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The present language is "The magnitudes (lengths) of the vectors are not shown accurately; only the directions are significant." which is accurate but vague. The proposed language is "In reality, the electric and magnetic field strengths decrease as with distance r from the wire, so . Hence, most of the energy flow takes place in the close vicinity around the wires." which is more precise, but in accurate in detail. The fields only drop off as 1/r close to the wires. So describing it that way over the entire surface is inaccurate. Secondly, if it were true, then power density would drop off as 1/r², but area increases as r², so total power flow would be roughly constant versus distance from the wire. Constant314 (talk) 18:56, 21 November 2021 (UTC)
"The fields only drop off as 1/r close to the wires." − actually, when you go far away (exceeding the distance between the two wires), they fall off as 1/r^3 − much faster. See an article on the dipole field. And of course, this is the reason why electric wires always go in pairs - to minimize fields and, hence, interference effects. "so total power flow would be roughly constant versus distance from the wire". What is your "total power flow"? Please give a definition. Something "total" is a constant, so I don't understand how it can depend on anything... I guess you meant "cumulative" - an integral from zero (actually, from - the radius of the wire, since inside a perfect conductor the electric field is zero) to some finite r. In that case, it is . Now, let's assume the wire is 0.5 mm in diameter and the distance between the wires is 10 cm (based on the picture). Then the power flow within a "tube" of 5 mm from the wire will carry , to be compared to the total of about (up to a same constant factor for both values) - that is, within 1% of the cross-section, more than 50% of the energy flows. But really, is the article about practical advises on home wiring or its intent is to teach physics? When you study physics, idealized models are used all the time − like point-like bodies etc. Similarly, an ideal conducting wire has zero (negligible) diameter. In which case, the whole power flow is concentrated in the infinitesimally thin surface of the wire. Evgeny (talk) 23:41, 21 November 2021 (UTC)
Listen, your math looks correct but I'd be careful how you describe it. What you're saying is that about half the power flow is within a distance which is the geometric mean of the wire radius and separation. So yes, it is more concentrated close to the wire. But not close in terms of the wire's size. So when you do let r_wire -> 0 then the power is closer to the wire, but the, let's say "median" power position, within which half the power flows, isn't closer to the wire (in terms of the wire's diameter) but further from it -- so I wouldn't describe that as "close vicinity" of the wire. Anyway talking about wires having zero diameter even as an idealization isn't appropriate for this type of problem -- for instance the characteristic impedance of coax which follows a similar log law needs a finite wire diameter and similarly does not just approach a limit when you let r->0 (which would be a perfectly good idealization for some other types of problems). What I wrote before may have been misleading too and I apologize for my hasty revert, and I guess you could say the power is concentrated near the wire when they are quite separated. Such a statement could more apply to the conductors in a high voltage overhead line, whereas I just measured a typical cable and its separation to wire diameter ratio was more like 2.5, and if you ask people what they think of when you talk about 2 wires delivering power many would think of a cable rather than an overhead line. So I wouldn't make any such statement without specifying the conditions sufficiently, and in this case it would be distracting to do so unless it were a worked-out problem (illustrating how you'd compute S).
But back to the subject of editing Wikipedia, I think a diagram with arrows showing a vector quantity should really be drawn correctly and not just with arrows having the same direction as the vectors being considered. A diagram, for instance, showing the (uneven) power flow inside a coax cable (mathematics simpler with circular symmetry!) might be useful. Interferometrist (talk) 00:49, 22 November 2021 (UTC)
And actually I see now we DO have a diagram for coax, but it could be made better by having a larger R_2/R_1 ratio and with red crosses of different sizes (etc.) illustrating the nonuniformity of S over the cross-section.Interferometrist (talk) 00:56, 22 November 2021 (UTC)
"What you're saying is that about half the power flow is within a distance which is the geometric mean of the wire radius and separation. So yes, it is more concentrated close to the wire. But not close in terms of the wire's size." C'mon, this picture is not a scientific drawing, but a cartoon. So I'm talking about normal, visual perception. Ask any kid on the block whether the mark of 10, on the scale from 1 to 100 is a) low grade, b) average grade, c) high grade. You know what the answer would be. Furthermore, what matter is indeed the area (or volume), in which case the ratio would be 1:100:10,000. It's the same geometric mean, of course, but our eyes don't work in the log scale. Also, I don't understand how this discussion is related to a coaxial transmission line (or indeed a realistic, twisted or not, pair). I never claimed anything like that. If you don't like the word "vicinity", please choose a better one, preferably without making the caption too long. Maybe this topic indeed deserves a separate section instead of an anonymous floating picture. But meantime − please, let's fix this misconception. It's being widely spread. E.g., see https://www.youtube.com/watch?v=bHIhgxav9LY, around 8:00 (there are other, plain wrong claims there). In just 2 days, it has more than 3 million views and ~200k likes. I cannot exclude that this specific cartoon inspired it. Evgeny (talk) 08:46, 22 November 2021 (UTC)
What misconception? The caption says the length of the PV illustrated is inaccurate. That does not promote any misconception. The coax case is mentioned because it has a known closed form solution that is simple enough to integrate in cylindrical coordinates. If you want to make a mathematical calculation of power versus distance, both Interferometrist and I could validate the math. Any statements about other geometry would need a reliable source. Constant314 (talk) 17:55, 22 November 2021 (UTC)
The misconception is that the power flows more or less homogeneously through the space, as implied by the cartoon. And no, the caption does not say the Poynting vector magnitude is inaccurate. It says it is unimportant − which is exactly the source of misconception. If it is unimportant, one doesn't care. And I believe when something changes by orders of magnitude, it is important. Evgeny (talk) 18:58, 22 November 2021 (UTC)
The caption says "The magnitudes (lengths) of the vectors are not shown accurately; only the directions are significant."Constant314 (talk) 22:42, 22 November 2021 (UTC)
I fail to see how this contradicts to what I said. I also fail to see why you strongly refuse to improve the clarity of the text. Evgeny (talk) 08:56, 23 November 2021 (UTC)
I am not against clarification. I just don’t believe that the picture of the battery and resister can be fixed with words. Certainly, the attempts so far have failed. I am fine with saying that the PV is strongest near the wires, but I am against anything more quantitative without a reliable source or a calculation that I can verify. Constant314 (talk) 17:24, 23 November 2021 (UTC)
What exactly do you want to verify? The 1/r dependence of B or E near a wire? Evgeny (talk) 17:55, 23 November 2021 (UTC)
I don't know what is being argued about here anymore and I don't have time to watch youtube nor is it relevant to Wikipedia. I don't think the solution for a wire pair is or ever was part of the Poynting vector page and this all had to do with a few words in the caption of an included figure, and mainly I think that figure needs to be redone so that the disclaimer about the (mis-)representation of vectors can be omitted. The only reason I mentioned coax cable, 2 conductors which are coaxial rather than separated as in this picture, is that it's a MUCH easier problem to handle which is probably why someone DID include a section on coax as an example, but regrettably didn't go on to just compute S(r) for it which would have been pretty straight-forward and which I think WOULD be a useful example (and which could be displayed graphically much easier than the 2-wire diagram with properly scaled vectors). The only other simple example (actually simpler, almost trivial) would be two ribbon conductors with a gap in between them much smaller than the ribbons' width (-> constant S within the gap, zero well outside the gap).
But apart from editing Wikipedia, I'm happy Evgeny brought up the issue because now I see that around a wire of radius 1mm (and with the return conductor at a large distance) that the same amount of power is conducted between the wire surface and 2mm radius, as between 2mm and 4mm radius and as between 4mm and 8mm radius etc. And that this progression continues until the E and H of the return wire start interfering at which point S increases beyond the 1/r^2 by up to a factor of 4 (at the midpoint between the two conductors where both E and H are doubled) or decreased in regions where the directions of the two fields start to cancel. And that half the power flows within a distance of the conductors which is the geometric mean of the wire diameter and separation times a factor of order 1. So I learned something, which is one nice by-product of Wikipedia editing and interacting with others doing so as well :-) Interferometrist (talk) 21:50, 22 November 2021 (UTC)
