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Since the advent of relativity in 1905, physicists have understood electromagnetism as the first example of a unified field theory. However, the relativistic insight is often ignored in freshman physics classes, so that students learn the subject as it was understood in the 19th century. An excellent and influential exception to this is the treatment in Purcell, Electricity and Magnetism, Berkeley Physics Course vol. 2.

It seems that many people have encountered Purcell's pedagogy only in watered-down form, as a loose heuristic or motivation for electromagnetism. There is nothing wrong with such a nonrigorous treatment (it's what I do in my own courses), but there seems to be an impression on the part of many physicists that it is inherently nonrigorous. That's not true. Worse yet, one can find various garbled or incompetent presentations of the ideas, and some people seem to get the impression that this indicates a problem with the whole approach.

In online discussions, I've found that many people are happy to debate the merits of Purcell's approach without ever having taken the trouble to read what he wrote or carefully consider the lengthy and sometimes subtle thread of the argument. The first edition of the book is in fact available for free online, which is right and proper, since it was developed with NSF support and carries a message on its copyright page stating that it would "be available for use by authors and publishers on a royalty-free basis on or after April 30, 1970." Unfortunately, the copyright has passed through several hands since then, leaving a fog of legal confusion and causing the book not to be widely and freely distributed. There is an excellent third edition, Purcell and Morin, from Cambridge University Press, and it's quite inexpensive, but it does cost money. This has perhaps contributed to the tendency to debate the book without having read it. For these reasons, I've written up the following outline of what Purcell actually says.

Section 5.1 contains some historical introductory material, including this:

Section 5.2 gives empirical evidence leading to the Lorentz force law, but then says:

Sections 5.3 and 5.4 deal with how to define charge in a case where the charges may be in motion. The electrical neutrality of atoms is given as empirical evidnce that charge is invariant. This is then expressed mathematically as the frame-invariance of Gauss's law in integral form.

He finishes section 5.4 with the following dramatic promise:

This leads to a crucial foundational issue that seems to be ignored or not understood by many people discussing Purcell's pedagogy. Purcell's method is to reason from the specific to the general, but to do so in a logically rigorous way, so that the outcome of the argument is not a mere heuristic. He uses this approach in section 5.5 by invoking a scenario with two parallel infinite sheets of charge, with uniform and opposite charges. He applies Lorentz transformations in the directions parallel and perpendicular to the sheets, applies Gauss's law, makes a symmetry argument, and infers the transformed field. He then addresses the crucial logical point:

This establishes that the transformation of the fields found from this particular example is of general validity.

In section 5.6 he applies the transformation in order to find the field of a moving point charge, and in section 5.7 that of a charge that abruptly starts moving or stops moving.

In section 5.8 he recapitulates the transformation of the three-force from volume 1 of the Berkeley physics series. He finds a relation between the force acting on a particle in the particle's instantaneously comoving frame and the force acting on it in some other frame.

Section 5.9 is the meat of the argument. After walking through the historical development, from before the advent of relativity, he then says:

He then discusses an interaction between a current-carrying wire and a nearby free charge. In the lab frame, which I'll call K, the wire is electrically neutral and the charge has a nonzero instantaneous velocity parallel to the wire. In the charge's rest frame K', differential length contraction of the positive and negative charges in the wire causes the wire to have a nonzero net charge. The discussion is organized as it would have been if the historical order had been reversed, and physicists had known of relativity before discovering magnetism empirically. But to summarize the final results, we find that in K the charge experiences a force that is purely magnetic, while in K' it is purely electrical. The field in K is purely magnetic, while in K' it is both electrical and magnetic.

After this, Purcell integrates the above result to find the force between two current-carrying wires.

Here he arrives at far-reaching conclusions based on the consideration of one particular physical scenario. For the same reasons discussed above in the previous example of the parallel planes of charge, this does not imply a lack of rigor or that this is merely a heuristic. This is a common source of confusion in online discussions. I'll briefly discuss some other common confusions about section 5.9.

We are not transforming from a frame in which the field is purely electric to one in which it is purely magnetic. That would be impossible.

Purcell does not claim that all magnetic forces can be made into purely electrical forces by transforming into a different frame of reference. This does hold for one special system that he considers, but it is false in general. In particular, it does not hold for the force of one wire on another wire, nor does he claim that it does.

We are not integrating the equations of motion or claiming that the charge will hit the wire. We are only finding the instantaneous force on the charge at one moment.

Purcell makes four main assumptions: (1) that we know about Gauss's law and electrostatics; (2) that charge is invariant; (3) that fields must, for the reasons discussed above, have definite transformation laws; and (4) that we know some standard kinematical and dynamical facts about special relativity.

Purcell does not claim to derive Maxwell's equations from these assumptions. That would obviously be impossible. For example, one of Maxwell's equations states that the divergence of the magnetic field is zero. There is no way to derive this fact from his assumptions, and in fact it's quite possible that magnetic monopoles do exist.

It seems that many people have encountered Purcell's pedagogy only in watered-down form, as a loose heuristic or motivation for electromagnetism. There is nothing wrong with such a nonrigorous treatment (it's what I do in my own courses), but there seems to be an impression on the part of many physicists that it is inherently nonrigorous. That's not true. Worse yet, one can find various garbled or incompetent presentations of the ideas, and some people seem to get the impression that this indicates a problem with the whole approach.

In online discussions, I've found that many people are happy to debate the merits of Purcell's approach without ever having taken the trouble to read what he wrote or carefully consider the lengthy and sometimes subtle thread of the argument. The first edition of the book is in fact available for free online, which is right and proper, since it was developed with NSF support and carries a message on its copyright page stating that it would "be available for use by authors and publishers on a royalty-free basis on or after April 30, 1970." Unfortunately, the copyright has passed through several hands since then, leaving a fog of legal confusion and causing the book not to be widely and freely distributed. There is an excellent third edition, Purcell and Morin, from Cambridge University Press, and it's quite inexpensive, but it does cost money. This has perhaps contributed to the tendency to debate the book without having read it. For these reasons, I've written up the following outline of what Purcell actually says.

Section 5.1 contains some historical introductory material, including this:

Special relativity has its historical roots in electromagnetism. Lorentz, exploring the electrodynamics of moving charges, was led very close to the final formulation of Einstein. And Einstein's great paper of 1905 was entitled not ``Theory of Relativity,'' but rather ``On the Electrodynamics of Moving Bodies.'' Today we see in the postulates of relativity and their implications a wide framework, one that embraces all physical laws and not solely those of electromagnetism. We expect any complete physical theory to be relativistically invariant. It ought to tell the same story in all inertial frames of reference. As it happened, physics already *had* one relativistically invariant theory---Maxwell's electromagnetic theory---long before the significance of relativistic invariance was recognized. Whether the ideas of special relativity could have evolved in the absence of a complete theory of the electromagnetic field is a question for the historian of science to speculate about; probably it can't be answered. We can only say that the actual history shows rather plainly a path running from Oersted's compass needle to Einstein's postulates.

In this chapter and Chap. 6 we are going to follow that path *almost in reverse*. This implies no disrespect for history. Indeed, we think a student of the history of those magnificent discoveries will not be handicapped by a clear view of the essential relation between electricity and magnetism. That relation can be exposed very directly and simply by looking, in the light of special relativity, at what we have already learned about electric charge and the electric field.

Section 5.2 gives empirical evidence leading to the Lorentz force law, but then says:

Clearly this doesn't explain anything. Why does Eq. 1 work? Why can we always find a B that is consistent with this simple relation, for all possible velocities? We want to understand why there is a velocity-proportional force. It is really most remarkable that this force is strictly proportional to v, and that the effect of the electric field does not depend on v at all! In the following pages we'll see how this comes about.

Sections 5.3 and 5.4 deal with how to define charge in a case where the charges may be in motion. The electrical neutrality of atoms is given as empirical evidnce that charge is invariant. This is then expressed mathematically as the frame-invariance of Gauss's law in integral form.

He finishes section 5.4 with the following dramatic promise:

In the language of relativity theory ... charge is a scalar, an invariant number, with respect to the Lorentz transformation. This is an observed fact with far-reaching implications. It completely determines the nature of the field of moving charges.

This leads to a crucial foundational issue that seems to be ignored or not understood by many people discussing Purcell's pedagogy. Purcell's method is to reason from the specific to the general, but to do so in a logically rigorous way, so that the outcome of the argument is not a mere heuristic. He uses this approach in section 5.5 by invoking a scenario with two parallel infinite sheets of charge, with uniform and opposite charges. He applies Lorentz transformations in the directions parallel and perpendicular to the sheets, applies Gauss's law, makes a symmetry argument, and infers the transformed field. He then addresses the crucial logical point:

That is all very well for the particularly simple arrangement of charges here pictured; do our conclusions have more general validity? This question takes us to the heart of the meaning of *field*. If the electric field E at a point in space-time is to have a unique meaning, then the way E appears in other frames of reference, in the same space-time neighborhood, cannot depend on the nature of the sources, wherever they may be, that produced E. In other words, the observer in F, having measured the field in his neighborhood at some time, ought to be able to predict *from these measurements alone* what observers in other frames of reference would measure at the same space-time point. Were this not true, *field* would be a useless concept. The evidence that it is true is the eventual agreement of our field theory with experiment.

This establishes that the transformation of the fields found from this particular example is of general validity.

In section 5.6 he applies the transformation in order to find the field of a moving point charge, and in section 5.7 that of a charge that abruptly starts moving or stops moving.

In section 5.8 he recapitulates the transformation of the three-force from volume 1 of the Berkeley physics series. He finds a relation between the force acting on a particle in the particle's instantaneously comoving frame and the force acting on it in some other frame.

Section 5.9 is the meat of the argument. After walking through the historical development, from before the advent of relativity, he then says:

From our present vantage point, the magnetic interaction of electric currents can be recognized as an inevitable corollary of Coulomb's law. If the postulates of relativity are valid, if electric charge is invariant, and if Coulomb's law holds, then the effects we commonly call ``magnetic'' are bound to occur. They will emerge as soon as we examine the electric interaction between a moving charge and other moving charges. A very simple system will illustrate this.

He then discusses an interaction between a current-carrying wire and a nearby free charge. In the lab frame, which I'll call K, the wire is electrically neutral and the charge has a nonzero instantaneous velocity parallel to the wire. In the charge's rest frame K', differential length contraction of the positive and negative charges in the wire causes the wire to have a nonzero net charge. The discussion is organized as it would have been if the historical order had been reversed, and physicists had known of relativity before discovering magnetism empirically. But to summarize the final results, we find that in K the charge experiences a force that is purely magnetic, while in K' it is purely electrical. The field in K is purely magnetic, while in K' it is both electrical and magnetic.

After this, Purcell integrates the above result to find the force between two current-carrying wires.

In this chapter we have seen how the fact of charge invariance implies forces between electric currents. That does not oblige us to look on one fact as the cause of the other. These are simply two aspects of electromagnetism whose relationship beautifully illustrates the more general law: physics is the same in all inertial frames of reference.

If we had to analyze every system of moving charges by transforming back and forth among various coordinate systems, our task would grow both tedious and confusing. There is a better way. The overall effect of one current on another, or of a current on a moving charge, can be described completely and concisely by introducing a new field, the *magnetic* field.

Here he arrives at far-reaching conclusions based on the consideration of one particular physical scenario. For the same reasons discussed above in the previous example of the parallel planes of charge, this does not imply a lack of rigor or that this is merely a heuristic. This is a common source of confusion in online discussions. I'll briefly discuss some other common confusions about section 5.9.

We are not transforming from a frame in which the field is purely electric to one in which it is purely magnetic. That would be impossible.

Purcell does not claim that all magnetic forces can be made into purely electrical forces by transforming into a different frame of reference. This does hold for one special system that he considers, but it is false in general. In particular, it does not hold for the force of one wire on another wire, nor does he claim that it does.

We are not integrating the equations of motion or claiming that the charge will hit the wire. We are only finding the instantaneous force on the charge at one moment.

Purcell makes four main assumptions: (1) that we know about Gauss's law and electrostatics; (2) that charge is invariant; (3) that fields must, for the reasons discussed above, have definite transformation laws; and (4) that we know some standard kinematical and dynamical facts about special relativity.

Purcell does not claim to derive Maxwell's equations from these assumptions. That would obviously be impossible. For example, one of Maxwell's equations states that the divergence of the magnetic field is zero. There is no way to derive this fact from his assumptions, and in fact it's quite possible that magnetic monopoles do exist.

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