I would like to get expressions for the time-dependent electric and magnetic fields in the region around two moving point particles.

I can't see how this is possible. Clearly you can't use Gauss' Law or the Biot-Savart Law because the situation is not static.

You can't just sum the two field solutions for particles in vacuum, as this describes a non-interacting system.

If you try to use the retarded potentials then you need to know the past history of the particles, which in turn depends on the electromagnetic fields, which is the thing you're trying to calculate! You might say if the particles are sufficiently far apart, then they travel in straight lines...but their trajectories would only be straight in the limit that they are an infinite distance apart, in which case they would take an infinite amount of time to begin to interact. Perhaps you can somehow integrate from minus infinity in time up to the moment you're interested in.

  • $\begingroup$ I think it would be plausible to say that each particle is defined by some parameters. Let's say charge, mass, location and speed(sorry Heisenberg). If we have this info we can use the Lorentz force to calculate the force that is exerted on each of the particles. You can try it with linear time steps(each of them being static), and later try to move onto a continuous solution. The formula would be $\mathbf F=q(\mathbf v \times \mathbf B +\mathbf E)$, from which you can get $\mathbf a$. You can use the listed laws, you just need use the right form of them. $\endgroup$ – WalyKu Feb 23 '15 at 13:47
  • $\begingroup$ If you have initial conditions it is easy. But how do you show that your initial conditions are consistent with the laws of classical physics? i.e. how can you show that for given position and speed you choose, it is even possible to have the initial fields you choose? Seems like it's hard to prove a set of initial conditions is self-consistent. $\endgroup$ – kotozna Feb 23 '15 at 14:23
  • $\begingroup$ Hmm, what about letting time flow backwards? Going back the background of the question: The system was created by an external perturbation, you have to make some assumptions as to what was this perturbation. Where do the charged particles come from, how did they pop into existence? You can go pretty far asking this kind of questions. You could look at each problem separately, assuming that you solved the previous problems. This way you can work up the ladder and revise the previous solutions if they happen to be wrong because of new findings in an earlier layer. $\endgroup$ – WalyKu Feb 23 '15 at 15:04

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