It's one of these ideas that are universally useful, but get a different name in each context.

Of course it isn't, as any transformer or transmitter/radio would show you. But that's not what this is about. It's about as useful as telling architects to use quantum mechanics because that's the fundamentals of what they are doing or having every electronics design model each and every conductor as a transmission line. That's what Spice is for, to ensure that from an EM point of view the design is valid (and to ensure that you're not radiating EM).

But it's in the end a discussion about fundamentals and as you pointed out circuit theory is so damn useful that it gets you 'close enough for government work'. That there is a whole pile of field theory that you could use to model the same circuit is true but it isn't so damn useful and in practice would just complicate matters considerably requiring every electronics engineer to engage is high level math. Just like we're not going to ask carpenters and bricklayers (and architects) to take into account the fact that Newton 'got it wrong' and we should all be using Einstein's equations instead. But when you are designing satellites you should.

Engineering uses the tools appropriate for the task. If that tool is Maxwell's equations because we're dealing with stuff that should be modeled as a transmission line and the field component is the dominant one then so be it. But for most practical electronics circuitry you can use V/A/R just fine. When modeling capacitors and coils Maxwell's equations are applicable but you can likely still get away with approximations as long as you realize that that is what you are doing. When modeling a complex high frequency circuit things change rapidly and modeling your interconnects as transmission lines makes good sense because that is ultimately what they are and ignoring that aspect will make it much harder to design something that actually works.

So the statement that 'energy does not flow through a wire' makes sense in an EM theory view of electricity, which is fine for pedantry but won't get you places if you are moving bulk charge from point 'A' to point 'B' to get some useful work done. Just like EM theory in turn isn't correct either, the quantum physicist would tell you that your theory is 'just an approximation' and that you are 'doing it wrong'.

For the subject matter, antenna design the field is obviously the important part otherwise you're not getting anywhere at all. For DC to low KHz circuits that do not contain large inductors you will be able to keep things fairly simple. As soon as you start working with inductors you will have to 'level up' in your view of how the circuit works and if you don't you'll probably end up with something that is either sub-optimal or that subtly differs in its actual operation from what you think you've designed.

If you want to build GHz circuitry there is no way that you're going to avoid getting to know mr. Maxwell better, fact by then you'll be pretty intimately familiar with the underlying electromagnetic theory, it's unavoidable.

But for bulk energy transfer at very low frequencies it definitely isn't the skin that carries the energy, if it were you could replace all of your copper with foil and call it a day.

Actually it's the electric field that causes the flow of electrons. The flow of electrons does not create the electric field.

It can't be the other way around, as the movement of electrons does not create an electric field, only a magnetic field (which is then crossed with the electric field outside of the conductor to create a flow of energy).

Electrons do not carry energy in a circuit. They use up some of the energy from around the conductor when they bump into atoms of the conductor, we call this process resistance. In particular they can be used purposefully to convert EM energy into heat, like in a light bulb.

PS. I keep bringing the circuit theory in because it is the reason why people can't believe how electricity actually works. But you can forget about this if it doesn't help, it's not that important.

I literally said "this field is induced by the flow of electrons." (Specifically, the magnetic field. The electric field is due to charge difference between the wires.)

> for real world wires the bulk of the field is constrained within the wire.

No, there is none - zero - nada - electric field perpendicular to the wire within said wire. If there were, the electrons would all flow to one side of the wire -- but, as you agree, they don't for DC, they are distributed throughout the bulk.

(There is a small electric field within the wire parallel to the wire; this is associated with the resistivity and energy loss in the wire, which is not related to power transfer. Its Poynting vector points outward from the wire, not to or from the source.)

Ok so far that is in accordance with how I understand things to work, assuming a 'good conductor' and low impedance for AC or varying current.

> But along the wire's radius, within the wire, there's no net E field.

Because there are no electrons flowing in that direction, and even if there were there would be others canceling that out (this holds on average for thermal noise as well, which would show up as minor current fluctuations), to get an E field at right angles to the wire you'd have to to see a voltage difference between the center of the wire and the outside of the wire, and barring local effects there will be no such thing. Essentially because there is a dead short between those two points (a very short piece of a section of the wire) Right?

> Each electron (and each proton!) contributes a small field (which does extend immediately from the electron-point-source as you say), but the electrons arrange themselves within the metal such that the net field along these axes in the wire is zero. (If they were not arranged so, the nonzero field would push them toward this arrangement. Since the electrons aren't moving along these axes, the field along these axes is zero.) In the intuitive sense -- uniform static distribution of electrons begets a uniform static field among them.

Except for the limit case of a single electron moving in a conductor, that presumably would result in a net (extremely small) field, but for larger numbers of electrons (any practical measurable current) that would all cancel out within the wire perimeter.

> Outside the wire -- assuming the wire has a net charge (positive or negative) -- we do see a radial electric field, because the electrons are not free to move to/from the wire through free space to neutralize this field.

Because the wire perimeter is defined as the area where the conductor ends (which makes me wonder how that transition would work in a material that consists of an outer core that gradually becomes higher impedance).

> This field is much larger,

Right, because it can't neutralize because it is in a dielectricum.

> particularly if the wires are physically near each other

Yes, but that's not a given and any old wire loop has its wires as far away from each other as you want them to be. Coaxial cable is the way it is to get the fields to cancel out so that the amount of spurious radiation from the cable is kept to a minimum (you can also twist the two wires around each other for some of the same effect).

> since it derives from the voltage difference between the wires, rather than across them.

You mean 'wires on both sides of the load'? Somewhere along the line we switched from discussing a single wire in isolation to two wires.

> (And most importantly, the Poynting vector of this field points toward the load.)

So we're now talking about the electrical field between the two wires on either side of the load?

> You may be right that the region outside the wire where energy flows may be small and near the wire

I don't see how it could be any other way but I'm curious as to the results of your calculations.

> but it's definitely outside the wire for the reasons given above

Ok

> and flows between sources and sinks according to the Poynting field, not along any path that's visually represented in a circuit diagram.

Ok, I think I'm finally seeing what you are getting at, you are talking about the net flow of energy from a 'source' (say a battery or a generator) to a 'load' and that energy clearly flows in one direction even if the electrons themselves make a round trip from one side of the source to the other side of the source (well, technically they don't even do that, they may well reverse direction before they get there), and that the field that carries that energy is the field at right angles to the wire rather than the field that you see across the length of the wire.

edit: thinking about this some more: essentially the two wires you are talking about have some of the properties of a charged capacitor, and the electric field is passing between those two 'plates'. That makes me wonder if the field is mostly concentrated between the 'plane' (it can be quite a complex shape) between the two conductors rather than that most of it is in free space radiating outward. After all an electric field normally does the same thing for a charged capacitor, the bulk of the field sits between the plates with only a small fraction moving outside of the envelope of the plates.

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.