# How do I know if a solution of Maxwell Equations is caused solely by the charges?

My question is very general and it isn't limited to the system that I calculated the fields. I need a general answer or a particular answer for my problem. I found two fields $$\mathbf E,\mathbf B$$ solutions for a specific system, and I know that these field plus electromagnetic waves are solutions too. So which has to be my criteria? Because, an option would be Jefimenko's Equations, but these are horrendous.

In specific. I need a hint or something. I have a charged spherycal shell with variable radius $$R_{(t)}$$ but with charge $$Q$$ constant, so in CGS:

$$\rho(\mathbf r, t) = \frac{Q}{4\pi r^2}\delta(r-R_{(t)})$$

With arguments of symmetry I can say that $$\mathbf E(\mathbf r,t)=E(r,t)\mathbf{\hat r}$$, then with the 1st Maxwell equation:

$$\int_{r'=r} \mathbf E(r',t)\cdot d\mathbf A'= 4\pi\int_{r'$$\implies E(r,t)= \frac{Q}{r^2}\Theta(r-R_{(t)})$$

Where $$\Theta$$ is the Heaviside function. We can see that a valid solution for the magnetic field is:

$$\mathbf B(r,t)=0$$

How do I know if this solution is caused solely by the charges with no waves included?

• Very far away from the shell the electric field wouldn't change with time, since it's spherically symmetric and you could just use Gauss's law to show that the field remains the same provided you choose a surface sufficiently large so that all the charge remains enclosed in it. Commented Jun 19, 2020 at 0:11
• What's your argument to say that $\mathbf{B}(r,t)=0$? Commented Jun 19, 2020 at 0:13
• @Philip I know that $B=0$ is solution, that's all. I suppose that I can use symmetry to have a good argument, but I'm not sure. Commented Jun 19, 2020 at 0:22
• @Philip I know that the solution has sense, but I don't know if it exists a criteria for discard a solution. Commented Jun 19, 2020 at 0:46
• Are you looking for this? Birkhoff's theorem (electromagnetism) Commented Jun 19, 2020 at 0:58