[en] Eddy current heating of thin conducting plates of various shapes by a perpendicular field is studied, assuming that the magnetic field created by the eddy currents is negligible in comparison with the external field. The method is to introduce the stream function of the eddy currents, which is shown to satisfy Poisson's equation, and then employ a pair of complementary variational principles (i.e., a minimum principle and a maximum principle), the extrema of which equal the eddy current heating. Two such complementary principles give not only an estimate of the eddy current heating, but a bound on the error of the estimate as well