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We have developed a set of differential equations which describe the hydrodynamics of a conducting liquid crystal subjected to external electromagnetic fields. The mechanical forces resulting from the interaction of the fields with the liquid are expressed as operations on the Maxwell stress tensor. The use of this description allows a concise statement of the equations of motion. As an example of the validity of the formalism, we rigorously solve the boundaryvalue problem associated with the Williams domain (vortex) mode in nematic liquids. Using standard constitutive relations, physical boundary conditions, and experimentally measured material p-azoxylanisole constants, we quantitatively reproduce the significant experimental observations.

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