Quote by Vanadium 50
Indeed, I would say that the Lorentz force law defines E and B.

Exactly  and that's how it's traditionally taught, but it's also plainly inadequate...we all agree that fields have existence
independent of a charged particle to act on via the Lorentz force, right? So while you could provisionally define fields in terms of how they act on charged particles, that says nothing, for example, about how they act in the absence of charged particles.
Of course, you'll say that's what the Maxwell equations are for  but then we're back to the circular definition.
One example of a 'first principle' derivation might be something contrived like deducing the form that form of interaction equations of a spin 1 field must take, but I'm convinced there are more elegant ones: the covariant formulation is so simple that it's amazing that we
don't see these equations arising all over the place in many and various ways...
So, I'm just looking for one of these many possible derivations / explanations that doesn't get at the Lorentz force by just taking it as an axiom or as "empirical"  particularly since it when all we had was empirical evidence, minds as great as JJ Thompson
got it wrong by a factor of two. Empirical first principles aren't things that you get wrong by a factor of two...the correct form was eventually derived
theoretically by Heaviside  so it would be interesting to see, for example, how he did so.