Talk:Maxwell's equations

Relationship between differential and integral formulations?
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&and;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)

Accepting that they are equivalent, then the first sentence: "Maxwell's equations are a set of coupled partial differential equations " is, well, not so much wrong, as misleading. I first learnt (and tried to understand) them in line integral form. I am proposing that this sentence be altered to include the line integral form. Let's face it, T-shirts seem to offer both forms in roughly equal amounts :D — Preceding unsigned comment added by 2001:8003:E48C:E601:A408:ED0C:B596:882 (talk) 11:49, 24 June 2022 (UTC)

"Microscopic" versus "macroscopic"
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)
 * It seems way more pertinent to emphasize first that this is vacuum vs matter then say matter formulation does not hold at microscopic level in matter while E B formulation does (which yeilds name macroscopic fields). The statement that formulation are equivalent should be removed either way. (basing purposed changes off of Zhangwill Modern Electrodynamics). Someone should let me know what they think. Ehaarer (talk) 21:18, 2 May 2023 (UTC)

Classical
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&and;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)
 * I was thinking about this not so long ago. In the early days of quantum mechanics, around the time that Einstein explained the photoelectric effect, and even though Plank showed that his constant was needed to explain black body radiation, Einstein was one of the few believing in quantization of the electromagnetic field and photons. Partly that was because Maxwell's equations worked so well. Many believed that it was only a mathematical trick to generate the black body spectrum. Others that electron energy levels changed in quantum jumps, but that the EM field was still not quantized.  Gah4 (talk) 02:40, 12 February 2020 (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)
 * I was thinking about this not so long ago. In the early days of quantum mechanics, around the time that Einstein explained the photoelectric effect, and even though Plank showed that his constant was needed to explain black body radiation, Einstein was one of the few believing in quantization of the electromagnetic field and photons. Partly that was because Maxwell's equations worked so well. Many believed that it was only a mathematical trick to generate the black body spectrum. Others that electron energy levels changed in quantum jumps, but that the EM field was still not quantized.  Gah4 (talk) 02:40, 12 February 2020 (UTC)
 * I was thinking about this not so long ago. In the early days of quantum mechanics, around the time that Einstein explained the photoelectric effect, and even though Plank showed that his constant was needed to explain black body radiation, Einstein was one of the few believing in quantization of the electromagnetic field and photons. Partly that was because Maxwell's equations worked so well. Many believed that it was only a mathematical trick to generate the black body spectrum. Others that electron energy levels changed in quantum jumps, but that the EM field was still not quantized.  Gah4 (talk) 02:40, 12 February 2020 (UTC)

Phase
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)
 * 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  &#124; 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  &#124; 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) (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  &#124; 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  &#124; 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  &#124; contribs) 09:22, 22 August 2015 (UTC)

B is the magnetic field?
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. The definition of H is given in this and other articles (e.g. Maxwell's equations ), so if readers want to convert B to H, they can. 'M'&and;Ŝc2ħεИτlk 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)

Agree with YohanN7. In Jackson, over his 3 editions, that $H$ field is to be used in Amperes law instead of total magnetic induction $B$. Otherwise materials with high inductance mu are underestimated.

Alternative formulation section
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 94.196.243.2 (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
 * {|class="wikitable" style="text-align: center;"

! scope="column" | Formalism ! scope="column" | Formulation ! scope="column" | Homogeneous equations ! scope="column" | Non-homogeneous equations
 * }
 * "where ... $A$ is the vector potential "
 * --Ancheta Wis   (talk  &#124; contribs) 12:43, 13 March 2016 (UTC)
 * --Ancheta Wis   (talk  &#124; contribs) 12:43, 13 March 2016 (UTC)

Student query I could not answer
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) http://www.ivorcatt.co.uk/x18j184.pdf - Ivor Catt, 1.5.2022 — Preceding unsigned comment added by 2.24.141.28 (talk) 00:28, 1 May 2022 (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)
 * For running waves E and B acquire a common phase factor of π/2. For standing waves E acquires a temporal and B a spatial phase factor of π/2. Aoosten (talk) 16:07, 19 January 2024 (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)

http://www.ivorcatt.co.uk/x0102em.htm 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.). http://www.ivorcatt.co.uk/x267.pdf. 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 86.169.30.218 (talk) 13:30, 27 February 2018 (UTC)

Answer by Ivor Catt; Maxwell's Treatise, volume 2, page 439, article 790, Fig. 67, correctly has E and H in phase. E and H do not cause each other. If they did, we could only have monochromatic light. Einstein and Feynman (and all text books and Wikipedia entries) are wrong when they say the one causes the other. http://www.ivorcatt.co.uk/x18j51.pdf ; http://www.ivorcatt.co.uk/x0102em.htm - Ivor Catt, 9.8.2021


 * I think a better explanation is that the term Maxwell added allowed him to derive a wave equation that had a propagating solution. If you chase the math, it looks like this:
 * Causal relationships between fields in a propagating plane wave.jpg
 * From E you can derive D. From D you can derive ∂D/∂t ( the electric displacement current).  From that, you derive H.  From H you derive B.  From B you derive dB/dt  (the magnetic displacement current).  Notice that the arrows mean "is derived from" and do not mean "causes".  However, when speaking casually, it is common to interchange the notion of "is derived from" with "causes".  As an aside, the two displacement current terms are legitimate fields that can be drawn and plotted just like any other field.  So, if you want to intuitively understand how E causes H and H causes E, it is easier if you use four fields.  Notice that the two displacement current terms involve differentiation.  In a monochromatic wave, that causes 90 degrees of phase shift.  The Maxwell–Faraday equation includes a minus sign that provides another 180 degrees of phase shift.  If you chase your way around the loop then, you get 360 degrees of phase shift. The gain is "unity". It is exactly the condition for self-sustaining oscillation. Constant314 (talk) 21:09, 9 August 2021 (UTC)
 * From the horse's mouth. http://www.ivorcatt.co.uk/maxwell8.pdf Maxwell's Figure 67 shows E and H in phase. I hope the Wikipedia Thought Police don't remove this. That would be defending peer revued error against the truth. Does Wales want this; peer revued material or correct material? - Ivor Catt — Preceding unsigned comment added by 2.24.141.17 (talk • contribs) 15:47, 10 August 2021 (UTC)
 * Please put new comments at the bottom, and sign all your talk page messages with four tildes ( ~ ) — See Help:Using talk pages. Thanks. DVdm (talk) 10:08, 11 August 2021 (UTC)
 * I am not trying to derive the whole thing on the talk page. If I add a few more steps, dD/dt causes curl{H}.  The curl operator adds 90 degrees of spacial phase shift that accounts for E and H aligning in a propagating plane wave. Constant314 (talk) 14:47, 11 August 2021 (UTC)
 * It is interesting to remember that Maxwell's equations were the original inspiration for special relativity. Lorentz transformation was needed to make it work. One reason to expect different phase for E and B is that in other wave systems, such as vibrations on a string, the displacement and velocity are, for a sine moving one direction on a string, 90 degrees out of phase. Note that it gets more complicated with other shapes. On the other hand, consider a standing wave on a string, such as a violin mode. In that case, the displacement nodes match the velocity nodes. Next, the same nodes for E and B is only true for an unmodulated sine going one direction. (I didn't do the math, though.) Modulation will shift the nodes. And finally, consider EM standing waves. In that case, the E nodes are 90 degrees from the B nodes. Gah4 (talk) 21:03, 11 August 2021 (UTC)
 * Funny story. In my undergrad E&M class, we had a lecture demonstration showing the similarity between sound waves (in an air column) and EM waves (in a coaxial cable). It was meant to show the connection between nodes and boundary conditions (close/open tube end, short/open end of the coaxial cable). But then it came out wrong. There is a voltage node at the end of a shorted cable, but a pressure antinode, as measured by a microphone. That should have been fine, but in the next lecture the same setup returned, but with a current probe on the oscilloscope. Now antinodes agree! Gah4 (talk) 21:03, 11 August 2021 (UTC)
 * Another thought, though, is to look at in in terms of Φ and A. That is, scalar and vector potential, which are components of a four-vector in special relativity. Gah4 (talk) 21:03, 11 August 2021 (UTC)
 * E relates to electric displacement current (dD/dt) like pressure relates to velocity (or maybe the other way). Constant314 (talk) 21:29, 11 August 2021 (UTC)
 * E relates to electric displacement current (dD/dt) like pressure relates to velocity (or maybe the other way). Constant314 (talk) 21:29, 11 August 2021 (UTC)

Derivation from Quantum Mechanics
It may be useful to discuss the derivation of Maxwell's equations from quantum mechanics. Some material on this topic is being gathered at: https://www.quora.com/Can-Maxwell%E2%80%99s-equations-be-derived-from-quantum-mechanics Including a paper at: http://www.cft.edu.pl/~birula/publ/PhotonAPP.pdf 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
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
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.101.189.62.166 (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  &#124; contribs) 09:18, 15 February 2018 (UTC)
 * Might this paper be found on arXiv.org? 09:21, 15 February 2018 (UTC)

Table format
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.

&mdash;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.

&mdash;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.-- JohnBlackburne wordsdeeds 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.
 * &mdash;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  &#124; 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).

&mdash;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  &#124; contribs) 20:22, 24 June 2018 (UTC)

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

&mdash;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.

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

Equations in caption
Equations in caption to figure in "Vacuum equations, electromagnetic waves and speed of light" look wrong. Please check. —DIV (120.17.54.174 (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  &#124; contribs) 02:14, 6 August 2018 (UTC)
 * Poynting vector, plane wave, and Maxwell's equations are all consistent. What is it that you see? --Ancheta Wis   (talk  &#124; contribs) 02:40, 6 August 2018 (UTC)
 * Specifically, the equations $E = E_{0} sin(−ωt + k ⋅ r)$ and $B = B_{0} sin(−ωt + k ⋅ r)$ imply $E/E_{0} = B/B_{0}$.  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 (120.17.127.14 (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 (120.17.127.14 (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. 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  &#124; contribs) 13:33, 9 August 2018 (UTC)
 * I rather think that B0 and E0 are mistakenly formatted as scalars, where they should be vectors.&minus;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 (120.17.168.67 (talk) 05:29, 10 August 2018 (UTC))

Definition of asterisk (*)
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 (120.18.180.5 (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  &#124; contribs) 02:42, 12 August 2018 (UTC)
 * Hodge star is mentioned. The Laplace–Beltrami_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  &#124; 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 (120.17.161.143 (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  &#124; 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 (120.17.18.193 (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  &#124; contribs) 11:52, 13 August 2018 (UTC)
 * Caution: there is another article (stemming from quantum mechanics, from 1940 to 1949) with another star operator (phase-space star product) that seems to have appeared in the 1970s. Moyal bracket and Moyal product   use a five-pointed star for an associative, non-commutative product. --Ancheta Wis    (talk  &#124; contribs) 09:44, 12 April 2023 (UTC)

Notation for "d" in derivative
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 (120.18.180.5 (talk) 02:34, 12 August 2018 (UTC))

Fig. in section "Bound charge and current"
Shouldn't the microscopic dipoles in the figure have opposite polarization (positive up, negative down)? — Preceding unsigned comment added by 207.251.102.114 (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)

3RR
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  &#124; contribs) 16:44, 9 May 2019 (UTC)
 * Indeed, ! 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  &#124; contribs) 15:30, 11 May 2019 (UTC)

Units
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 &mdash; 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  &#124; 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  &#124; 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
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  &#124; contribs) 14:24, 15 May 2019 (UTC)

Faraday's Law
I recently made two edits to the section on Faraday's Law, both of which were reverted by User:Woodstone. The first edit was to change what I consider to be an inaccurate interpretation of Faraday's law. The article says that a time-varying magnetic field creates an electric field. In terms of Ampere's equation, this is true, but in terms of Faraday's equation, we are confusing cause with effect. The electric field causes the magnetic field to change over time, not the other way around. I should acknowledge that I am not an expert, so if you have a good reason for why the electric field will magically be set to exactly what the change of the magnetic field was over the past moment of time, please let me know, but for now I want my edit to stay.

My second edit is just as important for providing an accurate, robust intuition- I note that a clockwise motion of a particle around a loop causes a proportion counter-clockwise change in the orientation of the magnetic field. I concede that the concept, and potentially my wording, are confusing, so I will happily consider alternative ways to present the intuition of a magnetic field representing rotation, while the electric field represents direction - maybe earlier in the article, maybe as a picture, maybe with better wording. But if the reader does not appreciate that the magnetic field describes the strength and orientation of a rotation, while the electric field is the strength and direction of a push, and if they don't appreciate the relationship between a particle's movement in one direction with the unwinding of the magnetic field in the other, then they simply are unable to truly appreciate the nature of Maxwell's equations. I have left this part unchanged for now, since I would like to open this to further deliberation before changing it back.

Finally, I would like to note my intention to make somewhat similar edits regarding Ampere's law. My same reasoning for why I interpret Faraday's Law as describing the electric field causing a change in the magnetic field over time, I interpret Ampere's law as saying that a magnetic field causes a change in the electric field over time, modified by the experienced currents, not that the change in the electric field causes a change in the magnetic field. If you see things differently, again please let me know. Similarly, I also feel that since the rotational nature of the magnetic field is not adequately described, a reader cannot have a good understanding of how this evolution happens.

All the best, Ramzuiv (talk) 20:53, 18 November 2019 (UTC)


 * Causation goes both ways. The fields are linked by the formulas. How do you explain that in a dynamo, a moving magnet creates an electric potential difference in a wire? &minus;Woodstone (talk) 09:21, 19 November 2019 (UTC).


 * Thank you for your good faith contribution. Based on your argument, I am reverting your contribution as incomplete, and not considering the interlocked nature of the fields. The right hand rule (both links) may help you in your efforts. We need to work this out. --Ancheta Wis   (talk  &#124; contribs) 23:29, 18 November 2019 (UTC)
 * I'm well aware of the right-hand rule. It is not a law of physics, but a way of representing information (in my view, in a manner that can cause even well-versed people to forget what it is supposed to represent, see Surrogation). There is no positive or negative when talking about pseudovectors. There is only clockwise and counterclockwise. Pseudovectors do not describe motions along an axis, but rotation around an axis. Note that both the RH rule and LH rule give the same axis, which exists, but disagree on the direction along the axis, since nothing happens in either direction along the axis. Magnetism and the curl of a vector field are both pseudovector phenomena, which are best understood in terms of rotations, not directions
 * Regarding the interlocked nature of the fields, I appreciate that E and M fields are just different parts of the same whole, and everything about them is in relation to that whole, not actually the individual parts. But in that light, my wording (regarding the relationship between E and M, not the orientation) appreciates this fact better than the previous wording -Ramzuiv (talk) 00:07, 19 November 2019 (UTC)
 * I switched an image. But please clarify how a current meter operates given your good faith edit. Per your 00:07, 19 November 2019 reply, the axis of the current is inside-out. --Ancheta Wis   (talk  &#124; contribs) 00:59, 19 November 2019 (UTC)
 * The article now reflects what Ørsted observed Hans_Christian_Ørsted without rehearsal (Faraday's law) almost two hundred years ago. --Ancheta Wis   (talk  &#124; contribs) 01:42, 19 November 2019 (UTC)

Mathematical conditions
Twice I undid ( and ) the edit by anon,, referring to some journal article Evaporating Black-Holes, Wormholes, and Vacuum Polarisation: Must they Always Conserve Charge? Reasons for my reverts: Comments welcome - DVdm (talk) 12:39, 23 April 2020 (UTC)
 * 1) The reference has formatting errors.
 * 2) Wikipedia does not WP:NOTETHAT.
 * 3) MOS prefers not to refer to the "above derivation".
 * 4) Of course in every derivation mathematical conditions apply. I don't think that the conditions of 2.2 are relevant in the context of this article.


 * Anon:188 response:


 * 1) your first three points are all style ones, and style can be fixed; why revert instead of fix?
 * 2) The derivation in this section refers to Gauss's Law, which is a special case of Stokes. The conditions on Stokes are therefore directly relevant to Gauss, and thus relevant to the current content of the "Charge Conservation" section. The "corollary" claim that Maxwell is guaranteed to enforce global charge conservation is not true (it is just usually true in typical circumstances).
 * 3) You need be clearer as to whether your reversion is about (i) your claim to understanding of the physics, about (ii) issues of style, or (iii) something else. It seems to me that (i) a peer-reviewed journal article written by professional physicists has a greater authority on matters of mathematical physics than some pseudononymous wiki editor, leaving your reversion poorly justified; and (ii) style can be fixed (and which you are no doubt eminently qualified for), and again not really providing strong reasons for reversion.  So what, if anything, is the "something else"?
 * — Preceding unsigned comment added by 82.71.19.188 (talk) 13:50, 1 May 2020 (UTC)
 * I agree that formatting and style issues should not, in themselves, be cause for reverting, and bringing them up here has unnecessarily complicated the discussion. The essential issue is that the sentence added by 82.71, It is worth noting that the derivation presented above depends on some assumptions, is so vague as to be meaningless and not worth adding to the article.  No doubt the source is a sterling piece of scholarship, but it has very few citations so anything it has to say on charge conservation cannot be taken as mainstream.  Only two citations are shown on gscholar that are not by the original authors, and neither of those seem to be primarily, if at all, about charge conservation. SpinningSpark 15:03, 1 May 2020 (UTC)
 * Indeed, it is the it-is-worth-noting-that clause (together with the obscurity of the source), that had triggered my eyebrows in the first place. - DVdm (talk) 15:20, 1 May 2020 (UTC)

Quantum cryptography comment
THe article states: "E.g. quantum cryptography cannot be described by Maxwell theory, not even approximately" without citation. I'm guessing they're talking about the usual implementation of the BB84 key exchange using photon detectors and such, though this is not clear at all. Seems like the sentence might have been written by somebody who didn't know what they were talking about, otherwise they would have said something more precise. 2607:9880:1A18:33:94A2:47BB:F046:3A2B (talk) 20:48, 18 June 2020 (UTC)
 * Anything that depends on measurement of single photons can't be described by Maxwell's equations. One of the reasons it took so long before the idea of photons (that is, quantized EM field) to be understood is that Maxwell's equations work so well. (Even Planck didn't believe in photons!) Gah4 (talk) 22:15, 18 June 2020 (UTC)
 * One has to distinguish averred or avowed belief from private notions. It should perhaps not be this way, but the best of us learn to keep one's most advanced convictions to ourselves. 2A01:CB0C:CD:D800:B99A:E7DA:547E:3DD9 (talk) 07:26, 17 October 2022 (UTC)
 * One has to distinguish averred or avowed belief from private notions. It should perhaps not be this way, but the best of us learn to keep one's most advanced convictions to ourselves. 2A01:CB0C:CD:D800:B99A:E7DA:547E:3DD9 (talk) 07:26, 17 October 2022 (UTC)

Geomagnetic storm
Here are some resources:
 * NASA Video (May 20, 2013) Space Weather and Earth's Aurora -- This video from forskning.no —credits beginning at minute 4:27— depicts a solar storm event which show how some magnetic flux lines within the Sun's plasma can free themselves from the Sun's own weather, and propagate to Earth in an 18-hour journey, then funnel around Earth, converging at the polar regions of the atmosphere, manifesting as Aurora borealis at minute 3:45 in the video. Those of us in the northern US can sometimes view these silent atmospheric displays at night.
 * [[file:Aurora_australis_ISS.jpg|thumb|left|Aurorae seen from the [[International Space Station]].]] But if an event is severe enough, it can affect the power distribution grid.
 * See also American Museum of Natural History (Mar 9, 2012) Science Bulletins: Space Weather—Storms from the Sun's plasma at minute 2:15
 * NASA Video (May 17, 2013) Hidden Magnetic Portals Around Earth describes X-points, which pinpoint magnetic reconnections when/where solar magnetic flux lines merge with Earth's magnetic flux lines in the magnetosphere. Minute 0:45 is already in the article.
 * --Ancheta Wis   (talk  &#124; contribs) 04:12, 28 June 2020 (UTC)

Picture for Gauss's law
For years, I just assumed there would be an illustration for Gauss's law. Perhaps http://www.physics.udel.edu/~bnikolic/teaching/phys208/lectures/gauss_law.pdf is a possible start for picturing the calculation which is needed. --Ancheta Wis   (talk  &#124; contribs) 12:28, 27 July 2020 (UTC)
 * We have Graphics Lab where illustrations can be requested (they're usually quite efficient), but you need to be specific on exactly what you want, the volunteers there do not necessarily have a technical understanding of the subject. SpinningSpark</b> 16:34, 27 July 2020 (UTC)

Lines of force for a solenoidal field
This page on Maxwell's equations (but the same happens with the pages on "Field line" and "Solenoidal vector field") supports the wrong idea that the lines of force of a solenoidal field always from closed curves (or extend to infinity). Actually, solenoidal fields may exhibit (and often have, indeed) an ergodic behavior. Moreover, in many examples used to teach, the lines of force have "singular" points, where they simply originate or end. The mistake is widely common, but it has been already pointed out by many researchers, including authoritative people like Purcell, Rosser, Tamm and Van Bladel. In their books (that can be considered as primary sources) the claim about this error is made. Hence, I believe it's time to correct this situation. The most recent and comprehensive review of the problem (from both the technical and historic viewpoints) is Zilberti's paper "The Misconception of Closed Magnetic Flux Lines" (DOI: 10.1109/LMAG.2017.2698038), that should be a reliable reference. I already applied the correction, but it was rejected because this reference was not accepted as a secondary source. Actually, it is a review paper that recaps the story and refers to many primary sources (including the books mentioned above), thus satisfying the definition of "secondary source".47.53.123.202 (talk) 10:19, 5 February 2021 (UTC)


 * Thanks for bringing it here. Note to others: shallowly discussed at User talk:DVdm. - DVdm (talk) 10:29, 5 February 2021 (UTC)
 * Hello. Would you mind saying which book by Purcell.  I have his Electricity and Magnetism book.  If it is that one, would you mind helping me find it by giving page number or chapter and section? Constant<b style="color: #1100cc;">314</b> (talk) 13:02, 5 February 2021 (UTC)
 * I refer to that book, Volume 2, Edition 2. The discussion is in chapter 6.5 FIELDS OF RINGS AND COILS. In particular, you can refer to Fig.6.20 (i.e. filed lines of a realistic -helical- solenoidal coil). There, in pag. 231, you can find the sentence "If you follow a looping field line from inside to outside to inside again, you will discover that it does not close on itself. Field lines generally don't.". It's incredible that, today, students still learn something wrong about a basic property of vector fields. It's quite easy to show, mathematically, that solenoidal fields in general do not have closed field lines.47.53.123.202 (talk) 13:20, 5 February 2021 (UTC)
 * Yes. Right there in the book, it says “If you follow a looping field line from inside to outside to inside again, you will discover that it does not close on itself.  Field lines generally don’t.”  I have to agree that this is a reliable secondary source that supports your assertion.  It is a college textbook that has been used for many years.  The figure shows that  the lines normally shown as closed, in fact miss closing and continue to cycle around the solenoid.  Constant<b style="color: #1100cc;">314</b> (talk) 13:43, 5 February 2021 (UTC)
 * Thanks. Purcell was not the first to recognize the problem. To my knowledge, Tamm was the first, followed by Slepian. If you want to recapitulate the whole story (from Faraday on) you can refer to the paper "The Misconception of Closed Magnetic Flux Lines" (DOI: 10.1109/LMAG.2017.2698038).47.53.123.202 (talk) 13:47, 5 February 2021 (UTC)
 * Magnetic_field_produced_by_an_electric_current_in_a_solenoid.png I became aware of this while working on field line visualizations of real coils (one pictured on the right) so yes - it's real ;) not fringe. However, I am very much against this being discussed in any article other than Field line, because field lines are just the way some of us like to visualize fields and there's really no deep physics in whether they close on themselves or not (afaik). In the case of the magnetic field, the way they're constructed is more artificial than for some other fields; for those one can at least imagine some kind of movement of a particle along the field lines, but here we do what - place tiny little magnetic needles one after another? Let's not confuse our readers by giving too much weight to this little nuisance (unless you can show it has real physical consequences). Ponor (talk) 14:29, 5 February 2021 (UTC)
 * I fully agree. Maxwell's equations are the right way to describe an electromagnetic field. Flux lines are pictorial representations, appealing and sometimes useful, but do not have physical properties. Slepian gave an outstanding discussion of this. In my opinion, the page on Maxwell's equations should not rely too much on the concept of field lines (especially supporting wrong beliefs!). That was the intention of the changes that I applied a few days ago (but they were rejected...). May we restore them? I had also applied corrections to the page on Field line.47.53.123.202 (talk) 14:45, 5 February 2021 (UTC)
 * If I'd change anything that would be removing all unnecessary/unhelpful mentions of the field lines.
 * We have:"Picturing the electric field by its field lines, this means the field lines begin at positive electric charges and end at negative electric charges" - this is half-true, and was discussed at Field line talk page; as it's stated here, one would think any field line that starts at + has to end at −.
 * You then added "(with the exception of singular points)" which I'm not sure I understand, not without checking the cited article.
 * The article then said "'Counting' the number of field lines passing through a closed surface yields the total charge..." which says nothing because we are allowed do draw as many field lines as we want and the enclosed charge would depend on our choice of the number of field lines we start with. I believe this is the language they used some 150 years ago.
 * Then you inserted "The dynamically induced electric field has closed field lines similar to a magnetic field..." and the article further said "This aspect of electromagnetic induction is the operating principle behind many electric generator" - I am not sure I understand what "this" refers to? Again, it'd be more helpful not to mention field lines because picturing those becomes more and more difficult. Ponor (talk) 15:20, 5 February 2021 (UTC)
 * Yes. The problem does not involve magnetic fields only, but also electric fields, of course. In general, whenever you have a divergence-free field, you cannot state that its flux lines are closed. The electric field "generated" during electromagnetic induction is not an exception.
 * "Singular points" is the name that some authors used to indicate points where field lines begin or end. Despite what many people think, there is nothing strange in them. For instance, if you consider two identical positive point charges and the segment that connects them, in the middle of the segment the field lines that start from the charges "die" (i.e. the field vanishes in that point and you cannot go on drawing the lines, unless you move a little bit from this "singular point"). A detailed description of singular points is given in the paper that I used as a reference.
 * "Counting the number of field lines" is really an anachronistic way of thinking. I would avoid expressions of this kind.
 * I did not iserted "The dynamically induced...". That sentence was written before; I had removed it, but (alas!) the previous version has been restored. I simplified the paragraph as "The electromagnetic induction is the operating principle behind many electric generators: for example, a rotating bar magnet creates a changing magnetic field, which in turn generates an electric field in a nearby wire".47.53.123.202 (talk) 15:41, 5 February 2021 (UTC)
 * My sincere apologies, I was reading the wrong diff.
 * Let's not rush anything yet, but you have my support for most changes you previously introduced; I only think some should go into different articles or different parts of the articles:
 * As for the statement "(with the exception of singular points)" - to me, singular points are the points where point charges reside. I'd say this is not that important for this article. It can be added to Field line if it's not already there. Just please be more verbose, and (maybe) use some other name for these points to avoid confusion.
 * I agree with the removal of "...this equation states that magnetic field lines neither begin nor end but make loops or extend to infinity and back...". I'd not add "Despite a very common idea, this does not mean that the flux lines of the magnetic fields always form closed curves" - this will be said in Field line and Solenoidal vector field ; Maxwell's eqs are not about the field lines anyway. (striking out SVF... no need to mention field lines in that article at all))
 * I agree with your removal of "The dynamically induced electric field has closed field lines similar to a magnetic field..." (apologies again!)
 * When it comes to Solenoidal vector field: would you mind adding a whole new section about this? I could provide you with a nice illustration of the "effect" similar to the one I posted above. I'd prefer not to have this discussed in the lead section because, as we said, field lines are not that important (and the lead should only sumarize what's said in the article) (striking out, no need for field lines a mathematical article; Solenoid probably better)
 * You could also be more verbose about this in Field line especially because there is no open access version of Zilberti's article (is there?). Ponor (talk) 16:38, 5 February 2021 (UTC)
 * (later corrections ) Ponor (talk) 17:03, 5 February 2021 (UTC)
 * Many thanks Ponor. I will work on the pages as soon as possible. Please, note that "singular points" are not points where the charges reside. Consider, for instance, two identical cylindrical permanent magnets, facing each other with the same polarity (e.g. north pole against north pole). If you start drawing a line from one magnet, while moving towards the point in the middle between the two magnets the field vanishes. When you are exactly in the middle, your line ends, because the field magnitude is zero in that point and therefore you cannot find the direction to go on drawing. As you can see, this is an example of a "singular point" and there are no charges (indeed, this is a magnetic example). If, from the singular point, you slightly move transversally, the field magnitude is small but no longer null and you can therefore start a new line. Hence, singular points are points where some line ends and other lines start. This means that the continuity of each single line is broken (and indeed, the continuity of a single line has nothing to do with the physical properties of the field, properly described by Maxwell's equations) but, globally, the flux around the singular point keeps null (even if you shrink the volume as you like). The big issue that many people do not understand, is that, when drawing field lines, you try to "connect" different field points, but this is completely against the concept of "field" (i.e. what happens in a given point, must be related to the value of the field in the point itself; no matter what happens in the neighbouring points). Open access versions of Zilberti's paper are available on ResearchGate and on Zenodo.93.66.102.253 (talk) 11:19, 8 February 2021 (UTC)
 * If there are no monopoles, eventually the flux lines will close, but it might take a long time. I suppose one way to see it is that there could be only one flux line in the universe, so eventually. An ideal solenoid with a phi direction current (and no other currents) will have them close. A slightly less ideal one, made up of a coil of very thin wire, will have a small axial current, and so the lines will come around slightly offset. On the other hand, if you wind the solenoid so that the end and beginning are at the same place, then the axial current will be zero. And there are imperfections in any real solenoid. But eventually it has to close, unless there is a monopole somewhere in the universe. Gah4 (talk) 17:05, 5 February 2021 (UTC)
 * If you could see the picture in Purcell, you would see why the flux lines do not have to close. It is too complicated for me to draw, but I will try to describe it.  You start out with closed field lines as in Ponor's illustration, only instead of closing, the ends fail to meet, missing each other slightly.  The lines goes around the wires again.  Each time it goes around, it precesses (probably not the right word) around the coil.  Unless the amount of precession is a rational fraction of 2 pi radians, i will never close, but it will get quite close. Constant<b style="color: #1100cc;">314</b> (talk) 17:24, 5 February 2021 (UTC)
 * Yes. Well, even though we actually had Purcell as a text 40 years ago, I don't remember the picture. But that is what I tried to say above. Eventually it will meet again, though after an infinite number of loops if it isn't rational. But within the resolution of ink on paper, they can meet. Or within the wavelength of an electron that might be following them. Gah4 (talk) 20:57, 5 February 2021 (UTC)
 * Dear Gah4, please see my reply to Ponor above. Lines must not necessarily close (and monopoles are no needed to justify this). In my opinion, this becomes clear if you try to implement an automatic procedure to draw field lines. At a first view, this seems to be a trivial task, but actually you must be careful to avoid numerical artifacts.93.66.102.253 (talk) 11:19, 8 February 2021 (UTC)
 * OK, field line density is a visual approximation to flux density. That is, sort of like contours on a 2D map, but in 3D. In 2D, you give specific contour values and draw them. It is less obvious in 3D, but it should be possible. Similar to contour maps, there can be large areas with no lines through them. In the case of a 3D saddle point, if you increase the line density enough, you will find that they avoid the saddle point. In the case of magnetic flux lines, electrons will follow a helical path along them, so they do have an important physical significance. But they don't follow the infinitely narrow lines that we might imagine, but some finite helical radius. As far as the discussion, I suspect I would rather minimize the use of them in the article, but if they help explain flux density, they might work in that section. Gah4 (talk) 02:51, 9 February 2021 (UTC)
 * See: magnetic mirror for an example using flux lines. Gah4 (talk) 04:02, 9 February 2021 (UTC)
 * Dear Gah4, please see my reply to Ponor above. Lines must not necessarily close (and monopoles are no needed to justify this). In my opinion, this becomes clear if you try to implement an automatic procedure to draw field lines. At a first view, this seems to be a trivial task, but actually you must be careful to avoid numerical artifacts.93.66.102.253 (talk) 11:19, 8 February 2021 (UTC)
 * OK, field line density is a visual approximation to flux density. That is, sort of like contours on a 2D map, but in 3D. In 2D, you give specific contour values and draw them. It is less obvious in 3D, but it should be possible. Similar to contour maps, there can be large areas with no lines through them. In the case of a 3D saddle point, if you increase the line density enough, you will find that they avoid the saddle point. In the case of magnetic flux lines, electrons will follow a helical path along them, so they do have an important physical significance. But they don't follow the infinitely narrow lines that we might imagine, but some finite helical radius. As far as the discussion, I suspect I would rather minimize the use of them in the article, but if they help explain flux density, they might work in that section. Gah4 (talk) 02:51, 9 February 2021 (UTC)
 * See: magnetic mirror for an example using flux lines. Gah4 (talk) 04:02, 9 February 2021 (UTC)
 * See: magnetic mirror for an example using flux lines. Gah4 (talk) 04:02, 9 February 2021 (UTC)

Magnetic Flux, not Field
Under the "Key to the notation" heading B is labeled the magnetic field, however the curl of the electric field is the negative time differential of magnetic flux. This is easy to see through units:

Del cross E = V/m^2

dB/dt != A/ms

dB/dt = Tesla/s = V s/m^2 s = V/m^2

Bghagar (talk) 22:31, 23 June 2021 (UTC)


 * For most people nowadays (physicists at least) B is the magnetic field, measured in tesla. See Feynman. Ponor (talk) 23:51, 23 June 2021 (UTC)
 * The unit of B is tesla, yes. What its name is is less clear.  Much of the world does not call it the magnetic field (strength).  In the SI, B is called the magnetic flux density, and H is called the magnetic field strength.  —Quondum 03:07, 24 June 2021 (UTC)
 * I'm afraid Wikipedia will be the last to admit that the magnetic flux (Field*Area) density (Field*Area/Area) is a silly name for B and that *everyone* (*relevant*) in physics calls B the magnetic field. Start from this lab where they produce some of the strongest fields... in tesla: nationalmaglab. Ponor (talk) 03:48, 24 June 2021 (UTC)
 * Bghagar expressed a valid concern about something that appears to need correcting in the article: something that is confusing and contradictory to the way a large portion of the world uses the terminology. An article's talk page is for discussing improvements to an article, not for advocating a particular perspective.  —Quondum 10:58, 24 June 2021 (UTC)

Quondum, I'm saying that nothing needs to be fixed. I gave two references and I can give a hundred more. But go ahead, change all "magnetic field B” with "magnetic flux density B" and watch the response. Try not to use yr 1887 references along the way. Ponor (talk) 12:43, 24 June 2021 (UTC)
 * Your implication is ironic. An article such as this is aimed at a broader audience than particle and quantum physicists, and so adhering to the archaic Gaussian system and its associated terminology in WP is inappropriate.  References for modern, widely used and standardized terminology are easy to find, but resistance to using such in WP is a perennial problem: precisely the response (by who I regard as traditionalists) to which you refer.  —Quondum 14:10, 24 June 2021 (UTC)
 * I am sorry, but I don't know what you're talking about. Tesla is not a Gaussian system unit. I'd think US national lab people, EU magnetic field lab people, Japanese high field lab, Nature, MRI manufacturers know their fields. The prevalent name for B nowadays is "the magnetic field", and when we say "the magnetic field" we should assume it's the B-field. That's modern. The name "magnetic flux density" is old and is disappearing. That's why it's okay to call B the field, and say that the field is measured in tesla and not A/m. It's the H-field/magnetizing field/ that's measured in A/m. We can only continue this discussion if you provide references to the claim that "much of the world does not call B the magnetic field". So far I've seen none. Ponor (talk) 15:00, 24 June 2021 (UTC)
 * You are evidently unfamiliar with the SI, which defines the tesla, and also incidently gives the names for quantities such as these (read the 2019 SI Brochure: ; your "So far I've seen none" suggests that you are unaware of it). Your claim that 'The name "magnetic flux density" is old and is disappearing is simply nonsense.  Public facing websites are not references for technical use.  —Quondum 15:49, 24 June 2021 (UTC)

I ran into this by working with units from the WP article on magnetic fields: https://en.wikipedia.org/wiki/Magnetic_field Perhaps the introduction of the magnetic fields WP page can mention that Flux and Field can be used interchangeably depending on context? Bghagar (talk) 14:53, 24 June 2021 (UTC)


 * Bghagar, the full 'old' name for B is "magnetic flux density", but most people call it simply "the magnetic field". The Magnetic field article is confusing and inconsistent (check the bottom section where the strongest fields, and not flux densities are measured in tesla), but all attempts to fix it fail when people start fighting about the right names. My advice, when the field is measured in tesla, call it the B-field of just the magnetic field. When it is measured in A/m call it the H-field. The flux Φ of B through a small area S is B·S (bold is for vectors); using the old confusing terminology this would be the flux of the magnetic flux density. Ponor (talk) 15:20, 24 June 2021 (UTC)
 * Bghagar, the article Magnetic field seems to be fairly clear about the variation of usage of terminology, including that the SI name for B is magnetic flux density and for H is magnetic field strength. Characterizing usage more specifically by each context is too much, so I'm unclear what you are suggesting with respect to that article.  —Quondum 15:49, 24 June 2021 (UTC)
 * The preamble defines magnetic field in units of A/m and magnetic flux density in terms of Tesla. A line there stating that both the magnetic field and the magnetic flux density may be referred to as "magnetic field" which Ponor initially pointed out might help. Bghagar (talk) 17:46, 24 June 2021 (UTC)
 * The second paragraph of the lead starts with "In electromagnetics, the term "magnetic field" is used for two distinct but closely related vector fields denoted by the symbols $B$ and $H$", which seems to me to be what you are asking for. It then goes on to speak of $B$ and $H$ and their names in the SI context.  Perhaps this juxtaposition of switching from the general to more specific is what is confusing?  The situation is actually a bit more complex: there are several different contexts: SI quantities, Gaussian-CGS, and several more, all of which use the symbols $B$ and $H$ for corresponding but different quantities and with different units and varying names (e.g. in SI quantities and Gaussian-CGS, the "$B$-field" refer to different quantities that are related by a constant factor).  The sentence that I have just quoted refers to this collection of contexts.  Unfortunately, the historical development has left us with a confusion of modern usages now, although as far as I am aware, only the SI conventions are endorsed by the major standards bodies.  In non-technical contexts, one is generally not unduly concerned about being precise or even cares about which of $B$ and $H$ are meant, and the term "magnetic field" is commonly used for either (maybe "magnetic flux density" is a bit of a mouthful for casual use?).  —Quondum 18:21, 24 June 2021 (UTC)
 * I must have read that a dozen times, I'm blind and you are correct. Bghagar (talk) 18:11, 1 July 2021 (UTC)

Compatible Units of Meter and Second
I would like to reword the following two sentences which are confusing:"The equations are particularly readable when length and time are measured in compatible units like seconds and lightseconds i.e. in units such that c = 1 unit of length/unit of time. Ever since 1983 (see International System of Units), metres and seconds are compatible except for historical legacy since by definition c = 299 792 458 m/s (≈ 1.0 feet/nanosecond)." The SI Unit system is designed with the intent of coherence between units. I do not understand why meters and seconds are not compatible. Please add more details to what this means. The SI Unit system uses the kilogram and not the gram as the base unit for coherence between mechanical units of mass, length, and time and electric physical quantities such as charge, current, resistance, inductance, and capacitance as related by Power=R*I^2=Mechanical Energy/time=Force*velocity. ScientistBuilder (talk) 01:55, 28 January 2022 (UTC)ScientistBuilderScientistBuilder (talk) 01:55, 28 January 2022 (UTC)
 * The statement applies particularly to a specific section of the article, for those physicists who work in Gaussian units. For others who work in other branches of science or technology, Maxwell's equations do not include the factor of c (they use Maxwell's equations in the forms of the other sections). But even in the other sections, it is useful to remember that meters and seconds of System International are arbitrarily selected units, for convenience, dating back to the French Revolution. The units could just as well have been units of feet and nanoseconds, and the speed of light could have been proclaimed to be 1 foot per nanosecond, for ease of calculation and visualization of the flow of electrons in an electrical application. Not everyone works in Meters and Seconds. Some students are taught Dimensional analysis for this reason, to break free of arbitrary units in their thinking. It is true that when people from multiple disciplines work together, they have to take care that their work is translated to remove incompatibilities, a situation like the railroad gauges that vary from country to country. See also natural units. --Ancheta Wis   (talk  &#124; contribs) 05:39, 28 January 2022 (UTC)

beauty
Someone wondered about beauty in Maxwell's equations. From Feynman Lectures, vol II, chapter 18, (before equation 18.2): we find that Maxwell’s beautiful edifice stands on its own, and He brought together all of the laws of electricity and magnetism and made one complete and beautiful theory. Then after equation 18.25: What a beautiful set of equations! Gah4 (talk) 07:03, 18 February 2022 (UTC)


 * The New York Times and the Feynmann Lectures are a reliable source for the statement:
 * James Clerk Maxwell's unification of the fields of electricity and magnetism into electromagnetism in the form of Maxwell's equations is an example of mathematical beauty. ScientistBuilder (talk) 19:26, 18 February 2022 (UTC)
 * I don't understand the reason for reverting this edit. ScientistBuilder (talk) 19:27, 18 February 2022 (UTC)
 * There are misstatements: No magnetic monopoles is a slogan for Gauss's law for magnetism (one of Maxwell's equations), which your edit contradicted in the name of greater symmetry, but which would not have appeared in a textbook. For example Paul Dirac spent years of his life on one equation. But beauty was not his motive. --Ancheta Wis    (talk  &#124; contribs) 19:46, 18 February 2022 (UTC)
 * I am not concerned about the magnetic monopoles argument but about the reliability of The New York Times and the Feymann Lectures being taken down. ScientistBuilder (talk) 20:12, 18 February 2022 (UTC)
 * You used the NYT column as a source for “Maxwell's equations is an example of mathematical beauty”.  But the source does not say that.  It tells us that someone conducted a poll of the Readers of Physics World magazine for the greatest equation and that Maxwell’s equation were among the top vote getters.   You are citing that as evidence of beauty.  It is flawed synthesis.  The poll was not about beauty.  There is no WP:RS to show that the population polled is representative of the rest of the world, and the winner of a beauty poll may still be ugly, if all the contestants were also ugly.
 * Feynman is a reliable source for quoting Feynman, but we need a RS to tell us that Feynmann's opinion is representative of the rest of the world. Constant<b style="color: #1100cc;">314</b> (talk) 20:38, 18 February 2022 (UTC)
 * You are not wrong about RS etc., but in the academic folklore there does undeniably exist a certain appreciation for the intellectual achievement (at the level of all mankind) which these equations represent. I cannot give an RS for that; it is very much in the air and one absorbs it by osmosis. If one is an academic and genuinely fails to notice this, then I would suggest one is more than a bit peculiar. Calling these equations a flawed synthesis is not incorrect, but a bit like saying that the pyramids are not "up to code."
 * As for the particular term used, beauty seems to be more popular these days then the one I heard in my younger days, which was elegance. The precise word does not matter at all, since as I have pointed out, we all get the idea. It is quite possible that the much-appreciated Feynman lectures played a role in this. 2A01:CB0C:CD:D800:B99A:E7DA:547E:3DD9 (talk) 07:21, 17 October 2022 (UTC)
 * I am also confused why a book with a chapter devoted to Maxwell's equations 17 Equations that changed the world as a source was reverted. ScientistBuilder (talk) 20:14, 18 February 2022 (UTC)
 * I incorrectly cited the magnetic monopoles page. The page I meant to add is https://www.maxwells-equations.com/zero.php. ScientistBuilder (talk) 20:15, 18 February 2022 (UTC)

Magnetohydrodynamics (MHD), and Extension to nonlinear phenomena
I propose to revert the Nonlinear phenomena section and take it to a talk page. One part that bothers me includes the sentence "Linear relations are much more easy to model and understand than nonlinear models such as partial differential equations". Maxwell's equations are partial differential equations, so the quoted sentence does not follow. If the sentence is really about MHD, then we might use Magnetohydrodynamics as the target for discussion. One approach might be to model the fluids of MHD with the constitutive equations D=εE and B=μH, but treating permittivity ε and permeability μ as tensor quantities which interact with E and H. Then the Navier-Stokes equations, which can be used to describe the Sun's coronal plasma as the plasma would flow to Earth, would have to be simultaneously solved along with Maxwell's equations for D and B, or for charges and currents etc. on the MHD plasma.

If we were to model the reversals of Earth's magnetic field, as described in magnetohydrodynamics we might discuss the permittivity ε and permeability μ of the plasma surrounding Earth's magnetic field. Similar discussions might also be had about the nonthermal radio filaments (NTFs) which surround the center of our Milky Way galaxy.

The Navier-Stokes equations would need to be decoupled from a discussion of Maxwell's equations until any phenomenological aspects of MHD are disentangled from Maxwell's equations, such as the speed of magnetic reconnection (not currently explained from known models).

My point is that this kind of discussion would belong on the MHD talk page, as it is an application of, not about Maxwell's equations per se. --Ancheta Wis   (talk  &#124; contribs) 12:40, 21 February 2022 (UTC)

When reading the magnetic reconnection page just now, "The confinement of plasma in devices such as tokamaks, spherical tokamaks, and reversed field pinches requires the presence of closed magnetic flux surfaces. By changing the magnetic topology, magnetic reconnection degrades confinement by disrupting these closed flux surfaces, allowing the hot central plasma to mix with cooler plasma closer to the wall.". But we know from, that magnetic flux lines do not have to close, so we have more to learn, perhaps about an appropriate representation for a topology of magnetic phenomena. --Ancheta Wis   (talk  &#124; contribs) 13:09, 21 February 2022 (UTC)
 * I have no problem with removing the section. Constant<b style="color: #1100cc;">314</b> (talk) 17:16, 21 February 2022 (UTC)
 * The Good faith contribution belongs on a talk page, as an adjunct to Constitutive_equation, showing up as an application of say, Magnetohydrodynamics, but not this article, at the current state of our collective understanding. We need a talk page location; which is yet To Be Determined. --Ancheta Wis   (talk  &#124; contribs) 10:38, 22 February 2022 (UTC)

Table for SI units is not really specific to SI Units
I think it is deeply misleading for students to believe that most physics equations are dependent upon a system of units, because this is really only true for rationalized equations in systems of natural units. Most generalized physical equations apply for all systems of orthogonal curvilinear coordinates, and this is the case for the equations in the SI Units table, and the cited sources ought to support it. The table should be called "General Formulation" or something to that effect. I think this harms reader understanding more than people appreciate. I'd also like to suggest having the differential formulation from this table in a more convenient position at the top of the page where it helps to serve well as reference material. 71.32.81.195 (talk) 15:53, 6 May 2022 (UTC)
 * I am not sure about the most. Other than E&M, equations should be independent of units, though aren't always for some different reasons. For E&M (at least before 2019), there are two ways of generating a unit system, one based on electrostatics, the other on magnetostatics. SI units are (traditionally) magnetism based, and Gaussian units electrostatic based. It isn't so obvious that one is better than the other, but SI has gotten popular, at least for experimental physics. Gah4 (talk) 05:32, 7 May 2022 (UTC)
 * Reminds me of something that I have noticed in the past, though never did enough statistics to show. It seems like physics mostly writes equations independent of units, while engineers often factor out the units. That is, physics will say F=ma, where once you supply units to two of the terms, you know the units for the third. Engineers will say F(Newtons) = m(kg) * a (m/s/s). To use an equation this way, you convert from whatever units you have to the given units, apply the equation, and convert the result. (Not always, as there is much mixing between physics and engineering, but often enough.) As to equations depending on units, consider the system used by US chemical engineers, based on pound mass, Gah4 (talk) 05:32, 7 May 2022 (UTC)
 * Well, we try to do it independent of the strongest systems of units we can account for. It's impossible to write an equation which is completely independent of all possible systems of units because some are nonlinear or uncomputable or arbitrarily badly behaved. This is why i have called out systems of orthogonal curvilinear coordinates in particular. This is the usual bar for physics because of several reasons. However, the systems of units for which a given equation system does apply ought to be stated as broadly as possible. The equations in Gaussian units are almost completely identical apart from the physical quantities which appear in them (only a slight rationalization has been made) and most physical laws can be restated in a few different terms like this. One should be able to derive the Gaussian units equations from the "SI Units" equations by making the appropriate substitutions and rationalizations.
 * So, now that we know we need to figure out over what systems of units an equation applies, we should see what the sources say. Wikipedia is actually probably just *out of date* in terms of source material on this matter.
 * 71.32.81.195 (talk) 07:29, 7 May 2022 (UTC)
 * So, now that we know we need to figure out over what systems of units an equation applies, we should see what the sources say. Wikipedia is actually probably just *out of date* in terms of source material on this matter.
 * 71.32.81.195 (talk) 07:29, 7 May 2022 (UTC)

Cause before effect!
Two of the laws are usually stated:
 * $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$$
 * $$\nabla \times \mathbf{B} = \mu_0\left(\mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t} \right) $$.

And accordingly, the text says that changes in the fields over time causes rotation in the fields over space. This seems wrong to me. Causes should proceed effects! So we should rewrite the equations as:
 * $$ \frac{\partial \mathbf{B}}{\partial t} = - \nabla \times \mathbf{E} $$
 * $$ \frac{\partial \mathbf{E}} {\partial t} = \frac{1}{\varepsilon_0 \mu_0} \nabla \times \mathbf{B} - \frac{1}{\varepsilon_0} \mathbf{J} $$.

And we should change the text to say that the rotation of the fields and the current causes the changes in the fields over time. JRSpriggs (talk) 21:05, 9 May 2022 (UTC)
 * None of these equations state cause and effect. They simply state relationships. Constant<b style="color: #4400bb;">314</b> (talk) 21:43, 9 May 2022 (UTC)

Using the d'Alembert operator and modifying Maxwell's equations to retain the source terms, we get:
 * $$ \Box \mathbf{B} = \mu_0 \nabla \times \mathbf{J} $$
 * $$ \Box \mathbf{E} = - \left( \mu_0 \frac{\partial \mathbf{J}}{\partial t} + \frac{1}{\varepsilon_0} \nabla \rho \right) $$.

This shows that the magnetic field is caused exclusively by the rotation of the (total) current, and that the electric field is caused exclusively by the change in the (total) current over time and the gradient of the (total) charge. Thus it is an error to say (as the article does) that the rotation of the electric field is caused by the change in the magnetic field with time or that the rotation in the magnetic field is caused by the change in the electric field over time. JRSpriggs (talk) 18:28, 12 May 2022 (UTC)
 * These equations do not say that something causes something else. What they say is that whatever causes, E. B and currents, it must do so in accordance with these equations. Constant<b style="color: #4400bb;">314</b> (talk) 18:37, 12 May 2022 (UTC)

See Green's function. Integrating and taking the derivatives outside, we get:
 * $$ {\mathbf B} = \nabla\times{\mathbf A} $$
 * $$ \mathbf{A} [\mathbf{r}, t] = \frac{\mu_0}{4 \pi} \iiint_{R^3} \frac{ \mathbf{J} [\mathbf{r} + \mathbf{s}, t - \frac{s}{c}] }{s} d^3 \mathbf{s} $$
 * $$ {\mathbf E} = -\nabla\varphi - \frac{\partial{\mathbf A}}{\partial t} $$
 * $$ \varphi [\mathbf{r}, t] = \frac{1}{4 \pi \varepsilon_0} \iiint_{R^3} \frac{ \rho [\mathbf{r} + \mathbf{s}, t - \frac{s}{c}] }{s} d^3 \mathbf{s} $$.

OK? JRSpriggs (talk) 18:33, 21 May 2022 (UTC)


 * As far as I can tell, there is nothing wrong with your math. You can impose your notion of cause and effect on them, but the equations themselves say nothing about cause and effect. Constant<b style="color: #4400bb;">314</b> (talk) 20:14, 21 May 2022 (UTC)

If nothing can be inferred from Maxwell's equations about cause and effect (as you appear to be claiming), then should not all mention of cause and effect be removed from the article? JRSpriggs (talk) 21:17, 21 May 2022 (UTC)


 * Consider this: ρ is a charge distribution and J is a current distribution. How do you cause a particular charge and current distribution?  You have to push chare around.  How do you do that?  You use an E field.  So, E must be the cause of J and ρ.  That is the problem with trying to assign cause to one of the actors and effect to another.  The equations do not confer cause on any of the actors.  Rather, they constrain the actors to conform with the equations.  You can use E to push charge around, but the E you use better conform with the equations that you have written.  In a particular engineering situation, it is help to assign some of the actors to the role of cause and others to effect.  But that is a convenience. As for the rest of the article, it would be great to regularize it in this manner, but I suppose it will have to be case by case.Constant<b style="color: #4400bb;">314</b> (talk) 21:57, 21 May 2022 (UTC)
 * Cause and effect occur nowhere in physics --- there are few instances where we do use "causality" in very technical senses that only on the surface seem connected to the idea of cause and effect. But this just how we humans like to talk about things and if it helps, go right ahead! Quite similar and related caveats apply to "why" "because" and "explains" --- if I had to purge all such words and turns of phrase from my language, my lectures would become even more impenetrable to my students. So speak freely of cause and effect, and perhaps learn to appreciate that a greater ontological commitment is suggested by common sense and usage than is warranted by the substance of our theories. In a sense, our equations are smarter than we are and we have yet to realise this. 2A01:CB0C:CD:D800:B99A:E7DA:547E:3DD9 (talk) 07:11, 17 October 2022 (UTC)
 * Your comments are insightful. Perhaps you would consider opening an account.  If you do, please ping me. Constant<b style="color: #4400bb;">314</b> (talk) 18:56, 17 October 2022 (UTC)

1. There are forces other than electromagnetism. Even leaving out the weak and nuclear forces, we have gravity and mechanical forces (pressure, tension, and shear stress from touching). Mechanical forces are not primarily electromagnetic in nature, they are largely manifestations of Pauli force which arises from the unwillingness of electrons to occupy the same state as other electrons. So we can turn a magnet or connect a battery into a circuit without having to use some pre-existing Lorentz force.

2. So you do not object to my plan to remove the claims about causality from Maxwell's equations? JRSpriggs (talk) 22:59, 21 May 2022 (UTC)
 * I was not so long ago (not in WP) answering a question about where nuclear fission energy comes from. As Lise Meitner first figure out, the nucleus is balanced between electrostatic and nuclear binding energy. It is not so easy to say which is which. It is, I believe, not so easy to explain mechanical energy. When you compress materials, yes, there is exchange interaction, but that pushes electrons into different energy levels, storing electrostatic potential energy. That is why you are not supposed to call it exchange force, as it isn't a force in the usual sense. But as above, it is complicated to explain where energy is. Or as also came up recently, where does the water molecule binding angle come from? The p orbitals would make it 90 degrees, but hydrogen repulsion increases it. That moves electrons to, at least partly, different energy levels, that allow for a larger angle. Again, electrostatic energy. Gah4 (talk) 08:11, 22 May 2022 (UTC)
 * I was not so long ago (not in WP) answering a question about where nuclear fission energy comes from. As Lise Meitner first figure out, the nucleus is balanced between electrostatic and nuclear binding energy. It is not so easy to say which is which. It is, I believe, not so easy to explain mechanical energy. When you compress materials, yes, there is exchange interaction, but that pushes electrons into different energy levels, storing electrostatic potential energy. That is why you are not supposed to call it exchange force, as it isn't a force in the usual sense. But as above, it is complicated to explain where energy is. Or as also came up recently, where does the water molecule binding angle come from? The p orbitals would make it 90 degrees, but hydrogen repulsion increases it. That moves electrons to, at least partly, different energy levels, that allow for a larger angle. Again, electrostatic energy. Gah4 (talk) 08:11, 22 May 2022 (UTC)

Sources support using EMF in place of claim about work.
In electromagnetic induction, emf can be defined around a closed loop of conductor as the electromagnetic work that would be done on an electric charge (an electron in this instance) if it travels once around the loop.

The above sentence is taken directly from Wikipedia. Belaboring the point doesn't help. Allude to EMF when presenting Faraday's law. Jackson explains this.

71.32.81.195 (talk) 09:51, 10 May 2022 (UTC)


 * It does not matter what other Wikipedia articles say because Wikipedia is not a reliable source. My objection is to the language proposed. It is fine for a practical subject like describing how a generator works.  However, it is too casual for a theoretical subject like Maxwell’s equations (ME).  ME do imply that the line integral of the E field around a closed path is proportional to the rate of change of the enclosed flux.  I do not object to parenthetically saying that the line integral is also known as the electromotive force.  I do object to saying that the changing flux induces the EMF.  ME do not say that.   They only say that there is a relationship between the EMF and the changing flux. They do not imply that one is the cause and the other is the effect.
 * I also object to saying that a bar magnet has a magnetic field. That is fine for practical subjects.  In the context of Maxwell’s equations, nothing has a magnetic field.  The one and only magnetic field is simply a convenient artifice.  It is made of nothing but numbers.  It fills all space.  It does not move.  A spinning bar magnet causes the numbers that compose the magnetic field in its vicinity to change. Constant<b style="color: #4400bb;">314</b> (talk) 14:21, 10 May 2022 (UTC)
 * Induction is not a causal claim. It would be wrong to say that the changing flux causes the EMF. It doesn't. It induces it. Induction *is* the relationship. This is how the literature treats the word. The EMF article uses the same sources.
 * If a bar magnet doesn't have a field then it doesn't create one either. They go together. A non-constant polynomial has a root. A bar magnet has a field. This is absolutely supported and less misleading.
 * Magnetic fields are not composed of numbers.
 * TheodoricStier (talk) 17:33, 10 May 2022 (UTC)
 * This is an example of a lie-to-children. It is useful to explain the operation of a motor, but it interferes with a deeper understanding.  For whatever reason, only physics majors are typically taught more.  You can read what Feynman had to say here .  See section 15-4.  Here Feynman says: A “real” field is then a set of numbers we specify in such a way that what happens at a point depends only on the numbers at that point. Constant<b style="color: #4400bb;">314</b> (talk) 03:23, 12 May 2022 (UTC)

Do not make Gaussian rationalization when presenting Faraday's law
Gaussian units formulations of the E field rationalize its dimension to conflate it with that of changing magnetic flux. This is inappropriate because they are differently dimensioned physical quantities. The previous article claimed that an amount of magnetic flux is equal to an amount of work. This is confusing for both a physicist and a student! It is a coordinate rationalization based on a system of natural units and should not be incorporated into a presentation in order to be glossed over.

71.32.81.195 (talk) 09:55, 10 May 2022 (UTC)


 * We present equations in the way they appear in reliable sources. If there is more than one way, we go with the predominant way.  If no predominant way is apparent, we go with the consensus. If you want to make specific proposals, please show the equations here in the discussion, otherwise we have to guess about which equations you want to change.  Constant<b style="color: #4400bb;">314</b> (talk) 13:17, 10 May 2022 (UTC)
 * The predominant presentation of these equations does not make that rationalization! I don't know what else to say. Just use correct language and stop conflating things. The books explain this.
 * TheodoricStier (talk) 17:30, 10 May 2022 (UTC)
 * Here, I will explain the exact problem with the language involved.
 * > Integral form, it states that the work per unit charge required to move a charge around a closed loop equals the rate of change of the magnetic flux through the enclosed surface.
 * "work per unit charge required to move a charge around a closed loop".
 * This has dimension of *electric potential* i.e. the same as *volts* i.e. *joules per coulomb*.
 * "rate of change of the magnetic flux" this has dimension of *magnetic flux per time* i.e. something else entirely.
 * The equations are fine. You present rationalized equations by explaining what rationalization you have made. You do not present them by saying "the things are equal". They are not. They are just equal in the equations. They are proportional in the real world.
 * TheodoricStier (talk) 17:41, 10 May 2022 (UTC)
 * The equal sign in the equations mean that the number on the left is the same as the number on the right. It does not mean that the thing on the left is the same as the thing on the right and it does not mean that the thing on the left is caused by the thing on the right.  I am not sure what you mean by "rationalized equations" unless that is just the name you use for the form that you advocate. Constant<b style="color: #4400bb;">314</b> (talk) 04:06, 12 May 2022 (UTC)
 * See Gaussian units --Ancheta Wis   (talk  &#124; contribs) 05:37, 12 May 2022 (UTC)
 * It appears that the Impedance of free space then becomes subject to experimental determination. --Ancheta Wis   (talk  &#124; contribs)
 * I did not see any rationalized equations. Maybe I missed them.  So, which equations are rationalized that should not be rationalized ? Constant<b style="color: #4400bb;">314</b> (talk) 02:59, 13 May 2022 (UTC)
 * The units are the issue for students and practitioners alike: See Gaussian, SI and Other Systems of Units in Electromagnetic Theory, Appendix A, http://bohr.physics.berkeley.edu/classes/221/1112/notes/emunits.pdf © Robert G. Littlejohn (2020) Physics 221A. Littlejohn's Appendix A, especially page 1, of 12. It cites Jackson’s discussion of units (an appendix in Jackson’s Classical Electrodynamics) The equations are discussed only in terms of the notation (G vs. SI), which compare their units side by side. That would mean, of course, to pick one version of units (G, or SI) and to not mix G with SI in one's version of choice. User talk:71.32.81.195's interest apparently involves a clear presentation of Faraday's law. But if one's interest were in relativity, for example, one might select gaussian units for convenience in calculation or computation. --Ancheta Wis    (talk  &#124; contribs) 12:59, 15 May 2022 (UTC)
 * If it is just about the units, I support keeping both forms. There is a lot of 20th century scientific literature that use Gaussian units.  I don't see any harm in exposing the reader to both systems. Constant<b style="color: #4400bb;">314</b> (talk) 13:34, 16 May 2022 (UTC)

can read article in dark mode
when I look at the article in dark mode on my phone half the equations are just black on black 89.27.237.58 (talk) 13:39, 15 June 2022 (UTC)

Field or flux
There is a recent edit and edit summary regarding field or flux in electrical generators. It seems to me that different generators work in different ways, and some are better explained through flux, and others through field. For some, the explanation is based on changing flux (integral of field) through a loop. For others, it is a wire moving through a magnetic field. In some cases, the difference depends on the reference frame. Can we say field or flux, or is that too confusing? Gah4 (talk) 04:54, 13 July 2022 (UTC)
 * Flux is the surface integral of "magnetic flux density" which is usually taken to be the the meaning of "magnetic field". The Maxwell–Faraday equation is an equation about fields at  a point. So, field should be preferred. Constant<b style="color: #4400bb;">314</b> (talk) 05:17, 13 July 2022 (UTC)
 * After posting that, I was reading Homopolar generator which is described in term of flux. For many generators, it is the change in flux, as you note integral of the flux density or field, that determines the output. I suppose this goes back to using differential or integral form of Maxwell's equations. In any case, I am tending to think that flux is better. Gah4 (talk) 05:37, 13 July 2022 (UTC)
 * If you want to use flux then relate it to the induced EMF, not the curl of the electric field. But the article is about Maxwell's equations and not generators. Constant<b style="color: #4400bb;">314</b> (talk) 06:25, 13 July 2022 (UTC)
 * Hmm, then maybe it isn't a good example. Actually applying Maxwell's equations isn't always easy. I was first thinking about more ordinary generators, a loop rotating in a uniform field, which can be either the change in flux through the loop, or the wire cutting through field lines. Then I thought about the Homopolar generator, which I thought would be in terms of field, but instead is flux. Interesting, though. It seems that the Homopolar generator works even if you stick permanent magnets onto the rotor. I didn't try to figure out why. In any case, Maxwell's equations still have to work. Gah4 (talk) 20:08, 13 July 2022 (UTC)
 * If you want to use flux then relate it to the induced EMF, not the curl of the electric field. But the article is about Maxwell's equations and not generators. Constant<b style="color: #4400bb;">314</b> (talk) 06:25, 13 July 2022 (UTC)
 * Hmm, then maybe it isn't a good example. Actually applying Maxwell's equations isn't always easy. I was first thinking about more ordinary generators, a loop rotating in a uniform field, which can be either the change in flux through the loop, or the wire cutting through field lines. Then I thought about the Homopolar generator, which I thought would be in terms of field, but instead is flux. Interesting, though. It seems that the Homopolar generator works even if you stick permanent magnets onto the rotor. I didn't try to figure out why. In any case, Maxwell's equations still have to work. Gah4 (talk) 20:08, 13 July 2022 (UTC)
 * Hmm, then maybe it isn't a good example. Actually applying Maxwell's equations isn't always easy. I was first thinking about more ordinary generators, a loop rotating in a uniform field, which can be either the change in flux through the loop, or the wire cutting through field lines. Then I thought about the Homopolar generator, which I thought would be in terms of field, but instead is flux. Interesting, though. It seems that the Homopolar generator works even if you stick permanent magnets onto the rotor. I didn't try to figure out why. In any case, Maxwell's equations still have to work. Gah4 (talk) 20:08, 13 July 2022 (UTC)

"Or definitions of linear dependence for PDE can be referred."
This is neither grammatical English nor very clear, even for an expert. 2A01:CB0C:CD:D800:B99A:E7DA:547E:3DD9 (talk) 07:04, 17 October 2022 (UTC)


 * Agree. It has been removed.
 * Constant<b style="color: #4400bb;">314</b> (talk) 18:54, 17 October 2022 (UTC)

quote
A recent edit removed a Hertz quote. I do believe it is an interesting quote that might go in the article. I don't know where it should go, and especially don't know if it should go where it was removed. In case anyone is interested: Gah4 (talk) 22:54, 13 December 2022 (UTC)

''Heinrich Hertz said of Maxwell's equations, "It is impossible to study this wonderful theory without feeling as if the mathematical equations had an independent life and intelligence of their own, as if they were wiser than ourselves, indeed wiser than their discoverer, as if they gave forth more than he put into them." Hertz used Maxwell's equations to produce radio waves, leading to radar and much else.''


 * Yes, Hertz's opinion is mildly interesting, but not notable. It is shared by at least 100,000,000 people over history.  Did his proclamation of his opinion have anything to do with the acceptance of Maxwell's Equations?  Was it important? If so, who says it is important?  Was there a body of opposing opinions?  In the end, this quote is just a decoration that is unneeded in a concise encyclopedic article.  If I was writing a textbook about ME, I might include it, since I share the opinion.   It certainly doesn't need to clutter up the intro.  The intro is supposed to summarize the body.  If we had a section on the cultural impact of ME, maybe it could go there. Constant<b style="color: #4400bb;">314</b> (talk) 00:00, 14 December 2022 (UTC)

Reactance
I just noticed Maxwell's equations is credited to Heaviside who developed vector physics. Heaviside took his ideas from Tait and Gibbs who he credits. Why then is Heaviside not noted for reactance.Electrical reactance It credited that the french coined the term reactance, and the complex part was developed by Steinmetz from Heaviside's operational calculus. It seems that this disqualifies Heaviside's name in the article. But by that same note Heaviside should be disqualified from the Maxwell's equations article. I think people who are reading this and object should head over the the electrical reactance article. K00la1dx (talk) 23:38, 19 December 2022 (UTC)


 * I can hardly follow your logic. There were different contributions from different people, and Heaviside was a primary person in expressing Maxwell's work in the form of vector calculus, thus producing what we call "Maxwell's equations".  In any event, it is not the role of Wikipedia to deal with matters of credit.  —Quondum 23:46, 19 December 2022 (UTC)
 * Because the topic of Maxwell's equations includes a lot more than reactance. Heaviside gets credit for boiling Me down to four variables from the previous 20 or so. Constant<b style="color: #4400bb;">314</b> (talk) 23:47, 19 December 2022 (UTC)
 * Heaviside used Reactance to describe Maxwell... (Maxwell's equations and reactance go hand and hand) BTW Heaviside never got credit for Maxwell's equations. Just open up a physics textbook. It is not there K00la1dx (talk) 23:51, 29 December 2022 (UTC)

Shouldn't the article mention the Faraday's law of induction when the surface is time-dependent?
I mean,

$$ \oint_{\partial \Sigma(t)}{\mathbf{E} \cdot d\boldsymbol{\ell}} = - \frac{d}{dt} \iint_{\Sigma(t)}{\mathbf{B} \cdot d\mathbf{S}} $$

(a) applying the Leibniz Integral Rule:

$$ \oint_{\partial \Sigma(t)}{\mathbf{E} \cdot d\boldsymbol{\ell}} = \oint_{\partial \Sigma (t)} \left( \mathbf{v} \times \mathbf{B} \right) \cdot d\boldsymbol{\ell} - \iint_{\Sigma (t)} \left( \frac{\partial \mathbf{B}}{\partial t} + \left[\nabla \cdot \mathbf{B} \right] \mathbf{v} \right) \cdot d \mathbf{S} $$

(b) from the Gauss's law for magnetism we know that $$\nabla \cdot \mathbf{B} = 0$$:

$$ \oint_{\partial \Sigma(t)}{\mathbf{E} \cdot d\boldsymbol{\ell}} = \oint_{\partial \Sigma (t)} \left( \mathbf{v} \times \mathbf{B} \right) \cdot d\boldsymbol{\ell} - \iint_{\Sigma (t)} \frac{\partial \mathbf{B}}{\partial t} \cdot d \mathbf{S} $$

(motional induction) (transformer induction)

' 2804:14D:5CA1:905E:301A:4CE5:BBE9:ABE7 (talk) 22:58, 8 March 2023 (UTC)
 * This might be related to what I mentioned above, in Field or flux, or, as noted above, depends on frame of reference. The effect is the same, moving a loop in a field, or moving the field through a loop, even though it seems different. Gah4 (talk) 00:37, 9 March 2023 (UTC)
 * Well, yes, but I think you are assuming that it is always easy to choose a reference in which you will perceive the total induction as being motional only ("moving a loop in a field") or transformer-like only ("moving the field through a loop").
 * In any case, I read the article again, carefully this time, and noticed that the article starts this discussion already:
 * $$ \frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{S} = \iint_{\Sigma} \frac{\partial \mathbf{B}}{\partial t} \cdot \mathrm{d}\mathbf{S}\,$$
 * But it also states that . And this is wrong. The most widespread form of the differential version of Faraday's law, $$\nabla \times \mathbf{E} = - \partial \mathbf{B} / \partial t$$, would not hold for a moving surface (looking at my first post you can see that the equality turns out to be $$\nabla \times (\mathbf{E} - \mathbf{v} \times \mathbf{B}) = - \partial \mathbf{B} / \partial t$$).
 * In any case, I started this topic because I think that it is worth explicitly mentioning the Faraday's law when the surface is time-dependent. Just like some old textbooks in electric machine.
 * ' 2804:14D:5CA1:905E:A1AF:E56:F329:1DFB (talk) 20:11, 9 March 2023 (UTC)
 * And then you get to inductors, where it is often written V = L dI/dt, where it should be V = d(LI)/dt. It seems this mostly comes out for solenoids, where L changes when the core moves. Maybe less often, but then there is I = d(CV)/dt also. Gah4 (talk) 05:48, 10 March 2023 (UTC)
 * Yes. We can't use $$v = L \, di/dt $$ to analyze inductances that are time-dependent. And we can't come back to $$v = d(Li)/dt $$ starting from $$v = L \, di/dt $$.
 * However, adapting to our original context, that isn't my understanding of the statement . Since the only differential version shown in the article is $$\nabla \times \mathbf{E} = - \partial \mathbf{B} / \partial t$$, I understand that it is possible to obtain $$ \oint_{\partial \Sigma(t)}{\mathbf{E} \cdot d\boldsymbol{\ell}} = \oint_{\partial \Sigma (t)} \left( \mathbf{v} \times \mathbf{B} \right) \cdot d\boldsymbol{\ell} - \iint_{\Sigma (t)} \frac{\partial \mathbf{B}}{\partial t} \cdot d \mathbf{S} $$ solely from $$\nabla \times \mathbf{E} = - \partial \mathbf{B} / \partial t$$. But it is not. The presented differential version is restricted to the assumption of time-independent surface.
 * ' 2804:14D:5CA1:905E:84E6:695A:5BBA:12F2 (talk) 17:03, 10 March 2023 (UTC)
 * However, adapting to our original context, that isn't my understanding of the statement . Since the only differential version shown in the article is $$\nabla \times \mathbf{E} = - \partial \mathbf{B} / \partial t$$, I understand that it is possible to obtain $$ \oint_{\partial \Sigma(t)}{\mathbf{E} \cdot d\boldsymbol{\ell}} = \oint_{\partial \Sigma (t)} \left( \mathbf{v} \times \mathbf{B} \right) \cdot d\boldsymbol{\ell} - \iint_{\Sigma (t)} \frac{\partial \mathbf{B}}{\partial t} \cdot d \mathbf{S} $$ solely from $$\nabla \times \mathbf{E} = - \partial \mathbf{B} / \partial t$$. But it is not. The presented differential version is restricted to the assumption of time-independent surface.
 * ' 2804:14D:5CA1:905E:84E6:695A:5BBA:12F2 (talk) 17:03, 10 March 2023 (UTC)

units
I just found a paper that has interesting suggestions on units and Maxwell's equations. Some that have been discussed here, but not so well. Gah4 (talk) 22:12, 1 June 2023 (UTC)
 * Yes, when I learned E&M circa 1970, we used Purcell's Berkley Physics book which used gaussian units. There was no pesky permittivity or permiability.  B=H so there is no debate about which is the real field.  We didn't even bother with D and B in vacuum.  The impedance of free space was unity.  Etc.  Hey, Purcell was a Nobel Laureate who worked on radar in WW2, so he ought to know a thing or two.  But over in engineering, they used the MKS system.  When I became one of them, I had to relearn a thing or two and now MKS seems "natural". <b style="color: #4400bb;">Constant314</b> (talk) 00:14, 2 June 2023 (UTC)
 * Yes, we had Purcell, too. First edition in 1977. But now there is 3rd edition Purcell in SI units. I believe with an explanation of Gaussian units at the end. Since I didn't have my original Purcell, I bought a 2nd edition last year. Purcell is one of the few books that treat magnetism through special relativity, and also problems with charge moving one way at high speed, and you (the field measurer) moving the other way at high speed. But anyway, the idea of the paper is to write them in unit independent form, with the right constants added for each unit system. Gah4 (talk) 01:55, 2 June 2023 (UTC)
 * Yes, we had Purcell, too. First edition in 1977. But now there is 3rd edition Purcell in SI units. I believe with an explanation of Gaussian units at the end. Since I didn't have my original Purcell, I bought a 2nd edition last year. Purcell is one of the few books that treat magnetism through special relativity, and also problems with charge moving one way at high speed, and you (the field measurer) moving the other way at high speed. But anyway, the idea of the paper is to write them in unit independent form, with the right constants added for each unit system. Gah4 (talk) 01:55, 2 June 2023 (UTC)
 * Yes, we had Purcell, too. First edition in 1977. But now there is 3rd edition Purcell in SI units. I believe with an explanation of Gaussian units at the end. Since I didn't have my original Purcell, I bought a 2nd edition last year. Purcell is one of the few books that treat magnetism through special relativity, and also problems with charge moving one way at high speed, and you (the field measurer) moving the other way at high speed. But anyway, the idea of the paper is to write them in unit independent form, with the right constants added for each unit system. Gah4 (talk) 01:55, 2 June 2023 (UTC)

Gauss's Law
Note: this is my first attempt to contribute to a wikipedia article so your patience is appreciated. Comment: In the article's section on Gauss's Law (https://en.wikipedia.org/wiki/Gauss%27s_law#Deriving_Gauss's_law_from_Coulomb's_law) it is written "Gauss's law describes the relationship between a static electric field and electric charges" I believe the word "static" should be removed here. Using "static" would be correct for "Coulomb's Law" but not "Gauss's Law". This can be seen from the main article on Gauss's Law in the section "Deriving Gauss's law from Coulomb's law" (https://en.wikipedia.org/wiki/Gauss%27s_law#Deriving_Gauss's_law_from_Coulomb's_law). In that article, they write "Since Coulomb's law only applies to stationary charges, there is no reason to expect Gauss's law to hold for moving charges based on this derivation alone. In fact, Gauss's law does hold for moving charges, and in this respect Gauss's law is more general than Coulomb's law." EMclarity (talk) 16:32, 26 July 2023 (UTC)


 * I believe you're correct about the word static; I'll go remove it right now.
 * -Proxima Centari (talk) 04:16, 15 August 2023 (UTC)
 * Done. -Proxima Centari (talk) 04:22, 15 August 2023 (UTC)

Jefimenko
Why the discussion of Jefimenko in the Solutions section? Lienard-Wiechert were over 60 years earlier, and Jefimenko really does not add to their work as far as I can see. Having the LW work here would be appropriate, instead of the paragraph there now.

Relativistic, Tensor calculus table
Here an esoteric antisymmetrisation notation is used, that is not explained in the text (you have to follow a link). This introduces an ugly factor of 2 (actually 2!) into some equations and not others. Besides it makes the article less self contained. Just write $$F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$$ etc.

An explicit form of the field tensor would make this article more self contained. Aoosten (talk) 16:19, 19 January 2024 (UTC)