Oops getting late, partial reply for now...
...energy is force times distance, so if we differentiate energy with respect to distance, we get something force-like. ...
Sounds nice (and it is - well described by the way, I can understand it!), but there is something in that, as you hint at, which confuses me. Energy isn't force. Potential energy is. Energy is the integral of power. Power relates to time and distance (and velocity obviously). It's all consistent and obvious, but my mind feels like it is having to make a leap of faith somewhere. Could just be me.
...(indeed, over short time scales, nothing can due to skin effect)...
Ouch - is an example of something I knew to the point of second nature, but as soon as I start to reformulate my knowledge with an admittedly hackish dig into the physical nature of things, I completely miss it. Such is the mental disconnect. It may not alter the outcome awfully much (thin wire, fast pulse), but it shows the magnetic effects I was trying so hard to partially ignore, are crucial to the behaviour of the physical system. Depositing a patch of charge (and bringing one near), including how the shielding might work (thanks for the Debye pointer) is exactly what I was trying to consider.
Somewhere amidst the statistical mechanics article on Wikipedia by brain checked out. I was never destined to be a theoretical physicist (terrible at maths), but I like to think I've never had a serious problem understanding the concept of any system (even QED doesn't look toooo hard, I say very optimistically). But this 'field' just explodes, even if it were possible to understand it all - there's just so very much. And I'm probably (as in, obviously) too old.
My confusion, instilled from an early-ish age, is with the physical reality of charge, pressure, current, and energy. Poynting might agree that energy is a concept, not something that anyone can take a picture of (drawing a diagram is not the same as taking a photo). Similar for time, and pressure. Consider a pipe with water at 1000 psi in it, versus a cylindrical hole in an infinite solid made of the same thing: Despite the mechanical configuration of water being the same (compressed to the same degree), the former has potential energy, for the latter it has something that does not exist. The maths is the same, the model is correct, the concept exists the same in both cases, but a seemingly irrelevant change makes it physically implausible in one case.
Ah but does it, really? A pipe at only 1000 PSI is a vacuum compared to deep underground. Energy is relative!
I'm not sure what you're getting at with the infinite solid? Mind, it's not perfectly rigid, the hole expands somewhat under internal pressure, tangentially stretching at the inner surface while radially compressing the surrounding material (how much, is given by the elastic modulus). In terms of the pipe's stretchiness adding to the compressibility of the fluid and thus affecting wave velocity/impedance, the two situations will be different, but the latter will certainly not be the same as an ideal (truly incompressible, perfectly rigid) pipe, there will always be some effect.
Yes, I think I'm saying it does (make it physically implausible). The infinite solid does not permit a "deep underground" (being the point). Nor gravity.
The 1000 psi is absolute, otherwise it would be arbitrary. By that I mean the water is compressed away from its vacuum state, so would be possible for it to know that there is something going on: It would possible to build a vacuum in there (I think). It could be -1000 psi for that matter (well under its tensile strength), but is still a known configuration.
I thought about the various ways it could expand differently, but in this picture the pressure is set to that fixed amount and won't change. What is at the ends is deliberately undefined, which may make that difficult.
My intention is to break not so much relativity, but the ways conservation of energy are handled, specifically that if the energy can't go anywhere, then treating it as "potential" might break (or be unnecessary).
Some commenters (here or on YT - can't find now) have boiled the "energy flows outside the wires" down to nicely intuitive statements that basically go; the space between the conductors is where the potential difference exists, the conductors are where the charge carriers flow, so it has to be a combination of wire and space that "energy" traverses. (I was going to add the example of a PCB with power and ground planes, and say the only place you need go looking for power is in the gap - but that's kind of redundant.)
Except it's worse than that - a location for the potential difference isn't needed, nor its "field strength", it just has to exist. And that is clear from the language, the focus is on force and movement. No one seems to question why power in a chain drive flows "outside the chain" (which is the same kind of situation).
I'd go one step further though; and infer that because energy seems to take a path that occupies either all or no space, and seems to transmit as if there were nothing in its way, it seems not to flow in spacetime at all. There is just distance and time. Kind of like it goes in a straight line but without direction, and chooses where to go based on external constraints. Photons show this behaviour.
Well, hold on a moment. Energy is certainly flowing in the chain -- it might not be obvious how much is there, from just looking at one side of the drive, but considering the complete chain, we can take its velocity (which will be, on average, equal for both up and down sides), and the total tension (i.e., the difference -- the total with respect to a consistent direction, as one side is pulling up, the other down), and there's the power. Clearly the power is contained within the chain!
Or for a more mathematical treatment: say we slice the system in half, between pulleys. One side of the chain flows into the cutting plane, the other side out. Integrate the tension over the chain cross-section (well, it'll be pressure at this point), and multiply by velocity. Now we don't need to look at chains or belts under tension, we can do it for any mass flow: the crack of a whip, or fluids in a pipe (or not, like a waterfall). And, as long as our cutting surface is closed (an infinite plane can be seen as a facet of an infinite sphere, or we can make a smaller box around a source or load of interest), we'll always have the correct total; we'll never miss the return path of a hydraulic pump for example, or when fluid is spraying out onto the floor. (Not that it's necessarily easy to account for such flows, like evaporation and ground-seepage of water in the environment -- just that, in principle, it will be in this way.)
And, voila, that's how you use a Gaussian surface, you look at the total flux in/out of the surface, and that corresponds to the total contained within.
Well, if we do the same thing with the circuit, we find a superposition of two things:
1. DC flow in the wire,
2. AC flow around the wire (and along its surface).
The Poynting vector is just the quantity we integrate when we want to find total power flow. How it's distributed spatially, depends on which case we're checking; both are valid in general!
In my first post I also agree energy is flowing in the chain (in that case, a string). But then I don't. It is this latter case.
Without providing for the return force (which is in the axles etc, had to add a "kind of" to cater for that), the system can't function at DC. Power delivery is via transverse pressure (potential energy). But the integral you propose (over the chain cross section) misses this.
Integration is saying that we don’t know (or choose to ignore) the locations of the components of this flow, if over infinite distance along the potential gradient (1D) then this location is everywhere or nowhere if it misses it in a particular reference frame, or if 'heading off at the pass' over a smaller closed curve like a circle, then it's impossible to remove the return path (from having its specific location ignored).
Truncating (whatever the word is) this integral is a little synthetic, because integrating pressure*velocity at each location builds in an assumption that the contributions to total energy flow are coincident in this way (I know they are mathematically, but in principle the point of a closed integral is to avoid these exceptions). It can only ever show energy going through a point where there is flow in some reference frame, and if my claim is that a component of that energy can go elsewhere, then this can never falsify that claim (or principle). All it can do is show half the power flowing in the chain and half flowing in the structure (in my mind it was horizontal and bottom part loose). I guess I've never trusted mathematics. You differentiate energy to get potential, then integrate it again to get energy (or power) - what could go wrong?!
Anyway, I am trying (and failing) to stick solely to DC. I'm trying to pretend Maxwell's equations don't exist, which I was aware is a losing battle to start with. So my attempt to say energy flows outside the wires is purely this principle of pressure difference, and not what Maxwell and Poynting would say. I brush that aside in a "well they would say that, wouldn't they?" manner because it's a self-fulfilling prophecy (the theory provides an exact location for the energy flow) as the only reasonable solution.
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