Talk:Maxwell's equations

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Relationship between differential and integral formulations?[edit]

The article has the following quote:

"The differential and integral formulations of the equations are mathematically equivalent"

I do not believe this is correct because the integral formulation carries boundary conditions while the differential formulation doesn't. Any thoughts? --Frozenport (talk) 10:27, 19 February 2015 (UTC)

Whenever you solve partial differential equations involving space and/or time, you need corresponding boundary and/or initial conditions. And yes, the set of equations are mathematically equivalent, you get from the differential forms to the integral forms by the relevant vector calculus identities. M∧Ŝc2ħεИτlk 10:46, 19 February 2015 (UTC)
Yes, equivalent in their shared domains of applicability. One can pick mathematical nits and say that there are cases where the integral formulation is defined and the differential formulation is not, where the fields are integrable but not differentiable (e.g. in the classical context when the charge and current is confined to a surface). But this does not relate to the original question with regard to boundary conditions, only to a strict interpretation of general mathematical equivalence. I am not advocating a change, though. —Quondum 16:54, 19 February 2015 (UTC)

"Microscopic" versus "macroscopic"[edit]

In the section ""Microscopic" versus "macroscopic"" E and B look like they are both "microscopic" and "macroscopic" fields. There is an averaging and therefore I think the last statement in "Auxiliary fields, polarization and magnetization" is not pertinent, or is misleading, or need an explanation on what it shows.Ludo987 (talk) 09:50, 28 April 2015 (UTC)

D and H are the macroscopic fields, which include permittivity and permeability of macroscopic objects. Fundamentally, it is the electrons in atoms that cause these effects. Gah4 (talk) 18:19, 13 August 2015 (UTC)


Since the mid-20th century, it has been understood that Maxwell's equations are not exact but are a classical approximation to the more accurate and fundamental theory of quantum electrodynamics.

As well as I know it, Maxwell's equations satisfy special relativity. Classical is often used in descriptions that satisfy Newton but not Einstein. Should this say "relativistic approximation" or "non-quantum approximation"? Gah4 (talk) 18:22, 13 August 2015 (UTC)

Not necessarily. "Classical" more generally sometimes means "not quantum": Newtonian mechanics or Einstein's special/general relativity. Maxwell's equations are not "relativistic approximation"s because special relativity by design is already consistent with (you even pointed this out). "Non-quantum approximation" is better, feel free to go ahead and change. M∧Ŝc2ħεИτlk 19:15, 13 August 2015 (UTC)
In mechanics "classical" means pre-relativity. In field theory, classical means non-quantum, i.e. it includes general relativity (and of course EM). (I'm not sure about topics like statistical mechanics tbh.) We actually have an article, Classical field theory, that could be linked under "clasical". YohanN7 (talk) 19:40, 13 August 2015 (UTC)


In addition, E and B are mutually perpendicular to each other and the direction of wave propagation, and are in phase with each other. A sinusoidal plane wave is one special solution of these equations.

In the sinusoidal case, special solution as it says, E and B are in phase. That is, they are both sinusoids with a constant ratio. But for other waveforms, it isn't so easy to define phase. Should the in phase comment apply only to sinusoids? Gah4 (talk) 23:24, 19 August 2015 (UTC)

circularly polarized wave, E alone is 90 degrees out of phase with is shown. B not shown would be at right angles to E, and also at right angles to the direction of propagation.
For propagation in vacuum, phase is a useful concept in general, since we can analyze waves into sums of sinusoids. Another solution is for the circularly polarized wave which is the animated sinusoid, a spiral, shifting from E only and gradually transferring to B only, and back to E only, again in the picture. --Ancheta Wis   (talk | contribs) 02:24, 20 August 2015 (UTC)
The circular polarized case needs an Ex, Ey, Bx, By, which that diagram doesn't show. Gah4 (talk) 18:09, 20 August 2015 (UTC)
Meaning the range of the blue and red projections on the x & y axes are but half the story, I presume. --Ancheta Wis   (talk | contribs) 18:49, 20 August 2015 (UTC)
Even more, I don't know which half. First I thought it was Ex and By, but maybe Ex and Ey. I don't know how to make these diagrams, and maybe one with Ex, Ey, Bx, By would be too hard to understand when looking at it. Gah4 (talk) 20:23, 20 August 2015 (UTC)
Um, actually in circularly polarized light the spiral in the animation is just a single field, either E or B but not both. If it's E, then the animation shows it shifting from purely Ex to purely Ey and so on, so that the magnitude of E is constant. The magnitude of B is also constant and B is always at a right angle to E. If you only look at the field components along a single axis, then it looks E and B are out of phase. For example, looking along x we will see that when Ex is at a maximum (or minimum), Bx is zero, and visa versa. --FyzixFighter (talk) 13:14, 21 August 2015 (UTC)
Yes. But notice that the article animation has a red E and blue B, and that the caption here mentions E and B. Would it be too much to have a circularly polarized version, with rotating E and B? (That is, not components of E and B, but the actual vector E and B in perspective?) But I don't know how to make one. Gah4 (talk) 17:40, 21 August 2015 (UTC)
Ancheta's caption of the animation is incorrect. The red and blue are not different field but are orthogonal components (red=y and blue=x) of the single field, either E or B. See also circular polarization and polarization (waves)#Polarization state (which has a correct caption for the animation). An equivalent animation with both E and B would look similar but would have a double-helix like structure rotating around the axis of propagation. --FyzixFighter (talk) 18:08, 21 August 2015 (UTC)
Conveniently, the caption doesn't actually say that one is E and the other B, but does seem to suggest it. Yes, the double-helix is what I was thinking about. Gah4 (talk) 19:51, 21 August 2015 (UTC)
I apologize if the caption is incorrect. But what does the rotating arrow alternately red and blue signify to you? It seems that the blue and red projections alternately apply to the arrow... --Ancheta Wis   (talk | contribs) 21:31, 21 August 2015 (UTC)
I'm holding my fingers in the poynting vector S=ExH mnemonic we learned in school: right hand, forefinger poynting in the direction of propagation forward S, thumb sticking upward E, middle finger projecting to the left H. I rotate my thumb to the right 90 deg , poynting finger still points forward, and now the middle finger sticks upward, replacing the direction formerly held by my thumb. Now I compare to the animation. The arrow seems to change color 4 times in one cycle at the corresponding changes of orientation of E and H. --Ancheta Wis   (talk | contribs) 22:54, 21 August 2015 (UTC)
You asked what the rotating arrow alternately red and blue signify to me - if this is describing the electric field for circular polarization, then how red the arrow is corresponds to the magnitude its y-axis projection at that instant, and how blue to its x-axis projection. Another way to describe this is that circular polarization is the superposition of horizontal and vertical polarizations with equal amplitude and a 90° phase delay between the two. The colors then correspond to the contribution of each polarization to the total E (red=vertical, blue=horizontal) at that instant in time. The result is that the E vector has a constant magnitude but changes direction in a rotary manner. H would show up as a second vector orthogonal to E and also of constant magnitude, which would also trace out a second helix so that each instance ExH would give you the correct Poynting vector and direction of propagation. --FyzixFighter (talk) 23:31, 21 August 2015 (UTC)
And the reason why E and B have to be in phase ... otherwise the Poynting vector averages to zero. We could have one with red E field, x and y components, blue B field, x and y components. That would match the article diagram for circular polarization. Gah4 (talk) 00:23, 22 August 2015 (UTC)
Fixed caption. Danke gut, as we say in Spanish German. --Ancheta Wis   (talk | contribs) 09:22, 22 August 2015 (UTC)

B is the magnetic field?[edit]

That honor belongs to the H field according to (at least some of) my books. The field B is there the called the magnetic induction or the magnetic flow density. I thought that that order of business was the most common. YohanN7 (talk) 13:52, 24 November 2015 (UTC)

There have been extensive discussions on terminology, see talk:Magnetic field#Definition. The definition of H is given in this and other articles (e.g. Maxwell's equations #Constitutive relations), so if readers want to convert B to H, they can. MŜc2ħεИτlk 14:07, 24 November 2015 (UTC)
See A Treatise on Electricity and Magnetism for the ultimate reference. — Rgdboer (talk) 01:58, 25 November 2015 (UTC)

Alternative formulation section[edit]

What is A ? You use it, but you don't define it anywhere on the page. Non-expert readers (that is to say, most people reading the page) won't have a clue what this is on about, so that's bad writing. — Preceding unsigned comment added by (talk) 12:07, 13 March 2016 (UTC)

A is defined below the table, in the first item; look below the last line of the wikitable at Maxwell's equations #Alternative formulations
Formalism Formulation Homogeneous equations Non-homogeneous equations
"where ... A is the vector potential "
--Ancheta Wis   (talk | contribs) 12:43, 13 March 2016 (UTC)

Student query I could not answer[edit]

This text comes from the article.

"Maxwell's addition to Ampère's law is particularly important: it shows that not only does a changing magnetic field induce an electric field, but also a changing electric field induces a magnetic field."

I could not answer this question from a school age student.

If each field is induced by a change in the other, why do all the text book diagrams show the magnetic and electric fields in phase? When the E field is changing fastest (passing the zero line) the B field should be maximum. Are all the text books wrong? --Neil (talk) 11:36, 20 June 2016 (UTC)

This is something for our WP:Reference desk/Science, not for article talk pages, where we should discuss the article, not the content—see wp:Talk page guidelines. Good luck at the ref desk! - DVdm (talk) 11:39, 20 June 2016 (UTC)
OK, but if the article isn't clear about something, then we can discuss it here to see if it can be fixed. Gah4 (talk) 20:33, 27 July 2016 (UTC)
If all the text books are wrong, this article will need correcting too. If the text books are correct, a simple explanation in this article would be nice to have. --Neil (talk) 11:48, 20 June 2016 (UTC)
If all the text books are wrong, then—by design—Wikipedia will (and must) be wrong too. DVdm (talk) 12:10, 20 June 2016 (UTC)
This common confusion is caused by the ambiguity of the English language. You should use mathematical equations rather than words to examine this question. A slightly better translation of the equations into words would be "a (shear) change in the magnetic field over space causes a change in the electric field over time, just as a (shear) change in the electric field over space causes a change in the magnetic field over time". JRSpriggs (talk) 18:59, 20 June 2016 (UTC)

The easy answer is to say "special relativity" and leave it at that. If you want to ask where the E and B are, where the energy is, you have to specify the reference frame. If you consider a wave on a spring, it is not so hard to derive the wave equation, which has energy moving between kinetic (motion of the spring), and potential (stretched spring). In the case of mechanical waves in pretty much any system (springs, strings, sound through air) at any point, energy moves between kinetic and potential. In the EM case, it is usual to equate one of E and B with kinetic, and the other with potential, though it doesn't matter which. (Equate E with moving electrons, or with the field that causes them to move.) In any case, energy does move between E and B, but where is that energy? Consider 1/4 cycle, when a changing E is creating B, and also that, at the speed of light, the wave has moved on 1/4 of the wavelength. This means you can't ignore special relativity, which we already knew, but now you can see why. In the spring case, the spring has a fixed reference frame. In EM case, there is no fixed frame to look at it in. E and B are in phase in any frame. Gah4 (talk) 20:33, 27 July 2016 (UTC) Einstein and Feynman wrongly say changing E causes H and changing H causes E. These ideas are derived from Oersted and Faraday’s experiments, which are misinterpreted (by them and everyone else.). . I expect the Wikipedia Thought Police to rapidly remove this (dirty secret) paragraph. Ivor Catt 13.30 GMT, 27 Feb 2018 — Preceding unsigned comment added by (talk) 13:30, 27 February 2018 (UTC)

Derivation from Quantum Mechanics[edit]

It may be useful to discuss the derivation of Maxwell's equations from quantum mechanics. Some material on this topic is being gathered at: Including a paper at: Thanks! --Lbeaumont (talk) 12:00, 27 July 2016 (UTC)

Rather than describing the photon with a complex combination of E and B, it is more usual to describe the photon with the electromagnetic four-potential. JRSpriggs (talk) 18:05, 27 July 2016 (UTC)

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Light-by-light scattering[edit]

Is it worth referring to an article suggesting an exception to the equations?

Yes, but not here. There. YohanN7 (talk) 09:52, 6 September 2017 (UTC)
The fourth paragraph in the lead covers the "exceptions" (as you put it) decently. YohanN7 (talk) 09:54, 6 September 2017 (UTC)

Maxwell's field equations can be formulated in the form of Dirac equation[edit]

Besides those formulations given in this article, Maxwell's field equations can also be formulated in the same form as Dirac equation. Please refer to an article entitled FORMULATION OF MAXWELL FIELD EQUATIONS FROM A SYSTEM OF LINEAR FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS posted on ResearchGate by Vu B Ho for more details. (talk) 09:01, 15 February 2018 (UTC)

But see ResearchGate, which is criticized as a social networking site for those with institutional affiliations, and which lists predatory journals. --Ancheta Wis   (talk | contribs) 09:18, 15 February 2018 (UTC)
Might this paper be found on 09:21, 15 February 2018 (UTC)

Table format[edit]

In the Formulation in SI units section, the table as of a few days ago was too big. It didn't fit in the standard Wikipedia page width, and the text was awkwardly shoehorned into narrow columns.

The current format fits into the standard width, which is a big help for readers using mobile devices. The headings are simple and obvious, and the whole table fits in one or two pages.

The meanings should absolutely NOT be deleted, because they do a great job of explaining the equations to readers.

Coder Dan (talk) 05:13, 24 June 2018 (UTC)

User:JohnBlackburne has reverted my changes to the table format twice, even after I simplified the format. I understand that not everyone is interested in editing tables, but the original table was too big. The compact format makes the table more accessible to readers who use mobile devices or large font sizes, and the formula meanings in the table are a valuable aid in helping readers understand the formulas.

Coder Dan (talk) 16:18, 24 June 2018 (UTC)

The problem with your preferred format is it is largely incoprehensible. In particular the central column contains both the names and the meanings, jumbled up so it is unclear which goes with which; e.g. ""The electric flux through a closed surface is proportional to the charge inside the enclosed volume." is right next to "Gauss's law for magnetism", but they do not go together. The formulae are similarly broken up and disconnected from their headings. It’s only possible to interpret it if you already know the formulae and meanings, so know which go with which.
Looking to address the width problem I removed the meanings from the second table as there is no point repeating them. This then looked so much better with them removed that I did the same to the first table. Again the table is vastly improved, and readers can find fuller descriptions of the meanings at #Conceptual descriptions as well as in the individual articles by clicking on the links. I think this is much better that including the meanings in the table where they don’t easily fit.--JohnBlackburnewordsdeeds 16:20, 24 June 2018 (UTC)
> your preferred format ... is largely incoprehensible.
I don't believe that for a second. The two-row format is simple and obvious.
> the central column contains both the names and the meanings
There is no "central column".
> "The electric flux ..." is right next to "Gauss's law for magnetism", but they do not go together.
There was a blank line between them in the original version, but you reverted that. I would be delighted to discuss minor variations in the format.
> The formulae are similarly broken up and disconnected from their headings.
"broken up and disconnected" is exaggeration. The added complexity is minimal.
> It’s only possible to interpret it if you already know the formulae and meanings.
This is pure gibberish. It sounds bad, but there's no truth to it at all.
> I removed the meanings from the second table as there is no point repeating them.
That's fine.
> This then looked so much better with them removed that I did the same to the first table.
The purpose of Wikipedia is to provide useful information, not pretty pictures.
> readers can find fuller descriptions of the meanings at #Conceptual descriptions
Readers also like to have convenient summaries. The meanings in the table are very helpful to readers.
> I think this is much better that including the meanings in the table where they don’t easily fit.
The meanings fit in the compact format just fine.
Coder Dan (talk) 17:13, 24 June 2018 (UTC)
I agree with John Blackburne that this table is confusing/badly designed. Headbomb {t · c · p · b} 18:06, 24 June 2018 (UTC)
On the other hand, I have found the meaning in words, when placed directly below the mathematical notation by Coder Dan, restate and reinforce the notation. This is especially meaningful to me when I mentally compare the equations, in their various formulations, to other other articles, such as Stokes' theorem. That said, the verbal descriptions which are currently elided in the John Blackburne version actually correspond to the integrals column. The two columns side by side really say the same thing to those of us who studied this subject in school. That suggests we might replace the partial differential column with the verbal restatement for the readers who need it (which is most readers).
I notice that the columns currently have a minimum width at which we can no longer see both columns.--Ancheta Wis   (talk | contribs) 19:00, 24 June 2018 (UTC)

Now that Ancheta Wis mentions it, the original table didn't really make sense, since the meanings are specific to one column.

I understand the logic behind the table format, and many technical workers display large amounts of data with large screens and small fonts, so it might not seem like a big deal. But other people use large fonts or mobile devices, so it's a huge problem for some of us.

The problem is that text is primarily horizontal in extent, whereas web pages are mostly vertical. There's a fundamental mismatch between those two shapes, and tables just makes it worse (and the meanings are too good to delete).

Coder Dan (talk) 19:50, 24 June 2018 (UTC)

Thank you for your rapid response to the comments. Lest we swing to a text-only format, I have to say that the tables for the relativistic and gauge formulations have their place in the article, as they serve to compress a ton of information that can be ignored at the verbal level. But the article is very suggestive when we 'read between the lines' but cannot write about here because we are editors only. I hope others get this same impression about the encyclopedia article, and are inspired to write these original thoughts elsewhere. --Ancheta Wis   (talk | contribs) 20:22, 24 June 2018 (UTC)

For comparison, there's a version of the page with compact tables here.

Coder Dan (talk) 16:07, 25 June 2018 (UTC)

Hi Andy. Other editors have been reverting tables with more than one row per entry, so I'm not sure how your most recent version will be received. Also, as Ancheta Wis pointed out, the meaning only applies to the integral form of the equations. There's a cleaned up version of the table here if you're interested.

Coder Dan (talk) 16:15, 25 June 2018 (UTC)

Equations in caption[edit]

Equations in caption to figure in "Vacuum equations, electromagnetic waves and speed of light" look wrong. Please check. —DIV ( (talk) 14:08, 5 August 2018 (UTC))

I asked this question some years ago and the OP convinced me. What exactly is your question? --Ancheta Wis   (talk | contribs) 02:14, 6 August 2018 (UTC)
Poynting vector, plane wave, and Maxwell's equations#Vacuum equations, electromagnetic waves and speed of light are all consistent. What is it that you see? --Ancheta Wis   (talk | contribs) 02:40, 6 August 2018 (UTC)
Specifically, the equations E = E0 sin(−ωt + kr) and B = B0 sin(−ωt + kr) imply E/E0 = B/B0. But the figure shows the two fields perpendicular — as seems to be generally accepted. How is the perpendicularity accounted for in those equations? (Or if they're truly overlaid, then the figure is wrong.)
—DIV ( (talk) 11:46, 9 August 2018 (UTC))
And furthermore, the equations appear to set a vector field equal to the product of a scalar and the output of sine, for which the result is a scalar . —DIV ( (talk) 12:07, 9 August 2018 (UTC))
Yes, Maxwell's equations are functional equations, whose solutions are functions. Your observation about E and B is baked-in to the history of electromagnetic theory. There are experiments from the nineteenth century to measure the ratio of E to B, observed to be a constant, 377 ohms, the impedance of free space, for plane waves. Formulations where E and B are of the same type (per relativity) are in Maxwell's equations#Relativistic formulations. And the graphical solution for the plane wave shown is a specific case with a restricted geometry, but which models linearly polarized light waves from the fixed stars to our planet very well. --Ancheta Wis   (talk | contribs) 13:33, 9 August 2018 (UTC)
I rather think that B0 and E0 are mistakenly formatted as scalars, where they should be vectors.−Woodstone (talk) 14:22, 9 August 2018 (UTC)
Woodstone is right. I modified the caption accordingly. JRSpriggs (talk) 18:45, 9 August 2018 (UTC)
Thanks for clarifying. —DIV ( (talk) 05:29, 10 August 2018 (UTC))

Notation in Differential forms[edit]

Definition of asterisk (*)[edit]

In the "Differential forms" given under Alternative formulations, an asterisk (*) is used repeatedly, and there is no explanation of its interpretation there or elsewhere in the article, nor in the "main article" mentioned which links to Mathematical descriptions of the electromagnetic field. The following section on Relativistic formulations provides more helpful explanation of 'special' notation, but the asterisk doesn't appear in that section at all. —DIV ( (talk) 02:27, 12 August 2018 (UTC))

I'm not sure when it was omitted but the Hodge star operator must have been mentioned at one time. --Ancheta Wis   (talk | contribs) 02:42, 12 August 2018 (UTC)
Hodge star is mentioned. The Laplace–Beltrami_operator#Laplace–de Rham operator has explanation for the asterisk usage. So the haphazard notation for Hodge star must have been inserted by multiple editors, since there is also a 5-pointed star notation for asterisk in the article. --Ancheta Wis   (talk | contribs) 03:00, 12 August 2018 (UTC)
Yes, the Hodge star operator is defined and used elsewhere in this article. The problem is that the asterisk is not consistent with that defined "star" notation, if that's what it was supposed to represent. Rather, it looks more like a convolution operator or a complex conjugate.
The Laplace–de Rham operator article (section) defines the asterisk as the Hodge star operator, and uses it as such, so it's self-consistent (albeit not ideal). The Hodge star operator article uses a star symbol (not an asterisk) for the operator — it also seems to use asterisks for some other (unexplained) purpose.
—DIV ( (talk) 13:14, 12 August 2018 (UTC))
The asterisk notation for Hodge star came first. The five-pointed star is proposed here with some examples for its use, as well as appearing in the encyclopedia article; versions of the encyclopedia article also used the wedge notation associated with Hodge. Ancheta Wis   (talk | contribs) 22:01, 12 August 2018 (UTC)
Hi, Ancheta Wis. In what sense did the asterisk notation come first? If that was what was first used in the WP article a decade or so ago, I don't think I'd take that as a reliable primary reference — I'd guess it could also have been due to technology limitations, lack of mathematical typesetting options, or convenience/laziness. I had a quick look at the link you provided to The Hodge Operator Revisited (dated circa 2015): I can't clearly see that they've proposed using a star instead of an asterisk. From my lay skimming it seemed that they proposed using an existing operator for a new application. Star notation was also used in The Geometry of Supermanifolds and New Supersymmetric Actions (dated 2015), with no statement about using a new symbol. It's not my area, so I don't know who pioneered the symbols in the printed literature. So I don't dispute the history, I'm just saying that I couldn't see convincing evidence in the information you provided.
Being that it's not my area, I also don't know about the "wedge" symbol, but it wasn't clear how it could have been used instead of the star (or asterisk), given that presently the wedge and star seem to be used concurrently with distinct meanings (as in the Hodge star operator article and the arxiv manuscript linked above (e.g. equations 1.15, 1.16, 4.16, 4.17). Or were you trying to say something else about the "wedge"?
Anyway, the main point I'm making is that within each article the nomenclature and symbols should be consistent: they should not vary from section to section. Even in an extraordinary case where there might be a reason to use differing notation in one section, it must be explicitly defined (and probably justified too) in that section of the article.
—DIV ( (talk) 10:35, 13 August 2018 (UTC))
This paper on Hodge theory, p.2 has an explicit statement that asterisk (*) is Hodge star. The wedge ∧ and the external derivative/ differential (italic d) are introduced on p.1. I can now see why you are so cautious about assigning notation, considering the incomplete List of things named after W. V. D. Hodge. So the Maxwell's equations article is a jumping-off point for mathematicians. --Ancheta Wis   (talk | contribs) 11:52, 13 August 2018 (UTC)

Notation for "d" in derivative[edit]

In the "Differential forms" given under Alternative formulations, the differential "d" is set italic, making it look like a variable (such as diameter, distance, ...). I strongly recommend that the convention be followed that all variables be set italic, and everything else (text, labels, and operators) be set roman, as has been done in Relativistic formulations. —DIV ( (talk) 02:34, 12 August 2018 (UTC))

Fig. in section "Bound charge and current"[edit]

Shouldn't the microscopic dipoles in the figure have opposite polarization (positive up, negative down)? — Preceding unsigned comment added by (talk) 14:43, 3 October 2018 (UTC)

The figure is showing how the microscopic polarization aggregates to form the macroscopic effect of apparent surface charges. You are confusing that with how an externally imposed field might induce a polarization in the material (which is a different effect). JRSpriggs (talk) 05:06, 4 October 2018 (UTC)


It is rare that a well-established article gets to this stage; there is a protocol, wp:BRD that editors adhere to, to stay out of trouble: Wp:3RR is the bright-line we cannot cross, as editors. So we need to discuss the changes on this talk page. Please post your points here if this message is not clear to you or if you need help. Otherwise ... --Ancheta Wis   (talk | contribs) 16:44, 9 May 2019 (UTC)

Indeed, Ancheta Wis! I've protected the page for a short while in the hope that everyone involved will engage in some discussion – which I'm relieved to see has already begun below. Justlettersandnumbers (talk) 12:58, 11 May 2019 (UTC)
@User talk: Co-scienza, Based from a reading of this diff there is a very clear implication, namely that there is an experimental outcome which could be observed astronomically. We need at least one citation for that implication (ie, a prediction of an experimental outcome [meaning not yet observed]). Failing that citation, I urge you to self-revert your edits, about which you clearly believe. But the article was stable before your edits, so there needs to be a justification for the newest changes. I can explain my reasoning if you like, but I am waiting for your good-faith response. --Ancheta Wis   (talk | contribs) 15:30, 11 May 2019 (UTC)


There are recent edits with the edit summary mentioning the electromagnetic tensor. There is a convenience of Gaussian units, in that the components of the EM tensor are components of E and B, with no factors of 1/c needed. Recent edit summaries mention the components, but I didn't figure out what they actually did. It seems that the electromagnetic tensor page uses only SI units, and so does not show this. (And it mentions the use of SI units in a very tiny font.) Gah4 (talk) 19:05, 10 May 2019 (UTC)

In the note #1 which Co-scienza added to the lead, he says "... are the components of a unique field, as well highlighted by the their formulation in Gaussian Units where E and B have the same units ...". I am not disputing the fact that those Gaussian units are the same. I object to the note on the grounds that it is misleading — it encourages the dangerous myth that all the components of a tensor must have the same units. JRSpriggs (talk) 00:45, 11 May 2019 (UTC)
I didn't figure out what Co-scienza did, but it didn't seem to match the edit summary. I suppose the components don't have to have the same units, but the result of expanding the tensor, in the places that it is used, have to be dimensionally consistent. Otherwise, the EM tensor is supposed to show the symmetry of electromagnetism, which is easier to see when they are dimensionally the same. Gah4 (talk) 07:08, 11 May 2019 (UTC)
The stance of the formulation of the EM tensor is symmetric (so probably time-symmetric). But there have been quantum computations on the IBM quantum computer that explain the observability of the arrow of time. (Feynman's point that playing a movie backward makes us laugh.)
From the developmental point of view for the equations of physics, the conservation of charge and mass seem to state observations about the classical time-scale, and yet the mathematicians (such as Hilbert in 1915) were concerned about these conservation laws (that they do not seem inescapable).
Also, from the point of view of the history of the equations of physics, the equations embody experimental observations. But to use GR as the justification of the EM tensor is ahistorical; the use of GR is non-intuitive, from the developmental view of physics, unless mass, charge, and spin are taken as givens. And yet the EM tensor seems to make no statement for the physical evolution of our universe. --Ancheta Wis   (talk | contribs) 16:24, 11 May 2019 (UTC)

In note 1 I don't speak about tensors, but simply of E and B. Tensors (electromagnetic tensor or energy-momentum for the electromagnetic field) have not the same units (if we use time in seconds and not measured in ct). In nmy note 1 I anticipate what said in the Gaussian formulation chapter. If you want to erase it I agree, but speaking of fields (and not of a unique field) at the beginning is to adopt an "engineer" approach and not a physical one, that naturally has been strongly influenced by Einstein's work.Co-scienza (talk) 14:16, 14 May 2019 (UTC)

To answer to Ancheta, to tell that Maxwell's equations are good also in general relativity, means do not recognize the limits of Maxwell's equations that are linear and not non-linear as in the cuved spacetime of the GR equations (also in the ideal absence in the universe of other energy-momentum fields (so also without mass, spin, etc.), strong pure electromagnetic field is not well described by Maxwell's equations that are exact only in the ideal Minkowski spacetime. Co-scienza (talk) 14:16, 14 May 2019 (UTC)

To Co-scienza: It is true that E and B (with appropriate scaling factors) are parts of one field, that is, the same tensor. But everything in the note after "unique field" is irrelevant (a mere coincidence) and having it in the note highlights your subservience to the myth.
By the way, please sign your comments with four tildes. JRSpriggs (talk) 08:28, 12 May 2019 (UTC)
Well, there is no citation for Co-scienza's statements; what if I were to wait til 15 May 2019 UTC, and in the absence of a response, restore the page? --Ancheta Wis   (talk | contribs) 12:45, 12 May 2019 (UTC)

To Ancheta: in note 1 I added a simply method to verify what is said after, in "Formulation Gaussian units convention": I read, textually:"[...] These definitions are often preferred in theoretical and high energy physics where it is natural to take the electric and magnetic field with the same units". In my note I wrote to go to page 819 of "D. Jackson, Classical Electrodynamics, 2nd edition" and, knowing that [E]=c[B] in SI units, verify, using the Jackson's Table, that E and B have the same units in Gaussian formulation. Probably this very simple calculation is superfluos.Co-scienza (talk) 14:16, 14 May 2019 (UTC)

Regarding my note 2, I would cite: "W. R. Davis, Classical Fields, particles and the Theory of Relativity", Gordon and Breach, 1970" that on page 176, Note 3, he said that the necessity of introducing the Lorentz force density equation (I would say: or the Maxwell stress tensor), an equation specifying the interaction of the field with its sources, in addition to the field Maxwell's equations, is characteristic of linear field theories. In my opinion for this reason it is necessary to highlight (note 2) that Maxwell's equations are for a "weak field" and that GR extend this model to a non-linear model.Co-scienza (talk) 14:16, 14 May 2019 (UTC)

I found interesting what Gah4 says. To resume, in my opinion: - In Gaussian units E,H,D and B have the same units (see for example, "J. Franklin, Classical Electromagnetism, Dover (2017)", page 253).

- In Gaussian formulation, Gauss (for B), Oersted (for H) have the same units [cm^(−1/2)g^(1/2)s^(−1)] as also D and E.

- The electromagnetic tensor in Gaussian formulation is simpler (as Gah4 says: without 1/c) and is used in much texts

- Tensors are constructed to have the same units for all components, but obviously, the "uniformisation" by c or 1/c, it is clear the dimension difference between time and space components (see the energy-momentum tensor). But for the electromagnetic tensor, F, I prefer the homogeneous Gaussian that is preferrable also in the Lorentz force density to highlight the split (by relative speeds or accelerations) of a unique field. Co-scienza (talk) 14:16, 14 May 2019 (UTC)

I learned this mostly from the first edition of Purcell's Electricity and Magnetism which, unlike most other books, explains it through special relativity. As above, there is no need to use general relativity for it. If you use four-vectors for vector quantities, then it comes out naturally to use the EM tensor with them. It seems that the second edition continues using Gaussian units, but the third is updated (sic) to use SI units. I remember homework problems with charges moving at some speed like c/2, and also an observer moving at a similar speed, either in the same or opposite direction. Learning in both unit systems, you get used to switching between them, and to see the advantage and disadvantage of each. It seems to me that WP should fairly explain these advantages and disadvantages, in describing both systems. Gah4 (talk) 13:47, 13 May 2019 (UTC)
From reading the preface to different editions, it seems that earlier Purcell did not ignore SI. Problems were given in both Gaussian and SI units, and equations were explained in both systems. The 3rd edition, also, does not ignore Gaussian units, but seems to put most of the explanation into an appendix. Gah4 (talk) 15:09, 13 May 2019 (UTC)

Yes, Gah4, WP well explain what I said and what you are saying. Co-scienza (talk) 14:16, 14 May 2019 (UTC)

Back to Hodge star[edit]

Re-reading " The topological condition is again that the second real cohomology group is trivial", I propose casting this sentence to be more congenial for physicists by rewording the usage "trivial" to "compatible with the definitions". Of course, the problem then shifts to providing evidence for the definitions. -- 12:20, 15 May 2019 (UTC)

Reading exterior derivative, the statement above applies to 2-forms, meaning Stokes' theorem applies. My problem would then be "how is matter (or charge, or spin) to be incorporated into this formulation?". Citations needed, of course. --Ancheta Wis   (talk | contribs) 14:24, 15 May 2019 (UTC)