing, beeping, or chirping emitted by most phone sets. The “gong” sound is less irritating to some people than other attention-demanding signals.


In some electronic devices, it is inconvenient to place a switch exactly where it should be. For example, you might want to switch a communica-
CHAPTER 14 Magnetism 363

tions line from one branch to another from a long distance away. In wire-
less transmitters, some of the wiring carries high-frequency alternating cur- rents that must be kept within certain parts of the circuit and not routed out
to the front panel for switching. A relay makes use of a solenoid to allow remote-control switching.
A drawing and a diagram of a relay are shown in Fig. 14-7. The movable lever, called the armature, is held to one side by a spring when there is no current flowing through the electromagnet. Under these conditions, termi- nal X is connected to terminal Y but not to terminal Z. When a sufficient

Fig. 14-7. (a) Pictorial drawing of a simple relay. (b) Schematic symbol for the same relay.

PART 2 Electricity, Magnetism, and Electronics

current is applied, the armature is pulled over to the other side. This dis-
connects terminal X from terminal Y and connects X to Z.
There are numerous types of relays, each used for a different purpose. Some are meant for use with dc, and others are for ac; some will work with either ac or dc. A normally closed relay completes a circuit when there is no current flowing in its electromagnet and breaks the circuit when current flows. A normally open relay is just the opposite. (Normal in this sense means “no current in the coil.”) The relay shown in Fig. 14-7 can be used either as a normally open or normally closed relay depending on which contacts are selected. It also can be used to switch a line between two dif- ferent circuits.
These days, relays are used only in circuits and systems carrying extreme currents or voltages. In most ordinary applications, electronic semiconduc- tor switches, which have no moving parts and can last far longer than relays, are preferred.


Magnetic fields can produce considerable mechanical forces. These forces can be harnessed to do work. The device that converts dc energy into rotat- ing mechanical energy is a dc motor. In this sense, a dc motor is a form of transducer. Motors can be microscopic in size or as big as a house. Some tiny motors are being considered for use in medical devices that actually can circulate in the bloodstream or be installed in body organs. Others can pull a train at freeway speeds.
In a dc motor, the source of electricity is connected to a set of coils producing magnetic fields. The attraction of opposite poles, and the repulsion of like poles, is switched in such a way that a constant torque, or rotational force, results. The greater the current that flows in the coils, the stronger is the torque, and the more electrical energy is needed. One set of coils, called the armature coil, goes around with the motor shaft. The other set of coils, called the field coil, is stationary (Fig. 14-8). In some motors, the field coils are replaced by a pair of permanent mag- nets. The current direction in the armature coil is reversed every half- rotation by the commutator. This keeps the force going in the same angular direction. The shaft is carried along by its own angular momen- tum so that it doesn’t come to a stop during those instants when the cur- rent is being switched in polarity.
CHAPTER 14 Magnetism

Fig. 14-8. Simplified drawing of a dc electric motor. Straight lines represent wires. Intersecting
lines indicate connections only when there is a dot at the point where the lines cross.


An electric generator is constructed somewhat like a conventional motor, although it functions in the opposite sense. Some generators also can oper- ate as motors; they are called motor/generators. Generators, like motors, are energy transducers of a special sort.
A typical generator produces ac when a coil is rotated rapidly in a strong magnetic field. The magnetic field can be provided by a pair of permanent magnets

(Fig. 14-9). The rotating shaft is driven by a gasoline-powered motor, a turbine, or some other source of mechanical energy. A commutator

PART 2 Electricity, Magnetism, and Electronics

can be used with a generator to produce pulsating dc output, which can be
filtered to obtain pure dc for use with precision equipment.

Magnetic Data Storage

Magnetic fields can be used to store data in various forms. Common media for data storage include magnetic tape and the magnetic disk.


Recording tape is the stuff you find in cassette players. These days, mag- netic tape is largely obsolete, but it is still sometimes used for home enter-

Fig. 14-9. A simple type of ac generator.
Magnetism 367

tainment, especially high-fidelity (hi-fi) music and home video. It also can
be found in some high-capacity computer data storage systems.
The tape consists of millions of particles of iron oxide attached to a plastic or nonferromagnetic metal strip. A fluctuating magnetic field, produced by the recording head, polarizes these particles. As the field changes in strength next
to the recording head, the tape passes by at a constant, controlled speed. This produces regions in which the iron oxide particles are polarized in either direc- tion. When the tape is run at the same speed through the recorder in the play- back mode, the magnetic fields around the individual particles cause a fluctuating field that is detected by a pickup head. This field has the same pat- tern of variations as the original field from the recording head.
Magnetic tape is available in various widths and thicknesses for differ- ent applications. Thick-tape cassettes don’t play as long as thin-tape ones, but the thicker tape is more resistant to stretching. The speed of the tape determines the fidelity of the recording. Higher speeds are preferred for music and video and lower speeds for voice.
The data on a magnetic tape can be distorted or erased by external mag- netic fields. Therefore, tapes should be protected from such fields. Keep magnetic tape away from permanent magnets or electromagnets. Extreme heat also can damage the data on magnetic tape, and if the temperature is high enough, physical damage occurs as well.


The era of the personal computer has seen the development of ever-more- compact data storage systems. One of the most versatile is the magnetic disk. Such a disk can be either rigid or flexible. Disks are available in various sizes. Hard disks (also called hard drives) store the most data and generally are found inside computer units. Diskettes are usually 3.5 inches (8.9 cm) in diameter and can be inserted and removed from digital recording/playback machines called disk drives.
The principle of the magnetic disk, on the micro scale, is the same as that of magnetic tape. But disk data is stored in binary form; that is, there are only two different ways that the particles are magnetized. This results
in almost perfect, error-free storage. On a larger scale, the disk works dif- ferently than tape because of the difference in geometry. On a tape, the information is spread out over a long span, and some bits of data are far away from others. On a disk, no two bits are ever farther apart than the diameter of the disk. Therefore, data can be transferred to or from a disk more rapidly than is possible with tape.

PART 2 Electricity, Magnetism, and Electronics

A typical diskette can store an amount of digital information equivalent
to a short novel. Specialized high-capacity diskettes can store the equiva- lent of hundreds of long novels or even a complete encyclopedia.
The same precautions should be observed when handling and storing magnetic disks as are necessary with magnetic tape.


Refer to the text in this chapter if necessary. A good score is eight correct. Answers are in the back of the book.
1. The geomagnetic field
(a) makes the Earth like a huge horseshoe magnet.
(b) runs exactly through the geographic poles.
(c) makes a compass work.
(d) makes an electromagnet work.
2. A material that can be permanently magnetized is generally said to be
(a) magnetic.
(b) electromagnetic.
(c) permanently magnetic.
(d) ferromagnetic.
3. The magnetic flux around a straight current-carrying wire
(a) gets stronger with increasing distance from the wire.
(b) is strongest near the wire.
(c) does not vary in strength with distance from the wire.
(d) consists of straight lines parallel to the wire.
4. The gauss is a unit of
(a) overall magnetic field strength.
(b) ampere-turns.
(c) magnetic flux density.
(d) magnetic power.
5. If a wire coil has 10 turns and carries 500 mA of current, what is the magne- tomotive force in ampere-turns?
(a) 5,000
(b) 50
(c) 5.0
(d) 0.02
CHAPTER 14 Magnetism 369

6. Which of the following is not generally observed in a geomagnetic storm?
(a) Charged particles streaming out from the Sun
(b) Fluctuations in the Earth’s magnetic field
(c) Disruption of electrical power transmission
(d) Disruption of microwave propagation
7. An ac electromagnet
(a) will attract only other magnetized objects.
(b) will attract iron filings.
(c) will repel other magnetized objects.
(d) will either attract or repel permanent magnets depending on the polarity.
8. A substance with high retentivity is best suited for making
(a) an ac electromagnet.
(b) a dc electromagnet.
(c) an electrostatic shield.
(d) a permanent magnet.
9. A device that reverses magnetic field polarity to keep a dc motor rotating is
(a) a solenoid.
(b) an armature coil.
(c) a commutator.
(d) a field coil.
10. An advantage of a magnetic disk, as compared with magnetic tape, for data storage and retrieval is that
(a) a disk lasts longer.
(b) data can be stored and retrieved more quickly with disks than with tapes.
(c) disks look better.
(d) disks are less susceptible to magnetic fields.

Alternating Current


Alternating Current

Direct current (dc) can be expressed in terms of two variables: the polarity
(or direction) and the amplitude. Alternating current (ac) is more compli- cated. There are additional variables: the period (and its reciprocal, the frequency), the waveform, and the phase.

Definition of Alternating Current

Direct current has a polarity, or direction, that stays the same over a long period of time. Although the amplitude can vary—the number of amperes, volts, or watts can fluctuate—the charge carriers always flow in the same direction through the circuit. In ac, the polarity reverses repeatedly.


In a periodic ac wave, the kind discussed in this chapter, the mathematical function of amplitude versus time repeats precisely and indefinitely; the same pattern recurs countless times. The period is the length of time between one repetition of the pattern, or one wave cycle, and the next. This
is illustrated in Fig. 13-1 for a simple ac wave.
The period of a wave, in theory, can be anywhere from a minuscule frac- tion of a second to many centuries. Some electromagnetic (EM) fields have periods measured in quadrillionths of a second or smaller. The charged par- ticles held captive by the magnetic field of the Sun reverse their direction


PART 2 Electricity, Magnetism, and Electronics

Period Period


Fig. 13-1. A sine wave. The period is the length of time required for one cycle to be completed.

over periods measured in years. Period, when measured in seconds, is
symbolized T.


The frequency, denoted f, of a wave is the reciprocal of the period. That is,
f 1/T, and T 1/f. In the olden days (prior to the 1970s), frequency was specified in cycles per second, abbreviated cps. High frequencies were expressed in kilocycles, megacycles, or gigacycles, representing thousands, millions, or billions of cycles per second. Nowadays, the standard unit of frequency is known as the hertz, abbreviated Hz. Thus 1 Hz 1 cps, 10 Hz
10 cps, and so on.
Higher frequencies are given in kilohertz (kHz), megahertz (MHz),
gigahertz (GHz), and terahertz (THz). The relationships are
1 kHz 1,000 Hz 103 Hz
1 MHz 1,000 kHz 106 Hz
1 GHz
1,000 MHz
109 Hz
1 THz
1,000 GHz
1012 Hz
CHAPTER 13 Alternating Current 325

The period of an ac wave is 5.000 10 6 s. What is the frequency in hertz? In kilohertz? In megahertz?

First, find the frequency fHz in hertz by taking the reciprocal of the period in seconds:
fHz 1/(5.000 10
6 ) 2.000 105 Hz
Next, divide fHz by 1,000 or 10
to get the frequency fkHz in kilohertz:
3fkHz fHz/10
2.000 105/103
200.0 kHz
Finally, divide fkHz by 1,000 or 10
to get the frequency fMHz in megahertz:
fMHz fkHz/10
0.2000 MHz


If you graph the instantaneous current or voltage in an ac system as a function of time, you get a waveform. Alternating currents can manifest themselves in an infinite variety of waveforms. Here are some of the simplest ones.


In its purest form, alternating current has a sine-wave, or sinusoidal, nature. The waveform in Fig. 13-1 is a sine wave. Any ac wave that consists of a single frequency has a perfect sine-wave shape. Any perfect sine-wave cur- rent contains one, and only one, component frequency.
In practice, a wave can be so close to a sine wave that it looks exactly like the sine function on an oscilloscope when in reality there are traces of other frequencies present. Imperfections are often too small to see. Utility ac in the United States has an almost perfect sine-wave shape, with a fre- quency of 60 Hz. However, there are slight aberrations.


On an oscilloscope, a theoretically perfect square wave would look like a pair of parallel dotted lines, one having positive polarity and the other hav- ing negative polarity (Fig. 13-2a). In real life, the transitions often can be seen as vertical lines (see Fig. 13-2b).

PART 2 Electricity, Magnetism, and Electronics




Fig. 13-2. (a) A theoretically perfect square wave. (b) The more common rendition.

A square wave might have equal negative and positive peaks. Then the
absolute amplitude of the wave is constant at a certain voltage, current, or power level. Half the time the amplitude is x, and the other half it is
x volts, amperes, or watts.
Some square waves are asymmetrical, with the positive and negative magnitudes differing. If the length of time for which the amplitude is positive differs from the length of time for which the amplitude is negative, the wave is not truly square but is described by the more general term rectangular wave.


Some ac waves reverse their polarity at constant but not instantaneous rates. The slope of the amplitude-versus-time line indicates how fast the magnitude is changing. Such waves are called sawtooth waves because of their appearance.
In Fig. 13-3, one form of sawtooth wave is shown. The positive-going slope (rise) is extremely steep, as with a square wave, but the negative-
CHAPTER 13 Alternating Current 327


Fig. 13-3. A fast-rise, slow-decay sawtooth wave.

going slope (fall or decay) is gradual. The period of the wave is the time
between points at identical positions on two successive pulses.
Another form of sawtooth wave is just the opposite, with a gradual positive-going slope and a vertical negative-going transition. This type of wave is sometimes called a ramp (Fig. 13-4). This waveform is used for scanning in cathode-ray-tube (CRT) television sets and oscilloscopes.
Sawtooth waves can have rise and decay slopes in an infinite number of different combinations. One example is shown in Fig. 13-5. In this case, the positive-going slope is the same as the negative-going slope. This is a triangular wave.

Suppose that each horizontal division in Fig. 13-5 represents 1.0 microsecond
(1.0 s or 1.0 10 6 s). What is the period of this triangular wave? What is the frequency?

The easiest way to look at this is to evaluate the wave from a point where it
crosses the time axis going upward and then find the next point (to the right or left) where the wave crosses the time axis going upward. This is four hori- zontal divisions, at least within the limit of our ability to tell by looking at it. The period T is therefore 4.0 s or 4.0 10 6 s. The frequency is the reciprocal
of this: f 1/T 1/(4.0 10 6) 2.5 105 Hz.

PART 2 Electricity, Magnetism, and Electronics


Fig. 13-4. A slow-rise, fast-decay sawtooth wave, also called a ramp wave.


Fig. 13-5. A triangular wave.
CHAPTER 13 Alternating Current 329

Fractions of a Cycle

Scientists and engineers break the ac cycle down into small parts for analysis and reference. One complete cycle can be likened to a single revolution around a circle.


Suppose that you swing a glowing ball around and around at the end of a string at a rate of one revolution per second. The ball thus describes a circle
in space (Fig. 13-6a). Imagine that you swing the ball around so that it is always at the same level; it takes a path that lies in a horizontal plane. Imagine that you do this in a pitch-dark gymnasium. If a friend stands some distance away with his or her eyes in the plane of the ball’s path, what does your friend see? Only the glowing ball, oscillating back and forth. The ball seems to move toward the right, slow down, and then reverse its direction,

Top view




Side view


Fig. 13-6. Swinging ball and string. (a) as seen from above; (b) as seen from some distance away in the plane of the ball’s circular path.

PART 2 Electricity, Magnetism, and Electronics

going back toward the left (see Fig. 13-6b). Then it moves faster and faster and
then slower again, reaching its left-most point, at which it turns around again. This goes on and on, with a frequency of 1 Hz, or a complete cycle per sec- ond, because you are swinging the ball around at one revolution per second.
If you graph the position of the ball as seen by your friend with respect to time, the result will be a sine wave (Fig. 13-7). This wave has the same char- acteristic shape as all sine waves. The standard, or basic, sine wave is described by the mathematical function y sin x in the (x, y) coordinate plane. The general form is y a sin bx, where a and b are real-number constants.



Left Right

Position of ball
Fig. 13-7. Position of ball as seen edge-on as a function of time.


One method of specifying fractions of an ac cycle is to divide it into 360 equal increments called degrees, symbolized ° or deg (but it’s okay to write out the whole word). The value 0° is assigned to the point in the cycle where the magnitude is zero and positive-going. The same point on the next cycle
is given the value 360°. Halfway through the cycle is 180°; a quarter cycle is 90°; an eighth cycle is 45°. This is illustrated in Fig. 13-8.
CHAPTER 13 Alternating Current 331


0° 90° 270°

Fig. 13-8. A wave cycle can be divided into 360 degrees.


The other method of specifying fractions of an ac cycle is to divide it into exactly 2 , or approximately 6.2832, equal parts. This is the number of radii of a circle that can be laid end to end around the circumference. One radian, symbolized rad (although you can write out the whole word), is equal to about 57.296°. Physicists use the radian more often than the degree when talking about fractional parts of an ac cycle.
Sometimes the frequency of an ac wave is measured in radians per second
(rad/s) rather than in hertz (cycles per second). Because there are 2 radians
in a complete cycle of 360°, the angular frequency of a wave, in radians per second, is equal to 2 times the frequency in hertz. Angular frequency is symbolized by the lowercase italicized Greek letter omega ( ).

What is the angular frequency of household ac? Assume that the frequency
of utility ac is 60.0 Hz.

Multiply the frequency in hertz by 2 . If this value is taken as 6.2832, then the
angular frequency is

6.2832 60.0 376.992 rad/s

PART 2 Electricity, Magnetism, and Electronics

This should be rounded off to 377 rad/s because our input data are given only
to three significant figures.

A certain wave has an angular frequency of 3.8865 105 rad/s. What is the frequency in kilohertz? Express the answer to three significant figures.

To solve this, first find the frequency in hertz. This requires that the angular
frequency, in radians per second, be divided by 2 , which is approximately
6.2832. The frequency fHz is therefore
fHz (3.8865 105)/6.2832
6.1855 104 Hz

To obtain the frequency in kilohertz, divide by 103, and then round off to three significant figures:

fkHz 6.1855 104/103
61.855 kHz » 61.9 kHz


Amplitude also can be called magnitude, level, strength, or intensity. Depending on the quantity being measured, the amplitude of an ac wave can be specified in amperes (for current), volts (for voltage), or watts (for power).


The instantaneous amplitude of an ac wave is the voltage, current, or power
at some precise moment in time. This constantly changes. The manner in which it varies depends on the waveform. Instantaneous amplitudes are represented by individual points on the wave curves.


The average amplitude of an ac wave is the mathematical average (or mean) instantaneous voltage, current, or power evaluated over exactly one wave cycle or any exact whole number of wave cycles. A pure ac sine wave always has an average amplitude of zero. The same is true of a pure ac square wave or triangular wave. It is not generally the case for sawtooth
CHAPTER 13 Alternating Current 333

waves. You can get an idea of why these things are true by carefully looking
at the waveforms illustrated by Figs. 13-1 through 13-5. If you know calculus, you know that the average amplitude is the integral of the waveform eval- uated over one full cycle.


The peak amplitude of an ac wave is the maximum extent, either positive or negative, that the instantaneous amplitude attains. In many waves, the positive and negative peak amplitudes are the same. Sometimes they differ, however. Figure 13-9 is an example of a wave in which the positive peak amplitude
is the same as the negative peak amplitude. Figure 13-10 is an illustration of a wave that has different positive and negative peak amplitudes.


The peak-to-peak (pk-pk) amplitude of a wave is the net difference between the positive peak amplitude and the negative peak amplitude (Fig. 13-11). Another way of saying this is that the pk-pk amplitude is equal to the positive peak amplitude plus the absolute value of the negative peak amplitude.




Fig. 13-9. Positive and negative peak amplitudes. In this case, they are equal.

PART 2 Electricity, Magnetism, and Electronics




Fig. 13-10. A wave in which the positive and negative peak amplitudes differ.

to- peak

Fig. 13-11. Peak-to-peak amplitude.
CHAPTER 13 Alternating Current 335

Peak to peak is a way of expressing how much the wave level “swings”
during the cycle.
In many waves, the pk-pk amplitude is twice the peak amplitude. This is the case when the positive and negative peak amplitudes are the same.


Often it is necessary to express the effective amplitude of an ac wave. This
is the voltage, current, or power that a dc source would produce to have the same general effect in a real circuit or system. When you say a wall outlet has 117 V, you mean 117 effective volts. The most common figure for effective ac levels is called the root-mean-square, or rms, value.
The expression root mean square means that the waveform is mathemat- ically “operated on” by taking the square root of the mean of the square of all its instantaneous values. The rms amplitude is not the same thing as the average amplitude. For a perfect sine wave, the rms value is equal to 0.707 times the peak value, or 0.354 times the pk-pk value. Conversely, the peak value is 1.414 times the rms value, and the pk-pk value is 2.828 times the rms value. The rms figures often are quoted for perfect sine-wave sources of voltage, such as the utility voltage or the effective voltage of a radio signal. For a perfect square wave, the rms value is the same as the peak value,
and the pk-pk value is twice the rms value and twice the peak value. For sawtooth and irregular waves, the relationship between the rms value and the peak value depends on the exact shape of the wave. The rms value is never more than the peak value for any waveshape.


Sometimes a wave can have components of both ac and dc. The simplest example of an ac/dc combination is illustrated by the connection of a dc source, such as a battery, in series with an ac source, such as the utility main. Any ac wave can have a dc component along with it. If the dc compo-
nent exceeds the peak value of the ac wave, then fluctuating or pulsating dc will result. This would happen, for example, if a 200-V dc source were connected in series with the utility output. Pulsating dc would appear, with an average value of 200 V but with instantaneous values much higher and lower. The waveshape in this case is illustrated by Fig. 13-12.

An ac sine wave measures 60 V pk-pk. There is no dc component. What is
the peak voltage?

PART 2 Electricity, Magnetism, and Electronics


ac Component





Fig. 13-12. Composite ac/dc wave resulting from 117-V rms ac in series with 200-V dc.

In this case, the peak voltage is exactly half the peak-to-peak value, or 30 V pk.
Half the peaks are 30 V; half are 30 V.

Suppose that a dc component of 10 V is superimposed on the sine wave
described in Problem 13-5. What is the peak voltage?

This can’t be answered simply, because the absolute values of the positive
peak and negative peak voltages differ. In the case of Problem 13-5, the pos- itive peak is 30 V and the negative peak is 30 V, so their absolute values are the same. However, when a dc component of 10 V is superimposed on the wave, both the positive peak and the negative peak voltages change by
10 V. The positive peak voltage thus becomes 40 V, and the negative peak voltage becomes 20 V.

Phase Angle

Phase angle is an expression of the displacement between two waves having identical frequencies. There are various ways of defining this. Phase angles are usually expressed as values such that 0° 360°. In radians, this

Alternating Current

range is 0

2 . Once in awhile you will hear about phase angles

specified over a range of 180° 180°. In radians, this range is
. Phase angle figures can be defined only for pairs of waves whose frequencies are the same. If the frequencies differ, the phase changes from moment to moment and cannot be denoted as a specific number.


Phase coincidence means that two waves begin at exactly the same moment. They are “lined up.” This is shown in Fig. 13-13 for two waves having different amplitudes. (If the amplitudes were the same, you would see only one wave.) The phase difference in this case is 0°.
If two sine waves are in phase coincidence, the peak amplitude of the resulting wave, which also will be a sine wave, is equal to the sum of the peak amplitudes of the two composite waves. The phase of the resultant is the same as that of the composite waves.


When two sine waves begin exactly one-half cycle, or 180°, apart, they are said to be in phase opposition. This is illustrated by the drawing of Fig. 13-14.


0° 90° 270°

Fig. 13-13. Two sine waves in phase coincidence.

PART 2 Electricity, Magnetism, and Electronics

0° 90° 270°

Fig. 13-14. Two sine waves in phase opposition.

If two sine waves have the same amplitude and are in phase opposition,
they cancel each other out because the instantaneous amplitudes of the two waves are equal and opposite at every moment in time.
If two sine waves have different amplitudes and are in phase opposition, the peak value of the resulting wave, which is a sine wave, is equal to the difference between the peak values of the two composite waves. The phase of the resultant is the same as the phase of the stronger of the two composite waves.


Suppose that there are two sine waves, wave X and wave Y, with identical frequencies. If wave X begins a fraction of a cycle earlier than wave Y, then wave X is said to be leading wave Y in phase. For this to be true, X must begin its cycle less than 180° before Y. Figure 13-15 shows wave X leading wave Y by 90°. The difference can be anything greater than 0°, up to but not including 180°.
Leading phase is sometimes expressed as a phase angle such that 0°
180°. In radians, this is 0 . If we say that wave X has a phase of /2 rad relative to wave Y, we mean that wave X leads wave Y by /2 rad.
CHAPTER 13 Alternating Current 339


0° 90° 270°

Wave X

Wave Y

Fig. 13-15. Wave X leads wave Y by 90°.


Suppose that wave X begins its cycle more than 180° but less than 360° ahead of wave Y. In this situation, it is easier to imagine that wave X starts its cycle later than wave Y by some value between but not including 0° and
180°. Then wave X is lagging wave Y. Figure 13-16 shows wave X lagging wave Y by 90°. The difference can be anything between but not including
0° and 180°.
Lagging phase is sometimes expressed as a negative angle such that
180° 0°. In radians, this is 0. If we say that wave X has a phase of 45° relative to wave Y, we mean that wave X lags wave Y by 45°.


If a sine wave X is leading a sine wave Y by x degrees, then the two waves can be drawn as vectors, with vector X oriented x degrees counterclockwise from vector Y. If wave X lags Y by y degrees, then X is oriented y degrees clockwise from Y. If two waves are in phase, their vectors overlap (line up).
If they are in phase opposition, they point in exactly opposite directions.

PART 2 Electricity, Magnetism, and Electronics

Wave Y Wave X

0° 90°



Fig. 13-16. Wave X lags wave Y by 90°.

Figure 13-17 shows four phase relationships between waves X and Y.
Wave X always has twice the amplitude of wave Y, so vector X is always twice as long as vector Y. In part a, wave X is in phase with wave Y. In part b, wave X leads wave Y by 90°. In part c, waves X and Y are 180° opposite in phase. In part d, wave X lags wave Y by 90°.
In all cases, the vectors rotate counterclockwise at the rate of one complete circle per wave cycle. Mathematically, a sine wave is a vector that goes around and around, just like the ball goes around and around your head when you put it on a string and whirl it.
In a sine wave, the vector magnitude stays the same at all times. If the waveform is not sinusoidal, the vector magnitude is greater in some direc- tions than in others. As you can guess, there exist an infinite number of variations on this theme, and some of them can get complicated.

Suppose that there are three waves, called X, Y, and Z. Wave X leads wave
Y by 0.5000 rad; wave Y leads wave Z by precisely one-eighth cycle. By how many degrees does wave X lead or lag wave Z?

To solve this, let’s convert all phase-angle measures to degrees. One radian
is approximately equal to 57.296°; therefore, 0.5000 rad 57.296° 0.5000
CHAPTER 13 Alternating Current 341

90 90

0 180 0


90 90

0 180 0

Fig. 13-17. Vector representations of phase. (a) waves X and Y are in phase;
(b) wave X leads wave Y by 90 degrees; (c) waves X and Y are in phase opposition; (d) wave X lags wave Y by 90 degrees.

28.65° (to four significant figures). One-eighth of a cycle is equal to 45.00°
(that is 360°/8.000). The phase angles therefore add up, so wave X leads wave Z by 28.65° 45.00°, or 73.65°.

Suppose that there are three waves X, Y, and Z. Wave X leads wave Y by
0.5000 rad; wave Y lags wave Z by precisely one-eighth cycle. By how many degrees does wave X lead or lag wave Z?

The difference in phase between X and Y in this problem is the same as that
in the preceding problem, namely, 28.65°. The difference between Y and Z is also the same, but in the opposite sense. Wave Y lags wave Z by 45.00°. This is the same as saying that wave Y leads wave Z by 45.00°. Thus wave X

PART 2 Electricity, Magnetism, and Electronics

leads wave Z by 28.65° ( 45.00°), which is equivalent to 28.65° 45.00°,
or 16.35°. It is better in this case to say that wave X lags wave Z by 16.35°
or that wave Z leads wave X by 16.35°.

As you can see, phase relationships can get confusing. It’s the same sort of thing that happens when you talk about negative numbers. Which number
is larger than which? It depends on point of view. If it helps you to draw pictures of waves when thinking about phase, then by all means go ahead.


1. Approximately how many radians are in a quarter of a cycle?
(a) 0.7854
(b) 1.571
(c) 3.142
(d) 6.284
2. Refer to Fig. 13-18. Suppose that each horizontal division represents 1.0 ns
(1.0 10 9 s) and that each vertical division represents 1 mV (1.0 10 3 V). What is the approximate rms voltage? Assume the wave is sinusoidal.
(a) 4.8 mV
(b) 9.6 mV
(c) 3.4 mV
(d) 6.8 mV
3. In the wave illustrated by Fig. 13-18, given the same specifications as those for the previous problem, what is the approximate frequency of this wave?
(a) 330 MHz
(b) 660 MHz
(c) 4.1 109 rad/s
(d) It cannot be determined from this information.
4. In the wave illustrated by Fig. 13-18, what fraction of a cycle, in degrees, is represented by one horizontal division?
(a) 60
(b) 90
CHAPTER 13 Alternating Current 343


Fig. 13-18. Illustration for quiz questions 2, 3, and 4.

(c) 120
(d) 180
5. The maximum instantaneous current in a fluctuating dc wave is 543 mA over several cycles. The minimum instantaneous current is 105 mA, also over sev- eral cycles. What is the peak-to-peak current in this wave?
(a) 438 mA
(b) 648 mA
(c) 543 mA
(d) It cannot be calculated from this information.
6. The pk-pk voltage in a square wave is 5.50 V. The wave is ac, but it has a dc component of 1.00 V. What is the instantaneous voltage?
(a) More information is needed to answer this question.
(b) 3.25 V
(c) 1.25 V
(d) 1.00 V
7. Given the situation in the preceding question, what is the average voltage?
(a) More information is needed to answer this question.
(b) 3.25 V
(c) 1.25 V
(d) 1.00 V

PART 2 Electricity, Magnetism, and Electronics

8. Suppose that there are two sine waves having identical frequency and that their
vector representations are at right angles to each other. What is the difference in phase?
(a) More information is needed to answer this question.
(b) 90°
(c) 180°
(d) 2 rad
9. A square wave is a special form of
(a) sine wave.
(b) sawtooth wave.
(c) ramp wave.
(d) rectangular wave.
10. An ac wave has a constant frequency f. Its peak voltage Vpk is doubled. What happens to the period T?
(a) It doubles to 2T.
(b) It is reduced to T/2.
(c) It is reduced to 0.707T.
(d) It remains at T.

Direct Current


Direct Current

You now have a solid grasp of physics math, and you know the basics of classical physics. It is time to delve into the workings of things that can’t be observed directly. These include particles, and forces among them, that make it possible for you to light your home, communicate instantly with people on the other side of the world, and in general do things that would have been considered magical a few generations ago.

What Does Electricity Do?

When I took physics in middle school, they used 16-millimeter celluloid film projectors. Our teacher showed us several films made by a well-known professor. I’ll never forget the end of one of these lectures, in which the professor said, “We evaluate electricity not by knowing what it is, but by scrutinizing what it does.” This was a great statement. It really expresses the whole philosophy of modern physics, not only for electricity but also for all phenomena that aren’t directly tangible. Let’s look at some of the things electricity does.


In some materials, electrons move easily from atom to atom. In others, the electrons move with difficulty. And in some materials, it is almost impos- sible to get them to move. An electrical conductor is a substance in which the electrons are highly mobile.
The best conductor, at least among common materials, at room tempera- ture is pure elemental silver. Copper and aluminum are also excellent electrical


PART 2 Electricity, Magnetism, and Electronics

conductors. Iron, steel, and various other metals are fair to good conductors of
electricity. Some liquids are good conductors. Mercury is one example. Salt water is a fair conductor. Gases are, in general, poor conductors because the atoms or molecules are too far apart to allow a free exchange of electrons. However, if a gas becomes ionized, it can be a fair conductor of electricity.
Electrons in a conductor do not move in a steady stream like molecules of water through a garden hose. They pass from atom to atom (Fig. 12-1). This happens to countless atoms all the time. As a result, trillions of elec- trons pass a given point each second in a typical electric circuit.

Outer electron shell

Outer electron shell
Fig. 12-1. In an electrical conductor, electrons pass easily from atom to atom. This drawing is greatly simplified.

Imagine a long line of people, each one constantly passing a ball to his or her neighbor on the right. If there are plenty of balls all along the line, and if everyone keeps passing balls along as they come, the result is a steady stream of balls moving along the line. This represents a good conductor. If the peo- ple become tired or lazy and do not feel much like passing the balls along, the rate of flow decreases. The conductor is no longer very good.


If the people refuse to pass balls along the line in the preceding example, the line represents an electrical insulator. Such substances prevent electric cur- rents from flowing, except in very small amounts under certain circumstances.
CHAPTER 12 Direct Current 299

Most gases are good electrical insulators (because they are poor con-
ductors). Glass, dry wood, paper, and plastics are other examples. Pure water is a good electrical insulator, although it conducts some current when minerals are dissolved in it. Metal oxides can be good insulators, even though the metal in pure form is a good conductor.
An insulating material is sometimes called a dielectric. This term arises from the fact that it keeps electric charges apart, preventing the flow of electrons that would equalize a charge difference between two places. Excellent insulating materials can be used to advantage in certain electrical components such as capacitors, where it is important that electrons not be able to flow steadily. When there are two separate regions of electric charge having opposite polarity (called plus and minus, positive and negative, or and ) that are close to each other but kept apart by an insulating mate- rial, that pair of charges is called an electric dipole.


Some substances, such as carbon, conduct electricity fairly well but not very well. The conductivity can be changed by adding impurities such as clay to a carbon paste. Electrical components made in this way are called resistors. They are important in electronic circuits because they allow for the control of current flow. The better a resistor conducts, the lower is its resistance; the worse it conducts, the higher is the resistance.
Electrical resistance is measured in ohms, sometimes symbolized by the uppercase Greek letter omega (W). In this book we’ll sometimes use the symbol W and sometimes spell out the word ohm or ohms, so that you’ll get used to both expressions. The higher the value in ohms, the greater is the resistance, and the more difficult it is for current to flow. In an electrical system, it is usually desirable to have as low a resistance, or ohmic value,
as possible because resistance converts electrical energy into heat. This heat is called resistance loss and in most cases represents energy wasted. Thick wires and high voltages reduce the resistance loss in long-distance electrical lines. This is why gigantic towers, with dangerous voltages, are employed in large utility systems.


Whenever there is movement of charge carriers in a substance, there is an elec- tric current. Current is measured in terms of the number of charge carriers, or

PART 2 Electricity, Magnetism, and Electronics

particles containing a unit electric charge, passing a single point in 1
Charge carriers come in two main forms: electrons, which have a unit negative charge, and holes, which are electron absences within atoms and which carry a unit positive charge. Ions can act as charge carriers, and in some cases, atomic nuclei can too. These types of particles carry whole- number multiples of a unit electric charge. Ions can be positive or negative
in polarity, but atomic nuclei are always positive.
Usually, a great many charge carriers go past any given point in 1 sec- ond, even if the current is small. In a household electric circuit, a 100-W light bulb draws a current of about 6 quintillion (6 1018) charge carriers per second. Even the smallest minibulb carries a huge number of charge carriers every second. It is ridiculous to speak of a current in terms of charge carriers per second, so usually it is measured in coulombs per sec- ond instead. A coulomb (symbolized C) is equal to approximately 6.24
1018 electrons or holes. A current of 1 coulomb per second (1 C/s) is called an ampere (symbolized A), and this is the standard unit of electric current.
A 60-W bulb in a common table lamp draws about 0.5 A of current.
When a current flows through a resistance—and this is always the case, because even the best conductors have resistance—heat is generated. Sometimes visible light and other forms of energy are emitted as well. A light bulb is deliberately designed so that the resistance causes visible light
to be generated. However, even the best incandescent lamp is inefficient, creating more heat than light energy. Fluorescent lamps are better; they produce more light for a given amount of current. To put this another way, they need less current to give off a certain amount of light.
In physics, electric current is theoretically considered to flow from the positive to the negative pole. This is known as conventional current. If you connect a light bulb to a battery, therefore, the conventional current flows out of the positive terminal and into the negative terminal. However, the electrons, which are the primary type of charge carrier in the wire and the bulb, flow in the opposite direction, from negative to positive. This is the way engineers usually think about current.


Charge carriers, particularly electrons, can build up or become deficient on objects without flowing anywhere. You’ve experienced this when walking on a carpeted floor during the winter or in a place where the humidity is
CHAPTER 12 Direct Current 301

low. An excess or shortage of electrons is created on and in your body. You
acquire a charge of static electricity. It’s called static because it doesn’t go anywhere. You don’t feel this until you touch some metallic object that is connected to an electrical ground or to some large fixture, but then there is
a discharge, accompanied by a spark and a small electric shock. It is the current, during this discharge, that causes the sensation.
If you were to become much more charged, your hair would stand on end because every hair would repel every other one. Objects that carry the same electric charge, caused by either an excess or a deficiency of electrons, repel each other. If you were massively charged, the spark might jump several cen- timeters. Such a charge is dangerous. Static electric (also called electrostatic) charge buildup of this magnitude does not happen with ordinary carpet and shoes, fortunately. However, a device called a Van de Graaff generator, found
in some high-school physics labs, can cause a spark this large. You have to be careful when using this device for physics experiments.
On the grand scale of the Earth’s atmosphere, lightning occurs between clouds and between clouds and the surface. This spark is a greatly magni- fied version of the little spark you get after shuffling around on a carpet. Until the spark occurs, there is an electrostatic charge in the clouds, between different clouds, or between parts of a cloud and the ground. In Fig. 12-2, four types of lightning are shown. The discharge can occur with-
in a single cloud (intracloud lightning, part a), between two different clouds (intercloud lightning, part b), or from a cloud to the surface (cloud-
to-ground lightning, part c), or from the surface to a cloud (ground-to-cloud lightning, part d). The direction of the current flow in these cases is con- sidered to be the same as the direction in which the electrons move. In cloud-to-ground or ground-to-cloud lightning, the charge on the Earth’s surface follows along beneath the thunderstorm cloud like a shadow as the storm is blown along by the prevailing winds.
The current in a lightning stroke can approach 1 million A. However, it takes place only for a fraction of a second. Still, many coulombs of charge are displaced in a single bolt of lightning.


Current can flow only if it gets a “push.” This push can be provided by a buildup of electrostatic charges, as in the case of a lightning stroke. When the charge builds up, with positive polarity (shortage of electrons) in one place and negative polarity (excess of electrons) in another place, a powerful

PART 2 Electricity, Magnetism, and Electronics

+ +
+ + +
+ + +
+ + A

_ _ _

_ _ _ _
_ _ + +
_ _ _ _ _
+ + +

+ + +
+ + + +
+ + +
_ _ _ _ _ _


Fig. 12-2. (a) Lightning can occur within a single cloud (intracloud),
(b) between clouds (intercloud), or between a cloud and the surface
(c) cloud to ground or (d) ground to cloud.

electromotive force (emf) exists. This effect, also known as voltage or electri-
cal potential, is measured in volts (symbolized V).
Ordinary household electricity has an effective voltage of between 110
and 130 V; usually it is about 117 V. A car battery has an emf of 12 V (6 V
in some older systems). The static charge that you acquire when walking on a carpet with hard-soled shoes can be several thousand volts. Before a discharge of lightning, millions of volts exist.
An emf of 1 V, across a resistance of 1 W, will cause a current of 1 A to
flow. This is a classic relationship in electricity and is stated generally as
CHAPTER 12 Direct Current 303

Ohm’s law. If the emf is doubled, the current is doubled. If the resistance is
doubled, the current is cut in half. This law of electricity will be covered in detail a little later.
It is possible to have an emf without having current flow. This is the case just before a lightning bolt occurs and before you touch a metallic object after walking on the carpet. It is also true between the two prongs of a lamp plug when the lamp switch is turned off. It is true of a dry cell when there
is nothing connected to it. There is no current, but a current can flow if there is a conductive path between the two points.
Even a large emf might not drive much current through a conductor or resistance. A good example is your body after walking around on the carpet. Although the voltage seems deadly in terms of numbers (thousands), not many coulombs of charge normally can accumulate on an object the size of your body. Therefore, not many electrons flow through your finger, in relative terms, when you touch the metallic object. Thus you don’t get a severe shock. Conversely, if there are plenty of coulombs available, a moderate voltage,
such as 117 V (or even less), can result in a lethal flow of current. This is why
it is dangerous to repair an electrical device with the power on. The utility power source can pump an unlimited number of coulombs of charge through your body if you are foolish enough to get caught in this kind of situation.

Electrical Diagrams

To understand how electric circuits work, you should be able to read electri- cal wiring diagrams, called schematic diagrams. These diagrams use schematic symbols. Here are the basic symbols. Think of them as something like an alphabet in a language such as Chinese or Japanese, where things are represented by little pictures. However, before you get intimidated by this comparison, rest assured that it will be easier for you to learn schematic sym- bology than it would be to learn Chinese (unless you already know Chinese!).


The simplest schematic symbol is the one representing a wire or electrical conductor: a straight solid line. Sometimes dashed lines are used to repre- sent conductors, but usually, broken lines are drawn to partition diagrams into constituent circuits or to indicate that certain components interact with

PART 2 Electricity, Magnetism, and Electronics

each other or operate in step with each other. Conductor lines are almost
always drawn either horizontally across or vertically up and down the page so that the imaginary charge carriers are forced to march in formation like soldiers. This keeps the diagram neat and easy to read.
When two conductor lines cross, they are not connected at the crossing point unless a heavy black dot is placed where the two lines meet. The dot always should be clearly visible wherever conductors are to be connected, no matter how many of them meet at the junction.
A resistor is indicated by a zigzaggy line. A variable resistor, such as a rheostat or potentiometer, is indicated by a zigzaggy line with an arrow through it or by a zigzaggy line with an arrow pointing at it. These symbols are shown in Fig. 12-3.

(a) (b) (c)

Fig. 12-3. (a) A fixed resistor.
(b) A two-terminal variable resistor.
(c) A three-terminal potentiometer.

An electrochemical cell is shown by two parallel lines, one longer than the other. The longer line represents the positive terminal. A battery, or combination of cells in series, is indicated by an alternating sequence of parallel lines, long-short-long-short. The symbols for a cell and a battery are shown in Fig. 12-4.


Meters are indicated as circles. Sometimes the circle has an arrow inside it, and the meter type, such as mA (milliammeter) or V (voltmeter), is written alongside the circle, as shown in Fig. 12-5a. Sometimes the meter type is indicated inside the circle, and there is no arrow (see Fig. 12-5b). It doesn’t
CHAPTER 12 Direct Current 305



(a) (b)

Fig. 12-4. (a) An electrochemical cell. (b) A battery.



(a) (b)

Fig. 12-5. Meter symbols: (a) designator outside; (b) designator inside.

matter which way it’s done as long as you are consistent everywhere in a
given diagram.
Some other common symbols include the lamp, the capacitor, the air-core coil, the iron-core coil, the chassis ground, the earth ground, the alternating- current (AC) source, the set of terminals, and the black box (which can stand for almost anything), a rectangle with the designator written inside. These are shown in Fig. 12-6.

Voltage/Current/Resistance Circuits

Most direct current (dc) circuits can be boiled down ultimately to three major components: a voltage source, a set of conductors, and a resistance. This is shown in the schematic diagram of Fig. 12-7. The voltage of the emf source

PART 2 Electricity, Magnetism, and Electronics

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)
Fig. 12-6. More common schematic symbols:
(a) incandescent lamp; (b) fixed-value capacitor;
(c) air-core coil; (d) iron-core coil; (e) chassis ground;
(f) earth ground; (g) ac source; (h) terminals;
and (i) and black box.

is called E (or sometimes V); the current in the conductor is called I; the
resistance is called R. The standard units for these components are the volt
(V), the ampere (A), and the ohm (W), respectively. Note which characters here are italicized and which are not. Italicized characters represent mathe- matical variables; nonitalicized characters represent symbols for units.
You already know that there is a relationship among these three quanti- ties. If one of them changes, then one or both of the others also will change.
If you make the resistance smaller, the current will get larger. If you make
CHAPTER 12 Direct Current 307




Fig. 12-7. A simple dc circuit. The voltage is E, the current is I,
and the resistance is R.

the emf source smaller, the current will decrease. If the current in the cir-
cuit increases, the voltage across the resistor will increase. There is a sim- ple arithmetic relationship between these three quantities.


The interdependence among current, voltage, and resistance in dc circuits
is called Ohm’s law, named after the scientist who supposedly first expressed it. Three formulas denote this law:

You need only remember the first of these formulas to be able to derive the others. The easiest way to remember it is to learn the abbreviations E for emf, I for current, and R for resistance; then remember that they appear in alphabetical order with the equals sign after the E. Thus E IR.
It is important to remember that you must use units of volts, amperes, and ohms in order for Ohm’s law to work right. If you use volts, mil- liamperes (mA), and ohms or kilovolts (kV), microamperes ( A), and megohms (MW), you cannot expect to get the right answers. If the initial quantities are given in units other than volts, amperes, and ohms, you must convert to these units and then calculate. After that, you can convert the units back again to whatever you like. For example, if you get 13.5 million ohms as a calculated resistance, you might prefer to say that it is 13.5 megohms. However, in the calculation, you should use the number 13.5
million (or 1.35 107) and stick to ohms for the units.

PART 2 Electricity, Magnetism, and Electronics


The first way to use Ohm’s law is to find current values in dc circuits. In order to find the current, you must know the voltage and the resistance or be able to deduce them.
Refer to the schematic diagram of Fig. 12-8. It consists of a variable dc generator, a voltmeter, some wire, an ammeter, and a calibrated wide-range potentiometer. Actual component values are not shown here, but they can be assigned for the purpose of creating sample Ohm’s law problems. While calculating the current in the following problems, it is necessary to mentally
“cover up” the meter.

Suppose that the dc generator (see Fig. 12-8) produces 10 V and that the potentiometer is set to a value of 10 W. What is the current?

This is solved easily by the formula I E/R. Plug in the values for E and
R; they are both 10, because the units are given in volts and ohms. Then
I 10/10 1.0 A.

The dc generator (see Fig. 12-8) produces 100 V, and the potentiometer is set to 10.0 kW. What is the current?



dc generator



Fig. 12-8. Circuit for working Ohm’s law problems.
CHAPTER 12 Direct Current 309

First, convert the resistance to ohms: 10.0 kW 10,000 W. Then plug the val-
ues in: I 100/10,000 0.0100 A.


The second use of Ohm’s law is to find unknown voltages when the current and the resistance are known. For the following problems, uncover the ammeter and cover the voltmeter scale in your mind.

Suppose that the potentiometer (see Fig. 12-8) is set to 100 W, and the meas-
ured current is 10.0 mA. What is the dc voltage?

Use the formula E IR. First, convert the current to amperes: 10.0 mA
0.0100 A. Then multiply: E 0.0100 100 1.00 V. This is a low, safe volt- age, a little less than what is produced by a flashlight cell.


Ohm’s law can be used to find a resistance between two points in a dc cir- cuit when the voltage and the current are known. For the following prob- lems, imagine that both the voltmeter and ammeter scales in Fig. 12-8 are visible but that the potentiometer is uncalibrated.

If the voltmeter reads 24 V and the ammeter shows 3.0 A, what is the value
of the potentiometer?

Use the formula R E/I, and plug in the values directly because they are expressed in volts and amperes: R 24/3.0 8.0 W.


You can calculate the power P (in watts, symbolized W) in a dc circuit such as that shown in Fig. 12-8 using the following formula:


where E is the voltage in volts and I is the current in amperes. You may not be given the voltage directly, but you can calculate it if you know the cur- rent and the resistance.

PART 2 Electricity, Magnetism, and Electronics

Remember the Ohm’s law formula for obtaining voltage: E IR. If you
know I and R but don’t know E, you can get the power P by means of this formula:
P (IR) I I 2R

That is, take the current in amperes, multiply this figure by itself, and then multiply the result by the resistance in ohms.
You also can get the power if you aren’t given the current directly. Suppose that you’re given only the voltage and the resistance. Remember the Ohm’s law formula for obtaining current: I E/R. Therefore, you can calculate power using this formula:
P E (E/R) E 2/R

That is, take the voltage, multiply it by itself, and divide by the resistance. Stated all together, these power formulas are
P EI I 2R E 2/R

Now we are all ready to do power calculations. Refer once again to
Fig. 12-8.

Suppose that the voltmeter reads 12 V and the ammeter shows 50 mA. What
is the power dissipated by the potentiometer?

Use the formula P EI. First, convert the current to amperes, getting I
0.050 A. Then P EI 12 0.050 0.60 W.

How Resistances Combine

When electrical components or devices containing dc resistance are con- nected together, their resistances combine according to specific rules. Sometimes the combined resistance is more than that of any of the compo- nents or devices alone. In other cases the combined resistance is less than that of any of the components or devices all by itself.


When you place resistances in series, their ohmic values add up to get the total resistance. This is intuitively simple, and it’s easy to remember.
CHAPTER 12 Direct Current 311

Suppose that the following resistances are hooked up in series with each
other: 112 ohms, 470 ohms, and 680 ohms (Fig. 12-9). What is the total resistance of the series combination?

112 470 680


Fig. 12-9. An example of three specific resistances in series.

Just add the values, getting a total of 112 470 680 1,262 ohms. You
can round this off to 1,260 ohms. It depends on the tolerances of the compo- nents—how much their actual values are allowed to vary, as a result of man- ufacturing processes, from the values specified by the vendor. Tolerance is more of an engineering concern than a physics concern, so we won’t worry about that here.

When resistances are placed in parallel, they behave differently than they do in series. In general, if you have a resistor of a certain value and you place other resistors in parallel with it, the overall resistance decreases. Mathematically, the rule is straightforward, but it can get a little messy.
One way to evaluate resistances in parallel is to consider them as con- ductances instead. Conductance is measured in units called siemens, some- times symbolized S. (The word siemens serves both in the singular and the plural sense). In older documents, the word mho (ohm spelled backwards)
is used instead. In parallel, conductances add up in the same way as resist- ances add in series. If you change all the ohmic values to siemens, you can add these figures up and convert the final answer back to ohms.
The symbol for conductance is G. Conductance in siemens is the recip- rocal of resistance in ohms. This can be expressed neatly in the following two formulas. It is assumed that neither R nor G is ever equal to zero:

G 1/R R 1/G

PART 2 Electricity, Magnetism, and Electronics

Consider five resistors in parallel. Call them R1 through R5, and call the total resistance R, as shown in the diagram of Fig. 12-10. Let R1 100 ohms, R2
200 ohms, R3 300 ohms, R4 400 ohms, and R5 500 ohms, respec- tively. What is the total resistance R of this parallel combination?

R R 1 R 2 R 3 R 4 R 5

Fig. 12-10. Five general resistances in parallel.

Converting the resistances to conductance values, you get G1 1/100
0.0100 siemens, G2 1/200 0.00500 siemens, G3 1/300 0.00333
siemens, G4 1/400 0.00250 siemens, and G5 1/500 0.00200
siemens. Adding these gives G 0.0100 0.00500 0.00333 0.00250
0.00200 0.02283 siemens. The total resistance is therefore R 1/G
1/0.02283 43.80 ohms. Because we’re given the input numbers to only three significant figures, we should round this off to 43.8 ohms.

When you have resistances in parallel and their values are all equal, the total resistance is equal to the resistance of any one component divided by the num- ber of components. In a more general sense, the resistances in Fig. 12-10 combine like this:

R 1/(1/R1 1/R2 1/R3 1/R4 1/R5) If you prefer to use exponents, the formula looks like this:
5R (R1 2
3 R4
R 1) 1
These resistance formulas are cumbersome for some people to work with, but mathematically they represent the same thing we just did in Prob- lem 12-7.


Have you ever used those tiny holiday lights that come in strings? If one bulb burns out, the whole set of bulbs goes dark. Then you have to find out which bulb is bad and replace it to get the lights working again. Each bulb
CHAPTER 12 Direct Current 313

works with something like 10 V, and there are about a dozen bulbs in the
string. You plug in the whole bunch, and the 120-V utility mains drive just the right amount of current through each bulb.
In a series circuit such as a string of light bulbs, the current at any given point is the same as the current at any other point. An ammeter can be connected in series at any point in the circuit, and it will always show the same reading. This is true in any series dc circuit, no matter what the components actually are and regardless of whether or not they all have the same resistance.
If the bulbs in a string are of different resistances, some of them will consume more power than others. In case one of the bulbs burns out and its socket is shorted out instead of filled with a replacement bulb, the current through the whole chain will increase because the overall resistance of the string will go down. This will force each of the remaining bulbs to carry too much current. Another bulb will burn out before long as a result of this excess current. If it, too, is replaced by a short circuit, the current will be increased still further. A third bulb will blow out almost right away. At this point it would be wise to buy some new bulbs!


In a series circuit, the voltage is divided up among the components. The sum total of the potential differences across each resistance is equal to the dc power-supply or battery voltage. This is always true, no matter how large or how small the resistances and whether or not they’re all the same value.
If you think about this for a moment, it’s easy to see why this is true. Look at the schematic diagram of Fig. 12-11. Each resistor carries the same current. Each resistor Rn has a potential difference En across it equal to the product of the current and the resistance of that particular resistor. These En values are in series, like cells in a battery, so they add together. What if the En values across all the resistors added up to something more or less than the supply voltage E? Then there would be a “phantom emf” someplace, adding or taking away voltage. However, there can be no such thing. An emf cannot come out of nowhere.
Look at this another way. The voltmeter V in Fig. 12-11 shows the volt- age E of the battery because the meter is hooked up across the battery. The meter V also shows the sum of the En values across the set of resistors sim- ply because the meter is connected across the set of resistors. The meter

PART 2 Electricity, Magnetism, and Electronics

E n

R n


Fig. 12-11. Analysis of voltage in a series dc circuit. See text for discussion.

says the same thing whether you think of it as measuring the battery volt-
age E or as measuring the sum of the En values across the series combina- tion of resistors. Therefore, E is equal to the sum of the En values.
This is a fundamental rule in series dc circuits. It also holds for common utility ac circuits almost all the time.
How do you find the voltage across any particular resistor Rn in a cir- cuit like the one in Fig. 12-11? Remember Ohm’s law for finding voltage:
E IR. The voltage is equal to the product of the current and the resist- ance. Remember, too, that you must use volts, ohms, and amperes when making calculations. In order to find the current in the circuit I, you need to know the total resistance and the supply voltage. Then I E/R. First find the current in the whole circuit; then find the voltage across any par- ticular resistor.

In Fig. 12-11, suppose that there are 10 resistors. Five of them have values
of 10 ohms, and the other 5 have values of 20 ohms. The power source is 15
V dc. What is the voltage across one of the 10-ohm resistors? Across one of the 20-ohm resistors?

First, find the total resistance: R (10 5) (20 5) 50 100 150
ohms. Then find the current: I E/R 15/150 0.10 A. This is the current
through each of the resistors in the circuit. If Rn 10 ohms, then
CHAPTER 12 Direct Current 315

En I (Rn) 0.10 10 1.0 V
If Rn 20 ohms, then
En I (Rn) 0.10 20 2.0 V
You can check to see whether all these voltages add up to the supply volt- age. There are 5 resistors with 1.0 V across each, for a total of 5.0 V; there are also 5 resistors with 2.0 V across each, for a total of 10 V. Thus the sum of the voltages across the 10 resistors is 5.0 V 10 V 15 V.


Imagine now a set of ornamental light bulbs connected in parallel. This is the method used for outdoor holiday lighting or for bright indoor lighting. You know that it’s much easier to fix a parallel-wired string of holiday lights if one bulb should burn out than it is to fix a series-wired string. The failure of one bulb does not cause catastrophic system failure. In fact, it might be awhile before you notice that the bulb is dark because all the other ones will stay lit, and their brightness will not change.
In a parallel circuit, the voltage across each component is always the same and is always equal to the supply or battery voltage. The current drawn by each component depends only on the resistance of that particular device. In this sense, the components in a parallel-wired circuit work inde- pendently, as opposed to the series-wired circuit, in which they all interact.
If any branch of a parallel circuit is taken away, the conditions in the other branches remain the same. If new branches are added, assuming that the power supply can handle the load, conditions in previously existing branches are not affected.


Refer to the schematic diagram of Fig. 12-12. The total parallel resistance
in the circuit is R. The battery voltage is E. The current in branch n, con- taining resistance Rn, is measured by ammeter A and is called In.
The sum of all the In values in the circuit is equal to the total current I
drawn from the source. That is, the current is divided up in the parallel cir-
cuit, similarly to the way that voltage is divided up in a series circuit.

Suppose that the battery in Fig. 12-12 delivers 12 V. Further suppose that
there are 12 resistors, each with a value of 120 ohms in the parallel circuit.

PART 2 Electricity, Magnetism, and Electronics


R n

Fig. 12-12. Analysis of current in a parallel dc circuit. See text for discussion.

What is the total current I drawn from the battery?

First, find the total resistance. This is easy because all the resistors have the
same value. Divide Rn 120 by 12 to get R 10 ohms. Then the current I
is found by Ohm’s law:

I E/R 12/10 1.2 A

In the circuit of Fig. 12-12, what does the ammeter A say, given the same
component values as exist in the scenario of the preceding problem?

This involves finding the current in any given branch. The voltage is 12 V
across every branch; Rn 120. Therefore, In, the ammeter reading, is found by Ohm’s law:

In E/Rn 12/120 0.10 A
Let’s check to be sure all the In values add to get the total current I. There are
12 identical branches, each carrying 0.10 A; therefore, the sum is 0.10 12
1.2 A. It checks out.
CHAPTER 12 Direct Current 317


n nLet’s switch back now to series circuits. When calculating the power in a circuit containing resistors in series, all you need to do is find out the cur- rent I, in amperes, that the circuit is carrying. Then it’s easy to calculate the power Pn, in watts, dissipated by any particular resistor of value Rn , in ohms, based on the formula P I 2R .
The total power dissipated in a series circuit is equal to the sum of the wattages dissipated in each resistor. In this way, the distribution of power
in a series circuit is like the distribution of the voltage.

Suppose that we have a series circuit with a supply of 150 V and three resis-
tors: R1 330 ohms, R2 680 ohms, and R3 910 ohms. What is the power dissipated by R2?

Find the current in the circuit. To do this, calculate the total resistance first.
Because the resistors are in series, the total is resistance is R 330 680
910 1920 ohms. Therefore, the current is I 150/1920 0.07813 A
78.1 mA. The power dissipated by R2 is
2 2P I 2R 0.07813 0.07813 680 4.151 W

We must round this off to three significant figures, getting 4.15 W.


n n nWhen resistances are wired in parallel, they each consume power accord- ing to the same formula, P I2R. However, the current is not the same in each resistance. An easier method to find the power Pn dissipated by resis- tor of value R is by using the formula P E2/R , where E is the voltage
of the supply. This voltage is the same across every resistor.
In a parallel circuit, the total power consumed is equal to the sum of the wattages dissipated by the individual resistances. This is, in fact, true for any dc circuit containing resistances. Power cannot come out of nowhere, nor can it vanish.

A circuit contains three resistances R1 22 ohms, R2 47 ohms, and R3
68 ohms, all in parallel across a voltage E 3.0 V. Find the power dissipated
by each resistor.

PART 2 Electricity, Magnetism, and Electronics

First find E 2, the square of the supply voltage: E 2 3.0 3.0 9.0. Then
P1 9.0/22 0.4091 W, P2 9.0/47 0.1915 W, and P3 9.0/68 0.1324
W. These should be rounded off to P1 0.41 W, P2 0.19 W, and P3 0.13 W,

Kirchhoff’s Laws

The physicist Gustav Robert Kirchhoff (1824–1887) was a researcher and experimentalist in electricity, back in the time before radio, before electric lighting, and before much was understood about how electric cur- rents flow.


Kirchhoff reasoned that current must work something like water in a net- work of pipes and that the current going into any point has to be the same as the current going out. This is true for any point in a circuit, no matter how many branches lead into or out of the point (Fig. 12-13).
In a network of water pipes that does not leak and into which no water
is added along the way, the total number of cubic meters going in has to be the same as the total volume going out. Water cannot form from nothing, nor can it disappear, inside a closed system of pipes. Charge carriers, thought Kirchhoff, must act the same way in an electric circuit.

In Fig. 12-13, suppose that each of the two resistors below point Z has a value
of 100 ohms and that all three resistors above Z have values of 10.0 ohms. The current through each 100-ohm resistor is 500 mA (0.500 A). What is the current through any of the 10.0-ohm resistors, assuming that the current is equally distributed? What is the voltage, then, across any of the 10.0-ohm resistors?

The total current into Z is 500 mA 500 mA 1.00 A. This must be divided
three ways equally among the 10-ohm resistors. Therefore, the current through any one of them is 1.00/3 A 0.333 A 333 mA. The voltage across any one of the 10.0-ohm resistors is found by Ohm’s law: E IR 0.333
10.0 3.33 V.
CHAPTER 12 Direct Current 319


3I I


1 2
Fig. 12-13. Kirchhoff’s current law. The current enter- ing point Z is equal to the current leaving point Z. In this case, I1 I2 I3 I4 I5.


The sum of all the voltages, as you go around a circuit from some fixed point and return there from the opposite direction, and taking polarity into account, is always zero. At first thought, some people find this strange. Certainly there is voltage in your electric hair dryer, radio, or computer! Yes, there is—between different points in the circuit. However, no single point can have an electrical potential with respect to itself. This is so sim- ple that it’s trivial. A point in a circuit is always shorted out to itself.
What Kirchhoff was saying when he wrote his voltage law is that volt- age cannot appear out of nowhere, nor can it vanish. All the potential dif- ferences must balance out in any circuit, no matter how complicated and no matter how many branches there are.
Consider the rule you’ve already learned about series circuits: The volt- ages across all the resistors add up to the supply voltage. However, the polarities of the emfs across the resistors are opposite to that of the battery. This is shown in Fig. 12-14. It is a subtle thing, but it becomes clear when

PART 2 Electricity, Magnetism, and Electronics

a series circuit is drawn with all the components, including the battery or
other emf source, in line with each other, as in Fig. 12-14.

+ 2 _
+ 3 _

_ + _ +
1 E 4
Fig. 12-14. Kirchhoff’s voltage law. The sum of the voltages
E E1 E2 E3 E4 0, taking polarity into account.

Refer to the diagram of Fig. 12-14. Suppose that the four resistors have val-
ues of 50, 60, 70, and 80 ohms and that the current through them is 500 mA
(0.500 A). What is the supply voltage E?

Find the voltages E1, E2, E3, and E4 across each of the resistors. This is done using Ohm’s law. In the case of E1, say, with the 50-ohm resistor, calculate E1 0.500 50 25 V. In the same way, you can calculate E2 30 V, E3 35 V, and E4 40 V. The supply voltage is the sum E1 E2 E3 E4
25 30 35 40 V 130 V.


1. Suppose that 5.00 1017 electrical charge carriers flow past a point in 1.00 s. What is the electrical voltage?
(a) 0.080 V
(b) 12.5 V
CHAPTER 12 Direct Current 321

(c) 5.00 V
(d) It cannot be calculated from this information.
2. An ampere also can be regarded as
(a) an ohm per volt.
(b) an ohm per watt.
(c) a volt per ohm.
(d) a volt-ohm.
3. Suppose that there are two resistances in a series circuit. One of the resistors has a value of 33 kW (that is, 33,000 or 3.3 104 ohms). The value of the other resistor is not known. The power dissipated by the 33-kW resistor is 3.3 W. What is the current through the unknown resistor?
(a) 0.11 A
(b) 10 mA
(c) 0.33 mA
(d) It cannot be calculated from this information.
4. If the voltage across a resistor is E (in volts) and the current through that resis- tor is I (in milliamperes), then the power P (in watts) is given by the following formula:
(a) P EI.
(b) P EI 103.
(c) P EI 10 3.
(d) P E/I.
5. Suppose that you have a set of five 0.5-W flashlight bulbs connected in paral- lel across a dc source of 3.0 V. If one of the bulbs is removed or blows out, what will happen to the current through the other four bulbs?
(a) It will remain the same.
(b) It will increase.
(c) It will decrease.
(d) It will drop to zero.
6. A good dielectric is characterized by
(a) excellent conductivity.
(b) fair conductivity.
(c) poor conductivity.
(d) variable conductivity.
7. Suppose that there are two resistances in a parallel circuit. One of the resistors has a value of 100 ohms. The value of the other resistor is not known. The power dissipated by the 100-ohm resistor is 500 mW (that is, 0.500 W). What
is the current through the unknown resistor?
(a) 71 mA
(b) 25 A
(c) 200 A
(d) It cannot be calculated from this information.

PART 2 Electricity, Magnetism, and Electronics

8. Conventional current flows
(a) from the positive pole to the negative pole.
(b) from the negative pole to the positive pole.
(c) in either direction; it doesn’t matter.
(d) nowhere; current does not flow.
9. Suppose that a circuit contains 620 ohms of resistance and that the current in the circuit is 50.0 mA. What is the voltage across this resistance?
(a) 12.4 kV
(b) 31.0 V
(c) 8.06 10 5 V
(d) It cannot be calculated from this information.
10. Which of the following cannot be an electric charge carrier?
(a) A neutron
(b) An electron
(c) A hole
(d) An ion