This is basically problem 64 from your homework. Our archetypical model for a resistor is drawn above: two circular conducting plates (metal contacts) with radius , separated at a distance by a material with resistivity .
a) In a steady state situation where a DC voltage is applied as shown, find the field inside the resistive material.
b) Find the current density inside the resistive material.
c) From Ampere's law, find the magnetic field as a function of in the region between the plates.
d) From your answers to a) and c), find the Poynting vector (magnitude and direction) as a function of in the region in between the plates.
e) NOW show that:
where is the outer surface of the resistor and is its outward-directed normal unit vector.
Thus the heat that appears in the resistor can be thought of as the electromagnetic field energy that flows in through its outer surface!