Talk:Maxwell's equations/Archive 1

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Archive 1

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Personally, I like Gauss' law better if the ε₀ is moved over to the other side, under the integral. That way, it still works if the permitivity isn't constant; you just replace ε₀ with a function ε --AxelBoldt

What do you mean if ε_o isn't constant? Do you mean for cases where there is matter between the surface and the charge, and thus you need to account for a drop in the E field due to the permittivity of that matter? --BlackGriffen

Yes, that's what I meant. If there's matter with varying permittivity ε around, you need to integrate ε multiplied with E in Gauss's law. --AxelBoldt

Ok, I'm adding a note about it now.

Upon further reflection, that is wrong. The dielectric constant of matter doesn't just magically reduce the electric field. The dielectric constant (I had the wrong name previously) is a measure of how easy it is to separate the molecules of matter in to a dipole. To show why is relatively easy (now that I think about it properly). Consider a small positively charged sphere. The electric field outside this sphere is is the same is if it was a point charge: kq/r². Stick a neutrally charged spherical shell around it. The electric field of the sphere creates dipoles within the shell that surrounds it. The net effect is like two thin shells of charge have formed; a negative one on the inner surface of the shell and a positive one on the outer surface. The charges of these shells have to be precisely equal due to conservation of charge. The net effect? Everywhere but on the inside of the walls of the neutral shell, the electric field still looks like kq/r². Within the walls of the shell the electric field is weaker, but as long as the surface entering that region removes no net charge, the decrease in the electric field is compensated for by two factors: a change in the area of integration, the fact that the charge shells are approximations of microscopic dipoles means that there is still a net surface charge that compensates for the inner charge. Even in the limiting case, metals, where the surface charges are great enough to reduce the electric field in the body of the metal to zero, Gauss's law holds. --BlackGriffen

Is it worth mentioning that the elegant formulations of Maxwell's equations were not developed by Maxwell, but by another man ? Maxwell had the right idea, but he was definitely not elegant in his math.... I've done a bit of web-searching to validate this idea and currently cannot vouchsafe it. I recall a history of science teacher describing it in great detail when I was younger, but have no way to verify/validate it.

Okay, I found a page that claims: 1884: Oliver Heaviside expresses Maxwell's Equations as we know them today ie: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Heaviside.html therefore this validates my recollection. (it said Maxwell's equations were originally 20 equations in 20 variables instead of two equations in two variables) Now I can go to sleep...

I deliberately left the history empty because I did not know it. By all means, add a history section. I do know that the wave equations for light that can be derived from them led to relativity. (the term describing the velocity of the wave was 1/(μ_oε_o)^.5 which is equal to c, and the fact that it didn't contain a term for the velocity of the observer is what sparked Einstein's imagination/lead to his postulate that c is a constant to any observer.

Also, 4 Maxwell's equations with 4 variables (time, charge density, the electric field, and the magnetic field). Where do you get two? --BlackGriffen

I think it would be nice to mention in the main article how ε_o, μ_o and c are related, so that people realize that the speed of light occurs in Maxwell's equations and that therefore the conjecture that electromagnetic radiation is light is not too abstruse. --unknown

Nice idea, but this article needs to remain focused on these equations because that is all it is for. A better place for that connection would be in an electromagnetic radiation/waves/light article. --BlackGriffen

Oh, there's also a minor oversight: ε is used as the permittivity and also as the electromotive force around a loop. --unknown

EMF is supposed to be a scripty E. Anyone know how to do one of those? --unknown

I understand that, but there are only so many symbols in the english language. I used ε instead of ΔV or Δφ_E for three reasons: first, φ and/or V are used in electrostatics to represent the electric potential as a scalar function in space, and any closed loop integral over a continuous scalar function in space has to be zero; second, ε is the closest thing (almost exactly the same, in fact, to the scripty thing described above); and third, the limited number of symbols means that what the symbol represents has to be labeled each time anyway. To give you another couple of overloaded characters in physics: p represents both momentum and pressure (in mathematics p also represents the period of the wave); v is used for velocity, volume, and voltage, velocity is generally lower case, volume is upper, and voltage is usually upper if it's constant and lower if it's time varying. I've really beaten that horse to death, but I wanted to make it crystal clear that I had considered the conflict when I wrote the article. --BlackGriffen

And one last thing: I don't quite understand why the last paragraph mentions cgs versus mks units? How could the units possible change the equations? --AxelBoldt

If you use kg for mass, m/s² for acceleration, and lbs for force, Newton's second law takes on the form F=kma, k a constant. Choosing a better system makes k go away, simplifying the equation. It's the same deal with CGS and MKS, a lot of the constants go away in the former system. --Unknown

Precisely, I'll add more to the main page presently, but it's all about clairity. --BlackGriffen

I would love it if someone more knowledgeable than I would add concrete examples to each equation description, to make the descriptions more accessible to non-physicists. Such an example might perhaps be: as you move away from a sphere charged with static electricity, the charge density in space drops by four for every doubling in the distance from the center of the sphere (just as gravity drops when you move away from the Earth, due to the equidistant spreading of lines of force from a sphere). [Please, please forgive me for the mistakes in this--I just wanted to illustrate what I meant by a 'concrete example'.] David 16:31 Sep 17, 2002 (UTC)

On my computer (Windows 98/IE/Arial), the symbolic manipulations in the subject page all show a character that looks like a vertically-oriented rectangle. Here is an example: ∇×E = - 1/c ∂B/∂t. The rectangles are before the E, B, and t. The browser appears to have translated them to Unicode, which is not supported by Windows 95 and 98.

I have created and uploaded an image for the Del symbol and have edited this page to reflect the change. Here is an example: $\nabla$ E = 0. David 16:27 Oct 7, 2002 (UTC)

    I use lynx to browse, so I would prefer the word epsilon to a picture of a squiggly e.  As long as there is text saying what each variable stands for, using the plain letter e for epsilon is clear as well.
         --- Urushiol

The math formulae had had great big \bullets added to them: I have removed them, and cleaned up the layout. The Anome 17:59 10 Jun 2003 (UTC)

Maxwell's Ether exists!

The Ether's impedance z and Planck's Constant h are related, making them both Quantum Constants. z= m/q and h=mq where m is the ether magnetic charge in webers(volt seconds) and q is the ether electrical charge in coulombs. Knowing the value of h and z , m=500 atto webers and q = 1.326 atto coulombs or 8.28 electrons.

The three constants, c, z and h unify Quantum, Relativity and Electric Theory. Wardell Linday

Maxwell's Equations Derived and Revised!

This is an excellent article on Maxwell's Equations. However the entire discussion of Maxwell's Equations and electricity and Magnetism would be much simpler and more correct using quaternions.

The complete and correct Equations of Electricity and Magnetism is given by the Homeostasis Condition: 0=XE

where E = Es + IEx + JEy + KEz = Es + Ev is a quaternion electric field and

where X = d/cdt + Id/dx + Jd/dy + Kd/dz = d/cdt + DEL

is my Quaternion Change operator, a quaternion extension of Hamilton's DEL.

"c' is the speed of light and E is related to c and "z" the free space impedance by E = cB = zH = zcD.

X and E are quaternions and follow quaternion multiplication. "Maxwell's " Equations completely are given by:

0 = XE = (dEs/cdt - DEL.Ev) + (dEv/cdt + DEL Es + DELxEv)

substituing E/c=B gives the traditional terms.

0 = XE = (dBs/dt - DEL.Ev) + (dBv/dt + DEL Es + DELxEv)

The observation here is that the first term is scalar of the quaternion and the second term is the vector of the quaternion.

0= XE requires both the scalar term and the vector term to be zero, thus

0 = (dBs/dt - DEL.Ev) and 0 =(dBv/dt + DEL Es + DELxEv)

If I had started with 0=XB = (dEs/cdt - DEL.Ev) + (dEv/cdt + DEL Es + DELxEv) then

0 = (dBs/cdt - DEL.Bv) and 0= (dBv/dt + DEL Es + DELxEv)

Thus one quaternion equation gives Maxwell's four and corrects them.

Notice that dBs/cdt = DEL.Bv, or the divergence or growth of the magnetic field, is not zero, but zdDs/cdt = z rho or z times the charge density rho!

I recommend revising this article to reflect this view.

Wardell Lindsay

Hello Wardell: have you considered using four-vectors? They neatly wrap up Maxwell's equations in a way that is very similar to what you propose. -- The Anome 17:57 2 Jul 2003 (UTC)

The Anome,

Yes I do use four vectors see my webpage:

http://www.geocities.com/wardelllindsay/unification.html

My Interval is a natural fallout of quaternions without introducing "imaginary time". The difference is in the mathematics. Only quaternions provide an associative (a(bc) = (ab)c) division algebra ( ax=b is solvable).

I have not seen a derivation of Maxwell's Equations similar to mine, which poses the stationary condition of the electric field and "one" equation.

A similar equation also describes Quantum Theory, using the "Life" variable L=hc.

Thanks for your comment and interest.

Lindsy

Please read: http://www.innerx.net/personal/tsmith/QOphys.html Maxwell's Quaternions were thrown away from Electromagnetism by Josiah Willard Gibbs at Yale and Oliver Heaviside in England.

and this page is informative http://www.ott.doe.gov/electromagnetic/history.shtml [notice it's a .gov site]

[That depends on one's definition of "informative:" This "history" page is written by the crackpot disciples of the infamous and equally crackpot Lt. Col. Thomas Bearden (USAF, Ret.), who believes in everything from "overunity" free-energy machines, to "scalar-wave" cancer cures, to "Tesla Death Rays," to UFOs. Hence, the fact that this is a ".gov" site simply proves that some UNBELIEVABLY moronic and insane people can manage to get into the civil service, from whence they become nearly impossible to fire...]

Electromagnetic History

more later ... reddi 03:47 7 Jul 2003 (UTC)

First of all, I have no dispute with the historical fact that Gibbs and Heaviside, along with the rest of practicing physicists and engineers, abandoned the quaternion notation. However, you seem to be under the misapprehension that by doing so, they "threw away" something profound, or that Gibbs didn't "understand" it. Quaternions, as proposed by Maxwell (some years after his initial work), were only notation, they contained no special physical content and are mathematically equivalent to the modern formulation. As notation, even Maxwell himself found them inconvenient: in his two-volume treatise on electromagnetism, he devotes all of five pages or so to quaternion notation; he lauds it as a promising notation, but admits that he finds it inconvenient for practical calculations and doesn't use it in the rest of the books. (I just checked this evening.) As a completely separate issue (independent of quaternions), Maxwell used the vector potential explicitly and picked a particular gauge choice (the Coulomb gauge, I think it was); this sort of thing can be (and is) done all of the time in the ordinary vector notation as well, and in "ordinary" physics this vector-potential gauge is not observable. As to the papers on the web site that you mention, you'll notice the telling fact that they're not published in respected peer-reviewed journals (e.g. Phys. Rev.). -- [User:Stevenj|Steven G. Johnson]

Howdy .... if you read in this link .... http://216.239.53.104/search?q=cache:Yc5OJrDDTX8J:www.nku.edu/~curtin/crowe_oresme.do ... or ....http://www.nku.edu/~curtin/crowe_oresme.doc ... it's titled 'A History of Vector Analysis' ... I kinda AM under the "misapprehension" that by doing so, they did "throw away" something profound. As fars as i can tell Gibbs didn't "understand" it. I saw that there were two important functions (or products) called the vector part & the scalar part of the product, but that the union of the two to form what was called the (whole) product did not advance the theory as an instrument of geom. investigation. (gibb's words) .... Heaviside didn't either .... I had the same difficulties as the deceased youth, but by *skipping* them, was able to see that quaternions could be explored consistently in vectorial form. But on proceeding to apply quaternionics to the development of electrical theory, I found it very inconvenient. ... So I dropped out the quaternions altogether, and kept to pure scalars and vectors.... [heaviside's words] .... Quaternions is one of the simpliest ways to describe space-time. x * y * z * t = 4D ... simple .... maxwell did want to unify EM with it eventually (as you said) ... and i like Einstien when he say, "I like to make things simple, but not one bit simpler" ... (it's also describe this reality ... not a 3D or 2D imaginary world) ... moving on ..... Quaternions are just notation .... no special physical content .... I do doubt that though they are equilivant (but i am not a math guy so until such time i can find any substantial proof ... i'll take your word for it) to modern notation ... though i thoughT, IIRC, that the modern notation cannot handle the _non-linear_ electromagnetic phenonomen that quaternions are naturally suited for (such as scalar waves) ... reguarding "quaternions are burdensome" ...no pain no gain 'eh? =-] ... Finally ... Do all links and book citations on wiki need to be published in respected peer-reviewed journals? From the limited knowledge i have about math, it looks correct to me, no worse than Sweetser quaternions .... mabey we can let the reader decide? more later (bedtime for bonzo here soon) be safe .... reddi 05:42 7 Jul 2003 (UTC)

- ("Sweetser quaternions" probably refers to this which is/was linked at quaternions -- fmr Kwantus)

Your quote from Gibbs just shows that he didn't like the notation, and that he felt it didn't add anything: writing vector fields as quaternions with no scalar part and using the quaternion product with del to get curl and divergence (as Maxwell did) neither simplifies the algebra (arguably) nor exposes any new symmetries, mainly because Maxwell was always forced to treat the scalar and vector parts separately (in which case, why combine them at all?). (The real use of four-component objects comes from 4-vectors and the relativistic inner products of the Lorentz group, which came later and is quite distinct from quaternion multiplication, as well as being different from the component grouping used by Maxwell) Similarly, Heaviside found them inconvenient; there's no reason to think that they failed to "understand" them. Vector notation can handle nonlinear media just fine, by the way (see e.g. Agarwal & Cooley's nonlinear optics book). Regarding citations, I would go further: science articles should stick to generally accepted results (i.e. peer-reviewed stuff that has stood the test of time, with special skepticism when it comes to speculative work on new physical law that is not yet tested). The problem with "letting the reader decide" is that the reader does not have the evidence to do so in a short summary, nor can the casual reader distinguish between the consistent/well-supported/widely-accepted and the crackpot. (And if you don't have sufficient maths to understand something, you should be wary about writing on it yourself.) Steven G. Johnson

[quote from Gibbs] didn't like the notation? Why didn't he like it? hmmm ... probably because he didn't understand them ... you disliek what you don't understand ... basic human nature ... too bad he doesn't see how it add things ... writing vector fields as quaternions as Maxwell did, was NOT the ultimate intention for these equation .... it does simplify the algebra [it mirrors the 4D world we live in, as Hamilton realized) .... Maxwell saw the promise of treating the scalar and vector parts together to refelct nature more elegantly (that's why combine them) ... [snip vectors diversion] .... Similarly, Heaviside didn't understand them either [didn't see the elegance]; if they did, what other reason can there be then? mabey it's a reverse hanlon's razor ... now as to the ability of vectors .... Vector notation can handle nonlinear media ? wha'? yea .. that's why they figured out the geomagnetic nonlinear phenomenon [among others] and [begin] they have that GUF already [/end sarcasm] .... Now over citations, science articles should stick to generally accepted results? Wha'? ... why is progress made in science? ... becasue information conventional AND unconventional is given to ppl [thankfully ppl were able to hear about a college graduate with this totally unconventional idea that the earth surface was on plates and they shifted around ... then it caught the attention of other scientists that realized that yea ... the earth had techtonic plates!] ... (now ... peer-reviewed stuff is good ... but the free flow of ideas is VITALLY important .... isn't that the part of this encyclopedia concept? GPL et al.?] ... skepticism comes with BOTH speculative works and conventional works .... physical laws are only a law till they are broken (or does Newton still trump einstien?)) ... I think i see what the real problem is ... "letting the reader decide" .... we wikipedians are giveing them the evidence, are you proposing that only one side of the evidence is presented? [doesn't seem NPOV to me] ... and it can be done in a short summary [that's a strawman arguement that it can't] ...as to can the casual reader distinguish between "consistent" "well-supported" "widely-accepted" or the "anomolous" "narrowly-accepted" "fringe"? May NEVER know if they dont's see BOTH sides .... (AND i do comprehend some math, I'm just not a mathematician and understand everything. "Why should I refuse a good dinner simply because I don't understand the digestive processes involved" - Heaviside =-) more later ... reddi 00:17 8 Jul 2003 (UTC)

this is a page with an interesting spin to say the least: http://www.cheniere.org/books/analysis/history.htm My degree is in maths with a good dose of physics, and I know enough to attest quaternions are essentially despised in the west; they were used only to provide illustrations for algebra theory, and their use by Maxwell was never mentioned. (Until I happened on that page, and then checked here, my hedgykashun left me the impression quaternions had never been applied to anything.) I know enough of quaternions to know they have subtle differences from vectors, and I know enogh of about how physics is done to know it's quite possible something was simplified away Heaviside's conversion to vectors. Remember that we habitually bash Newton's definition into F=ma even though he originally said F=dp/dt - and we got away with it until relativity gave us time-dependent mass. Remember that physicists habitually throw away the interestig parts of their equations until they cane be solved and then pretend that result answers their original question - that's exactly how what's studied under the chaos rubrik got ignored for so long. I'd have to see a step-by-step review of Heaviside's logic before I'll decide whether his version is exact or simplified. -- user formerly known as Kwantus (PS It also seems odd to read things like "superstring theory is free from quantum anomalies if the spacetime dimension is 10 and the quantum gauge symmetry is SO(32) or E8×E8" or "string theory in a background of 5-dimensional anti-de Sitter space times a 5-sphere obeys a duality relationship with superconformal field theory in 4 spacetime dimensions"[1] but encounter claims there's no physical significance to algebraic structures (somewhere up above)...)

This PDF appears to list the original 20 equations in reals, and reductions to 6 vector+2 real. (1.6) is Ohm's law, (1.4) the Faraday force, and (1.8) the continuity equation, leaving five. Comparing with the General Case (GC) on the main page, (1.1, 1.3) are combined into GC4, and (1.7) matches GC1 except for sign (!). That leaves (1.2) and (1.5) which don't obviously match GC2 and GC3, esp since a variable appears in (1.2) which does not appear in GC. (There may be some way of stirring up Maxwell's eight to get Heaviside's four without loss; I'd just like to see what it is.)

Reddi, Thanks for the references. I had previously read smith.

Here is my take on quaternions the math and quaternions the physics. Math: I think Hamilton was a better mathematician than his contemporaries. But that doen's make his math the end all. I think in the late 1800s, the idea of a fourth dimension was novel and considered useless in a three dimensional world. Thus the useless scalar dimension was jetisoned and the three vectors found work. Maxwell and others were upset over Hamilton's Rules (II= minus 1). Gibbs and others "fixed" this and made II= +1, and voila we have vector Algebra. In a way, the physicists created a "mathematics" that has serious defects. Associativity (AB.C =A.BC) and Closure ( II is not a member of the set of vectors)is lacking. In a physics sense Maxwell complained that when he thought he was computing a maximum, he got a minimum. For example when a rock is displaced in the direction of gravity the sign is negative in quaternions. This sign told Maxwell that this was exergy (outenergy), when he was expecting enegy. Energy is when you displace the rock against gravity!

My point is that mathematics is a very useful tool and physicists need to understand the mathematics they use and should not select defective mathematics. In a sense quaternions represent the only Associative Division Algebra. This means that if physicists want to solve AX=B, the only algebra competent to do this is quaternions or systems isomorphic to quaternions! (Real algebra and complex algebra being sub-algebras of quaternions.)

Tony Smith and others (John Conway and Derek Smith: On Quaternions and Octonions) have shown quaternions to be isomorphic (do the same thing) to the Group Theory view of physics.

PHYSICS: Planck's and Einstein's Quantum Equations and "Theory" are also seen to be derivable from the same quaternion equation as Maxwell's Equation. The variable I call Life L = Ls + Lv, where Ls is the scalar and Lv the vector of Life. I believe Life is the most important variable in the universe and came into being when God said "Let there be Light", and there was Life. Life is related to action by the speed of light, L=ch:

Work = XL = (dLs/cdt - DEL.Lv) + (dLv/cdt + DEL Ls + DELxLv)

work = XL = (dhs/dt - DEL.Lv) + (dLv/cdt + DEL Ls + DELxLv)

Planck and Einstein only considered the scalar equation "(dhs/dt - DEL.Lv)" and physicists have not "discovered" the vector equation.

Planck's Law is the Boundary/conservation condition 0 =XL. Einstein, deals with the internal/non-boundary condition, kinetic Energy = (dhs/dt - DEL.Lv).

The Boundary Condition vector equation is the widely discussed but seldom shown "action reaction equation "0=(dLv/cdt + DEL Ls + DELxLv)".

These I believe are physical facts that have not been "discovered". For example I have not seen the "work function, phi" in Quantum theory described as the Divergence of a vector function with units energy-distance, Lv. I have not seen discovered the vector equation of dependency of the vector radiation the gradient of a scalar and the curl of the Lv. Or to put it in conventional action h terms,

0 = dhv/dt + (DEL Ls) + c(DELxLv) = (dBv/dt + DEL Es + DELxEv)

It may be that this is not true for actio h, but the same equation is true in electromagnetism for the the E field! Maybe the experimentalist should look at this relationship.

Reddi and Steven thanks for your points. I appreciate the critical thinking. Physics is alive abd Life is beautiful.

--- reddi I have these ideas laid out fuller in : http://www.geocities.com/wardelllindsay/unification.html

There is not a lot of text but the math and ideas are there.

It is my understanding that Theodore Kaluza unified electromagnetism with Albert Einstein's theory of general relativity. Apparently, Einstein did not like Kaluza's assumption that the universe is invariant in the 5th dimension he invoked. Therefore, Einstein tried to redo the theory with curled up dimensions. But, he never succeeded. I refer to Chapter 18 of the _Introduction to the Theory of Relativity_ by Peter Gabriel Bergmann.

Joseph D. Rudmin

I saw a an old version of this article at http://www.rare-earth-magnets.com/magnet_university/maxwells_equations.htm I'm just curious, did he copy Wikipedia, or did we copy them? There's is copyright GPL, so I assume they got it from here. Are they obliged to provide a link to Wikipedia or to mention that they got it from Wikipedia? Actually, I just read Wikipedia:Copyrights and it says that Wikipedia must be referenced. What can we do about this minor breach? dave 19:43 16 Jul 2003 (UTC)

One of us should write a curteous request to the owner of the webpage that credit be given, and perhaps a link to this page be provided, since it is occasionally edited. To avoid multiple requests and for consistency, one of the managers of this site should have a designated task of reconciling copyright violations. I personally would look very unfavorably on any punitive action on the part of wikipedia. The wikipedia page http://www.wikipedia.org/wiki/Wikipedia:Copyrights discusses what to do in case of copyright infringement. Your notice is sufficient, I think.Rudminjd 15:15 20 Jul 2003 (UTC) Joseph D. Rudmin

It has been taken care of, and there is such a page: Wikipedia:Sites that use Wikipedia for content. I have sent out a standard letter, which asks for them to provide a link to the original article, and a link to GFDL. I'm still waiting for a response. I'll wait a few weeks and then try contacting them again, perhaps by phone. dave 15:52 20 Jul 2003 (UTC)

I permitted myself to edit "The Source of the Magnetic Field" section. I replaced the vector for the electric displacement field (D) with that of the electric field (E), for 2nd and 3rd equation of this section. Anyone feeling confident enough please counter check. Jerome Peeters (08.08.2003)

Both the previous revision and your correction were incorrect. D should appear in the law, or \epsilon E, but not \epsilon_0 E (except in vacuum) and not not \epsilon_0 D. —Steven G. Johnson

This article needs to clarify the relationship between the microscopic Maxwell's equations (in terms of E and B) and the macroscopic Maxwell's equations (in terms of D and H, which involve macroscopically averaged quantities like the dielectric constant of a material). I find the whole discussion to be currently slightly confused (e.g. I just noticed that the discussion of Gauss's law contained several errors before I fixed it just now) and in need of a much more careful rewriting. Sigh, not that I have the time to do it myself right now. —Steven G. Johnson

Do we need to jump all the way to differential geometry and differential forms to do relativistic Maxwell's eqns? Surely a gentle introduction with 4-vectors first would help. See this treatment -- The Anome 19:20, 10 Aug 2003 (UTC)

I think there was an error in saying that div(mu*B)=0 since the mu is in the wrong place. I allowed myself to remove the mu from the "linear media" equations.

The first tensor equation covers only Conservation of Charge, Coulomb's Law and Ampere's Law. My reference for the 2nd tensor equation (which expresses Faraday's Law and No Magnetic Monopoles) in the tensor version of Maxwell's equations is: Charles F. Stevens 1995, Six Core Theories of Modern Physics p.199, MIT Press ISBN 0-262-69188-4. 169.207.115.28 03:28, 31 May 2004 (UTC)

It's all well-covered in Jackson, which is already referenced. Strictly speaking, the first equation does does not directly express conservation of charge, which is the equation

\partial _{\beta }J^{\beta }=0

, although this can be derived from the first equation by taking the 4-gradient

\partial _{\beta }

of both sides. (I'm not sure if all of the sign conventions are consistent with those in four-vector, by the way, since Jackson uses the opposite sign for g.)

Maxwell's equations or Maxwell equations

I think modern usage would have the Maxwell equations rather than Maxwell's equations. Here I quote J. D. Jackson in the preface to the 2nd edition of "Classical Electrodynamics"

Of minor note is the change from Maxwell's equations and a Green's function to the Maxwell equations and a Green function. The latter boggles some minds, but is in conformity with other usage (Bessel function, for example).

He then uses the term Maxwell equations throughout this book, which is pretty authoritative. Thus in my opinion the article should be changed to reflect this modern usage. DMB 13th December 2005

As much as I respect Jackson, neither he nor any authority can dictate the English language, and "Maxwell's equations" is still far more widespread as far as I can tell. When it comes to things like this (where there is no "right" or "wrong", only convention), the safest thing is for Wikipedia to simply follow the most common usage and to note widespread alternate usages. —Steven G. Johnson 16:55, 13 December 2005 (UTC)

You have some heavyweights on the side of the apostrophe.

"The special theory of relativity owes its origins to Maxwell's equations of the electromagnetic field" — Albert Einstein

The Feynman Lectures on Physics chapter headings have 1 heading for Maxwell Equations and 2 headings for Maxwell's equations --Ancheta Wis 01:55, 14 December 2005 (UTC)

Albert einstein might have been a genius, but how could he use the modern form when writing 100 years ago DMB 14th Dec 2005

This entire discussion is irrelevant and a complete waste of time. Wikipedia is an encyclopedia, not a cutting edge journal. The purpose of Wikipedia is not to make new propositions or to change existing conventions. The purpose is to report things the way they are, not the way they ought to be. Right, wrong, or indifferent, the currently accepted and only correct terminology as of today is Maxwell's equations and not the Maxwell equations. If you don't think that is correct or proper, then write an article and publish it on Wikibooks or the Wiki Commons, but not on Wikipedia. -- Metacomet 17:48, 14 December 2005 (UTC)

Metacomet, you're overstating the case. "The Maxwell equations", while not the most common terminology currently, is certainly used by some prominent authorities (e.g. Jackson), and you can find it in journal articles, etcetera — it is also accepted as "correct". While I don't think we should change the title of the Wikipedia page, we should probably also "report things the way they are", in your words, and say that this is a common alternative name. In fact, I'll do that now. —Steven G. Johnson 23:19, 14 December 2005 (UTC)

Some more explanations please, Cleanup

I think this page is not accessible for people who don't know the subject already. Please help to explain the equations for lay people with basic knowledge of mathematics and physics. Andries 19:44, 2 Jun 2004 (UTC

I presume you don't think that "basic knowledge of mathematics" includes partial differential equations, and that "basic knowledge of physics" does not include electromagnetism. In that case, see my comment below for all the articles that need work. Maxwell's equations are not the laws of electromagnetism, they are a particular mathematical formulation of them. Being technical, anyone who knows the mathematical and physical prerequisites (of multivariate calculus and electromagnetism) should have no problem with them. But the prerequisites should not be explained here but linked. That's what's great about hypertext: not everything has to be in the same article. Miguel 18:11, 5 Jun 2004 (UTC)

I have put the page on cleanup, not because the article bad in itself but it needs to be made more accessible. Andries 12:20, 5 Jun 2004 (UTC)

Could you be specific about what parts you think are unclear? or do you think the whole article should be rewritten? Bear in mind that the article is not meant to be a course in electrodynamics, but rather just an explanation of Maxwells equations. Lethe

At least the symbols should be explained e.g. B in the equations. It is not that I can't find out but one should not have to go other articles to find them. Wikipedia is not written by experts for experts but by experts for everybody with basic knowledge of physics and mathematics. I can do it too but I need some time , if you could help then that would be great Andries 15:23, 5 Jun 2004 (UTC)

We could insert little wiki links before each of the 4 equations, which point directly to the article which is in the right side-bar on Electromagnetism; would that help you out? Ancheta Wis 16:09, 5 Jun 2004 (UTC)

Yes, I think that will be sufficient.Andries 16:16, 5 Jun 2004 (UTC)

Maxwell's equations really cannot be understood without a lot of math and a physical intuition for the laws of electromagnetism that they encode: Gauss' law, the Faraday-Lenz law, Ampère's law (or Maxwell-Ampère law) and the absence of magnetic charge. Those four laws of electromagnetism can and should be discussed in intuitive physical and geometrical terms without the need for partial differential equations. Lay readers should be directed to those articles, assuming they are accessible. Do you think they are? Miguel 18:05, 5 Jun 2004 (UTC)

One can still try to improve the introduction, to make it a little clearer what quantities and concepts the equations describe, as well as their broad implications and historical significance. I've rewritten the introduction accordingly. —Steven G. Johnson 20:56, Jun 5, 2004 (UTC)

Note: The partial differential equations expressed in the article represent centuries of human thought and development. Researchers have literally given their lives investigating the phenomena they represent. Please spend a little time on this article if you wish to gain some insight into the physical phenomena and the attendant notation which represents the phenomena.

StevenJ, Miguel and Ancheta_Wis, thanks for all your help. The article is quite okay now. I think I got a bit spoilt by the accessibility of encarta. Andries

I don't think there is any reason that we shouldn't be as accessible or more so than encarta. if you have more suggestions, i guess we should hear them. Lethe

This division of Maxwell's equations is derived from tensor representation, it is also relativistic expression. Therefore, this division is essential and clearer.

e.g. Landau "The Classical Theory of Fields".L-H

This article is not for speculative physics

User:Reddi insists on including, without explanation, the following citation:

Jack, Peter Michael, "Maxwell-equations: A Brief Note". Physical space as a quaternion structure - I.

This is an unpublished, unrefereed article, on a site (apparently) by the author, advancing a speculative "new theory [that] now links thermal, electric, and magnetic phenomena alltogether in one set of elementary equations. This result is based on an initial hypothesis, named 'The Quaternion Axiom,' that postulates physical space is a quaternion structure."

This sort of speculative pseudo or proto-science is not appropriate for an encyclopedia article on Maxwell's equations, where a reader looking at the citations for places to go for more information should expect to find only authoritative and well-established material.

—Steven G. Johnson 02:37, Jun 22, 2004 (UTC)

i think Stevenj is correct, i removed the linkLethe 02:51, Jun 22, 2004 (UTC)

Do all external lnks need to be published and refereed article for articles in wikipedia? It may be a site on a speculative "new theory", but the link deals specifically with one topic. Just because you don't "approve" of the information [you POV is pretty clear from "speculative pseudo or proto-science"] does not mean the iinformation is not appropriate for A EXTERNAL LINK (not a reference) in an encyclopedia article on Maxwell's equations. The citations section should be for the reference that the article uses to construct that article. I'll be adding more external links for places to go for more information. JDR

Author's Note: Given that the "accepted vector analysis" originally derived from Hamilton's Theory of Quaternions, and these same quaternions motivated James C. Maxwell in his mathematical formulation of Electromagnetic Theory, (vector analysis did not exist in Maxwell's time in 1873, only Quaternions) it is probably appropriate to cite some sources that lead researchers to consider what sort of ideas Maxwell himself needed to grapple with at the time his Electromagnetic Equations were being formulated. From this point of view, the Quaternion paper could be considered very relevant. A revised version of the html article can also be officially referenced through the ARXIV archives here [math-ph/0307038],—pmj 02:35 pm, Aug 27, 2004 (EST)

This is false. Maxwell's original 1864 work (I have the paper) did not use quaternions in any shape or form. Second, your paper is not a merely historical discussion, nor do you use simply the 1870's quaternion formalism — you introduce a new scalar component "T" into the field quaternion, introducing speculative new physics beyond the standard Maxwell's equations. Wikipedia is not for promotion of original research, especially unpublished and non-peer-reviewed research. —Steven G. Johnson 18:58, Aug 27, 2004 (UTC)

I think you misinterpreted what I said. Or, perhaps I wasn't clear enough. So let me clarify a few points. First of all, many of Maxwell's private papers and letters for the period 1860-1879 were lost soon after his death. So, nobody knows today exactly what Maxwell's original thoughts were on his formulation of EM, since this was the period when he was working on the mathematical formulation (He dies in 1879). Maxwell was discouraged from publishing his Quaternion ideas, so much of what he thought about this would mostly be found only in his private papers which are now lost. Harman discusses this in his book, and tells us that it is believed these papers were destroyed in a fire that burnt down the Maxwell's House at Glenlair (sometime around 1886). We know that by 1873 Maxwell had decided to re-cast the formulation of EM using Hamilton's Quaternions, and he published "some" of his ideas in that year in his now famous text. We know that Maxwell was thinking about Quaternions at least as early as 1867, because he wrote to Tait asking him about the reasons for the shape of the nabla symbol in a letter dated 11th December of that year. This was an important question, which Tait seems not to have understood at all from his response. Reading many of the other surviving letters and publications written by Maxwell, suggests to us that Maxwell himself did in deed "get it," while Tait did not. However, there's a hint in Maxwell's writings that the ideas of left and right nablas may have originated with Hamilton, although there is no published work from Hamilton, nor anyone else of that period, that show both forms of nabla. Indeed, Hamilton himself had to "insist" that some of his ideas on the applications of Quaternions be included in the records. As far back as the June 1845 meeting of the British Association for The Advancement of Science (the organization that held discussions on these things back then) we find Hamilton making a "special request" to have his idea (His Quote--"Is there not an analogy between the fundamental pair of equations ij=k ji=-k, and the facts of opposite currents of electricity corresponding to opposite rotations?") put in the records. This suggests to us that many things were discussed and presented "off the record," but obviously Hamilton wanted the future generations to know that he'd thought of it first, he'd seen the beautiful connection between EM and quaternions, but he too was discouraged from publishing his ideas. This mystery continues up to today. People are still being discouraged from publishing their quaternion application ideas. However, the ideas are so simple and clear, that anyone who understands a bit of math and physics can see that there is a hole in the historical literature of physics. And that hole is right around these ideas that utilize Quaternions in EM theory. To give another example, of how mysterious this phenomenon is, we note that all the private letters between Maxwell and Stokes and Maxwell and Thomson (Lord Kelvin), of that critical formative period, are missing. To understand the significance of this, recall that Stokes was the Cambridge Professor who was the leading expert on Hydrodynamics in his time, the same hydrodynamics that Maxwell borrows mathematical ideas from to formulate his theory. And Thomson was the leading expert on Thermodynamics and Heat at that same time. These were the very two men, to whom Maxwell would turn to discuss any ideas that linked Thermoelectricity with Electromagnetism (Maxwell was well aquainted with both men). Indeed, Thomson had succeeded in uniting the two phenomena, 1822 Seebeck Effect and 1834 Peltier Effect, and had proposed and discovered another Thermoelectric Effect, now called the Thomson Effect, all by the year 1854. So, the subject of thermolectricity was well known by 1873 when Maxwell brings together "Electricity" and "Magnetism" under one beautiful simple mathematical theory. So, if Maxwell was preoccupied with the question of uniting "Electric" and "Magnetic" phenomena under one roof, how is it that he missed considering the inclusion of "Thermal" phenomena into the one simple beautiful theory, when Thermoelectricity was sitting right there telling everyone that "Electric" and "Thermal" phenomena were also connected too? Where are the discussions between Maxwell and his contemporaries on the "idea" of linking Thermoelectric and Electromagnetic effects? "Electric" effects are "vector" effects, "Magnetic" effects are "vector" effects, but "Thermal" effects are "scalar". And what is a Quaternion? The union of a "vector" and a "scalar." Note, that the very first publication by Hamilton on the application of Quaternion nabla was to thermal phemonena. (again, this paper mysteriously, wasn't published...the record says it was "misdirected" by accident, and so Hamilton has to mention it in another brief footnote (see Graves, discussion of the year 1846)) so, this mysterious hole in physics continues to propagate through time. It is true that I introduce a new scalar term "T", but even using the 3-vector quaternion form of nabla back then, Maxwell must have dealt with the related "scalar" term that is just a special case of what I call "T". This is because the product of two vectors, in quaterion theory, is a 4-dimentional quaternion, (a1.i+a2.j+a3.k)(b1.i+b2.j+b3.k) = -a.b + axb, (in modern notation), that includes a scalar part. So you have to include this scalar term in the mathematical formulation. I just extend it and give it a meaning related to the current physics we already know. It's simple. Now, I would like to think that I am the original discoverer of this elementary theory that links thermal and electromagnetic phenomena, that would be nice. However, after I had discovered these things, the simplicity of it puzzled me, since someone must have done this before. So, I started to search the historical literature to find out more information, but drew blank! Yet, I could see from what is in the historical records, the hints and suggestions of thoughts having traversed the path before me. But still, there are no official writings that clearly discuss the links my paper does. Not even to shoot it down! So, I'm not just promoting "my" idea, but those of "Hamilton" and "Maxwell" too. Not many people are aware that Hamilton was the first to see the links between Quaternions and Electromagnetism, and that was way back in 1845-6, long before Maxwell thought about these things. Maxwell was really completing Hamiltons work, when he sought to cast Elecromagnetic Theory in Quaternions, it wasn't Maxwell's original idea to do this. ]],—pmj 11:04 pm EST, Sep 30, 2004 (EST)

Meaning of citations

If you think that "external links" are significantly different from references, then you don't understand how citations are used in practice. Regardless of whether you call them references, a bibliography, or external links, they are places the reader will go to better understand the material, as well as to have additional sources with which the reader can convince herself of the veracity of the material. This is why I think that a separate "external links" heading here is misguided. —Steven G. Johnson 03:04, Jun 22, 2004 (UTC)

(If you are writing in your field, you may not need to consult any references at all to write something. You still look for references, though, to give the reader pointers on places to go for more depth/breadth.)

If you think that "External links" don't "significantly differ" from references but there some differences. Your "attack" on my understanding does nothing to the point. Citations are used in practice to cite material used. IT IS IMPORTANT to call them references, a bibliography, or external links ... pending on the links and how it related to that article. How can a reader can convince herself of the veracity of the material if all the information is not provided? A separate "external links" heading here is not misguided (only if you want to exclude information that does't fits you POV). A reader will go to better understand the material if the reader is given all the information [not half of it). JDR 03:38, 22 Jun 2004 (UTC) (BTW, this isn't NuPedia)

Citations are used for far more than to list "material used" in writing an article. In fact, for most articles using citations in the real world (not homework assignments), the vast majority of citations are not material used so much as pointers to sources of more information and breadth, as I said. —Steven G. Johnson 03:44, Jun 22, 2004 (UTC)

Encyclopedias are not for new research, they are for well-established material. Especially on a well-established subject like Maxwell's equations, it is misleading to the reader to direct him/her at such a speculative article, especially when the link is right next to an authoritative reference like Landau. —Steven G. Johnson 03:44, Jun 22, 2004 (UTC)

Wikipedia is a secondary or a tertiary source. Encyclopedias are not for new research (I did write alil about that article on that, IIRC). If you don't understant what kinds of sources should be used in Wikipedia ... then mabey tyou should read up on the types.

Wikipedia is for established material ... as well as current research and so-called "speculative" information. Providing information on the "well-established" subject like Maxwell's equations [more appropriately the "Heaviside-Gibbs equations"] is not misleading the reader ... it's providing the reader information (you seem to have a problem with that). Directing the reader to external articles is not "harmful", but I gues YMMV on that. JDR

Introduction before TOC

Please keep the high-level introduction before the table of contents. It is important to distinguish it from the later material, which is much more technical.

—Steven G. Johnson 03:04, Jun 22, 2004 (UTC)

The introduction after the table of contents helps the readability of the page. To distinguish it from the more technical material ... nest the subheadings. JDR 03:10, 22 Jun 2004 (UTC)

Nested subheadings do not improve readability either. I don't feel too strongly about this formatting issue, but I would like to hear other opinions besides yours. (PS. Write what you want in the talk section, but please don't edit my text. Including my headings.) —Steven G. Johnson 03:17, Jun 22, 2004 (UTC)

Whoa, there seems to be an edit war going on here! -Lethe 03:11, Jun 22, 2004 (UTC)

Welcome to the wonderful world of Reddi's edits. He's been periodically garbling physics articles he doesn't understand for ages now, leaving it to the rest of us to clean up his messes, and he usually fights with insistent reverts/insertions, with scant answer to criticisms, until several people start to complain. —Steven G. Johnson 03:17, Jun 22, 2004 (UTC)

Nice attack, Stevie. I respond [not "fight"]. I also provide information and links (more than most do; eg., your "scant answers") ... if the factual information and references are ignored, I guess I can't do anything about that. JDR 03:31, 22 Jun 2004 (UTC) (Disdains pompous donkeys)

Reddi, I initially tried to be understanding to you, way back when, but you persist in editing articles you don't have the technical or mathematical background to understand, and you ignore the fact that essentially everyone disagrees with your arguments regarding technical additions. On balance, you do more harm than good on Wikipedia. —Steven G. Johnson 03:36, Jun 22, 2004 (UTC)

Stevenj, you don't "understand" me nor have we agreed on many things ... and I try to avoid articles you edit [if they interest me]. I will continue to do this where I can ... but sometimes that is not possible. I will continue edit articles, adding information ... I'm glad you know what my understanding of the technical or mathematical background is [/end sacasm]. I do not ignore the facts ... and mob rule does not make the state of wikipedia "better".

I do more harm than good on Wikipedia? IYNSHO ...

JDR 03:44, 22 Jun 2004 (UTC)

Anyone object to restoring the tensor equations

The Revision as of 23:18, 24 Jul 2004 removed the tensor equations. Anyone object if we restore them Ancheta Wis 21:42, 10 Aug 2004 (UTC)

Yeah, It looks like User:205.188.116.206 deleted the special relativistic formulation and the differential forms formulation without justification. Let's get those back into the article - Lethe | Talk

Inaccuracy in the equations

I'm not a physicist or a mathematician, however, I do know that Otto Schmitt stated privately if not publicly in his last year of life that Maxwell's equations are inaccurate. Please consider this statement hearsay as I was not told directly by Otto, rather a friend was. I would hate to be responsible for marking Mr. Schmitt's name in the negative. However, if there is truth to the statement it would be an important scientific piece of information.

I wanted to find reference to this statement before posting and could find none. I did find reference to Otto Schmitt and Maxwell in a paper:

http://www.nebic.org/icebi/schwan1.htm

Specifically, the "Maxwell-Wagner effect".

Digging a bit more I found a Russian publication of 1995 which may or may not be relevant:

http://eos.wdcb.rssi.ru/transl/izve/9509/pap05.ps

"Geoelectrical problems covering sonic and infrasonic frequencies commonly deal with a quasi-stationary approximation, which essentially simplifies solving the Maxwell equations. To what extent does this approximation ignore the displacement current applicable to the model involving macroanisotropic media?"

I haven't read this paper, but the quote describes the accuracy of an approximate solution of Maxwell's equations (by dropping the displacement-current term), not the accuracy of the full equations themselves. —Steven G. Johnson

My guess is that Otto Schmitt was seeing some indications of a poor fit for Maxwell's equations in the research he was doing. Experimental was not matching theoretical with a high enough degree of correlation. As a result he knew either that his measurements were wrong or that Maxwell's equations were wrong. Considering the weight Otto's life's work carries, it is probable that Otto had good reason to believe problems exist within the equations.

Wikipedia does not report unpublished speculations, much less rumors of unpublished deathbed speculations. If you find an unrefuted paper in a mainstream scientific journal that seriously questions the validity of Maxwell's equations (other than known corrections for quantum effects and GR), please post it here. —Steven G. Johnson 14:59, Dec 1, 2004 (UTC)

Examples section?

Is it okay to add an examples section straight after the detailed formulas one? With some simple stuff like a point charge and the infinite wire with current in it?

That is what would be in a textbook, but not under Maxwell's equations, rather as applications in electrodynamics or electricity and magnetism. How about creating a new article with links to the electromagnetism series, as well as the appropriate link to the new page in this article? It might be useful to electrical and electronics engineering students as well, and as examples in applied mathematics, as the solutions to equations. Ancheta Wis 02:26, 22 Dec 2004 (UTC)

Textbook-style tutorials are more appropriate for WikiBooks than for an encyclopedia. Please add examples to the physics textbook there. The only exceptions might be some example that is relevant to a particular article (e.g. capacitor or light) or some problem that has historical importance for other reasons. —Steven G. Johnson 02:44, Dec 22, 2004 (UTC)

Maxwell's equations in terms of differential forms

This section looks to me like the key for a deeper understanding, but it is very very short at the moment. Does anyone have more information about that? 84.160.214.150 13:40, 19 Feb 2005 (UTC)

Like you, we are eagerly awaiting mathematicians who can develop more structure which expresses the differential forms which may be applied to the manifolds currently contemplated for use in physics. For example, we need forms which can handle the presence of matter in all its forms (probably a pun, but maybe there is some physics in it), not just vacuum. Ancheta Wis 00:57, 16 Mar 2005 (UTC)

old four-vector notation

Any chance of having my favourite four-vector Maxwell's equation, namely

d'Alembertian of (four potential (A,phi) = four current (j,rho)

in the older notation that most of us physicists learnt a few years back.Linuxlad 18:33, 9 Apr 2005 (UTC)

see Electromagnetic four-potential for more links. Ancheta Wis 10:25, 11 Apr 2005 (UTC)

Thanks (I had in fact just found it! and had come to post a link here :-)). I've added a few tiny mods for those of us used to the older vector operators. Linuxlad 10:48, 11 Apr 2005 (UTC)

Equation Form

can someone who is familiar with the mathtype used on wikipedia please add the constants onto maxwell's equations? This would be very useful as the current form of the equations is totally unusable for any practical application. Cpl.Luke 19:59, 25 Jun 2005 (UTC)

also the differentials are mislabeled, for instance in gauss's law it should be da not ds Cpl.Luke 20:02, 25 Jun 2005 (UTC)

Currently looks good to me. If you scroll down a bit you get the simple version

\nabla \cdot \mathbf {E} ={\frac {\rho }{\epsilon _{0}}}

which is only good in free space, but

\nabla \cdot \mathbf {D} =\rho

is the general form and therefore belongs in the box near the top as it is more generally useful. Also, ds is a rather standard alternative to da as a is used for many things such as acceleration. See the notation in surface integral. --Laura Scudder | Talk 20:16, 25 Jun 2005 (UTC)

However in every physics textbook I've read (only talking about the integral version here) da is used (also missed the version of the equations with constants apparently)

Also it would agree more with the gauss's law article to use da in the gauss's law entry, also looking in a textbook right now I see da used for the gauss's law entries and ds used for faradays and ampere's law. It would be wise to avoid confusion by changing it to da. Also we can say what the differentials mean in the "where:" entry directly below the table. since D and H seem to be used souly to make the equations appear simpler we should attempt to avoid confusion by reducing the number of variables down to 2 E, and B, later on in the article it is explained what the relationship between B,E and H,D is and thus we can continue using them for the rest of the article, but it would seem that it is far simpler to only use B and E in the original table. I made the relevant edits however they were promptly reverted, I would appreciate some feedback as to why it was reverted so that we can come to a compromise. Cpl.Luke 05:19, 12 July 2005 (UTC)

See the comment for 18:06, 27 May 2005 by Stevenj:

"(the equations can include interactions with non-charged matter as well (e.g. spin effects), although in any case epsilon and mu have to come from experiment (or quantum theory) and aren't given by ME)". That is what LauraScudder meant above. BTW, the thread should be in temporal order. Right now, this 2005 entry is on top of the 2004 entries below. After this all settles down, we should cut and paste this entry to the bottom of the thread. Ancheta Wis 05:50, 12 July 2005 (UTC)

I still don't see the relevence of wheather or not mu and epsilon come from experiment or not, as all your doing by using D, and H is moving the mu or epsilon over to the other side of the equation. Also just in case I'm missing something in the above statement, this still doesn't disclude us from using da instead of dsCpl.Luke 06:43, 12 July 2005 (UTC)

You often can't move the μ and ε through the derivative as they can be spatially dependent. Think of a non-uniform material or a boundary between materials. There's also special cases for nonlinear and anisotropic materials where in general

D_{i}=\epsilon _{i},jE_{j}

. So until you know the problem you're working with, you cannot move ε and μ. --Laura Scudder | Talk 14:40, 12 July 2005 (UTC)

The choice of ds for line integrals I've always found very unfortunate, but I think it arises from wanting to generalize length as the surface area of a line. --Laura Scudder | Talk 14:40, 12 July 2005 (UTC)

yes that would seem to make sense, but since we are talking about area here specifically, and the gauss's law article already uses da we should make the switchCpl.Luke 16:53, 12 July 2005 (UTC)

A can also mean magnetic vector potential. Thus Area could be confused with an important electromagnetic concept. S has the handy mnemonic Surface, with lowercase s for surface element. Ancheta Wis 17:09, 12 July 2005 (UTC)

however we can say that its the differential of area in the "where:" entry directly below the table, we actually should already have what the differentials mean in that entry.

also whats the problem with using flux in faradays law?Cpl.Luke 18:55, 12 July 2005 (UTC)

Again, the problem with Faraday's Law is whether the derivative can commute, in this case with the integral. It is possible to have a changing surface of integration (i.e. a railgun, a closing loop of wire, etc.). So the time derivative can only pass through the integral sign if the particular problem consists of a non-changing surface. This is actually also a problem with Ampere's Law, but the changing surface there comes up much less often. --Laura Scudder | Talk 19:49, 12 July 2005 (UTC)

ok, but we should at least change the differential term to agree with the gauss's law article, da. Then just label what the da stands forCpl.Luke 23:17, 12 July 2005 (UTC)

Clifford Algebra Notation

Maybe it's interesting to add the notation in Clifford Algebra, it's very concise. Andy 13:48, 5 April 2006 (UTC)

Meaning of symbols

Please note that the table in the General case section contains symbols that are not defined in the key underneath. In particular, Q_encl, I_encl, and Φ_D are not explained (until the Detail section, where slightly different notation is used). - dcljr (talk) 05:08, 25 July 2005 (UTC)

Solutions ???

Is there a list of some important solutions of Maxwell's equations on WP ? I think there should be. The general relativity pages have a page on exact solutions of Einstein's field equations and I think it would be good if the same was done here. :)--Mpatel 13:28, 31 July 2005 (UTC)

As Feynman said, the same equations have the same solutions - what do you say to a set of links to equations of the form of xxx. Then it becomes a reference in mathematical physics. Probably the electrical engineers have a say in this as well. Ancheta Wis 13:37, 31 July 2005 (UTC)

To illustrate, every electrical engineer has to learn Maxwell's equations, but the situation is not symmetrical: not every physics student has to learn about the practical issues involved in a circuit.

Oops. I just remembered how Kurt Lehovec (one of the 4 founders of the integrated circuit) would derive the fundamental physics for a flash memory. He would start from electrostatics. But that is a nit in the machinery for Maxwell's equations. Maybe I should withdraw the suggestion. Ancheta Wis 14:01, 31 July 2005 (UTC)

Proca's Extended Maxwell Equations with magnetic monopoles?

Proca's Extended Maxwell Equations can be found here http://www.innopro.de/maxwell_equations.htm#maxwell_quantum .

Could someone make a page about Proca's Extended Maxwell Equations with magnetic monopoles? I mean Maxwell equations with Proca's extensions and hypothetical magnetic monopoles. Henri Tapani Heinonen 14:07, 6 November 2005 (UTC)

Perhaps you are thinking of Dirac monopoles. The general theory of electromagnetic fields in the presence of monopoles is that of cohomology and Hodge theory. linas 23:33, 14 December 2005 (UTC)

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://wiki.riteme.site/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://wiki.riteme.site/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 φ is the electric potential and A is the magnetic vector potential. If you take the curl of this relation you get Faraday's law. —Steven G. Johnson

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

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

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

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

Integral vector notation

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

\oint {\vec {E}}\cdot d{\vec {A}}={\frac {Q_{encl}}{\epsilon _{0}}}

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

\oint {\vec {E}}\cdot d{\vec {\ell }}=\varepsilon =-{\frac {d\Phi _{B}}{dt}}

\oint {\vec {B}}\cdot d{\vec {\ell }}=\mu _{0}I_{encl}=\mu _{0}\epsilon _{0}{\frac {d\Phi _{E}}{dt}}

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

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

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, “the vacuum”), as Maxwell states at one point in the treatise. The term G×B is a compensation for Galilean invariance. Maxwell's E is equivalent to what we would now write as E + G×B (and his μH is our B, and is written as B in the Treatise). Consequently, the constitutive law, written in equivalent modern form, would read D = εE_Maxwell = ε(E + G×B). Though it is true that the Lorentz force involves a similar expression, E + v×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 “Maxwell's equations”. 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 “stationary theory”, 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 “shutting off” the vacuum — i.e., making ε constant, ε₀. This, then, gets you the Lorentz relation D = ε₀E, without the G×B term. In contrast, Maxwell insisted that ε (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" — if two sets of equations are mathematically equivalent, with at most trivial rearrangements, then they are at most superficially different. Moreover, it is universal in modern science and engineering to refer to Heaviside's formulation as "Maxwell's equations", despite the superficial differences from the way Maxwell expressed the same mathematical/physical ideas (plus some auxiliary equations like the Lorentz force law which we now group separately). Overemphasizing these superficial differences in a general overview article does a disservice to a novice reader.
(You state that "the curl A = B equation tells us more than the div B equation does," but this is dubious: there is an elementary mathematical theorem that any divergence-free vector field can be written as the curl of some other vector field.)
(By the way, it looks like we've had much the same experience as you, here on Wikipedia — if you look at the history if this Talk page, you'll see we've had the same problem with specious arguments about the supposed "lost" physics of Maxwell's original formulation, or his quaternion-based formulation, compared to the modern formulation.)
I also strongly encourage you to get a Wikipedia username (click the "Log in / create account" button at the top-right). It is extremely helpful to other editors if you use a consistent username so that we know who we are dealing with when we see your edits/comments.
—Steven G. Johnson 18:12, 22 February 2007 (UTC)
Point taken. Yes it seems that we hold basically the same point of view but that we were differing only on strategy. For years I used to argue that the so called 'Heaviside Four' carried in substance exactly what was in Maxwell's original papers.
But recently these specious arguments about quaternions seem to have been surfacing alot on the internet and so I thought that a clear exposition of the original '20' needed to be made.
I have moved the original eight now well down the page to section 8. What I don't understand is that the edits only ever show up when I re-save everytime that I log on. That might be something to do with the cookies on this computer. Are you currently getting the orignal eight Maxwell's equations at section 8?
Anyway, by all means move them to a new indexed historical link. As long as they are accessible to the readers, that is all that is important. I am quite hapy to refer to the 'Heavisde Four' as Maxwell's equations. I always found it so annoying everytime I talked about Maxwell's equations, and somebody would totally duck the point I was making and correct me and say 'You mean Heaviside's Equations!'.
There are so many people out there who are steeped in some belief that Maxwell's original equations carried some hyperdimensional secrets and that Heaviside's modifications are some kind of cover story.
Yes, I think I will get a username. I'm quite knew to this and I was simply browsing over the electromagnetism topics. (203.115.188.254 01:30, 23 February 2007 (UTC))
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)
References
There seem to be an awful lot of links to crackpot sites like zpenergy.com and vacuum-physics.com. What gives? —The preceding unsigned comment was added by 164.55.254.106 (talk) 19:07, 23 February 2007 (UTC).
The ZPenergy links are only photocopies of Maxwell's 1865 paper 'A Dynamical Theory of the Electromagnetic Field'. There is nothing crackpot about that. Don't shoot the messenger. (203.189.11.2 11:52, 24 February 2007 (UTC))
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. [2]
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 Wikipedia: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 Wikipedia: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))