Current in Wire and Poynting Vector

A long straight wire of radius carries a current I in response to a voltage V between the ends of the wire.

a) Calculate the Poynting vector S for this DC voltage.

b) Obtain the energy flux per unit length at the surface of the wire. Compare this result with the Joule heating of the wire and comment on the physical significance.

SOLUTION

a) Let us calculate the flux of the Poynting vector. Introduce cylindrical coordinates with unit vectors e_ρ, e_θ, and ẑ. Current flows along the wire in the z direction and the electric field E = Eẑ. Using one of Maxwell’s equations in vacuum, the fact that conditions are stationary, and Stokes’ theorem,

(1)

(2)

where J is the current density and A is the surface. At any given radius r, B_θ is constant, so we have

(3)

(4)

Using the relation between current density and total current J = I/(πb²):

b) The Poynting flux per unit length is then S ^. 2πb = -IE. So the flux enters the wire, and we see that the dissipated power per unit length IE is equal to the total incoming S-flux, in agreement with Poynting’s theorem:

(5)

where u is the energy density. Under stationary conditions such as ours

and we have