Chapter 11, Problem 11.16P

Consider the power dissipation term. $\int E.jdv,$ in Painting’s theorem [Eq. (70)]. This gives the power lost to heat within a. volume into which electromagnetic waves enter. The term p_d = E. J is thus the power dissipation per unit volume in W/m³. Following the same reasoning that resulted in Eq. (77), the time-average power dissipation per volume will be < P (l/2) R.e{E _B. J }. (a) Show that in a conducting medium, through which a uniform plane wave of amplitude Ec propagates in the forward z direction, < P _d > = (o"/2) \E₀ |² e-2az. (b) Confirm this result for the special case of a good conductor by using the left-hand side of Eq. (70), and consider a very small volume.