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In SI units, Maxwell's equations (in vacuo) seem to have two free "parameters" or "constants". The vacuum permittivity, however, can be eliminated by properly redefining the electric charge, leaving just one parameter, which can be taken to be the speed of electromagnetic waves (as is done in cgs units). Of course, that's the speed of light, $c$, which is the only constant in special relativity. I find this quite fascinating, as EM, as a physical theory, has no free parameters if it is assumed to be consistent with relativity. Alternatively, one can think of EM+SR as having one single parameter, $c$. (Of course, you can also eliminate $c$ by making it equal to 1, but you still have it in your new time/space conversion definition, so it's not really "gone", unlike the electric charge unit redefinition that can eliminate the vaccum permittivity.) This is unlike Newtonian gravity or GR, for which a constant $G$ is needed (plus potentially $\Lambda$ in GR), not to mention the Standard Model of particle physics.

I would be really grateful if you could comment on this reflection. Should I really be amazed at the fact that EM has no free parameter, whereas gravity (Newtonian or GR) or the SM do? Does this mean anything deep about EM, or EM+SR, or am I missing something?

Thanks!

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Maxwell's equations can be rewritten in terms of some other quantities in order to hide unit-dependent constants such as permittivity or permeability of vacuum, or speed of light. But these equations are not the whole EM theory.

EM theory has also elementary charge $1.6\text{e}-19$ C, characteristic length of electrons $1\text{e}-15$ m where things go weird regarding EM mass of composite systems. If we include material media into EM theory, many more constants and properties have to be included.

The more you want the EM theory to explain, the more constants and measured quantities have to be included.

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  • $\begingroup$ Thanks for your reply. However, I don't think writing Maxwell's equations in cgs units leads to any loss of information in the theory; if anything, it shows that defining both a vacuum permittivity and a permeability is redundant (though a useful redundancy, that I do not deny). Thus, my comment, I believe, is still valid. Regarding the electron charge, and other quantities, Maxwell's equations by themselves, without any reference to the electron charge, etc., (plus a set of boundary conditions and Lorentz force) comprise a self-consistant classical physical theory in their own right. (...) $\endgroup$
    – Astro137
    Commented Oct 21, 2021 at 17:02
  • $\begingroup$ [Continuation] This is analogous to Newtonian gravity being a physical theory in its own right, with a wide range of applicability, regardless of what the masses of the planets, etc. are. Thus, my reflection is still valid. Finally, about EM in media other than vacuum, more parameters are indeed needed, but this can be seen as an effective theory, not as the fundamental one, which is Maxwell's equations in vacuo (classically at least). $\endgroup$
    – Astro137
    Commented Oct 21, 2021 at 17:11
  • $\begingroup$ [Continuation] Since my reflexion was about the fundamental theory, I think, again, that my point about there not being free "fundamental" parameters/constants (unlike gravity, which does have one "fundamental" parameter, G) is valid, and is still awaiting "clarification" (if that's possible at all). I appreciate your reply though, thanks! $\endgroup$
    – Astro137
    Commented Oct 21, 2021 at 17:12
  • $\begingroup$ If you think gravity has "fundamental parameter" $G$, why don't you think electromagnetic force has similar "fundamental parameter" $\epsilon_0$? Both $G$ and $\epsilon_0$ can be hidden via change of units of mass and charge, but then the "fundamental parameter" is in those units, no? $\endgroup$ Commented Oct 22, 2021 at 12:27

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