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Interpretation and application of Poynting's theorem?

  1. Sep 1, 2012 #1
    In several recent threads the Poynting's theorem was brought up, and the discussion there became a distraction from the questions at hand without really being solved:
    https://www.physicsforums.com/showthread.php?t=621018 (starting #74)
    https://www.physicsforums.com/showthread.php?t=628896 (this post continues from the discussion starting with post #198)

    Roughly, the theorem states (I think) that the energy that dissipates as heat inside a volume equals the energy that enters it minus the stored energy that is extracted from it.

    A basic issue is the meaning there of the term E.j.
    According to some, the energy transferred from EM fields to matter (work) is E.j.
    However, the texts that I consulted specify that it is the Ohmic heating rate, or the dissipated energy inside the volume under consideration. Clearly there is some ambiguity here.

    A paper that appeared a few years ago in Europhysics Letters 81 (6): 67005 (thanks Wikipedia!)
    - http://arxiv.org/abs/0710.0515 could be helpful.

    Clarifications are welcome! :smile:
    Last edited: Sep 1, 2012
  2. jcsd
  3. Sep 1, 2012 #2


    Staff: Mentor

    In the reference you cited please see page 2, second column, first paragraph, in the description of equations 7, 8, and 9. E.j clearly refers to mechanical energy in that equation, which is appropriate at the microscopic level.

    I believe that you are simply misreading your other texts. Most likely they discuss Ohmic dissipation, but don't even mention mechanical work in their derivation of the E.j term. You are interpreting that to mean that the E.j term applies only to resistive losses. The absense of an assertion about mechanical work should not be taken as an assertion of the absence of mechanical work.

    Please see point 3 in the derivation of Poynting's theorem in the Wikipedia entry I have posted multiple times now:
  4. Sep 1, 2012 #3
    Sorry I forgot to include the Wikipedia reference to that paper:
  5. Sep 5, 2012 #4
    In the other thread still more on this topic was added:
    That looks interesting, and it also refers to "ambiguities". :smile:
    I have in mind to check out the references and then I'll give feedback of how I understand those.
    Last edited by a moderator: May 6, 2017
  6. Sep 7, 2012 #5
    OK, I now have two precise questions about this.

    1. It appears to me that the distinction that http://arxiv.org/abs/0710.0515 makes between Eext and Eind could be essential for a good understanding. I assume that Eext refers to externally applied electric field strength and Eind to induced electric field strength.
    It's the first time that I see that distinction being made, and it was also not brought up in the discussions from which this topic is a spin-off.

    That same distinction is made in http://my.ece.ucsb.edu/bobsclass/201C/Handouts/Chap1.pdf [Broken] : There the author uses Einc for incident or applied field, and Escat for "a component produced by the induced currents". This agrees in essences with my interpretation of the meaning of the symbols that I saw earlier. And the second text adds the clarification:

    "The distinction between impressed and induced currents is therefore a natural breakdown
    in terms of “cause and effect”."

    In the earlier threads in which Poynting was brought up for an analysis of "what does work", this distinction between Eext and Eind was not made. I think that such a distinction between cause and effect must be made if one wants to use Poynting's theorem for inferring what does work and what does not.

    Does anyone disagree? And if so, why?

    2. The particular question that came up in the thread on magnets, was how we should describe Lenz-Faraday magnetic induction in a zero resistance current carrying wireloop with the use of Poyntings theorem.

    While definitely work is done under displacement, it suddenly strikes me that here the term E.j can only be zero. How does Poynting's theorem look like for this particular case?

    At first sight, following eq.1.31 of "bobs" handout, it looks to me that the energy per volume that is extracted from the magnetic field for doing work equals M.H for the case without external source (like a permanent magnet).

    Is that correct?
    Last edited by a moderator: May 6, 2017
  7. Sep 8, 2012 #6


    Staff: Mentor

    I have been through the whole paper, but not in super detail, but I couldn't find Eext and Eind. I apologize if I need new glasses, but could you point out which part you are refering to. I did find a reference to jext and jind, could that be what you are refering to?

    However, I don't understand your point in general. The paper you cited re-derives and confirms the usual microscopic Poynting formula. It is not a contradiction of the standard formula, but a confirmation of it. In particular, he explicitly contradicts your idea that the E.j term represents only Ohmic heating.
    Last edited: Sep 8, 2012
  8. Sep 8, 2012 #7
    Against my better judgement, dive in to imo cut to the chase, and offer yet another reference: http://arxiv.org/abs/1208.0873 - see esp. sect. 8. Quite along the lines I argued 'back there'. (another paper by that author generated quite some flack awhile back, but that was a different paper.)
  9. Sep 9, 2012 #8
    You are right - so where the one text makes the distinction in the J, the other makes it in the E!
    I interpret E.J as a whole, in which case it doesn't matter. Does it matter? If so, why/how?
    I just discovered that there are at two E.j terms that are distinguished; it's thus important to know what someone means with "the" E.j term, and it may well be that some discussions (+derivations) only consider one of the two. What does Ohmic heating correspond to?

    PS. my point in general here is that "Poynting", at least the generalised version, must be consistent with any direct case analysis such as the one to which I referred in my second question. And, as I mentioned, this seems to work fine. If so, then there remain only a few details about Poynting's theorem to clarify for me.
    Last edited: Sep 9, 2012
  10. Sep 9, 2012 #9


    Staff: Mentor

    I don't know. In principle if you had Eind and Eext and jind and jext then you could have up to 4 distinct E.j terms. So it is just good to be clear what you are talking about.

    The author of this paper shows that "the" E.j term is the sum of the ind and ext terms. That is discussed in the paragraph immediately preceding eq 10. I think you are correct that some derivations only discuss one (implicitly assuming the other is 0), but it appears that the j in Poynting's (microscopic) theorem is equal to the sum of both in general.
  11. Sep 10, 2012 #10
    Yes, it would be good if anyone can clarify if there are two or even four E.j to distinguish.
    I understood from the summary in Wikipedia that the generalization was done later; OK I'll have a second look at that (later!).
  12. Sep 10, 2012 #11
    Thanks, that's an interesting paper.
    Sect. 8: "a static E-field [..] acts on the loop current". That is very different from the ideal permanent magnet case that I discussed, but these examples may still provide guidance. So I'll have look at that (later).
  13. Sep 10, 2012 #12
    My pleasure. And not only interesting, but more directly relevant than the other articles earlier cited.
    Depends what you mean by ideal permanent magnet. If you mean one composed of perfectly conducting loop currents (that was your model?), just remember a permanent magnet made from such could not be a permanent magnet! A real PM is necessarily of radically different nature. In addition to sect. 8, consider carefully the last two paragraphs in sect. 3, all of sect. 9, and sect. 11.
    [It has occurred to me your main concern is possibly the term 'static E-field' - implying an electrostatic field. No - just a poor choice of words there. Further down it is made clear the field has non-zero curl owing to association with a time-changing B. So evidently the sense was 'a steady-state curl E', implying a constant ramp current dI/dt as source of applied E, which can be true only for a limited time span.]

    The main value of that article is in it's confronting the issue of just how different in character is the 'bound current density' Jm = 1/μ0∇×M to that of the other two - free current Jfree and bound polarization current Je_bound. Latter two are real currents and respond as such to an applied E, while the former is not, and it shows as admirably demonstrated in the article. Something Jackson and I presume Griffiths fail to do. I do not agree with the author that hidden momentum is an absurdity. It's quite real but using his preferred formalism it is 'hidden from view'. 'Hidden energy' is an interesting issue all by itself and there is a further aspect not covered by the author there. But let's not go too far astray here.
    Last edited: Sep 10, 2012
  14. Sep 11, 2012 #13
    As you certainly remember, I calculated the classical case of two single zero Ohm current carrying loops as it is the minimal configuration of two permanent magnets.
    I cannot remember that for the simple reason that I have never seen such a claim in a classical physics textbook (and not even in Quantum physics either).
    Once more: I found that an E-field in a circle-symmetric closed loop configuration is an impossibility. If you disagree, please try to sketch one.
    I'll certainly look at it in more detail, thanks!
  15. Sep 11, 2012 #14
    Thanks for clarifying - yes I do remember that one. I disagree that it is a proper representation of two PM's - rather that of two electromagnets, and there is an important difference. As I explained in effect beginning with the #5 post back there and as Mansuripur does in the linked article - energy exchanges are totally different in the two cases. In the electromagnets case, in order to simulate genuine PM's, a source of emf in series with each loop current must act to keep the currents constant - and that means pumping energy into or our out of those coils, something PM's do 'for free'. I want to be clear here the series emf's (via batteries or similar) are exactly countering that owing to the -dA/dt emf induced in each coil owing to relative motions between them. We assumed perfect conductivity so no emf is consumed overcoming ohmic resistance.
    Possibly because they are not concerned with the issue as it has arisen here and particularly in that other thread. We are distinguishing between acting as a source of EM field (and a current loop will do for that purpose - provided unrealistic constraints are imposed), and responding to such fields (very different between classical loop current and intrinsic moment). I made the point explicitly re impossibility of perfect loop currents being able to respond as though intrinsic moments in a number of posts, but this one is good enough. Lenz's law and thus intrinsic diamagnetism applies to classical loop currents - it does not and cannot do so in the case of genuine intrinsic moments. Putting this around - are you aware of any serious text on magnetism that claims it would be possible to create a PM out of classical loop currents? Also, have a look at:
    sect'n. 12.2 under 'What spin is not' here: "folk.ntnu.no/ioverbo/TFY4250/til12eng.pdf" [Broken]
    Sect'n. 23.1 here: "web.mit.edu/sahughes/www/8.022/lec23.pdf" [Broken]
    I'm frankly amazed you believe that! No need to sketch. Transformer action! That would be ∇×E=-∂B/∂t, or it's close cousin E = -∂A/∂t. You are aware lines of A tend to hug the contours of a conducting wire current-carrying circuit? A little closer to the issue here; if one moves a cylindrical bar magnet along it's axis, there is induced perfectly circular closed-loop lines of E, centred on the bar axis. That's why an emf appears in a coil when a bar magnet is moved in-and-out of the coil. I'm starting to see why my earlier inputs, over in that other thread were not particularly impressing you!
    [this is the best link re intuitive feel I can find just now, on the link between lines of E associated with motion of a magnet and associated B field: http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html]
    Last edited by a moderator: May 6, 2017
  16. Sep 11, 2012 #15
    I called case 1 an electromagnet (again the simplest one that I could imagine); an electromagnet has open windings with an external source connected to them. However, the added complexity of pumping energy from outside into the magnetic fields could distract from the basics of "Poynting" that I brought up here. It's irrelevant for the discussion what you call case 2, as long as you understand what case 2 is - but even that appears not to be the case, see next!
    Yes of course: assuming that you are like me discussing classical physics, I already linked to Ampere's classical model (no Quantum here please!). And even your references pretend that they are magnetic, which is all that matters here.
    I only needed to make the sketch in my head - it took a mere 2 seconds. A transformer has open loops like an electromagnet, which breaks the symmetry. Why did you do such a big effort to not make a simple sketch of an E-field along a perfect loop without openings? You'll see that you can't do it, just as you can't sketch an ever increasing time zone shift around the Earth, or stairways with the top connecting to the bottom (you could if your name is Escher!).

    And in order to reduce noise, for your sake I'll rephrase my post #11 as follows (this discussion did not progress for me since that post!):

    Thanks, that's an interesting paper.
    Sect. 8: "a static E-field [..] acts on the loop current". That is very different from the stand-alone closed current loop case 2 that I discussed, but these examples may still provide guidance. So I'll have look at that (later).
    Last edited: Sep 11, 2012
  17. Sep 11, 2012 #16
    Went back to that #216, to the case 2 there, and honestly it never quite made sense then or now - possibly just the grammatical construction. Here it is again:
    No argument here. But then....
    Where the hell do wires suddenly enter? Aren't we talking about PM's?! Or have you switched back to case 1 unannounced? :confused: I guess so. In which case, yes to that point.
    Getting the hang now of your casually switching back-and-forth between Cases 1 and 2, I get your picture sort of: Work is magnetic because of the magnetic field energy changes. True to a point but only part of the picture. As said last time, energy exchanges are different between electromagnet case (genuine E.j work is involved), and true PM's (no or negligible actual E.j work involved).
    But the use of Amperian loop currents in all such serious texts make it plain they are not to be considered what really applies. I refer you to the links supplied last time.
    Did you check out that last HyperPhysics link I gave? Please think about the examples shown there - in particular involving axial motion between coil and bar magnet. What then is your idea of the E field generated when a cylindrical bar magnet moves along it's axis of symmetry? How could there be other than concentric circular E field lines?

    On the matter of transformers having 'open' windings; sure there must be an opening somewhere in the primary if a current is to be driven through it. And an opening somewhere in the secondary if voltage/current/power is to be extracted. But that matters? Lets' suppose for the moment there is something special about that opening that makes it different to the rest of the circuit. There can be 100, 1000, 10,000 turns or any number N turns in principle. As N goes to a large value, how relevant is that there is some 'opening'? Clearly it's relative importance diminishes towards zero. But beyond that, the 'opening', unless a genuine open-circuit (irrelevant because then there is no current to consider) is in fact a continuation of the circuit. Current flows through the opening (as source - generator etc., or load - resistor etc.) to complete the circuit. If there is any current, there is always a complete circuit. Well there is the exception of open geometry conducting objects driven at or near self-resonance being able to carry appreciable currents, but that is far from what we are talking about here. At low frequencies, and moving magnets certainly qualify there, all current-carrying circuits are closed ones. [What's more, even a single turn coil can be easily formed in such a way that the input/output feed wires have no appreciable EM effect - use a twisted-wire configuration or similar; e.g. coaxial.]
    [just caught your edit - yes, please give that article by Mansuripur some further thought.]
    Last edited: Sep 11, 2012
  18. Sep 11, 2012 #17
    Looking at this in a possibly simplistic way I see E.j as being a more detailed variation of the power equation (VI) but being representative of power per unit volume rather than just power.By considering it as a power equation and one which expresses the conservation of energy I find it easier to apply it to real situations.
    Consider the power equation (VI) as applied to a basic motor.At switch on and just before the motor starts to turn Ohms law applies and we can write:

    Vs=IR (Vs= supply voltage)

    As the motor turns and picks up speed an increasing back EMF(Vb) and reducing current(I') results and for any value of Vb and corresponding current I' we can write

    Vs-Vb=I'R (R= circuit resistance) From this:

    VsI'-VbI'=I' squared R

    Ignoring other energy losses eg those due to friction,the above equation expresses the conservation of energy quite nicely.VsI' represents input power VbI' represents mechanical output power and VsI'-VbI' represents resistive power losses.
    If the equation is reformulated in terms of E.j it is easy to see that E and j need to be properly defined.Taking E,for example,there would be an E value which corresponds to the supply voltage,an E value which corresponds to back EMF voltage and an E value which corresponds to the difference between the two and which is instrumental in heating losses.
    At present I look at E.j as being an equation which is just one link in an energy chain.I don't consider it as being a prime source of energy,if there is such a thing,because energy changes can be traced back in time.
    Last edited: Sep 11, 2012
  19. Sep 11, 2012 #18
    Yes of course that is true and Poynting theorem makes that explicitly and implicitly plain. But the seemingly tricky part of interest here arises in interpreting the meaning and application of the 'J' in E.J that applies to the case of so called Amperian currents in magnetic media, in particular permanent magnets. Although the character of the E part has also become under scrutiny of late, but hopefully not for long.
  20. Sep 11, 2012 #19
    How true. Now it appears that those funny little pieces of metal are least understood of all. They probably took us to study magnetism/electricity in the first place! I also realise how little I know about them and how little there’s on the net.

    For example say we have 2 permanent magnets a good distance apart. We then let them collide. The only thing I know for sure is that the combined energy content of the magnets has gone down. But eg what happened to: the flux φ, field strength H, flux density B, inductance L, possible current I, possible spin magnetic moment?

    On the other hand, I am fairly confident I know what happens to their electrical equivalents in case of 2 opposite charged metal plates. I suppose lack of understanding is why harrylin started this thread in the first place.
  21. Sep 11, 2012 #20
    Good questions. A few quick takes: Maximum net flux, as defined as the maximum number of field lines moving in a given general direction within the total system of two magnets, will be greater after colliding than before. As an equivalent solenoid lengthens = magnets join together, so the maximum internal field grows. So therefore greater maximum values of H and B also. But this is over a reduced effective net volume, hence the total magnetic field energy has diminished, the difference chiefly being in mechanical energy extracted, with a generally tiny additional eddy current component. If there are just permanent magnets involved, inductance doesn't apply. The only appreciable 'induced currents' will be of the fictitious 1/μ0∇×M kind - and that assuming each magnet was less than fully magnetized to start with. As for spin magnetic moments, intrinsic spin moments are invariant and rigidly aligned within each of the many magnetic domains comprising each magnet. I'm no expert on the complexities of all possible spin-orbit interactions, but empirically, spin contribution is overwhelmingly dominant in ferromagnets, and once saturation (full domain alignment) occurs, no more can be appreciably wrung-out. Must go. :zzz:
  22. Sep 11, 2012 #21
    Now I understand what went wrong. The context is a wire loop. I cite (with added emphasis in bold, and corrections):

    "[..] For the following analysis I will restrict myself to mere high school classical physics.

    Perhaps the simplest case to model two permanent magnets is two single current loops 1 and 2. Let's assume current loops of 1 m circumference, made of 1 mm diameter wire; distance d between the loops ca.1 cm (diameter << d). As pictured, the currents flow through the wires (---) into the screen on the left (x), and flow out of the screen on the right (0):

    1 x----0

    2 x----0

    Here the wires are attracted to each other by the Lorentz force: F = IxB . L

    However, for calculation we can approximate this particular example with two straight wires of 1 m length by opening the loops above the screen and straightening them out:

    1 --------->--------- I1

    2 --------->--------- I2

    Now we have the basic configuration of the definition of the Ampere:

    F = 2.10-7 I1 I2 L / d

    For this case I obtain F= 2.10-7 * 100 * 1 / 0.01 = 2 mN.

    Suppose that the top magnet is connected to an external mechanism that is actuated by it. We allow the top magnet to move towards the bottom magnet over a small distance, for example from 11 mm to 9 mm. We will assume that the average currents are approximately 1 A (more follows).

    Then the work done by the Lorentz force on the top magnet and whatever it is driving, according to [1] and [2]:

    F.d ≈ 2.10-3 * 2.10-3 J = 4 μJ.


    In that context, we can distinguish two main cases for the current flow.

    1. A common electromagnet, let's say with a current source [...]

    2. For a [STRIKE]true [/STRIKE]sourceless [STRIKE]permanent [/STRIKE]non-fading magnet there is no external current source.
    Following Lenz-Faraday-Maxwell we find that the motion will induce a[STRIKE]n electric field [/STRIKE]EMF in each wire that opposes the motion by reducing the currents and the corresponding magnetic fields.[..]

    Note the mistake which I now corrected: contrary to case 1, in case 2 the loop must be closed. I'll repeat once more: I found that an E-field along a circle-symmetric closed loop configuration is an impossibility. If you disagree, please try to sketch one with an E-field.

    I checked out the Hyperphysics site 10 days ago while writing my above example. And I hope to find time in the coming week for reading more related to Poynting.
    Sorry, a closed wire loop has no opening, and "a current driven through" corresponds to case 1. Making it more complex adds noise to the discussion, and in this discussion I strive for a high S/N ratio.

    BTW I see that you want to discuss a more complex matter related to J, but I think that that is contra productive as long as this simpler case related to E is not understood first.
    Last edited: Sep 11, 2012
  23. Sep 12, 2012 #22
    Ah; at last it becomes clear (I think). Sorry about previous confusion - a case of words getting in the way of what was meant. :rolleyes: So you were not denying the possibility of generating a circular E field, just saying the emf in a shorted perfectly conducting loop is necessarily zero - right? Well sure then, no argument! And if you look at my #61 from back there, that same point was made clearly then - Lenz's law applied and one has perfect diamagnetism. But it's from that point on that we seem to fundamentally differ. I went on to argue there that such diamagnetic behavour is the very antithesis of permanent magnetism, which has no respect for Lenz's law at all!

    Anyway if I now get your main point of 'zero emf in Amperian loop current' to rephrase it, you are saying there is thus zero E.J type work being done owing to E being zero everywhere in those loop currents. And thus all the work must be magnetic. That's it in a nutshell? Well I agree that is the case if such Amperian loop currents actually existed in permanent magnets. But in actuality the overwhelmingly dominant source of B field in ferromagnetism is intrinsic moments, and their character is opposite to the diamagnetism of an Amperian loop current. Ferromagnetism and diamagnetism are opposites!

    I dug up those links in #14 for a reason. Let me now quote from both. First, from 12.2, p10 in http://folk.ntnu.no/ioverbo/TFY4250/til12eng.pdf
    And from 23.1, p214, and 23.1.2, p214-5 in http://web.mit.edu/sahughes/www/8.022/lec23.pdf
    (bold emphasis added)
    And if you care to study it, that article by Mansuripur only reinforces and expands on this greatly. I emphasized all this from the start back in that other thread, but it collectively went in one ear and out the other. So with this emphasized contrast between intrinsic moments and classical loop currents - given by authority figures - it's a case of accept or reject, but don't complain the issue has not been plainly stated for all to see. :grumpy:
    We now have your intended meaning straight I believe, so that bit is now moot. But please Harald - in the interests of that low S/N ratio objective, strive for clarity!
    Said my piece on that above. Your simpler case is fine - just as long as there is no confusing that diamagnetic loop(s) scenario with the situation for real PM's - which as per quoted text - in respect of magnetic response are wholly governed by quantum mechanics. That has to be faced.
  24. Sep 12, 2012 #23
    :bugeye: Oops I made a colossal blooper here:
    As I realise now, that challenge was based on a secondary definition which does not hold here; the electrical potential approach with a field from high to low potential simply isn't valid for such a case. In other words, my correction was wrong! Sorry for the confusion :redface:
    Regretfully I brought you off track, as there certainly is not only an EMF but consequently also an E-field according to the fundamental definition of electric field: https://en.wikipedia.org/wiki/Electric_field
    See also: http://farside.ph.utexas.edu/teaching/316/lectures/node87.html

    Once more my excuses - but making that blooper and correcting it was useful for me, and may be useful for some others. :smile:
    Last edited: Sep 12, 2012
  25. Sep 12, 2012 #24
    Yes, thanks for the elaboration. I also read an article about VI which highlighted the complexity of it all, and helped to understand it better.
  26. Sep 12, 2012 #25
    That is very right. In the thread on magnets, some claims were made concerning Poynting's theorem that appeared to contradict my analysis there. However, that analysis was based on the fundamental equations on which also Poynting's theorem was based. I suspect that the disagreement was due to a misapplication of Poynting, and I would like to get this straightened out.
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