Which treatable general theory or mathematical tool is best suited to deal with multiple electrical charges in 2D and 3D spaces in terms of static analysis and in equilibrium condition and also its dynamics showing paths independently of the number of particles?

  • $\begingroup$ This is too vague a question. Maxwell's equations, Lorentz formula, differential calculus? What specifically are you trying to understand? $\endgroup$ – Ján Lalinský Mar 16 at 15:25
  • $\begingroup$ Imagine that we are talking about an atomic model where we have N electrons (negative charges) orbiting around a nucleus containing N protons (positive charges). Electrodynamically how to deal with the problem of the trajectories of each electron? $\endgroup$ – M. Tredinnick Mar 18 at 15:29
  • $\begingroup$ Write down all equations of motion for all particles (net force is sum of forces due to all other particles) and integrate them numerically. The simplest case for proton+electron in retarded variant of the EM theory was analyzed by J. L. Synge, On the electromagnetic two–body problem., Proc. Roy. Soc. A 177 118–39 (1940) royalsocietypublishing.org/doi/pdf/10.1098/rspa.1940.0114 $\endgroup$ – Ján Lalinský Mar 18 at 15:55
  • $\begingroup$ Thank you for sharing the Synge's work. It is an interesting begin to my development. Do you imagine that from this work could be possible generalize to many charged bodies? $\endgroup$ – M. Tredinnick Mar 19 at 1:52
  • $\begingroup$ Yes, the generalization of the equations to $N$ particles is obvious, however solving them becomes a very difficult task, best solved numerically using computer. It is similar to gravitational n-body problem, but even more difficult because of retardation. $\endgroup$ – Ján Lalinský Mar 19 at 3:13

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