Does an oscillating electrical monopole radiate? For example, the displacement R(r,t) of the single charge q is given by
$$R = R_0\cos(\omega t)$$
In the view of multipole expansion, what is the monopole moment and the dipole moment for this configuration? Why the monopole does not contribute to the final EM field?
 A: There is a famous nonradiating configuration, which is not exactly what you asked, but it is very interesting, since it is also true in GR. If you have a sphere of fixed total charge whose radius oscillates arbitrarily, there is no electromagnetic radiation emitted. The solution is just given by Gauss's law outside the sphere, and you can verify that this satisfies Maxwell's equations even in the presence of everywhere equal radial currents, since the magnetic fields produced by these currents cancel out.
The deep reason that this doesn't radiate is that photons are spin 1, so they can't be emitted by an always spherically symmetric source distribution. The analogous Birkhoff theorem in GR tells you that a radially oscillating massive sphere doesn't radiate, although there it's because gravitons are spin 2.
A: I believe that to get radiation you need a "jerk," that is a non-zero third time-derivative of the displacement. Hence, this will radiate.
A: An electrical monopole is just charged particle. Any charged particle that accelerates will definitely radiate an electromagnetic wave.  A sinusoidal oscillation you gave as an example will produce radiation at the frequency of the sinusoidal acceleration.  But any acceleration of any kind will produce some kind of electromagnetic wave.
