What does Purcell actually say about electromagnetism and relativity?

[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.