> I think we're mostly on the same page, yes, sorry I wasn't clear, I've been talking about two wires, linking a power source with a load.

Ah that makes a big difference. Note how all of this started:

> The energy doesn't flow inside of a wire, it flows around the wire.

As though all of the energy flows outside of a single wire, which doesn't make any sense at all (and it still doesn't!), if it did then the wire itself would never heat up, which given the fact that there is a voltage across the wire (low, but still, it is there) and current flowing through it shows that there definitely is 'work' being done inside a wire. So there is a non-zero length Poynting vector pointing (heh) to the inside of the wire! (as a result of the resistance of the wire), and that energy is subtracted from the amount of energy available for the load.

The 'B' component of the fields created by a current carrying wire is in fact the magnetic field (and a magnetic field as I'm sure you are aware doesn't need a 'return wire' as such, as any magnet demonstrates (it's circulating 'virtual photons' for a bit of physics weirdness), and there too the field is much stronger at the poles of the magnet), these fieldlines would be best visualized as concentric circles around the wire ( https://www.researchgate.net/figure/The-magnetic-field-lines... ) with the spacing dropping off as you get further away from the wire ( http://physics.bu.edu/~duffy/semester2/c15_inside.html ), and it definitely would be present inside the (single!) wire. Its presence gives rise to the 'Hall effect', which shows up as a voltage induced in a conductor placed across the field lines.

The electrical field by itself doesn't do any work, but has the potential to do so and this potential diminishes the further along the path from 'source' to 'sink' and on the return leg through the wire. The magnetic field of the free electrons in a conductor also doesn't do any work. But when you start moving those electrons using an electric field their magnetic field is doing work in the medium that they are moving through. The electrons themselves only move a fraction (on the order of a millimeter per second) while they move the magnetic field lines 'along' with them in the direction of their travel and the strength of the electrical field between the two conductors is a measure of the potential energy available to do work.

[This also neatly explains why electrical signals propagate at the speed of light: the electrons themselves may move slowly but their magnetic fields propagate at the speed of light, and if you think about it a bit longer it also explains why voltage and current in an AC circuit can be different than the real power drawn by the load because the load itself can temporarily store energy (power factor!). This can even cause zero apparent power while real work is done in the load. This shows up as the angle of phase difference between current and voltage, and is why the electrical company really doesn't like you if you use large inductive loads because they have a very low power factor, all the way down to zero for 'perfect' inductors.]

In a pure DC setup the interactions between the moving electrons and the stationary material shows up as the voltage across the wires relative to a reference ground (say, a sense wire on the - pole of the battery in the simplest possible circuit, a power source coupled to a single loop of wire without any visible load, the wire is the load). As you move around the circuit you'll see a nice and predictable drop in voltage as the potential of the fields to do work diminishes until it reaches zero at the point of reference. The current in such a setup would be mostly limited by the internal resistance of the source (the wire loop is essentially a short). You'd see the same drop in voltage if you picked the + as the reference, only the magnitude would be reversed, but the product of voltage and current across any given section of wire of a particular length would be constant.

Adding a load is then a slight modification of that circuit, a localized increase in resistance relative to the resistance of the wires, but work is done in all parts of the circuit (including the source!) proportional to the resistance (and hence the voltage drop) across that fraction of the circuit.

In an AC setup some of the radiated energy would 'escape' from the wire because the changing magnetic fields radiated by the wire would induce currents elsewhere, and if you engineer your AC setup carefully (resonance) to maximize such loss you have an antenna. Conversely, that loop antenna would be able to capture an external magnetic field and turn it into electrical energy again (technically: potential energy, the circuit would have to be closed somehow for current to actually flow), such as in an AM radio (which uses a coil of wire across a ferrite core, essentially an electromagnet).

If you carefully engineer this to transfer energy using magnetic fields between two such circuits with minimal losses you end up with a closely coupled set of two loops of wire aka a transformer.

In each of these cases the part that you try to engineer away shows up as loss, imperfect transfer in a transformer warms up the transformer (eddy currents in the laminations for instance) and imperfect transfer for an antenna system shows up as heat due to reflected power (to the point where a very simple way to blow up a radio transmitter of any serious power is to disconnect the antenna).

I think what is happening here is that OP mistook the Poynting vector (a mathematical construct derived from the cross product of electrical and magnetic fields) for 'reality' in the same way that the map is not the territory, it is a very useful tool to show the flow of energy in a system, essentially it allows you to decompose the 'A' and 'B' fields of a system of wires, for instance a current carrying loop and a load.

But it's wrong to say that the Poynting vector moves the energy outside (or even mostly outside) of the wire because you are now talking about the whole system rather than just a single wire and in that system the plane described by the loop of wire is still mostly 'inside' the wire (say: a standard powercord where the conductors are close to each other, but they definitely do not have to be, in fact the wires could be of unequal length all the way down to zero). All the Poynting vector does is to show how strong the cross product of electrical and magnetic fields are at any given point of a system and what the magnitude (power delivered) and direction (where the source is and where the sink is). But that's a simplified view that does not take into account the physical lay-out of the system, the interior resistance of the source and the interior resistance of the wire and if you don't take these into account you end up with a system that is not realistic and which leaves a lot of real world phenomenon unexplained (conductor losses, source losses, radiated EM field), each of which would require their own miniature Poynting vector to account for all of the imperfections.

So by switching from one wire to two wires and a load the picture became needlessly complicated, and technically incorrect. You don't actually need a separate load to demonstrate any of this (the resistance of the wire is a load!). And then all of the rest of the bits fall into place much easier, after all the work done by the magnetic field does not depend on in which direction the current flows and doesn't even care about the direction of the field itself, it will simply excite the atoms of the material that the carrier is made out of by virtue of moving the free electrons around and it will do so proportionally to the resistance of the carrier resulting in heating the material up (which in turn shows up as excited atoms with electrons in a higher oribit, and gives rise to new photons when they fall back, IR ones...).

This is is self evident: if it didn't you would violate the conservation of energy rules, you can't excite the atoms in the material without energy transfer of some sort, and in this case the energy transfer at the quantum level is mediated by the EM fields of the free electrons interacting ('bumping into', if you want to use a simplified view) with the EM fields of the electrons in the material itself. The more such moving free electrons (the larger the current) the more such interactions and the higher the voltage the taller the stack of such interactions, hence P = I * V (momentary) and over time that translates into energy delivered. If the energy really did flow 'mostly outside of the wire' then the wire wouldn't heat up, it heats up because of the fields of all of those electrons interacting with the fields of the captive electrons in the material just like a mirror absorbs light and then spits out brand new photons (the basis for the photo electric effect, think of a solar panel as a 'one way' mirror that turns those absorbed photons into free electrons which are then captured by a diode resulting in a potential difference across that diode). It also explains superconductivity: in a superconductor the current flows unimpeded because the number of interactions between the fields of the electrons moving through the material and the fields of the atoms of the material itself is extremely low, so there is no 'work' done in the conductor itself and hence no potential across the superconductor. If you use a superconductor to short a power source then all of the energy will go into the magnetic field, rather than that it gets converted into heat (which creates all kinds of interesting complications, the fields are so strong that if the electromagnet isn't constructed very carefully it will tear itself apart).

Interesting discussion. Main takeaway: precision matters.