Talk:Maxwell's equations

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Archives Archive 1 ( ? - 2003) Archive 2 ( 2004 - 2005)

1 Magnetic Monopoles and Complete and Correct Equations of Electromagnetism (Maxwell's Equations)
2 Maxwell relations
3 SI Verses CGS Units
4 Meaning of "S" and "V" and "C" on the integrals
5 Balancing the view on Maxwell's equation.
6 Possible correction
7 Integral vector notation
8 Matter of global structure
9 Stable version now
10 Macro vs Micro
11 Original Maxwell Equations
12 Last occurrences of boldface vectors
13 Link to simple explanation
14 Another formation
15 History of Maxwell's equations
16 References

[edit] Magnetic Monopoles and Complete and Correct Equations of Electromagnetism (Maxwell's Equations)

The above equations are given in the International System of Units, or SI for short.

$\nabla \cdot \mathbf{E} = \frac{1}{c} \frac{\partial E} {\partial t}$

$\nabla \cdot \mathbf{B} = \frac{1}{c} \frac{\partial B} {\partial t}$

$\nabla \times \mathbf{E} + \frac{1}{c} \frac{\partial \mathbf{E}} {\partial t} + \nabla E = 0$

$\nabla \times \mathbf{B} + \frac{1}{c} \frac{ \partial \mathbf{B}} {\partial t} + \nabla B = 0$

Maxwell's Equations are really just one Quaternion Equation where E=cB=zH=czD

Where c is the speed of light in a vacuum. For the electromagnetic field in a "vacuum" or "free space", the equations become: Notice that the scalar, non-vector fields E and B are constant in "free space or the vacuum". These fields are not constant where "matter or charge is present", thus there are "magnetic monopoles", wherever there is charge. This is due to the relation between magnetic charge and electric charge W=zC, where W is Webers and C is Coulomb and z is the "free space" resistance/impedance = 375 Ohms!

Notice that there is a gradient of the electric field E added to the Electric Vector Equation.

$\nabla \cdot \mathbf{E} = 0$

$\nabla \cdot \mathbf{B} = 0$

$\nabla \times \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{E}} {\partial t}$

$\nabla \times \mathbf{B} = -\frac{1}{c} \frac{\partial \mathbf{B}}{\partial t}$

Yaw 19:19, 23 December 2005 (UTC)

Yaw, thanks for putting that here, instead of the article, because some of it is wrong (if there is $\frac{1}{c} \frac{\partial \mathbf{E}} {\partial t}$ and $\frac{1}{c} \frac{\partial \mathbf{B}}{\partial t}$ , the units are not SI but are cgs. moreover the sign on one or the other cannot be the same. one has a + sign and the other - (which one is a matter of convention - essentially the right hand rule). This has the appearance of original research (and thus doesn't belong in WP), but i'll let others decide. r b-j 22:34, 23 December 2005 (UTC)

I see that User:Yaw has just created Laws_of_electromagnetism; I don't want to bring back nightmares by clawing through the physics, so may I ask that one of you folks from Maxwell's equations take a look and figure out what to do with it? I am guessing that it will need to be merged (or not) and redirected here. Thanks. bikeable (talk) 01:58, 31 December 2005 (UTC)

Yaw is basing this on the characteristic impedance of free space (which can be derived from the constancy of the speed of light). It's a math exercise, non-standard, but looks self-consistent. It would be unfair to spring on others as standard; and probably would not survive AFD. So Yaw has an uphill climb to acceptance in the larger community. --Ancheta Wis 12:06, 1 January 2006 (UTC)

[edit] Maxwell relations

Is there any chance of getting the maxwell relations page (http://en.wikipedia.org/wiki/Maxwell_relations) linked to this page? In P-chem, we referred to these also as maxwell's equations, and it seem like linking the page for those would be a nice improvement. Thanks.

$\nabla \times \mathbf{H} = \mathbf{J} + \frac {\partial \mathbf{D}} {\partial t}$

then

$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0\varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}$

then

$\nabla \times \mathbf{B} = \frac{1}{c} \frac{ \partial \mathbf{E}} {\partial t} + \frac{4\pi}{c} \mathbf{J}$

Might be right?

[edit] SI Verses CGS Units

Why do the equations change when you switch from kilograms to grams and meters to centimeters? Or are there other changes as well? That is, are there various arbitrary definitions for units of D, E, H, B, etc., and the constants mu and epsilon, that vary when we shift from one system to another?

Consider, as an example, Einstein's equation relating energy to mass. If we let the number E be the energy in Joules = kg * (meters)^2 / sec^2 , and let the number M be the rest mass in kg, then the ratio E/M equals the value c^2 , where c is the number equal to the the speed of light in meters/sec. That is, E = M * c^2.

Now, suppose we represent distances in terms of "light-seconds". Suppose we let E' be the energy in terms of the new system, that is, in terms of kg * (light sec)^2 / sec^2. Then the number E' = E/(c^2). Hence, under the new system of units, the ratio of energy to mass is E'/M = [ E/(c^2) ] /M = E/(M*c^2) = E/E = 1. That is, if we measure distances in terms of light seconds and energy in terms of kg * (light sec)^2 / sec^2 , then E = m.

This raises another question: What would physics equations look like if we used light seconds, and altered measurement units to match this in a nice fashion?

please see http://en.wikipedia.org/wiki/Planck_units for your answer.

[edit] Meaning of "S" and "V" and "C" on the integrals

I think that it would be very useful to explain exactly what the $\oint_S$ or $\oint_c$ is integrating over. I would assume that "S" stands for surface, "V" stands for volume, and "C" stands for .. Closed path? In any case, it should be explained to what extend the surfaces, paths, of volumes can be changed, and the meaning behind it. Fresheneesz 07:21, 9 February 2006 (UTC)

It might be helpful to put explanations of them in that table where all the main variables are explained. Fresheneesz 07:24, 9 February 2006 (UTC)

Perhaps a link to Green's theorem or Stokes' theorem in the explanatory text would suffice. --Ancheta Wis 11:20, 9 February 2006 (UTC)

I see that the 3rd, 4th, and 5th boxes from the bottom explain the S C and V. 11:25, 9 February 2006 (UTC)

I suppse it is explained, a bit. But I think it would be more consistant to give the integral notations their own box (after all, the divergence and curl operators get their own box - and somehow.. units?). Also I just have a gut feeling that it could be more clear how the contours, Surfaces, and volumes connect with the rest of the equation. Maybe I'm just expecting too much. Fresheneesz 20:18, 9 February 2006 (UTC)

Here is where Green's theorem comes into its own because Green assumed the existence of the indefinite integrals on a surface (the sums of E, B etc) extending to +/- infinity (think a set of mountain ranges, one mountain range for each integral). Then all we have to to do is take the contours and read out the values (the altitudes of the mountain) of the integrals at each point along the contour, and voila the answer. This method is far more general than only for Maxwell's equations. I think the additional explanation which you might be looking for belongs in the Green's theorem article rather than cluttering up the physics page. However, you are indeed correct that physicists would have a better feel for these integrals because of the hands-on experience. Same concept for volume integrals, only it is an enclosing surface, etc. --Ancheta Wis 00:35, 10 February 2006 (UTC)

[edit] Balancing the view on Maxwell's equation.

To follow wikipedias neutrality standard I think we should make a sektion where we describe the most important objections to Maxwell's. Equanimous2 22:05, 24 February 2006 (UTC)

Maxwell's equations are well established; they document the research picture of Michael Faraday. They are the basis of special relativity. They form part of the triad Newtonian mechanics / Maxwell's equations / special relativity any two of which can derive the third (See, for example, Landau and Lifshitz, Classical theory of fields ). Lots has been written about Newton and Einstein but I have never seen the same fundamental criticisms for Maxwell's equations. I hope you can see why -- they simply document Faraday (with Maxwell's correction). --Ancheta Wis 10:29, 25 February 2006 (UTC)

You illustrate the problem very well when you write that you never seen fundamental criticisms for Maxwell's equations. That is exactly why I think we should have such a section. What page in Landau and Lifshitz do you find that prof ? It could maybe be a good counter argument for use in the section. Maxwell himself didn't believe that his equations where correct for high frequencies. Another critic is that Maxwell's don't agree with Amperes force law and there is some experiments which seems to show that Ampere where correct. See Peter Graneau and Neal Graneau, "Newtonian Electrodynamics" ISDN: 981022284X --Equanimous2 15:42, 27 February 2006 (UTC)

Maxwell didn't predict the electric motor either. That happened by accident when a generator was hooked up in the motor configuration. The electric motor was the greatest invention of Maxwell's century, in his estimation. That doesn't invalidate his equations. I refer you to electromagnetic field where you might get some grist for your mill. It's not likely that his equations are wrong, because the field is a very successful concept. On the triad of theories, if you can't find Landau and Lifshitz, try Corson and Lorrain. Landau and Lifshitz are classics and I would have to dig thru paper to get a page number. But at least you know a book title which you could get at a U. lib. and search the index. --Ancheta Wis 21:35, 27 February 2006 (UTC)

I personally have never seen a valid criticism of Maxwell's equations, however I am aware that critics of Maxwell do exist.

The most famous objection to Maxwell came at around the turn of the 19th century from a French positivist called Pierre Duhem. This objection came in relation to the elasticity section in part III of Maxwell's 1861 paper On Physical Lines of Force - 1861, and not in relation to 'Maxwell's Equations'.

Duhem's allegation, echoed more recently by Chalmers and Siegel, concerned Maxwell's use of Newton's equation for the speed of sound at equation (132). Duhem alleged that Maxwell should have inserted a factor of 1/2 inside the square root term and hence obtained the wrong value for the speed of light. Duhem alleged that in getting the correct value for the speed of light, that Maxwell had in fact cheated.

Duhem's allegation was based on the notion that Maxwell hadn't taken dispersion into consideration. However, we all know nowadays that a light ray doesn't disperse. Extreme coherence is a peculiar property of electromagnetic radiation. Whether or not Maxwell was explicitly aware of this, it is now retrospectively clear that Maxwell did indeed use the correct equation and that it was Pierre Duhem that made the error. (203.115.188.254 07:58, 20 February 2007 (UTC))

[edit] Possible correction

Please, check out the Historical Development where it says :

" the relationship between electric field and the scalar and vector potentials (three component equations, which imply Faraday's law), the relationship between the electric and displacement fields (three component equations)".

I think there is a mistake there because Faraday's law relates the electric field with the variable magnetic field density(B), as I have studied it in the book "Fundamentals of Engineering Electromagnetics" by David K. Cheng.

The text is correct as written. What Maxwell gives is essentially the relation $\mathbf{E} = -\nabla \phi - \partial\mathbf{A}/\partial t$ , where φ is the electric potential and A is the magnetic vector potential. If you take the curl of this relation you get Faraday's law. —Steven G. Johnson

Another thing is that it says "displacement fields", but that has no sense because it doesn't say whether it is an electric or magnetic field which it displaces. I think that a possible correction could be:

" the relationship between electric field and the scalar and vector potentials (three component equations), the relationship between the electric field and displacement magnetic fields (three component equations, which imply Faraday's law)".

I'd appreciate if someone could check whether this correction could be made or not. Thank you.

No, the term displacement field in electromagnetism always refers to a specific quantity (D). It doesn't really "displace" the electric or magnetic fields. —Steven G. Johnson 05:58, 28 February 2006 (UTC)

[edit] Integral vector notation

I'll admit that I don't know much about tensors, but I do recall Maxwell's equations in vector (first order tensor) form:

$\oint \vec{E} \cdot d\vec{A} = \frac{Q_{encl}}{\epsilon_0}$

$\oint \vec{B} \cdot d\vec{A} = 0$

$\oint \vec{E} \cdot d\vec{\ell} = \varepsilon = -\frac{d\Phi_B}{dt}$

$\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{encl} = \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$

How does this fit in with all those other tensor variables this article uses? —Matt 04:17, 7 May 2006 (UTC)

Stokes theorem let's you change from the differential form to the integral form. Both forms are already listed in the article. The integral equations you mention are in the article in the right-hand column of the first table. -lethe ^{talk +} 04:32, 7 May 2006 (UTC)

[edit] Matter of global structure

Scientific reliability often rests with two points :

* the experimental checking
* the explanation of the phenomena as caused by more fundamental laws

In the case of Maxwell's equations, the link with relativity, discovered by Einstein, has fulfiled the second clause.

Moreover, the expression of these equations using only E and B instead of D and H is often used in the first place to teach electromagnetism, because it seems a more self-consistent whole and is easily linkable to other domains like optics, radio or relativity. The case is that the relations expressed with D and H are more useful by those who use them practicaly : the E and B ones are limited to the vacuum.

Note: The set of Maxwell's Equations that use E and B are not limited to vacuum. They are completely general, so long as as the charge density refers to the net charge density (not free charge density) and similarly for the current density. In matter it is easier to sweep the bound charge and current densities under the rug by introducing D and H, but the original equations continue to apply in matter. See Griffiths's Introduction to Electrodynamics for a similar argument.--Jewtemplar 05:48, 17 July 2006 (UTC)

Last, the relations with D and H are true only under one hypothesis : that of the medium is continuous. Then we make a model that permits to write generalized relations, but that must therefore be completed by the relations that show what happens when we cross from a medium to another, in order to remain coherent and utterly useful (there isn't any infinite continuous medium in the universe excepted that of the vacuum):

$\vec{n}_{12} \wedge (\vec{E}_{2} - \vec{E}_{1}) \ = \ 0$

$\vec{n}_{12} \wedge (\vec{H}_{2} - \vec{H}_{1}) \ = \ \vec{J}_s$

$\vec{n}_{12} \ . \ (\vec{D}_{2} - \vec{D}_{1}) \ = \ \rho_{s}$

$\vec{n}_{12} \ . \ (\vec{B}_{2} - \vec{B}_{1}) \ = \ 0$

I propose thus a constructive plan, starting from the expression with E and B, then introducing the model of continuous materials, then finally writing and presenting in a pretty frame the expressions with D and H and the relations just above.

What's your opinion ?

Almeo 08:04, 25 May 2006 (UTC)

Your suggestion would fit very well in the Electromagnetic field article. --Ancheta Wis 08:31, 25 May 2006 (UTC)

In that article is a red link to the Maxwell-Hertz equations which were named by Einstein. Perhaps your suggested article might fit there. --Ancheta Wis 08:45, 25 May 2006 (UTC)

I disagree with your premise. The boundary conditions at discontinuous material interfaces are already perfectly derivable from the macroscopic Maxwell equations (see any textbook); there is no special problem with discontinuous media as long as you are willing to deal with delta functions and the like. It's reasonable to derive and supply the field boundary conditions somewhere on Wikipedia (probably in a separate article), but there's no need to start with them here. —Steven G. Johnson 08:48, 25 May 2006 (UTC)

Unless you are referring to the distinction between microscopic and macroscopic Maxwell equations? That is already referred to in the article, but it might make a reasonable separate article to derive this relationship more precisely, following e.g. the treatment in Jackson or some similar graduate-level text. Although it is rather fundamental, it certainly doesn't belong in a basic introduction to electromagnetism. —Steven G. Johnson 08:48, 25 May 2006 (UTC)

Yes I agree, it's more a problem of distinction between the microscopic and the macroscopic model (that included, in my view, the relations for discontinuity, but you are right they may also be derivated from the generalized Maxwell equations). Because of the way I've been taught Maxwell's equations, it seemed more natural for me to introduce them with the microscopic form -- that is the reason why I am tempted to change the structure of the article. I will probably follow your advice -- link, from Electromagnetic Field article, an article exposing how macroscopic is deduced from microscopic. Then I would make it more noticeable on the Maxwell Equation article that there is a equivalent microscopic form in the Electromagnetisme Field article (I didn't know!). Thanks. Almeo 09:31, 25 May 2006 (UTC)

[edit] Stable version now

Let's begin the discussion per the protocol. What say you? --Ancheta Wis 05:08, 11 July 2006 (UTC)

HOw about "stop adding this to bunch of articles when the proposal is matter of days old, in flux, under discussion, not at all widely accepted and generally obviously not ready for such rapid, rather forceful, use. -Splash - tk 20:12, 12 July 2006 (UTC)

[edit] Macro vs Micro

Please change your discussion of the microscopic versus macroscopic Maxwell equations. It is wrong and misleading but you are in good company. This mistake is made often in the literature. I see it regularly in bioelectromagnetics literature (as a matter of fact today) accompanied by lengthy and awkward discussions. As far as we presently know Maxwell's equations, as written with charge and currents, are valid at all scales (in the absence of quantum considerations). There is no need to make a distinction between microscopic and macroscopic Maxwell equations. You simply pick a scale and then include all charges and all currents in a manner that is mathematically consistent with the choosen scale. So at macroscopic levels atomic current and charges take on a mathematical description in terms of generalized distributions (ie delta functions). ---- Daniel

[edit] Original Maxwell Equations

I think it would be a good idea, for completeness, to also include the original versions of Maxwells equations. From what I gather, there were the 1865 versions and the 1873 versions, if I am not mistaken. All previous versions should be included here for historical and reference purposes. Also does anyone have a link to the original 1865 paper by Maxwell on electromagnetism, this would be a good link to be included on this page, and as well links to other relevant documents from Maxwell.

Millueradfa 18:36, 5 August 2006 (UTC)

They are the same equations. The notation differs. I propose that the other equations which are not the canonical 4 (or 2 in Tensor notation) can be listed by link name (such as conservation of charge). This links strongly to the set in the history of physics. --Ancheta Wis 19:47, 5 August 2006 (UTC)

Gauss's law is the only equation which occurs both in the original eight 'Maxwell's Equations' of 1864 and the modified 'Heaviside Four' of 1884. (203.115.188.254 08:08, 20 February 2007 (UTC))

It would be more accurate to say that they are mathematically equivalent equations; even when the notation is modernised, the arrangement of the equations is somewhat different. The equations in the arrangement that Maxwell gave them (but in modern vector notation) are listed in the article: A Dynamical Theory of the Electromagnetic Field. —Steven G. Johnson 16:22, 6 August 2006 (UTC)

Sorry, my english isn´t as good I want and this is my first edition. I think that the curl and divergence operator have not units, there are a diferential operators.

[edit] Last occurrences of boldface vectors

This notes that the edit as of 05:21, 25 November 2006 is one of the last occurrences of boldface to denote vectors, with italic to denote scalars. Boldface has been the convention for vectors in the textbooks, in contrast with the current (10:30, 14 December 2006 (UTC)) article's → notation for vectors, as used on blackboard lectures. Feynman would also use blackboard bold to denote vectors when lecturing, if it wasn't perfectly clear from the context.

The current look is jarring, but readable, to me. --Ancheta Wis 10:30, 14 December 2006 (UTC)

Thanks to the anon. The look has reverted to the textbook appearance for the equations. --Ancheta Wis 13:26, 6 January 2007 (UTC)

[edit] Link to simple explanation

http://www.irregularwebcomic.net/1420.html has a simple English explanation of the equations and their physical implications. I don't know how accurate it is, but I think it's close enough. --71.204.251.243 15:07, 16 December 2006 (UTC)

I think it's accurate enough, although there are couple of shortcuts but they're needed to make it simple enough. So I added that link to the article. --Enok.cc 21:35, 17 December 2006 (UTC)

The article says the same thing. The boldface for vectors in your link is how the equations have looked in past versions of the article. --Ancheta Wis 17:53, 16 December 2006 (UTC)

[edit] Another formation

Isn't there another formation were you take the modified Schroedinger equation and assume gauge invariance, and you solve it, and out pop Maxwell's Equations, almost magically? I am no expert in the field, but I remember a professor mentioning how remarkable it is. IS that notable enough for mention here? Danski14 00:43, 2 February 2007 (UTC)

[edit] History of Maxwell's equations

I propose that these newest changes to the article be placed in another article History of Maxwell's equations, and a link to them be included in this article. With thanks to the contributor, --Ancheta Wis 09:02, 16 February 2007 (UTC)

This diff ought to help you in that article. --Ancheta Wis 03:38, 18 February 2007 (UTC)

Ancheta Wis, The idea of a special historical section is fine. However, your reversion contains a number of serious factual inaccuracies which can be checked simply by looking up both the 1861 and the 1864 papers. Web Links for both were supplied.

Take for example your paragraph "Maxwell, in his 1864 paper A Dynamical Theory of the Electromagnetic Field, was the first to put all four equations together and to notice that a correction was required to Ampere's law: changing electric fields act like currents, likewise producing magnetic fields. (This additional term is called the displacement current.) The most common modern notation for these equations was developed by Oliver Heaviside."

This is not true. Maxwell put a completely different set of 'eight equations' together in his 1864 paper. The set of four that you are talking about was complied by Oliver Heavisde in 1884 and they were all taken from Maxwell's 1861 paper. Also, the correcton to Ampère's Circuital Law occurred in Maxwell's 1861 paper, and not in the 1864 paper.

Your quote of Maxwell's regarding electromagnetic waves was wrong also. The correct quote can be found, exactly as referenced in the 1864 paper.

Also, you restored the vXB term into the integral form of Faraday's law. That term is correct, but only if we have a total time derivative in the differential form. The Heaviside four use partial time derivatives, and the Lorentz force F = qvXB sits outside this it as a separate equation. (203.115.188.254 06:13, 18 February 2007 (UTC))

The equations that Maxwell put forth in his 1865 paper were equivalent to the modern ones plus some equations that are now considered auxiliary (such as Ohm's law and the Lorentz force law), the only substantive non-notational difference being that since Maxwell wrote them in terms of the vector and scalar potentials he had to make a gauge choice. And whether it is 8 equations or 20 depends on how you count. Maxwell labelled them A-H, but several of these were written as three separate equations, due to the lack of vector notation. Maxwell himself wrote, on page 465 of the 1865 paper, that There are twenty of these equations in all, involving twenty variable quantities.

On the issue of the 20 equations, I am fully aware of everything that you have said above. But to call it 20 equations is like talking about Newton's 'Nine' Laws of Motion, ie. three for the X- direction, three for the Y- direction, and three for the Z- direction. I wish that these people who insist on emphasizing the issue of the 20 equations would make their point.

The original eight equations are indeed as you say, equivalent to the 'Heaviside Four'. Faraday's law in the 'Heaviside Four' is the one that corresponds most closely to the Lorentz Force in the original eight. I was never disputing whether they were physically equivalent or not. The fact is nevertheless that only one equation exactly overlaps between the two sets and as such we need to be clear and accurate as to which set we are talking about. The 'Heaviside Four' are the commonly used set that appear in most modern textbooks, and as such it is right that they should take precedence in the article. It is still very convenient however to be able to view the historical 'Eight' further down the page. (222.126.43.98 13:49, 21 February 2007 (UTC))

The quotation that we had regarding the speed of light was:

This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.

This quotation is correct. It appears on page 466 of the 1865 paper.

You are correct. This quote does indeed appear on page 466 of the 1864/1865 paper. I had forgotten about it. However the quote that I replaced it with appears on page 499 of the same paper in immediate connection with his electromagnetic theory of light. The page 466 quote in some respects is a quote out of context because it ommits the very important sentences that follow on from it and that expose Maxwell's thinking on the matter. The page 499 quote is concise and completely conveys Maxwell's thoughts on that particular issue. (222.126.43.98 13:57, 21 February 2007 (UTC))

There was no "1864 paper" as far as I can tell. In 1864 he gave a presentation to the Royal Society, only the abstract of which was published in 1864. The main paper was published in Philosophical Transactions of the Royal Society of London, volume 155, p. 459-512, in 1865. (This paper was "read" on December 8, 1864, which refers to the oral presentation. The Philosophical Transactions list the publication date as 1865.)

—Steven G. Johnson 16:39, 20 February 2007 (UTC)

OK then, refer to it as the 1865 paper. But this is an extremely pedantic point. He wrote it in 1864 and he dated it 1864 so I would have thought that 1864 was a more accurate way of describing it. But change it to 1865 if you like. The web links are available anyway if anybody wants to read it. (222.126.43.98 13:50, 21 February 2007 (UTC))

A little more humility on your part would be nice. You anonymously went through this article with a sledgehammer, carelessly calling lots of things "wrong" when they were not wrong at all, as you reluctantly acknowledge above.

It is important to emphasize that Maxwell's original equations are mathematically equivalent to the present-day understanding of classical electromagnetism, since Wikipedia readers can (and have been, on several occasions) confused on this very point. And calling them 20 equations, as Maxwell himself did, is important to emphasize the debt that we owe to modern notation; calling them "8 equations" obscures the historical fact that Maxwell had to work with each component separately. (Your analogy with Newton's laws seems off, since I would guess that Newton phrased them in a coordinate-free fashion.)

The p. 466 quote is much clearer and more evocative than your quote about "The agreement of the results ...," in my opinion, and your vague complaint about it being out of context seems baseless. As a general principle, one expects to find this kind of general summarizing quote in the introduction of the paper, and passages buried in the middle of the paper (such as your quote) tend to be more technical and less accessible.

When citing publications, it is the standard scholarly convention to cite the actual publication date, not the date the manuscript was written or sent to the publisher. I'm surprised you don't know this, or call it "pedantic"...citing the incorrect year makes it significantly harder to look up the publication in a library.

I'm inclined to revert the article to something close to the state it was in a few days ago, before you hacked it up. A detailed description of Maxwell's historical formulation should go into a separate History of classical electromagnetism article (probably merged with A Dynamical Theory of the Electromagnetic Field), since it is only marginally useful to present-day readers trying to understand the physical laws and their consequences. (Look at any present-day EM textbook.) —Steven G. Johnson 18:23, 21 February 2007 (UTC)

The physical equivalence of the two sets of equations is certainly a very interesting topic. I'm actually much more sympathetic to you on that point than you realize. I have never been able to have a rational discussion on the equivalence of the two sets because I am endlessly having to counteract people who claim that the two sets represent completely different physics and that the modern 'Heaviside Four' have removed all the vital ingredients from the 'Twenty' in the 1865 paper.

I have been trying to argue that the two sets are essentially equivalent. That is why I wanted to have the two sets clearly laid out, so as everybody can see them and make their own minds up.

However, I am still correct when I say that they are a completely different set of equations. Gauss's law alone appears in both sets. The Ampère/Maxwell equation in the 'Heaviside Four' is an amalgamation of equations (A) and (C) in the 1865 paper. Faraday's law of electromagnetic induction occurs as a partial time derivative quation in the 'Heaviside Four'. This means that it excludes the convective vXB term of the Lorentz force. The Lorentz force therefore has to be introduced nowadays as a separate auxilliary equation beside the 'Heaviside Four'. In the 1865 paper, we actually have the Lorentz force in full as equation (D).

The div B equation of the 'Heaviside Four' as you correctly state is already implied in the 1865 paper by the curl A = B equation. However, the curl A = B equation tells us more than the div B equation does.

Overall, I agree with you that the physical differences between the two sets are very minor and that they most certainly do not contradict each other in any manner whatsoever. But both sets need to be made available in order to counteract specious suggestions from certain quarters that the modern set has taken very important physics out of the original set.

As it stands now, the original set are well down the page. I can see no problem with this. If you want to take them out altogether, you will have to explain that the modern 'Heaviside Four' all appeared in Maxwell's 1861/62 paper and not in his 1865 paper, and that they were selected by Oliver Heaviside. The physical equivalence argument is no basis for allowing confusion to set in regarding the finer details. The facts must be clearly laid out for all to see. (203.115.188.254 06:56, 22 February 2007 (UTC))

I have no objection to clearly laying out the historical facts and explaining precisely how Maxwell's formulation can be transformed into the modern formulation. I just agree with Ancheta that it does not belong in this article, which should be an introduction to the equations governing electromagnetism as they are now named and understood. The article should have a brief summary of the history and link to another Wikipedia article for a detailed exposition.

Most practicing mathematicians and scientists would disagree with your assertion that the equations are "completely different" — if two sets of equations are mathematically equivalent, with at most trivial rearrangements, then they are at most superficially different. Moreover, it is universal in modern science and engineering to refer to Heaviside's formulation as "Maxwell's equations", despite the superficial differences from the way Maxwell expressed the same mathematical/physical ideas (plus some auxiliary equations like the Lorentz force law which we now group separately). Overemphasizing these superficial differences in a general overview article does a disservice to a novice reader.

(You state that "the curl A = B equation tells us more than the div B equation does," but this is dubious: there is an elementary mathematical theorem that any divergence-free vector field can be written as the curl of some other vector field.)

(By the way, it looks like we've had much the same experience as you, here on Wikipedia — if you look at the history if this Talk page, you'll see we've had the same problem with specious arguments about the supposed "lost" physics of Maxwell's original formulation, or his quaternion-based formulation, compared to the modern formulation.)

I also strongly encourage you to get a Wikipedia username (click the "Log in / create account" button at the top-right). It is extremely helpful to other editors if you use a consistent username so that we know who we are dealing with when we see your edits/comments.

—Steven G. Johnson 18:12, 22 February 2007 (UTC)

Point taken. Yes it seems that we hold basically the same point of view but that we were differing only on strategy. For years I used to argue that the so called 'Heaviside Four' carried in substance exactly what was in Maxwell's original papers.

But recently these specious arguments about quaternions seem to have been surfacing alot on the internet and so I thought that a clear exposition of the original '20' needed to be made.

I have moved the original eight now well down the page to section 8. What I don't understand is that the edits only ever show up when I re-save everytime that I log on. That might be something to do with the cookies on this computer. Are you currently getting the orignal eight Maxwell's equations at section 8?

Anyway, by all means move them to a new indexed historical link. As long as they are accessible to the readers, that is all that is important. I am quite hapy to refer to the 'Heavisde Four' as Maxwell's equations. I always found it so annoying everytime I talked about Maxwell's equations, and somebody would totally duck the point I was making and correct me and say 'You mean Heaviside's Equations!'.

There are so many people out there who are steeped in some belief that Maxwell's original equations carried some hyperdimensional secrets and that Heaviside's modifications are some kind of cover story.

Yes, I think I will get a username. I'm quite knew to this and I was simply browsing over the electromagnetism topics. (203.115.188.254 01:30, 23 February 2007 (UTC))

[edit] References

There seem to be an awful lot of links to crackpot sites like zpenergy.com and vacuum-physics.com. What gives? —The preceding unsigned comment was added by 164.55.254.106 (talk) 19:07, 23 February 2007 (UTC).

The ZPenergy links are only photocopies of Maxwell's 1865 paper 'A Dynamical Theory of the Electromagnetic Field'. There is nothing crackpot about that. Don't shoot the messenger. (203.189.11.2 11:52, 24 February 2007 (UTC))

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