Electromagnets are a relativistic phenom?

[vanhees71]

Well, as we discuss here, there are situations, where you cannot use a Lorentz transformation, where the electromagnetic field has only electric components. In fact that's the case for almost all electromagnetic fields. Only for electrostatics that's the case, i.e., if you can find an inertial frame, where there are only charges at rest at all times. Already in situations, where in one frame there is a stationary current (be it one of moving charges or the equivalent of a magnetized ferromagnet) with vanishing charge density you cannot transform away the magnetic field in any reference frame (magnetostatics). This is clear from my previous argument. If there is a frame of reference, where you have $\vec{E}=0$ and $\vec{B} \neq 0$, then the invariants $\vec{E} \cdot \vec{B}=0$ and $\vec{E}^2-\vec{B}^2<0$. This implies that in any frame either $\vec{E}=0$ (which is the one from which we started modulo rotations) or you have both electric and magnetic components, the electric and magnetic fields are perpendicular. But you cannot have $\vec{B}=0$, because otherwise the latter invariant would have to be positive.

This very simple argument implies that you cannot derive to complete set of Maxwell equations simply from electrostatics and Poincare invariance. You can only make an educated guess, most easily when you have Hamilton's principle for fields at hand, using Poincare invariance, gauge invariance, and invariance under spatial reflections. Of course, this is very much to assume and not suitable for an introductory em. theory lecture. I think, Maxwell's equations cannot be derived but are just the essence of about a century of empirical work by many great physicists, culminating finally in Faraday (who discovered the concept of fields) and Maxwell (who brought everything in a mathematical form).

 

[harrylin]

[..]
I guess I was responding to your previous post, which didn't seem to relate to the discussion of the Purcell pedagogy. But maybe it was just intended to be an interesting tangent...? [...]

I consider "Purcell pedagogy" a useful and interesting tangent, but my replies relate more to the subject matter of the OP - which does not directly refer to Purcell.

[...] I don't understand what this has to do with the discussion, and I don't understand why you want to compare the effect of a single electron to the effect of currents in wires. I also don't think this is true as a real-world statement, since, e.g., we could have the magnetic field created by a beam of electrons in vacuum, with no wires involved.

Inversely, I had the impression that it was you who brought up a single charge. According to me, the discussion here is (or was?) about currents in a wire and electromagnets; and I clarified why the effect of a single electron - or a beam of electrons - is a special case.

I don't understand what this has to do with the discussion, since nobody has claimed that [..] magnetic fields don't exist. [..]

The quote of the OP suggests to me that the electromagnetic field of electromagnets can be explained away as in fact due to an electric field and length contraction - the author claims that it is thanks to special relativity that this phenomenon (of magnetic force) is possible. He pretends that magnetic force is just the effect of length contraction: "the metal exhibits a positive charge" which causes the object to be attracted or repelled. I do not swallow that. Assuming that Purcell also disagreed, it may be useful to cite him (I do not have his book at hand).

 

[Nugatory]

The quote of the OP suggests to me that the electromagnetic field of electromagnets can be explained away as in fact due to an electric field and length contraction...

And maybe a half-dozen posts in, we had agreement that that the text OP quoted was wrong and misleading...

 

[harrylin]

And maybe a half-dozen posts in, we had agreement that that the text OP quoted was wrong and misleading...

Yes indeed; and we elaborated some more. Do we now all agree?

 

[bcrowell]

Well, as we discuss here, there are situations, where you cannot use a Lorentz transformation, where the electromagnetic field has only electric components.

This sentence seems to have been garbled...? Use a Lorentz transformation to do what?

In fact that's the case for almost all electromagnetic fields. Only for electrostatics that's the case, i.e., if you can find an inertial frame, where there are only charges at rest at all times.

Because of the garbled initial sentence, I'm not sure what you mean by this.

Already in situations, where in one frame there is a stationary current (be it one of moving charges or the equivalent of a magnetized ferromagnet) with vanishing charge density you cannot transform away the magnetic field in any reference frame (magnetostatics). This is clear from my previous argument. If there is a frame of reference, where you have $\vec{E}=0$ and $\vec{B} \neq 0$, then the invariants $\vec{E} \cdot \vec{B}=0$ and $\vec{E}^2-\vec{B}^2<0$. This implies that in any frame either $\vec{E}=0$ (which is the one from which we started modulo rotations) or you have both electric and magnetic components, the electric and magnetic fields are perpendicular. But you cannot have $\vec{B}=0$, because otherwise the latter invariant would have to be positive.

Nothing to disagree with here, but that has nothing to do with the Purcell pedagogy.

This very simple argument implies that you cannot derive to complete set of Maxwell equations simply from electrostatics and Poincare invariance.

This is a non sequitur. Furthermore, nobody, including Purcell, has made a claim to the contrary. That would obviously be false since, for example, there would be no way to infer the zero divergence of the magnetic field.

I'm finding this discussion to be extremely unproductive, for the reasons described at the top of #35. I have this quaint idea that if you're going to criticize a particular book, you should read the book, and if you're going to criticize a particular argument in a book, you should read the argument and discuss what it actually says.

 

[bcrowell]

I consider "Purcell pedagogy" a useful and interesting tangent, but my replies relate more to the subject matter of the OP - which does not directly refer to Purcell.

I thought we had long since established that the link from the original post was a garbled and incompetent presentation. If you thought that was still in doubt, I can see why we weren't understanding each other.

This is why Nugatory pointed to Purcell in #6. Purcell popularized this pedagogy, and his presentation is a model of rigor. Therefore if people want to make claims that there is something nonrigorous about the approach, the burden is on them to point out a specific error in Purcell.

 

[harrylin]

I thought we had long since established that the link from the original post was a garbled and incompetent presentation. If you thought that was still in doubt, I can see why we weren't understanding each other.

This is why Nugatory pointed to Purcell in #6. Purcell popularized this pedagogy, and his presentation is a model of rigor. Therefore if people want to make claims that there is something nonrigorous about the approach, the burden is on them to point out a specific error in Purcell.

Good! I see that just now a new thread about that topic was started: [/PLAIN]
https://www.physicsforums.com/threads/problem-with-purcells-model-of-electromagnetism.837274/[/URL]

 

[vanhees71]

I try again. The idea that you can "derive" Maxwell's equations from electrostatics and special relativity is flawed. From the full Maxwell equations (which in my opinion are not derivable from simpler ideas but are fundamental laws concerning (all) phenomena of electromagnetism modulo quantization) it is clear that this cannot be the case, because for almost all fields you cannot find a Lorentz transformation into a reference frame, where the electromagnetic field has only electric components. This is the case if and only if the situation is electrostatic, i.e., if there's a reference frame, where all charges are at rest and where no dynamical em. fields are present. The reason, I've given in the previous posting, which is very clear and simple.

Not to get me wrong. I like the principle heuristics, but I don't like the Berkeley Physics Course vol. II. I prefer

Schwartz, Principles of Electrodynamics, Dover

for this heuristical approach. My favorite E&M book is still Landau&Lifshitz vol. II, because there you get a straight-forward approach based on relativity and the Maxwell equations. A more modern approach is given in

F. Scheck, Classical Field Theory: On Electrodynamics, Non-Abelian Gauge Theories and Gravitation, Springer

 

[SlowThinker]

I try again. The idea that you can "derive" Maxwell's equations from electrostatics and special relativity is flawed.

But... suppose Purcell's method can find the force between 2 charged particles in any configuration.
If you throw in conservation of charge too, then that's pretty much everything you need to reinvent all 4 basic laws. Because, in reality, interaction between particle pairs is what's really happening.
Not saying that the full derivation would be easy.

Am I wrong?

 

[vanhees71]

I've never seen this, and it's not clear to me, how you can do this. Why should you get out only electromagnetic interactions and not others due to different fields (scalar, tensor,...)? Besides, a fully self-consistent equation of motion for an interacting-point-particle system seems also not to be achieved, although there are famous attempts like The Wheeler-Feynman "absorber model".

 

[Demystifier]

That's a great book (also bit plagued with typos)

We are just discussing one error (more than a typo?) in this book:
https://www.physicsforums.com/threads/problem-understanding-baruts-book.837633/

 

[bcrowell]

I try again. The idea that you can "derive" Maxwell's equations from electrostatics and special relativity is flawed.

As I pointed out in #40, Purcell does not claim to do so.

From the full Maxwell equations (which in my opinion are not derivable from simpler ideas but are fundamental laws concerning (all) phenomena of electromagnetism modulo quantization) it is clear that this cannot be the case, because for almost all fields you cannot find a Lorentz transformation into a reference frame, where the electromagnetic field has only electric components.

As we've discussed extensively in this thread, this is true, and Purcell does not make any claim to the contrary.

Nobody can force you to read and think about the details of a sophisticated argument if you refuse to make the effort.

I nominate this thread for worst ever on PF. Next let's have a literary discussion of Huckleberry Finn in which nobody has actually read the book. Let's start off by criticizing the book because of its unrealistic depiction of the relationship between Jim, who is a used car salesman, and Huck, his hunting spaniel.

 

[PeterDonis]

Thread closed for moderation.