Talk:Maxwell's equations

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Archive 1 ( ? - 2003)

Archive 2 ( 2004 - 2005)

1 Magnetic Monopoles and Complete and Correct Equations of Electromagnetism (Maxwell's Equations)
2 Maxwell relations
3 SI Verses CGS Units
4 Meaning of "S" and "V" and "C" on the integrals
5 Balancing the view on Maxwell's equation.
6 Possible correction
7 Integral vector notation
8 Matter of global structure
9 Stable version now
10 Original Maxwell Equations

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

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

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

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

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

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

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

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

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

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

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

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

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

Yaw 19:19, 23 December 2005 (UTC)

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

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

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

[edit] Maxwell relations

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

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

then

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

then

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

Might be right?

[edit] SI Verses CGS Units

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[edit] Possible correction

Please, check out the Historical Development where it says :

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

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

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

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

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

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

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

[edit] Integral vector notation

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

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

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

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

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

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

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

[edit] Matter of global structure

Scientific reliability often rests with two points :

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

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

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

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

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

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

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

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

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

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

What's your opinion ?

Almeo 08:04, 25 May 2006 (UTC)

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

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

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

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

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

[edit] Stable version now

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

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

[edit] 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)

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

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

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