An infinitely long straight wire on the z-axis has a circular cross section and obeys j(ω) = σ₀E(ω) for all ρ ≤ a. After initial transients, one finds the charge density ρ(r, t) ≡ 0 and the current I (t) = I₀ cos ωt everywhere inside the wire.
(a) Solve an appropriate Helmholtz equation and find the exact E(r, t) inside the wire. Express the amplitude of the field in terms of I₀.
(b) Solve an appropriate Helmholtz equation and find E and B exactly outside the wire. 

(c) Use Poynting’s theorem to show that the normal component of the time-averaged Poynting vector (S) evaluated on any cylindrical surface concentric with the wire always points toward the z-axis.
(d) Use the Poynting vector to calculate the rate at which energy is lost to ohmic heating per unit length of wire.
(e) Use the Poynting vector to calculate the rate at which energy is lost to radiation per unit length of wire.
How is this result consistent with conservation of energy and the answer to part (c)?