Consider the power dissipation term, E Jdv, in Poyntings theorem (Eq. (70)). This gives the power lost to heat within a volume into

Consider the power dissipation term, ʃ E · Jdv, in Poynting’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 < pd >= (1/2)Re {E_s · J^∗_s}.

(a) Show that in a conducting medium, through which a uniform plane wave of amplitude E0 propagates in the forward z direction, < p_d >= (σ/2)|E₀|2e^−2αz.

(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.