  Systems involving nematic liquid crystals subjected to magnetic fields or
electric fields are modeled using the Oseen-Frank macroscopic continuum theory,
and general criteria are developed to assess the local stability of equilibrium
solutions. The criteria take into account the inhomogeneity of the electric
field and the mutual influence of the electric field and the liquid-crystal
director field on each other. The criteria show that formulas for the
instability thresholds of electric-field Freedericksz transitions cannot in all
cases be obtained from those for the analogous magnetic-field transitions by
simply replacing the magnetic parameters by the corresponding electric
parameters, contrary to claims in standard references. This finding is
consistent with [Arakelyan, Karayan, Chilingaryan, Sov. Phys. Dokl., 29 (1984)
202-204]. A simple analytical test is provided to determine when an
electric-field-induced instability can differ qualitatively from the analogous
magnetic field-induced instability. For the systems we study, it is found that
taking into account the full coupling between the electric field and the
director field can either elevate or leave unchanged an instability threshold
(never lower it), compared to the threshold provided by the magnetic-field
analogy. The physical mechanism that underlies the effect of elevating an
instability threshold is the added free energy associated with a first-order
change in the ground-state electric field caused by a perturbation of the
ground-state director field. Examples are given that involve classical
Freedericksz transitions and also periodic instabilities. The inclusion of
flexoelectric terms in the theory is studied, and it is found that these terms
are not capable of altering the instability thresholds of any of the classical
Freedericksz transitions, consistent with known results for splay transitions.
