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In this way, Dirac got the radiation resistance force and none of the inertial forces. There is no electromagnetic mass, and the classical theory is saved—but at the expense of an arbitrary assumption about the self-force. The arbitrariness of the extra assumption of Dirac was removed, to some extent at least, by Wheeler and Feynman, who proposed a still stranger theory.

They suggest that point charges interact only with other charges, but that the interaction is half through the advanced and half through the retarded waves. The radiation resistance is not due to the electron acting on itself, but from the following peculiar effect. You see what tight knots people have gotten into in trying to get a theory of the electron! This is another modification of the laws of electrodynamics, proposed by Bopp. You realize that once you decide to change the equations of electromagnetism you can start anywhere you want.

You can change the force law for an electron, or you can change the Maxwell equations as we saw in the examples we have described , or you can make a change somewhere else. One possibility is to change the formulas that give the potentials in terms of the charges and currents. One of our formulas has been that the potentials at some point are given by the current density or charge at each other point at an earlier time.

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No differential equation, nothing else. Well, one more thing. We also ask that the result should be relativistically invariant.


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Then Eq. What is the significance?

In other words, this theory of Bopp approaches the Maxwell theory—so long as we are far away from any particular charge—in the sense that it gives the retarded wave effects. We can, in fact, see approximately what the integral of Eq. And it all came out of the simple proposition that the potential at one point in space-time depends on the current density at all other points in space-time, but with a weighting factor that is some narrow function of the four-dimensional distance between the two points.

This theory again predicts a finite electromagnetic mass for the electron, and the energy and mass have the right relation for the relativity theory. They must, because the theory is relativistically invariant from the start, and everything seems to be all right. There is, however, one fundamental objection to this theory and to all the other theories we have described. All particles we know obey the laws of quantum mechanics, so a quantum-mechanical modification of electrodynamics has to be made.

Light behaves like photons. So the electrodynamic theory has to be changed. We have already mentioned that it might be a waste of time to work so hard to straighten out the classical theory, because it could turn out that in quantum electrodynamics the difficulties will disappear or may be resolved in some other fashion. But the difficulties do not disappear in quantum electrodynamics. That is one of the reasons that people have spent so much effort trying to straighten out the classical difficulties, hoping that if they could straighten out the classical difficulty and then make the quantum modifications, everything would be straightened out.

The Maxwell theory still has the difficulties after the quantum mechanics modifications are made. And the answers depend on how you stop the integrals. We cannot, unfortunately, demonstrate for you here that the difficulties are really basically the same, because we have developed so little of the theory of quantum mechanics and even less of quantum electrodynamics.

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It turns out, however, that nobody has ever succeeded in making a self-consistent quantum theory out of any of the modified theories. The theories with the advanced and retarded waves of Dirac, or of Wheeler and Feynman, have never been made into a satisfactory quantum theory.

The theory of Bopp has never been made into a satisfactory quantum theory. So today, there is no known solution to this problem. We do not know how to make a consistent theory—including the quantum mechanics—which does not produce an infinity for the self-energy of an electron, or any point charge. And at the same time, there is no satisfactory theory that describes a non-point charge.

In case you are deciding to rush off to make a theory in which the action of an electron on itself is completely removed, so that electromagnetic mass is no longer meaningful, and then to make a quantum theory of it, you should be warned that you are certain to be in trouble.

There is definite experimental evidence of the existence of electromagnetic inertia—there is evidence that some of the mass of charged particles is electromagnetic in origin. It used to be said in the older books that since Nature will obviously not present us with two particles—one neutral and the other charged, but otherwise the same—we will never be able to tell how much of the mass is electromagnetic and how much is mechanical.

But it turns out that Nature has been kind enough to present us with just such objects, so that by comparing the observed mass of the charged one with the observed mass of the neutral one, we can tell whether there is any electromagnetic mass. For example, there are the neutrons and protons. They interact with tremendous forces—the nuclear forces—whose origin is unknown.

However, as we have already described, the nuclear forces have one remarkable property. So far as they are concerned, the neutron and proton are exactly the same. The nuclear forces between neutron and neutron, neutron and proton, and proton and proton are all identical as far as we can tell. Only the little electromagnetic forces are different; electrically the proton and neutron are as different as night and day. This is just what we wanted.

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There are two particles, identical from the point of view of the strong interactions, but different electrically. And they have a small difference in mass. The only trouble is that the sign is wrong!

Field, Force, Energy and Momentum in Classical Electrodynamics (Revised Edition)

The neutron is heavier than the proton. Nature has also given us several other pairs—or triplets—of particles which appear to be exactly the same except for their electrical charge. You will see from the table that the mass differences of the other particles are usually of the same general size. Now the size of these particles can be determined by other methods, for instance by the diameters they appear to have in high-energy collisions. So the electromagnetic mass seems to be in general agreement with electromagnetic theory, if we stop our integrals of the field energy at the same radius obtained by these other methods.

You are no doubt worried about the different signs of the mass differences in the table. It is easy to see why the charged ones should be heavier than the neutral ones. But what about those pairs like the proton and the neutron, where the measured mass comes out the other way? Well, it turns out that these particles are complicated, and the computation of the electromagnetic mass must be more elaborate for them. For instance, although the neutron has no net charge, it does have a charge distribution inside it—it is only the net charge that is zero.

We must mention one more piece of information, which is the most annoying. There is another particle in the world called a muon which, so far as we can tell, differs in no way whatsoever from an electron except for its mass.

The Classical Theory of Fields. Electromagnetism

It acts in every way like an electron: it interacts with neutrinos and with the electromagnetic field, and it has no nuclear forces. Therefore, whenever someone finally gets the explanation of the mass of an electron, he will then have the puzzle of where a muon gets its mass. Because whatever the electron does, the muon does the same—so the mass ought to come out the same. There are those who believe faithfully in the idea that the muon and the electron are the same particle and that, in the final theory of the mass, the formula for the mass will be a quadratic equation with two roots—one for each particle.

We would like to make some further remarks about the part of the mass of nuclear particles that is not electromagnetic. Where does this other large fraction come from? There are other forces besides electrodynamics—like nuclear forces—that have their own field theories, although no one knows whether the current theories are right. The main focus of the book is to provide the reader with the fundamentals of the classical theory of radiation. This aim is well complemented by examples from a variety of fields. Since the purpose of the book is not to provide a general treatment of electromagnetism or electrodynamics, the reader cannot expect to find some of the topics usual in other electrodynamics texts, such as relativistic transforms of electromagnetic fields although the Lorentz condition is mentioned or a discussion of the causality principle in the derivation of the Green function.

However, within the scope of the book, these topics are not essential for an understanding of electromagnetic radiation and can therefore be omitted. I feel that the author has covered the essentials of the subject of electromagnetic radiation. The reader, even the advanced undergraduate, will be able to follow the mathematical steps, as well as the physical insights. The large number of figures and the numerous examples also facilitate the reading of the book.

This book appears to me to be a very useful textbook that is accessible to a very broad category of readers, from advanced undergraduates to researchers, since it starts with the basic physics and covers quite extensively the field of electromagnetic radiation. For a reader who is interested in electromagnetism and electrodynamics, it is an excellent companion to the classical and more fundamental text books, such as Jackson's Classical Electrodynamics , Landau and Lifshitz's The Classical Theory of Fields or Stratton's Electromagnetics Theory.

In summary, I heartily recommend this book, An Introduction of Classical Electromagnetic Radiation , to all my colleagues who teach the subject of electromagnetic radiation, to students who follow such courses and to researchers active in this field. FAQ Policy. About this Textbook The study of classical electromagnetic fields is an adventure. Show all. Carl S. Mathematical Background Pages Helrich, Prof. Electrostatics Pages Helrich, Prof.


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Magnetostatics Pages Helrich, Prof. Applications of Magnetostatics Pages Helrich, Prof. Particle Motion Pages Helrich, Prof. Time Dependence Pages Helrich, Prof.

Electromagnetic Waves Pages Helrich, Prof. Energy and Momentum Pages Helrich, Prof. Special Relativity Pages Helrich, Prof.