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Consider a force like gravitation which varies predominantlyinversely as the square of the distance, but which is about abillion-billion-billion-billion times stronger. And with anotherdifference. There are two kinds of “matter,” which we can callpositive and negative. Like kinds repel and unlike kindsattract—unlike gravity where there is only attraction. What wouldhappen?

A bunch of positives would repel with an enormous force and spread outin all directions. A bunch of negatives would do the same. But anevenly mixed bunch of positives and negatives would do somethingcompletely different. The opposite pieces would be pulled together bythe enormous attractions. The net result would be that the terrificforces would balance themselves out almost perfectly, by formingtight, fine mixtures of the positive and the negative, and between twoseparate bunches of such mixtures there would be practically noattraction or repulsion at all.

There is such a force: the electrical force. And all matter is a mixtureof positive protons and negative electrons which are attracting andrepelling with this great force. So perfect is the balance, however,that when you stand near someone else you don’t feel any force at all.If there were even a little bit of unbalance you would know it. If youwere standing at arm’s length from someone and each of you hadonepercent more electrons than protons, the repelling force would beincredible. How great? Enough to lift the Empire State Building? No! Tolift Mount Everest? No! The repulsion would be enough to lift a“weight” equal to that of the entire earth!

With such enormous forces so perfectly balanced in this intimatemixture, it is not hard to understand that matter, trying to keep itspositive and negative charges in the finest balance, can have a greatstiffness and strength. The Empire State Building, for example, swingsless than one inch in the wind because the electrical forces hold everyelectron and proton more or less in its proper place. On the otherhand, if we look at matter on a scale small enough that we see only afew atoms, any small piece will not, usually, have an equal number ofpositive and negative charges, and so there will be strong residualelectrical forces. Even when there are equal numbers of both chargesin two neighboring small pieces, there may still be large netelectrical forces because the forces between individual charges varyinversely as the square of the distance. A net force can arise if anegative charge of one piece is closer to the positive than to thenegative charges of the other piece. The attractive forces can then belarger than the repulsive ones and there can be a net attractionbetween two small pieces with no excess charges. The force that holdsthe atoms together, and the chemical forces that hold moleculestogether, are really electrical forces acting in regions where thebalance of charge is not perfect, or where the distances are verysmall.

You know, of course, that atoms are made with positive protons in thenucleus and with electrons outside. You may ask: “If this electricalforce is so terrific, why don’t the protons and electrons just get ontop of each other? If they want to be in an intimate mixture, whyisn’t it still more intimate?” The answer has to do with the quantumeffects. If we try to confine our electrons in a region that is veryclose to the protons, then according to the uncertaintyprinciple theymust have some mean square momentum which is larger the more we try toconfine them. It is this motion, required by the laws of quantummechanics, that keeps the electrical attraction from bringing thecharges any closer together.

There is another question: “What holds the nucleus together”? In anucleus there are several protons, all of which are positive. Why don’tthey push themselves apart? It turns out that in nuclei there are, inaddition to electrical forces, nonelectrical forces, called nuclearforces, which are greater than the electrical forces and which are ableto hold the protons together in spite of the electrical repulsion. Thenuclear forces, however, have a short range—their force falls off muchmore rapidly than $1/r^2$. And this has an important consequence. If anucleus has too many protons in it, it gets too big, and it will notstay together. An example is uranium, with 92 protons. The nuclearforces act mainly between each proton (or neutron) and its nearestneighbor, while the electrical forces act over larger distances, givinga repulsion between each proton and all of the others in the nucleus.The more protons in a nucleus, the stronger is the electrical repulsion,until, as in the case of uranium, the balance is so delicate that thenucleus is almost ready to fly apart from the repulsive electricalforce. If such a nucleus is just “tapped” lightly (as can be done bysending in a slow neutron), it breaks into two pieces, each withpositive charge, and these pieces fly apart by electrical repulsion. Theenergy which is liberated is the energy of the atomic bomb. This energyis usually called “nuclear” energy, but it is really “electrical”energy released when electrical forces have overcome the attractivenuclear forces.

We may ask, finally, what holds a negatively charged electron together(since it has no nuclear forces). If an electron is all made of onekind of substance, each part should repel the other parts. Why, then,doesn’t it fly apart? But does the electron have “parts”? Perhaps weshould say that the electron is just a point and that electricalforces only act betweendifferent point charges, so that the electron does not actupon itself. Perhaps. All we can say is that the question of what holdsthe electron together has produced many difficulties in the attempts toform a complete theory of electromagnetism. The question has never beenanswered. We will entertain ourselves by discussing this subject somemore in later chapters.

As we have seen, we should expect that it is a combination of electricalforces and quantum-mechanical effects that will determine the detailedstructure of materials in bulk, and, therefore, their properties. Somematerials are hard, some are soft. Some are electrical“conductors”—because their electrons are free tomove about; others are “insulators”—because theirelectrons are held tightly to individual atoms. We shall consider laterhow some of these properties come about, but that is a very complicatedsubject, so we will begin by looking at the electrical forces only insimple situations. We begin by treating only the laws ofelectricity—including magnetism, which is really a part of the samesubject.

We have said that the electrical force, like a gravitational force,decreases inversely as the square of the distance between charges. Thisrelationship is called Coulomb’s law. But it is not precisely true when charges are moving—theelectrical forces depend also on the motions of the charges in acomplicated way. One part of the force between moving charges we callthemagnetic force. It is really one aspect of an electrical effect. That is why wecall the subject “electromagnetism.”

There is an important general principle that makes it possible totreat electromagnetic forces in a relatively simple way. We find, fromexperiment, that the force that acts on a particular charge—nomatter how many other charges there are or how they aremoving—depends only on the position of that particular charge, onthe velocity of the charge, and on the amount of charge. We can writethe force $\FLPF$ on a charge $q$ moving with a velocity $\FLPv$ as\begin{equation}\label{Eq:II:1:1}\FLPF=q(\FLPE+\FLPv\times\FLPB).\end{equation}We call $\FLPE$ theelectric field and $\FLPB$ themagneticfield at thelocation of the charge. The important thing is that the electricalforces from all the other charges in the universe can be summarized bygiving just these two vectors. Their values will depend onwherethe charge is, and may change withtime. Furthermore, if wereplace that charge with another charge, the force on the new chargewill be just in proportion to the amount of charge so long as all therest of the charges in the world do not change their positions ormotions. (In real situations, of course, each charge produces forces onall other charges in the neighborhood and may cause these other chargesto move, and so in some cases the fieldscan change if we replaceour particular charge by another.)

We know from Vol. I how to find the motion of a particle if we knowthe force on it. Equation (1.1) can be combined with theequation of motion to give\begin{equation}\label{Eq:II:1:2}\ddt{}{t}\biggl[\frac{m\FLPv}{(1-v^2/c^2)^{1/2}}\biggr]=\FLPF=q(\FLPE+\FLPv\times\FLPB).\end{equation}So if $\FLPE$ and $\FLPB$ are given, we can find the motions. Now weneed to know how the $\FLPE$’s and $\FLPB$’s are produced.

One of the most important simplifying principles about the way thefields are produced is this: Suppose a number of charges moving insome manner would produce a field $\FLPE_1$, and another set ofcharges would produce $\FLPE_2$. If both sets of charges are in placeat the same time (keeping the same locations and motions they had whenconsidered separately), then the field produced is just the sum\begin{equation}\label{Eq:II:1:3}\FLPE=\FLPE_1+\FLPE_2.\end{equation}This fact is calledthe principle ofsuperposition of fields. Itholds also for magnetic fields.

This principle means that if we know the law for the electric andmagnetic fields produced by asingle charge moving in anarbitrary way, then all the laws ofelectrodynamics are complete. If we want to knowthe force on charge $A$ we need only calculate the $\FLPE$ and $\FLPB$produced by each of the charges $B$, $C$, $D$, etc., and then add the$\FLPE$’s and $\FLPB$’s from all the charges to find the fields, andfrom them the forces acting on charge $A$. If it had only turned outthat the field produced by a single charge was simple, this would be theneatest way to describe the laws of electrodynamics. We have alreadygiven a description of this law (Chapter 28, Vol. I) andit is, unfortunately, rather complicated.

It turns out that the forms in which the laws of electrodynamics aresimplest are not what you might expect. It isnot simplest togive a formula for the force that one charge produces on another. Itis true that when charges are standing still the Coulomb forcelaw is simple, but when charges are moving about the relations arecomplicated by delays in time and by the effects of acceleration,among others. As a result, we do not wish to present electrodynamicsonly through the force laws between charges; we find it moreconvenient to consider another point of view—a point of view inwhich the laws of electrodynamics appear to be the most easilymanageable.

First, we must extend, somewhat, our ideas of the electric andmagnetic vectors, $\FLPE$ and $\FLPB$. We have defined them in termsof the forces that are felt by a charge. We wish now to speak ofelectric and magnetic fieldsat a point even when there is nocharge present. We are saying, in effect, that since there are forces“acting on” the charge, there is still “something” there when thecharge is removed. If a charge located at the point $(x,y,z)$ at thetime $t$ feels the force $\FLPF$ given by Eq. (1.1) weassociate the vectors $\FLPE$ and $\FLPB$ withthe point inspace $(x,y,z)$. We may think of $\FLPE(x,y,z,t)$ and $\FLPB(x,y,z,t)$as giving the forces thatwould be experienced at the time $t$by a charge located at $(x,y,z)$,with the condition thatplacing the charge theredid not disturb the positions ormotions of all the other charges responsible for the fields.

Following this idea, we associate withevery point $(x,y,z)$ inspace two vectors $\FLPE$ and $\FLPB$, which may be changing withtime. The electric and magnetic fields are, then, viewed asvector functions of $x$, $y$, $z$, and $t$. Since a vector isspecified by its components, each of the fields $\FLPE(x,y,z,t)$and $\FLPB(x,y,z,t)$ represents three mathematical functions of$x$, $y$, $z$, and $t$.

It is precisely because $\FLPE$ (or $\FLPB$) can be specified at everypoint in space that it is called a “field.” A “field” is anyphysical quantity which takes on different values at different pointsin space. Temperature, for example, is a field—in this case a scalarfield, which we write as $T(x,y,z)$. The temperature could also varyin time, and we would say the temperature field is time-dependent, andwrite $T(x,y,z,t)$. Another example is the “velocity field” of aflowing liquid. We write $\FLPv(x,y,z,t)$ for the velocity of theliquid at each point in space at the time $t$. It is a vector field.

Returning to the electromagnetic fields—although they are producedby charges according to complicated formulas, they have the followingimportant characteristic: the relationships between the values of thefields atone point and the values at anearby point arevery simple. With only a few such relationships in the form ofdifferential equations we can describe the fields completely. It is interms of such equations that the laws of electrodynamics are mostsimply written.

There have been various inventions to help the mind visualize thebehavior of fields. The most correct is also the most abstract: wesimply consider the fields as mathematical functions of position andtime. We can also attempt to get a mental picture of the field bydrawing vectors at many points in space, each of which gives the fieldstrength and direction at that point. Such a representation is shownin Fig. 1–1. We can go further, however, and draw lineswhich are everywhere tangent to the vectors—which, so to speak,follow the arrows and keep track of the direction of the field. Whenwe do this we lose track of thelengths of the vectors, but wecan keep track of the strength of the field by drawing the lines farapart when the field is weak and close together when it is strong. Weadopt the convention that thenumber of lines per unit area atright angles to the lines is proportional to thefieldstrength. This is, of course, only anapproximation, and it will require, in general, that new lines sometimesstart up in order to keep the number up to the strength of the field.The field of Fig. 1–1 is represented by field lines inFig. 1–2.

There are two mathematically important properties of a vector fieldwhich we will use in our description of the laws of electricity fromthe field point of view. Suppose we imagine a closed surface of somekind and ask whether we are losing “something” from the inside; thatis, does the field have a quality of “outflow”? For instance, for avelocity field we might ask whether the velocity is always outward onthe surface or, more generally, whether more fluid flows out (per unittime) than comes in. We call the net amount of fluid going out throughthe surface per unit time the “flux of velocity” through thesurface. The flow through an element of a surface is just equal to thecomponent of the velocity perpendicular to the surface times the areaof the surface. For an arbitrary closed surface, thenet outwardflow—orflux—is the average outward normal component ofthe velocity, times the area of the surface:\begin{equation}\label{Eq:II:1:4}\text{Flux}=(\text{average normal component})\cdot(\text{surfacearea}).\end{equation}\begin{equation}\label{Eq:II:1:4}\text{Flux}=\begin{pmatrix}\text{average}\\[-.75ex]\text{normal}\\[-.75ex]\text{component}\end{pmatrix}\cdot\begin{pmatrix}\text{surface}\\[-.75ex]\text{area}\end{pmatrix}.\end{equation}

In the case of an electric field, we can mathematically definesomething analogous to an outflow, and we again call it the flux, butof course it is not the flow of any substance, because the electricfield is not the velocity of anything. It turns out, however, that themathematical quantity which is the average normal component of thefield still has a useful significance. We speak, then, of theelectric flux—also defined by Eq. (1.4). Finally, it is alsouseful to speak of the flux not only through a completely closedsurface, but through any bounded surface. As before, the flux throughsuch a surface is defined as the average normal component of a vectortimes the area of the surface. These ideas are illustrated inFig. 1–3.

There is a second property of a vector field that has to do with aline, rather than a surface. Suppose again that we think of a velocityfield that describes the flow of a liquid. We might ask thisinteresting question: Is the liquid circulating? By that we mean: Isthere a net rotational motion around some loop? Suppose that weinstantaneously freeze the liquid everywhere except inside of a tubewhich is of uniform bore, and which goes in a loop that closes back onitself as in Fig. 1–4. Outside of the tube the liquidstops moving, but inside the tube it may keep on moving because of themomentum in the trapped liquid—that is, if there is more momentumheading one way around the tube than the other. We define a quantitycalled thecirculation as the resulting speed of theliquid inthe tube times its circumference. We can again extend our ideas anddefine the “circulation” for any vector field (even when there isn’tanything moving). For any vector field thecirculation aroundany imagined closed curve is defined as the average tangentialcomponent of the vector (in a consistent sense) multiplied by thecircumference of the loop (Fig. 1–5):\begin{equation}\label{Eq:II:1:5}\text{Circulation}=(\text{average tangentialcomponent})\cdot(\text{distance around}).\end{equation}\begin{equation}\label{Eq:II:1:5}\text{Circulation}=\begin{pmatrix}\text{average}\\[-.75ex]\text{tangential}\\[-.75ex]\text{component}\end{pmatrix}\cdot\begin{pmatrix}\text{distance}\\[-.75ex]\text{around}\end{pmatrix}\end{equation}You will see that this definition does indeed give a number which isproportional to the circulation velocity in the quickly frozen tubedescribed above.

With just these two ideas—flux and circulation—we can describe allthe laws of electricity and magnetism at once. You may not understandthe significance of the laws right away, but they will give you someidea of the way the physics of electromagnetism will be ultimatelydescribed.

The first law of electromagnetism describes the flux of the electricfield:\begin{equation}\label{Eq:II:1:6}\text{The flux of $\FLPE$ through any closed surface}=\frac{\text{the net charge inside}}{\epsO},\end{equation}\begin{equation}\label{Eq:II:1:6}\begin{pmatrix}\text{Flux of $\FLPE$}\\[-.5ex]\text{through any}\\[-.5ex]\text{closed surface}\end{pmatrix}=\frac{\begin{pmatrix}\text{net charge}\\[-.5ex]\text{inside}\end{pmatrix}}{\epsO},\end{equation}where $\epsO$ is a convenient constant. (The constant $\epsO$ isusually read as “epsilon-zero” or “epsilon-naught”.) If there areno charges inside the surface, even though there are charges nearbyoutside the surface, theaverage normal component of $\FLPE$ iszero, so there is no net flux through the surface. To show the powerof this type of statement, we can show that Eq. (1.6) isthe same as Coulomb’s law,provided only that we also add the ideathat the field from a single charge is spherically symmetric. For apoint charge, we draw a sphere around the charge. Then the averagenormal component is just the value of the magnitude of $\FLPE$ at anypoint, since the field must be directed radially and have the samestrength for all points on the sphere. Our rule now says that thefield at the surface of the sphere, times the area of thesphere—that is, the outgoing flux—is proportional to the chargeinside. If we were to make the radius of the sphere bigger, the areawould increase as the square of the radius. The average normalcomponent of the electric field times that area must still be equal tothe same charge inside, and so the field must decrease as the squareof the distance—we get an “inverse square” field.

If we have an arbitrary stationary curve in space and measure thecirculation of the electric field around the curve, we will find thatit is not, in general, zero (although it is for the Coulombfield). Rather, for electricity there is a second law that states: forany surface $S$ (not closed) whose edge is the curve $C$,\begin{equation}\label{Eq:II:1:7}\text{Circulation of $\FLPE$ around $C$}=-\ddt{}{t}(\text{flux of$\FLPB$ through $S$}).\end{equation}\begin{equation}\label{Eq:II:1:7}\begin{pmatrix}\text{Circulation of $\FLPE$}\\[-.5ex]\text{around $C$}\end{pmatrix}=-\ddt{}{t}\begin{pmatrix}\text{flux of $\FLPB$}\\[-.5ex]\text{through $S$}\end{pmatrix}.\end{equation}

We can complete the laws of the electromagnetic field by writing twocorresponding equations for the magnetic field $\FLPB$:\begin{equation}\label{Eq:II:1:8}\text{Flux of $\FLPB$ through any closed surface}=0.\end{equation}\begin{equation}\label{Eq:II:1:8}\begin{pmatrix}\text{Flux of $\FLPB$}\\[-.5ex]\text{through any}\\[-.5ex]\text{closed surface}\end{pmatrix}=0.\end{equation}For a surface $S$ bounded by the curve $C$,\begin{align}c^2(\text{circulation of $\FLPB$ around $C$})=&\ddt{}{t}(\text{flux of$\FLPE$ through $S$})\notag\\\label{Eq:II:1:9}&+\frac{\text{flux of electric current through$S$}}{\epsO}.\end{align}\begin{gather}\label{Eq:II:1:9}c^2\begin{pmatrix}\text{circulation of $\FLPB$}\\[-.5ex]\text{around $C$}\end{pmatrix}=\\[1.5ex]\ddt{}{t}\begin{pmatrix}\text{flux of $\FLPE$}\\[-.5ex]\text{through $S$}\end{pmatrix}+\frac{\begin{pmatrix}\text{flux of}\\[-.75ex]\text{electric current}\\[-.5ex]\text{through $S$}\end{pmatrix}}{\epsO}.\notag\end{gather}

The constant $c^2$ that appears in Eq. (1.9) is thesquare of the velocity of light. It appears because magnetism is inreality a relativistic effect of electricity. The constant $\epsO$ hasbeen stuck in to make the units of electric current come out in aconvenient way.

Equations (1.6) through (1.9), together withEq. (1.1), are all the laws ofelectrodynamics¹. As you remember,the laws of Newton were very simple to write down, but they had a lotof complicated consequences and it took us a long time to learn aboutthem all. These laws are not nearly as simple to write down, whichmeans that the consequences are going to be more elaborate and it willtake us quite a lot of time to figure them all out.

We can illustrate some of the laws of electrodynamics by a series ofsmall experiments which show qualitatively the interrelationships ofelectric and magnetic fields. You have experienced the first term ofEq. (1.1) when combing your hair, so we won’t show thatone. The second part of Eq. (1.1) can be demonstrated bypassing a current through a wire which hangs above a bar magnet, asshown in Fig. 1–6. The wire will move when a current isturned on because of the force $\FLPF=q\FLPv\times\FLPB$. When acurrent exists, the charges inside the wire are moving, so they have avelocity $\FLPv$, and the magnetic field from the magnet exerts aforce on them, which results in pushing the wire sideways.

When the wire is pushed to the left, we would expect that the magnetmust feel a push to the right. (Otherwise we could put the whole thingon a wagon and have a propulsion system that didn’t conservemomentum!) Although the force is too small to make movement of the barmagnet visible, a more sensitively supported magnet, like a compassneedle, will show the movement.

How does the wire push on the magnet? The current in the wire producesa magnetic field of its own that exerts forces on themagnet. According to the last term in Eq. (1.9), acurrent must have acirculation of $\FLPB$—in this case, thelines of $\FLPB$ are loops around the wire, as shown inFig. 1–7. This $\FLPB$-field is responsible for the forceon the magnet.

Equation (1.9) tells us that for a fixed current throughthe wire the circulation of $\FLPB$ is the same forany curvethat surrounds the wire. For curves—say circles—that are fartheraway from the wire, the circumference is larger, so the tangentialcomponent of $\FLPB$ must decrease. You can see that we would, infact, expect $\FLPB$ to decrease linearly with the distance from along straight wire.

Now, we have said that a current through a wire produces a magneticfield, and that when there is a magnetic field present there is aforce on a wire carrying a current. Then we should also expect that ifwe make a magnetic field with a current in one wire, it should exert aforce on another wire which also carries a current. This can be shownby using two hanging wires as shown in Fig. 1–8. Whenthe currents are in the same direction, the two wires attract, butwhen the currents are opposite, they repel.

In short, electrical currents, as well as magnets, make magneticfields. But wait, what is a magnet, anyway? If magnetic fields areproduced by moving charges, is it not possible that the magnetic fieldfrom a piece of iron is really the result of currents? It appears tobe so. We can replace the bar magnet of our experiment with a coil ofwire, as shown in Fig. 1–9. When a current is passedthrough the coil—as well as through the straight wire above it—weobserve a motion of the wire exactly as before, when we had a magnetinstead of a coil. In other words, the current in the coil imitates amagnet. It appears, then, that a piece of iron acts as though itcontains a perpetual circulating current. We can, in fact, understandmagnets in terms of permanent currents in the atoms of the iron. Theforce on the magnet in Fig. 1–7 is due to the secondterm in Eq. (1.1).

Where do the currents come from? One possibility would be from themotion of the electrons in atomic orbits.Actually, that is not the case for iron, although it is forsome materials. In addition to moving around in an atom, an electronalso spins about on its own axis—something like the spin of theearth—and it is the current from this spin that gives the magneticfield in iron. (We say “something like the spin of the earth”because the question is so deep in quantum mechanics that theclassical ideas do not really describe things too well.) In mostsubstances, some electrons spin one way and some spin the other, sothe magnetism cancels out, but in iron—for a mysterious reason whichwe will discuss later—many of the electrons are spinning with theiraxes lined up, and that is the source of the magnetism.

Since the fields of magnets are from currents, we do not have to add any extraterm to Eqs. (1.8) or (1.9) to take care ofmagnets. We just takeall currents, including the circulating currents ofthe spinning electrons, and then the law is right. You should also notice thatEq. (1.8) says that there are no magnetic “charges” analogous tothe electrical charges appearing on the right side of Eq. (1.6).None has been found.

The first term on the right-hand side of Eq. (1.9) wasdiscovered theoretically byMaxwell and is of great importance.Itsays that changingelectric fields produce magnetic effects. Infact, without this term the equation would not make sense, becausewithout it there could be no currents in circuits that are notcomplete loops. But such currents do exist, as we can see in thefollowing example. Imagine a capacitor made of two flat plates. It isbeing charged by a current that flows toward one plate and away fromthe other, as shown in Fig. 1–10. We draw a curve $C$around one of the wires and fill it in with a surface which crossesthe wire, as shown by the surface $S_1$ in the figure. According toEq. (1.9), the circulation of $\FLPB$ around $C$(times $c^2$) is given by the current in the wire (divided by $\epsO$). Butwhat if we fill in the curve with adifferent surface $S_2$,which is shaped like a bowl and passes between the plates of thecapacitor, staying always away from the wire? There is certainly nocurrent through this surface. But, surely, just changing the locationof an imaginary surface is not going to change a real magnetic field!The circulation of $\FLPB$ must be what it was before. The first termon the right-hand side of Eq. (1.9) does, indeed,combine with the second term to give the same result for the twosurfaces $S_1$ and $S_2$. For $S_2$ the circulation of $\FLPB$ isgiven in terms of the rate of change of the flux of $\FLPE$ betweenthe plates of the capacitor. And it works out that the changing $\FLPE$is related to the current in just the way required forEq. (1.9) to be correct.Maxwell saw that it was needed, andhe was the first to write the complete equation.

With the setup shown in Fig. 1–6 we can demonstrateanother of the laws of electromagnetism. We disconnect the ends of thehanging wire from the battery and connect them to agalvanometer whichtells us when there is a current through the wire. When wepushthe wire sideways through the magnetic field of the magnet, we observea current. Such an effect is again just another consequence ofEq. (1.1)—the electrons in the wire feel theforce $\FLPF=q\FLPv\times\FLPB$. The electrons have a sidewise velocitybecause they move with the wire. This $\FLPv$ with a vertical $\FLPB$from the magnet results in a force on the electrons directedalong the wire, which starts the electrons moving toward thegalvanometer.

Suppose, however, that we leave the wire alone and move the magnet. Weguess from relativity that it should make no difference, and indeed,we observe a similar current in the galvanometer. How does themagnetic field produce forces on charges at rest? According toEq. (1.1) there must be an electric field. A moving magnetmust make an electric field. How that happens is said quantitatively byEq. (1.7). This equation describes many phenomena of greatpractical interest, such as those that occur in electric generators andtransformers.

The most remarkable consequence of our equations is that thecombination of Eq. (1.7) and Eq. (1.9)contains the explanation of the radiation of electromagnetic effectsover large distances. The reason is roughly something like this:suppose that somewhere we have a magnetic field which is increasingbecause, say, a current is turned on suddenly in a wire. Then byEq. (1.7) there must be a circulation of an electric field.As the electric field builds up to produce its circulation, thenaccording to Eq. (1.9) a magnetic circulation will begenerated. But the building up ofthis magnetic field willproduce a new circulation of the electric field, and so on. In this wayfields work their way through space without the need of charges orcurrents except at their source. That is the way wesee eachother! It is all in the equations of the electromagnetic fields.

We now make a few remarks on our way of looking at this subject. Youmay be saying: “All this business of fluxes and circulations ispretty abstract. There are electric fields at every point in space;then there are these ‘laws.’ But what isactually happening?Why can’t you explain it, for instance, by whatever itis thatgoes between the charges.” Well, it depends on your prejudices. Manyphysicists used to say that direct action with nothing in between wasinconceivable. (How could they find an idea inconceivable when it hadalready been conceived?) They would say: “Look, the only forces weknow are the direct action of one piece of matter on another. It isimpossible that there can be a force with nothing to transmit it.”But what really happens when we study the “direct action” of onepiece of matter right against another? We discover that it is not onepiece right against the other; they are slightly separated, and thereare electrical forces acting on a tiny scale. Thus we find that we aregoing to explain so-called direct-contact action in terms of thepicture for electrical forces. It is certainly not sensible to try toinsist that an electrical force has to look like the old, familiar,muscular push or pull, when it will turn out that the muscular pushesand pulls are going to be interpreted as electrical forces! The onlysensible question is what is themost convenient way to look atelectrical effects. Some people prefer to represent them as theinteraction at a distance of charges, and to use a complicatedlaw. Others love the field lines. They draw field lines all the time,and feel that writing $\FLPE$’s and $\FLPB$’s is too abstract. Thefield lines, however, are only a crude way of describing a field, andit is very difficult to give the correct, quantitative laws directlyin terms of field lines. Also, the ideas of the field lines do notcontain the deepest principle of electrodynamics, which is thesuperposition principle. Even though we know how the field lines lookfor one set of charges and what the field lines look like for anotherset of charges, we don’t get any idea about what the field linepatterns will look like when both sets are present together. From themathematical standpoint, on the other hand, superposition is easy—wesimply add the two vectors. The field lines have some advantage ingiving a vivid picture, but they also have some disadvantages. Thedirect interaction way of thinking has great advantages when thinkingof electrical charges at rest, but has great disadvantages whendealing with charges in rapid motion.

The best way is to use the abstract field idea. That it is abstract isunfortunate, but necessary. The attempts to try to represent theelectric field as the motion of some kind of gear wheels, or in termsof lines, or of stresses in some kind of material have used up moreeffort of physicists than it would have taken simply to get the rightanswers about electrodynamics. It is interesting that the correctequations for the behavior of light were worked out byMacCullagh in 1839.But people said to him: “Yes, but there is no real materialwhose mechanical properties could possibly satisfy those equations,and since light is an oscillation that must vibrate insomething, we cannot believe this abstract equation business.”If people had been more open-minded, they might have believed in theright equations for the behavior of light a lot earlier than they did.

In the case of the magnetic field we can make the following point:Suppose that you finally succeeded in making up a picture of themagnetic field in terms of some kind of lines or of gear wheelsrunning through space. Then you try to explain what happens to twocharges moving in space, both at the same speed and parallel to eachother. Because they are moving, they will behave like two currents andwill have a magnetic field associated with them (like the currents inthe wires of Fig. 1–8). An observer who was ridingalong with the two charges, however, would see both charges asstationary, and would say that there isno magnetic field. The“gear wheels” or “lines” disappear when you ride along with theobject! All we have done is to invent anew problem. How canthe gear wheels disappear?! The people who draw field lines are in asimilar difficulty. Not only is it not possible to say whether thefield lines move or do not move with charges—they may disappearcompletely in certain coordinate frames.

What we are saying, then, is that magnetism is really a relativisticeffect. In the case of the two charges we just considered, travellingparallel to each other, we would expect to have to make relativisticcorrections to their motion, with terms of order $v^2/c^2$. Thesecorrections must correspond to the magnetic force. But what about theforce between the two wires in our experiment (Fig. 1–8).There the magnetic force is thewhole force. It didn’t look likea “relativistic correction.” Also, if we estimate the velocities ofthe electrons in the wire (you can do this yourself), we find that theiraverage speed along the wire is about $0.01$ centimeter per second.So $v^2/c^2$ is about $10^{-25}$. Surely a negligible “correction.” Butno! Although the magnetic force is, in this case, $10^{-25}$ of the“normal” electrical force between the moving electrons, remember thatthe “normal” electrical forces have disappeared because of the almostperfect balancing out—because the wires have the same number ofprotons as electrons. The balance is much more precise than one partin $10^{25}$, and the small relativistic term which we call the magneticforce is the only term left. It becomes the dominant term.

It is the near-perfect cancellation of electrical effects whichallowed relativity effects (that is, magnetism) to be studied and thecorrect equations—to order $v^2/c^2$—to be discovered, even thoughphysicists didn’tknow that’s what was happening. And that iswhy, when relativity was discovered, the electromagnetic laws didn’tneed to be changed. They—unlike mechanics—were already correct toa precision of $v^2/c^2$.

Let us end this chapter by pointing out that among the many phenomenastudied by the Greeks there were two very strange ones: that if yourubbed a piece of amber you could lift up little piecesof papyrus, and that there was a strange rock from the land of Magnesiawhich attracted iron. It is amazing to think that these werethe only phenomena known to the Greeks in which the effects ofelectricity or magnetism were apparent. The reason that these were theonly phenomena that appeared is due primarily to the fantasticprecision of the balancing of charges that we mentioned earlier. Studyby scientists who came after the Greeks uncovered one new phenomenonafter another that were really some aspect of these amberand/or lodestone effects. Now we realize that the phenomena ofchemical interaction and, ultimately, of life itself are to beunderstood in terms of electromagnetism.

At the same time that an understanding of the subject ofelectromagnetism was being developed, technical possibilities thatdefied the imagination of the people that came before were appearing:it became possible to signal by telegraph over long distances, and totalk to another person miles away without any connections between, andto run huge power systems—a great water wheel, connected byfilaments over hundreds of miles to another engine that turns inresponse to the master wheel—many thousands of branchingfilaments—ten thousand engines in ten thousand places running themachines of industries and homes—all turning because of theknowledge of the laws of electromagnetism.

Today we are applying even more subtle effects. The electrical forces,enormous as they are, can also be very tiny, and we can control themand use them in very many ways. So delicate are our instruments thatwe can tell what a man is doing by the way he affects the electrons ina thin metal rod hundreds of miles away. All we need to do is to usethe rod as an antenna for a television receiver!

From a long view of the history of mankind—seen from, say, tenthousand years from now—there can be little doubt that the mostsignificant event of the 19th century will be judged asMaxwell’sdiscovery of the laws of electrodynamics. The American Civil War willpale into provincial insignificance in comparison with this importantscientific event of the same decade.