  In this work we consider the Landau-de Gennes model for liquid crystals with
an external electromagnetic field to model the occurrence of the saturn ring
effect under the assumption of rotational equivariance. After a rescaling of
the energy, a variational limit is derived. Our analysis relies on precise
estimates around the singularities and the study of a radial auxiliary problem
in regions, where a continuous director field exists. Studying the limit
problem, we explain the transition between the dipole and saturn ring
configuration and the occurrence of a hysteresis phenomenon, giving a rigorous
explanation of what was conjectured previously by [H. Stark, Eur. Phys. J. B
10, 311-321 (1999)].
