Mr hamster_nz
The poynting Vector isn't a magical field that exists in reality.
Agreed! It is the cross product of E (the electric field vector) and H (the magnetizing field).
E really does exist. H is sort-of real (as it depends on the nature of material the B field is in, so is in some way derived), but it is close enough to real that I would call it such.
It is nothing more than a derivative imaginary third field for visualizing the positions of sources and loads in electrical and other systems.
It doesn't say anything about the nature of electricity itself. It also doesn't tell you anything about how/why/where or when electrical energy occurs.
But it does represent "the directional energy flux (the energy transfer per unit area per unit time)" to quote Wikipedia. So things like surface integrals can have a real physical interoperation.
Electricity actually flows from a source to a drain or a place of different potential. The poynting vector doesn't actually show the direction of energy flow or the amount of energy.
It points from an area of lower field power density to an area of higher field power density. The key word is density. NOT total energy.
So a direct derivative of the positions of electrostatic and magnetic field lines. It is a tool for analysis. Nothing more, nothing less.
I could just as easily say that the electrostatic and magnetic field lines do not exist, just as isobars on the weather map don't exist either.
A poynting vector drawn in a circuit a nanosecond after it has been turned on will show the distance electricity has travelled in a nanosecond and ignore the rest of the loop because electrons haven't started moving yet in the rest of the wire loop.
The poynting vector also doesn't take into account radio waves emitted by the circuit in all directions.
No movement of charged particles = no magnetic or electric fields = no poynting vector.
Electric fields still exist when there is no movement of charged particles (electrostatic). But as you say, without magnetic fields the Poynting vector is zero.
My rejecting of power flowing in wires has very tittle to do with the Poynting Vector. Here is some of my various thoughts:
1A flowing in a wire looks the same regardless if 1kW or 100mW of power is being transferred into the load. The only way to tell is to cut the wire, and measure the potential between the two end.
If there was a significant electric field gradient within a wire, then any mobile charges in the wire would be accelerated. That would be doing work in the wire. The wire would tip over to having significant resistance, have a significant potential difference along it, and generate heat.
When current is flowing the charges are drifting along very slowly, and when no current is flowing the charges aren't moving at all - even if the wire is considered to be at a high potential! So whatever the conditions are that enabling an electron to do work are not present in a wire, (which I guess is what makes it a good wire).
If you have a difference in potential between two plates and add a test charge between them it will experience a force. Whatever is providing the energy is present outside of the wires - the force has to be transferred some how. This is unlike the water pressure is voltage in the "electricity is water" analogy, where you have to be in contact with the water make use of the energy. You can measure the pressure difference between two hoses, but if you put a glass of water between two hoses won't feel a force towards the hose with the lowest pressure.
I think the underlying truth is "the conductors provide the charges, the fields supply the energy", and there are most likely deeper truths under that. But for day-to-day electronics where stupidly high potentials and energies are not involved it makes no difference - the Lumped Element model works fine, until your PCB designer forgets to put two inductors at right angles, or you get unexpected parasitic capacitance between traces, or some other reason that reality ruins your day.