electromagnetism - What is the Poynting vector of an electron?

On chapter 27 of "The Feynman Lectures on Physics Volume II: Mainly Electromagnetism and Matter", Richard Feynman says:

Finally, in order to really convince you that this theory is obviously nuts, we will take one more example — an example in which an electric charge and a magnet are at rest near each other — both sitting quite still. Suppose we take the example of a point charge sitting near the center of a bar magnet, as shown in Fig. 27-6 Everything is at rest, so the energy is not changing with time. Also, E and B are quite static. But the Poynting vector says that there is a flow of energy, because there is an E X B that is not zero. If you look at the energy flow, you find that it just circulates around and around. There isn't any change in the energy anywhere — everything which flows into one volume flows out again It is like incompressible water flowing around. So there is a circulation of energy in this so-called static condition. How absurd it gets!

Perhaps it isn't so terribly puzzling, though, when you remember that what we called a "static" magnet is really a circulating permanent current. In a permanent magnet the electrons are spinning permanently inside. So maybe a circulation of the energy outside isn't so queer after all.

I was wondering whether it would be possible to calculate the Poynting vector of an electron, considering its static electric field and the magnetic field associated to its magnetic dipole moment, caused by its intrinsic property of spin.

In case it is possible to do such calculation, what would be the interpretation of the Poynting vector?