How Maxwell's equations predict how sources are affected by the EM fields?

Question

Sources in classical electrodynamics, described by Maxwell's equations, are inmutable entities, that feel external; and most of the times, work as a mathematical tool for describing particles which are not affected by the laws.

$$ \partial_\mu F^{\mu \nu} = \mu_0 J^\nu $$ The sources are: $$ J^\nu = \left (c\rho, \vec J \right) $$ Which, by the moment works by giving the input to the equations, but the particles themselves aren't affected by the fields.

But, particles trajectories are describe by the Lorentz force: $$ \frac{d p_\mu}{d\tau} = qF_{\mu \nu}u^\nu $$ This means that source particles, also should be affected by the fields (and not just be an useful mathematical tool).

Could you relate the $J^\nu$ location, with the EM fields (using the Lorentz law) and then plug everything into Maxwell's equations, to see how particles autoinduce themselves? Does the Maxwell's equations take this into account (as far as I know, no, right?).

Answer 1

This is a difficult issue, and an impossible one when you consider matter to be composed of particles.

Maxwell's equations describe how the electromagnetic fields react to sources. The sources, though, are also affected by the electromagnetic field and must react to it somehow. This is described by either the Lorentz force law or by more complicated equation of motion.

To solve the complete problem, one must solves the coupled system of PDE's involving both Maxwell's equations and the equations of motion for the source. This is no easy task, and for a particle source it is an ill-posed problem (the electric field of the particle at the particle diverges, for example). However, it can be done for other matter systems (such as a charged fluid or a charged scalar field).

Wald's Advanced Classical Electromagnetism provides an excellent introductory discussion on these topics. Check especially Chaps. 1, 9, and 10.

Answer 2

You've hit upon the fact that a set of physics equations, on its own, doesn't tell you which values should be the "inputs" (coming in from outside the problem you're solving), and which values should be the "outputs" (solved by the equations, given the specific inputs). Physicists usually have a context, along with causal intuitions, which tell them which values should be "inputted". Also, sometimes the choice is one of calculational convenience.

If the context is an experimenter controlling a bunch of charges, wiggling them at will to create chosen currents, then our causal intuition tells us that the full spacetime history of $J^\nu$ should be an input, and the first equation you give can be used to solve for the fields generated in this manner. But consider a different context, where the experimenter is controlling the fields (say, a laser operator), wiggling the fields at will, while merely setting up the initial charges. Now the natural way to use these equations is to input the fields and the initial charge distribution, and then solve for the future charge distribution, using your final equation. The common thread is that the "input" tends to be what we imagine controlling, from outside the modelled system. (This follows the modern "interventionist" view of causation, where the "cause" is where we imagine we are able to intervene from outside the system.)

Remarkably, there's no one right answer here. One could claim that the "right" way to do it is to start with the initial charge and field distribution as an input, but that doesn't work well if you're externally controlling anything at all, manipulating charges and/or fields as time goes on. (Not to mention that self-consistently solving such equations is profoundly difficult!) In practice, how one uses these equations really is context-dependent. You need more than just the bare equations to solve physics problems. Along with the equations, to make them useful, one generally needs a causal model, supplying some knowledge about where the interventions are happening in any given context.