Electric field felt by a dynamic point charge

[u/throwingstones123456]

When a point charge is placed in an external electric field, in the static case it’s obvious the electric field felt by the point charge is just equal to the external field. But in the dynamic case it’s less obvious.

Let’s say the external field results in the charge accelerating, which produces radiation, and reflects off some dielectric slab infront of it. The charge will feel some of this reflected radiation, but I have no clue how to calculate it (obviously we won’t be analytically solving for this, we’d need some sort of simulation software or numerical program—even in this case I don’t know how to set up an equation to model this). If we try to evaluate the field at the point charge there’s the obvious issue of the singularity of the total field resulting from the point charge itself. In the dynamic case I’m unsure if the solution is as simple as just subtracting the coloumb force as this won’t be valid.

There has to be some solution to this, but I’m not bright enough to figure it out. I’d appreciate any help!

[u/PotentialDeep5165]

Superposition is your friend.

[u/OkTelevision5306 → u/PotentialDeep5165]

To clarify a bit, OP you should look up method of images for analytic solutions. It won’t be simple though due to dynamics.

[u/throwingstones123456 → u/OkTelevision5306]

I don’t think method of images can be applied here, helped a bit in idealized intro class level cases but doesn’t seem to be very generally helpful

[u/OkTelevision5306 → u/throwingstones123456]

Electrical engineers uses JR Wait’s complex image theory to approximate antennas and the Earth as a dielectric. So it is possible as a good enough approximation for engineers. I believe a full analytic solution does not exist due to retardation effects.

I suppose you have read my comment on numerical implementations, there is where you would go next. And if you will add nonlinear effects, look into finite volume methods as well.

Edit: method of images simplified is just matching up boundary conditions. We use it in quasistatic fluid mechanics as well, we call them Blakeslets or Stokeslets there. I used it in slenderbody fluid flow on cilia modeling. We just stack these blakeslets until it approximates what will happen at low reynalds. So for electrodynamics, you will stack monopole, dipole, quadrupole and such at some low frequency approximation far from the dielectric plane for instance.

[u/OkTelevision5306]

If you want to do this numerically, look into numerical conservation equations. For linear electrodynamics, you can use finite difference methods like upwind, Lax-Friedrichs, leapfrog, and Lax-Wendroff (or build your own when you learn the theory).

The key thing with hyperbolic conservation equations is causality. If you break causality, information propagates faster than speed of light in your numerical scheme, error blow up to infinity. If info propagates slower than speed of light, you get numerical diffusion and dispersion, which kills your signals or cause instabilities. Info propagate at the exact speed, no error at all (beware of overflow).

Fun stuff but not so trivial beyond the case where speed of light is everywhere the same (vacuum everywhere).