The dissipation of electromagnetic energy from an ultrawideband plane wave pulse that is propagating through a causally dispersive, attenuative dielectric is described. Because the precursor fields, which are a characteristic of the dispersive medium, propagate with the least attenuation in the medium, the peak in the Brillouin precursor decaying only as z-1/2, they carry a significant fraction of the input electromagnetic energy much deeper into the lossy material. These precursor fields then generate heat at propagation depths where virtually none is generated by the main signal at the input carrier frequency. The implications of this ultrawideband phenomenon to bioelectromagnetics and radiation dosimetry may be significant. ¿ 1998 American Geophysical Union