## Electromagnetic Energy

Action at a Distance

In this chapter we look into a form of energy so far mentioned only in passing. To begin, it is well to repeat, in different words and with less detail, three statements from chapter 2.

First, applying a force to an object—pushing it or pulling it—changes the object's motion; it either accelerates it or decelerates it in the direction of the force. If the object was originally stationary, the force causes it to start moving—accelerates it.

Second, gravity is a force. Specifically, it is the force that causes any mass to attract toward itself any other mass at a distance from it. The example known to virtually everybody is the traditional tale of Newton and the apple; gravity between the gigantically massive earth and a comparatively tiny apple (growing on Newton's apple tree) caused the apple, which wasn't firmly attached to the tree, to fall to earth. Simultaneously, gravity also caused the earth to move an imperceptibly small distance toward the apple, but this point is rarely mentioned because the effect is far too minute to be measurable. Note that gravity acts at a distance—the masses that attract each other may be widely separated: more on this below.

Third, a force acting through a distance performs work (in the physicist's sense of the word) or, equivalently, expends energy (again in the physicist's sense). This is what defines both work and energy.

These statements immediately invite two questions: Do any other forces resemble gravity in acting at a distance? And exactly how does action at a distance operate?

To answer the first question, Yes, there are other familiar forces that act at a distance, namely, electric force and magnetic force. At one time they were thought to be separate and unrelated, but nineteenth-century physics showed that in fact the two seemingly different forces are the outcome of a single physical process. Everybody is familiar with them even if they haven't thought much about them. Anyone who has experienced static cling has seen electric force in action, and anyone who has seen a refrigerator magnet in use has observed magnetic force in action.

We go further into these matters after tackling the second question: How does action at a distance work?

Take gravity as a specific example. Its action can be explained in various ways. One explanation is that any piece of matter—any mass—is surrounded by a field of gravitational force or, more briefly, a gravitational field. This means that at any chosen point in the space surrounding the given mass a force of specifiable strength and direction will act upon some other mass, used as a "test" mass, if you place the test mass at the chosen point (see fig. 16.1). This happens because an unseen gravitational force pervades the space surrounding the given mass in the form of a force field reaching out, in theory, to infinite distance. If this is an acceptable "explanation" of gravity, it leads to the conclusion that the given mass doesn't really act at a distance. Rather, it is the force field that affects the test mass, by acting on it directly at the precise spot where field and test mass touch. Inspecting this explanation shows that in fact it isn't one. It merely replaces the notion of a force acting at a distance with a force field acting wherever you want, leaving the latter concept still undefined.

One model of the way a force field acts follows from Einstein's general theory of relativity: it is that forces act "downhill" within a "space-time" having a "geometry" modified by the presence of masses scattered here and there within it. This is all very well, but the only outcome of the foregoing arguments are (fairly) easily visualizable mental images of how gravity acts. More modern images, designed to explain gravity in the context of modern quan-

Figure 16.1. The arrows show a few representative lines of force of the gravitational field surrounding a chunk of rock, M, isolated in space.

tum theory, are more sophisticated than their predecessors, but they are still only mental images, as physicists readily concede.1 It seems futile to keep refining them before one has arrived at a satisfactory definition of what is meant by a body's mass, a term we have not yet defined. Here is a modern definition: the mass of a body is a measure of its resistance to being accelerated or, what comes to the same thing, its resistance to a force.2 The circularity is seamless: a force is something that acts on a mass, while a mass is something that responds to a force.

It therefore seems best to treat both "mass" and "force" as terms labeling fundamental, intrinsically undefinable concepts that must be accepted a priori in order to make further discussion of the material world possible: one has to start somewhere. In what follows, we let the two terms have their obvious, intuitive meanings.

To return to the first question we asked above, Do any other forces resemble gravity in acting at a distance? As already remarked, two other forces behave like this—electric force and magnetic force. To begin with we consider them separately, deferring for now the linkage between them.

### Electric Force

A weak electric force is easy to generate. Most children have seen it done as a party game. An effective method is to rub a plastic rod (a ballpoint pen or the handle of a plastic spoon) on a piece of fur (human hair serves well); rub vigorously for a minute or two and then promptly hold the end of the rod just above a few small scraps of torn paper. One of the scraps will rise and stick to the tip of the rod, and other scraps will follow until several are lined up as if strung together, end to end. Evidently a force is at work strong enough to overcome the force of gravity that held the scraps down on the table; and the force acts at a distance. Admittedly the force is slight (the scraps of paper aren't heavy), and the distance over which it acts is a centimeter or two at most; all the same, a force acting at a distance has been created.

What has happened is that the brisk rubbing has dislodged some outer electrons from atoms at the surface of the fur and left them adhering (temporarily and weakly) to atoms at the surface of the plastic. The adhesion is a frail version of an ionic chemical bond as described in chapter 10. In acquiring extra electrons, the plastic has acquired a negative electric charge; simultaneously, in losing these electrons, the fur has acquired a positive electric charge.

The plastic is now capable of exerting an electric force that attracts positively charged objects and repels negatively charged ones. But the scraps of paper have not had electrons rubbed off them or stuck to them—they are electrically neutral: Why should they be attracted to the negatively charged plastic rod? The answer (see fig. 16.2) is that the negative charge on the tip of the rod repels electrons from the near edge of the closest scrap, giving this edge a positive charge so that it is attracted to the rod. The farther edge of the same scrap receives the repelled electrons, giving it a negative charge: thereupon it acts on the second scrap of paper in the same way that the plastic rod acted on the first scrap. This chain reaction can seldom be made to reach beyond three or four scraps of paper because stray electrons "leak" between the charged paper and the surrounding air: the small electric charges soon fade away.

This desktop experiment allows you to produce and examine one of the fundamental forces of nature, the electric force that holds atoms and molecules together (but not atomic nuclei). The force "acts at a distance" as gravity does, but it is wholly unlike gravity in acting as both an attractive force and a repulsive force. Gravity has an identical effect on all masses: it is always a force of attraction; it never causes one mass to repel another. In contrast, electric force acts only on electrically charged bodies, and it acts in two ways: as a force

of attraction between two bodies having unlike charges (one positive and one negative) or as a force of repulsion between two bodies having like charges (both positive or both negative). Note also that gravity acts on all material bodies, for all of them have mass, whereas electric force acts only on electrically charged bodies; it has no effect on uncharged, electrically neutral ones. An electrically charged object has either an excess of electrons, giving it a negative charge, or a shortage of electrons, leaving it positively charged.

Now consider what happens to the surplus electrons on a negatively charged object. Each one repels all the others. Can they "escape"? The answer depends on whether the object is connected to the ground, however indirectly, by materials that conduct electricity (conductors), or that do not (insulators).3

If the object is separated from the ground by good insulators, then it will retain its charge—the surplus electrons—for quite a long time (not forever: no insulator is perfect). Because the electrons repel each other, they will come to be evenly spread over the surface of the object and will remain there; in a word, they will be static. This is why the electric charges, forces, and fields we have been considering are often called electrostatic charges, forces, and fields. While it is charged, the object will be surrounded by an electric field in the same way that an object with mass—any object, in fact—is surrounded by a gravitational field.

Figure 16.3. Electrostatic force fields surrounding (a and b) a single charge; (c and d) a pair of charges. The charges of the bodies are shown. In each case the arrows point in the direction a negatively charged test charge would move.

Figure 16.3. Electrostatic force fields surrounding (a and b) a single charge; (c and d) a pair of charges. The charges of the bodies are shown. In each case the arrows point in the direction a negatively charged test charge would move.

Figure 16.3 shows some typical electric fields. The lines are lines of force. Each line is the direction (shown by the arrows) that a small, negatively charged body—a test charge—would move if placed on the line. In theory, infinitely many lines could be drawn; in practice, only enough are shown to illustrate the form of the field without cluttering the drawing.

Imagine next that the negatively charged object is connected to the ground c by a conductor—say a metal rod. The surplus electrons it carries will quickly flow to the ground through the metal, where they will be instantly absorbed and neutralized. All metals are good conductors: a characteristic of metals is that their electrons "are not held permanently in orbits related to particular atoms but can rove freely____They form what is sometimes known as an "electron sea.'"4 Consequently, electrons can travel quickly and easily through them, creating an electric current. Rapidly flowing electrons and an electric current are, indeed, the same thing. The passage of about 1018 electrons per second through a conductor is a current of one ampere, or amp, the unit in which current is measured.5 Note that the number is only approximate: an ampere is not defined in this way, but rather in terms of the force the electrons exert. A precise definition appears in a later section.

We have now reached a point where it is possible to consider the energy provided by an electric current. The force that acts whenever an electric current flows generates energy, and the energy can be dissipated in a variety of ways. Often it is given off as heat: the filament in a lightbulb heats up when current flows through it, likewise the element in a toaster or an electric radiator (the way heat and light are radiated is discussed in chapter 18).

The rate at which an electric current yields energy—its power—doesn't depend only on the current (the number of amperes flowing). The force driving the current—the voltage—is equally important.6 Imagine a waterfall and regard the water as analogous to an electric current flowing along a conductor. The power of the waterfall depends on both the volume of falling water and the distance it falls. With an electric current, the number of amps corresponds to the volume of water, while the voltage corresponds to the height through which it falls.

The voltage between two points on a current-carrying conductor—a wire, for instance—is said to be one volt if a current of one ampere yields power equal to one watt (one joule per second). This leads to the well-known formula amps x volts = watts, which can equally well be written volts = watts + amps.

The latter formula leads to the definition of a volt: one volt is the voltage between two points on a wire carrying a current of one ampere, when the power dissipated between the points is one watt. As promised, the definition of an ampere will appear later. One other unit can conveniently be defined here—

the electronvolt (symbol eV). It is the energy gained by a single electron in "falling" (recall the waterfall analogy) through one volt.

The concept of potential energy—energy ready to be dissipated when circumstances make it possible—is as relevant in the context of electricity as it is in the context of gravity, described in chapter 2. Think of the waterfall analogy again. We know that gravitational potential energy (PE) resides in a mass that would fall if it were able to, for example, a massive body of water prevented by a dam from flowing down a valley. Similarly, electric PE resides in an electrical charge prevented by insulators from going anywhere.

### Lightning

As illustrative examples of electric fields, we have so far considered only small-scale (very small scale!) fields that are artificially created indoors. Now we turn to the electric fields occurring outdoors in the natural world. They give unmistakable evidence of their presence every time lightning strikes.

In calm weather, the surface of the earth is negatively charged: a permanent electric field is believed to encase the whole earth.7 The voltage across the gap between the ionosphere and the ground is estimated to be anywhere between 200,000 and 1,000,000 volts. The ionosphere is the electrified upper atmosphere starting about 100 km up. A current of one or two picoamperes per square meter is thought to flow continuously across the gap (one picoampere is 10-12 amperes); although air is a very good insulator, it is not so perfect as to prevent the flow of a current as small as this.

Lightning strikes when high voltages develop in the lower atmosphere, between a cloud base and the ground and between adjacent clouds (fig. 16.4).The way these strong electric fields develop is the subject of ongoing research; collecting data is both difficult and dangerous. All that needs to be said here is that it is hardly surprising that electric charges are apt to develop on the myriad tiny particles always suspended in the air. They are in a medium that is both turbulent, causing the particles to collide, and insulating, enabling them to retain the charges they acquire by losing or gaining electrons when they strike each other. Voltages high enough to produce lightning develop best in clouds with abundant ice particles.

Finely divided dust in a turbulent matrix is also found in the plumes of erupting volcanoes; the dust particles acquire electric charges, and lightning flashes are sometimes seen within the plumes.

In thunderstorms, lightning strikes when the voltage becomes so high that

Figure 16.4. The arrangement of electric charges, in a thunder cloud and on the ground and buildings below, when lightning is about to strike. The negative charge at the bottom of the cloud repels the surplus electrons on the surfaces directly below (ground, trees, buildings, and the like), leaving them positively charged.

Figure 16.4. The arrangement of electric charges, in a thunder cloud and on the ground and buildings below, when lightning is about to strike. The negative charge at the bottom of the cloud repels the surplus electrons on the surfaces directly below (ground, trees, buildings, and the like), leaving them positively charged.

the air fails to function as an insulator.8 A large current flows, momentarily, through a fraction of a kilometer and produces a blinding flash. The current may flow between a cloud and the ground or between neighboring clouds; in either case the charge that created the voltage is discharged (neutralized), whereupon new charges quickly develop. Details on the exact behavior of lightning can be found in any book on meteorology, but it is worth remarking that the reported quantities—volts, amps, watts, and joules—are wildly inconsistent in the different accounts, because of the difficulty of making the appropriate measurements. A lightning flash comes unexpectedly, it is over in a fraction of a second, and its peak power would overwhelm ordinary measur ing instruments; currents and voltages have to be inferred from indirect measurements. Often the quantities reported are not comparable: for instance, it is impossible to convert from joules to watts (joules per second) or vice versa without knowing how long a flash takes to complete. Currents are said to range from several thousand amperes to several hundred thousand.9

A quantity on which there seems to be some agreement is the temperature to which a lightning flash heats the air around it—about 20,000 to 30,000°C, which is three to four times the temperature at the surface of the sun.10 The sudden heating causes an explosion of the heated air as it expands. This explosion and its reverberations produce a sharp clap of thunder and the rumbling that follows it; we consider the energy in sound waves in chapter 17.

### Magnetic Force

Anybody who has refrigerator magnets uses magnetic force every day without thinking about it. To focus one's thoughts, it helps to do a simple desktop experiment like the one shown in figure 16.2, but this time employing magnetic, rather than electric, force. The instruments needed are a few sewing pins and a straight bar magnet (most refrigerator magnets are poorly shaped for the test and too strong; an adequate bar magnet can be made by repeatedly stroking an iron nail with a strong refrigerator magnet, always in the same direction). You can then pick up a series of two or three pins as shown in figure 16.5; note the strong resemblance to figure 16.2. It is evident that the tips of the nail and the pins acquire a property akin to electric charges, of two kinds.

Every magnet has two dissimilar poles, and as with positive and negative electric charges, unlike poles attract each other whereas like poles repel each other. This is easily tested using two straight magnets. The poles of a magnet are known as its north and south poles, and every magnet, without exception, has one of each. There's no need to know which is which to carry out the test if you use one end of a handheld magnet as a probe to "explore" a second magnet; you find that the probe attracts one end of the second magnet and repels the other end. The test can conveniently be done using a compass needle as the second magnet. The compass needle is itself a magnet mounted so that it can swivel freely; it spontaneously aligns itself with the magnetic field of the earth. This is why a magnet's poles are labeled north and south, or N and S for short.11 What makes the whole earth into a single huge magnet is a topic we come to later in this chapter.

We now consider the differences between the phenomena of figures 16.2

Figure 16.5. A magnetized nail picking up two pins. North and south poles are labeled N and S. (See text for details.)

and 16.5.The most obvious is that in the experiment with plastic and paper the materials are insulators, whereas in the experiment with nail and pins they are conductors. This shows that the same force cannot be responsible for both phenomena. Being good conductors, iron nails and pins cannot hold electrostatic charges and therefore cannot exert electrostatic forces.

Another, less obvious difference is that whereas electric charges can exist independently of each other, magnetic poles cannot. To demonstrate this requires more materials than the desktop experiments described so far; you may need to accept the following descriptions on faith. Suppose two plastic beads are rubbed with fur and two glass beads with silk (it is known, from separate tests, that the plastic beads become charged negatively, the glass ones positively).12 Each bead is hung from a length of thread, and pairs of beads are brought close to each other. The two plastic beads repel each other; likewise the two glass beads. But if one of the glass beads is brought close to one of the plastic ones, they attract each other. This is more than simply a demonstration of the phenomenon already noted, that like charges repel each other and unlike charges attract each other. In this respect they behave exactly like magnetic poles.

The noteworthy difference between the two experiments is this:electrically charged bodies can exist separately and independently of each other; magnetic poles cannot. Thus any one of the charged beads in the experiments just described can be carried into another room and then brought back without its charge disappearing or being altered. In contrast, the poles of a magnet, cannot be separated from each other by any means whatever. All magnets have a pole of each kind, a north pole and a south pole. If you cut a bar magnet in half, new poles will appear at the cut ends, so that the original magnet has become two half-length magnets, each with a pair of opposite poles, like this:

The final difference to note is that rubbing glass or plastic creates an electric charge, and with it an electric field, where none existed before. The energy of the rubbing is stored in the electric field. No analogous method will create a magnet: magnetic poles cannot be made to appear just by using muscle power.

Where, then, do magnets come from? How are they created? Weak magnets—lodestones—occur naturally and have been known for at least 2,500 years; a lodestone will attract other lodestones and also pieces of iron. A compass constructed from a sliver of lodestone was probably first used for overland navigation about 1,000 years ago, and for ocean navigation not long after. Lodestones consist of the mineral magnetite, an oxide of iron that is nowadays used as an iron ore in places where it is abundant.13

The problem of why some materials are strongly magnetic, or in technical terms, ferromagnetic, is one that would take us—if we were to follow it—deep into modern quantum theory. The elementary (and very incomplete) answer is that every electron is a miniature magnet.14 Therefore any chunk of material, of any kind, contains countless hordes of subatomic "electron-magnets." In most materials these electron-magnets are all aligned independently, pointing in every possible direction, so that taken together they cancel each other out, leaving the material as a whole nonmagnetic.

In ferromagnetic materials the electron-magnets, instead of acting independently, line up with each other spontaneously in microscopically small "packets" known as domains.15 This makes the material behave as an ordinary iron nail does in the presence of a ready-made magnet: although it is attracted by the magnet, the nail will not act as a magnet itself unless it is in contact with a ready-made one.

Ferromagnetic material can be turned into a magnet proper by putting it into a magnetic field strong enough to line up all the little domains so that they become parallel with each other (hitherto they had been pointing randomly in all directions). Once they are aligned, they stay aligned: the material has been converted into a "permanent" magnet. This is what happens when you stroke an iron nail with a magnet as described above: the nail, though not a magnet, originally consisted of a vast number of unaligned magnetic domains. Stroking it with a strong magnet turns all the domains in the nail so that they are aligned along it, all pointing in the same direction. This means that the nail has itself become a magnet; it will continue to be one, if not permanently, then until it is melted, violently hammered, or otherwise mistreated.

### The Link between Electricity and Magnetism

It isn't necessary to have a magnet to create a magnetic field. it can easily be done by connecting the ends of a length of copper wire to the terminals of a six-volt electric battery (cautionary note: this short-circuiting of a battery should be done only momentarily, to avoid overheating). Before closing the circuit, lay the wire flat on a table and place a compass on top of it. Before the circuit is closed, the compass needle points to magnetic north in the usual way: it is unaffected by the nonmagnetic copper wire touching it. But as soon as the circuit is closed, the needle will align itself at right angles to the wire, leading to the conclusion that the flowing current is creating a magnetic field.

Figure 16.6 contrasts the ambient magnetic field due to the earth's magnetism far from electrical disturbances with the field near an electric current. In figure 16.6a a collection of compasses (only their needles are shown) is arrayed on a table; they all align themselves parallel with one another, pointing toward magnetic north. Figure 16.6b shows the same setup, but this time a current-carrying wire passes through a hole in the middle of the table; the wire is at right angles to the tabletop, and its cross section is the black dot. Now the compass needles align themselves with the magnetic field around the live wire. As they clearly indicate, the lines of magnetic force form concentric circles centered on the wire.

Note that the lines of force are closed loops, with no end points. Contrast them with the lines of force of a gravitational field, which all terminate on a mass (fig. 16.1), and with the lines of force of an electric field, which all terminate on an electric charge (fig. 16.3). This suggests that there must be some connection between the impossibility of separating the poles of a magnet and the

 / / zr—^ s \ i 1 \ 1 v \ 1 / \ \ bV / /

Figure 16.6. An array of compasses (a) in the earth's natural magnetic field and (b) in the field surrounding a current-carrying wire. The wire enters the page at right angles; its cross section is shown by the black dot.

fact that the magnetic lines of force around a current-carrying wire form closed loops; they do this invariably, however coiled or tangled the wire may be.

To explain why this should be so without going into details (which would take us too far from the subject of energy), we need only consider the field of force of an iron bar magnet, shown in two forms in figure 16.7. Figure 16.7a shows how the lines of force would be interpreted if there were indeed such a thing as a "magnetic charge" at each end of the magnet, where the lines of force appear to terminate. Figure 16.7b shows the lines of force as closed loops; from the point where each line seems to end at the magnet's south pole, it is assumed to continue, through the interior of the iron bar, and emerge at the north pole. The lines are believed to "thread through tiny circulating currents on an atomic scale."16 Indeed, these are the currents that cause the magnet to be a magnet.

Because a current-carrying wire creates a magnetic field around itself, it functions as a magnet. Two current-carrying wires ranged side by side exert magnetic force on each other. If the current flows in the same direction in both wires, they attract each other; if the currents are antiparallel (oppositely directed), they repel each other.

Now for the promised definition of an ampere, the unit used for measuring an electric current: if the magnetic force between two identical straight, parallel, current-carrying wires separated by a gap of one meter is 2 x 10-7

Figure 16.7. (a) The observed field of force of a bar magnet. (b) The (inferred) complete field of force. All lines of force are closed loops passing, for part of their length, through the solid iron of the magnet.

newtons per meter of length, then the current in each wire is defined as one ampere (recall from chapter 2 that one newton is the force required to give a mass of one kilogram an acceleration of one meter per second per second). An ampere is a measure of current or, equivalently, of moving electric charge. Next, we need a quantitative measure of electric charge itself. The unit devised for this is the coulomb: one coulomb is the amount of electric charge transported in one second by a current of one ampere. These units are named for two French scientists whose researches, in the eighteenth and nineteenth centuries, helped unravel the connection between electricity and magnetism.17 That electric charge is measured in terms of electric current, which is measured in terms of magnetic force, gives some idea of the order in which different topics were developed.

The knowledge that flowing electric charge (a current) creates a magnetic field leads to the suspicion that a moving magnetic field might create a current. It does. The most convenient way to make it happen is to move a conductor (a wire or a coil of wire) through a stationary magnetic field; this causes an electric current to flow through the conductor. The kinetic energy provided to the conductor by whatever is moving it becomes converted to electrical energy. This is how an electrical generator (sometimes called a dynamo) works.

The close relation between electric and magnetic forces should now be clear.

A motionless electric charge (an electrostatic charge) creates an electrostatic field, while a stream of electric charges (an electric current) creates a magnetic field. The discovery that electric fields and magnetic fields are actually two manifestations of a single phenomenon, now known as an electromagnetic field, was one of the greatest scientific advances of the nineteenth century. We return to the topic in chapter 18. Before doing so, we look at the magnetic field of the whole earth, which enfolds all of us, everywhere and all the time.

### The Earth as a Magnet

A hiker using a compass is benefiting from the fact that the whole earth is a magnet. The compass needle, because it is a magnet, spontaneously aligns itself with the earth's magnetic field.

This raises the question, Why should the earth be a magnet? That it is one implies, as we saw in the preceding section, that electric currents must be flowing somewhere around or within the earth. Where and why? And what is the source of the energy driving the currents?

Theoretical arguments show that the currents must be within the earth. Moreover, they must be confined to the iron of the core, because the mantle consists of silicate minerals, which are electrical insulators.18

It is believed that convection currents in the liquid layer of the core rotate in a manner that generates a magnetic field that is almost (but not quite) parallel with the earth's axis of rotation. At the same time, electric currents flow in the liquid iron because it is moving in the magnetic field. Positive feedback is always in progress: the electric currents boost the magnetic field, and the magnetic field intensifies the electric currents.

The whole subject—magnetohydrodynamics—is made excruciatingly complicated by, among other things, the reciprocal interactions between the currents of liquid iron flowing convectively and the electric currents that flow in the iron. The words "current" and "flow" must both be used in two senses to describe what is going on. Suffice it to say that the energy generating all this action comes from the radioactivity that heats the earth's core, plus the residual heat still remaining from the time of the earth's formation.As we saw in chapter 15, heat from these sources is what keeps the outer core molten and convection happening; ultimately, it keeps the earth's magnetic field in existence.