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This page had the naming of the B and H field wrong in a couple locations. I changed them so that they are consistent with both the rest of the page and the magnetic field page. Hopefully this will reduce confusion of their naming in the future.
Commendable work. For an extra flourish, you could assume the load is space cloth with a resistivity of Ω/□ and show that the power dissipated in each is the same as predicted by the Poynting vector. Constant314 (talk) 00:21, 24 November 2021 (UTC)[reply]
Nice work indeed. But I think there is no need to mention permittivity and permeability of the materials - they influence nothing (and don't appear in any formula used). Evgeny (talk) 17:09, 24 November 2021 (UTC)[reply]
You're right, they don't appear in relation to the main form (ExH) but are still hidden in the definitions of H and E relative to "free" currents and charges. And also (more on this later) there is an entire section on the ExB form which goes so far as to call it the general expression of the Poynting vector! And also a mention (needed?) of the constitutive relations between EDBH under totally unrealistic (or just for low frequencies) conditions (I take non-dispersive to mean "instantaneous"). So it's already mentioned, and hidden, but yes, needn't muddy the text where it can be ignored. I should mention I created the figure before writing the text at which point I realized that epsilon of the dielectric pointed to in the figure had no place in the explanation and proof I wrote up.
This also touches on Constant314's mention of "space cloth" termination, and I think I understand what he had in mind in reference to waves along a transmission line, but which he didn't fully think through in the present context. Nor would I have seen this exactly before writing the analysis and let me confess that just because I'm writing about a subject doesn't mean that I jumped into this knowing everything about it just needing to transfer my knowledge to paper. Although I always approach editing a Wikipedia page thinking that I'm going to (and hopefully do!) help educate the next 10,000 readers of that article, especially with a subject like this I am also learning at the same time (and even if not learning something new, at least increasing my familiarity with the subject and writing down equations I might not have looked at for many many years).
But what I realized quickly when writing this, is after I limited the analysis (for AC signals at frequency f) to instants << 1/f and lengths of coax << λ=c/f, the concept of a transmission line's characteristic impedance fades into nothingness. Everything you see, applying to some instant, is in terms of the instantaneous V and I with total disregard to how these change in time either individually or with respect to each other, or in space as you go along the transmission line not <<λ. Indeed it works at DC (like the figure it replaced with the battery). It is only an analysis which DOES consider full wave periods and wavelengths when you can talk about "reflected power", standing waves, propagation constants, and characteristic impedance. These DO change depending on epsilon (and mu) but the validity of the Poynting vector can be easily ignored under the above stated restrictions. (I hope I gave you something to think about. I have!)
Page still needs a lot of work to flow well, not to mention completeness and (in a couple cases) accuracy, but I tried to put the more important and solid subtopics toward the top. Interferometrist (talk) 19:53, 24 November 2021 (UTC)[reply]
Yes, AC makes everything more complicated. But all my comments/concerns so far were strictly with DC in mind. For AC, and even more so for pulsed power, permittivity is very important. Maybe the title of the example section should mention DC? Evgeny (talk) 20:58, 24 November 2021 (UTC)[reply]
Look, you're thinking in electronic terms: that a capacitor in a DC circuit doesn't change anything in the steady-state. But for the Poynting vector even in DC it makes a difference if you use the DxB (or DxH) form (as we don't, but the idea is still out there). The 4 possible forms are a big complication I'd rather ignore, but can't really. My eyes are tired, but could someone else look at the ExB section? It seems wrong. I looked at a paper long ago which goes through all 4 forms and says exactly what each is good for and more importantly how the energy density & dissipation are defined in each case -- not very interesting for me right now but it's within the subject matter of this page. Interferometrist (talk) 21:37, 24 November 2021 (UTC)[reply]
My understanding is ExH is the right form. If you take D instead of E and repeat the same calculations, you'll end up with a bogus extra energy. Also, I don't think B vs H would make any difference here anyway (unless you mean that the dielectric has non-trivial magnetic permeability). The electric field inside the conductors is zero, so is the Poynting vector... Evgeny (talk) 22:04, 24 November 2021 (UTC)[reply]
I never felt like this discussion was bickering. It might have been tedious. That tends to happen when the participants have different sleep cycles. I regard it as a healthy, productive, respectful discussion that led to an improvement in the article. Every comment whether incorporated into the article or rejected helped to lead to a better article.Constant314 (talk) 14:58, 25 November 2021 (UTC)[reply]
@Constant314: Sure, sure, I wasn't being totally serious! Indeed the discussions are generally helpful not only for the original purpose of agreeing on the wording of the encyclopedia but also is educational or at least thought-provoking for the editors, whether or not that gets reflected in the article seen by the public. I guess by "bickering" I was more thinking of discussions which degenerate into people repeating the same points without converging, or disputing the philosophical meaning of a scientific principle or equation without being able to identify an experiment that would settle it. I guess I might have been thinking more of the Loop antenna page which I will look at again sometime soon after I've completely forgotten what we were arguing about (and fortunately I have a short memory ;-) Interferometrist (talk) 18:39, 25 November 2021 (UTC)[reply]
I don’t think that I have ever been guilty of thinking something through entirely. However, I believe my suggestion about space cloth is spot on, except for the incorrect subscript on the permittivity. When a lossless parallel transmission line is terminated by space cloth, it will be terminated by its characteristic impedance at all frequencies including DC. That means that there can be no reflection which means that the PV must be absorbed entirely, with none reflected of allowed to pass through. The energy absorbed when computed by the PV times the area of each ρdρdθ element must be the same when computed by the product of the voltage drop across the element and the current through the element. This works, in part, because once you specify the termination, you establish a relationship between V and I.Constant314 (talk) 15:13, 25 November 2021 (UTC)[reply]
@Constant314:Well I understand what you have in mind and we've both worked on articles where wave termination by space cloth (or just a resistor for a cable) has relevance. But no, I don't think you thought through the difference between those topics and the issues involved here which concern the local continuity of energy flow on arbitrarily small time and distance scales. That is why the example with coax involved simply a single voltage and current over a tiny time interval and tiny length of coax without any specification of the frequency of the oscillation (including DC) or wavelength (what's happening further down the cable). On that scale, the "wave" picture (which does depend on frequency, wavelength, and propagation velocity) loses its meaning and we are left with E field/voltage and H field/current. There is NO given relationship between V and I implied; these are independent. Sure, you want to and can resolve these into the sum of a forward and backward propagating wave, since that's the way coax is normally used and is evident in standing waves when you look further down the cable or in a ratio and phase between V and I when you look forward in time. And I believe the power of those waves are required if you think you need to express the energy in terms of photons traveling in both directions, but of course ALL the articles in electromagnetics are within the realm of classical physics and photons (however you conceive them, I could tell you how I feel about the subject but then I'd be defying my own dictum in the above paragraph!) do not enter into the picture. So discussing waves (in terms of V and I) along the coax is an unneeded and distracting complication for a problem concisely specified by V and I applying to a small space and time interval. Yes you could put space cloth, or a resistor, of ANY value at the right side of the coax segment and all you have done now is constrained V and I to that resistance but there is nothing in the analysis that ties that space cloth or resistor to what you would normally compute as the characteristic impedance of the cable or dielectric medium. At DC (where the coax analysis is equally valid, as the previous figure had specified) the constraints I mentioned on the allowed spatial and temporal extent of the analysis disappear and the concept of characteristic impedance becomes irrelevant, you can transfer power using arbitrary I and V. (And in case you're thinking of the surge response, note that a step function is NOT DC, it's a mixture of frequencies as you well know). Agree? Interferometrist (talk) 18:47, 25 November 2021 (UTC)[reply]
I agree with most of that. I am suggesting adding a special case to your fine work and that is the case where the load is space cloth. The nice thing in this special case is that your expressions for E, H, and PV apply all the way to the load since a lossless parallel transmission line terminated with space cloth is the same as the same transmission line terminated with and infinite line of the same type. Part of the old diagram that was lost showed the PV being absorbed by the load. This special case would bring that back. It also nicely ties classical EM to circuit theory by showing that the power distribution computed from PV is the same as computed by circuit theory, in this special case. Of course, that should be true in all cases, but in most cases, calculating the PV in the vicinity of the load is intractable. Constant314 (talk) 19:53, 25 November 2021 (UTC)[reply]
Well I haven't thought this through. My initial reaction, identical to what I was saying earlier, is that this would only make a difference if you were to turn the problem into a longer length of coax driven with AC over more than a tiny fraction of a wavelength. Now I can see how space cloth would absorb a plane wave in free space of any frequency, but the wave inside the coax is planer but not (part of) a plane wave since its amplitude varies over the cross section. But indeed its wave impedance E/H matches space cloth at every point in the cross section. I'm trying to see (in the DC case, for instance) what difference it would make if you used space cloth that did not match that impedance. Maybe it'll come to me in the shower or something... Interferometrist (talk) 21:15, 25 November 2021 (UTC)[reply]
Actually, space cloth will not absorb a wave in free space. However, when you use it to terminate a parallel conductor lossless transmission line (TEM mode) it will be a perfect termination. If you use space cloth of a different resistivity in the DC case, I'll have to think about that. Crawford in Waves, Berkeley Physics Course Volume 3, around p. 230 discusses why space cloth will not absorb a wave in free space. Constant314 (talk) 21:25, 25 November 2021 (UTC)[reply]
So you're saying that "space cloth" is just a clever way of finding the correct terminating resistor for a conductor geometry when you know the wave impedance inside/around the conductors is eta_0/sqrt(eps_r)? A different way of computing a cable's characteristic impedance? But geometrically more symmetrical than a single resistor from the center conductor to an arbitrary point on the shield, so it would make a nicer figure? Is what you think would be nice to show is the Poynting vectors at each point impinging on the space cloth and ending there? Interferometrist (talk) 21:47, 25 November 2021 (UTC)[reply]
Yes, to all of that. The space cloth idealization reduces the calculation of characteristic impedance to the calculation of the resistance of a two-dimensional surface, which is easily solved numerically by the Relaxation (iterative method). But the reason I am suggesting it as a special case is that it would be easy to draw an accurate depiction. Constant314 (talk) 22:00, 25 November 2021 (UTC)[reply]
It would be useful if somebody would add a line of text describing the application of the right (if that's correct) -hand rule to the E, H and S vectors. This would fit nicely in the "Definition" section 90.241.239.175 (talk) 14:25, 21 March 2023 (UTC)[reply]
@Constant314: Thanks for having a look at my edits https://en.wikipedia.org/w/index.php?title=Poynting_vector&diff=prev&oldid=1250154748 :) You mentioned that the new section is redundant with the coaxial cable. Could you lay this fact out for the reader in the article? To be honest adding a second coaxial conductor around the DC wire changes quite a lot (we know that coaxial wires behave different to simple wires which is obvious with AC). For me the direct redundancy to a coaxial cable is not clear. I therefore would like to see a discussion of either a) the DC wire or b) how the DC wire is directly and undoubtedly the same as the coaxial cable. Please keep assuming my good faith, because I really just do not see the equivalence in b) but I am happy to learn.
I hope I did not read over the explaination, but also in this case I find the image of the above circuit quite good to catch attention of the unfamiliar reader. Please consider these thoughts with the best intentions :) Biggerj1 (talk) 06:13, 9 October 2024 (UTC)[reply]
@Biggerj1 I noticed your commit when it happened, and I noticed it being removed. I personally have a strong interest in the magnetic field of a bare DC wire. It might be relevant to dendrite currents in the brain, or something like that. So I thought your text and diagram were a really good idea.
However, I'm just an ordinary editor without extra powers. So I can't do anything about it. However, I encourage you to persevere. Cases which are apparently too trivial often give great insight into more complex cases. Alan U. Kennington (talk) 06:22, 9 October 2024 (UTC)[reply]
You let the radius of the outer conductor become arbitrarily large and what is left is essentially a single wire. Also, the coax case includes DC. Constant314 (talk) 10:49, 9 October 2024 (UTC)[reply]
I think for the reader it would be helpful to add this statement explicitly in the article, it is not that obvious. Would you do it? Also what about the image of a DC circuit? I do not feel the Poynting vector construction with surface charges etc. is not so intuitive ..., so explicitly showing it is helpful Biggerj1 (talk) 12:23, 9 October 2024 (UTC)[reply]
@Biggerj1 I agree with you 100%. It takes just a few words to state the facts rather than making the reader spend time and effort trying to work out what the limit of an infinite diameter coax cable does. When parameters of systems are made infinite, they don't always converge to the limit you expect. It's non-trivial to show correct convergence. Alan U. Kennington (talk) 12:29, 9 October 2024 (UTC)[reply]
@Constant314 Since I am a mathematician who has worked a lot in the area of analysis of limiting processes for boundary value problems, which the EM fields of a coax cable are, I'm very aware of how necessary it is to give formal proofs. (Only mathematicians will read the proofs though!) I used to talk with engineers who refused to prove things which they considered to be intuitively obvious. But some "obvious" theorems turn out to be false. (This happens with Fourier analysis for example.)
But anyway, on the subject of this particular case, I think it would be very instructive to the reader if they were told (a) what the fields and Poynting vector field are for the simple cylindrical wire with DC current, derived from the boundary value problem in the steady state. (Such computations don't need to be presented in detail or at all.) And then (b) an insightful comment could be made that you get the same answer for the cylindrical wire case as you get in the limit for a coax cable with outer sleeve diameter tending to infinity. For experienced readers, this would be sort of obvious. But it should stimulate the beginner to think about whether infinite limits always give the right answer. (There are many situations in theoretical physics where direct computation is impossible, and only the infinite limit method is possible.)
So I'm suggesting that only two little sentences are required. Once sentence states what the fields are in the DC wire case. And the second sentence says that this can also be obtained as the limit when the diameter of the coax sleeve tends to infinity. This would not be an obvious equivalence to beginners. And encyclopedias are supposed to be readable by beginners.
Perhaps go ahead and add the sentences. My concern is that you cannot define the field around a wire unless you define the boundary conditions. The only easy boundary condition is when the wire is surrounded by a coaxially oriented cylindrical conductor for the return current (a coaxial cable).
I realize that when you take limits to infinity that sometimes you get an unexpected result. I did compute the limits a long time ago and found no surprises. I certainly could have made a mistake when I did that. I will look for a reference.
I disagree with "necessary it is to give formal proofs." That usually is not done in Wikipedia articles, except articles about a particular proof or theorem. However, I do not mind discussing it here on the talk page. Constant314 (talk) 14:26, 9 October 2024 (UTC)[reply]
@Constant314 Yes that's a good point about what the boundary conditions are in the infinite case with just the bare wire. That's what I was alluding to regarding physics models (e.g. quantum field theory) where it is not actually possible to solve the problem for an infinite universe. The tricks they get up to are best not discussed here, except to say that they are weird and wonderful. E.g. pretending that the world is a sphere, renormalisation etc.
However, I think that for a piece of wire, one can assume at the mythical "points at infinity" that the field tends to zero, and that the field is constant in time. I haven't solved such things recently. So I can't say much about it specifically.
The PDF document which @Biggerj1 cited is really very interesting and worth reading. There's a long quote from Feynman, and this short comment.
However, the result of such an application and the resulting energy transfer in the circuit apparently did not satisfy Feynman. He wrote: ‘‘this theory is obviously nuts, somehow energy flows from the battery to infinity and then back into the load, is really strange.’
The whole PDF document [1] seems to be about the craziness of the infinite case. I think it so fascinating, in fact, that it deserves a whole section on the wikipedia page. The very important lesson for the reader (including for me) is that in unbounded spaces, crazy things happen. It raises important questions about what exactly the Poynting vector is physically. And that, I think, would be a great addition to the article.
So in my opinion, the best thing would be for the text and image of @Biggerj1 to be reinstated, and then a summary of the "craziness" should be added, with possible resolutions.
@Constant314 Sorry I disappeared off the map. I had to get some sleep because I'm in Australia. I think I've written enough about this issue, except to say that in wikipedia one should spell things out explicitly, unlike in advanced textbooks and research papers. A large proportion, maybe the majority, of readers have expertise in different areas and just want some information quickly. So I'm in favor of spelling things out, even if there is some repetition. Recently I've seen huge amounts of repetition in the neurophysiology wikipedia pages, but I think that's a good thing. I'm a beginner in that area.
In my mathematics area, we give the general theory followed by utterly trivial cases as a kind of "sanity check", even in research papers. In the case of the Poynting vector field for DC current in an infinite wire, at least the field has a simple form, as in Electromagnet#Physics. So it's a great example for checking out the "sanity" of the definitions.
Therefore I think the simple DC current in an infinite wire case should be spelled out explicitly. Just my opinion! (Now I have to drink my coffee and have breakfast.) Alan U. Kennington (talk) 01:21, 10 October 2024 (UTC)[reply]
Not disagreeing, but explicitly what do you want to add? @Biggerj1: added one sentence: "The concept of the Poynting-Vector can also be applied to direct current flowing through a wire." That is not enough to be helpful. And this picture is not correct since E outside the wire should be radially directed. While I appreciate the intent of the edit, as it stands it is not an improvement. The article can be improved, but it is going to take some work. Constant314 (talk) 02:39, 11 October 2024 (UTC)[reply]
Hi Constant314 :) I essentially created the image in 1:1 analogy to the cited paper which discusses it in this way. The paper contains more information about the reasoning - since this is the simplest model it has to have some flaws. I would prefer to increase the modelling complexity from simple to more and more involved: the reasons why you needed to come up with a more sophisticated model is typically a very good leaning (and I think you brought up a good point). Still: simplistic models can be a good starting point for discussion - so I would still add it and say why it is simplistic and needs to be improved. Also Feynman said: "obviously this is nuts"... Do you know what I mean? Since many people would think that the flow of electrons is responsible for energy transfer, bringing up this simplistic model which demonstrates energy transfer through the fields is super important (if you ask me) Biggerj1 (talk) 05:56, 11 October 2024 (UTC)[reply]
@Biggerj1@Constant314 This whole discussion has made me lose confidence in the meaningfulness of the Poynting vector. So I think the simple case of the DC current in a linear wire must be presented, and the consequences explained. It does seem doubtful that energy is somehow travelling through a vacuum when the electric and magnetic fields are both constant. If it can't be made to sound reasonable in a very simple case, one can't have much confidence in the meaning in a general case. The mathematical theorem is doubtless correct. It's just the interpretation that needs explaining, and a trivial example is the best way to do that. (By the way, I've forgotten most of my 3rd year Uni physics. So I'm struggling with the Poynting vector. That's why I looked it up in wikipedia in the first place.) Alan U. Kennington (talk) 06:15, 11 October 2024 (UTC)[reply]
With regard to Feynman, you need to read more of the lectures to know how he thought. It is important, that the Feynman Lectures are not a textbook. They are literally lectures presented to a live audience and transcribed into a book. Unfortunately, we don't get to see his face, hear the lilt in his voice, or experience the other non-verbal communication. Feynman would occasionally make provocative and sometimes incorrect statements to engage the audience. Usually, by the end of the lectures, he would resolve the provocative statement or give you enough to resolve it yourself. You cannot just lift a quote from the Lectures and count on it being literally correct. In this case when Feynman says, "this is nuts," I think he is empathizing with the typical undergraduate's reaction to the material. Later, he goes on to elaborate that there is no experimental proof that there is energy out there flowing in the field, but states that it is pragmatic to assume that the energy is out in the field.
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