Talk:Maxwell's equations/Archive 3

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)

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?

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.


 * The reason for the change is that CGS was defined only for mechanics, and was later extended to electrodynamics is a more intuitive manner. Indeed, one sees that in SI the coulombic force is $$ F = \frac{Q_1 Q_2}{4\pi \epsilon_0 r^2}$$, with the constant factor there because the unit of charge was defined elsewhere. In CGS, we know that the expression will be of the same form, so $$ F' \propto \frac{q_1 q_2}{r^2}$$. Since we have only defined length mass and time, charge has yet to be set. We can now just use this expression to define charge, without any prefactor, since there is no reason not to. Then F' is no longer proportional, but equal to $$\frac{q_1 q_2}{r^2}$$. It follows from there that resultant equations in E&M will be different. Redoubts (talk) 16:12, 3 April 2008 (UTC)


 * The SI and CGS units for the same quantities have different dimensions, so they are not interchangeable. CGS sets certain constants (that have dimensions) to 1. --Spoon! (talk) 00:10, 10 April 2008 (UTC)

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)

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))

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 &phi; 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)

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? &mdash;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)

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)

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)


 * The characterization of the equation for E in terms of the potential needs to be changed. This is not a reflection of the Lorentz force. The velocity vector v (which Maxwell calls G) is the velocity with respect to the stationary reference (that is, &ldquo;the vacuum&rdquo;), as Maxwell states at one point in the treatise. The term G&times;B is a compensation for Galilean invariance. Maxwell's E is equivalent to what we would now write as E + G&times;B (and his &mu;H is our B, and is written as B in the Treatise). Consequently, the constitutive law, written in equivalent modern form, would read D = &epsilon;EMaxwell = &epsilon;(E + G&times;B). Though it is true that the Lorentz force involves a similar expression, E + v&times;B, the velocity vector v is relative, and not fixed to the putatitive stationary reference, as G is.


 * The equations listed by Maxwell are most definitely not mathematically equivalent to what currently goes under the name of &ldquo;Maxwell's equations&rdquo;. The modern formulation was formerly called the Maxwell-Lorentz theory (e.g. A.O. Barut in his 1964 Electrodynamics and the Classical Theory of Fields and Particles). They are what were originally designated as the &ldquo;stationary theory&rdquo;, where G = 0. Einstein used this term in the abstract of the 1905 landmark paper on Relativity, and made reference to the G vector in the abstract, though not by its letter name.


 * To get to the modern theory from Maxwell's formulation requires both setting G = 0 and &ldquo;shutting off&rdquo; the vacuum &mdash; i.e., making &epsilon; constant, &epsilon;0. This, then, gets you the Lorentz relation D = &epsilon;0E, without the G&times;B term. In contrast, Maxwell insisted that &epsilon; (which he denoted K) could not be a fixed constant, since making it constant would then entail self-energy and self-force divergence for point-like and line-like sources. Therefore, besides pulling the rug of Galilean relativity out from under the old Maxwell equations and replacing it by a Lorentz-invariant theory, the other innovation of the Maxwell-Lorentz reformulation was to reintroduce the very divergences into the field dynamics, that Maxwell had sought to excise, and which eventually were passed on to the quantized theory.

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.

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)

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)

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)

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))

ANSWER: in the article as written now (2007-Nov) the derivative is with the straight d, hence a TOTAL derivative. The article is WRONG here. But I am not going to change it again. just think: if B is constant in time E is irrotational. So the LHS is zero, the RHS is not zero as soon as the circuit moves. —Preceding unsigned comment added by 74.15.226.33 (talk) 04:54, 10 November 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" &mdash; 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 &mdash; 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))


 * My understanding is that "Hertz-Heaviside equations" was originally in general usage instead of "Maxwell's equations." I have read that it was Einstein who in fact changed it for his own usage to "Maxwell-Hertz equations," which in time was adopted and truncated to the current form of "Maxwell's equations."  see pages 110-112 of Nahin's book --Firefly322 (talk) 01:24, 27 March 2008 (UTC)


 * I wonder who actually called Maxwell's equations "Hertz-Heaviside" or other variations, as Nahin claims. Certainly Hertz did not, nor would he have accepted others doing so.  In his theory paper (quoted by Nahin on page 111) where he acknowledges Heaviside's priority, he also explicitly calls them "Maxwell's equations", giving the argument that "any system of equations that produces the same conclusions and describes the same results as Maxwell's I would call Maxwell's equations" (or words to that effect).  In other words, his argument is that the reformulations (his own and Heaviside's, which are essentially the same) only change the form and not the essence; the essence is that of Maxwell so his name belongs on the equations.  I can dig up the exact text and cite it in the article. Paul Koning (talk) 18:57, 5 August 2008 (UTC)


 * Silberstein 1914 calls them that, and says a bit more about it. Dicklyon (talk) 06:30, 7 August 2008 (UTC)

Some Suggestions
I don't think the Maxwell's equations should be labeled (1) (2) (3) & (4) like that. Firstly, it's very artificial. Secondly, I think it's like a duplicate to the numbering already present in the TOC (4.1, 4.2, 4.3 & 4.4). Now I don't know why Heaviside is so emphasized (in the titles) in this article, but is there a reason to this? (And add this to the article preferably.) —The preceding unsigned comment was added by Freiddy (talk • contribs) 20:27, 21 April 2007 (UTC).


 * Regarding Heaviside, I would agree with you that his role has been over played. The current set of Maxwell's equations as appear in modern textbooks are actually Heaviside's modifications to the original eight. The only significant difference between the two sets as far as any physics is concerned, is that Heaviside managed to lose the vXH term from Maxwell's original fourth equation by making his equations have partial time derivatives.


 * There is now a little bit of a dilemma. If credit wasn't given to Heaviside for having re-formulated Maxwell's equations, then we would be open to accusations that we were covering up Heaviside's involvement. There are certain quarters that have latched on to the fact that Heaviside re-formulated Maxwell's equations in 1884 and they like to make out that Maxwell's equations are really Heaviside's equations. The only answer seems to be to openly acknowledge that the modern textbook versions are Heaviside's re-writes and to leave both sets open for examination so as everybody can make up their own minds as to where they differ in any important regards. (58.10.103.145 10:07, 2 May 2007 (UTC))

Maxwell's Equations under Lorentz Transformation
To Steve Weston. You are getting confused here. I have enclosed the textbook method for applying the Lorentz transformation to Maxwell's equations. Here is the link. 

Relativity adds relativistic effects to the electric and magnetic fields. The Lorentz transformations have to act on Maxwell's equations to do this. I don't know where you got the idea that relativity can produce Maxwell's equations from the Coulomb force. Can you please give us all a demonstration. (58.10.103.145 09:52, 2 May 2007 (UTC))
 * I believe it is appropriate to revert that 3rd paragraph (introduced by  07:00, 2 May 2007 by 193.198.16.211 (STR)) until a citation for it is in the article. --Ancheta Wis 11:07, 2 May 2007 (UTC)

Hamilton's Principle
In order to invoke Hamilton's principle, it is necessary to know the Lorentz force. The Biot-Savart law is also needed to complete Maxwell's equations. The Lorentz force and the Biot-Savart law provide solutions to the two curl equations of Maxwell's equations. The Biot-Savart law introduces the magnetic permeability. ("""") —The preceding unsigned comment was added by 201.252.200.196 (talk) 01:15, 12 May 2007 (UTC).


 * The Lorentz force and the Biot-Savart law are only relativistic consequences of Coloumb's law. Hamilton's principle is also more fundamental than Lorentz force and the Biot-Savart law. --83.131.31.193 10:34, 12 May 2007 (UTC)

I think you would need to give a citation for this assertion. Hamilton´s principle involves having to know the Lorentz Force. Hamilton´s principle concerns the alternation between kinetic and potential energy. In order to apply Hamilton´s principle to electromagnetism, we need to obtain a Lagrangian type of expression. The Lagrangian for electromagnetism is obtained by deriving an A.v term from the Lorentz force.

It is not possible to apply Hamilton´s principle without first knowing the Lorentz force. To say that it is the other way around would be the same as saying that the Lorentz force falls out of the law of conservation of energy, and we know that this is not so.

Even less so can we ascertain that the Biot-Savart law falls out of Hamilton´s principle. The Biot-Savart law defines a B field. I think that you would need to demonstrate how a B field can be derived from the law of conservation of energy. (ññññ)


 * Magnetic field is consequence of the Lorentz transformations of the electric field, and so is Biot-Savart law (and Lorentz force) such consequence of more fundamental Coloumb's law. If there would be electric field but no special relativity (if Galilean transformations would be absolutely correct), then there would be no magnetic field.


 * Also, Hamilton's principle is one thing, while Biot-Savart law and Lorentz force are two things. Entities should not be multiplied beyond necessity.---antiXt 09:40, 13 May 2007 (UTC)

Let´s go over Maxwell´s equations one by one. First of all we have Gauss´s law. A solution to Guass´s law is Coulomb´s law. Coulomb´s law is irrotational and is commensurate with both Hamilton´s principle and the law of conservation of energy.

Then we have the two curl equations. These are Ampère´s law and Faraday´s law. The solutions are respectively the Biot-Savart law and the Lorentz force.

The two curl equations relate to rotational phenomena and as such they cannot possibly relate to Hamilton´s principle. Hamilton´s principle embodies the entire concept of irrotationality.

As for relativity, it only comes into play at very high speeds approaching the speed of light.

A magnetic field is a curled phenomenon and it can be created by electric currents with drift velocities as slow as two centimetres per second.

There is absolutely no question of the two magnetic curl equations or their solutions being derivable from either Hamilton´s principle or relativity or both.

Ampère´s law by is very nature bears no relationship with either relativity or Hamilton´s principle. It relates to electric currents flowing in electric circuits. (201.53.36.28 19:25, 14 May 2007 (UTC))


 * To whoever it is that keeps insisting on putting the misinformation into the introduction, I suggest that if you believe in what you are saying then you should have absolutely no difficulty whatsoever in explaining your position so that everybody else can understand it.


 * I suggest that you should explain, line by line, how we can obtain Maxwell´s equations using only Coulomb´s law, Hamilton´s principle and special relativity.


 * I know that it can´t be done. I know for a fact that we need to know the Lorentz force in order to derive a Lagrangian for electromagnetism. I´ve seen how it is done. The derivation for the electromagnetic Lagrangian is in Goldstein´s `Classical Mechanics`. It begins with the Lorentz force.


 * When relativity is applied to electromagnetism, it begins by applying the Lorentz transformation to Maxwell´s equations.


 * Relativity is only a linear transformation. Magnetism is a rotational effect. You are trying to fit a square peg into a round hole.


 * If you believe in what you are saying, then let´s all see how its done. I am pretty sure that you haven´t got a clue what you are talking about and that you are merely reciting some nonsense that you have read in a science fiction comic. (201.53.36.28 00:08, 16 May 2007 (UTC))


 * See Special Relativity and Maxwell's Equations from page 39. —The preceding unsigned comment was added by 193.198.16.211 (talk) 17:50, 16 May 2007 (UTC).

Original Research
The introduction to this article is designed to give an overview of Maxwell´s equations. The bit which you keep adding in is original research and it totally contradicts modern physics.

The official position is that the Lorentz transformation acts on Faraday´s law and Ampère´s law to produce the vXB component of the Lorentz force.

You are trying to tell us all that the Lorentz transformation can derive the Lorentz force and the Biot-Savart law directly from Coulomb´s law.

This is totally wrong, and you have attempted to justify this assertion using an unsourced article that constitutes original research. The flaw in the article begins at your transformation law. Your transformation law is a creation of your own making and has got no place in modern physics.

The application of your transformation law on Coulomb´s law is total gibberish nonsense.

Coulomb´s law is irrotational. The Lorentz force is rotational. You cannot derive a rotational force from an irrotational force using a linear transformation.

You are merely using the wikipedia article on Maxwell´s equations to advertise your own private research and you are spreading misinformation. (201.19.158.235 23:05, 17 May 2007 (UTC))


 * This is not my research. If you wish I might find other source, but this one is best I could find so far. In case you didn't read it (I wouldn't wonder if this is true), here is direct link to paper in pdf format.


 * And Magnetic field is not an (true) vector field, but pseudovector field, and because it have zero divergence (no magnetic monopoles) it can be expressed as curl of more fundamental magnetic vector potential: $$\mathbf{B} = \nabla \times \mathbf{A}$$, (where $$\mathbf{A}$$ is magnetic vector potential) so magnetic part of Lorentz force would be $$\mathbf{F} = q\mathbf{v} \times \mathbf{B} = q\mathbf{v} \times (\nabla \times \mathbf{A})=q (\nabla (\mathbf{v} \cdot \mathbf{A}) - \mathbf{A} \cdot (\nabla \cdot \mathbf{v})) = \nabla (q\mathbf{v} \cdot \mathbf{A})$$. In case of magnetic field around the infinite wire, magnitude of \mathbf{A} drops linearly with the distance and direction is parallel to the wire. So there is nothing curved in there and nothing that Lorentz transformations with Coloumb's law (assuming invariance of charge) couldn't produce alone.


 * And again, you failed to explain where exactly (in which step) do you think that derivation is flawed, you are just saying that it is wrong, probably without reading it. --193.198.16.211 10:02, 18 May 2007 (UTC)


 * I am taking no position on the subject under discussion, but you should realize that class notes, no matter how nicely formatted, do not constitute an adequate reference for a controversial physics topic in Wikipedia. The Scientific citation guidelines state that "When writing a new article or adding references to an existing article that has none, follow the established practice for the appropriate profession or discipline that the article is concerning..." In both physics and math, the established practice is to prefer peer-reviewed articles in respected academic journals.


 * If you are Richard Hanson, the author of those class notes, then you also need to be aware that introducing your own unpublished and unconfirmed research ideas into a Wikipedia article is a blatant violation of the Conflict of interest policy, no matter how correct these ideas are. If your ideas are good, then pass them through the scientific peer review process first, then write them up for Wikipedia.


 * Finally, both you and your critics really should register with Wikipedia. This has many advantages; not least of these is that you can identify yourself and present your professional qualifications in your User page. This goes a long way towards preventing accusations of original research, and it allows other Wikipedia editors to gain some idea of who they are talking to. In my humble opinion, hiding your identity behind an anonymous IP number is not a good way to gain respect and credibility.


 * —Aetheling 15:20, 20 May 2007 (UTC).

I´m surprised that the wikipedia editors have been so complacent. They are normally very zealous about obstructing original research and banning people who breach the three revert rule.

Your example of the infinitely long straight wire is to no avail. That situation never occurs in nature. A magnetic field only occurs when we have a closed electric circuit. Ampère´s circuital law requires a closed electric circuit. That means a curled situation. The two curl equations imply that we have a rotational situation.

In due course I will point out exactly where the flaws lie in your referenced article. However, I ought to point out that one cannot assume that because a research paper is complicated and incomprehensible that it must necessarily be correct. Often its falsity is manifestly obvious by virtue of the fact that its implications contradict already known theory that is easily demonstrated. This is the case with your article.

People don´t always have the time to weed through the writings of crackpots to expose where the flaws are, especially in articles where there are about ten flaws on every line. (201.53.10.180 22:55, 18 May 2007 (UTC))
 * The articleis comprehensible. We ought to refrain from pejoratives if an article is understandable and mainstream. Have you read 'Corson and Lorrain,Electromagnetic Fields and Waves ' or 'Landau & Lifschitz, Classical Theory of Fields'? These books are structured around the controversial paragraphs. My professor made similar statements, so the controversial paragraphs might even be considered part of the lore of physics from 100 years ago. There was the Erlangen program back then which had the agenda of unification in mathematics. This would have fit right in at the time. The technique of making abstractions to simplify a problem (action at a distance, adiabatic expansion, rigid bodies, point masses, et cetera) has been used for 400 years in physics. An infinite medium or wire is a device for abstracting away the problem of boundary conditions. --Ancheta Wis 11:43, 19 May 2007 (UTC)


 * I´ve seen the official position on this, and that is that the Lorentz transformation acts on the electromagnetic field tensor in order to produce the Lorentz force. The elctromagnetic field tensor arises out of the symmetry of the two curl equations in Maxwell´s equations.


 * It is therefore impossible that the Lorentz transformation could also act on the irrotational Coulomb´s law alone to produce the same result.


 * The controversial paragraph does not form part of mainstream physics. I suggest that if you wish to push this original research then you should at least remove it from the introductory paragraph and create a special discussion paragraph. (201.19.151.50 17:32, 19 May 2007 (UTC))

The Flaws in the Original Research
One of the key flaws in this original research lies between equations (91) and (94). The author has manufactured the magnetic induction vector along with the implied magnnetic permeability and the Biot-Savart law. He has manufactured it out of thin air. He has arbitrarily decided that an assymetric vector called C should just happen to correspond to the magnetic induction vector. He has pulled an entire Maxwell equation out of nothing. Think very carefully before you decide to promote this as orthodox theory in your introductory paragraph.

You cannot conclude that because a vector is asymmetric that Ampère´s law must exist. (201.19.151.50 18:05, 19 May 2007 (UTC))


 * The final result at equation (94) yields the non-relativistic version of the Lorentz force despite the fact that the entire purported derivation was a relativistic derivation.


 * In the official textbook method in which the Lorentz transformation is applied to the two curl equations (in the form of the electromagnetic field tensor), the final result comes out to a version of the Lorentz force that is amended relativistically. I have a link to the official textbook version here . The official relativistic solutions can be seen at equation (19).


 * In the unorthodox version which is being supported by the wikipedia editors, they define the magnetic induction vector B just before equation (94) in the unsourced original research article. The definition neither conforms to the classical definition of B as per the Biot-Savart law or to the relativistically amended version as per equation (19).


 * Normally the wikipedia editors are very swift to scotch original research and to block persons who breach the three revert rule.


 * In this case we are looking at something very interesting. The zeal with which they continue to insert this misinformation in the introductory paragraph indicates that there is in existence within wikipedia, a group of persons who possess some vested interest in advertising and promoting the lie that a magnetic field is a relativistic effect.


 * A magnetic field can be created by electric currents with extremely low drift velocities. It is clearly not a relativistic effect as wikipedia is trying to tell us. A magnetic field is a solenoidal field whereas an electric field is a radial field. There is no transformation law that allows a radial field in one reference frame to be viewed as a solenoidal field in another reference frame. The wikipedia editors are clearly promoting false science in their own interest. This is further confirmed by their insistence on inserting their heresy in the introductory paragraph of Maxwell´s equations when in normal circumstances, such an insertion would have its own paragraph further down the article.


 * Are we dealing with a group of anarchists who sit guarding this article twenty four hours a day in order to deliberately confuse the general public? (201.53.10.180 14:39, 20 May 2007 (UTC))


 * Now about connection of relativity and magnetism. You think that relativistic effects of electric field couldn't possibly be the cause of magnetic field because drift velocity of currents producing magnetic field is extremely low. But amount of electric charge involved is extremely high, so extremely low velocities are not good argument. Nobody here claims that magnetic field is electric field in some reference frame.
 * This cannot be for two reasons:
 * Dimensions are different
 * Magnetic field is pseudovector, while electric field is a true vector
 * Now in second reason lies also invalidates argument that magnetic field cannot be obtained from Lorentz transformations of electric field because it have solenoidal shape while electric field have radial shape: magnetic field isn't directly* responsible for force it causes, this cannot be because force is a true vector, and true vector cannot be directly* obtained from  pseudovector. (*in this context, "directly" means "only by multiplying with scalar") Field that is directly* responsible for force is $$ \mathbf{v} \times \mathbf{B} $$ and not $$ \mathbf{B} $$ itself. It can be seen that $$ \mathbf{v} \times \mathbf{B} $$ not solenoidal, and $$ \mathbf{v} \times \mathbf{B} $$ is electric field in reference frame of charge upon which force is exerted (and which moves with speed $$ \mathbf{v} $$). Simple way to see that relativity is responsible for existence of magnetic fields is that $$ \mu_0 $$ is obtained from fundamental constants from equation
 * $$ \mu_0 = \frac{1}{\epsilon_0 c^2}$$
 * If Galilean transformations would be true instead of Lorentz transformations, c would be infinite and $$ \mu_0 $$ would be zero, hence there would be no magnetic phenomena. --antiXt 19:10, 20 May 2007 (UTC)
 * In support of antiXt's statements, see L. D. Landau and E. M. Lifshitz 1962 (translated from the Russian by Morton Hamermesh), The Classical Theory of Fields Revised Second Edition, (Chapters 1-4). The 3rd edition is ISBN 0080160190. Landau and Lifshitz start with the Lorentz transformation and the principle of least action. They then trace the trajectory of charges in an electromagnetic field and recover Maxwell's equations (assuming conservation of charge). Landau & Lifshitz are well-known and mainstream physicists. Note that Noether's theorem, like Maxwell's equations,  implies conservation of charge.
 * I would like to thank anon 201.x.y.z for forcing me to look up the Landau & Lifshitz citation. It would help us all if 201.x.y.z selected a User accountname so that we might participate in a more equitable discussion. --Ancheta Wis 21:51, 20 May 2007 (UTC)
 * Thanks for the reference. I'm putting it in the article. --antiXt 22:50, 20 May 2007 (UTC)

Ancheta Wis, Haskell´s derivation is flawed to the backbone and you cannot see it. I pointed out exactly where one of the major flaws lies but you have totally ignored it.

Haskell produces an outward form of the Lorentz force by fudging the coefficient in the Biot-Savart law. The coefficient in the Biot-Savart law depends totally on the choice of units. In SI units the coefficient happens to be 1/c^2. Haskell has made this be the case by building 1/c^2 into his ad hoc transformation law. The transformation law itself is independent of the system of units used and so Haskell would have had a very hard job making it work for every system of units.

I note that in the controversial paragraph, you also state that Haskell´s transformation can be applied to gravity as well, so as to obtain a gravitomagnetic equivalent to the Lorentz force. In that case, the correct equivalent gravitomagnetic Biot-Savart law should not have a coefficient of 1/c^2 since that coefficient is linked to the coefficient in Coulomb´s law and not to the coefficient in Newton´s law of gravitation. Yet Haskell´s transformation would give it the coefficient of 1/c^2 irrespective of what system of units was chosen.

Now let´s look at Haskell´s transformation itself. Leaving coefficients aside, what Haskell is trying to do is to obtain an expression of the form E´= vX(uXE).

His transformation law is tailor made to do exactly that. But Haskell´s transformation is not the Lorentz transformation. It is something completely different that is of Haskell´s own creation. Haskell´s derivation is a total fraud and you cannot see it. And yet you are claiming in the introductory paragraph that Haskell´s reference is evidence that the Lorentz transformation can produce the Lorentz force directly from the Coulomb law. And this on top of the fact that we already know that it produces part of the Lorentz force by acting on the two curl equations in Maxwell´s equations!

As for what username AntiXt says above, I am not even going to reply because it is quite clear that he doesn´t have the first clue regarding what he is talking about.

And you Ancheta Wis are a total fool for coming in to back up somebody that uses a username such as AntiXt. Had you had any common sense at all you would have known immediately that anybody that masquerades anonymously behind a username such as AntiXt is merely a wretched liar who is doing what he is doing for no other reason than to pervert the article on Maxwell´s equations.

One shouldn´t have to decypher nonsense such as that written by Haskell in order to justify why it shouldn´t be included in Wikipedia. The fact that it is original research should be sufficient grounds alone.

Haskell has concocted his own transformation law with a curl in it and then fudged the coefficients deceptively in order to make it appear as if he has derived the Lorentz force from Coulomb´s law.

I showed you the correct relativistic approach to EM theory but you have totally ignored it in favour of a bogus reference supplied by somebody with username AntiXt. (201.37.32.230 20:16, 21 May 2007 (UTC))


 * Consider creating an account and reading WP:CIVIL. And please do not make personal attacks. Thank you. --antiXt 21:21, 21 May 2007 (UTC)

Poynting vector
Is there any expert on electrodynamics who would like to comment on a content dispute on Poynting vector? See history and Talk. Thanks. Han-Kwang 08:09, 9 July 2007 (UTC)

Magnetic field vs Lorentz transformations
I'd like to see here (or in another related article) an explanation and some equations of what happens to electric or magnetic field in an inertial frame moving in relation to us at high speed. For example, E and B in frame S are known, what would they be in a frame T.

This may also serve as explanation why a static charge in S may become source of magnetic field in T etc.

212.179.248.33 17:32, 14 July 2007 (UTC)

Hi there. See Mathematical descriptions of the electromagnetic field; a stub I created but never got round to tidying up. MP  (talk) 11:36, 15 July 2007 (UTC)

layout of the page
I was just crossing the page, and the first two full-screen pages are occupied by this huge "menu". The font is incredibly small, everything is centered. The problem does not appear when using IE6 instead of Firefox. Please correct! (I tried to, but I didn't succeed)Jakob.scholbach 02:21, 26 July 2007 (UTC)

Greek characters on keyboard
I personally prefer equations to words in physics articles, but there is a practical consideration for editing the encyclopedia, which is our keyboards. Might we please refrain from renaming an article, say on vacuum permittivity, to epsilon nought ($$\epsilon_0$$)? Perhaps one day when we can render equations with a WYSIWYG editor, then the encyclopedia might entertain this style... This article is fairly stable right now, which I think the majority of the editors appreciate. Might we discuss these types of changes on the talk page first? --Ancheta Wis 09:43, 12 August 2007 (UTC)
 * The article page is not called that. I had deleted content there and changed ε0 to be a redirect to vacuum permittivity a few days ago. The redirect is useful, I think. /Pieter Kuiper 10:27, 12 August 2007 (UTC)

The Introduction
I very much doubt that paragraph claiming that Maxwell's equations can be derived from Coulomb's law and charge invariance. It totally contradicts Purcell's derivation of magnetism from electrostatics since that is based on the principle that charge must vary.

I don't think that this is a suitable paragraph for the introduction. Even my textbooks admit that this idea is highly speculative and not fully proven.

Maxwell's equations are curl equations. Coulomb's law is irrotational. Where does the curl suddenly come from? (****) —Preceding unsigned comment added by 203.150.119.212 (talk) 14:19, 26 September 2007 (UTC)


 * You might try Paul Lorrain, Dale R Corson,  François Lorrain Electromagnetic fields and waves : including electric circuits, which is an undergrad text as well as Landau & Lifshitz, Classical Theory of Fields a graduate-level text. The idea has been around a while. You might also try Steven's book, The Six Core Theories of Modern Physics for a corrected derivation by G.W. Hammett et. al., based on an identity which I remember as BAC-CAB, to get the Lorentz force on a moving charge. Have fun. --Ancheta Wis 21:37, 26 September 2007 (UTC)

The encyclopedia has a list of vector identities which show how to get curl out of div. --Ancheta Wis 23:18, 26 September 2007 (UTC)


 * Yes, the idea was mentioned in my undergraduate textbook 'Electromagnetism' by Grant and Philipps. It also said that it is only an idea and that what follows falls short of being a proof, as it depends on certain unproven assumptions. It also contradicts Purcell's proof that the magnetic field is the relativistic component of the electric field, since that proof demands charge variance.


 * I am quite familiar with the vector identities which you referenced. But they don't explain how an irrotational force can become a rotational force under linear transformation, as would be implied by the ambiguous assertion in the introduction.


 * I think that it should be moved to the section on relativity, and away from the introduction. (124.157.247.234 15:21, 28 September 2007 (UTC))

I think I see the point. In fact, and I must admit that is quite non-intuitive, in relativity we consider that the electromagnetic field is an antisymmetric linear map of a 4-dimensional real vector space. And then, the irrotational 3-dimensional electric field corresponds to the coefficients of first column and first row, and the rotational magnetic field corresponds to the others coefficients.

Thus, the Coulomb law concerns only these first row and column (a coulombian field would keep the other coefficients to be zero), and this 4x4 matrix (which is like a vector in 16 dimensions) can undergo the action of a linear transformation, for example the Lorentz one when there is a change of referential, that would make appearing other non-null coefficients that mean rotational fields.

So, the hidden thing, is that in fact we consider to be in a sort of vector space of linear combinations of 3-dimensional rotational and irrotational fields. So that the transformation you criticised do exists.

You know, you should create a user account. And then I think it would be wise to relocate our discussions on your own user-discussion page.Almeo 20:55, 28 September 2007 (UTC)


 * You can see the matter more clearly when you apply the Lorentz transformation to the electromagnetic stress tensor. This produces the vXB force. However, that stress tensor already contains the two curl equations to begin with. Hence we cannot obtain the magnetic force by applying relativity purely to the Coulomb force. (^^^^) —Preceding unsigned comment added by 125.25.183.50 (talk) 12:02, 29 September 2007 (UTC)

The controversial clause is actually about the vXB force and not about Maxwell's equations. The vXB force is in the original eight Maxwell's equations and so I suppose it does have relevence. At any rate, I have moved it to the relativity section because it is still too speculative to be included in the introduction. Jordan Sweet (61.7.166.223 15:31, 29 September 2007 (UTC))

General Version of the Equations
Hi. I noticed, that the equations given under the paragraph "General case" are not the general case equations! They are valid only in media, which are isotropic, instantnous, and linear. Check out the german wikipedia entry to see how the general equations look like. --89.50.45.220 23:42, 13 October 2007 (UTC)

Derivation from Relativity
Purcells' derivation of the Biot-Savart law from the Coulomb law depends totally on the fact that charge will vary. It totally contradicts the other standard derivation which depends on charge invariance and is highly dubious. In fact some standard textbooks say that it is not actually a proof at all but rather a mere suggestion. As such, these controversial issues ought to be left to the electromagnetism section in the relativity article. 210.4.100.115 (talk) 20:51, 25 January 2008 (UTC)

Wow!
(moved from above) This article really blows me away! As a non-physicist trying to use Wikipedia to understand physics I have a couple of observations: doctorwolfie (talk) 09:40, 13 March 2008 (UTC)
 * 1) Since the article is called Maxwell's equations, it should state the equations at the top
 * 2) The history section is beautiful!  I'm thinking that a lot of modern controversy surrounds a misunderstanding of how particular terms were first derived. However, the purpose of this encyclopedia article entitled "Maxwell's equations" is simply to state what Maxwell said.  The subsequent controversies and misinterpretations can be listed elsewhere.  In medicine (my day job) we're confronted with a very immediate reality (ie: illness) and are forced to change our concepts to meet constantly-changing demands....every time we write an equation to describe reality, the equation becomes obsolete very quickly.  The point of using an eponym is that it references the person that derived it and places it in historical context.  When the concept changes, a new eponym should emerge.
 * 3) The table that lists the definitions of symbols is fantastic; NONE of the other articles on basic electrical physics that I looked at defined the inverted triangle....which may be quite basic to physicists, but even with several college courses in physics and mathematics I was left in the dark.  This table could stand separately in another article and should be cross-referenced by ALL the articles on electromagnetism, physics, etc.
 * 4) One of the critics on this page mentioned a preference for equations to text in a physics article.  The purpose of an encyclopedia article is to have a much broader appeal...to be inclusive, rather than exclusive.  Text, equation, pictures, links to useful videos,and references all enrich an article and each method of presenting the same material will speak to a different subset of readers; all methods are valid because they should say the same thing in different ways.
 * 5) Registering with Wikipedia is awesome in many ways; one's privacy is completely protected (I have yet to receive a single piece of spam....or even an E mail inquiry on any of my few posts.....what happens on Wikipedia seems to stay on Wikipedia!)  Furthermore if anyone updates a page that you find interesting, you can mark it on you're watch page so that you can track the articles that matter most to you.  I know this is all explained on Wikipedia, but it took me a while to find some of this information; as these anonymous writers are obvious highly educated and probably very busy I would implore them to just take half an hour to explore the organizational structure of Wikipedia.  Anonymity can be completely maintained if it is desired, but following the rules can really help the Wiki community to work.  I've found it an indispensible resource for my own students.  I've also seen that it is a way for people from all over the world (from students, to professors, to casual readers) to participate in a global dialogue.  Please join us so that we can tidy up this important page, spread some of this rich information across several articles (so that some of the tables can be more widely enjoyed), and by all means continue this fascinating (albeit obscure....to me...an average reader) controversy in the appropriate venues (talk pages, etc)


 * Doctorwolfie, I took the liberty of adding some markup to your posting. It makes it easier to read. The nabla symbol (the inverted triangle) is a kind of derivative. As the article states, Maxwell simply reformulated Michael Faraday's lines of force into a field notation and added a term (the displacement current) to make the equations more symmetric. I agree that pictures and equations can be converted from one to the other, back and forth. In fact, the solenoidal diagram at the top of the article illustrates one of Maxwell's equations. He is respected because he unified a lot of equations (Ampere, Gauss, Lenz, etc) into a larger picture, so his name is associated with the larger view. --Ancheta Wis (talk) 11:01, 13 March 2008 (UTC)

Start with "general" Maxwell's eqns?
What do people think of starting Section 2 of the article with "General case" (instead of "Case without dielectric or magnetic materials"):

Of course, bound charge and free charge and bound current and free current would be defined in the box below. One advantage would be that the most general form would be right at the top, benefiting from the chart of symbols immediately below. Another advantage would be that a reader who sees it this way could very easily (a) make the connection to the "Case without dielectric or magnetic materials" version which is there now (and which would be moved to a separate, short section), (b) make the connection to the version with D and H, (c) make the connection to what's going on microscopically. On the other hand, it's kinda wide [any way around that?], and also a little bit less conventional. What do people think? --Steve (talk) 16:43, 24 March 2008 (UTC)


 * I fully support starting with the general case. But then it should be done right. In my view it is not proper to talk about div E or curl B. These are ill defined at material boundaries. It should be expanded to div D, where D=&epsilon;E+P and curl H, where B=&mu;(H+M). &minus;Woodstone (talk) 17:45, 24 March 2008 (UTC)


 * I tend to write the macroscopic equations in terms of D, H, E, and B myself (similar to most textbooks, e.g. Jackson), but writing things in terms of E and B only and expressing bound charges explicitly is not ill-defined at boundaries. In practice, physicists always describe things like electromagnetic fields and charge densities by generalized functions, so there is no problem differentiating at a discontinuity. You just get a delta function, corresponding to a surface charge density.  In any case, you don't avoid the "problem" of singularities by using D and H, because it is extremely common to have delta-function distributions of free charges (e.g. a surface charge density for a charged metal object, or a point charge for that matter). —Steven G. Johnson (talk) 18:58, 24 March 2008 (UTC)


 * Nevertheless, the form above still ignores values of &mu; and &epsilon; other than &mu;0 and &epsilon;0, so is still limited to the case without diamagnetic or dielectric materials. Jumps in &mu; or &epsilon; at material boundaries in the general case cause the ill-defined behaviour of some of the operators. Using the right ones does not suffer from this problem. You might want to have a look at the German version. &minus;Woodstone (talk) 19:13, 24 March 2008 (UTC)


 * No, that's not correct: general &mu; and &epsilon; are implicit in the above equations because they are what determine the bound charge and current densities. And again I have to tell you that you are simply wrong in maintaining that jump discontinuities lead to "ill-defined" behavior; there is no problem as long as one talks about generalized functions.


 * A more reasonable objection to the above form of the equations is that they don't give any indication regarding how to determine the bound charge and current densities. Writing the equations in terms of the permittivity and permeability (or in terms of the corresponding susceptibilities) tells you how the bound charge densities (which include surface charges/currents at interfaces) arise from the macroscopic material properties. —Steven G. Johnson (talk) 19:43, 24 March 2008 (UTC)


 * Could you then show from these forms how the velocity of the EM waves could be anything else than c? We know it is less in many materials. &minus;Woodstone (talk) 19:58, 24 March 2008 (UTC)


 * Just to make clear what these equations are saying, if you plug in:
 * $$ \rho_{bound} = -\nabla\cdot\mathbf{P} $$
 * $$\mathbf{J}_{bound} = \nabla\times\mathbf{M} + \frac{\partial\mathbf{P}}{\partial t}$$
 * $$\mathbf{D} = \epsilon_0\mathbf{E} + \mathbf{P}$$
 * $$\mathbf{B} = \mu_0\mathbf{(H + M)}$$
 * then you can check explicitly that the equations I put at the top of this section are completely equivalent to:
 * $$\nabla \cdot \mathbf{D} = \rho_{\text{free}}  $$
 * $$\nabla \cdot \mathbf{B} = 0$$
 * $$\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}} {\partial t}$$
 * $$\nabla \times \mathbf{H} = \mathbf{J}_{\text{free}} + \frac{\partial \mathbf{D}} {\partial t}$$


 * Either the version I wrote before, or this one with D and H, are completely general and correct, and do not assume linear or isotropic materials. (See Jackson Section 6.6.) Certainly, if these equations were going to be put down, it would also have to be included how bound charges and currents are calculated (as in the above equations). But...it's becoming clearer to me that bound charges and currents are a little too involved for the start of the article. So I now, instead, propose starting with the (completely general) version with D and H (see just above), which currently isn't even given in the article! (The article only has a close variant which does assume linear materials.) So...what do people think of that? (Note: I still think it would be worthwhile to spell out a little more clearly how bound charges and currents relate the versions with D's and H's to the versions without them...but I'm no longer arguing that it needs to be right at the top of section 2.) --Steve (talk) 21:06, 24 March 2008 (UTC)


 * The set of the last 4 equations above has a pleasant consistency and avoids singularities. It needs to be combined with the 2 just above them to be complete, and will need some explanation of where M and P come from and how they are determined (linearity, permanent magnetism, bound charge). &minus;Woodstone (talk) 22:19, 24 March 2008 (UTC)
 * Simpler route, maybe?
 * I'd suggest using the equations for free space with all charges and current explicit - no "free" and "bound" distinctions. That is a very basic starting point. Then you can go ahead and try to calculate the charge densities and currents in whatever approximation you desire - for example using Kubo formalism or Ohm's law or plasma physics or whatever suits you. You can even introduce "free" and "bound" if you like that sort of thing. :-)  Brews ohare (talk) 22:57, 24 March 2008 (UTC)


 * That would be, I guess, the equations I wrote at the top of this section, but with "ρtotal" instead of "ρbound+ρfree" and "Jtotal" instead of "Jbound+ρfree". I don't think that's an improvement: Even if we explicitly explain that the "total" also includes the "bound", people are still likely to find it confusing and objectionable, simply because people are so strongly in the habit of thinking about only free charge and free current. That's why I advocate writing "bound" and "free" explicitly in the equations, if we're going to use those. By the way, there's no approximation in either of these sets of equations; they're both perfectly exact classically (and equivalent to each other), unlike, say, Ohm's law. I'm not sure what you're getting at by bringing up Ohm's law and such things...Approximate dynamical solutions to Maxwell's equations are certainly interesting, but do not belong in a section which presents the equations themselves.


 * Woodstone, I don't think it would be necessary to explain M and P right there; after all, "D" and "H" are well-known concepts, which we'll wikilink. It would be worth putting in for pedagogical purposes at some point, but I don't think it's strictly necessary in the first presentation of the equations. --Steve (talk) 02:51, 25 March 2008 (UTC)


 * First table proposed by Steve (one with bound and free charges and currents) would be the best solution, since it shows the way E and B are related to charges and currents, and usually it is E and B one has most interest to find out since they are appearing in the Lorentz force while D and H depend on what portion of total charges and currents are free.


 * Perhaps it might be good to put in both ways, so most of the readers will find it easy to understand. --193.198.16.211 (talk) 01:25, 25 March 2008 (UTC)


 * Free vs. bound: I'm no expert here. However, here's my two cents on this. Please feel free to educate me. My take is that Maxwell's equation relate fields to current and charge. They do not speak to the issue of where the current and charge originate, nor upon whether the current and charge themselves depend upon the fields in some way. Thus, in a synchrotron (for example) maybe the currents and charges come close to being very simply related to the fields. In a plasma, maybe you have to solve the Vlasov equation or the Fokker-Planck equation. Or, you can invoke the notion of "conductivity" and "dielectric constant", which could lead you a simple approach based on J = σ E, say, or to Linear response theory. My take on using "free" vs. "bound" is that it is an approximation method that heuristically divides charges, while a more basic approach would find this distinction was artificial and its intention to segregate charges would be enforced by real physics in a basic calculation.


 * Thus, I favor beginning in the vacuum. Brews ohare (talk) 15:49, 25 March 2008 (UTC)


 * I guess there are rare situations where the distinction between free and bound charge and current is a little bit arbitrary, but that doesn't mean it's an approximation. You can choose a cutoff however you want (e.g. an electron within 10.0 nanometers of its nucleus is "bound", outside that it is "free"), but if you choose a cutoff and stick to it, you have perfectly precise formulation of classical electromagnetism. This fact is quite transparent, I think, from the formulation at the top of this section, where bound charge and free charge are simply added up, as are bound current and free current. If there's no material, then D=ε0 E and B=μ0 H, and you get the version currently at the top of the article; in other words, that's a special case. I'd like to see the version with D and H right at the top, as "general Maxwell's equations", since it's clearly the most common way to write the totally-general equations. Then it could be followed by "Case without dialectric or magnetic materials", which is currently at the top. In that section, it could be stated (briefly) that this version relates to the above one (with D and H) by the splitting of charge and current into bound and free components (which I see as a worthwhile pedagogical point to make.) --Steve (talk) 16:55, 25 March 2008 (UTC)

(unident) That seems like an excellent plan. It absorbs the section on linear materials and you can delete the strange section on "Maxwell's equations in CGS units" (as if the fundamental equations depend on the choice of units). I would have done it myself if my Maxwell wasn't so rusty. &minus;Woodstone (talk) 17:08, 25 March 2008 (UTC)


 * About CGS: That section should definitely stay. The fundamental equations of electromagnetism do depend on the choice of units. See, for example here, or the appendix to Jackson, or many other places. --Steve (talk) 18:31, 25 March 2008 (UTC)
 * I'd guess that "bound" charges are treated in a dielectric approach, subsumed under a dielectric constant, no? That definitely would be an approximation to the contribution of these charges. If one instead used a linear response theory approach, the dielectric constant would be calculated without need for this distinction, and, for example, valence-band electrons would contribute differently than conduction-band electrons. Or, in a more complex case, the theory would predict whether some carriers were in Cooper pairs, and some not. Brews ohare (talk) 19:06, 25 March 2008 (UTC)


 * Writing Maxwell's equations in terms of D and H, as above, is not an approximation. Writing D=εE (for example) is an approximation (unless ε can be an arbitrary function of frequency, location, field strength, field direction, history, etc.). All the material-dependent properties, which are usually treated approximately, are tied up in the constitutive equations that relate D and E, and B and H (or in weirder cases, D might also be a function of B, etc.). Writing down Maxwell's equations in terms of D and H doesn't say or imply anything about what the constitutive relationships are, or how they would be (approximately) computed. We can make that clear, and it would also be mentioned again in the "linear materials" section later on. --Steve (talk) 21:24, 25 March 2008 (UTC)
 * What is the value of D and P and M and H apart from preparation to use μ and ε or tensor versions of same? From polarization density I gather that bound charge is defined by −divP and P has no meaning independent of a constitutive relation like $$\stackrel {P_i = \sum_j \epsilon_0 \chi_{ij} E_j}{} \!$$, which comes from a Kubo formula or some substitute. I find I am very happy with the article as it stands, with the intro referring to the case "without magnetic or dielectric materials" and then proceeding into the treatment of such materials in the order of growing complexity of treatment of the materials.Brews ohare (talk) 22:46, 25 March 2008 (UTC)

Note that there is a standard name for Maxwell's equations "without magnetic or dielectric materials" &mdash; they are the microscopic Maxwell equations, whereas once you include continuum material approximations (even very complicated approximations, including nonlinearities, material dispersion, nonlocality in space, ...) they are the macroscopic Maxwell equations. (See e.g. Jackson.)  In practice, using D and H always imply the macroscopic equations with some continuous-material approximation; there is no point to using them (versus E and B only) otherwise.

There is a reasonable argument to have the macroscopic equations only briefly mentioned here and mostly covered in a separate, more advanced article. Most introductory treatments of electromagnetism (e.g. freshman EM courses, not to mention high school) barely deal with the macroscopic equations at all (with the possible exception of the very simplified case of a homogeneous/uniform, isotropic, linear, nondispersive, local medium). (Well, in high school you do Snell's law in ray optics, but I've never seen a high school or freshman course derive this from the macroscopic Maxwell equations; and in any case, Snell's law is really a consequence of symmetry mostly independent of the particular wave equation.)

Note also that all of the relativistic descriptions, the differential geometry formulations, etcetera, are only for the microscopic/vacuum Maxwell equations, which is another argument for breaking the article in two&mdash;right now the article is somewhat schizophrenic. As soon as you introduce a material, it sets a preferred frame of reference for the equations, so trying to write them in a covariant form no longer makes sense. I've seen one or two attempts at writing the macroscopic Maxwell equations in differential geometry terms and they were a hopeless mess. (There are beautiful statements about coordinate transformations that one can make for the macroscopic equations, as well as higher-level algebraic descriptions, but they don't involve invariance in the traditional differential-geometry or Lorentz-group sense.)

—Steven G. Johnson (talk) 03:32, 26 March 2008 (UTC)
 * Hi Steve: Maybe a new article would be useful, though I'm unsure what it would contain. It seems that there are already many articles dealing with things like many-body Green's functions, quantum field theory of condensed matter, plasma physics, nonlinear optics, spontaneous emission, etc.
 * You state:
 * In practice, using D and H always imply the macroscopic equations with some continuous-material approximation; there is no point to using them (versus E and B only) otherwise. 
 * That sounds like my own opinion as well. I think the article in its present form deals with this issue very well. What's not to like? Brews ohare (talk) 05:29, 26 March 2008 (UTC)


 * The macroscopic Maxwell equations are just the Maxwell equations with macroscopic material properties like the permittivity and permeability (or more general, possibly nonlinear, susceptibilities). One doesn't need to get into quantum field theory etcetera (and indeed, the macroscopic Maxwell equations considerably predate these concepts); we need not confuse the issue by bringing these things in! (A discussion of the macroscopic equations should not get into a discussion of ab initio computation of the susceptibilities, if that's what you are implying.) It is complicated enough to discuss the constitutive equations, the relationship to the microscopic equations, point out the various approaches for solving the equations (either analtically, in a handful of cases, or numerically), and the properties of the solutions (many of which, like reciprocity, would be described in more detail their own articles). —Steven G. Johnson (talk) 05:55, 26 March 2008 (UTC)


 * I think the terminology "macroscopic" and "microscopic" is unfortunate, since both are true microscopically (assuming you don't spatially-average the fields, in both cases), and both are true macroscopically (assuming you do). Of course, they're not equally useful, so there is something to the name :-) Regardless, common notation is common notation, and I'd be fine incorporating that terminology. I understand that the macroscopic equations are less often taught in introductory courses, but I'd bet they're more often used by actual engineers. In any case, I'm very happy with both versions at the top, as it is now, so that a reader can very easily find the appropriate version, and understand how they differ and how they're both true. Regarding the macroscopic equation, we already have a modest section on the constitutive relations. What else needs to be said about it? Is there really enough content for a spin-off article? Is it that much more "advanced"?


 * Brews, you ask "What is the value of D and P and M and H apart from preparation to use μ and ε or tensor versions of same?" The answer is that you pick some electrons to be "free charge" and all the other charge is "bound charge". The choice of which electrons are "free" is usually pretty clear-cut and well-defined, but I suppose may be a bit arbitrary in certain weird cases. From that definition, you can define P and D and everything else, as explained in detail in Jackson Section 6.6, for example. The equations are true even if the susceptibilities don't exist, but they do become less useful. Not useless though: For example, magnet hysteresis curves are usually in terms of B-versus-H, so H is being used in a situation where B=μH isn't a particularly helpful equation. --Steve (talk) 04:13, 31 March 2008 (UTC)

The Maxwell-Faraday Law
As a point of interest, Maxwell did not include Faraday's law in his original eight equations. Equation (D) of the original eight covers electromagnetic induction.

It was Heaviside who used a restricted partial time derivative version of Faraday's law in his symmetrical set. Therefore I am not sure about the merits behind the term 'Maxwell-Faraday' equation. I have never seen it used before in a textbook. Normally we talk about Maxwell's equations for the Heaviside four and simply use the term Faraday's law when talking about this restricted version of Faraday's law. 222.127.247.207 (talk) 11:08, 25 March 2008 (UTC)
 * Why use term: Maxwell-Faraday equation ? See here for some books that use the term "Maxwell-Faraday equation". Common usage may well be the simple phrase "Faraday's law", as you suggest. However, Faraday's law links to a disambiguation page, indicating it has even more meanings. Selection of the term "Maxwell-Faraday equation" was not based upon a misconception about being in most common use, however, but on the practical issue that Faraday's law, which describes how to find an EMF, is different from the Maxwell-Faraday equation, which is a relation between fields (which can be related to EMF using Lorentz force law to connect the fields to EMF, but does not itself refer to EMF). To use the same name for both relationships is awkward - articles must distinguish between them in some way. By using the less common but unique name this distinction is accomplished without inventing some new way to separate the two, a way that probably would be less clear and less widespread than "Maxwell-Faraday equation", which is descriptive and can claim some acceptance. Brews ohare (talk) 15:28, 25 March 2008 (UTC)
 * I agree with 222.127.247.207 that the most common term (by far) for the "restricted partial time derivative version" of Faraday's law is "Faraday's law of induction". Unfortunately, this is also the most common term for the "unrestricted" version, which says that the EMF is the total time derivative of magnetic flux. In the context of the page "Faraday's law of induction", where both versions need to be repeatedly and unambiguously referenced, it made sense to go out of the way for clarity, using a less-common (but not nonexistent) term for one of the two laws...thus "Maxwell-Faraday". On this page, though, there is little risk of ambiguity, so I would agree that we should just call it "Faraday's law". We lose intra-Wikipedia consistency, but gain consistency with all the textbooks. Anyway, a reader who clicks through to Faraday's law of induction will have no problem figuring out what's going on, even despite the inconsistent terminology from that page to this. --Steve (talk) 15:38, 25 March 2008 (UTC)
 * Evidently, one does not gain consistency with  all  the textbooks, and as said, one does lose consistency with other Wikipedia articles where a distinction is needed. Forcing the reader to "click through" to Faraday's law of induction is a nuisance for the reader, but more importantly, it forces the confused reader to recognize they are confused, which may not be as simple a matter as the versed reader may imagine it to be, and then straighten things out themselves. That is not easy reading, especially if one is not confident about what is going on, and I don't see any gain in clarity as the term "Maxwell-Faraday equation" is clear. Maybe the article could use a footnote; for example, upon occurrence of Maxwell-Faraday equation
 * Brews ohare (talk) 18:10, 25 March 2008 (UTC)


 * I'd also be okay with that, as long as the word "sometimes" in the footnote is replaced by "usually", or "ubiquitously", or something like that. Yes, there are textbooks that use the term "Maxwell-Faraday law", but these are a tiny minority compared to the ones that call it "Faraday's law" or "Faraday's law of induction". We don't want a reader to get a misleading idea about the frequency with which different terminologies are used. --Steve (talk) 21:33, 25 March 2008 (UTC)
 * Here's what I put in this article as a note:


 * Brews ohare (talk) 00:22, 26 March 2008 (UTC)

My main concern is the fact that Maxwell had absolutely nothing to do with the equation in question. It is a Heaviside equation. It is the restricted partial time derivative version of the full Faraday's law that appeared at equation (54) in Maxwell's 1861 paper. That's why I think that it's better just to call it Faraday's law. The distinction between the full and the restricted versions of Faraday's law is explained on the Faraday's law page.

There would actually be much more merit in referring to the next equation down as the Maxwell-Ampère law since Maxwell was definitely involved in modifying Ampère's law. In fact that is Maxwell's crowning achievement.

So why is Brews O'Hare so keen to overlook this fact, but yet to stamp Maxwell's name on a Faraday equation that he had nothing to do with? 203.177.241.5 (talk) 00:39, 26 March 2008 (UTC)
 * I see your concern is from an historian's viewpoint. Mine is simply an expository viewpoint, and the descriptor "Maxwell" in "Maxwell-Faraday" in my mind is just to point out that it is part of the standard four Maxwell equations, as opposed to Faraday's law of induction, which is not part of theses standard four equations. There is a trade-off here, and I'd suggest that the historical record be set straight in the historical sections of the article. Brews ohare (talk) 01:09, 26 March 2008 (UTC)

The article is already about Maxwell's equations and nobody is disputing that this particular limited form of Faraday's law is one of the modern Maxwell's equations. But if you are wanting to label it more precisely, you should technically be calling it the Heaviside-Faraday law. Maxwell had nothing to do with Faraday's law in this restricted form. Maxwell did however contribute in a very important way to the next equation down which might be more accurately be called the Maxwell-Ampère law.

In summary, the modern Heaviside versions of Maxwell's equations contain a couple of Gauss's laws, a Heaviside-Faraday law, and a Maxwell-Ampère law.

Heaviside contributed negatively to Faraday's law by removing the vXB aspect, whereas Maxwell contributed positively to Ampère's law by adding the displacement current term.

These facts are already fully reflected in the history section. I don't think that the term Maxwell-Faraday law should ever be used for this restricted Heaviside equation. George Smyth XI (talk) 03:14, 26 March 2008 (UTC)


 * George: It doesn't seem your labeling is common. The term "Maxwell-Faraday equation" is in use. The term "Heaviside-Faraday law" gets no Google Book Search hits. The term "Maxwell-Faraday equation" is only descriptive, and not a testimonial for the Heaviside version of Maxwell's equations. Brews ohare (talk) 05:20, 26 March 2008 (UTC)


 * I'm still ambivalent between "Maxwell-Faraday" and "Faraday's law", but I'm strongly opposed to "Heaviside-Faraday". There are many names of laws in science that are historically inaccurate (e.g. "Maxwell's equations" instead of "Maxwell-Heaviside equations", or "Lorentz force" instead of "Maxwell-Lorentz force" or whatever). That doesn't mean that Wikipedia should invent new names out of thin air. --Steve (talk) 18:18, 26 March 2008 (UTC)

EMF
There seems to at least be agreement that the Faraday's law that appears in modern sets of Maxwell's equations is only a restricted version which deals with situations in which the test charge is stationary whereas the magnetic field is time varying. However, Brews O'Hare seems to think that the E in this restricted version does not constitute an EMF. Of course it is an EMF. The E in this restricted version is the exact same E as the E in the Lorentz force.

Therefore, any explanations in the Faraday's law section should not be trying to tell us that EMF is not involved until we consider the Lorentz force. The Lorentz force adds to the EMF by adding the term qvXB. That extra term was in Maxwell's original eight equations anyway at equation (D). 203.177.241.5 (talk) 00:46, 26 March 2008 (UTC)


 * To some extent there is a semantical issue here that came up in the Lorentz force article. I was not party to that discussion, but I've agreed to live with it. The decision was that the Lorentz force is defined as F = q (E + v × B) and that the two terms would be referred to as the "electric" and the "magnetic" force components.


 * The EMF issue is simply that E is a field, not a force. It becomes a force when a charge is present, via the Lorentz force law. The EMF is work, and requires force × distance.


 * This item and the name confusion are why EMF does not exist without Lorentz force. Brews ohare (talk) 01:18, 26 March 2008 (UTC)

EMF is a force quantity. It is definitely not a work/energy quantity. The E term in the restricted version of Faraday's law as appears in modern Maxwell's equations is an EMF.George Smyth XI (talk) 03:16, 26 March 2008 (UTC)
 * George: You don't agree with the article on electromotive force. Brews ohare (talk) 05:09, 26 March 2008 (UTC)


 * According to Griffiths (p293), EMF is the "line integral of force per unit charge", or "work done, per unit charge". If we lived in some weird universe where the Lorentz force law didn't hold, and in particular the electric force was not equal to qE, then I think we would have to conclude that the E term in the restricted version of Faraday's law was not an EMF. But we don't live in that universe.


 * Certainly, it would be unnecessary and misleading to say explicitly that the law has nothing whatsoever to do with EMFs. Why can't we just say that changing magnetic fields create electric fields, state the law, and leave the term "EMF" out of it altogether? --Steve (talk) 05:11, 26 March 2008 (UTC)


 * I'm not understanding you, Steve. From a purely axiomatic viewpoint, Maxwell's four equations relate fields to currents and charges. Without adding the Lorentz force law, there is no way to connect the fields to forces, and therefore, no way to connect the fields to work on charges. And hence, no way to get EMF's. What is wrong with this picture? Brews ohare (talk) 06:01, 26 March 2008 (UTC)

Yes indeed. The modern usage of EMF does rather equate to voltage and hence work done per unit charge. I had been looking at Maxwell's 1865 paper equation (D) where he uses the term EMF to apply to force. So there is clearly ambiguity surrounding the meaning of the term. I therefore tend to agree with Steve that we should drop the use of the term EMF altogether as we don't need to use it. The E in the restricted form of Faraday's law as it appears in the modern Maxwell's equation is clearly electric field which equates to force per unit charge acting on stationary particles. The full version of Faraday's law adds an additional convective component which corresponds physically to the vXB term in the Lorentz force. Mathematically, vXB is also an E term but modern textbooks don't ever use the E term for moving charges.

I'll re-word that bit in the main article again so as to remove the term EMF because the term EMF is not generally used in modern treatments of Maxwell's equations.

In fact, I am beginning to wonder why this article is even dealing with integral forms at all. They may be correct physically and they can be equated to the differential forms through the vector field theorems. But are they actually Maxwell's equations? Certainly Maxwell's original eight didn't use integral notation and I don't think that Heaviside did either.

Heavisde truncated Faraday's law because he wasn't interested in the convective part as it isn't important when it comes to deriving the EM wave equation. George Smyth XI (talk) 07:53, 26 March 2008 (UTC)


 * Brews, I agree with what you're saying. All I'm saying is that it's misleading to say that the law has nothing whatsoever to do with EMFs. If you know nothing else, than you can't compute an EMF using it, but it's certainly related to EMFs--if I want to know how to compute EMFs, it would help to know that law, among others. Likewise, Newton's second law F=ma is not sufficient to derive the motion of a ball rolling down a hill, but no one would say that a ball rolling down a hill has nothing whatsoever to do with Newton's second law. :-)


 * George, tons of reliable sources refer to the integral forms as "Maxwell's equations". Many readers looking up this article will be specifically looking for Maxwell's equations in integral form. We don't have to dwell on it, but it should certainly be there. (For example, some people understand line and surface-integrals, but not the grad operator, and for them, the integral forms are the only comprehensible ones.) --Steve (talk) 18:07, 26 March 2008 (UTC)


 * I concur. Also, a general point: Maxwell's 1865 paper used lots of terminology differently from modern scientists.  (e.g. no modern scientist or engineer uses "EMF" for a force, or refers to the "electrotonic [sic] state".)  (If you think there is any ambiguity in modern usage, find a published reference from the last 50 years to support your point.)  While Wikipedia should certainly have information on the historical development of Maxwell's equations (by Maxwell and others) and the historical evolution of the terminology, the article on "Maxwell's equations" should be primarily about what are now universally called "Maxwell's equations."  Our terminology and notation should be dictated by current usage.  (Note that the integral and differential forms are mathematically equivalent and the equations are nowadays commonly written and named as such in both forms.)  —Steven G. Johnson (talk) 18:15, 26 March 2008 (UTC)
 * Not being content to leave well enough alone, I raise the following quotation from above:
 * Certainly, it would be unnecessary and misleading to say explicitly that the law has nothing whatsoever to do with EMFs. Why can't we just say that changing magnetic fields create electric fields, state the law, and leave the term "EMF" out of it altogether?
 * Are you suggesting changes to the existing Faraday's law of induction, for example, "The Maxwell-Faraday equation makes no reference to EMF, and refers to only one aspect of Faraday's law of induction" ? I do believe this quoted statement to be 100% accurate.
 * And later on "At this point, the right-hand side of the EMF version of Faraday's law has been found using the Maxwell-Faraday equation. Finding the left side, namely the EMF $$\stackrel{\mathcal{E}}{} $$   (that is, the work required to bring one unit of charge around the loop) in Faraday's law, requires addition of the Lorentz force law to the Maxwell-Faraday equation, inasmuch as work is force × distance."
 * Any changes here?? Brews ohare (talk) 20:01, 26 March 2008 (UTC)


 * Nope, sounds basically fine to me. --Steve (talk) 20:10, 26 March 2008 (UTC)

Convective term
Quote: The curl of v × B is −( v•∇ ) B which is the convective term of the full Faraday's law. I'm left waiting for the other shoe to drop. Can a few words of explanation be added here, or does it take a paragraph or two to explain what the "convective term" means and why it is related to the curl? And why do we care? Maybe this point could be made more easily in English (excuse my French)? Brews ohare (talk) 20:01, 26 March 2008 (UTC)


 * Brews, it is important because it goes right to the heart of the difference between the full version of Faraday's law and the partial time derivative version of Faraday's law.


 * A total time derivative can be split into a partial time derivative and a convective term of the form v.grad


 * Maxwell didn't include Faraday's law at all in his original eight equations. Instead he used equation (D) for electromagnetic induction. That equation was derived from Faraday's law between equations (54) and (77) in his 1861 paper. It is in effect the Lorentz force and it contains all the effects covered by Faraday's law.


 * If we ignore the vXB term and take the curl of equation (D), we end up with the partial Faraday's law that appears in the Heaviside versions of Maxwell's equations.


 * But if we include the vXB term and take the curl, we will additionally get the convective (v.grad)B term. Added together they sum to the full Faraday's law.


 * See http://www.answers.com/topic/convective-derivative?cat=technology


 * and also see section 9 in this web link for the curl of a cross product. The result when applied to vXB comes out to be the convective term of Faraday's law. https://www.math.gatech.edu/~harrell/pde/vectorid.html George Smyth XI (talk) 01:08, 27 March 2008 (UTC)


 * I was not complaining about meaning, but exposition. I don't think most readers will know what you mean or why it's important. It should be written for a broader audience. Brews ohare (talk) 07:39, 27 March 2008 (UTC)

OK. I've removed those deatils from the main article. George Smyth XI (talk) 12:02, 27 March 2008 (UTC)

Maxwell's Original Works
Steve, I wasn't trying to introduce archaic terminologies on the main page. However, I had been reading Maxwell's original works and found them to clarify alot of the existing confusion surrounding the fact that the Faraday's law in the modern (Heaviside) Maxwell's equations does not cater for all aspects of electromagnetic induction.

I accept the fact that Maxwell used EMF as a force whereas modern textbooks use it as a voltage (energy/work done per unit charge).

The important thing is to get the readers to understand the relationship between the Lorentz force, the partial time derivative Faraday's law and the full Faraday's law.

Equation (D) in Maxwell's 1865 paper is in effect both the Lorentz force and the full Faraday's law. Every aspect of electromagnetic induction that is covered by the full Faraday's law is also covered by the Lorentz Force and vica-versa.George Smyth XI (talk) 01:58, 27 March 2008 (UTC)


 * I'm familiar with Maxwell's paper, and the fact that he combined the Lorentz force with what most people now call Faraday's law (formulated in terms of a vector potential). Nowadays, however, it's considered mathematically more convenient to separate the two equations for most purposes.  I'm not sure why you think there is a lot of confusion here (at least in terms of the application of the equations, rather than their history).  It's not as if modern practitioners don't know how to compute the forces (or the emf) for a moving conductor.  —Steven G. Johnson (talk) 02:06, 27 March 2008 (UTC)

Steve, You've only got to read back through the talk pages to see the confusion. There are people here who think that Faraday's law and the Lorentz force are different physics.

When they understand the underlying link between Faraday's law and the Lorentz force, then they will be in a position to write the main article in modern format and explain how the Lorentz force covers all aspects of EM induction, as does Faraday's law, but that the particular form of Faraday's law that is found in modern Maxwell's equations only covers the aspect of EM induction that occurs for stationary charges in time varying magnetic fields.

At the moment, we are witnessing no end of confusion. George Smyth XI (talk) 02:59, 27 March 2008 (UTC)


 * The "full version of Faraday's law" is not one of "Maxwell's equations", and neither is the Lorentz force. I'm not exactly sure what clarification you want to include, but you should be sure that it's on-topic for this article, which is already fairly long and hard to navigate. A full detailed explanation would certainly be appropriate at Faraday's law of induction (and is already there), and it would also be appropriate to mention briefly here, maybe in a sentence, that the Maxwell-Faraday law (or whatever we're calling it) is not the same as what you call the "full version of Faraday's law". Also, a comment would make sense in the history sections -- where it is, in fact, already discussed. --Steve (talk) 15:42, 27 March 2008 (UTC)

Steve, I think there was a bit of confusion over purpose here. It began over the issue of EMF. We are agreed that the issue of EMF is not related to the difference between the full Faraday's law and the partial time derivative version. But then the issue got side tracked to the fact that the meaning of EMF effectively meant electric field in Maxwell's papers, whereas it means voltage in modern textbooks.

For the purposes of clarity, Faraday's law was not part of Maxwell's original eight equations at all, and only the partial time derivative version of Faraday's law that ommits motional EMF is involved in the modern Maxwell's equations. I think we are agreed on that. The Lorentz force is effectively one of Maxwell's original eight equations, catering for EM induction, whereas it sits alongside the modern four Maxwell's equations as an additional equation, since it is needed to supply the motional vXB effect which is absent from the Heaviside partial time derivative version of Faraday's law. The E in the Lorentz force is a duplicate of the E in the partial time derivative Faraday's law. George Smyth XI (talk) 10:40, 28 March 2008 (UTC)


 * Well if we're not disagreeing about specific things to be included or not included in the article, then I don't want this conversation to go on too long. For what it's worth, though, I think I agree with everything in the second paragraph you wrote. I probably disagree with your claim that "EMF means voltage", insofar as the concepts of electrostatic potential and electromotive force are quite different, and I'm not sure exactly what you mean. And I sorta-agree with your sentence "EMF is not related to the difference between the full Faraday's law and the partial time derivative version": If you're saying that the partial version of Faraday's law accounts for one component of the total EMF, or one way to create an EMF, while the full version of Faraday's law is an expression for total EMF, or every way possible to create an EMF in a conducting loop, then we're basically in agreement, apart from some relatively minor (pedantic) issues related to the role of the Lorentz force. --Steve (talk) 17:14, 28 March 2008 (UTC)

Steve, see what I wrote to Brews below. EMF began historically with force in mind. But nowadays it is accurately applied to voltage, but still losely applied to force.

The total time derivative Faraday's law, which appears in no sets of Maxwell's equations, caters for both aspects of EM induction. The partial time derivative bit caters for the time varying field/static test charge aspect. The convective (v.grad)B bit is the curl of vXB and caters for motionally induced EMF.George Smyth XI (talk) 04:42, 29 March 2008 (UTC)

Summary Section
Sections 2 and 3 begin with the title 'Summary of Maxwell's equations'. They look more like a major encyclopaedia article. I don't doubt that these sections contain some useful information but should they not be moved to further down the article and a title of "In-depth discussion on Maxwell's equations". The section 'Heaviside versions in detail' was already intended to be an introduction to the modern textbook versions. Now it has been moved down the page. I think that the positions should be reversed. 58.69.106.22 (talk) 05:29, 27 March 2008 (UTC)


 * I see the problem. It seems to me like the best solution would be to rename the top section, "Formulation of Maxwell's equations", and shorten the "Heaviside versions in detail" section, the content of which is all already covered in separate articles. (Apart from Gauss's law for magnetism, which should have its own article.) I think the details about individual equations should be left for the individual articles (even more than they are now), while this article can focus more on what happens you put it all together. --Steve (talk) 15:22, 27 March 2008 (UTC)

Too much deleted
Many useful links and discussion removed for no apparent reason. Brews ohare (talk) 07:46, 27 March 2008 (UTC)
 * If you're referring to my edit yesterday, I moved some stuff around but if you read it through you'll see that I didn't delete much content or links (I think). If you're referring to 121.97.233.43's shortening of the "Maxwell-Faraday law" section a couple days ago, it may have been a bit extreme, but I do think we should try harder to keep those sections short, since the topics already have their own articles. --Steve (talk) 15:51, 27 March 2008 (UTC)
 * Steve: I did find that material, thanks. There are two major items that require a subsection: something about applications, linking to the many articles on applications like waveguides, antennas, filters; and a subsection on boundary conditions - linking to standing waves, modes, plasma oscillations, jump conditions at boundaries etc. Brews ohare (talk) 16:33, 27 March 2008 (UTC)

Brews, I decided to put in a reference to the fact that the Faraday law in the Maxwell-Heaviside equations does not cater for motionally induced EMF. I avoided the term force in order to avoid a conflict over terminologies. I should however point out to you that electric field is force per unit charge and that hence E is to all intents and purposes a force. Also, although Maxwell used EMF as a force, whereas modern textbooks use it as a voltage, it is to all practical intents and purposes a force in relation to EM induction.

When making analogies between mechanical situations and electric circuits, force is always equated with voltage/EMF, mass with inductance, spring constant with inverse capacitance, and air resistance to electrical resistance.

But I am fuly aware that once we start the accurate mathematical analysis then we have to treat voltage as a force times distance. George Smyth XI (talk) 04:35, 29 March 2008 (UTC)

To Do: Combine the two sections on history
There are two sections about the historical development of these equations: The "History" section and the "Maxwell's original equations" section. Right now, they're full of redundancies and even a few contradictions. I think they ought to be combined, or at the very least, the scope of each should be better specified. I'm neither interested in this topic nor knowledgeable about it, but if someone wants to do that (or even spin off "History of Maxwell's equations" and/or "Maxwell's original 8 equations" into separate articles), I think that would be a great thing for this article.

Thanks! --Steve (talk) 17:47, 27 March 2008 (UTC)


 * Done. One should never lose track of the importance of Maxwell's original works when it comes to matters relating to Maxwell's equations. George Smyth XI (talk) 10:45, 28 March 2008 (UTC)


 * Great! I decided to chip in after all and tried to organize it a little bit better. I apologize if I made any mistakes. --Steve (talk) 16:57, 28 March 2008 (UTC)

Steve, it was a nice tidy up job that you did. I did however re-emphasize the fact that the Faraday equation is not closely connected to Maxwell. George Smyth XI (talk) 04:27, 29 March 2008 (UTC)

Reasons for the removal of the extra two paragraphs in the introducion
I removed the paragraphs about the Lorentz force in the introduction for a number of reasons.

Firstly this is an aticle about Maxwell's equations. There is no need to add in that kind of extra information in the introduction. The relevence of the Lorentz force is well discussed in the main body of the article.

Also, the substance of those paragraphs was wrong and misleading. It claimed that Maxwell's equations don't deal with force. That is not true. Maxwell's equations are all about force. The electric field E is the force per unit charge acting on a particle in an electrostatic field or in a time varying magnetic field.

The only thing that the Lorentz force adds is the force per unit charge on a moving particle in a magnetic field given by qvXB. The E in the Lorentz force is already the same E that is in Maxwell's equations. The Lorentz force overlaps with Maxwell's equations in this respect.George Smyth XI (talk) 04:57, 29 March 2008 (UTC)
 * Hi George: Please fill in for me the gaps in the following logical (not historical) viewpoint.


 * 1. We have four Maxwell equations that relate the vectors E, B to j and ρ.
 * 2. Based upon these equations alone we have absolutely no clue why we want anything to do with same. They aren't physics. They cannot be connected to any actual experimental results. They are simply mathematical objects that can be explored. We also do not know how to find the sources j and ρ.
 * 3. To remedy this blank slate, we can add the Lorentz force equation F = q ( E + v × B ). We know what force and velocity are from our choice of mechanics; we can use them in Newton's laws, for instance, to get a meaning for same.
 * 4. The situation with j and ρ is left hanging. We have to have some definitions, which might be as simple as one of the constitutive equations approaches reviewed later in the article.


 * So I come to this rather categorical statement of position that I'd like you to rework: Without Lorentz force law, E and B have no connection to mechanics in any way, unless we add something else to Maxwell's equations, maybe some verbal context or some historical background. However, from a strictly axiomatic viewpoint à la Euclid, connection to experiment can be done with only the four Maxwell equations, the Lorentz force law, and some constitutive relations for j and ρ. No other historical context is necessary, although I do not discount its interest. (BTW, I added an historical link to the Lorentz force law history on electro-tronic state because I got swept up in the discussion there. I assume you had a hand in that?)


 * It's because of the above axiomatic viewpoint that I added the stuff you deleted. I felt that something needed to be said. I think that within the axiomatic viewpoint à la Euclid, my paragraphs make sense. If you disagree with me as far as an axiomatic à la Euclid meaning of all this (bare of historical context), that is one thing. But if you disagree with me because you come to this axiomatic approach with a surrounding context of history and example not found in the axiomatic à la Euclid  approach, that is something else. Which is it? Or is there more to it? Brews ohare (talk) 05:45, 29 March 2008 (UTC)

Brews, You are correct in saying that we need the Lorentz force to complete the picture when we are using the modern Heaviside equations. But this is solely for the reason that the modern Heaviside versions contain a Faraday's law which ommits the vXB effect.

In Maxwell's original eight equations, the Lorentz force, including vXB was already there. Maxwell actually derived the Lorentz force from the full version of Faraday's law between equations (54) and (77) of his 1861 paper.

The Lorentz force contains three components. There is,

(i) F/q = gradΨ

(ii)F/q = -(partial)dA/dt

(iii) F/q = vXB

Take the curl of (i) and we get zero. Take the curl of (ii) and we get -(partial)dB/dt Take the curl of (iii) and we get -(v.grad)B

The curl of (iii) is the convective term that you asked me about the other day. Add the curl of (ii) to the curl of (iii) and we get the total Faraday's law,

curl (F/q) = -(total)dB/dt

Maxwell did it the other way around using his vortex sea model. Hence the full Faraday's law is the differential form of the Lorentz force.George Smyth XI (talk) 08:58, 29 March 2008 (UTC)


 * Hi George: I understand that the Lorentz force can be expressed this way and examined for the source of its contributing terms. However, that is not helping me to understand why the deleted paragraphs are either wrong or misleading. That is particularly true of the paragraph on boundary conditions, omission of which is a very common oversight among students. Brews ohare (talk) 11:35, 29 March 2008 (UTC)

Brews, OK, the Heaviside-Maxwell's equations are basically differential equations. Hence E and B are not uniquely determined because of the issue of arbitrary constants.

I would agree with you that these equations can't do much for us. They only express relationships which enable us to see that light is an EM wave. But the article is on Maxwell's equations and so we have to tell the readers what we know about Maxwell's equations.

Maxwell himself preferred to use the Lorentz force over the differential Faraday's law as being the definitive equation which yields all sources of EMF. And it has since been accepted that the Lorentz force is necessary to complete the picture because the Heaviside equations don't deal with the convective vXB force.

But regarding the E term in the Lorentz force, yes it is uniquely determined and we can get the differential form (Faraday's law) simply by taking the curl, so yes, the Lorentz force is the more useful equation.

But I don't think that we add the Lorentz force to the modern Maxwell's equations for the sake of the E bit. It's not about whether or not we know what E actually is. It's all just about relationships.

It's the fact that the vXB concept is not catered for at all in the modern Maxwell's equations. That's the real importance of the Lorentz force. It completes the picture by bringing in the one outstanding electromagneic effect.

At any rate it is not an issue that should be discussed in the introduction of an article on Maxwell's equations. George Smyth XI (talk) 12:42, 29 March 2008 (UTC)


 * Part of the confusion may be that the term "Lorentz Force" is used in two different ways, sometimes just for the magnetic force qvXB, and sometimes for the whole thing q(E+vXB). Brews means the latter, but people who learn the former sometimes (improperly) regard F=qE as not a law, but rather something that "goes without saying".


 * I agree with George that those paragraphs aren't topical: as Jackson says (page 3), although E and B were originally defined just as an intermediary for force calculations via the Lorentz Force law, they're now regarded as substantive, physical entities in their own right. Therefore, "context" for E and B, meaning a connection to forces, isn't particularly necessary. Sure, the Lorentz Force is relevant enough to be written out somewhere in the article, but not in the opening. (I would, however, be very happy to have the following sentence in the opening: "Maxwell's equations, together with the Lorentz force law, form the basis of classical electromagnetism.") --Steve (talk) 15:10, 29 March 2008 (UTC)
 * I'll settle for "Maxwell's equations, together with the Lorentz force law, form the basis of classical electromagnetism." in a spirit of great compromise.:-) However, I'd like to add the line about boundary conditions - there is no way that can be said to be "non-topical" as the equations are actually fundamentally incomplete without the boundary conditions, and that should be said.


 * However, as a final two-cents, I'd say an exact logical analogy here would be to state the axioms of Euclidean geometry without the part on parallel lines (corresponding to Maxwell's equations without the Lorentz law) and then refusing to add any mention in the intro to Euclidean geometry that the "parallel line" hypothesis was missing.


 * George - you have not bit the bullet here on the logical role of the Lorentz law as the only source of interpretation of Maxwell's fields in the experimental arena. Even the Electromagnetic wave equation has no interpretation in experiment until we know how to detect E and B → it could refer instead to phonons, sound waves, vibrating strings or whatever. There is nothing in the four Maxwell equations to tell you how to interpret their predictions, or how to observe them. Brews ohare (talk) 16:36, 29 March 2008 (UTC)