I'm just a spectator in this shit flinging but I'll weigh in. The Poynting vector is only a part of Poyntings theorem, which states that for an arbitrary volume in space:
rate of energy transfer = rate of work done on a charge distribution + energy flux going out of the volume
>>13897843You integrate the Poynting vector to get the energy flux out of the volume, which is zero since there is no EM field inside a conductor. So the power (rate of energy transfer) is the rate of work done on a charge distribution, which in this case are the free electrons in the conductor. Power flows through conductors, there's a load on one end resisting the motion of electrons and a battery on the other trying to move them. The power that flows is the work done on the electrons moving them.
Technically, thinking about it this way only works when taking the arbitrary volume to include the conductor and the load (or battery). If you take it to be just the conductor, you're basically asking "how much power flows into this piece of metal", which is called waste heat and ideally its 0.
>>13897974Poyntings theorem is a very general result that comes directly from maxwells equations and conservation of energy. It is true in the near field and far field,
>>13898940 is correct. But the Poynting vector alone is not enough to tell you about power transfer, for that you also need to know the work done on the charges.
