Anomalous behavior of electric-field Fréedericksz transitions

Fr\'eedericksz transitions in nematic liquid crystals are re-examined with a focus on differences between systems with magnetic fields and those with electric fields. A magnetic field can be treated as uniform in a liquid-crystal medium; while a nonuniform director field will in general cause nonuniformity of the local electric field as well. Despite these differences, the widely held view is that the formula for the threshold of local instability in an electric-field Fr\'eedericksz transition can be obtained from that for the magnetic-field transition in the same geometry by simply replacing the magnetic parameters by their electric counterparts. However, it was shown in [Arakelyan, Karayan, and Chilingaryan, Sov. Phys. Dokl., 29 (1984) 202-204] that in two of the six classical electric-field Fr\'eedericksz transitions, the local-instability threshold should be strictly greater than that predicted by this magnetic-field analogy. Why this elevation of the threshold occurs is carefully examined, and a simple test to determine when it can happen is given. This "anomalous behavior" is not restricted to classical Fr\'eedericksz transitions and is shown to be present in certain layered systems (planar cholesterics, smectic-A) and in certain nematic systems that exhibit periodic instabilities.