It turns out that the forms in which the laws of electrodynamics are simplest are not what you might expect. It is not simplest to give a formula for the force that one charge produces on another. It is true that when charges are standing still the Coulomb force law is simple, but when charges are moving about the relations are complicated by delays in time and by the effects of acceleration, among others. As a result, we do not wish to present electrodynamics only through the force laws between charges; we find it more convenient to consider another point of view—a point of view in which the laws of electrodynamics appear to be the most easily manageable.

This principle means that if we know the law for the electric and magnetic fields produced by a single charge moving in an arbitrary way, then all the laws of electrodynamics are complete. If we want to know the 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, and from them the forces acting on charge $A$. If it had only turned out that the field produced by a single charge was simple, this would be the neatest way to describe the laws of electrodynamics. We have already given a description of this law (Chapter 28 , Vol. I) and it is, unfortunately, rather complicated.

One of the most important simplifying principles about the way the fields are produced is this: Suppose a number of charges moving in some manner would produce a field $\FLPE_1$, and another set of charges would produce $\FLPE_2$. If both sets of charges are in place at the same time (keeping the same locations and motions they had when considered 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 called the principle of superposition of fields. It holds also for magnetic fields.

We know from Vol. I how to find the motion of a particle if we know the force on it. Equation ( 1.1 ) can be combined with the equation 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 we need to know how the $\FLPE$’s and $\FLPB$’s are produced.

There is an important general principle that makes it possible to treat electromagnetic forces in a relatively simple way. We find, from experiment, that the force that acts on a particular charge—no matter how many other charges there are or how they are moving—depends only on the position of that particular charge, on the velocity of the charge, and on the amount of charge. We can write the 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 $\FLPE$ the electric field and $\FLPB$ the magnetic field at the location of the charge. The important thing is that the electrical forces from all the other charges in the universe can be summarized by giving just these two vectors. Their values will depend on where the charge is, and may change with time. Furthermore, if we replace that charge with another charge, the force on the new charge will be just in proportion to the amount of charge so long as all the rest of the charges in the world do not change their positions or motions. (In real situations, of course, each charge produces forces on all other charges in the neighborhood and may cause these other charges to move, and so in some cases the fields can change if we replace our particular charge by another.)

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

As we have seen, we should expect that it is a combination of electrical forces and quantum-mechanical effects that will determine the detailed structure of materials in bulk, and, therefore, their properties. Some materials are hard, some are soft. Some are electrical “conductors”—because their electrons are free to move about; others are “insulators”—because their electrons are held tightly to individual atoms. We shall consider later how some of these properties come about, but that is a very complicated subject, so we will begin by looking at the electrical forces only in simple situations. We begin by treating only the laws of electricity—including magnetism, which is really a part of the same subject.

We may ask, finally, what holds a negatively charged electron together (since it has no nuclear forces). If an electron is all made of one kind of substance, each part should repel the other parts. Why, then, doesn’t it fly apart? But does the electron have “parts”? Perhaps we should say that the electron is just a point and that electrical forces only act between different point charges, so that the electron does not act upon itself. Perhaps. All we can say is that the question of what holds the electron together has produced many difficulties in the attempts to form a complete theory of electromagnetism. The question has never been answered. We will entertain ourselves by discussing this subject some more in later chapters.

There is another question: “What holds the nucleus together”? In a nucleus there are several protons, all of which are positive. Why don’t they push themselves apart? It turns out that in nuclei there are, in addition to electrical forces, nonelectrical forces, called nuclear forces, which are greater than the electrical forces and which are able to hold the protons together in spite of the electrical repulsion. The nuclear forces, however, have a short range—their force falls off much more rapidly than $1/r^2$. And this has an important consequence. If a nucleus has too many protons in it, it gets too big, and it will not stay together. An example is uranium, with 92 protons. The nuclear forces act mainly between each proton (or neutron) and its nearest neighbor, while the electrical forces act over larger distances, giving a 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 the nucleus is almost ready to fly apart from the repulsive electrical force. If such a nucleus is just “tapped” lightly (as can be done by sending in a slow neutron), it breaks into two pieces, each with positive charge, and these pieces fly apart by electrical repulsion. The energy which is liberated is the energy of the atomic bomb. This energy is usually called “nuclear” energy, but it is really “electrical” energy released when electrical forces have overcome the attractive nuclear forces.

You know, of course, that atoms are made with positive protons in the nucleus and with electrons outside. You may ask: “If this electrical force is so terrific, why don’t the protons and electrons just get on top of each other? If they want to be in an intimate mixture, why isn’t it still more intimate?” The answer has to do with the quantum effects. If we try to confine our electrons in a region that is very close to the protons, then according to the uncertainty principle they must have some mean square momentum which is larger the more we try to confine them. It is this motion, required by the laws of quantum mechanics, that keeps the electrical attraction from bringing the charges any closer together.

With such enormous forces so perfectly balanced in this intimate mixture, it is not hard to understand that matter, trying to keep its positive and negative charges in the finest balance, can have a great stiffness and strength. The Empire State Building, for example, swings less than one inch in the wind because the electrical forces hold every electron and proton more or less in its proper place. On the other hand, if we look at matter on a scale small enough that we see only a few atoms, any small piece will not, usually, have an equal number of positive and negative charges, and so there will be strong residual electrical forces. Even when there are equal numbers of both charges in two neighboring small pieces, there may still be large net electrical forces because the forces between individual charges vary inversely as the square of the distance. A net force can arise if a negative charge of one piece is closer to the positive than to the negative charges of the other piece. The attractive forces can then be larger than the repulsive ones and there can be a net attraction between two small pieces with no excess charges. The force that holds the atoms together, and the chemical forces that hold molecules together, are really electrical forces acting in regions where the balance of charge is not perfect, or where the distances are very small.

There is such a force: the electrical force. And all matter is a mixture of positive protons and negative electrons which are attracting and repelling 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 you were standing at arm’s length from someone and each of you had one percent more electrons than protons, the repelling force would be incredible. How great? Enough to lift the Empire State Building? No! To lift Mount Everest? No! The repulsion would be enough to lift a “weight” equal to that of the entire earth!

A bunch of positives would repel with an enormous force and spread out in all directions. A bunch of negatives would do the same. But an evenly mixed bunch of positives and negatives would do something completely different. The opposite pieces would be pulled together by the enormous attractions. The net result would be that the terrific forces would balance themselves out almost perfectly, by forming tight, fine mixtures of the positive and the negative, and between two separate bunches of such mixtures there would be practically no attraction or repulsion at all.

Consider a force like gravitation which varies predominantly inversely as the square of the distance, but which is about a billion-billion-billion-billion times stronger. And with another difference. There are two kinds of “matter,” which we can call positive and negative. Like kinds repel and unlike kinds attract—unlike gravity where there is only attraction. What would happen?

There have been various inventions to help the mind visualize the behavior of fields. The most correct is also the most abstract: we simply consider the fields as mathematical functions of position and time. We can also attempt to get a mental picture of the field by drawing vectors at many points in space, each of which gives the field strength and direction at that point. Such a representation is shown in Fig. 1–1 . We can go further, however, and draw lines which are everywhere tangent to the vectors—which, so to speak, follow the arrows and keep track of the direction of the field. When we do this we lose track of the lengths of the vectors, but we can keep track of the strength of the field by drawing the lines far apart when the field is weak and close together when it is strong. We adopt the convention that the number of lines per unit area at right angles to the lines is proportional to the field strength. This is, of course, only an approximation, and it will require, in general, that new lines sometimes start 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 in Fig. 1–2 .

Returning to the electromagnetic fields—although they are produced by charges according to complicated formulas, they have the following important characteristic: the relationships between the values of the fields at one point and the values at a nearby point are very simple. With only a few such relationships in the form of differential equations we can describe the fields completely. It is in terms of such equations that the laws of electrodynamics are most simply written.

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

Following this idea, we associate with every point $(x,y,z)$ in space two vectors $\FLPE$ and $\FLPB$, which may be changing with time. The electric and magnetic fields are, then, viewed as vector functions of $x$, $y$, $z$, and $t$. Since a vector is specified 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 $t$.

First, we must extend, somewhat, our ideas of the electric and magnetic vectors, $\FLPE$ and $\FLPB$. We have defined them in terms of the forces that are felt by a charge. We wish now to speak of electric and magnetic fields at a point even when there is no charge present. We are saying, in effect, that since there are forces “acting on” the charge, there is still “something” there when the charge is removed. If a charge located at the point $(x,y,z)$ at the time $t$ feels the force $\FLPF$ given by Eq. ( 1.1 ) we associate the vectors $\FLPE$ and $\FLPB$ with the point in space $(x,y,z)$. We may think of $\FLPE(x,y,z,t)$ and $\FLPB(x,y,z,t)$ as giving the forces that would be experienced at the time $t$ by a charge located at $(x,y,z)$, with the condition that placing the charge there did not disturb the positions or motions of all the other charges responsible for the fields.

With just these two ideas—flux and circulation—we can describe all the laws of electricity and magnetism at once. You may not understand the significance of the laws right away, but they will give you some idea of the way the physics of electromagnetism will be ultimately described.

There is a second property of a vector field that has to do with a line, rather than a surface. Suppose again that we think of a velocity field that describes the flow of a liquid. We might ask this interesting question: Is the liquid circulating? By that we mean: Is there a net rotational motion around some loop? Suppose that we instantaneously freeze the liquid everywhere except inside of a tube which is of uniform bore, and which goes in a loop that closes back on itself as in Fig. 1–4 . Outside of the tube the liquid stops moving, but inside the tube it may keep on moving because of the momentum in the trapped liquid—that is, if there is more momentum heading one way around the tube than the other. We define a quantity called the circulation as the resulting speed of the liquid in the tube times its circumference. We can again extend our ideas and define the “circulation” for any vector field (even when there isn’t anything moving). For any vector field the circulation around any imagined closed curve is defined as the average tangential component of the vector (in a consistent sense) multiplied by the circumference of the loop (Fig. 1–5 ): \begin{equation} \label{Eq:II:1:5} \text{Circulation}=(\text{average tangential component})\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 is proportional to the circulation velocity in the quickly frozen tube described above.

In the case of an electric field, we can mathematically define something analogous to an outflow, and we again call it the flux, but of course it is not the flow of any substance, because the electric field is not the velocity of anything. It turns out, however, that the mathematical quantity which is the average normal component of the field still has a useful significance. We speak, then, of the electric flux—also defined by Eq. ( 1.4 ). Finally, it is also useful to speak of the flux not only through a completely closed surface, but through any bounded surface. As before, the flux through such a surface is defined as the average normal component of a vector times the area of the surface. These ideas are illustrated in Fig. 1–3 .

There are two mathematically important properties of a vector field which we will use in our description of the laws of electricity from the field point of view. Suppose we imagine a closed surface of some kind and ask whether we are losing “something” from the inside; that is, does the field have a quality of “outflow”? For instance, for a velocity field we might ask whether the velocity is always outward on the surface or, more generally, whether more fluid flows out (per unit time) than comes in. We call the net amount of fluid going out through the surface per unit time the “flux of velocity” through the surface. The flow through an element of a surface is just equal to the component of the velocity perpendicular to the surface times the area of the surface. For an arbitrary closed surface, the net outward flow—or flux—is the average outward normal component of the velocity, times the area of the surface: \begin{equation} \label{Eq:II:1:4} \text{Flux}=(\text{average normal component})\cdot(\text{surface area}). \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}

1–4 The laws of electromagnetism

The first law of electromagnetism describes the flux of the electric field: \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 $\epsO$ is usually read as “epsilon-zero” or “epsilon-naught”.) If there are no charges inside the surface, even though there are charges nearby outside the surface, the average normal component of $\FLPE$ is zero, so there is no net flux through the surface. To show the power of this type of statement, we can show that Eq. (1.6) is the same as Coulomb’s law, provided only that we also add the idea that the field from a single charge is spherically symmetric. For a point charge, we draw a sphere around the charge. Then the average normal component is just the value of the magnitude of $\FLPE$ at any point, since the field must be directed radially and have the same strength for all points on the sphere. Our rule now says that the field at the surface of the sphere, times the area of the sphere—that is, the outgoing flux—is proportional to the charge inside. If we were to make the radius of the sphere bigger, the area would increase as the square of the radius. The average normal component of the electric field times that area must still be equal to the same charge inside, and so the field must decrease as the square of the distance—we get an “inverse square” field.

If we have an arbitrary stationary curve in space and measure the circulation of the electric field around the curve, we will find that it is not, in general, zero (although it is for the Coulomb field). Rather, for electricity there is a second law that states: for any 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 two corresponding 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 $S$})

otag\\ \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}.

otag \end{gather}

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

Equations (1.6) through (1.9), together with Eq. (1.1), are all the laws of electrodynamics . As you remember, the laws of Newton were very simple to write down, but they had a lot of complicated consequences and it took us a long time to learn about them all. These laws are not nearly as simple to write down, which means that the consequences are going to be more elaborate and it will take us quite a lot of time to figure them all out.

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

When the wire is pushed to the left, we would expect that the magnet must feel a push to the right. (Otherwise we could put the whole thing on a wagon and have a propulsion system that didn’t conserve momentum!) Although the force is too small to make movement of the bar magnet visible, a more sensitively supported magnet, like a compass needle, will show the movement.

How does the wire push on the magnet? The current in the wire produces a magnetic field of its own that exerts forces on the magnet. According to the last term in Eq. (1.9), a current must have a circulation of $\FLPB$—in this case, the lines of $\FLPB$ are loops around the wire, as shown in Fig. 1–7. This $\FLPB$-field is responsible for the force on the magnet.

Equation (1.9) tells us that for a fixed current through the wire the circulation of $\FLPB$ is the same for any curve that surrounds the wire. For curves—say circles—that are farther away from the wire, the circumference is larger, so the tangential component of $\FLPB$ must decrease. You can see that we would, in fact, expect $\FLPB$ to decrease linearly with the distance from a long straight wire.

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

In short, electrical currents, as well as magnets, make magnetic fields. But wait, what is a magnet, anyway? If magnetic fields are produced by moving charges, is it not possible that the magnetic field from a piece of iron is really the result of currents? It appears to be so. We can replace the bar magnet of our experiment with a coil of wire, as shown in Fig. 1–9. When a current is passed through the coil—as well as through the straight wire above it—we observe a motion of the wire exactly as before, when we had a magnet instead of a coil. In other words, the current in the coil imitates a magnet. It appears, then, that a piece of iron acts as though it contains a perpetual circulating current. We can, in fact, understand magnets in terms of permanent currents in the atoms of the iron. The force on the magnet in Fig. 1–7 is due to the second term in Eq. (1.1).

Where do the currents come from? One possibility would be from the motion of the electrons in atomic orbits. Actually, that is not the case for iron, although it is for some materials. In addition to moving around in an atom, an electron also spins about on its own axis—something like the spin of the earth—and it is the current from this spin that gives the magnetic field in iron. (We say “something like the spin of the earth” because the question is so deep in quantum mechanics that the classical ideas do not really describe things too well.) In most substances, some electrons spin one way and some spin the other, so the magnetism cancels out, but in iron—for a mysterious reason which we will discuss later—many of the electrons are spinning with their axes 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 extra term to Eqs. (1.8) or (1.9) to take care of magnets. We just take all currents, including the circulating currents of the spinning electrons, and then the law is right. You should also notice that Eq. (1.8) says that there are no magnetic “charges” analogous to the 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) was discovered theoretically by Maxwell and is of great importance. It says that changing electric fields produce magnetic effects. In fact, without this term the equation would not make sense, because without it there could be no currents in circuits that are not complete loops. But such currents do exist, as we can see in the following example. Imagine a capacitor made of two flat plates. It is being charged by a current that flows toward one plate and away from the 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 crosses the wire, as shown by the surface $S_1$ in the figure. According to Eq. (1.9), the circulation of $\FLPB$ around $C$ (times $c^2$) is given by the current in the wire (divided by $\epsO$). But what if we fill in the curve with a different surface $S_2$, which is shaped like a bowl and passes between the plates of the capacitor, staying always away from the wire? There is certainly no current through this surface. But, surely, just changing the location of an imaginary surface is not going to change a real magnetic field! The circulation of $\FLPB$ must be what it was before. The first term on the right-hand side of Eq. (1.9) does, indeed, combine with the second term to give the same result for the two surfaces $S_1$ and $S_2$. For $S_2$ the circulation of $\FLPB$ is given in terms of the rate of change of the flux of $\FLPE$ between the plates of the capacitor. And it works out that the changing $\FLPE$ is related to the current in just the way required for Eq. (1.9) to be correct. Maxwell saw that it was needed, and he was the first to write the complete equation.

With the setup shown in Fig. 1–6 we can demonstrate another of the laws of electromagnetism. We disconnect the ends of the hanging wire from the battery and connect them to a galvanometer which tells us when there is a current through the wire. When we push the wire sideways through the magnetic field of the magnet, we observe a current. Such an effect is again just another consequence of Eq. (1.1)—the electrons in the wire feel the force $\FLPF=q\FLPv\times\FLPB$. The electrons have a sidewise velocity because they move with the wire. This $\FLPv$ with a vertical $\FLPB$ from the magnet results in a force on the electrons directed along the wire, which starts the electrons moving toward the galvanometer.

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