3.4. Degenerate Stars, White Dwarfs (~ 2000 km) (~ 106 Gauss)
A breakdown of classical theory is expected. A moving electron must spiral around a magnetic field line in a circle with radius rL; the stronger B is, the smaller rL is. The classical electromagnetic theory breaks down at small scales rL ~ 10-9 m, where quantum electromagnetic theory takes over. This occurs at a magnetic field strength Bt > 107 Gauss, where rL can take only certain definite quantized values. In ordinary classical conditions, a small external B does not affect the internal structure of an atom. But when immersed in a high external B, the electronic orbits around atomic nuclei become very oblate ellipses. The study of magnetic field > 107 Gauss is a relatively new area of physics, and it is difficult to create such magnetic fields in terrestrial laboratories, so the astronomical research on dwarfs and pulsars having huge magnetic fields may continue to inspire physicists for a while.
A dipolar magnetic field shape can be seen also in some degenerate stars.
Several white dwarf stars have a dipolar magnetic field strength ~ 107 Gauss and a mean gas density ~ 1031 cm-3 (e.g., Fig. 9.8 in Shi-Hui 1994), but most white dwarfs may have a smaller field strength 106 Gauss and a radius ~ 2000 km. Dwarfs have exhausted their nuclear-burning fuels, and they contract until gravity is balanced by the pressure of degenerate electrons.
Degenerate objects such as white dwarfs and neutron stars have highly conducting degenerate matter and do not require dynamo action to sustain their magnetic fields, and their magnetic fields have very long decay times. Thus a "fossil" magnetic field is predicted.