# I What Purcell actually said

1. Oct 15, 2015

### bcrowell

Staff Emeritus
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.

Last edited: Oct 18, 2015
2. Oct 15, 2015

### Staff: Mentor

3. Oct 16, 2015

### vanhees71

Perhaps I'm guilty as charged. I never liked Berkeley physics course vol. 2, because I find it is more obscuring than clarifying electromagnetism. Of course, this is totally subjective. Perhaps many other people benefit a lot from this book. A similar approach is followed by Schwartz, Principles of Electrodynamics, which I find a brillant book. The most clear exposition of the relativistic treatment is found in Landau, Lifshitz. Sometimes less pedagogy is good to enhance clarity!

4. Oct 16, 2015

### Demystifier

If by "less pedagogy" you mean less words and less examples of special cases, then I agree.

5. Oct 16, 2015

### vanhees71

With less pedagogy I mean not to obscure clear facts by pretending to derive something from unclear assumptions, which is very well known not to be derivable at all. In Purcell (and to be fair also in the Feynman lectures vol. II) the example with the straight DC wire is oversimplified, using non-relativistic approximations (see the other thread).

Simple examples, however, are very important and one should not be reluctant to use them to make a point, but as Einstein said, you should make them as simple as possible but not simpler!

6. Oct 16, 2015

### Demystifier

I read this book after finishing my high school and before going to college, during my serving for the Yugoslav Army* (a year before the war in former Yugoslavia, if someone wonders ). At that time, I loved that book very much. But today, when I try to read this book again, I find that I don't like it so much any more. I guess it only confirms what we already knew, that the book is written for beginning students of physics, not for mature theoretical physicists.

*There are some anecdotes related to my self-study of physics during the serving for the army, but that's another story.

7. Oct 16, 2015

### vanhees71

Well, it's the more important for beginning students to provide clear statements!

8. Oct 16, 2015

### Demystifier

Sure! But it is very interesting that at that time the book looked very clear to me, and that it does not longer look so to me today. My point is - the clarity is not the property of a book itself, but of the way how a book influences the reader. Or to use quantum terminology, clarity is contextual.

9. Oct 16, 2015

### atyy

10. Oct 16, 2015

### Demystifier

I see your point. But if such an impression is so common among physicists, there must be something about the book that creates such an impression. Sure, the readers are partially guilty for not reading it carefully enough, but perhaps the book is also partially guilty for requiring more careful reading than necessary. But on the other hand, see also my post #8.

11. Oct 16, 2015

### atyy

It can mean that it is merely a heuristic, if by "heuristic" one means that the argument only shows that magnetic effects are bound to occur.

12. Oct 16, 2015

### martinbn

One reason, pointed out in the first post, is that people who haven't read the book make that statement.

13. Oct 16, 2015

### Demystifier

This begs the question. Why would people who haven't read the book make such statement on it?

I am not defending such people. But clearly there are many other books that many people didn't read, and yet they don't tell that it's not rigorous. (Principa Mathematica by Whitehead and Russell comes to my mind, for which, without actually reading it, many would say that it is too much rigorous.) So there still must be something about the book itself which provokes certain statements about it, even when people didn't actually read it.

Last edited: Oct 16, 2015
14. Oct 16, 2015

### harrylin

Already the historical introduction is not quite right*, but that's without importance for the topic here. See my earlier remark in the other thread about his derivation that current carrying wires repel each other: #15 . Note that I have no opinion about Purcell's overall pedagogy; my issue is only with one calculation example in his book which appears to have been toxic for some readers. But it is of course possible that the misunderstanding with which these recent discussions started, was instead based on another book and that Purcell just receives all the blame!

*This recent post by ZapperZ about Maxwell's theory is quite right: https://www.physicsforums.com/threads/why-is-the-speed-of-light-the-same-in-all-frames-of-reference.534862/#post-5239151 [Broken]. As a matter of fact, Maxwell predicted in principle a positive result for MMX type experiments.

Last edited by a moderator: May 7, 2017
15. Oct 16, 2015

### bcrowell

Staff Emeritus
I can't tell what section in Purcell you're referring to. Could you give the section number and a specific pointer to what part of the section you're talking about? (Most of the section numbers are the same in all three editions of the book. Later editions just added new sections at the ends of chapters.) What you're describing doesn't actually sound to me like anything that Purcell even does in ch. 5.

16. Oct 16, 2015

### bcrowell

Staff Emeritus
You can find it on Library Genesis. The PF staff don't want me to post a link because Library Genesis also hosts things that violate copyright. I did a conversion to LaTeX of the 1st edition (which is supposed to be freely available, according to the information on its copyright page): https://github.com/bcrowell/purcell

By the way, the Library Genesis site hosts both a PDF created from my LaTeX and a version of that PDF, with my name on it, that illegally has the cover of the third edition added as a cover page. I'm not the person who posted that illegal version.

17. Oct 18, 2015

### harrylin

The section 5.9 on github in the link from John Duffield which I cited in my comment #15. I'm not totally sure that it's a legit link, which is why I did not propagate the link into this thread (but it's only a few clicks away!).
As I said, it's the part where he is discussing the force between current carrying wires. In that doc it's on p.196-197.

18. Oct 18, 2015

### vanhees71

This is an example for, why I don't like the book. I've given a very simple derivation of the content of this chapter, as I read it (perhaps the culprit is also the German translation, which I have here, because obviously many other people like the book, but at least this translation makes a simple Lorentz boost a pretty opaque business).

So let me do this example in the usual approximation, which neglects the Hall effect which is a correction of order $\beta^2$, where $\beta=v/c$ is the drift velocity of the conduction electrons.

In this approximation the question of the forces on a charge moving momentarily parallel to the wire is given here:

I still don't know, why I need this complicated description in Berkeley II, and also not why this criticism on a textbook is taken so emotionally. I know that Purcell is a Nobel laureate, and many people seem to like this book. This whole debate, on the other hand, shows that it's overcomplicating things which are not that diffcult in principle. Also in Minkowski's original article it's written in a very clear way. It's simply about the transformation laws of the electromagnetic quantities, where the electromagnetic field is discribed by Faraday's antisymmetric 2nd-rank four-tensor, the polarization fields in matter also as a four-tensor, and the charge and current density as a four-vector. That's all.

Interestingly, the full treatment even of the straight wire for DC in a fully relativistic way (including the Hall effect on the conduction electrons), I cannot find anywhere in the literature. I've derived it for a coaxial cable, but unfortunately it's written up only in German. For those who like to have a look, it's found here (there are many formulae, so perhaps you can see the principle even without understanding the German words in between):

http://theory.gsi.de/~vanhees/faq/coax/node7.html

http://theory.gsi.de/~vanhees/faq-pdf/coax.pdf (p. 11)

19. Oct 18, 2015

### bcrowell

Staff Emeritus
Just so you know, that's a copyright-violating copy of the second edition. It's only the first edition that is freely distributable (depending on your reading of the statement on the copyright page). Because of that, I'll refer to the first edition instead. The corresponding page numbers in the first edition are 177-178.

This doesn't make a lot of sense to me, and I can't help wondering whether you've taken the time to read the material carefully.

Most of section 5.9 consists of an analysis of the force of a wire on a single charge. Let's call this part 1. Only at the end, at the pages you refer to, does he apply that to find the force between two wires, which is a fairly trivial step. Call this part 2. Your complaint that he "fails to account for the magnetic force between moving charges in the two wires, only accounting for electric fields" would seem to apply to part 1. If part 1 is correct, then there is no further logical issue of this kind when moving on from part 1 to part 2, because in part 2 he is merely integrating the result from part 1.

"Does he pretend that in certain reference frames the forces between those wires are merely due to electric fields?" No, he makes no such statement. In the situation described in part 1, the force is purely electrical in one frame. However, once he has established an equation for the force in that situation, he simply integrates it in part 2, without making any claim (which would be erroneous) that there is a frame in which the force is purely electrical.

What an interesting way of putting it. Is "toxic" a synonym for "hard to understand?" Sophisticated ideas are often hard to understand.

Last edited: Oct 18, 2015
20. Oct 19, 2015

### Jano L.

Are there explicit rules here on PF regarding sites serving copyrighted documents?