| Then stop playing with words and say what you mean. Point charges make sense without qualification in EM, that much is true. But that doesn't mean that there is something strange or fishy going on here. You can formulate Lorentz forces using mass distributions and you're fine (it's right there in the next subsection of the wiki). The pathologies don't appear. Of course the resulting equations are non-linear, and that is the origin of the failure of point sources to make sense. If you then try to recover point sources by taking a limit, the details of the assumptions you make matter a great deal [1]. For a very easy example, your point source might have a dipole. This problem is kind of well known and studied in the General Relativity literature where the non-linearity of the field equations forces you to confront similar type of problems already when trying to derive geodesic motion [2]. I suspect all this is familiar to you. I would phrase the conclusion as such: The Newtonian idea of focusing exclusively on the motion of the centre of mass, which reflects the influence of distributed forces on a rigid body, breaks down in any relativistic theory. This is simply because the notion of a rigid body breaks down. As such the point mass, which was a conceptual cornerstone of Newtonian mechanics, along with the notion of a force acting on it, are relegated to a technical tool useful only when non-linearities are neglected. [1] https://arxiv.org/abs/0905.2391 When googling for this paper I found a nice set of slides by Wald about this whole business: http://www.math.utk.edu/~fernando/barrett/bwald1.pdf [2] https://arxiv.org/abs/gr-qc/0309074 |
Mass distributions don't solve the issue in a satisfactory way, in my opinion. If you replace a particle with a finite size sphere you've solved the infinity but lost relativistic invariance. You could perhaps come up with a way to hold the charge distribution together in a relativistically invariant way, but that can hardly be considered part of EM, and might involve arbitrary choices. That a dipole behaves differently than a monopole is clear. That's already the case even if you ignore the self interaction.
The question "What happens if I put an electron in a uniform magnetic field?" or "What happens if I have two electrons?" seems like it should be answered by EM. One can hardly ask a simpler question. I'm pretty sure that most physics students who've had an EM course are under the impression that they should be able to answer this question. When I was in such a class it was never explained that this was even an issue, and when I asked about it the answer I got was "just wait for QED".
If you don't like the word "schizophrenic" for this issue, that's cool. I think it's descriptive, but YMMV. Wald's slides say:
> Classical Electrodynamics as Taught in Courses
> At least 95% of what is taught in electrodynamics courses at all levels focuses on the following two separate problems: (i) Given a distribution of charges and/or currents, find the electric and magnetic fields (i.e., solve Maxwell’s equations with given source terms). (ii) Given the electric and magnetic fields, find the motion of a point charge (possibly with an electric and/or magnetic dipole moment) by solving the Lorentz force equation (possibly with additional dipole force terms).
That's all I meant by it.
Wald's slides are interesting, thanks :)