What does Purcell actually say about electromagnetism and relativity?

[bcrowell]

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:

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.

 

[Dale]

The first edition of the book is in fact available for free online,

Oh, do you have a link? I tend to use the "Purcell simplified" as a source.

http://physics.weber.edu/schroeder/mrr/MRR.html

 

[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!

 

[Demystifier]

Sometimes less pedagogy is good to enhance clarity!

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

 

[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!

 

[Demystifier]

I never liked Berkeley physics course vol. 2, because I find it is more obscuring than clarifying electromagnetism.

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.

 

[vanhees71]

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

 

[Demystifier]

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

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.

 

[atyy]

Another book that is interesting, maybe closer to vanhees71's taste is Ohanian - he too does something like derive something like Maxwell's equations from electrostatics and relativity, though I can't present the details off the top of my head.

https://books.google.com.sg/books/about/Classical_Electrodynamics.html?id=4OiFAAAACAAJ&redir_esc=y

 

[Demystifier]

but there seems to be an impression on the part of many physicists that it is inherently nonrigorous. That's not true.

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.

 

[atyy]

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.

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.

 

[martinbn]

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.

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

 

[Demystifier]

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

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.

 

[harrylin]

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.

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 . As a matter of fact, Maxwell predicted in principle a positive result for MMX type experiments.

 

[bcrowell]

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.

On a side note, but maybe in line with your thinking, I'm also puzzled by that section in Purcell's book; I get the impression that when discussing the force between current carrying wires, he apparently fails to account for the magnetic force between moving charges in the two wires, only accounting for electric fields. Does he pretend that in certain reference frames the forces between those wires are merely due to electric fields?
That is at odds with SR: in any inertial reference frame there are moving charges in both wires and thus also magnetic forces between the wires. The misunderstanding addressed in an earlier thread does seem to be caused by Purcell...

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.

 

[bcrowell]

Oh, do you have a link?

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.

 

[harrylin]

I can't tell what section in Purcell you're referring to.

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!).

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.

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.

 

[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:

https://www.physicsforums.com/threads/electromagnets-are-a-relativistic-phenom.836422/#post-5252755

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)

 

[bcrowell]

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.

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.

On a side note, but maybe in line with your thinking, I'm also puzzled by that section in Purcell's book; I get the impression that when discussing the force between current carrying wires, he apparently fails to account for the magnetic force between moving charges in the two wires, only accounting for electric fields. Does he pretend that in certain reference frames the forces between those wires are merely due to electric fields?

That is at odds with SR: in any inertial reference frame there are moving charges in both wires and thus also magnetic forces between the wires. The misunderstanding addressed in an earlier thread does seem to be caused by Purcell...

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.

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.

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

 

[Jano L.]

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. ...

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

 

[bcrowell]

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

https://www.physicsforums.com/threads/physics-forums-global-guidelines.414380/

Posting links to unauthorized downloads of copyrighted material will not be permitted. Advertisement of locations where copyrighted materials may be obtained will not be permitted.

This particular example seems to me to have been in a gray area, since the 1st edition of Purcell carries a notice saying it's free after 1970, but there are cloudy legal issues surrounding the meaning of that notice (see https://github.com/bcrowell/purcell/blob/master/README.md ), and the sites that are currently hosting pdfs are sites that also host copyright-violating stuff. I posted a link, but the PF mentors told me that wasn't allowed, and I want to respect their wishes.

 

[harrylin]

[..] The corresponding page numbers in the first edition are 177-178. [..]
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.

At first sight in part 1 only electrostatic forces are at play according to the used reference systems in which he only considers electrostatic forces, and therefore that part looks OK to me. Once more, if I'm not mistaken then it's not a trivial step but a big error; in fact, one of us must be mistaken here, and it will be interesting to elaborate.

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.

Now that is an interesting way of putting it: I do not use that book and so I have no complaint; however I offer my criticism for this discussion about what Purcell actually said. And I clearly referred to part 2 (where in part 1 does he discuss two wires?).

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.

It's very unlikely that I overlooked it, but who knows... he discusses the forces between the two wires and in the reference system under consideration, all ions are moving. Please cite the part in which he accounts for the magnetic forces between the moving ions of the two wires.

What an interesting way of putting it. Is "toxic" a synonym for "hard to understand?" [..]

Certainly not! I was thinking of the recent thread about a guy who clearly was put on the wrong leg by something, and the suspicion fell on Purcell as the possible source of his confusion. I zoomed in on the part here under consideration.

 

[bcrowell]

@harrylin: We're just going in circles here. I can't help you if you won't make an effort to read the book carefully and understand it.

 

[harrylin]

@harrylin: We're just going in circles here. I can't help you if you won't make an effort to read the book carefully and understand it.

Ben this is not my topic but yours, I responded and we were not going in circles. So it looks to me that you don't want to engage in your own topic... but never mind!

 

[vanhees71]

I think this thread proves that there is a problem with Purcell's didactics (I think the physics is correct). The presentation of the subject is simply confused by "too much pedagogics" ;-), but this is again circular ;-))).

 

[pervect]

I think the thread shows that there is a "problem", disagreement exists. How much we can blame "the problem" it on Purcell is debatable, and is being debated :-)

 

[stevendaryl]

It's very unlikely that I overlooked it, but who knows... he discusses the forces between the two wires and in the reference system under consideration, all ions are moving.

I think that there is a limitation in Purcell's approach, but it's not the one you're bringing up. If you have a charged particle, you can compute the instantaneous electromagnetic force on it by transforming to a frame in which that particle is instantaneously at rest. In that frame, the only forces are electrical. Furthermore, in a static situation (the electromagnetic fields are constant), the electrical forces can computed using just Coulomb's law. That's a pretty general recipe for static electromagnetic fields. However, I don't see how it generalizes to changing electromagnetic fields. In a changing electromagnetic field, the electrical forces are not due to Coulomb's law exclusively, but there is also a contribution by radiation (fluctuating $B$ fields give rise to $E$, and vice-versa). I don't see how Purcell's approach helps in the general case.

But getting back to the attraction/repulsion between wires: That's a case where the electromagnetic field can be treated as approximately constant, and so Purcell's approach definitely works. If you want to know what is the force of one current-carrying wire on another current-carrying wire, it is enough to consider the force on each charge in the second wire, and Purcell shows how to do that. If we model the second wire as two lines of charge--a stationary line of positive charge, and a moving line of negative charge, then:

• We stipulate that In the frame in which the positive charges are stationary (the "lab" frame), both wires are electrically neutral. So the positive charges of one wire, which are stationary in this frame, experience no electrical forces. Since they are not moving, they experience no magnetic forces, either.
• In the frame in which the negative charges are stationary, the charge density of the first wire is positive (due to length contraction). So the negative charges of one wire experience an attractive electrical force toward the other wire. In this frame, the negative charges are stationary, so they experience no magnetic forces.

So clearly, the net force between the wires is attractive.

The fuzzy part to me (besides the issue of nonstatic electromagnetic fields) is why you assume that the wire is neutral in the frame in which the positive charges are at rest.

 

[vanhees71]

That's also not completely correct. It is very hard to interpret the Maxwell equations in a way that a time-dependent magnetic field is the cause of an electric field and vice versa. Of course you can reformulate the Maxwell equations in somewhat awkward integral form such as if it looks like this. However, these rewritings are not manifestly causal in the correct relativistic sense, and they are highly non-local expressions.

Fortunately, since the electromagnetic theory is relativistic (right from the beginning when Maxwell wrote down his ingenious equations, which are the condensation of some 100 years of experimental and theoretical research about electric and magnetic phenomena and their relation), this awkward and complicated way to look at these equations is not necessary, and one can derive fully causal and local solutions of Maxwell's equations via the socalled retarded Green's function. This goes back to the mid-19th century (as far as I know the first to write down the retarded solutions was Ludvig Lorenz) but is usually known as "Jefimenko's equations". By chance we have another thread on the issue, where I gave some steps how to derive these formulas:

https://www.physicsforums.com/threads/why-is-faradays-law-one-sided.839240/#post-5268540

Of course, the math by Purcell is completely correct for this static situation (although it's not a fully relativistically consistent solution, but the deviations are of order $v^2/c^2$, where $v$ is the velocity of the conduction electrons which is of the order of $10^{-3} \text{m}/\text{s}$ and thus the deviations due to the relativistic corrections are totally immeasurable against the approximation of and infinite wire made to get simple analytic solutions for the field equations.

Again, the book is correct but didactically awkward. At least it was for me when I was a student studying E&M for the first time (in the experimental course in the 2nd and in the theory course in the 4th semester). I was using the Berkeley Course #2 only for a short time and then rather switched to Sommerfeld and Greiner. Also the relativistic electrodynamics is much more clear in these books than in Berkeley #2.

 

[stevendaryl]

That's also not completely correct. It is very hard to interpret the Maxwell equations in a way that a time-dependent magnetic field is the cause of an electric field and vice versa.

I only meant that in the case of fluctuating electromagnetic fields, you can't compute the electric field using just Coulomb's law, so the strategy of transforming to the frame in which a charge is at rest to compute the forces doesn't help--you don't know what the electric field is in that frame, unless you use the full Maxwell's equations.

 

[vanhees71]

That's of course very true and can be used as another criticism against Purcell's didactics, as discussed before.

 

[bcrowell]

I think that there is a limitation in Purcell's approach, but it's not the one you're bringing up. If you have a charged particle, you can compute the instantaneous electromagnetic force on it by transforming to a frame in which that particle is instantaneously at rest. In that frame, the only forces are electrical. Furthermore, in a static situation (the electromagnetic fields are constant), the electrical forces can computed using just Coulomb's law. That's a pretty general recipe for static electromagnetic fields. However, I don't see how it generalizes to changing electromagnetic fields. In a changing electromagnetic field, the electrical forces are not due to Coulomb's law exclusively, but there is also a contribution by radiation (fluctuating $B$ fields give rise to $E$, and vice-versa). I don't see how Purcell's approach helps in the general case.

Purcell neither claims to address radiation reaction nor attempts to address it. There is in fact no satisfactory theory of radiation reaction on a point charge in classical electromagnetism, since the theory is not self-consistent in its description of point charges, and cannot be made so. But this has nothing to do with static fields as opposed to varying fields. Purcell is finding the force on a test charge. All observable effects of radiation, such as the self-force and the radiated power, are proportional to the square of the charge, and therefore vanish in the case of a test charge.

It also seems to me that you're missing the point of Purcell's argument. The point is to establish that "the magnetic interaction of electric currents can be recognized as an inevitable corollary of Coulomb's law," and to determine the transformation of the fields between frames.

The fuzzy part to me (besides the issue of nonstatic electromagnetic fields) is why you assume that the wire is neutral in the frame in which the positive charges are at rest.

Have you read Purcell? He doesn't make any such assumption in section 5.9, and in fact your statement is false in the toy model he employs there.

I only meant that in the case of fluctuating electromagnetic fields, you can't compute the electric field using just Coulomb's law, so the strategy of transforming to the frame in which a charge is at rest to compute the forces doesn't help--you don't know what the electric field is in that frame, unless you use the full Maxwell's equations.

This is the point that seems to be widely misunderstood by people who haven't carefully read Purcell's argument. He addresses this near the point where he says, "This question takes us to the heart of the meaning of field." Although I've given a brief summary in #1 (including a quote from this part of the text), there is really no "royal road" here -- you have to read what he actually wrote in order to understand it.

 

[Dale]

Closed pending moderation.

 

[Dale]

After some discussion we have edited out some inflammatory comments, and removed the responses. This thread will remain closed, and the other thread which was opened in the same topic has been deleted.

Please do not open any new threads on this topic during the next month or so.