In a real spin ...
Magnetic moment of spinning classical charge, M = (1/2)qs/m
Where q is observed (shielded) electron charge, s is spin (discussed below a couple of posts back), and m is the observed mass of an electron.
Dirac's quantum field theory multiplies this by a factor of 2 to get the proper magnetic moment of an electron. But still it is not very accurate to meet experiment, so we use the 1st Feynman coupling term for virtual charge to give a more precise correction factor, 2[1 + 1/(2.Pi.137)]. (Let's ignore more coupling terms for the moment, as we are happy with accuracy to 6 s.f.)
I'm content now that the veil shield of polarised vacuum leptons (which shields the electric field from the electron core by a factor of just over 137) has no significant effect on the polar magnetic field from the electron core. The full task of identifying the causal process in the quantum 'foam' spacetime fabric which gives rise to each mathematical coupling correction remains ahead.
For the moment, let's worry about the size, shape, and range of the polarised virtual particle shells surrounding the core. An electric field line is defined by convention as an arrow from a positive charge towards a negative one. So we have an inward arrow from the positive virtual charge shell around the negative core, and an outward arrow between the outer negative virtual shell and the inner positive shell. Since the shielding factor is massive, 137, the arrows are pretty similar in length to look at visually. The polarised shells are stopping 99.27 % of the electric field from the core of the electron.
How long will it take to get around to working out the details, the sizes of the shells? No idea. There are actually quite a few unknowns to deal with. We can hope that the assumptions will be few due to the way in which mathematical unknowns can be cancelled out to give a definite answer. (See my home page for the gravity proof which uses this fact.) While there is no evidence that the Great Architect of the universe is a mathematician (J.B.S. Haldane said, by analogy, that God must be a beetle-lover, because of the vast number of beetle species on Earth), the use of mathematics for understanding things easily is essential. Without maths, you would have be be extremely clever to understand the atom or anything else realistically!
Update: Dr Motl reports discovery of extra dimensions in India: http://motls.blogspot.com/2005/10/news-on-string-theory-in-media.html
(Presumably this refers to the warping around the black hole of Calcutta?)