Electromagnetism
The most dramatic moments in the development of physics are those in                         which great syntheses take place, where phenomena which previously had                         appeared to be different are suddenly discovered to be but different                         aspects of the same thing. The history of physics is the history of                         such syntheses, and the basis of the success of physical science is                         mainly that we are able to synthesize. Perhaps the most dramatic moment in the development of physics during the                         19th century occurred to J. C. Maxwell one day in the 1860s, when he combined the laws of electricity and                         magnetism with the laws of the behavior of light. As a result, the properties of                         light were partly unravelled—that old and subtle stuff that is so important                         and mysterious that it was felt necessary to arrange a special creation for                         it when writing Genesis. Maxwell                         could say, when he was finished with his discovery, “Let there be electricity                         and magnetism, and there is light!” For this culminating moment there was a long preparation in the                         gradual discovery and unfolding of the laws of electricity and                         magnetism. This story we shall reserve for detailed study next                         year. However, the story is, briefly, as follows. The gradually                         discovered properties of electricity and magnetism, of electric forces                         of attraction and repulsion, and of magnetic forces, showed that                         although these forces were rather complex, they all fell off inversely                         as the square of the distance. We know, for example, that the simple                         Coulomb law                         for stationary charges is that the electric force field                         varies inversely as the square of the distance. As a consequence, for                         sufficiently great distances there is very little influence of one                         system of charges on another.                         Maxwell noted that the                         equations or the laws that had been discovered up to this time were                         mutually inconsistent when he tried to put them all together, and in                         order for the whole system to be consistent, he had to add another term                         to his equations. With this new term there came an amazing prediction,                         which was that a part of the electric and magnetic fields would fall off                         much more slowly with the distance than the inverse square, namely,                         inversely as the first power of the distance! And so he realized that                         electric currents in one place can affect other charges far away, and he                         predicted the basic effects with which we are familiar today—radio                         transmission, radar, and so on. It seems a miracle that someone talking in Europe can, with mere                         electrical influences, be heard thousands of miles away in Los                         Angeles. How is it possible? It is because the fields do not vary as                         the inverse square, but only inversely as the first power of the                         distance. Finally, then, even light itself was recognized to be                         electric and magnetic influences extending over vast distances,                         generated by an almost incredibly rapid oscillation of the electrons                         in the atoms. All these phenomena we summarize by the word                         radiation or, more specifically, electromagnetic                             radiation, there being one or two other kinds of radiation                         also. Almost always, radiation means electromagnetic radiation. And thus is the universe knit together. The atomic motions of a                         distant star still have sufficient influence at this great distance to                         set the electrons in our eye in motion, and so we know about the                         stars. If this law did not exist, we would all be literally in the                         dark about the exterior world! And the electric surgings in a galaxy                         five billion light years away—which is the farthest object we have                         found so far—can still influence in a significant and detectable way                         the currents in the great “dish” in front of a radio telescope. And                         so it is that we see the stars and the galaxies. This remarkable phenomenon is what we shall discuss in the present                         chapter. At the beginning of this course in physics we outlined a                         broad picture of the world, but we are now better prepared to                         understand some aspects of it, and so we shall now go over some parts                         of it again in greater detail. We begin by describing the position of                         physics at the end of the 19th century. All that was then known about                         the fundamental laws can be summarized as follows. First, there were laws of forces: one force was the law of                         gravitation, which we have written down several times; the force on an                         object of mass $m$, due to another of mass $M$, is given by                         \begin{equation}                         \label{Eq:I:28:1}                         \FLPF=GmM\FLPe_r/r^2,                         \end{equation}                         where $\FLPe_r$ is a unit vector directed from $m$ to $M$, and $r$ is                         the distance between them. Next, the laws of electricity and magnetism, as known at the end of                         the 19th century, are these: the electrical forces acting on a                         charge $q$ can be described by two fields, called $\FLPE$ and $\FLPB$, and                         the velocity $\FLPv$ of the charge $q$, by the equation                         \begin{equation}                         \label{Eq:I:28:2}                         \FLPF=q(\FLPE+\FLPv\times\FLPB).                         \end{equation}                         To complete this law, we have to say what the formulas for $\FLPE$                         and $\FLPB$ are in a given circumstance: if a number of charges are                         present, $\FLPE$ and the $\FLPB$ are each the sum of contributions,                         one from each individual charge. So if we can find the $\FLPE$                         and $\FLPB$ produced by a single charge, we need only to add all the                         effects from all the charges in the universe to get the total $\FLPE$                         and $\FLPB$!  This is the principle of                         superposition.                          What is the formula for the electric and magnetic field produced by                         one individual charge? It turns out that this is very complicated, and                         it takes a great deal of study and sophistication to appreciate                         it. But that is not the point. We write down the law now only to                         impress the reader with the beauty of nature, so to speak, i.e., that                         it is possible to summarize all the fundamental knowledge on                         one page, with notations that he is now familiar with. This law for                         the fields of an individual charge is complete and accurate, so far as                         we know (except for quantum mechanics) but it looks rather                         complicated. We shall not study all the pieces now; we only write it                         down to give an impression, to show that it can be written, and so                         that we can see ahead of time roughly what it looks like. As a matter                         of fact, the most useful way to write the correct laws of                         electricity and magnetism is not the way we shall now write them, but                         involves what are called field equations, which we shall learn                         about next year. But the mathematical notations for these are                         different and new, and so we write the law in an inconvenient form for                         calculation, but in notations that we now know.                          The electric field, $\FLPE$, is given by                         \begin{equation}                         \label{Eq:I:28:3}                         \FLPE=\frac{-q}{4\pi\epsO}\biggl[                         \frac{\FLPe_{r'}}{r'^2}+\frac{r'}{c}\,\ddt{}{t}\biggl(                         \frac{\FLPe_{r'}}{r'^2}\biggr)+\frac{1}{c^2}\,\frac{d^2}{dt^2}\,\FLPe_{r'}                         \biggr].                         \end{equation}                         What do the various terms tell us? Take the first term,                         $\FLPE=-q\FLPe_{r'}/4\pi\epsO r'^2$. That, of course, is                         Coulomb’s law, which we                         already know: $q$ is the charge that is producing the                         field; $\FLPe_{r}$ is the unit vector in the direction from the                         point $P$ where $\FLPE$ is measured, $r$ is the distance from $P$                         to $q$. But, Coulomb’s law is wrong.                         The discoveries of the 19th century                         showed that influences cannot travel faster than a certain fundamental                         speed $c$, which we now call the speed of light. It is not correct                         that the first term is Coulomb’s law,                         not only because it is not                         possible to know where the charge is now and at what distance                         it is now, but also because the only thing that can affect the                         field at a given place and time is the behavior of the charges in the                         past. How far in the past?  The time delay, or                         retarded time,                         so-called, is the time it takes, at speed $c$, to get from the charge to                         the field point $P$. The delay is $r'/c$. So to allow for this time delay, we put a little prime on $r$, meaning                         how far away it was when the information now arriving at $P$                         left $q$. Just for a moment suppose that the charge carried a light,                         and that the light could only come to $P$ at the speed $c$. Then when                         we look at $q$, we would not see where it is now, of course, but where                         it was at some earlier time. What appears in our formula is the                         apparent direction $\FLPe_{r'}$—the direction it used to                         be—the so-called retarded direction—and at the                         retarded distance $r'$. That would be easy enough to                         understand, too, but it is also wrong. The whole thing is much more                         complicated. There are several more terms. The next term is as though nature were                         trying to allow for the fact that the effect is retarded, if we might                         put it very crudely. It suggests that we should calculate the delayed                         Coulomb field and add a correction to it, which is its rate of change                         times the time delay that we use. Nature seems to be attempting to                         guess what the field at the present time is going to be, by taking the                         rate of change and multiplying by the time that is delayed. But we are                         not yet through. There is a third term—the second derivative, with                         respect to $t$, of the unit vector in the direction of the charge. Now                         the formula is finished, and that is all there is to the                         electric field from an arbitrarily moving charge. The magnetic field is given by                         \begin{equation}                         \label{Eq:I:28:4}                         \FLPB=-\FLPe_{r'}\times\FLPE/c.                         \end{equation}                         We have written these down only for the purpose of showing the beauty                         of nature or, in a way, the power of mathematics. We do not pretend to                         understand why it is possible to write so much in such a small                         space, but (28.3) and (28.4) contain the                         machinery by which electric generators work, how light operates, all                         the phenomena of electricity and magnetism. Of course, to complete the                         story we also need to know something about the behavior of the                         materials involved—the properties of matter—which are not                         described properly by (28.3). To finish with our description of the world of the 19th century we                         must mention one other great synthesis which occurred in that century,                         one with which Maxwell had a                         great deal to do also, and that was the synthesis of the phenomena of                         heat and mechanics. We shall study that subject soon. What had to be added in the 20th century was that the dynamical laws of                         Newton were found to be all wrong, and quantum mechanics had to be                         introduced to correct them. Newton’s laws                         are approximately                         valid when the scale of things is sufficiently large. These                         quantum-mechanical laws, combined with the laws of electricity, have                         only recently been combined to form a set of laws called quantum                             electrodynamics. In addition, there were                         discovered a number of new phenomena, of which the first was                         radioactivity, discovered by                         Becquerel in 1896—he just sneaked it in under the 19th century.                         This phenomenon of radioactivity was followed up to produce our                         knowledge of nuclei and new kinds of forces that are not gravitational                         and not electrical, but new particles with different interactions, a                         subject which has still not been unravelled. For those purists who know more (the professors who happen to be                         reading this), we should add that when we say that (28.3)                         is a complete expression of the knowledge of electrodynamics, we are                         not being entirely accurate. There was a problem that was not quite                         solved at the end of the 19th century. When we try to calculate the                         field from all the charges including the charge itself that we                             want the field to act on, we get into trouble trying to find the                         distance, for example, of a charge from itself, and dividing something                         by that distance, which is zero. The problem of how to handle the part                         of this field which is generated by the very charge on which we want                         the field to act is not yet solved today. So we leave it there; we do                         not have a complete solution to that puzzle yet, and so we shall avoid                         the puzzle for as long as we can.