Talk:Maxwell's equations

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: Missing image
Del.gif
Image:Del.gif

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 (http://world.std.com/~sweetser/quaternions/qindex/qindex.html) 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] (http://superstringtheory.com/history/history3.html) but encounter claims there's no physical significance to algebraic structures (somewhere up above)...)

This PDF (http://www.aw-verlag.ch/Documents/Notation%20of%20Maxwell%20Field%20Equations.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 (http://farside.ph.utexas.edu/~rfitzp/teaching/jk1/lectures/node17.html) -- 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 <math>\partial_\beta J^\beta = 0<math>, although this can be derived from the first equation by taking the 4-gradient <math>\partial_\beta<math> 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.)

Contents

1 Some more explanations please, Cleanup

2 This article is not for speculative physics

3 Meaning of citations

4 Introduction before TOC

5 Anyone object to restoring the tensor equations

6 Inaccuracy in the equations

7 Examples section?

8 Maxwell's equations in terms of differential forms

9 old four-vector notation

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 (http://www.hypercomplex.com/research/emgrav/hypcx-p20001015.html)". 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 (http://arxiv.org/abs/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)

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