Talk:Poynting vector/Archive 2

Deleted "Problems in Certain Cases" Section
Although technically not incorrect, I think this section is misleading. There are no problems with the interpretation of the Poynting vector as power flux in classical electrodynamics. Poynting Theorem is valid in all of these cases, and reading this section seems to give the feeling that Wikipedia is suggesting otherwise.

By the way, I think there needs to a section in this article dedicated to the relationship between the Poynting Vector and momentum, angular momentum, and the Maxwell Stress Tensor. --Armaetin (talk) 01:17, 12 September 2010 (UTC)


 * Disagree. I suggest reverting or creating such a section. (At least revert and discuss before deletion?) A notable property for any concept is the boundary of its validity. You are correct that the interpretation is valid, but you are mistaken to imply that it is the only valid (or natural) interpretation. As mentioned above, there are proofs that it suggests energy flows which are "nebulous" (devoid of physical measurable meaning), and that certain contradictory interpretations are equally valid. Moreover, the paradoxical-seeming cases are some of the most notable content of the physics literature on the topic. By removing such content completely, you tacitly present a viewpoint which is controversial and disputable, not neutral. (Furthermore, not warning with such examples encourages incorrect tangible results, such as confusion over whether an apparatus is radiating or not.) Imagine if someone rewrote the article on gravitational potential mgh (an example similar for involving the presence of energy that isn't trivial to completely localise) to assert it depends absolutely on heights relative to sealevel: it would be valid, but it would misrepresent that topic (and in a very few cases would encourage completely wrong results). Cesiumfrog (talk) 07:40, 28 September 2010 (UTC)


 * Agree with Armaetin. The section was misleading.  I had been thinking about how to fix it with minimal disturbance, but I agree that deleting the whole section was probably the best thing to do.  I think it might be replaced with a section called "Interesting Special Cases". Constant314 (talk) 18:07, 5 October 2010 (UTC)

I am quite late to this discussion here, I have been out of academia for some time. Well, I was the one who introduced that section there, and although it might seem misleading, it is there in major publications (Jordan-Balmain is a pretty standard book where I come from). It was also published in a notable journal, so it is not original research. I feel that a little rewording might help and it wouldn't be so unwise as to reintroduce the section to the article. Welt anschaunng 17:49, 31 August 2011 (UTC)


 * call it Interesting special cases instead of Problems with the Poynting vector and it will be more likely to survive. And which notable journal would it have been published in? Constant314 (talk) 04:51, 1 September 2011 (UTC)


 * And where should I look for it in Jordan-Balmain? Constant314 (talk) 04:53, 1 September 2011 (UTC)


 * I have a copy of Jordan-Balmain. On page 169 .  He says "The interpretation of E x B as the power flow per unit area ... never gives an answer known to be erroneous, ... "  That statement alone is enough lay rest to any claim that the Poynting vector has problems.  He further says that certain situations, involving static fields, produce a "picture that the engineer generally is not willing to accept."  He was writing in 1950.  We've come a long way since then. Constant314 (talk) 01:37, 3 September 2011 (UTC)


 * Here is the paper:

Speed of light inside a wire??
I have a problem with this sentence:


 * This is a consequence of Snell's law and the very slow speed of light inside a conductor. Inside the conductor, the Poynting vector represents energy flow from the electromagnetic field into the wire, producing resistive Joule heating in the wire.

In the first place, I don't know how anyone defines THE speed of light (what frequency??) INSIDE a wire, but probably what someone means is c/Re(n), that is, the "speed" that would correspond to the advancement of phase peaks inside the conductor, or c_wire = omega / k_r (radian frequency divided by the real part of the propagation constant). This isn't too meaningful since at most wavelengths, certainly below the plasma frequency, a conductor strongly attenuates the wave, that is, the imaginary part of the refractive index n is >= the real part of n. Yes, at frequencies << the plasma frequency (thus including the radio frequencies that coax would actually be used for) |n|>>1, but most of the field dies out in less than one of these (admittedly short) wavelengths, and to talk about Snells law is really misplaced. Inside a conductor or other strongly attenuating medium (where Im(n) is not << Re(n)) you obtain a so-called inhomogeneous wave where the energy flow (given by the Poynting vector, at least I hope!) is in a different direction than the normal to the phase peaks. I agree that this energy flows into the wire (and dissipates due to the current density and finite conductance), but talking about the "speed of light" inside the conductor is misleading, in an already confusing article! It should be sufficient to point out that the Poynting vector points (slightly) into the wire and that energy never comes out anywhere else, thus it is dissipated there. And as has been pointed out earlier on this talk page, it isn't too meaningful to discuss "where" the energy stored in an electric or magnetic field is "located," and I don't think that determining the Poynting vector inside the wire is useful in that regard. It's a lot clearer to talk about the distribution of currents based on the skin depth of the wire in conjunction with the metal's bulk conductance.

By the way, I looked up this page just as a quick reference regarding an unrelated issue, found it rather confusing and equivocal, and then read the talk page which I found much more interesting than the problem I had originally been working on! (But I really have to get back to my work now....). Interferometrist (talk) 17:02, 1 February 2011 (UTC)


 * The definition of the speed of light inside a good conductor, according to Hayt, pages 398-402 is


 * $$ v = \frac {\omega} { \beta }   \, $$


 * $$ \beta = Im(\gamma)    \, $$


 * $$ \gamma = \sqrt {  (j \omega \mu_0) ( j \omega \epsilon_0 + \sigma) }     \, $$


 * The velocity of light in copper at 60Hz is about 7.2 miles per hours, according to Hayt page 402.


 * As for Snell's law, it would probably be more accurate to say that this result (that EM waves enter the conductor almost exactly normal to the surface) and Snell's law arise as a consequence of the continuity requirements of the EM field where it crosses an interface between two different media. Or perhaps we might say that Snell's law has been generalized to include media with non-zero conductivity.
 * As for Snell's law, it would probably be more accurate to say that this result (that EM waves enter the conductor almost exactly normal to the surface) and Snell's law arise as a consequence of the continuity requirements of the EM field where it crosses an interface between two different media. Or perhaps we might say that Snell's law has been generalized to include media with non-zero conductivity.


 * The article is about the Poynting vector, so of course we want to talk about the Poynting vector inside the conductor.


 * It is true we cannot determine where the energy is when it is stored, but we can compute where it is converted to heat. If we known the Poynting vector inside the conductor, we can compute the power dissipation in each volume element of the wire.Constant314 (talk) 18:30, 1 February 2011 (UTC)


 * And I agree that the article is confusing. A consequence of many hands.Constant314 (talk) 18:35, 1 February 2011 (UTC)


 * in page 454 they begin the discussion the discussion of a wave impinging on a good conductor with Snell's law and indicate that both the index of refraction and the angle of propagation are complex.


 * in page 61 "For most practical purposes, the wave can be considered to propagate normally into the conductor regardless of the angle of incidence." Constant314 (talk) 05:08, 2 February 2011 (UTC)

Well thanks for responding (in detail) and I'm sorry I wasn't able to get back to you before you edited the article but let me just say that I'm not here especially to argue about wording in the WP article (though I still think my objection was valid); only a few words are at issue. The bigger issue about the quality of the article is stating clearly what THE poynting vector is (E or D, H or B, etc.) and what it exactly means, and I believe all of the confusion in the article and on the talk page (but I haven't dissected the discussion in detail) has to do with what portion of the energy is "from the EM wave" and what is attributed to the polarization, currents, magnetization, and/or (in some cases) space charge inside the medium. But let me just comment on what you have written.

Some of it has questionable or ambiguous meaning and I would still disagree with. The problem in discussion technically involves what are called (in a certain context) "inhomogeneous waves" which occur when a plane wave strikes at non-normal incidence the surface of an absorbing material (Im(n) not =0). I believe this is mentioned in Jackson etc. (but don't have a copy at hand). Then the "direction" of energy propagation is no longer normal to the phase wavefronts: there are 2 different directions involved.


 * Reitz discusses the fact that there is a plane of constant phase propagating at one angle and the planes of constant amplitude propagating at another direction. But when $$ \frac {g} {\epsilon_0 \omega} >> 1 \, $$ both wave fronts in the conductor are propagating normal to the surface.  They use g for dc conductivity.

You continue:


 * The definition of the speed of light inside a good conductor, according to Hayt....

In these equations gamma is what I call jk (if the wave propagates in the z direction as exp(-jkz)) and beta is the real part of k. The refractive index n = c/omega * k. But in the case of an inhomogeneous wave, you need to use the more general propagation equation E= exp(-j k.r) -- I'm trying to write k dot r -- where k is a vector and has a different direction for its real part (which governs the phase wavefronts) and the imaginary part (which governs the attenuation of the amplitude).

I agree with the first two equations you wrote but the third is only an approximation (even given the ideal model: of course real metals act quite differently at high frequencies anyway!) valid at frequencies << the plasma frequency. Otherwise the second term (sigma) needs to be divided by (1 + j omega tau) where tau is the mean time between collisions between free electrons and phonons (which accounts for the finite conductivity) and can be found from the conductivity (measurable) and the density of "free" electrons (not directly observable but deduced from the atom's electronic structure: I know for sure this model has very poor agreement with observation at IR wavelengths and beyond). Feynman includes this correction.


 * The particular section of the in question is about the PV in a coaxial transmission line. The frequencies are well below IR.


 * "The velocity of light in copper at 60Hz is about 7.2 miles per hours, according to Hayt"

Well, I don't usually call 60Hz radiation "light," but never mind! The number sounds right, I get 3.2 m/sec (but come on, let's keep it metric!). But think about what you are saying. To talk about a plane wave impinging on a surface (and "rays" and snells law etc.) in the usual sense, the surface of the material it is entering must be sort of flat over a surface wider than a wavelength. So at 60Hz, you'd need a sheet of copper larger than the earth. In order for a 2mm conductor to look much larger than a wavelength you'd need a frequency of say 1 THz. And then I calculate a phase velocity inside the copper of 4000km/sec, which indeed is "slow" for light.


 * Not sure what you are saying. The computation of skin depth comes out of the same analysis and it shows up at 60Hz when the wire diameter exceeds a couple of cm.

But again, that all has to do with normal incidence. If you examine this situation of grazing incidence, you'll find (I believe) that the energy propagation (yes, given by the poynting vector) is bent inward almost in the normal direction. But the phase velocity will have a small component into the metal, but a component along the wire equal to the phase velocity of the wave OUTSIDE the conductor (thus =c if no dielectric is present in the coax) which means that it is mainly in the direction of the wire, not into it, and it is not slow at all but ~=c.

Now here's my other disagreement, though I'm not sure this directly involves the WP article text:


 * It is true we cannot determine where the energy is when it is stored, but we can compute where it is converted to heat.

Sure..... I would have said J^2 / sigma, but.....


 * If we known the Poynting vector inside the conductor, we can compute the power dissipation in each volume element of the wire.

No, I don't think so! Not directly. Maybe you mean that by observing the ATTENUATION of the Poynting vector with depth, or the wave amplitude itself exp(- alpha x) where alpha = Im(k), then you can INFER that the energy was deposited in the intervening layer.


 * I mean the volume integral of the divergence of PV.

But there clearly is no direct formula for dissipation of energy dependent on the Poynting vector, and before you waste any time trying to figure it out let me just point out a simple counter-example (remember, I only need one!). A traveling wave through a weakly attenuating medium is reflected by a perfect mirror creating a standing wave. The Poynting vector in that region is zero, yet there certainly is dissipation in the medium due the oscillating electric field itself, carrying no energy in either direction. To compute the energy dissipation in a medium (assuming it interacts with the electric, not magnetic field of an EM wave) you need to know <|E|^2>, period. The Poynting vector (if any) supplies no additional information in this regard. It can tell you if there is a net power FLOW across a boundary (where knowing the E or M field alone would be insufficient), but you must admit that your statement, above, is incorrect (again, unless you mean the spatial derivative of the average Poynting vector).


 * If the medium is weakly attenuating then the reflected wave, having gone through more of the medium will be attenuated compared to the incident wave. The EM field will not be purely standing and the volume integral of the divergence of PV will be non-zero.

As far as the WP article goes, I think the wording employing Snell's law is wrong or misleading (there are two directions to consider), and:


 * Once the Poynting vector enters the conductor,

is meaningless, since after all, the Poynting "vector" doesn't enter the conductor (the poynting vector is not continuous across an interface); what you mean is that the WAVE enters the conductor,


 * Yes, I mean that the wave enters the conductor.

but only slightly: I believe the reflection of a 60Hz wave (your example!) by copper is > 99.9999% (which you could cite as one way of knowing that most of the energy propagates along the coax).


 * That is plausible to me, after all, the coax is a good transmission line at 60Hz, we certainly want as much of the energy as possible to stay out of the metal where it would be strongly attenuated. Reflecting off the metal is good.

Yes, you can QUANTIFY the poynting vector outside of the conductor, and even inside the conductor (though again that becomes tricky especially given the ambiguities present in the article itself and discussion too about E vs D, H vs B),


 * Yes, I would leave that out.

but overstating the utility of the poynting vector does a disservice to the reader.


 * Not sure what you mean. I don't have a point of view with respect to the utility of PV. I'm simply describing what the PV is (its magnitude, its direction, its attenuation, how it fits in with the other field quantities).  If I had to compute the power absorption I would use voltages and current densities.

But again, I don't think this one paragraph is the biggest problem in the article and I'll let you edit the corrections. I was mainly trying to get this all straight in my own mind :-) Interferometrist (talk) 19:19, 2 February 2011 (UTC)

Oh, I should have addressed these too:


 * in Reitz page 454 they begin the discussion the discussion of a wave impinging on a good conductor with Snell's law and indicate that both the index of refraction and the angle of propagation are complex.

n is certainly complex (with almost equal real and imaginary parts at electronic/radio frequencies) as I detailed above. A "complex angle of propagation" or a complex geometrical angle in general has absolutely no meaning whatsoever, and when you get such a number in a result of some trig problem (where you are describing a direction) that is nature's way of telling you that you didn't set the problem up in a useful way! In this case, he tried to compute a "direction" for an inhomogeneous wave which doesn't make sense.


 * Reitz says much the same thing regarding complex angles. But they go on to say that they have derived Snell's law entirely from algebra with no use of geometric arguments and that it should work for complex quantities.  They then go through a few pages of verification and conclude that it does.


 * in Harrington page 61 "For most practical purposes, the wave can be considered to propagate normally into the conductor regardless of the angle of incidence."

And again, it appears he is talking about the ENERGY but not the WAVEFRONTS (and thus the "ray" direction normal to those wavefronts, as most people understand "rays"). So I'd have to say that neither of these authors really did the subject justice. Interferometrist (talk) 19:52, 2 February 2011 (UTC)


 * Harrington is on Jackson's list of suggested reading.


 * I have dispersed my responses thoughout your comments. To make hem visible I will preceed them with **Constant314 (talk) 05:03, 3 February 2011 (UTC)

Listen, thanks for the responses, and I need to acknowledge that a lot of what I wrote wasn't well thought out and some was just wrong. In fact the direction of the wavefronts (in addition to the energy flow) inside the conductor at radio frequencies IS INDEED almost normal to the surface (even for an incoming wave at grazing incidence) exactly as you first wrote, and it was only out of confusion and haste that I challenged that. The intersection of that internal wavefront with the surface of the metal still propagates at about the speed of light, but that is only because the very slow (3 m/sec at 60Hz) propagation inside the metal is at such a very tiny angle from the normal theta that sin(theta) * c = 3m/sec. In principle it is an inhomogeneous wave (different direction for the phase and energy flow) but both are almost exactly 0 degrees in this case. You (and Reitz) were right, and I was all wrong: SORRY!

And yes, I should have known you were talking about the divergence of the poynting vector in computing the power dissipated in a volume element. I doubt that's the easiest way to compute it, since computing the Poynting vector itself requires solving for the physical responses (polarization, currents, etc.) which cause the dissipation in the material. No I don't have a "point of view" regarding the Poynting vector, but I guess that after I read about the 4 different ways that it can be defined and how these account differently for the energy contributed by the medium (as laid out in the Kinsler paper) I found that my original ideas about the Poynting vector representing "the" power flow were too naive to simply apply without further thought.

I still do not think that one can make physical sense of a complex (geometric) angle, and that problems where these come up as solutions should not have taken the arctangent (etc.) of something but left the answer in terms of direction cosines (for instance) which can well be complex. Thus the propagation constant k will be complex inside a metal, with its real part giving the solution of the phase and its imaginary part solving for the attenuation. Either of those could be resolved to a geometric angle, but trying to do that with the complex quantities gives a "complex angle." As far as I know, that complex angle isn't interpretable (but if you know of any physical interpretation for a complex angle, I'd really like to hear of it!).

Snell's law is useful for propagation from one dielectric into another, but is a simplification of the problem for this special (but most usual) case in terms of trig functions. Just because you CAN solve it for an angle = arcsin(a complex number) doesn't make it useful in that case, whereas the general solution is rather straight-forward. Namely that the tangential component of the k vectors, k_x and k_y on each side of the interface are the same, and that in each medium what I call the "complex magnitude" = k_x^2 + k_y^2 + k_z^2 (not |k_x|^2) is constrained to be (I believe, from memory) = omega * n / c where the index of refraction n might be complex. Then you solve for the propagation constant which contains the two directions as I mentioned. Snell's law is a simplification of that entire procedure when n (and the k) are real.

Anyway, I'm happy that from this you added references to the article, and that I got around some confusions on my own part, and I sincerely apologize if I wasted a bit of your time in the process of doing so! :-) Interferometrist (talk) 22:02, 3 February 2011 (UTC)

By the way, I don't think the article is bad at all! Nor is it actually confusing: I only felt a bit "confused" after reading through the old discussion on the talk page. (And I don't know why it is only rated "C" -- nor do I understand how that is determined or how to upgrade it). The only thing in the article that put me off (but I guess it had to be said) was when I hit the word "controversial" in an article about classical physics! But compared to the talk page, the article itself is pretty clear and carefully worded. Interferometrist (talk) 22:34, 3 February 2011 (UTC)


 * Expending a bit of time on mattters such as this is seldom a waste.Constant314 (talk) 23:05, 3 February 2011 (UTC)

notation
If its ok with people I changed S to N, since N = Poynting vector allows S = vector area (for surfaces) allowing A = magnetic potential, with no ambiguity/conflict. I know both are used in books (and added a comment at the beginning on that), but it will not hurt to use non-conflicting notation. N is not used for anything in EM, S and A are used each for two things...

In addition cleaned up a lot more messy editing with LaTeX, and re-organized slightly... Hope this is fine... =| F = q(E+v×B) ⇄ ∑ici 13:51, 18 May 2012 (UTC)


 * N also means the normal vector. Probably better to stick with Jackson's notation which uses S.Constant314 (talk) 22:16, 18 May 2012 (UTC)


 * It doesn't matter - either is fine so I'll change it back. I should add that in my experiance, normal vectors are usually lowercase n, rather than capital N, but this is sometimes used also. F = q(E+v×B) ⇄ ∑ici 23:26, 18 May 2012 (UTC)

Terminology
I have changed "maximum amplitude of elelctric field" to "peak value of electric field". I assume that this is what is intended since in this discussion the amplitude of E does not vary with time. — Preceding unsigned comment added by 82.32.50.178 (talk) 08:51, 28 June 2012 (UTC)

In static fields
What would be the behaviour of the circular (?) energy flow if the cylindrical capacitor is replaced by a parallel-plate capacitor in this setting.

Isn't this whole explanation a bit inconsistent ? It even mixes up classical physics and quantum mechanics ! — Preceding unsigned comment added by Basti Schneider (talk • contribs) 13:05, 11 September 2012 (UTC)


 * You'd have to be more specific with the replacement example. This is a purely classical interpretation, nothing related to quantum mechanics. And no, I doubt you'll be able to find an inconsistency with this interpretation. — Quondum 18:06, 11 September 2012 (UTC)

Setting with parallel-plate capacitor: H-field has the the same direction, capacitor is placed within this H-field. Right plate of the capacitor is charged positive, left plate negative. So there still is $$E \bot H $$. With $$S = E \times H $$ Poynting vector points out of the capacitor, parallel to the plates. In this case there is no circular flow of energy. So there is no angular momentum either.

The charges of the discharge current are electrons. As electrons are particles the laws of quantum mechanics apply for them.

See: Angular_momentum and Electron

Can you explain the physics of the setting described above ? Please explain exactly the path of the energy flow.

Basti Schneider (talk) 07:53, 12 September 2012 (UTC)


 * View the plates as finite, and the magnetic field as uniform throughout space (or at least in a large volume around it). The energy flow (and momentum) are vertically downwards between the plates, looping around the bottom edges, up on either side around the outside of the plates and back down between the plates due to the electric field around the outside of the plates. Due to the symmetry of the flow, there is no overall angular momentum. Energy–momentum is conserved exactly at every point in space; when the momentum due to S changes direction, the change of momentum is supplied by the stresses in the EM field. — Quondum 09:05, 12 September 2012 (UTC)

Ah okay, I expected this answer. Anticyclical 'vortexes' around the plates.

But you have to consider that this kind of energy flow isn't 'backed up' anymore by the classical E-field (and not by $$\phi$$ or $$A$$, either). On the upper and lower 'ending' of the parallel-plate capacitor the E-field becomes inhomogeneuous and on the outside of the capacitor the E-field is much weaker than between the plates.

But by definition the Poynting vector in any point of the setting is $$S = E \times H $$. But E is weaker outside of the capacitor than inside. Hence there is a difference between S within the plates and on the outside of the plates, so there are problems in this approach to maintain the conservation of energy in the flow -> there is a change in the strength of the E-field if compared inside and outside of the capacitor ($$ H = const$$).

Maybe you can argue, that the flow outside the capacitor is spatially more extended than in the inside, where it is more 'compressed'.

But in this case you should remember: In reality there are no spatially infinte H-fields.

Then the whole explanation fails once again as there exists the possibility that the E-field on the outside of the capacitor reaches in areas where $$H = 0$$. But in theses areas with $$H = 0$$ and $$ E \neq 0$$, applies $$S = 0$$, too. $$S = E \times H $$.

So in this case the conservation of energy in the S-flow is broken.

Basti Schneider (talk) 10:23, 12 September 2012 (UTC)


 * This is not the appropriate forum for personally disputing the standard interpretations, unless the article is incorrect. Any solution to Maxwell's equations automatically satisfy conservation of energy everywhere. The article references Poynting's theorem, which can also be used to understand the implication of a non-zero divergence of S such as you are postulating. — Quondum 10:53, 12 September 2012 (UTC)


 * Discussion moved to User talk:Basti Schneider — Quondum 16:25, 12 September 2012 (UTC)

Expand the section on Static Fields
I run across a lot of confusion with my students about the meaning of the Poynting vector of static fields, so I believe this section should be expanded. For dynamic fields, the Poynting vector represents both energy flux and field momentum propagation. For static fields, the Poynting vector only represents field momentum propagation. Therefore, for static fields there is no energy flowing despite there being a non-zero Poynting vector. But there is momentum carried by the fields. If you go back to how the Poynting vector S is derived in an energy flux context, only the divergence of S shows up in the equation and therefore only the divergence of S has physical meaning, from an energy-flux perspective. For all finite static field configurations, the divergence of S is zero and therefore there is no flow of energy, even locally. For infinite static systems, you can get S to diverge, but this is simply due to the infinite scale and is nonphysical. On the other hand, the derivation of S, the momentum propagation parameter, involves S itself and not the divergence of S, so S does retain its meaning as the flow of momentum even for static fields. In summary, static fields do not lead to energy flow, even locally, but can lead to field momentum flow. The Poynting vector of static fields therefore only has half the physical significance as it does for dynamic fields. A great reference for all this is in the American Journal of Physics: http://ajp.aapt.org/resource/1/ajpias/v35/i2/p153_s1. 129.63.129.196 (talk) 16:22, 22 February 2013 (UTC)
 * Just a point about terminology: I think we should merely say that momentum is stored in a static field rather than that there is a momentum flow.  Flowing mass has momentum, so the static field storing momentum has some of the characteristics of mass flow.  Given the equivalence of mass and energy you could say that the field has some of the characteristics of energy flow.  But, it's better to just say that momentum is stored in the field. Flow confuses people.Constant314 (talk) 07:18, 23 February 2013 (UTC)

Don't agree
hello, I don't agree with this section : " No energy flows in the conductors themselves, since the electric field strength is zero. No energy flows outside the cable, either, since there the magnetic fields of inner and outer conductors cancel to zero." first a current is the motion of charges in the classical picture, which carries energy. second there is a magnetic field inside both conductors, hence the so called skin effect, only outside the conductors and at the very center you have none. Remove it please, I don't want to fight with the author, he/she has to understand the mistakeKlinfran (talk) 23:33, 31 August 2013 (UTC)


 * This is pretty much the correct description of energy flow from the electromagnetic field point of view; however, there were a few unstated details and/or simplifying assumptions. I think that I have taken care of that.Constant314 (talk) 15:03, 1 September 2013 (UTC)

Badly Defined
Can someone please add what is $$\widetilde H$$ and $$H_c$$? — Preceding unsigned comment added by 78.128.194.120 (talk) 16:49, 17 February 2014 (UTC)


 * I believe $$H$$ is the three dimensional time domain magnetic field, $$\widetilde{H}$$ is a three dimensional vector whose components are the Hilbert transform of the components of $$H$$ and $$H_c$$ is a three dimensional vector whose components are phasors which are complex numbers that represent the magnitude and phase of the components of $$\tilde{H}$$. But I am guessing. Constant314 (talk) 01:36, 18 February 2014 (UTC)

H versus B
Shouldn't this article use B as the symbol for magnetic field? It is my understanding that H is corrected for magnetisation of matter, whereas B is the pure field. I remember Griffiths being very adamant about that. 2001:610:1908:8000:AC89:2716:3EA1:5AE (talk) 21:45, 26 June 2014 (UTC)


 * Yes, strictly speaking, H would only work on magnetic monopoles which don’t exist so H is a non-physical convenience. It is B that exerts a physical force on a moving charged particles and makes motors turn.  Physicists consider the A field (magnetic vector potential) as the most fundamental of all and that B is just a symbol for Curl (A).  You can find all that in volume two of the Feynman lectures.  There is probably an appropriate place to make note of it.  But, as I said, H and B are conveniences or even artifices that are commonly used.  No need to rewrite every use of H or B, especially in free space.  The encyclopedia expresses its subjects as they are, not as they should be.Constant314 (talk) 01:04, 27 June 2014 (UTC)


 * H is a convenient field to calculate with, but it has no rights to be called the magnetic field. That's just a confusing artifact from a long-outdated model (the model of physical magnetic poles, which we've replaced by currents). B is the magnetic field - magnetic flux density is a nonsensical term. Wikipedia should prioritize modern physics over historical physics. -- 2001:610:1908:8000:AC89:2716:3EA1:5AE (talk) 08:32, 27 June 2014 (UTC)


 * Magnetic Field is sufficiently vague to cover H, B, A and my favorite : $$ \frac{\partial H}{\partial t} $$, the magnetic displacement current. But in the end, Wikipedia documents the facts as they are presented in reliable secondary sources. Even Jackson uses H. See WP:RIGHTGREATWRONGS Constant314 (talk) 20:46, 27 June 2014 (UTC)

Poynting Vector drawing of DC circuit.
It's a nice drawing. You may want to show that the PV is stronger near the wires than in between them because the H field is stronger (a lot stronger) near the wires. Constant314 (talk) 18:13, 8 February 2015 (UTC)


 * The existence of drawings like this will go a long way towards developing intuitions around electromagnetism. Good work.  I concur that it can be tweaked to enhance detail (e.g. by also showing the field above and below), but the majority of the potential value of this drawing is already there. —Quondum 18:30, 8 February 2015 (UTC)

Complaint about the edit entitled "Nonisotropic -- is this correct?"
Cluebot picked up a syntax error in the change, but then didn't seem to understand my captcha even though I repeated it about 6 times. However, it seems to be respecting my revert of its revert, so I guess these robots aren't too insistent. 178.38.181.216 (talk) 23:38, 23 May 2015 (UTC)

Nonisotropic -- is this correct?
I added:
 * Here ε and μ are scalar, real-valued constants independent of position, direction, and frequency.

and also added the word "isotropic" in
 * Note that u can only be given if linear, nondispersive, homogeneous, and isotropic materials are involved...

Is this really correct?

Are the materials really required to be isotropic, so that ε and μ are scalars, not tensors?

I kind of threw this in based on the "precautionary principle", but it seems to partially contradict a few remarks in the article on birefringence, where the Poynting vector is mentioned and it's not parallel to the direction of motion, which can only happen if D is not parallel to E, and the article on birefringence still dares to refer to the Poynting vector in this situation and interpret it as the energy flux vector.

89.217.8.245 (talk) 22:00, 23 May 2015 (UTC)


 * It is true that nonisotropic materials will not be described correctly by a scalar permittivity and permeability. However, it is still possible to determine the Poynting vector.  —Quondum 23:22, 23 May 2015 (UTC)


 * Yes, I'm confident about the tensors too. Just not about the interpretation of Poynting vector. For example, which version of S does one use if both ε and μ are tensors? If that is too general to be palatable, then which formulation will match with the definitions given earlier? I am assuming that the S = E x H form that is favored in the introduction is compatible with ε a tensor or μ a tensor, but the asymmetry between electric and magnetic in the formula S = E x H makes me worry that something is not right if both ε and μ are tensors. In any case, the section under consideration needs to be tweaked a bit in order to be fully accurate. 178.38.181.216 (talk) 23:55, 23 May 2015 (UTC)


 * This is the kind of confusion that gives rise to nonsense such as the Abraham–Minkowski controversy. The microscopic Maxwell's equations are unassailable, and similarly the electromagnetic stress–energy tensor. It follows that, provided one can properly partition charge (and hence current) into "bound" and "free" forms, one can write Maxwell's equations and the electromagnetic stress–energy tensor equation (which incorporates the Poynting vector, the electromagnetic energy density and the electromagnetic stress tensor) in terms of this partition. The primary problem is trying to express the behaviour (and thus the bulk properties) of a material in an equation, since this inherently is merely an approximate model of the way bound charges react to applied fields – which is where the ε and μ tensors come in, only even they are simplistic (they only model frequency-independent linear behaviour).  So really, one should leave the permeability and permittivity out of it and express everything in terms of the partitioned quantities, only adding the bulk properties as an afterthought for models of the behaviour of specific materials.  So, yes, it would need to be tweaked to be fully accurate. I have not looked at what form the Poynting vector would take with a rigorous partitioning; like you, I expect that it does not apply as written. As a side note: a rigorous treatment would include another field that expresses the conversion between bound and free charge, which is typically entirely ignored (ionization is an example of such a conversion).  —Quondum 01:48, 24 May 2015 (UTC)


 * Thanks for the clarity!


 * Incidentally, the article Poynting's theorem seems to be a mess; it goes back and forth in its point of view and doesn't state the hypotheses that it is continually changing. 178.38.161.142 (talk) 12:39, 25 May 2015 (UTC)

Recent edits of interpretation section: " paradoxical results for certain cases"
Please say what the paradoxical results are and how the source resolves the paradoxical results. It is not informative to simply report that some source finds paradoxical results without saying what those results are and how they arrise. You may find that it is already covered under the static fields section. Constant314 (talk) 13:46, 31 May 2015 (UTC)


 * I think the statement should be entirely removed. There is nothing paradoxical in the Poynting vector and its interpretation as an energy flux density.  Side-tracking into historical dead-ends as though the definition itself is still not settled, or may give rise to uncertainties or paradoxes, simply does not belong in anything but a history section.  —Quondum 15:16, 31 May 2015 (UTC)


 * I agree.Constant314 (talk) 17:49, 31 May 2015 (UTC)


 * This statement came from the well-known textbook by Panofsky and Phillips:
 * “Paradoxical results may be obtained if one tries to identify the Poynting vector with the energy flow per unit area at any particular point. The energy term arose as the volume integral of $$\nabla \cdot(\mathbf{E}\times\mathbf{B})$$, and the net energy flow in the electromagnetic field will always vanish if the divergence of the Poynting vector is zero.  For example, in static superposed electric and magnetic fields in the absence of currents we may have nonzero values of the Poynting vector at various points in space, but its divergence vanishes everywhere.”


 * I think it may be of use to readers if we at least mention that Poynting vector may lead to strange results such as mentioned above.--LaoChen05:28, 4 June 2015 (UTC)


 * Without a sensible explanation of what is meant by "paradoxical", this is worthless. The only thing paradoxical is that a textbook might make a claim such as this.  In particular, in the statement "in static superposed electric and magnetic fields in the absence of currents we may have nonzero values of the Poynting vector at various points in space, but its divergence vanishes everywhere", there is nothing paradoxical: it is exactly what we expect.  I would be so far as to say that this disqualifies this textbook as a reliable source, or at least requires the replacement of the book's use of the term "paradoxical" with "perhaps surprising".  (Note: the surprise is only that of a novice who does not understand that a static EM field can carry momentum.)  —Quondum 13:06, 4 June 2015 (UTC)


 * I don’t have access to that text book, but, sometimes, in text books, the author, like Feynman, might introduce a subject with the suggestion that there is a paradox and then go on to explain why there is no paradox and why the theory is completely sound. The only paradoxes that might arise is in the mind of someone with unfounded expectations about what energy flow might be. Constant314 (talk) 21:12, 4 June 2015 (UTC)


 * I think this result may be useful to the readers. Is there someway we can rephrase the statement so that it is acceptable by everyone concerned about the problem?--LaoChen05:28, 4 June 2015 (UTC)


 * And what result is that? Please be clear.  Are we saying that the interpretation of the Poynting vector is unsettled, or that some people find it confusing?  The quote above appears to be the perpetuation of a genuine misconception, something that an encyclopaedia should avoid.  The author appears to be assuming that the energy flow density in a static electromagnetic field must be zero, which is easily demonstrated to be false.  The same goes for momentum density, which is subject to direct experimental measurement.  —Quondum 12:58, 5 June 2015 (UTC)


 * Quondom has said it well. I would just add that when you understand PV in its entirety, it makes perfect sense, there are no paradoxes, it calculates out to exactly what is needed to account for conservation of momentum and it is consistent with all theoretical outcomes.  I admit that someone without all the information may balk at the idea that the static field has energy whirling around in a closed path until you realize that that whirling energy exactly accounts for angular momentum stored in the field.  But, it is not an encyclopedic fact that some people are confused when they don’t have all the information.  We really want to emphasize that there are no paradoxes.  If you think there are, then we want to encourage you to dig a little deeper into the subject. Constant314 (talk) 21:14, 5 June 2015 (UTC)


 * Reading to the end of chap. 10 of the linked text, it seem that the author may be assuming that the reader has the mindset that "matter alone" [which in this conception does not include the electromagnetic field] carries momentum; he leaves it to the last paragraph of the chapter to point out that "we are thus forced to modify the law of conservation of momentum" to include the momentum density of the electromagnetic field. This is not the order of exposition that we have in an encyclopedia: we simply take it as a given that electromagnetic waves have a momentum density, and this excludes the use of terms such as paradoxical in this article.  —Quondum 22:27, 5 June 2015 (UTC)


 * I've simply removed the whole paragraph. —Quondum 17:21, 6 June 2015 (UTC)

Inaccurate statements attributed to Jackson
The Interpretation section contains the sentence “In other words, the Poynting vector is accurately defined only up to a term of the curl of an arbitrary vector.” And attributes it to Jackson. Jackson does not say this. Jackson says that the curl of any vector field could be added to E X H and still satisfy Poynting’s theorem. But he goes on to say that relativistic considerations show that S = E X H without any added terms is the unique solution. Constant314 (talk) 14:40, 6 June 2015 (UTC)

The Invariance to adding a curl of a field section contains the sentence “It is often thought that using a different vector than the classical Poynting vector will lead to inconsistencies in a relativistic description of electromagnetic fields where energy and momentum should be defined locally in terms of the stress–energy tensor.” The citation needed tag has been replaced with a citation to Jackson p258-260. In that page range, Jackson says nothing to support anything in that sentence.Constant314 (talk) 15:48, 6 June 2015 (UTC)


 * Consider revising the text accordingly; your description of Jackson appears to be what I would consider to be correct physics. I have some memory that requiring only energy conservation leaves some degree of freedom in defining a vector whose divergence has the correct value.  One alternative is the Slepian vector.  However, the further requirement of Lorentz invariance results in uniqueness.  I see no reason to give mention to the conclusions of authors who fail to consider the constraints imposed by special relativity when trying to define a physical quantity.  Your mention that "he goes on to say that relativistic considerations show" confirms this.  —Quondum 17:16, 6 June 2015 (UTC)


 * The discourse in seems to be totally overweight.  My feeling is that we should merely be stating that since adding the curl of any field to S does not change its divergence, Poynting's theorem would still be satisfied, but that the Poynting vector is the unique field that is Lorentz invariant, and is hence the only candidate for a physical field.  The whole section can be reduced to a statement.  —Quondum 17:34, 6 June 2015 (UTC)


 * The last sentence has had a citation-needed tag for about 9 months and the sentence before has had a tag since 2011. I think both of those sentences can be cut.  The fact that you can add the curl of a vector field to PV is a historical footnote.  Jackson says that any such field must be non-physical.  The magnetic free Poynting vector might be interesting, but there is nothing in Jackson about it.  It makes sense that between the plates of a parallel plate capacitor, the PV would point in the same direction as the displacement current.  Maybe the name of the section could be changed to magnetic free Poynting vector and everything not about that cut out.Constant314 (talk) 18:44, 6 June 2015 (UTC)


 * From the text, that concept (a magnetic-free Poynting vector) is premised on the field being both quasi-electrostatic and the medium being a dielectric. This is merely an approximate simplification for a very specialized context (the dielectric of a capacitor), and would only belong in an article on calculations relating to capacitors.  We must remove it as misleadingly being applied out of context.  The last thing we need in an article establishing what the Poynting vector is is equations that do not contribute to this understanding and are utterly incorrect in the general context.  The displacement current being considered to be "carrying" the power from one plate to the other is on par with the intuition that the power flow associated with an electric current is inside the conductor (as it would be for pressurized water, for example).  But it suffers from a similar flaw: the voltage is ill-defined.  Voltage is not a locally defined quantity.  The Poynting vector correctly describes the flow of energy into the dielectric from outside the capacitor as the field in the dielectric grows, with the bulk of the energy flow being around the exterior of the capacitor depending on the exterior electric field.  Delete the passage.  —Quondum 20:21, 6 June 2015 (UTC)


 * I see your point and agree.Constant314 (talk) 21:02, 6 June 2015 (UTC)


 * I've deleted the text that is particularly egregious in . What remains of the section is still horrible, but as you mention, it might be reshaped into some nod to history, and to help with understanding what probably still occurs in modern textbooks. It definitely still needs work. —Quondum 05:37, 7 June 2015 (UTC)


 * Given that the editor making recent good faith edits has been found to have misunderstood or misapplied meaning of the source in at least three instances, I suggest resetting the article back to its May 23 2015 version and then make these changes and then lets see what is left.Constant314 (talk) 14:43, 7 June 2015 (UTC)


 * It seems to be more complicated than that. The bit about quasistatic fields was added in this edit on 28 August 2013.  It seems to me some careful comparison is needed with versions back to that point, keeping the good edits.  A nontrivial job.  —Quondum 16:24, 7 June 2015 (UTC)

I made some of these changes. Maybe I was less respectful to the previous prose than might have been wished. The section is still wishy-washy. Unfortunately I don't have the third edition of Jackson. 84.226.185.221 (talk) 09:07, 16 October 2015 (UTC)

Vague wording
There are problems in the section, "Formulation in terms of microscopic fields". It seems that ExB is brought in without any clear rationale. Can't one always replace H=B/mu by H_0:=B/mu_0 if one is willing to consider all charges and currents as "free" -- which is the way nature made them, in any case? And is that transition all that's meant here? 84.226.185.221 (talk) 23:17, 15 October 2015 (UTC)
 * I changed the wording to reflect this. 84.226.185.221 (talk) 09:07, 16 October 2015 (UTC)

Poynting's Theorem
BTW the article on Poynting's theorem is an incomprehensible mess. It needs a total cleanup. 84.226.185.221 (talk) 09:09, 16 October 2015 (UTC)

Co-co-inventors
I changed the multiple credit to the (apparently 3) co-inventors so as not to sound self-contradictory. 84.226.185.221 (talk) 08:04, 17 October 2015 (UTC)

distinguish symbols for ExH and ExB
Is anyone opposed to adding a prime to the microscopic field definitions of S and u, in terms of B? I mean, change them to S' and u'. Otherwise the article can be a bit confusing in the comparisons. --Nanite (talk) 09:27, 26 October 2015 (UTC)


 * I think you should use the conventions used in a consensus of reliable sources. Constant314 (talk) 13:37, 26 October 2015 (UTC)

More precise version of why Poynting is unique
In the section "Adding the curl of a vector field", I wrote, following Jackson, that the Poynting vector is unique (should not have a curl added) because of relativistic invariance. Indeed, S^i appears as the T^0i components of the relativistic stress-energy tensor T^αΒ. This is found in Jackson in the indicated pages.

However, Jackson actually only derives this for the "microscopic" expression E x B / μ_0 which is suitable for the "microscopic" Maxwell equations. It is difficult to refer to the "microscopic" expression E x B / μ_0 in this section because, as the article is currently written, E x B / μ_0 is introduced only in a later section. For this reason, I think that the "microscopic" version should get more prominence at the start of the article.

Incidentally, if you try to investigate the behavior of the "macroscopic" Poynting vector under special relativistic transformations, you quickly run into the fact that the material matrix has a "rest frame" and therefore a spacetime velocity vector u^α. Therefore, you do not have actual invariance of the macroscopic Maxwell equations as written. Instead, the best you can hope for is to express everything in tensor notation and get invariance for reformulated Maxwell equations that involve u^α.

This can get very complicated, especially if the material is not homogeneous and isotropic. Or if its parts are moving with respect to each other!!

For this reason, I actually have some sympathy for the "alternative", minority viewpoint expressed in the second paragraph of the section "Adding the curl of a vector field". Something like this should stay in the article until a definitive source is found for "the full Monty".

84.226.185.221 (talk) 05:28, 17 October 2015 (UTC)


 * With regard to the sentence "A recent article claims ... the Poynting vector may not represent ...", the references appear to be primary sources instead of secondary sources. WP requires secondary sources which are considered more reliable.  You can challenge its inclusion on that basis.Constant314 (talk) 15:00, 17 October 2015 (UTC)

I added a few years ago (2013)in this section (adding a curl of a field), the possibility to define a "magnetic free" Poynting vector that is very helpful in situation where magnetic field is negligible (and is really used). Besides, this Poynting vector is more realistic to some extend as it crosses the plates of a capacitor instead of spreading radially as the usual Poynting does (this job was published and references inserted in my contribution). This contribution was removed as non compatible with the relative invariant definition. However who cares if it not relativistic if it does the job and if all measurable quantities are correctly depicted ? Henri BONDAR (talk) 12:56, 19 January 2016 (UTC)


 * I cannot tell if this belongs here of if it is just a one-off approximation that is close enough under a special circumstance. In statics, curl E and displacement current are zero.  In dynamics, in general, both are non-zero.  In quasi-statics, we usually take the displacement currents as zero.  In the regime of the magnetic free Poynting vector, curl E is taken to be zero and displacement current is non-zero.  That would seem to be limited domain of applicability. If it is going to stay in the article, it needs more specific information about the circumstance where it applies. There needs to be a justification as to why curl E can be taken as zero in this regime.  Does it apply when the dielectric is lossy or only when the dielectric is ideal?Constant314 (talk) 18:32, 21 January 2016 (UTC)

"Analytic signal" calculation is rather heavy, for what it is
The section entitled "Time-averaged Poynting vector" is rather symbol-intensive and lengthy for what it accomplishes -- which is just a very trivial computation with a cosine. It is the result that is of interest, not the method. Or the method should be illustrated with a less trivial example. I would trim this section considerably. 84.226.185.221 (talk) 07:58, 17 October 2015 (UTC)
 * I agree. I will try to simplify it. Interferometrist (talk) 23:52, 21 January 2016 (UTC)

Magnetic free Poynting vector
Hello ! I tried two times to add here the idea of a magnetic free Poynting vector. This quantity was derived, according cited authors, in the quasi-electrostatics approximation frame. The contribution was deleted twice mainly because such a vector is not a relativistic invariant. May be, I should have started by a reference to the fact that Quasi-electrostatics and Quasi-magnetostatics can be shown to be Galinean approximations of Maxwell equations (see for instance Levy-Lebond Galinean Electromagnetism). As a result none of energy density flux defined in these domains would be relativistic invariant. Nevertheless they can be useful tools in their respective domains, in the same way that Kepler laws are useful without being relativistic. Besides, the fact that Maxwell electromagnetism doesn't imply a specific choice for the Poynting vector (the localization of energy) if not satisfying in the relativity frame, is fully compatible with quantum electrodynamics where the photon cannot be followed during its flight and only has a probability of being emitted or adsorbed. See for instance Quantum electrodynamics: "Nothing is implied about how a particle gets from one point to another".Henri BONDAR (talk) —Preceding undated comment added 11:40, 22 January 2016 (UTC)
 * Relativistic invariance has nothing to do with my disagreement on the inclusion of this matter. It appears to only apply to static fields, which is much too special a case to need mentioning in conjunction with the Poynting vector. The result for energy dissipation in the static case appears correctly written as $$\nabla\cdot\mathbf{S} = -\mathbf{E}\cdot\frac{\partial \mathbf{D}} {\partial t}$$ which is just the ohmic dissipation due either to a free current in a non-ideal conductor or to polarization currents in a lossy dielectric WHILE the dielectric is relaxing (thus not a static case really). It still involves the magnetic field but isn't written out, and appears to exclude magnetic losses (such as due to hysteresis or saturation) which is usually the case. At least that's what I read from it. The fact that its divergence and the divergence of the Poynting vector are equivalent just means they can both be true at the same time (no mistake was made) but it isn't as general as the Poynting vector and doesn't appear to apply to any useful problems except for a conduction current through a resistive material. That doesn't really belong in the same discussion as the Poynting vector and doesn't help the reader. I will look at this further when I have more time but for now I'd leave this out, and as was pointed out this is formally justified according to Wikipedia rules unless secondary sources confirm not only that the result is valid but of sufficient significance. Interferometrist (talk) 12:36, 22 January 2016 (UTC)


 * Let me correct what I just wrote. I does apparently NOT apply to a constant conduction current through a resistance since in that case dD/dt =0. Conduction currents hadn't even been considered in this article but ARE handled correctly using ExH since H will reflect the current density. So I don't really see its applicability beyond dielectric losses in the static case, which as I pointed out isn't really a static case since there is only loss because the dielectric polarization is discharging presumably due to leakage. So it may be of theoretical interest (in terms of pure physics) in a way I can't appreciate (and my French isn't good enough to appreciate the paper I'm afraid) but not for engineering or solving useful problems (most of which involve AC fields). Interferometrist (talk) 12:59, 22 January 2016 (UTC)


 * I cannot tell if this belongs here of if it is just a one-off approximation that is close enough under a special circumstance. In statics, curl E and displacement current are zero.  In dynamics, in general, both are non-zero.  In quasi-statics, we usually take the displacement currents as zero.  In the regime of the magnetic free Poynting vector, as described, curl E is taken to be zero and displacement current is non-zero.  That would seem to be limited domain of applicability. If it is going to stay in the article, it needs more specific information about the circumstance where it applies. There needs to be a justification as to why curl E can be taken as zero in this regime.  Finally, does it apply when the dielectric is lossy or only when the dielectric is ideal?Constant314 (talk) 17:14, 22 January 2016 (UTC)


 * I agree that the frame of quasi-electrostatic (curl E zero with displacement currents) is often badly understood, however this frame is well described in some books such as Melcher. Quasi-electrostatics can be applied for instance to the capacitor even if working in the GHz frequencies (if the device is small enough). I planned to start a new page named "Galilean Electromagnetism" (many references are available) that will give a new sight on such quasi-static (= non relativistic) approximations. In the current page my idea was to clearly state that in the Maxwell frame, as well as in the QED frame,and in contrary to relativity, the local momentum P is not defined unambiguously. The interest of the :$$\mathbf{S}' = -\mathsf{V} \frac{\partial \mathbf{D}} {\partial t}$$ where V is the electrostatic potential is that this local vector leads to the same overall balance but with an energy flux oriented along the axis of the capacitor and then somehow more physical. However the question "is it important enough to be mentioned here" is a matter of debate. If you don't like the idea of "magnetic free poynting vector" why not add a section  Poynting vector in quasi-static approximations Henri BONDAR (talk)
 * Just submitted the  Galilean electromagnetism Draft:Galilean electromagnetism page with a link to a recent article "Forty Years of Galilean Electromagnetism", I guess this short content might have been added as a section in Electromagnetism or History of electromagnetic theory but both are already long and complex pages and not totally satisfying according to me(especially the second one), may be we may contribute to do the cleaning job.Henri BONDAR (talk)

Energy flux vs. photon energy
A nice addition to this subject could be the relation between the quantifiable energy of a photon (E=wh) to the Poynting vector and how one can be derived from and represented by the other. What do you think? — Preceding unsigned comment added by 62.145.70.245 (talk) 11:47, 21 June 2013 (UTC)
 * I second that 77.125.22.252 (talk) 00:38, 12 February 2017 (UTC)

To Wcherowi
“Formulation in terms of microscopic fields” section comes from primary sources; these statements required reliable secondary sources. Thus this section should be removed according to your policy. — Preceding unsigned comment added by 125.70.179.105 (talk) 20:09, 3 May 2017 (UTC)

Adding the curl of a vector field
The statement that adding the curl of a vector field to S is incompatible with relativity is not correct. The modification is a particular instance of the fact that the divergence ∂jTji of an energy-momentum tensor is unaffected by adding a term ∂kZkji to Tji where Zkji = -Zjki. (See, eg: https://www.google.com/search?client=firefox-b-d&q=belinfante+rosenfeld+tensor .) The asymmetric Noether energy-momentum tensor for electromagnetism and the symmetric tensor (in which energy flow and momentum density are identical) are related in this way. Ericlord (talk) 06:24, 8 February 2020 (UTC)