# How would you calculate (numerically) the shape of the electromagnetic waves emitted by an accelerating charge?

I am interested in trying to create an animation showing an accelerating charge emitting electromagnetic waves that is physically accurate. Writing the code is not what I am worried about, however, I have been having trouble determining what equations I would actually need to numerically solve.

It comes down to a few questions I have:

1. Which, if any, of Maxwell's Equations are valid for an accelerating charge? I understand that in special relativity, Maxwell's Equations still hold. Is it also true that, as long as we are viewing it from an inertial frame, Maxwell's Equations are also true in the vicinity of an accelerating charge?
2. I suppose this is more of a theoretical question, but it's not something I've talked about in class before; perhaps in my E&M class this fall. Do Maxwell's Equations guarantee a unique solution for the field shape in any given situation?
3. It seems like, in order to do this calculation, you might need to incorporate Lorentz Transformations somehow. After all, both the nonrelativistic and relativistic forms of Coulomb's Law, for example, obey Gauss's Law, so there is clearly more than one possible field shape that satisfies this law. How do you guarantee that you calculate a relativistically correct solution? Although I suppose that since the relativistic form of Coulomb's Law occurs when the charge is moving and there is also a magnetic field, there are the rest of Maxwell's Equations that need to be satisfied, and this might force the predicted $$\vec{E}$$ and $$\vec{B}$$ fields to be relativistically correct. After all, Maxwell's Equations are relativistic. But I want to verify this with someone else.
4. Somehow, you'd need to express the fact that EM waves don't propagate infinitely fast; rather, they travel at lightspeed. I think this is also covered by Maxwell's Equations in the general case, but, assuming it is, I'd like to get a better understanding of why.

Is the problem really as simple as just numerically solving Maxwell's Equations, regardless of how the charge moves? It seems like the acceleration would complicate things and require more physical laws to be involved. I am aware of the Lienard-Wiechert potential as a means of doing this, but as I haven't covered this material on my own or in class yet, let alone seen the derivation, I'd like to stick with what I know, at least for now.

• Why would you want to numerically solve multiple 3+1D partial differential equations when the exact analytic solution for an accelerating point charge is known? It would be 10x more work, at least. Commented Aug 5, 2023 at 5:15
• Maxwell’s equations — all of them — hold for any charge density $\rho(\mathbf r, t)$ and current density $\mathbf J(\mathbf r, t)$ as long as the charge and current density satisfy the continuity equation for charge. Commented Aug 5, 2023 at 5:24
• I guess, like I said, I want to stick with the physics I understand, although maybe that just means I'd be better off learning about the analytical solution. I don't know. I was a bit hesitant because I haven't worked with the vector potential much before, but maybe that doesn't justify doing it numerically. Commented Aug 5, 2023 at 9:44
• I usually like to try the derivation on my own before looking at how someone else does it, too, even if I'm not successful. I guess my only question here then is (4) in the original post. For some point $P$ in space, do I need to factor in by hand the fact that information about the EM disturbance propagates towards that point at speed $c$, or is this covered by Maxwell's Equations? Something is telling me it's not covered automatically, but I have yet to try and work that out on my own. Commented Aug 5, 2023 at 9:47
• I think I've got it, but correct me if I'm wrong: you can show with Maxwell's Equations that an EM disturbance propagates at the speed of light; this is a common exercise in introductory E&M texts. The problem you have to overcome is describing where the disturbance begins. If you naively assume the relativistic form of Coulomb's Law holds for an accelerated charge, then you are basically implying that the disturbance begins everywhere. To account for the fact that it begins in the immediate vicinity of the charge, you introduce these delayed potentials Wikipedia describes below: Commented Aug 5, 2023 at 11:41

It makes no sense to solve this problem numerically given that a full analytical solution is already known.

Specifically, the Liénard-Wiechert potentials provide an explicit form for both the potential and the fields associated with a point charge moving on a pre-specified trajectory.

The solutions are relatively simple, and can be quoted here. For the electric field, it reads $${\displaystyle \mathbf {E} (\mathbf {r} ,t)={\frac {1}{4\pi \varepsilon _{0}}}\left({\frac {q(\mathbf {n} _{s}-{\boldsymbol {\beta }}_{s})}{\gamma ^{2}(1-\mathbf {n} _{s}\cdot {\boldsymbol {\beta }}_{s})^{3}|\mathbf {r} -\mathbf {r} _{s}|^{2}}}+{\frac {q\mathbf {n} _{s}\times {\big (}(\mathbf {n} _{s}-{\boldsymbol {\beta }}_{s})\times {\dot {{\boldsymbol {\beta }}_{s}}}{\big )}}{c(1-\mathbf {n} _{s}\cdot {\boldsymbol {\beta }}_{s})^{3}|\mathbf {r} -\mathbf {r} _{s}|}}\right)_{t=t_{r}}},$$ where $$\mathbf {r} _{s}(t)$$ is the trajectory of the charge, $${\boldsymbol {\beta }}_{s}(t)={\frac {{\mathbf {v}}_{s}(t)}{c}}$$ is its normalized velocity, and $$\mathbf n_s$$ points from $$\mathbf {r} _{s}(t)$$ to the evaluation point $$\mathbf r$$.

The most crucial aspect of the solution is that everything inside the brackets must be evaluated at the retared time $$t_r$$, which captures the idea that you expressed as

Somehow, you'd need to express the fact that EM waves don't propagate infinitely fast; rather, they travel at lightspeed

and it is basically the time $$t_r$$ at which a signal starting from the particle's trajectory at $$\mathbf {r} _{s}(t_{r})$$ would reach your chosen observation point $$\mathbf r$$ at your chosen observation time $$t$$. This can be calculated through the implicit equation $${\displaystyle t_{r}=t-{\frac {1}{c}}|\mathbf {r} -\mathbf {r} _{s}(t_{r})|,}$$ which may or may not be awkward to solve analytically (or numerically) depending on what the trajectory looks like.

And, that's it. That's all you need for the explicit analytical solution.

I should also point out that if you tried to do this via a numerical solution of the Maxwell equations, you'd run into some strong headaches rather quickly. This is mostly because you're trying to calculate the radiation from a point charge, which means that the fields have singularities, and those will take a lot of dedicated handling if you want to describe them numerically.