The promise of increased signal-to-noise ratio and spatial/spectral resolution continues to drive MR technology toward higher magnetic field strengths. SAR management and B1 inhomogeneity correction become critical issues at the high frequencies associated with high field MR. In recent years, multiple coil excitation techniques have been recognized as potentially powerful tools for controlling specific absorption rate (SAR) while simultaneously compensating for B1 inhomogeneities. This work explores electrodynamic constraints on transmit homogeneity and SAR, for both fully parallel transmission and its time-independent special case known as radiofrequency shimming. Ultimate intrinsic SAR—the lowest possible SAR consistent with electrodynamics for a particular excitation profile but independent of transmit coil design—is studied for different field strengths, object sizes, and pulse acceleration factors. The approach to the ultimate intrinsic limit with increasing numbers of finite transmit coils is also studied, and the tradeoff between homogeneity and SAR is explored for various excitation strategies. In the case of fully parallel transmission, ultimate intrinsic SAR shows flattening or slight reduction with increasing field strength, in contradiction to the traditionally cited quadratic dependency, but consistent with established electrodynamic principles. Magn Reson Med 61:315–334, 2009. © 2009 Wiley-Liss, Inc.