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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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  • $\begingroup$ Maxwell equations only describe changes in EM fields for prescribed sources. How charged fluid moves and changes in time requires additional model for matter. This is very broad and complex topic. There are different models for that for conductors, superconductors, dielectrics, magnetic materials, plasma, and so on. $\endgroup$ – Ján Lalinský Dec 13 '15 at 5:59
  • $\begingroup$ You can, however, solve this in a meaningful way if you're talking about a distribution of charged particles in empty space and nothing else, provided the density is in a range where the discreteness of individual charges, and pairwise correlations, are unimportant. This basically is the vacuum space-charge problem, and it comes up in particle beam theory all the time. $\endgroup$ – elifino Dec 14 '15 at 1:19
  • $\begingroup$ @elifino Can you provide a reference? $\endgroup$ – Timaeus Dec 14 '15 at 3:28
  • $\begingroup$ @Timaeus Handbook of Charged Particle Optics, ed. Jon Orloff, chapter 7 by P. Kruitt and G. H. Jansen $\endgroup$ – elifino Dec 19 '15 at 18:48
  • $\begingroup$ For a continuous distribution, any of the plasma kinetic equations can be used. The Vlasov equation is the easiest to work with and so most commonly used. From there you can derive a fluid theory. $\endgroup$ – Robin Ekman Dec 20 '15 at 2:34
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It would certainly be more complicated. One issue is whether the charge distribution is solid or fluid. If solid (and unbreakable) it's not too bad. Using integration methods in Jackson, determine the fields then find the total force and torque on your object. If your object doesn't have much charge, only worth about the fields due to other charges.

If fluid, it's a combination of electrodynamics and fluids called plasma physics.

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  • $\begingroup$ I've actually looked into plasma physics and it seems like magnetohydrodynamics and relativistic magnetohydrodynamics are probably what I was looking for. I understand that a solid object with a fixed charge distribution would be impossible to describe relativistically, but I see how you could solve it in the way you describe approximately. I think you are almost spot on, however, it would be great if you could elaborate a little on plasma physics and magnetohydrodynamics in your answer, and how the coupled equations are usually solved. I'd really appreciate it! $\endgroup$ – Ignacio Dec 18 '15 at 3:45
  • $\begingroup$ I have to refer you to a textbook or course on plasma physics. It's not something I know off the top of my head. $\endgroup$ – Spirko Dec 18 '15 at 4:14
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Knowing the current and the charge doesn't tell you the fields. And knowing the fields doesn't tell you how the charge and current evolve. Even throwing in the mass of each species and the velocity of each species and the charge to mass ratio of each species doesn't entirely fix it and still wouldn't tell you the fields.

You instead have a coupled system of charges and fields. And all of it needs to be specified and then the evolution of the mutual system needs to be solved. And the continuum version is known to have catastrophes. And even the discrete version has radiation reaction issues.

And sometimes those problems only cause small issues and so aren't a big deal. And historically this was an issue, and most people abandoned it to study Quantum Electrodynamics they didn't abandon it becasue they solved it. So classical electrodynamics as normally done had flaws, and most people just walked away from it, which is fine if the current users know the limitations.

It is often stated that, in classical electrodynamics, an initial charge distribution along with an initial velocity field determines the electric and magnetic fields,

I don't know of a single example of anyone other than you ever saying that. But it is not true. For instance if you had two charges five lightyears apart one held at rest for all $t\lt t_0$ and the other held at rest for all $t$ with $|t-t_0-3\rm{yr}|\lt 1\rm{yr}$ and during that time it was forced to oscillate harmonically at a fixed small amplitude and a fixed frequency of say the frequency of red light.

Then at $t=t_0$ you can release both charges and they will act just like two charges that had always been at rest. For a while. For a year in fact. And then the radiation from the first charge will reach the second charge and it will start to move differently in the case described than the case where they had both always been at rest.

They had the same initial charge and current distribution at $t=t_0$ but had different fields (so that part wasn't true) and then they had different evolution (so that part didn't happen either).

In general, there are many possible fields given some initial charge and initial current. For instance, with no charges there are many possible vacuum solutions to Maxwell and you can add any one of them to an nonhomogenous solution to Maxwell and get another solution to Maxwell. So many solution to homogeneous Maxwell lead to many solutions to inhomogeneous Maxwell.

If you want to get a unique solution you should use Jefimenko or Liénard–Wiechert and both will require knowing the whole past history, not just the initial charge and initial current.

which in turn determine univocally the dynamics of said distribution (how it evolves in time).

Even if you somehow got the fields (such as being given them) then all this gives you is the force. Knowing how forces determine motions is not trivial. Already, Newton's second law by itself allows multiple solutions (such as Norton's Dome) and throwing in charged particles makes it even more complicated through radiation reaction and other complications.

Plus you'd either need the mass of the different charged particles species or else some other similar information.

So any dream that you could just have a $\rho (\vec r,0)$ and $\vec J(\vec r,0)$ and get dynamics (get $\rho (\vec r,t)$ and $\vec J(\vec r,t)$) is doomed for not specifying the fields (and to use Jefimenko and/or Liénard–Wiechert requires knowing the entire past not just the present). And even if you had that, the dynamics would be nontrivial because of radiation and other effects and you'd need the mass and such.

So you'd need a mass distribution and velocity field for each species with a fixed charge to mass ratio. And then you either would have to specify the fields (including any possible vacuum solutions) or you'd need a past history for the charges that includes the accelerations in the past. Or else you'd just have to be given the initial fields. And even then once you have the initial fields, the initial mass distribution of each species and the initial velocity field of each species. Then you'd have to deal with the mutual coupled dynamics of charges and fields, which is not trivial.

If a particle species has a tiny charge to mass ratio and there are no other forces then it move in basically straight lines. If they have a huge charge to mass ratio then the lines can be quite bent, and then radiation reaction and other complications become more relevant.

Those are all issues that come up for discrete particles. So let's get to the continuum situation. The whole fluid model has problems, and I'll reference *Inconsistency, Asymmetry, and Non-Locality: A Philosophical Investigation of Classical Electrodynamics" as a (flawed but) general source of many problems with classical electrodynamics. And in particular there is a famous issue where you have a spherical charge distribution so each shell only contributes to the electric field of outer shells and everything can move purely radially. And yet you can get initial shells to cross each other.

In general this phenomena is studied in the field called catastrophe theory. Which is merely the technical name. Basically it shows the fluid model breaks down. The fluid model means you break space into regions and for each region you assign a velocity vector that describes the collective motion of all the particles of that species. Then you take groups of regions and have different velocities for different regions in the group and effectively have something like a vector field. This breaks down if particles from one region end up crossing into particles from another region without mixing.

Imagine a sparse bunch of cars at 100kpm heading towards a sparsely parked parking lot, as they pass through there is a catastrophe (even though the sparsity means no collision, catastrophe is a technical term, not an emotional or colloquial tetm) the point is that even though the cars are the same species the model of one fluid with one velocity fails.

Particles with a low charge to mass ratio going in very straight lines. If they are sparse and a group with high velocity is heading towards a group with low velocity then they can mostly pass through unchanged (sparse so don't get close and low charge to mass, so they act similar to dust, a pressure free gas, again dust is another technical term, not a colloquial word). So later the fast ones should come out the other side pretty much unchanged and same with the slow bunch. A fluid model would try to assign a single velocity to the entire collection during the time they occupied the same region.

This would be fine if they were dense enough to interact and had enough time to form a shared collective velocity for each small region.

The catastrophe of the fluid theory comes from a continuous version of a similar problem, each shell is pushed outwards but at different rates. The charge density can vary radially. But the surface area also varies at different radii and so you can make it so that the inner surfaces feel a stronger force and thus there is a radial charge distribution and radial velocity field that is perfectly normal that develops over time to have an initially smaller radius shell with an initial velocity move outwards to cross (reach the same altitude) as an initially larger radius shell and so both shells end up at at the same radius at the same time (a finite amount of time) but with different velocities. So the fluid theory fails.

So you could try to hypothesize a force density given by $\rho\vec E+\vec J\times\vec B$ or an acceleration proportional to $\rho\vec E+\rho\vec v\times\vec B$ where $\vec v$ is the velocity field value of that species at that point and the proportionality depends on the charge to mass ratio. But since the whole existence of the fluid velocity field requires avoiding catastrophe and in general you can't this is a hopeless problem.

But it isn't extra hopeless, the original problem was bad. It is not the case that Newton's second law gives you dynamics. And the Lorentz Force Law is fatally flawed too.

What people do is consider situation where these flaws lead to minimally or unimportantly wrong results and then they honestly admit that their assumptions fail in the general situation. For instance using $$m\vec a=q\vec E+q\vec v\times\vec B$$ fails to take radiation reaction into account but for many situations that failure only leads to small incorrectness. If you pretend it is an exact result then you dishonesty. But you often are only trying to make a good enough prediction for a particular situation.

Some people deal with radiation reaction issues in an iterative manner. They use guessed extrapolations of the evolution of the charges to find the fields due to those charges and currents and then they use those fields to get the forces on the particles then use $\vec F=m\vec a$ and the Lorentz Force Law to attempt to get a new predicted charge evolution and a new predicted current evolution. Then from these new charge predictions and these new current predictions they use Maxwell to get new predictions for the fields.

And then they repeat: they use the new fields to find new forces and get even newer predictions for the charge and current. And then use those to get even newer predictions for the future fields.

And then they repeat: they use the even newer fields to find even newer forces and get even newer newer predictions of the charge and current.

And so on and so on. Alternating 1) using dynamical charges, initial fields, and Maxwell to get field dynamical fields and 2) using dynamical fields, initial charges, initial currents, masses, and Lorentz to get dynamical charges and currents.

And this isn't about finding iterative predictions for later and later times. It is about make iterations of predictions for all future times, even future times a short time in the future. And as far as I know there is no result saying this iterative process converges. But in many situations of practical interest each of these first few iterations produced very small corrections if the initial guess was good. And so we can call it quits after a finite number and hope it is good enough for practice.

This is not a satisfying theoretical framework, and people that tell you that classical electrodynamics as normally done is consistent and straightforward are mistaken or actively lying to you.

It is not simple, and the way many people do it is actually mathematically inconsistent. Pretending it is a more perfect theory than it is is dangerous and unwise, for instance you might fail to predict the radiation caused by a large magnetic field bending a high speed charged particle in a circle and that could result in damage to objects and/or cause injury or death. You need to know when you are using a theory with limitations, so that you can handle it with care.

Textbooks are going to hand you a toy theory in a special situation. For instance if you pull out a textbook with a test charge point particle in an external field, they are going to write down a Lagrangian that produces the Lorentz Force Law as the Euler-Lagrange equations because they expect you will like that. Or they might right a Lagrangian for fields with a fixed source and then get Maxwell as the Euler-Lagrange equations because they expect you will like that.

And then you can pretend $\vec F=m\vec a$ is a thing that always works even though examples are known where it isn't. And then you can pretend that knowing an external field and getting a force is all you need and ignore radiation caused by the particle feeling the force. But that would be ignoring the limitations instead of accepting and dealing with them.

But in reality you have initial fields and initial charges and you need to coevolve both. And that is a completely separate subject that isn't a standard textbook subject.

If you pick a particular problem, like plasma physics people do, then you can find approximate solutions that are good enough for your particular situation. That's what almost everyone does almost all of the time.

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