After a brief overview of electrohydrodynamics including Maxwell's electric stress tensor under AC fields where the medium has both conductive and dielectric characteristics, we focus on the problem of electrowetting actuation of sessile drops on a patterned array of electrodes with a thin dielectric coating. For both the case when the drop is electrically grounded from below and when it is floating, we compute the electric field in the vicinity of the drop over a range of frequencies and use the traction derived from the Maxwell stress tensor to calculate the effective electrowetting force on the drop. At low frequencies where the drop behaves like a perfect conductor, the results are compared with previously derived lumped parameter models for the electrowetting force. [Joint work with James Sterling and Maged Ismail.]