Tuesday, January 22, 2019

electromagnetism - How does a charge distribution evolve with time? (in classical electrodynamics)


It is often stated that, in classical electrodynamics, the electric and magnetic fields determine uni vocally the dynamics of a charge distribution distribution (how it evolves in time). I can more or less easily see how this applies to a charge distribution composed of point like particles. Basically the trajectories of the particles are given by the Euler-Lagrange equations of motion (or Hamilton's equations, or Newton's equations with the Lorentz force law, pick the ones you like the best). I can't understand how this can work for a continuous distribution, are there analogous equations that the charge distribution has to obey?


I can't come up with a satisfactory equation and not due to lack of trying, although I've never taken fluid mechanics or continuum mechanics so I'm kinda lost when dealing with continuous distributions and velocity fields.


In short: what is/are the equations that govern the dynamics of the charge distribution and its velocity field? Why haven't I been able to find this in Jackson's classical electrodynamics for example? Is it there in some other form? Is the answer something obvious that I'm missing?




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