Talk:Maxwell's equations/Archive 6

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epsilon0 vs epsilon in "Table of microscopic equations"

Gauss' law reads del(E) = rho/epsilon0

Isn't this the special case of vacuum, the more general case using the actual dielectric constant (epsilon), rather than that of vacuum (epsilon0)?

Someone who's sure of this, please fix! Thanks! Michi zh (talk) 15:47, 7 January 2012 (UTC)

The vacuum case refers to where there is no matter (and hence charge or current) present, and hence when ρ=0 and J=0. Do not confuse this with the different treatments, being the "microscopic" one where all bound current is treated as part of the current J and ε₀ is used, and the "macroscopic" one where the bound current is handled as the time-derivative of the polarization P and a "dielectric constant" ε applies. — Quondum^t_c 16:05, 7 January 2012 (UTC)

To be more specific, imagine that an extra electron gets shoved into the center of a block of glass. The electron polarizes the glass atoms nearby, drawing positive "bound charge" from the glass atoms towards itself (more specifically, the glass's electrons are pushed away but its nuclei stay in place, so there's positive charge from glass atoms in the vicinity of the extra electron). So you might think that there's a charge of -e at the spot where you had placed the electron, but actually the total charge there is much less: -0.1*e (assuming the dielectric constant of glass is 10), which is the sum of -e from the original electron and +0.9e from the glass atoms that were polarized. The terminology is that the "free charge" is -e, and the "total charge" is -0.1e. The equations are: del(E) = total charge density/epsilon0, and del(E) = free charge density/epsilon. (Both are correct.) :-)

The section is written to try to make this very clear. It does seem to me that most people see the links and definition...otherwise someone would have complained much sooner! The only other thing I can think of is to maybe use $\rho _{t}$ for total charge instead of $\rho$. Or put in an image... --Steve (talk) 17:34, 7 January 2012 (UTC)

Nice, clear description. And I see there is already a distinction made between J and J_f, and ρ and ρ_f. But I see there is no such distinction between the symbols for microscopic and macroscopic versions of the fields D and B, and this is perhaps an omission. I would have subscripted the macroscopic versions used in the article; the microscopic versions should be D=ε₀E and B=μ₀H. — Quondum^t_c 18:31, 7 January 2012 (UTC)

The result div E = ρ / ε₀ is correct. See this front cover and this textbook, Eq. 1.2.14. Brews ohare (talk) 00:42, 8 January 2012 (UTC)

Closed Double Integral

I saw that throughout the article the loops in the double integral symbol were not consistent. In the section Units and Summary of Equations, the loops were smaller in microscopic equations and bigger in macroscopic equations. I couldnt find any standard way to write this symbol as there are no such symbol in latex without packages. I don't think you can add packages though in wikipedia. Can someone fix this or give advise on how to write this symbol consistently? Pratyush Sarkar (talk) 03:50, 22 January 2012 (UTC)

I have made it consistent within the apparent constraints of the rendering system, and have elected to keep the spacing as for \iint ($\iint$). A smaller spacing may be more aesthetic, easily achieved by using \int twice and suitable spacing adjustment, but perhaps then the general \iint spacing should also be reduced for consistency (e.g. \iint → \int\!\int or similar). There is also another system that uses a pre-rendered PNG file: see {{oiint}}. — Quondum^☏✎ 09:37, 22 January 2012 (UTC)

I just left it the way it is since the spacing is different in the {{oiint}} package than \iint. Pratyush Sarkar (talk) 22:22, 22 January 2012 (UTC)

Late as it may be to reply, but maybe this could be the first article that actually uses this template? Its fine - so what if the spacing is slightly different yet constant for any repeated use (hence its form as a template)?? I'm not saying this as the initial author of those templates for sake of vanity, its because other editors have worked extremely hard to render them the quality they are now. Shall it be a waste for them?

Here are Maxwell's integral equations for total charges and currents:

$\scriptstyle S$ $\mathbf {E} \cdot {\rm {d}}\mathbf {S} =Q/\epsilon _{0}$

$\scriptstyle S$ $\mathbf {B} \cdot {\rm {d}}\mathbf {S} =0$

$\oint _{C}\mathbf {E} \cdot {\rm {d}}{\boldsymbol {\ell }}={\frac {\partial }{\partial t}}$ $\scriptstyle S$ $\mathbf {B} \cdot {\rm {d}}\mathbf {S}$

$\oint _{C}\mathbf {B} \cdot {\rm {d}}{\boldsymbol {\ell }}=\mu _{0}$ $\scriptstyle S$ $\left(\mathbf {J} +\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\cdot {\rm {d}}\mathbf {S}$

and for free charges and currents:

$\scriptstyle S$ $\mathbf {D} \cdot {\rm {d}}\mathbf {S} =Q$

$\scriptstyle S$ $\mathbf {B} \cdot {\rm {d}}\mathbf {S} =0$

$\oint _{C}\mathbf {E} \cdot {\rm {d}}{\boldsymbol {\ell }}={\frac {\partial }{\partial t}}$ $\scriptstyle S$ $\mathbf {B} \cdot {\rm {d}}\mathbf {S}$

$\oint _{C}\mathbf {H} \cdot {\rm {d}}{\boldsymbol {\ell }}=$ $\scriptstyle S$ $\left(\mathbf {J} +{\frac {\partial \mathbf {D} }{\partial t}}\right)\cdot {\rm {d}}\mathbf {S}$

Well, by no means am I forcing them into the article, its just my recommendation. =) -- F = q(E + v × B) 00:13, 30 January 2012 (UTC)

I should add: this article, continuity equation or any vector calculus articles, were the main motivation for creating this template. That’s why it exists. =)-- F = q(E + v × B) 01:19, 30 January 2012 (UTC)

I, for one, do not object to such a change: the new template is aesthetically more pleasing. The ideal solution would of course be to have suitable TeX rendering, but we do not have that. I did not introduce that change, since I felt that having a consistent non-template basis may be better as a reference point, but did anticipate that this suggestion would be made. I would simply ask that that we simultaneously think about (but do not necessarily change) the spacing in <math>\iint</math>. We will also have to be aware of the potential for display issues with some browsers when we make changes. — Quondum^☏✎ 04:31, 30 January 2012 (UTC)

Its nice to read this - by all means don't launch into changes if in doubt/think its best not to! =) If so, the \iiint may change to $\int \!\!\!\int _{S}\,\,\!$ (that is <math>\int\!\!\!\int_S\, \,\!</math>). I can't think of any issues with the display of templates, as long as they are as above (which most formulae are in the article) there should be no problem. Let’s see by adding them to a table (again some formulae in the article are in tables):

total charges and currents	free charges and currents
$\scriptstyle S$ $\mathbf {D} \cdot {\rm {d}}\mathbf {S} =Q$	$\scriptstyle S$ $\mathbf {D} \cdot {\rm {d}}\mathbf {S} =Q$
$\scriptstyle S$ $\mathbf {B} \cdot {\rm {d}}\mathbf {S} =0$	$\scriptstyle S$ $\mathbf {B} \cdot {\rm {d}}\mathbf {S} =0$
$\oint _{C}\mathbf {E} \cdot {\rm {d}}{\boldsymbol {\ell }}={\frac {\partial }{\partial t}}$ $\scriptstyle S$ $\mathbf {B} \cdot {\rm {d}}\mathbf {S}$	$\oint _{C}\mathbf {E} \cdot {\rm {d}}{\boldsymbol {\ell }}={\frac {\partial }{\partial t}}$ $\scriptstyle S$ $\mathbf {B} \cdot {\rm {d}}\mathbf {S}$
$\oint _{C}\mathbf {H} \cdot {\rm {d}}{\boldsymbol {\ell }}=$ $\scriptstyle S$ $\left(\mathbf {J} +{\frac {\partial \mathbf {D} }{\partial t}}\right)\cdot {\rm {d}}\mathbf {S}$	$\oint _{C}\mathbf {H} \cdot {\rm {d}}{\boldsymbol {\ell }}=$ $\scriptstyle S$ $\left(\mathbf {J} +{\frac {\partial \mathbf {D} }{\partial t}}\right)\cdot {\rm {d}}\mathbf {S}$

If we do come across any severe problems (unlikley), the transform to the new template can be decommissioned. -- F = q(E + v × B) 08:28, 30 January 2012 (UTC)

I will not do so for this article, but definitley agree the template should be added. I added it to the continuity equation article recently, and it seems fine. --Maschen (talk) 12:29, 7 February 2012 (UTC)

quaternions

The article should mention that the quaternion form of the equations is much simplified. The subject of quaternions is frequently neglected in mathematics. It should not be. — Preceding unsigned comment added by Skysong263 (talk • contribs) 02:49, 25 January 2012 (UTC)

As I understand it, the quaternion approach is absorbed by and superseded by the geometric algebra approach, which is mentioned in the article. As such, a mention of quaternions would only have historical relevance, and would have to be sourced. — Quondum^☏✎ 07:23, 25 January 2012 (UTC)

clarification (I hope)

I do not intend to extensivley edit this article (honest). However the section Table of 'microscopic' equations was frutterly confusing to understand what was supposed to be said about the integrals - so I tried to clarify the prosy wording. (Sureley I am not the only one???). Also added the oiint template there also for good measures (they are closed surfaces right? of course - as stated in the table just above them). -- F = q(E + v × B) 15:57, 25 February 2012 (UTC)

Another (minor) point on clarification, is it the right approach to use the "boundary notation" $\partial V=S,\,\partial S=C$ right from the beginning? I guess its no problem with explanation, but maybe readers can grab the concept better by just using S for surface and C for curve - more intuitive? less confusing? It doesn't matter, just pointing the obvious out.-- F = q(E + v × B) 16:17, 25 February 2012 (UTC)

$\mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P}$ and :$\mathbf {B} =\mu _{0}(\mathbf {H} +\mathbf {M} )$ are not called constitutive relations. Only $\mathbf {D} =\varepsilon \mathbf {E}$ and $\mathbf {B} =\mu \mathbf {H}$ are called constitutive relations. Big difference...The former two equations are always true by definition, the second two equations are empirical assumptions about materials, assumptions that may or may not be accurate.

The part you were complaining about--I think about whether S and V are changing in time--is altogether unnecessary. I deleted it. We can just say that S and V are not changing in time. The equations with that restriction are still correct and complete. "What happens if S and V might change?" is an interesting and worthwhile homework problem, but not at all essential for understanding Maxwell's equations.

I don't think many readers will guess that C means curve and S means surface, and it is also risky to have the symbol S referring to a closed surface in one equation and open in another. And even if they do understand that C means curve, what curve is it?? Anyway, it's a dangerous game for readers to be guessing what the symbols mean. They are liable to guess wrong. A few strange symbols are kinda nice insofar as they encourage readers to actually scroll down and take a look at the table. I think that when they see that $\partial$ can mean "boundary of" they will say "Oh, that's a nice and useful notation, maybe I'll start using it myself." Just my opinion :-) --Steve (talk) 04:01, 26 February 2012 (UTC)

Thanks for explaination etc, though I do not recall "complaining", only indicating. did anticipate this comment would be made about the intgerals now deleted as irrelavant altogether, or objection to the boundary notation. -- F = q(E + v × B) 09:00, 26 February 2012 (UTC)

integral and differential forms

I really think it might be worth to mention that the differential forms can be obtained from the integral forms by applying the integral forms to a differential volume element dx*dy*dz. — Preceding unsigned comment added by 117.193.106.245 (talk) 09:58, 20 March 2012 (UTC)

Is this not already adequately addressed by the section Relationship between differential and integral forms? — Quondum^☏✎ 12:48, 20 March 2012 (UTC)

Some integral versions of Maxwell's equations are not sound, because they are formulated for the special case of time-independent surfaces. I'll correct them and replace them by equations that are mathematically equivalent to the differential ones. Michael Lenz (talk) 15:52, 23 March 2012 (UTC)

Well, I disagree that the old versions "are not sound". They are correct, therefore they are "sound". I also disagree that they are not mathematically equivalent to the differential ones. In fact, you replaced something with something else mathematically equivalent to it (in the sense that one can derive either version from the other).

On the other hand, I don't have any major objection to your new versions. I do have minor objections, like the fact that you didn't update the table of definitions, the fact that the new versions do not make any use of the familiar concept of magnetic flux, and the fact that it will now look more unfamiliar to many people. But these are minor and I will not personally be changing it back. --Steve (talk) 17:49, 23 March 2012 (UTC)

Steve, we already had that topic in a different form, as far as I remember, and I learned a lot from the discussions with you.

The problem with the former integral versions becomes evident, when you allow time-dependent contour lines. I think, a universally valid equation should not have any restrictions on contour lines.

The thing is that the partial derivative $\partial /\partial t$ inside the integral solely affects the magnetic flux density B, whereas the d/dt in front of the integral considers both B and A. Therefore, the two equations obviously have different meanings.

For the special case of divB=0 (which is always true for the magnetic B-field), the relation between the two terms is:

${\frac {\mathrm {\mathrm {d} } }{\mathrm {d} t}}\int \limits _{A}{\vec {B}}\mathrm {d} {\vec {A}}=\int \limits _{A}{\frac {\partial {\vec {B}}}{\partial t}}\mathrm {d} {\vec {A}}-\oint \limits _{\partial a}({\vec {u}}\times {\vec {B}})\mathrm {d} {\vec {s}},$

where ${\vec {u}}$ is the speed of the contour line at the location of $\mathrm {d} {\vec {s}}$.

The general equation (allowing divB<>0) comprises an additional term that has some integral over divB in it. Unfortunatley, I do not memorize it properly. If you are interested, I can send you a copy of "Differentiation under the integral sign" by Flanders, Harley (American Mathematical Monthly 80 (6): 615–627.). The problem is discussed in more detail, there.

The discussion is not hypothetical. Look, for example, at the following example showing unipolar induction with a conductor bar sliding along two electrically conducting rails in a constant magnetic field $\partial {\vec {B}}/(\partial t)={\vec {0}}$.

The contour line goes through the (stationary) voltmeter, the two (stationary) rails and the moving conductor bar. (The connection between the conductor bar and the rails is electrically conductive.)

For clockwise integration, the former equation yields $\oint \limits _{\partial A}{\vec {E}}\mathrm {d} {\vec {s}}=-{\frac {\mathrm {d} \Phi }{\mathrm {d} t}}=-vBL$, which is wrong, because the vorticity of the electric field is equal to zero owing to $\mathrm {curl} {\vec {E}}=-{\frac {\partial {\vec {B}}}{\partial t}}=0$.

The current equations, however, lead to the correct result $\oint \limits _{\partial A}{\vec {E}}\mathrm {d} {\vec {s}}=0$. Michael Lenz (talk) 22:38, 23 March 2012 (UTC)

Hi Michael :-)

The previous version said "(In these equations, it is assumed that S and V are not changing as a function of time.)" You obviously noticed this sentence, because you deleted it. I imagine most readers would have noticed it too. It was not subtle or hidden! What I'm saying is (1) the previous versions were correct when S and V do not change as a function of time, (2) if you assume that the previous versions were correct when S and V do not change as a function of time, then you can prove that the new versions are correct whether or not S and V change as a function of time, (3) conversely, if you assume that the new versions are correct whether or not S and V change as a function of time, you can prove that the previous versions are correct when S and V do not change as a function of time. The new versions seem "more general" but that's only a superficial appearance. :-) --Steve (talk) 13:54, 24 March 2012 (UTC)

I think the new versions are strictly more general, in the sense that the old version is a special case of the new version. The converse is not true. What one can prove is not relevant. (Just my 2¢) — Quondum^☏✎ 16:02, 24 March 2012 (UTC)

Let S₁, S₂, ... be any number of surfaces. Then:

$\sum _{i}\oint _{\partial S_{i}}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =-\sum _{i}\iint _{S_{i}}{\frac {\partial \mathbf {B} }{\partial t}}\mathrm {d} \mathbf {A}$

This is "strictly more general" than the version in the article, because the version in the article was for only one surface, and this equation I wrote is for any number of surfaces. So, should we replace the version in the article with this one?

The point of this silly example is that it is not always, in every case, sensible to write things in the most transparently and superficially generalized way. --Steve (talk) 18:57, 24 March 2012 (UTC)

Hi Steve, what I suggest is that we use the Kelvin stokes theorem as it is: We take the differential form of Faraday's law $\mathrm {curl} {\vec {E}}=-{\frac {\partial {\vec {B}}}{\partial t}}$ and insert it into the Kelvin-Stokes-theorem $\int _{A}\mathrm {curl} {\vec {E}}\mathrm {d} {\vec {A}}=\oint _{\partial A}{\vec {E}}\mathrm {d} {\vec {s}}$. That's all. The time derivative is already inside the integral, without any extra effort.

The effort is on those who want to bring it out of the integral and maintain all the information that is comprised in the differential form. To do that, you need to: a) do a confinement on the accaptable contour lines and b) bring in an additional piece of information (divB=0) that indirectly retrieves the information that you have just lost by confining the accaptable contour lines. I understand that this zig-zag line of argument is done to rescue the traditional form of Faraday's law comprising the magnetic flux. It's ok to explain it, but don't let us overemphasise it. --Michael Lenz (talk) 01:30, 25 March 2012 (UTC)

I really fail to see the point of the new content in the section Relationship between differential and integral forms. It just re-states Faraday's law (why not all of them while at it?). There was a section previously [1] about the time derivatives in/outside the surface integrals for any flux - not just magnetic flux. Even before this thread started, in the previous discussion Steve had already stated that there is no need to worry about time-dependence of surfaces at all (at the time I "complained" and tried to clarify the flux integrals for time-independent/dependent surfaces before deletion in the box below, though its not essential - I'm not fussed about that deletion), so what is the point of this new thread and sequence of changes anyway??

dead section

Here the integration regions are assumed constant, in agreement with the common definitions. Equations containing surface integrals of the form (X can be E, D or B) ${\frac {\partial \Phi _{S}{(\mathbf {X} )}}{\partial t}}=$ $\scriptstyle S(t)$ ${\frac {\partial {(\mathbf {X} )}}{\partial t}}\cdot {\rm {d\mathbf {S} }}$ are generally true for any closed surface, time-dependent (arbitrary shape varies with time) or time-independent (arbitrary shape but fixed for calculation). Equations of the form ${\frac {\partial \Phi _{S}{(\mathbf {X} )}}{\partial t}}={\frac {\mathrm {d} }{\mathrm {d} t}}$ $\scriptstyle S$ $\mathbf {X} (t)\cdot {\rm {d\mathbf {S} }}$ are true only for time-independent surfaces.

I'm curious why the new addition is "allowed" and the previous one was not. Its enough to just state the div and curl theorems link the two forms, if and really keen, use a show/hide box for a derivation, but its not essential.

Furthermore some of the equations are technically wrong (though of course they are just typos): there are no dots for the dot products in some of the integrals, i.e. should be B•dA instead of BdA since two vectors juxtaposed (outer product) like this is a dyadic (as in AB i.e. rank-2 tensor). The same thing has happened above - where are the dots for the dot products as in

$\int _{A}\mathrm {curl} {\vec {E}}\mathrm {d} {\vec {A}}=\oint _{\partial A}{\vec {E}}\mathrm {d} {\vec {s}}$.

which should be

$\int _{A}\mathrm {curl} {\vec {E}}\cdot \mathrm {d} {\vec {A}}=\oint _{\partial A}{\vec {E}}\cdot \mathrm {d} {\vec {s}}$??

(If this was me - I'd definitley be told off, not that I care). Never mind - I have no intensions of deleting/reverting anything, just curious. F = q(E+v×B) ⇄ ∑_ic_i 13:19, 25 March 2012 (UTC)

I'll insert the dots now. F = q(E+v×B) ⇄ ∑_ic_i 13:22, 25 March 2012 (UTC)

As I mentioned, I don't have any major objection with the versions where there is a partial derivative inside the integral, even if it is not my personal preference. I will not keep arguing about this, it's fine. I do, however, object strongly to having both versions and describing their relationship in such detail, because I find it an off-topic diversion interrupting a reader's train of thought during the most important part of the article. We should just present one primary version, and people can always click Faraday's law of induction to learn about other ways the equation can be written and why. I will go ahead and delete this discussion (just like I recently did in F=q(E+v^B)'s version)...we can discuss if y'all object... :-) --Steve (talk) 20:00, 25 March 2012 (UTC)

I had no implication or intension to add the above box back, or delete things, or even to continue much more with this discussion. I only indicated that the initial version of the article was slightly better. Again not fussed at all about when you deleted my own modification, if anything I slightly agree.

Btw Mike Lenz - you are acting in good faith in trying to improve the article and I do appreciate your efforts, so if I'm de-motivating just forget me. F = q(E+v×B) ⇄ ∑_ic_i 20:21, 25 March 2012 (UTC)

I have no objections against deleting the discussion about the time derivative. It is not the main topic of the article, and since the equations are correct, there is no need for this kind of discussion. If everybody agrees, I would delete the comment about the "unchanging volumes/surfaces" in the Maxwell's_equations#Table_of_terms_used_in_Maxwell.27s_equations Table of Terms.

About the notation: In Germany (as far as I know), two vectors in a series ${\vec {a}}{\vec {b}}$ always means the scalar product ${\vec {a}}\cdot {\vec {b}}$ (mathematicians, however, prefer <a,b>), whereas the cross product ${\vec {a}}\times {\vec {b}}$ requires the $\times$-operator. I did not know that the conventions are different in the US or GB. To avoid any ambiguity, I added all the missing dots that I found. :-) Michael Lenz (talk) 18:12, 27 March 2012 (UTC)

You did indeed include extra dots where I didn't notice, so good job! =) Yes - over here in the UK we use dots in vector calculus, but in linear algebra the round (,) or angular <,> brackets are used for the more general and abstract inner product (same the USA I guess). I didn't know you were using a different convention in Germany, so apologies for making the accusation of typos. =( Just in case you haven’t already seen, there are guidelines to maths on WP: WP:MOSMATH, just flick through when you can, else never mind. It will appeal to other editors that they appear to be typos, so please be a little careful. F = q(E+v×B) ⇄ ∑_ic_i 19:35, 27 March 2012 (UTC)

Simplify

In the subsection Covariant formulation of Maxwell's equations, its hard to actually see Maxwell's equations in

the current format:

"With these ingredients, Maxwell's equations can be written: $\mu _{0}\,J^{\beta }\,=\,{\partial F^{\beta \alpha } \over {\partial x^{\alpha }}}\,{\stackrel {\mathrm {def} }{=}}\,\partial _{\alpha }F^{\beta \alpha }\,{\stackrel {\mathrm {def} }{=}}\,{F^{\beta \alpha }}_{,\alpha }\,$ and $0=\partial _{\gamma }F_{\alpha \beta }+\partial _{\beta }F_{\gamma \alpha }+\partial _{\alpha }F_{\beta \gamma }\ {\stackrel {\mathrm {def} }{=}}\ {F_{\alpha \beta }}_{,\gamma }+{F_{\gamma \alpha }}_{,\beta }+{F_{\beta \gamma }}_{,\alpha }.$ The first tensor equation is an expression of the two inhomogeneous Maxwell's equations, Gauss's law and Ampère's law with Maxwell's correction. The second equation is an expression of the two homogeneous equations, Faraday's law of induction and Gauss's law for magnetism. The second equation is equivalent to $0=\epsilon ^{\delta \alpha \beta \gamma }{F_{\beta \gamma }}_{,\alpha }$ where $\,\epsilon ^{\alpha \beta \gamma \delta }$ is the contravariant version of the Levi-Civita symbol, and ${\partial \over {\partial x^{\alpha }}}\ {\stackrel {\mathrm {def} }{=}}\ \partial _{\alpha }\ {\stackrel {\mathrm {def} }{=}}\ {}_{,\alpha }\ {\stackrel {\mathrm {def} }{=}}\ \left({\frac {1}{c}}{\frac {\partial }{\partial t}},\nabla \right)$ is the 4-gradient. In the tensor equations above, repeated indices are summed over according to Einstein summation convention. We have displayed the results in several common notations. Upper and lower components of a vector, vα and vα respectively, are interchanged with the fundamental tensor g, e.g., g = η = diag(−1, +1, +1, +1)."

Wouldn't it be better to write

this proposed format:

"With these ingredients, Maxwell's equations can be written: ${\dfrac {\partial F^{\beta \alpha }}{\partial x^{\alpha }}}=\mu _{0}J^{\beta }$ ${\dfrac {\partial F_{\alpha \beta }}{\partial x^{\gamma }}}+{\dfrac {\partial F_{\alpha \gamma }}{\partial x^{\beta }}}+{\dfrac {\partial F_{\beta \gamma }}{\partial x^{\alpha }}}=0$ Notice the cyclic permutation of indices in the second equation. The partial derivatives (constituting the 4-gradient) can also be written ${\dfrac {\partial }{\partial x^{\alpha }}}\,{\stackrel {\mathrm {def} }{=}}\,\left({\frac {1}{c}}{\frac {\partial }{\partial t}},\nabla \right)\,{\stackrel {\mathrm {def} }{=}}\,\partial _{\alpha }\,{\stackrel {\mathrm {def} }{=}}\,{}_{,\alpha }$ for simplicity (the comma is part of the index notation). In the tensor equations above, repeated indices are summed over according to Einstein summation convention. Upper and lower components of a vector, vα and vα respectively, are interchanged with the Minkowski metric tensor g, e.g., g = η = diag(−1, +1, +1, +1). The first tensor equation is an expression of the two inhomogeneous Maxwell's equations, Gauss's law and Ampère's law with Maxwell's correction. The second equation is an expression of the two homogeneous equations, Faraday's law of induction and Gauss's law for magnetism. The second equation is equivalent to $\epsilon ^{\delta \alpha \beta \gamma }F_{\beta \gamma \,,\alpha }=0$ where $\epsilon ^{\alpha \beta \gamma \delta }$ is the contravariant version of the Levi-Civita symbol."

to reduce the cluttered combination of equations and not repeat the alternative notations to $\partial x_{\alpha }$? The partial derivatives written as ${\dfrac {\partial }{\partial x^{\alpha }}}$ will be more familiar than $\partial _{\alpha }$, since the first is clearer what is being differentiated wrt what variable; the latter may be intimidating (maybe I'm wrong...). Also why "fundamental tensor"? Not sure if its standard terminology (I havn't come across it), perhaps it is, but why not be more direct and refer to the definitley common name "Minkowski metric tensor"?

One minor point is in the LaTeX coding, there were too many curly brackets { }, not that it affects the display of the equations, but the markup should be kept as simple as possible else editors can make typesetting errors and become lost in a flood of LaTeX syntax. In my proposed form they have been removed to the minimum number needed. F = q(E+v×B) ⇄ ∑_ic_i 10:13, 29 March 2012 (UTC)

Actually I'll just do it, why not? F = q(E+v×B) ⇄ ∑_ic_i 10:24, 29 March 2012 (UTC)

Other minor edits:

• add something moderatley mnemonical, ${\begin{array}{rc}&\scriptstyle {\alpha \,\,\longrightarrow \,\,\beta }\\&\nwarrow _{\gamma }\swarrow \end{array}}$ to the covairant form section (for the permuted indices)
• reduce more unnecessary LaTeX syntax
• exchange order of J and F 4-vectors/tensors, maybe more logical since $J^{\mu }$ is simpler, also changed 3-vector ${\vec {J}}$ to bold J for consistency with rest of article
• as ever - add colour boxes to grab the reader's attention to the main statements from the surrounding mathematical prose
• change a hyphen to a minus sign

=) F = q(E+v×B) ⇄ ∑_ic_i 11:08, 29 March 2012 (UTC)

Ok, so I ended up making more edits than originally intended. Thats all (for now). F = q(E+v×B) ⇄ ∑_ic_i 11:37, 29 March 2012 (UTC)

Nice. If it's OK, I made the notation discussion even shorter by eliminating altogether one of the specialized notations for derivatives. I also reorganized a bit and added another box. --Steve (talk) 18:23, 29 March 2012 (UTC)

Thanks for shortening the explanations without loss of continuity, though I thought it would help to keep just one line ${\dfrac {\partial }{\partial x^{\alpha }}}\,{\stackrel {\mathrm {def} }{=}}\,\left({\frac {1}{c}}{\frac {\partial }{\partial t}},\nabla \right)\,{\stackrel {\mathrm {def} }{=}}\,\partial _{\alpha }\,{\stackrel {\mathrm {def} }{=}}\,{}_{,\alpha }$, since readers who know partial differentiation will immediatley know the first - it will also allow them to realize other notations. It doesn’t seem to be at the d'Alembertian operator article either, but almost is at Four-gradient. I'll paste this expression in 4-gradient, since you added the link to there so notation shouldn't be a problem. You don't mind if I change the box to blue, analogous to the previous potential equations? =) F = q(E+v×B) ⇄ ∑_ic_i 20:08, 29 March 2012 (UTC)

It is over there now, better there than here. F = q(E+v×B) ⇄ ∑_ic_i 20:58, 29 March 2012 (UTC)

Differential Forms

Hi all,

I've changed the numerical factors in the differential forms section to maintain consistency with the pages regarding the Hodge dual and differential form. We start with the gauge curvature in flat space (it's easy to generalize to Lorentzian metrics):

$\mathbf {F} ={\frac {1}{2}}F_{\mu \nu }dx^{\mu }\wedge dx^{\nu }=-E_{x}dt\wedge dx-E_{y}dt\wedge dy-E_{z}dt\wedge dz+B_{z}dx\wedge dy+B_{x}dy\wedge dz+B_{y}dz\wedge dx$

The Hodge dual adds NO numerical prefactors (since we are in an orthonormal basis):

$*\mathbf {F} ={\frac {1}{2}}F_{\mu \nu }dx^{\mu }\wedge dx^{\nu }=E_{x}dy\wedge dz+E_{y}dz\wedge dx+E_{z}dy\wedge dx+B_{z}dt\wedge dz+B_{x}dt\wedge dx+B_{y}dy\wedge dt$

The exterior derivative also adds NO numerical prefactors. At the end we will have twelve (dependent) terms. However, we will also be in a four dimensional space. Thus to each form will be assigned three terms. These are precisely the three terms needed to equate to each component of the current. For example a term will arise of the following form

$(\partial _{x}E_{x}+\partial _{y}E_{y}+\partial _{z}E_{z})dx\wedge dy\wedge dz$

We would like to set this equal to $\rho \,dx\wedge dy\wedge dz$, which would be the zero component of $\mathbf {J}$. The original term written was $\epsilon _{\mu \nu \lambda \sigma }j^{\mu }dx^{\nu }\wedge dx^{\lambda }\wedge dx^{\sigma }$ The zero component then reads

${\begin{aligned}\epsilon _{0\nu \lambda \sigma }dx^{\nu }\wedge dx^{\lambda }\wedge dx^{\sigma }&=\epsilon _{0123}dx^{1}\wedge dx^{2}\wedge dx^{3}+\epsilon _{0132}dx^{1}\wedge dx^{3}\wedge dx^{2}\\&\quad +\epsilon _{0213}dx^{2}\wedge dx^{1}\wedge dx^{3}+\epsilon _{0231}dx^{2}\wedge dx^{1}\wedge dx^{3}\\&\quad +\epsilon _{0312}dx^{3}\wedge dx^{1}\wedge dx^{2}+\epsilon _{0321}dx^{3}\wedge dx^{2}\wedge dx^{1}\\&=6dx^{1}\wedge dx^{2}\wedge dx^{3}\end{aligned}}$

i.e., there's over-counting going on. To retain the beauty of $\partial _{\mu }F^{\mu \nu }=j^{\mu }$, the factor of $1/6$ is needed.

Note: I do admit minus signs may be off in the above, however, this does not affect the counting.

Eric.m.dzienkowski (talk) 04:25, 12 April 2012 (UTC)

macro/microscopic equations?

Any reason the tables in Maxwell's "macroscopic" equations and Maxwell's "microscopic" equations are repeated from the frontline summary? The titles "microscopic" and "macroscopic" in the first summary are enough for a reader to know where and what they are... F = q(E+v×B) ⇄ ∑_ic_i 20:55, 14 May 2012 (UTC)

I personally feel that this is excessive repetition; the article generally suffers from this. — Quondum^☏ 06:06, 15 May 2012 (UTC)

Lets just delete them for now. F = q(E+v×B) ⇄ ∑_ic_i 08:09, 15 May 2012 (UTC)

Overlap between section: Alternative formulations of Maxwell's equations and Maths of EM field

See Wikipedia talk:WikiProject Physics here

I would propose this summary in Maxwell's equations, and condensing all the Alternative formulations of Maxwell's equations sections into one shorter section after moving (only if so). The equation summary could be like this:

Formulation	Maxwell's equations
Algebra of physical space	$\left({\frac {1}{c}}{\dfrac {\partial }{\partial t}}+{\boldsymbol {\nabla }}\right)F=\mu _{0}cJ$
Geometric algebra	$\nabla F=\mu _{0}cJ$
Differential forms	$\mathrm {d} \mathbf {F} =0$		$\mathrm {d} \,{*\mathbf {F} }=\mathbf {J}$
Tensor calculus (fields)	${\dfrac {\partial F_{\alpha \beta }}{\partial x^{\gamma }}}+{\dfrac {\partial F_{\gamma \alpha }}{\partial x^{\beta }}}+{\dfrac {\partial F_{\beta \gamma }}{\partial x^{\alpha }}}=0$		${\dfrac {\partial F^{\beta \alpha }}{\partial x^{\alpha }}}=\mu _{0}J^{\beta }$
Vector calculus (fields)	$\nabla \cdot \mathbf {B} =0$	$\nabla \times \mathbf {E} +{\frac {\partial \mathbf {B} }{\partial t}}=0$	$\nabla \cdot \mathbf {E} ={\frac {\rho }{\epsilon _{0}}}$	$\nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}$
Vector calculus (potentials)	identites		$\nabla ^{2}\varphi +{\frac {\partial }{\partial t}}\left(\mathbf {\nabla } \cdot \mathbf {A} \right)=-{\frac {\rho }{\varepsilon _{0}}}$	$\left(\nabla ^{2}\mathbf {A} -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}\right)-\mathbf {\nabla } \left(\mathbf {\nabla } \cdot \mathbf {A} +{\frac {1}{c^{2}}}{\frac {\partial \varphi }{\partial t}}\right)=-\mu _{0}\mathbf {J}$
QED, vector calculus (potentials)	identites		$\nabla ^{2}\varphi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\varphi }{\partial t^{2}}}={\frac {1}{\varepsilon _{0}}}e\psi ^{\dagger }\psi$	$\nabla ^{2}\mathbf {A} -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}=\mu _{0}e\psi ^{\dagger }{\boldsymbol {\alpha }}\psi$
Tensor calculus (potentials)	Lorenz gauge:      $\Box A^{\mu }=\mu _{0}J^{\mu }$

(reading up the table are the accumulations of equations which become collected together). Nothing will be done at this point. Any other opinions? F = q(E+v×B) ⇄ ∑_ic_i 08:21, 15 May 2012 (UTC)

Continuing discussion from WT:Physics I second condensing that section into a more manageable form and keeping the bulk of the stuff in the other article. However, I'd prefer a brief (few para.) summary to a table, in this article. I'll try to work on it as and when I find time. SPat ^talk 17:39, 15 May 2012 (UTC)

Given that there are many forms of the equations, I hoped the table would collect everything together. This article could summarize for the reader the various forms the equations take, the other states every equation individually within the detailed mathematical prose (it does concentrate more on Maxwell's equations, without extension to the Lorentz force, continuity equation or the EM field in general). Certainly a few paragraphs should remain for this article though, but I shouldn't think they replace the table. Just my suggestion of course, not essential or necerssarily helpful. F = q(E+v×B) ⇄ ∑_ic_i 21:19, 15 May 2012 (UTC)

I prefer the direct correspondence made by the table between Maxwell's equations in the various forms. If they are separated, this is not so clear. No elaboration is needed, as this is available in the linked main article. So my preference is for F=q(E+v^B)'s suggestion. As I've commented elsewhere, the GA equation would be $\nabla F=\mu _{0}cJ$, not what is given in the table. This is easily sourced. — Quondum^☏ 12:04, 16 May 2012 (UTC)

Why not both (as just done now)? F = q(E+v×B) ⇄ ∑_ic_i 12:14, 16 May 2012 (UTC)

No problem with that – after all, this is about alternatives representations. Of course, the main articles should concerned should eventually also be updated so that the same formulae can be found there (they occur in sot-so-recognizibly similar form), with the detail explanation and definition of the variables. The APS might also be more understandable as $\left({\dfrac {1}{c}}{\dfrac {\partial }{\partial t}}+{\boldsymbol {\nabla }}\right)(\mathbf {E} +i\mathbf {B} )=\mu _{0}c\left(\rho +\mathbf {J} \right)$, though I'm not sure of the correctness of this. — Quondum^☏ 12:46, 16 May 2012 (UTC)

Splendid, although I leave it to those who understand GA and APS to update these formulae. Lets see how many more opinions arise between now and tommorow. If there are no objections I have a slot of time tommorow to make the first steps of reducing the alternative formalism section, inline with the currently strong consensus. =) F = q(E+v×B) ⇄ ∑_ic_i 18:00, 16 May 2012 (UTC)

I just modified the table to be "symmetric" in the potentials and fields, to remove the odd gaps and column widths. Also changed the Faraday-Maxwell equation to the homogeneous form, to match the other formulae just above and Gauss's magnetism law. F = q(E+v×B) ⇄ ∑_ic_i 18:37, 16 May 2012 (UTC)

I just took the first step in this edit. While the other article will now be tidied up, I'll leave this article for a while, anyone can edit/rewrite the section if they so desire... F = q(E+v×B) ⇄ ∑_ic_i 17:29, 17 May 2012 (UTC)

Surprisingly there wasn't much to it - just time-consuming to sort out the other article... The transfer doesn't seem to raise any problems - so it's more or less done. If anything else is to be moved perhaps the detailed material on constitutive equations to that main article, since these articles can cross link and that section isn't about Maxwell's equations, but electromagnetic properties of matter. I will not touch that subsection though, as its very well written and presumably there will be reaction from others if I did... =| F = q(E+v×B) ⇄ ∑_ic_i 22:08, 17 May 2012 (UTC)

I'm concerned that the links are inadequate. I imagine a reader scrolling down and seeing one of the equations and trying to find more information and failing, because they don't think of the "Main article" link. Not sure what to do about that ... maybe it's not a big deal. I also think more work could be done to improve the table, but it's OK as is. In other news, I drastically shortened the constitutive relations section, something I think is long overdue but others are welcome to disagree. --Steve (talk) 05:07, 18 May 2012 (UTC)

The alternative formalism section has the main article link Mathematical descriptions of the electromagnetic field, everything is in there and its the first link to notice in that section. Its good that you shortened the constitutive relation stuff though (6813 bytes lost). I'll try and make the table better soon if no-one else does... F = q(E+v×B) ⇄ ∑_ic_i 05:44, 18 May 2012 (UTC)

I can't improve the table currently in the article anymore... where are the links in the table supposed to go? They can't be linked to the matching sections in Maths of EM field, only to where the text actually says so (Wikipedia:Principle of least astonishment). The format of the collected equations was intensionally there to stay... F = q(E+v×B) ⇄ ∑_ic_i 17:23, 18 May 2012 (UTC)

surface/volume notation

If its ok with people I made the changes in this edit to the notation:

• change l / dl (letter ell) to ℓ / dℓ (curly ell) for length element, clearer
• change A/dA to S/dS for the area vector, since A conflicts with the magnetic potential (and N, not S, should be used for the Poynting vector, since N is Not used elsewhere in EM), in this way N = Poynting vector allows S = surface allowing A = magnetic potential, with no ambiguity/conflict,
• in doing so change V for volume to Ω, and surface S to Σ, to prevent further conflicts,
• remove subscripts from the current: change I_S / I_f,S to the simpler I / I_f, since its clear that the current must be flowing out of a surface due to the intricate definitions table, and this is the one-off mention of current (as opposed to current density J), there are no other different currents to deal with in the article (only current densities), so this should not be ambiguous anyway...

Hope this is fine... =| F = q(E+v×B) ⇄ ∑_ic_i 13:51, 18 May 2012 (UTC)

To me this change is unnecessary, but it is also fine. Of course if someone later changes to some other notation that is also relatively standard I will say the same thing. TStein (talk) 20:09, 18 May 2012 (UTC)

Keep up the good work!

There is not enough appreciation on wikipedia of good work only complaints when someone gets it 'wrong'. I must say that I am impressed by the work done on this article lately to get it under control. Kudos. TStein (talk) 20:13, 18 May 2012 (UTC)

You're responding to our edits - thank you! Of course there is Quondum and Sbyrnes321 to thank also! =) F = q(E+v×B) ⇄ ∑_ic_i 23:39, 18 May 2012 (UTC)

Maxwell's Equations over-specify the unknowns

The article states that Maxwell's equations overspecify the unknowns and that they contain redundancy. The reference attached to this statement to justify this clam says just about the opposite! I am changing the text to more accurately summarize the reference. 129.63.129.196 (talk) 17:20, 12 September 2012 (UTC)

129.63.129.196, thank you for your statement; can you please give the citation & page number? If the formatting is getting in the way, could you just give a text statement here, and another editor can format it for you. --Ancheta Wis   (talk | contribs) 20:42, 12 September 2012 (UTC)

Acheta -- 129.63.129.196 is presumably referring to Ref. 37.

129.63.129.196, let's start with the most basic fact: Given any plausible initial conditions and boundary conditions and charges and currents, there is only one possible E(t) and B(t) that satisfy the curl laws (Faraday's and Ampere's laws). You don't need to demand that E and B satisfy the div laws (Gauss's laws for E and B): This unique mathematical solution automatically satisfies the div laws . How do I know? Because if you take the divergence of both sides of the curl laws, you see that the div laws always have to be true, as long as you gave initial conditions in which they are true.

This basic fact is obvious and everyone universally agrees about it. You can check it in a few seconds. Ref 37 does not dispute it.

Your edit suggests that counting unknowns and counting equations is only appropriate for linear algebraic equations, not for differential equations. I don't know where you got that idea; it is perfectly appropriate for differential equations, and for the same reasons.

Ref. 37 is about how best to formulate Maxwell's equations to be solved numerically on a computer. Computers are not perfect, they have limited precision and storage. Therefore, the paper is not purely interested in whether a solution is mathematically unique. Mathematically, you can get the right answer by ignoring the div equations, but numerically, the paper argues that the best approach is to take both into account and do a least-squares finite-element method that tries to solve both at once as best as possible. (Note: This is very unusual in finite-elements. Normally, there are equal number of unknowns and equations in each step of finite elements, so all the equations in that step can be exactly satisfied. Here, there are fewer unknowns than equations, so you need least-squares.)

Ref. 37 says that this least-squares approach corresponds ultimately to solving Maxwell's equations with extra "dummy variables" (page 4). The dummy variables turn out to be zero everywhere. But by adding them into the system of equations, they can inflate the count of unknowns to equal the count of variables. Then the equations are not overdetermined anymore.

In any case, I don't think it's a problem to words like "seem overdetermined", as long as it is stated exactly what we're talking about. I just edited it, I hope you find it satisfactory! --Steve (talk) 13:56, 13 September 2012 (UTC)

I agree with you on the physics, I just think the wording can be misleading. I don't want my students to think the divergence equations in Maxwell's equations are redundant and therefore useless and should be thrown away, because they are not. The divergence equations are redundant if the initial conditions already satisfy them. But they are not completely redundant because you need them to ensure physical initial conditions. Faraday's law does not implicitly determine the divergence of the magnetic field and render the other equation redundant. Faraday's law only determines the time derivative of the divergence of the magnetic field. So, from a mathematical standpoint (maybe not a programming one), you need the divergence equations for a unique solution. This paper makes this clear:

Am. J. Phys. 48, 1071 (1980), http://dx.doi.org/10.1119/1.12289. Sorry, I don't have time to brush up on Wikipedia formatting or protocol (citing reference, etc.), so maybe somebody can digest my thoughts and tweak the wording to not be misleading. 129.63.129.196 (talk) 15:40, 13 September 2012 (UTC)

I added that reference, and made some small changes to shift emphasis, like "certain kind of redundancy" to "certain limited kind of redundancy", and "Gauss's laws can be ignored" into "Gauss's laws can be ignored (except in the initial conditions)", that kind of thing. I hope the description is at least moving in the right direction ... :-) --Steve (talk) 12:18, 14 September 2012 (UTC)

Should there be separate articles for Microscopic and Macroscopic sets of Maxwell's equations?

Please discuss. I think that we can have seperate articles which would do proper justice to the two seperate subjects. Sunshine Warrior04 (talk) 07:21, 1 September 2012 (UTC)

• Tentative support for separation – The concept of "bound charge" seems to be an engineering convenience (if I may call it that), and is probably not even rigorously definable. When does a bound charge become a free charge? Presumably the bound/free charge paradigm assumes that each remains unchanged (no ionisation etc.), or otherwise the equations might become quite complicated due to the non-conservation of each type of charge and current. Moving the treatment of the "macroscopic" case into a separate article that can deal with all this detail makes sense to me. — Quondum^☏ 08:21, 1 September 2012 (UTC)

• Oppose. Is the article too long? Maybe ... but if it is, a better solution IMO is to split off a main article for History of Maxwell's equations. Textbooks by physicists usually say that "Maxwell's equations" are the microscopic version, and textbooks for engineers always say that "Maxwell's equations" are the macroscopic version. Therefore if there is to be an article called "Maxwell's equations", it seems to me that it has to include both. Anyway, they are sufficiently intertwined in theory and practice that you can't consider yourself knowledgeable on the topic of macroscopic eqns without understanding microscopic, or vice-versa. Most of the article is indeed relevant to both (Conceptual Description, History, Solving, relation between micro and macro). I think the relation between the two is a sufficiently important aspect to devote substantial space in the main article to discussing it. If there is nevertheless a consensus that this discussion should be moved into a different article, I would rather it be structured differently ... maybe the two articles would be Maxwell's equations and Relation between microscopic and macroscopic Maxwell's equations or something like that. --Steve (talk) 13:02, 2 September 2012 (UTC)

  Note: I don't oppose your perspective, but I do expect this article to be rigorous. Can you respond to my point as to whether the macroscopic equations are correct in the context of incomplete separation between bound and unbound 4-currents? — Quondum^☏ 14:18, 2 September 2012 (UTC)

The macroscopic Maxwell equations correspond to an approximation of neutral materials as "continuous" polarizeable media (including piecewise continuous media, delta-function "media", etc.), in which case the definition of the bound charges is built into the the description of the polarizability/susceptibility of the media. This is valid for certain ranges of lengthscales (e.g. it breaks down at atomic lengthscales) and in practice any description of polarizability in only valid for certain ranges of fields (magnitudes not too large, and usually for frequencies only within a certain bandwidth). In particular, my understanding is that most macroscopic approximations of dielectrics will break down long before the fields are strong enough to actually ionize the material (although, once one has an ionized plasma, there are macroscopic approximations for that too). And one thing that is definitely "free" charge is any contribution of net charge, since the net bound charge is always zero.

Most of physical theory is an an approximation, valid in certain regimes, for a more accurate but more complicated theory; this doesn't make it less "rigorous" in the eyes of most physicists. In any case, echoing Steve's point, almost all important textbooks on Maxwell's equations cover both the microscopic and macroscopic variants, so Wikipedia should too (at least at a broad level, although of course there can be subarticles with more details on particular topics.

PS. On an unrelated note, it's a bit disappointing that apparently nowhere on Wikipedia can the relativistic formulation of the macroscopic Maxwell equations be found. This can be found e.g. in Landau's textbook, and gives you a way to treat moving macroscopic media by transforming from the rest frame to other frames. — Steven G. Johnson (talk) 20:24, 2 September 2012 (UTC)

It's a bit late to respond... Anyway there is also Griffiths' electrodynamics, p.563 (3rd edn), and this is the article Covariant formulation of classical electromagnetism in section Maxwell's equations in matter. Maschen (talk) 17:20, 21 September 2012 (UTC)

Quondum: Macroscopic and microscopic are equally "rigorous" - they're mathematically equivalent - but the microscopic equations are unique while macroscopic leave room for multiple different conventions (related to what charge is free versus bound). As long as you clearly specify a convention, you'll get the right answers and everything is fine, but you shouldn't expect that only one value of D or H or ε or μ is the only possible correct value; someone else can use a different convention and get a different value. Experts have always been aware that the macroscopic equations require you to be specific and explicit about which charge is free versus bound, in cases where it's not obvious. Two examples that spring to my mind are the value of D inside a non-centrosymmetric crystal and the description of AC current. --Steve (talk) 13:08, 3 September 2012 (UTC)

Oppose: Keep the micro/macroscopic formulations together, it’s easier to explain them together, and to compare them. Maschen (talk) 18:20, 4 September 2012 (UTC)

Delete monopoles section and move the statements + links + refs to the lead?

This article should concentrate on the non-monopole equations. The same section is in the monopoles article, everything is covered there. There would be no loss of continuity if:

these statements were moved to the end of the lead (or after introducing them first time in diff/int form)

"Maxwell's equations posit electric charge, but not magnetic charge, which has never been observed and may not exist. Nevertheless, Maxwell's equations can be generalized to include non-zero magnetic charges densities and currents, which become fully symmetric under interchange of electric and magnetic fields; see magnetic monopoles for detail. If every particle has the same ratio of electric to magnetic charge, then an E and B field can be defined that obeys the normal Maxwell equations (having only electric charges and currents - not magnetic)."

keeping the refs and linking to the main article. It doesn't add anything to just state the monopole equations as a "preview table" then link to the article where the same table is found. Also it cuts back on extra details which are elsewhere. Opposition? Maschen (talk) 22:55, 22 September 2012 (UTC)

I support the principle of trimming excess "preview" detail and cross-article duplication, and that this article should not directly address this topic. I don't really agree with it going into the lead; it is too tangential to be introduced there. Thus, some section (akin to the "Generalizations" section of many articles) is probably still appropriate. As such, trimming the section to what is in the box here makes sense to me. — Quondum 06:32, 23 September 2012 (UTC)

What about the end of section conceptual description, since after describing the equations in words, we extend to their modifications to monopoles? I thought to just do it so we can actually see, feel free to revert or move the paragraph somewhere else. Maschen (talk) 07:29, 23 September 2012 (UTC)

That works very nicely, IMO. — Quondum 07:38, 23 September 2012 (UTC)

I like it too. I just edited it a bit, hope that's OK. One thing was, I deleted about duality transformations. Not wrong, but I think it was too brief to be understandable. People are accustomed (for good reason) to believing that electrons have electric charge and the earth has a magnetic field, not the other way around. If I said "I believe that the electrons have both electric and magnetic charge", you could correctly respond to me, "No, you're wrong!". Therefore when you propose that people are free to redefine what is electric and what is magnetic, it could cause confusion ... anyway I think it's too subtle for one sentence. (In the magnetic monopole article OTOH a discussion of duality transformations is sorely missed!) --Steve (talk) 13:07, 23 September 2012 (UTC)

The statement about an arbitrary linear transformation provided, to me, a major epiphany. Perhaps it should be included in Magnetic monopole. It was saying that it was not simply a duality, but rather a U(1) symmetry, broken only by a constraint on the charge type. (I'm not disagreeing with the deletion.) — Quondum 13:25, 23 September 2012 (UTC)

Yes, "duality transformation" is the technical term for the continuous U(1) symmetry that you're referring to. I think you are assuming that the word "duality" can only refer to discrete things, but that's not the case here. (Sorry in advance if I'm not remembering this correctly ... it's discussed in Jackson's textbook, the section was cited by Maschen. But I don't have the book on hand right now.) Yes, this should be discussed in the magnetic monopole article. --Steve (talk) 18:23, 23 September 2012 (UTC)

...I didn't add any reference to the shortened paragraph, someone else did, and do not have Jackson's book, all refs are intended to be preserved. The sentence before Jackson's citation was:

"Further, if every particle has the same ratio of electric to magnetic charge, then an E and a B field can be defined that obeys the normal Maxwell's equation (having no magnetic charges or currents) with its own charge and current densities<ref>This is known as a duality transformation. See {{cite book|author=J.D. Jackson|title=Classical Electrodynamics|edition=3rd|chapter=6.12|isbn=0-471-43132-X}}.</ref>"

and I only trimmed that slightly... anyway thanks for your edits. Maschen (talk) 19:02, 23 September 2012 (UTC)

Oh, I didn't realize that it was already there. I guess I haven't read that section in a while. :-) --Steve (talk) 20:00, 23 September 2012 (UTC)

Okay, I've learned something now. You were right, I had "duality" pegged as a discrete exchange of some sort. — Quondum 20:59, 23 September 2012 (UTC)

Update: I added it as a new section in Magnetic monopole. --Steve (talk) 19:36, 24 September 2012 (UTC)

Magnetic fields do not induce Electric fields

This is a widespread errorneous interpretation of the Maxwells equations, the EM field is one object, the faraday tensor, its components are frame dependant. — Preceding unsigned comment added by 123.118.116.112 (talk) 12:50, 18 October 2012 (UTC)

Which part of the article are you unhappy about? I guess it's the part that says, "Faraday's law describes how a time varying magnetic field creates ("induces") an electric field." ... is that correct? --Steve (talk) 20:31, 18 October 2012 (UTC)

Faraday did not discover electromagnetic induction. http://www.ivorcatt.co.uk/x29j.htm . He discovered Crosstalk http://www.ivorcatt.co.uk/x0305.htm . Ivor Catt, 1 March 2014 — Preceding unsigned comment added by 86.164.171.64 (talk) 21:50, 1 March 2014 (UTC)

intro paragraph on space and time

This paragraph was just added to the intro by User:RogierBrussee

Underlying the Maxwell equations is the concept of space and time. The "classical" Maxwell equations can be considered as a special form of more symmetric equations defined on four dimensional space-time adapted to a split in space and time. These four dimensional equations are also called Maxwell equations. They are widely used in high energy and gravitational physics because they are manifestly invariant under Lorentz and even, when properly formulated, general coordinate transformations. In this formulation, the Maxwell equations restrict the Faraday tensor that combines the electric and magnetic field and relate it with the four-current that combines charge and current. The four dimensional invariance properties and four dimensional space-time formulation of the Maxwell equations were very important for the development of special and general relativity.

The "more symmetric equations defined on four dimensional space-time" is, I presume, ${\frac {\partial F^{\alpha \beta }}{\partial x^{\alpha }}}=\mu _{0}J^{\beta }$ and $\epsilon ^{\alpha \beta \gamma \delta }{\frac {\partial F_{\alpha \beta }}{\partial x^{\gamma }}}=0$. Right? It seems to me that these are exactly equivalent to "the "classical" Maxwell's equations" ... they are not a generalization, just a restatement using different variables and notation. Do you agree, RogierBrussee?

Moreover, the covariant formulation of the equations were not particularly important in the development of special relativity. The original formulation of the equations were the important ones for the historical development. How do I know? Because the covariant ones could not have possibly been written down until special relativity was already pretty well understood!!

Overall, I don't see how anything in this paragraph improves on the previous version ("It is often useful to write Maxwell's equations in other forms; these representations are still formally termed "Maxwell's equations". A relativistic formulation in terms of covariant field tensors is used in special relativity, while in quantum mechanics, a version based on the electric and magnetic potentials is preferred."). Except the mention that Maxwell's equations were important for the development of special relativity ... That is worth saying, but only requires one sentence. Other than that, it seems to me to add too much length and jargon and detail to the intro. I propose to restore the previous version --Steve (talk) 13:49, 20 October 2012 (UTC)

Agreed. SR (1905) came before the Faraday tensor (~1908) didn't it? [2] Anyway the intro needs to only say there are other convenient forms without any detail. This added paragraph may be good for Mathematical descriptions of the electromagnetic field or suchlike, not so much here. Maschen (talk) 15:18, 20 October 2012 (UTC)

Indeed I mean the equations ${\frac {\partial F^{\alpha \beta }}{\partial x^{\alpha }}}=\mu _{0}J^{\beta }$ and $\epsilon ^{\alpha \beta \gamma }{\frac {\partial F_{\alpha \beta }}{\partial x^{\gamma }}}=0$. I personally prefer the differential form formuation over the index notation but my personal taste is irrelevant. The heart of the matter is that the 4 dimensional equations are simpler, express the physical content of the theory more concisely, and do not incorporate a choice (the split in space and time) thereby revealing their 4 dimensional symmetry. From a conceptual point of view, the conventional 3D Maxwell equations have the same status as an expression in spherical coordinates, that is a convenient form to use when the physics has special symmetry. Thus, the 3d Maxwell equations are better though of a reformulation of the 4d equations rather than vice versa, in exactly the same sense that an equation in spherical coordinates is a reformulation of a vector equation rather than vice versa. This was the point I made. Of course, being earth bound creatures, rather than zipping through the universe at relativisic speeds, are used to treating space time differently and use plenty of non relativistic physics, but that is just our own restriction. For the same reason, historically, Maxwell historically obviously wrote down the 3d equations first. However, wherever relativistic formulations of the physics are used (high energy physics, quantum field theory, graviational physics) physicsts routinely call the 4D equations the Maxwell equations. RogierBrussee (talk) 09:24, 26 October 2012 (UTC)

Of course I agree that people doing relativistic physics are probably well-advised to use the formulation of Maxwell's equations that is written in an obviously-4D-symmetric way. And people doing non-relativistic quantum mechanics are probably well-advised to use the formulation written in terms of A and φ in the Coulomb gauge. And electrical engineers are probably well-advised to use the version with E, B, D, H as taught to undergrads.

I don't think you've justified your belief that the 4D formulation is so important that it warrants a large paragraph of discussion, as the third paragraph of the article. The intro, by the way, spends just two words mentioning the relation of Maxwell's equations to optics, two words about circuit theory, zero mention of "radios", "electromagnetic waves", "gauge invariance", Gauss or Hertz or Heaviside, "CGS units", etc. So I think it is far out of proportion to devote this large paragraph to the fact that Maxwell's equations can be formulated in an obviously-4D-symmetric way. (Since we live in a Lorentz-invariant universe, every correct law of physics can be formulated in an obviously-4D-symmetric way, including Newton's laws, conservation of energy, etc. So I don't think this fact is quite so remarkable!)

I think the relativistic formulation warrants one and only one sentence in the introduction. That was the status quo before your edit. I am unlikely to object if you edit that one sentence to your liking, as long as you don't significantly increase the length. More details are already in the body of the article and a lot more detail is in dedicated articles like Covariant formulation of classical electromagnetism. :-) --Steve (talk) 14:21, 26 October 2012 (UTC)

Look I really don't want to start an edit war, but you miss the point: the question is not why the 4D equations of a 4D Lorentz invariant phenomenon like EM warrant mention in the introduction but why the 3D equations do (and of course they do for historic and practical reasons). And no, you cannot formulate Newtons law in any sort of reasonable Lorentz invariant way because it implies action at a distance. And yes, electromagnetic waves should be mentioned in the introduction. And come to mention it, why does the intro even mention things like quantum entanglement, which is an important subject but clearly outside the scope of the Maxwell equations. So I guess the main reason I care is that I think it is a bad idea that after pages and pages of differences between CGS and SI units and microscopic and macroscopic formulations, integral and PDE formulations the main body of the article relegates the fundamental 4 dimensional formulation to "alternative" formulation and mathematical curiosity, mixing it with alternative 3D formulations, and the coupling of (classical) Dirac electrons with an external EM field that satisfies Maxwell equations i.e. not a formulation of the Maxwell equations at all).

What one can argue about, of course, is whether one has to mention the Faraday tensor in the introduction. RogierBrussee (talk) 19:27, 26 October 2012 (UTC)

Removing quantum entanglement from the intro is an excellent idea, I just did it. :-)

I will try to dramatically shorten your paragraph without taking away any important points. Here goes:

"Maxwell's equations are Lorentz-invariant, i.e. consistent with special relativity. In fact, investigation of the symmetry properties of Maxwell's equations helped Einstein formulate special relativity."

How did I do? I'm sure you think I ruined it. But the parts I left out, it seems to me, are so brief and jargony as to be incomprehensible to someone who isn't already familiar with those things. For example, even a long paragraph is insufficient to teach someone what is special relativity and why is it exciting. :-)

You answered the question of why the 3D equations are important: Very few people do covariant tensor calculations compared to how many people want to know how a transformer works or an EM wave or a transmission line. Also, all measurements, not to mention people's intuitions, are set in a particular inertial reference frame. It seems sensible to me to ensure that people understand the physics of a single reference frame first, and only later try to understand the relation between different reference frames. In other words, first learn the traditional Maxwell's equations, and second explore the relation to special relativity and four-dimensional spacetime. Usually special relativity is taught in that kind of way in my experience. :-)

In the Maxwell's equations and relativity section, there is certainly room (IMHO) for a few sentences like

"After special relativity was developed, it became clear that Maxwell's equations can be written in a way that is far more concise and convenient, by taking advantage of their Lorentz symmetry, i.e. writing everything in terms of four-dimensional tensors in four-dimensional spacetime, etc. etc."

In fact, in previous years, this article has been organized differently to group together (1) the historical discussion of Maxwell's equations helping in the discovery of SR, (2) the covariant formulation of Maxwell's equations, and also (3) a brief conceptual discussion of how an electric force in one frame can be a magnetic force in another frame. Without going all the way back to that previous organization, it might be possible to expand the Maxwell's equations and relativity section a bit with another paragraph or two along those lines. :-) --Steve (talk) 13:51, 27 October 2012 (UTC)

Rogier -- I like your newer round of intro edits :-) --Steve (talk) 23:44, 2 November 2012 (UTC)

Constitutive relations already has its own article

No objections to the additions to the constitutive relations (this article) section, but recently it was trimmed and transferred to Constitutive relations (main article). Why not just add the context there? Let's not allow this article to pile up with information again which could be in the main articles... (and eventually become unreadably long, not that I'm saying this is currently happening, the additions are in good faith, correct, etc.) M∧Ŝc²ħεИ_τlk 11:33, 23 December 2012 (UTC)

Rogier, apart from the obvious 'spac-time' typo, I can't understand what you are trying to say in your newest contribution to this section. It seems to me that nonlinear optics items like Kerr cells and Pockels cells, Hall effect devices, or the highly modulated magnetic devices, might also be subjects of this section. Or perhaps this discussion should be in the constitutive relations article. --Ancheta Wis   (talk | contribs) 12:07, 24 December 2012 (UTC)

I radically cut the constitutive relations section down. And did a bit of editing on the main article. Does that satisfy people's concern?

RogierBrussee (talk) 20:10, 1 January 2013 (UTC)

Looks better now, thanks. You could have transferred the non-linear materials subsection you extended to the constitutive relations article directly and just linked from here to there, but it's not essential right now... M∧Ŝc²ħεИ_τlk 09:43, 2 January 2013 (UTC)

About the sectioning...

I moved [3] the "potential formulations" material back into the main section "alternative formulations"; why have an entire section on potential forms then a compact section which says "alternative formulations" but only discusses the tensor/differential form equations... and fields, and with the potential equations there anyway? The potential equations were duplicated... M∧Ŝc²ħεИ_τlk 11:06, 2 January 2013 (UTC)

Well I was going to remove them from the alternative formulation section (which I don't like, allhough I do like having an overview table), but went to make a cup of coffee first.... The reason is I want to expand on gauge equivalence and gauge fixing, because they are somewhat subtle and quite essential even in an overview article, in my opinion. I can live with the current situation though.

RogierBrussee (talk) 12:25, 2 January 2013 (UTC)

It seems the section on alternative formulations is there so summarize them, linking to the main articles, achieved by a table of formulae with explanations following it...

Sadly the geometric algebra formulae have been removed [4][5] as "obscure", but I won't reinstate those, to prevent an edit war... M∧Ŝc²ħεИ_τlk 11:06, 2 January 2013 (UTC)

I really don't think that the geometric algebra formulation is important enough to warrant inclusion. The geometric algebra community is good at marketing, but $\nabla F=J$ is just a really confusing way to write $(d+\delta )F=J$(as a 1 form). Also using a Clifford algebra (which is what a geometric algebra is) mixes in the metric everywhere, and is bad if you want to understand invariance properties of the equations. More to the point, I am unaware of important applications and use of this Clifford algebra point of view by people outside of the geometric algebra community that would make this an important point of view to know. I did add a see also though.

This raises the legitimate question why do we have to keep forms (you probably all know the below, but I think it is good to have on the table). Mathematicians have been using forms, the exterior derivative and Hodge duality, as THE right formulation of higher dimensional generalisation of vector calculus since the forties (younger physicists are following since the 90's) because they behaves well with respect to curvilinear coordinate change, and therefore live on manifolds (with a metric if you need Hodge duality). The Faraday tensor is an alternating rank 2 tensor which is a two form by definition. The Maxwell equations are written completely naturally in terms of exterior derivative and Hodge duality, and this cleanly isolates the dependence of the Maxwell equations on the metric and gives you a generally covariant formulation for free. Finally, if you think of EM as a U(1) gauge theory and the Faraday tensor as the curvature of a connection, it comes out as a 2-form. In fact the form version of the 3 dimensionsional vector calculus formulation is increasingly used by engineers and numerical analysists and I would argue that it does warrant inclusion.

RogierBrussee (talk) 12:25, 2 January 2013 (UTC)

The reason is that some time ago the article did have separate sections, on each formalism, and over time people would keep piling material up which should be in the other main articles, recently this article was trimmed way back from 120kB to around 84kB. I'm not saying you're overflowing the article again, it seems only to fluctuate around 84kB while the text is generally improved when you're editing which is fine. The section was moved back for a uniform format in summary style, and although inessential the GA formulations deserve even the slightest mention by WP:NPOV (as they were in the table)...

I would propose that this article just concentrate fully on the microscopic/macroscopic formulations in terms of E, D, P, B, H, M, if you're going to write more about the potential formulation (or any), by all means do so, although it would be better expand all other articles

• Mathematical descriptions of the electromagnetic field
• Relativistic electromagnetism
• classical electromagnetism and special relativity
• Covariant formulation of classical electromagnetism
• Maxwell's equations in curved spacetime
• Magnetic monopoles
• etc...

and mention motivations and reasons for doing so in this article, which the original section "alternative formulations" did some way or another, and provide links in plenty... Maxwell's equations are throughout WP, not just this article.

Just my veiws, which everyone is welcome to disagree... I don't want to interrupt your good work so I'll stay out from now on... M∧Ŝc²ħεИ_τlk 13:18, 2 January 2013 (UTC)

Hi RogierBrussee, nice new table, a lot clearer in terminology. M∧Ŝc²ħεИ_τlk 16:00, 3 January 2013 (UTC)

Matrix rep and GA formulation in ‎Alternative formulations

These edits have reinstated the GA formalism, and this edit has introduced a link to a recent article Matrix representation of Maxwell's equations, with an excessive number of 6 refs.

I'm sorry, but yet again... the ‎Alternative formulations section has become too long (mainly due to the new GA formulation) when there are plenty of specialized articles and after this article has been trimmed.

What should we do about this trend? Probably nothing... M∧Ŝc²ħεИ_τlk 08:06, 3 March 2013 (UTC)

I agree that this is a problem. I've had to revert insertion of unpublished Matrix rep articles before. For the Matrix rep, this is likely a conflict of interest as the author of some of the refs is inserting those refs into the article; it seems promotional and POV pushing to me. The same author created Matrix representation of Maxwell's equations. --Mark viking (talk) 15:06, 3 March 2013 (UTC)

Taken to Wikipedia talk:WikiProject Physics. M∧Ŝc²ħεИ_τlk 08:34, 5 March 2013 (UTC)

The article 'Matrix representation of Maxwell's equations' currently has markup errors, as well. --Ancheta Wis   (talk | contribs) 15:43, 3 March 2013 (UTC)

Most likley to be mine but where? Can't find any... M∧Ŝc²ħεИ_τlk 15:59, 3 March 2013 (UTC)

After Resistance function, etc. In terms of these functions: Failed to parse (lexing error): ε= \frac{1}{v h}\,,\quad \mu = \frac{h}{v}

That didn't show up using mathjax... M∧Ŝc²ħεИ_τlk 17:44, 3 March 2013 (UTC)

You're right, it looked fine under IE & Firefox, but not Puffin. I'll ask at village pump Looks good. Thank you; I was worried that my everyday browser was XXX… __Ancheta Wis   (talk | contribs) 00:51, 4 March 2013 (UTC)

No problem! Best, M∧Ŝc²ħεИ_τlk 01:09, 4 March 2013 (UTC)

I removed the geometric algebra formulation but am guilty of extending the table with the tensor formulation on curved space time. Unlike the geometric algebra formulation, I think they are sufficiently mainstream and give enough physical insight to be included. Best RogierBrussee (talk) 10:20, 6 January 2014 (UTC)

Nothing to be guilty about. I like the table! ^_^ M∧Ŝc²ħεИ_τlk 17:36, 6 January 2014 (UTC)

Scope

The scope of this article is way too big.

EM is tricky, and if presented in the order of historical development (Gauss law, this law, that law, ...), it becomes even trickier. It all becomes a huge cookbook recipe with too many ingredients for anybody to have available at home. It's of course an utopia, but if EM is presented at the outset as one theory, and the immensely complicated macroscopic approximations are left out, then I believe it could be easier to grasp. This is unfortunately not the approach of most modern textbooks or this article, but the approach exists. See L&L for a masterful exposition of microscopic EM as compared to a merely very good one (like Jackson's). It is decidedly easier, cleaner, and much better, to go for the full set of equations right away, provided that one has a little of the the math background. Gauss law, this law, that law, ..., will follow. Lets face it, two screens full of equations and tables of terms is a mouthful. This reasoning of mine is only hypothetical, because it would be too much of a job to attempt anything, most people (incorrectly!) disagree, and the article is still pretty good as it is. YohanN7 (talk) 16:15, 7 March 2013 (UTC)

B t w, what is reference #36 (Myron Evans) doing here? He is not reliable, especially not when it comes to naming equations. YohanN7 (talk) 17:14, 7 March 2013 (UTC)

Above, I have been saying over and over that this is too long as people continuously add content which belongs to the main articles, explicitly listed in detail... Although microscopic/macroscopic formulations should be kept in. An overview table with links to the details is nice, but still the article drowns in advanced explanations of formalisms...

You know, maths of the EM field is basically about Maxwell's equations in various formalisms, so maybe we could move that page to a new name, and keep all formalisms there (from therein linking to advanced articles which is more or less the current case), restricting this article to just the E, B, D, H, P, M fields and no potentials, no differential forms, tensor fields, no geometric calculus? Nah - that wouldn't last long!... Someone will add! add! add!... (Don't get me wrong they are in good faith) M∧Ŝc²ħεИ_τlk 17:42, 7 March 2013 (UTC)

YohanN7, By L&L I assume you mean 'gasp' Landau & Lifshitz vol 2, Classical theory of fields. But there is an undergrad text, Corson & Lorrain 1962, Electromagnetic fields & waves, with the same program: start with Special relativity, and derive Maxwell's equations. Or take James Franck's suggestion to start with the constant of proportionality between electric & magnetic fields to derive SR, or start with moving electric field and shift frame to see magnetic field instead, etc., etc. Or start with continuity equation for electric charge.

But then there probably is an article about the interrelationships already. __Ancheta Wis   (talk | contribs) 18:31, 7 March 2013 (UTC)

Thank you for the book reference Ancheta. When I took an EM course for the first time, the approach was the common one. When we finally hit the full set of equations (page 400 or something in Roald Wangsness, forgot the name of the book) we all said pretty much that the whole course should have been taught backwards. But I guess it is much a matter of taste and emphasis. All of the physics lie in the microscopic equations, preferably presented in covariant form with rationalized Lorentz-Heaviside units. Most of he engineering lies in the macroscopic equations using SI units. (In my view, even the ε₀ and μ₀ are mainly complicating factors.) The macroscopic equations are a huge subject in their own right, and they present an additional layer of concepts. They really logically belong higher up in the food chain. But, I am not insisting on anything. YohanN7 (talk) 10:31, 8 March 2013 (UTC)

Fundamental natural quantities which are constant and observable are regarded as independent. Examples of these are the speed of light and the electronic charge.

In the often presented equation c = 1/sqrt(mu * epsilon) the result is an error about which quantity is the natural constant, and which are derived constants.

C, the speed of light, is the natural constant. The correct order is either that either mu = mu( c, epsilon ) or epsilon = epsilon ( c, mu ) or of course both. The problem is that mu and epsilon are arbitrary, if it is allowed they are represented by glass, diamond, plastic, or other noticeable dielectrics, or iron, cobalt, manganese, aluminum if they are ferromagnetic or diamagnetic, etc.

It appears the first definitions of permittivity were made assuming that glass, crystal or diamond, etc, were the substances of choice. This resulted in a value for permittivity being selected which was very far from the square root of c. The socio-economic causes of the selection cannot be discussed here except that permittivity was defined in the British Isles under the Royal Society.

With permittivity based on crystal and diamond, etc., the definition of the magnetic characteristics of the speed of light were left to Maxwell in Washington D.C., some seventy degrees of longitude to the West. Maxwell had no choice but to assume that the permittivity defined in Britain would have to be used, and the result was that the exponent for permeability was far smaller - and permeability itself was a much coarser quantity which was closely coordinated with iron such as is found in many constructions, tools and weapons.

Since Maxwell's work in the United States, the differences of opinion regarding optimum fundamental measures of permittivity and permeability have been lost in the longitudinal variation, that is, the 70 degree difference in time zones between London and Washington.

The particular example of the speed of light, presented as if it is a constant derived from two derived constants, is frequent. The correct order is that the speed of light is the observed natural constant. This fundamental natural constant is now operationally observed with high accuracy through communications between Earth and other planets such as Mars and Saturn and, of course with the GPS systems.

It is time to reconcile the wide disparity in magnitude of the permittivity and permeability, so that both have relatively close magnitudes. A ratio of 2*pi, or the Golden Mean, or some such constant would be useful as it would free users from confusion on the basis of absolute value alone.

The present, extreme difference appears makeshift, and it is quite possible they would naturally be close to equal, and different only in the topology of their existence - the one in linear fields, the other in curl-dominated fields.SyntheticET (talk) 00:37, 24 March 2013 (UTC)

Recent edits (26/03/2013) to Alternative formulations

Couldn't fit in summary [6], so...

• Deleted "clarification needed" tag in the classical electromagnetism and special relativity section - none was even needed anyway but still trimmed the end of that sentence which didn't add much.
• Applied the {{main}} template for main links to the articles, used as headings to make clear these are the main articles to read.
• Deleted the excessive number of references (discussed above) for the matrix representation of Maxwell's equations, move that link to a {{see also}} at the top.
• Removed the now-redundant {{main}} link to classical electromagnetism and special relativity at the top of the section.

M∧Ŝc²ħεИ_τlk 16:38, 26 March 2013 (UTC)

Removal of history section

I took the liberty of splitting off the history section to the article History of Maxwell's equations, to see how it would turn out. M∧Ŝc²ħεИ_τlk 07:56, 5 May 2013 (UTC)

List of textbooks in electromagnetism

Similarly to the above, c.f. this and here. M∧Ŝc²ħεИ_τlk 08:52, 7 May 2013 (UTC)

Table removed, restructured the lead of the article

Considering YohanN7's comments above, I tried to remove the tables (contrary to how much I really like tables) and rewrite in continuous prose.

• The links to the articles for each quantity should describe the units and alternative names. (The table of terms was in fact what motivated me to create the articles of equation tabulations, complete with units and quantities in physics equations).
• For this article, it was odd to bunch all the conceptual descriptions, micro/macro-scopic equations, then all the definitions, etc, in their own places, rather then in a (at least slightly more) continuously flowing prose. The article should just get to the equations as soon as possible, then describe their meaning.
• I split the micro/macro equations also, it makes sense since the macroscopic equations have the auxiliary fields defined and discussed later.
• Also removed the clumsy and pointless notation "Q_enc(V)" for "charge Q enclosed in volume V": since equations and quantities are supposed to be explained in words - the context should make the meaning clear. All that needs to be said is "Q is the charge enclosed in a volume" then the volume integral of ρ follows.

Apologies to suddenly "dominate" the article. Anyone is more than welcome to complain if I screwed up. M∧Ŝc²ħεИ_τlk 08:31, 12 May 2013 (UTC)

• I also rewrote the integral equations in terms of the volume integrals of charge and surface integral of current,

$\oint _{\partial \Omega }\mathbf {E} \cdot \mathrm {d} \mathbf {S} ={\frac {Q}{\varepsilon _{0}}}\,\rightarrow \,\oint _{\partial \Omega }\mathbf {E} \cdot \mathrm {d} \mathbf {S} ={\frac {1}{\varepsilon _{0}}}\iiint _{\Omega }\rho \,\mathrm {d} V$

$\oint _{\partial \Sigma }\mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}=\mu _{0}I+\mu _{0}\varepsilon _{0}\iint _{\Sigma }{\frac {\partial \mathbf {E} }{\partial t}}\cdot \mathrm {d} \mathbf {S} \,\rightarrow \,\oint _{\partial \Sigma }\mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}=\mu _{0}\iint _{\Sigma }\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\cdot \mathrm {d} \mathbf {S}$

since it's much easier to see the connection between the differential and integral equations this way (all that needs to be done is to change surface integrals into volume integrals, and line integrals into surface integrals, then equating integrands automatically, with no prior substitutions of definitions).

• Also equations are tidier when factorized and it's much easier to see the dimensions this way, for example we can immediately see ε₀∂E/∂t has dimensions of current density J.
• With the physical interpretations, there are "hints" on the connection between flux/divergence and circulation/curl without explicitly writing down the div/curl theorems in terms of E and B, complete with a fluid dynamics analogy to circulation/vorticity.

Hopefully section 1 is now a bit more stimulating for the reader to appreciate the physical nature of vector calculus inevitable in electromagnetism and contained within Maxwell's equations, and maybe even decide to pick up some vector calculus for him/her-self... M∧Ŝc²ħεИ_τlk 13:33, 12 May 2013 (UTC)

The restructuring looks reasonable to me, thanks for doing this. There remains the pedagogic problem of the table integral and differential formulations given without any explanation of the notation. It looks like much of the explanation is in the next section. Should we make a forward reference to the next section, or rearrange sections? --Mark viking (talk) 16:38, 12 May 2013 (UTC)

Thanks for feedback. I tried right now to resection, but since the integral notation fits in so well with the "relations between the integral and diff equations" I left them there for now, maybe a forward reference would be best? M∧Ŝc²ħεИ_τlk 16:51, 12 May 2013 (UTC)

I agree that a forward reference is a reasonable option--looking forward to the next section won't interrupt the flow of reading too much and anyone interested in the table will likely want to read the next section in any case. Thanks, --Mark viking (talk) 20:57, 13 May 2013 (UTC)

The forward reference was rather difficult, so I tried to smooth out the prose at the cost of introducing some repetition of stating the quantities and notations...

The "Relation between int/diff forms" is independent of units and should not be made a subsection or merged with the SI units, so it should stay it's own section, yet as close as possible to the first mention of the equations.

At this point it's better for someone else to continue further rewriting. M∧Ŝc²ħεИ_τlk 21:16, 14 May 2013 (UTC)

Maschen, The article now renders a ?mathjax? error in the Vacuum equations, electromagnetic waves and speed of light section

"which identify

   Failed to parse (lexing error): c = \frac{1}{\sqrt{ \mu_0 \varepsilon_0}} = 2.99792458 × 10^8 \, \mathrm{ms}^{-1}

with the speed of light in free space." I am using Firefox 21.0 --Ancheta Wis   (talk | contribs) 17:43, 15 May 2013 (UTC)

wave propagation picture/comment

I have a vage impression, that electromagnetic wave propagates as E=sin(), B=cos(); instead of E=sin(), B=sin(), as stated in the article, and depicted on the rdawing. Pls somebody comment. — Preceding unsigned comment added by 87.205.149.171 (talk) 06:38, 18 May 2013 (UTC)

That is correct, the one is sine, the other is cosine. See Landau & Lifshitz 1962 Classical Theory of Fields "Monochromatic plane waves" equation 48.9, p131. It would appropriate to alter the picture, or delete it. --Ancheta Wis   (talk | contribs) 11:39, 18 May 2013 (UTC)

Uhh... The E and B fields in a sinusoidal plane wave are in phase in time and space, though oriented at right angles in space (both perpendicular to the direction of propagation). Therefore E=sin(), B=sin() as in this diagram is correct. — Quondum 13:39, 18 May 2013 (UTC)

Then where does the energy go when both B & E are zero, and in phase? It's not as if the fields are in a box ... Anyway, take an infinite waveguide of rectangular cross-section in x & y, with z the infinite direction; Jackson 1962 Classical Electrodynamics p.247 equations 8.46 shows that B & E are indeed sines in x and y, but that $E_{x}$ is 90 degrees out of phase with $B_{y}$ a sine, and with $B_{z}$ a cosine down the infinite waveguide, in the ${TE}_{1,0}$ mode. More precisely, if you take away the box, and let a circularly polarized wave propagate in the z direction, the peak transverse E & B will rotate as they propagate, in an eternal braid. (I am not asking that the animation take this into account.) --Ancheta Wis   (talk | contribs) 15:41, 18 May 2013 (UTC)

Quondum is correct; they are in-phase when the wave is propagating in a vacuum. Taking turns is a inappropriate generalization from L-C circuits. As to where the energy goes, it travels with the wave in those parts where E and B are non-zero. JRSpriggs (talk) 16:21, 18 May 2013 (UTC)

Ancheta, the waveguide example is a bad place to start. Though I have not checked on the specific mode you are referring to, one tell-tale point will be that the propagation of the phase down the waveguide is at less than the speed of light. Every mode of propagation down a rectangular waveguide should be expressible as a superposition of several plane waves each travelling at an angle to the axis of the waveguide, none of which travel directly along the waveguide. In some (most?) waveguide modes, there are longitudinal components of E and/or H. In short: the waveguide example is far more complicated than one would think, and confuses the simple free-space plane wave example. — Quondum 18:54, 18 May 2013 (UTC)

Quondum, I apologize for misreading Landau & Lifshitz, p 131: the text mentioned x & y (x the direction of propagation, y and z transverse to x). The equation 48.9 was for y (sin) and z (cos), not x. The y & z are used for describing the components of the transverse fields. So my question is answered: the energy flux is in the direction of propagation (the Poynting vector, p. 126). --Ancheta Wis   (talk | contribs) 12:19, 19 May 2013 (UTC)

Missing factor in Gaussian equation?

To Maschen: Could you check again macroscopic version of Ampère's law (with Maxwell's extension) at Maxwell's equations#Equations in Gaussian units. I think you may be missing a factor of 4π. Or perhaps the error is in Gauss's law (an extra factor of 4π). JRSpriggs (talk) 10:56, 19 May 2013 (UTC)

Yes, there should be a factor of 4π multiplying the free current density J_f. Thanks, M∧Ŝc²ħεИ_τlk 11:01, 19 May 2013 (UTC)

Minus sign missing in alternative formulation, tensor calculus.

In alternative formulation, in the row "tensor calculus", "potentials, Lorentz gauge, flat spacetime" I think there should be a minus sign. Currently is says:

$F_{\alpha \beta }=\partial _{[\alpha }A_{\beta ]}$

$\partial _{\alpha }A^{\alpha }=0$

$\partial _{\alpha }\partial ^{\alpha }A^{\beta }=\mu _{0}J^{\beta }$

now if I take the last one:

$\partial _{\alpha }\partial ^{\alpha }A^{\beta }=\partial _{\alpha }\partial ^{\alpha }A^{\beta }-0=\partial _{\alpha }\partial ^{\alpha }A^{\beta }-\partial ^{\beta }\partial _{\alpha }A^{\alpha }=\partial _{\alpha }\partial ^{\alpha }A^{\beta }-\partial _{\alpha }\partial ^{\beta }A^{\alpha }=\partial _{\alpha }F^{\alpha \beta }=-\partial _{\alpha }F^{\beta \alpha }=-\mu _{0}J^{\beta }$

For now I will modify the table, if you find a mistake in my derivarion please correct the table and provide explanation here. — Preceding unsigned comment added by 89.65.5.6 (talk) 12:08, 16 June 2013 (UTC)

I reverted the edit. The sign was correct before:

$\partial _{\alpha }F^{\alpha \beta }=\mu _{0}J^{\beta }\Rightarrow \partial _{\alpha }F^{\beta \alpha }=-\mu _{0}J^{\beta }$

$\partial _{\alpha }(\partial ^{\beta }A^{\alpha }-\partial ^{\alpha }A^{\beta })=-\mu _{0}J^{\beta }$

$\partial _{\alpha }\partial ^{\beta }A^{\alpha }-\partial _{\alpha }\partial ^{\alpha }A^{\beta }=-\mu _{0}J^{\beta }$

$0-\partial _{\alpha }\partial ^{\alpha }A^{\beta }=-\mu _{0}J^{\beta }$

$\partial _{\alpha }\partial ^{\alpha }A^{\beta }=\mu _{0}J^{\beta }$

Seems there was a sign error at the end of your working, which should be $=\mu _{0}J^{\beta }$ because of the prior $\partial _{\alpha }F^{\alpha \beta }$. M∧Ŝc²ħεИ_τlk 12:24, 16 June 2013 (UTC)

Thank you for your reply. Please have a look two rows above in the same table, "Tensor calculus, Fields, Flat spacetime", there it is written that $\partial _{\alpha }F^{\beta \alpha }=\mu _{0}J^{\beta }$, it is in a contradiction with the first equation that you have written in your answer above. 89.65.5.6 (talk) 13:31, 16 June 2013 (UTC)

Sorry, yes, the equation $\partial _{\alpha }F^{\beta \alpha }=\mu _{0}J^{\beta }$ is the correct one, so you're sign was correct. Before checking in textbooks (MTW and Griffiths), I recalled the contraction with the first index on F, not the second. M∧Ŝc²ħεИ_τlk 13:50, 16 June 2013 (UTC)

Thank you. And sorry that I did not login before. Janek Kozicki (talk) 14:04, 16 June 2013 (UTC)

Memristors, the 4th 'solution' to the Equations is missing....why is this?

"In 2008, researchers at HP Labs published a paper in Nature reporting the realisation of a new basic circuit element that completes the missing link between charge and fluxlinkage, which was postulated by Leon Chua in 1971

Work has been progressing on the real and theoretical nature of memristors since then.

Memristors have the potential to up-end all of electronics and especially computer design....all of whose circuits are currently based on Von Neumann architecture which separates memory from operations. That separation may no longer be necessary.

An explanation of this development is a serious omission from this page.

Sadly, I do not feel technically able to start this work.

Artied (talk) 13:02, 21 September 2013 (UTC)

In the same way that Kirchhoff's circuit laws are a consequence of Maxwell's equations, so too are memristors. They do not represent new physics, but rather new engineering and materials science. What I mean is, memristors are worthy objects of study by electrical engineers, students, and applied researchers, for their own sake. There are in fact many such kinds of electrical, optical, electro-optical, magneto-optical, etc., objects out there, all of which can be understood from consequences of the basic equations of this article. However, these objects each have their own series of articles, even if their theory stems from this article. __Ancheta Wis   (talk | contribs) 15:36, 21 September 2013 (UTC)

This article is about physics, not engineering. So discussion of electronic components is inappropriate in this article. JRSpriggs (talk) 00:54, 22 September 2013 (UTC)

Roadmap to the thermodynamic web

Roadmap to the thermodynamic web

I just uploaded a "map" I created of thermodynamic equations. I wanted to convert it to SVD (e.g., using Inkscape), however, I am not really good at graphic design. If anyone has the time and know-how, please feel free to convert it to SVG. Also, I am not sure if this article is the best place for it. I am only including this comment on this article, as this "map" helped me a lot as an undergrad whilst learning thermodynamics, etc. --Thorwald (talk) 00:51, 7 December 2013 (UTC)

There is a tremendous difference between Maxwell's equations for electromagnetism (fundamental field equations) and Maxwell relations for thermodynamics (partial derivatives which follow from the definitions of thermodynamic potentials and symmetry in 2nd order partial derivatives). So these should be at Maxwell relations and not here. Nice picture though, will try and convert to SVG later but I'm tied up right now. M∧Ŝc²ħεИ_τlk 10:22, 7 December 2013 (UTC)

Oh. Wow. I meant to post this on the Maxwell relations. Note to self: Never post whilst up way past bedtime. Adding this thread to that article now. --Thorwald (talk) 22:11, 7 December 2013 (UTC)

Archive 1

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Archive 4

Archive 5

Archive 6

Maxwell's correction?

Is this the normal English terminology? 'Correction' suggest that it is a minor tweak, whilst it is essential to the wave equation. It also suggests to me that Ampere got it wrong, which is not really fair.

What about Maxwell's 'addition' or 'modification' ? Martin Hogbin (talk) 19:52, 10 February 2014 (UTC)Harmuth (talk) 21:23, 28 March 2014 (UTC) Reference the book "Modified Maxwell Equations in Quantum Electrodynamics" H.F.Harmuth. T.W.Barrett, B.Meffert, World Scientific Publishers, Singapore. It modifies Maxwell's equations by adding the magnetic dipole current density left out by Maxwell and produces equations that have transient solutions (Maxwell's equations have only steady state solutions).

I agree with "Maxwell's addition" as quantitatively and qualitatively more reflective of what Maxwell actually did, and will implement the changes. Good point, thanks, M∧Ŝc²ħεИ_τlk 18:26, 4 March 2014 (UTC)

Why list in Maxwell's equations#Conventional formulation in SI units?

I may be a hypocrite or curmudgeon (actually I'm 22)... while I understand the good faith motivations to add a list of variables after a set of equations, in rewriting parts of this article (including the section Maxwell's equations#Conventional formulation in SI units), I attempted to explain the symbols in the equations in words, directly under the table (and some even before the table, directly above). Also, are we not supposed to explain formulae in words rather than lists (Wikipedia:Manual of Style/Mathematics#Explanation of symbols in formulae)? So why do we have a list of all the variables which duplicates what is already explained?

We used to have a gigantic table of all the constants, variables, terminology, and units, which was converted into prose as it probably should for an encyclopedia (see the history in pre-2013). Editors (by all means in good faith) may add the units, alternative names and symbols, etc. to the list and make it longer and denser, then eventually there would be a good reason to resurrect the big table format again...

I will not delete the list, but there doesn't seem to be much benefit in keeping it there.

Finally, yes, admittedly I used to add bulleted lists to equations in articles, but that was years ago and as and when they have been or will be converted to prose. Best, M∧Ŝc²ħεИ_τlk 18:26, 4 March 2014 (UTC)

I agree that variables and mathematical notation are best explained in prose. It may be a font problem. For those of us not using MathJax, e.g., B and $\mathbf {B}$ look completely different, so it may not have been obvious that the prose variables corresponded to the table variables. For this reason I don't like the hack of using LaTex for display math and HTML/wiki markup for inline math. But converting the inline math to LaTeX runs afoul of WP:RETAIN. As a compromise, I'll convert the lists to prose and remove the unneeded bigmath modifiers. There will still be redundancy to deal with. --Mark viking (talk) 19:14, 4 March 2014 (UTC)

Hi Mark, I'm not sure if you misunderstood: all I meant was the list is redundant and could be deleted. The paragraphs below explain all the symbols in the default font (the only symbols which could be potentially bad are the curly partial d and nabla, but the math template has nothing to do with this, and fonts do not seem to be the motivation for the lists). Your efforts are appreciated, but converting the list into prose has only duplicated the explanations. Thanks, M∧Ŝc²ħεИ_τlk 20:10, 4 March 2014 (UTC)

Hi Maschen, sorry for not explaining my motivation better. When I saw the lists, I asked myself, why did the editor create these redundant lists? Is it a love of bulleted lists? Is it because they jumped straight to the equations and seeing no definition of variables next to the table, decided to add them? Or is it because, for instance, the prose B in the previous section and display $\mathbf {B}$ in the table look so different that they did not realize that these represented the same variable? I guessed that the last explanation was the possible one. Actually in my browser wiki bold B, math template B, and LaTeX mathbf $\mathbf {B}$ are all rendered in different fonts; the wiki bold is in a bold sans-serif font, the template is in a bold serif font and the Latex is in a much larger and not particularly bold serif font. I kept the math templated symbols as a compromise between the wiki bold and LaTeX mathbf versions. All the different fonts might have confused me if I was learning this stuff for the first time. It is a typographic train wreck.

Feel free to delete the now prose paragraph--I agree it is redundant in content, if not in form. But it would be good to figure out why the lists were added in the first place. --Mark viking (talk) 20:41, 4 March 2014 (UTC)

I disagree. I like having the variables explained in a list immediately after the equations. When someone is reading the equations and trying to understand them, he will most easily find the explanation that way, just a quick glance downward to the bulleted definition and then back to the equation. JRSpriggs (talk) 06:34, 5 March 2014 (UTC)

Yes, I realize that, but keeping the list still amounts to duplication and in all this time editors have been trying cut repetition from the article. The paragraphs explaining the symbols are as compressed as they need to be, it's not like they go on forever.

As a compromise: we could reinstate the list with a hidden note telling people not to make it too big, those inclined can do this. Thanks, M∧Ŝc²ħεИ_τlk 07:26, 5 March 2014 (UTC)

Hope people are happy with this change. M∧Ŝc²ħεИ_τlk 22:31, 8 March 2014 (UTC)

Those changes look good to me, thanks. --Mark viking (talk) 21:40, 28 March 2014 (UTC)

To update: I removed the pure list of symbols at the bottom of the table of equations (only to be followed by a more detailed sequence of symbols) and broke the prose into a list. Now we have a list of symbols with the neccersary explanations. Hope the change is not too controversial. M∧Ŝc²ħεИ_τlk 10:39, 2 October 2014 (UTC)

Three notes

"Paraphernalia, such as sweatshirts or T-shirts with "And God Said", followed by the Maxwell's equations, are extremely popular among physicists, and geek-types, because of the elegance of these equations that provide a bridge between classical physics and religion. The phrase refers to Genesis 1:3: and God Said "let there be light" and there was light, whereas the equations represent the essence of light, which is a form of electromagnetism." Three notes:

First, I don't understand what the word "extremely" means. How many of them? How much they feel about such T-shirts? There is a strong need to be concrete, otherwise it's not an encyclopaedia. For that matter, I don't understand what the word "geek" means, either.

Second, what is "the essence of light" depends on how you interpret the Bible, what this light is important for and how we are supposed to make judgements of it (is it the physical logic behind light? is it a sensation of light? is it an idea of light-like goodness? is it something else? all three are very different objects that have very little to do with one another). Also, it depends on how you interpret the ability of a human to know; after all, the equations are nothing more than a human understanding of light, a representation of some of our ways of reasoning of it, but they are not light itself. So this statement must have been discussed at length and with different outcomes. I don't see any links that lead to extra-Wikipedia discussions of what this section says, anyway.

Third, I don't see what this section has to do with the essence of Maxwell equations. Lots of things can be said à propos, does one need to have them all in an encyclopaedic article? Just imagine someone addresses the first point of mine and adds some links to statistical figures and quotes of famous physicists or "geeks", where they explain what they feel about such T-shirts… Without such links and discussions, it is not encyclopaedical. With such links, it is evidently out of place here. - 91.122.10.59 (talk) 19:00, 3 April 2014 (UTC)

Yes, good find. I have boldly removed the section. It is entirely unsourced. If someone insists that it really belongs in the article, I'm sure they will find a solid source. - DVdm (talk) 19:08, 3 April 2014 (UTC)

(ec – thanks, DVdm) You make some cogent points. The section and its content is IMO is extremely non-encyclopaedic, and really does not belong here. Besides, the interpretive phrasing "because ... provide a bridge between classical physics and religion" is potentially simply wrong: I see it as an in-joke poking fun at the wording of the Book of Genesis, in a sense via a pun. I'd hardly call this "providing a bridge". —Quondum 19:20, 3 April 2014 (UTC)

These kind of popular culture "mentions" abound in other wiki articles and I find them pretty enlightening. But for me Maxwell's equations are all business or should be. Good call to remove the T-shirt exegesis.

But do leave the actual image in. I've never found it disrespectful to religion; it well expresses a clockmaker paradigm that became popular during the Enlightenment and no doubt informs many people's thinking today. 84.227.250.178 (talk) 16:55, 8 April 2014 (UTC)

I had left the image in there because it doesn't do much harm—and it's a bit funny . I have now moved it next to the introduction of the section on alternative formulations, which looks like a better place. - DVdm (talk) 17:27, 8 April 2014 (UTC)

I'm fine with removing the religous OR, but these shirts have been around for at least 50 years. Check out the 1963 ad on page 3 of the MIT newspaper. It is reasonable to mention these shirts. --Mark viking (talk) 17:31, 8 April 2014 (UTC)

I've reworded the caption of the picture slightly, so that it makes only reference to the fact of the equations on the T-shirt. Highlighting the allusion to the passage in Genesis is unnecessary, and makes it specific to this example of the "bumper sticker"-like popularity of Maxwell's equations – for example, the advert in the MIT newspaper referred to above shows a sweat-shirt with only the equations on it. I think the picture serves as a light point (heh-heh) in the article, without making unreferenced claims or allusions. Most who have dealt with Maxwell's equations will need no more than the picture itself to appreciate the humour. I think too that it probably falls easily within the WP guidelines in that it is a related illustration, which typically does not carry such a burden of notability or relevance. —Quondum 17:35, 15 June 2014 (UTC)

Yes, I agree with this tweak. - DVdm (talk) 18:25, 15 June 2014 (UTC)

Italic vs. roman 'd' for derivative; distinguishing notation for distinct operations

This reverts an edit that was to make italic/roman font use consistent ; the revert does not take into account all the other equivalent uses of the same symbol and thus introduces inconsistency.

Two distinct concepts are used in the article: the derivative using Leibniz's notation, and the exterior derivative. For the former, we have had much discussion, and I would be in favour of using an italic d throughout for the Leibniz notation. The same discussion does not seem to have been as conclusive about the notation for the exterior derivative. However, I would like the two to be distinguished in this article to minimize confusion, so I propose explicitly mentioning the choice of notation and the distinguishing role of the font, and changing one to italic and keeping the other roman throughout. I nominate the italic d for the Leibniz notation, and roman d for the exterior derivative in this article. —Quondum 13:51, 29 August 2014 (UTC)

Why was my edit on Maxwell's equations undone?

Please read my edit on Maxwell's equations:

---

From a historical perspective, the equations that Maxwell arrived at are fairly distant and cumberous from the present form of four elegant equations. These set of four equations owe their present elegance to Oliver Heaviside, who reformulated Maxwell's field equations in terms of electric and magnetic forces and energy flux. Hence, for a time the equations were rightly called as Maxwell-Heaviside Equations.

---

This is historically correct. Maxwell's paper was scrutinised and the elegance of the modern form of the equations wasn't there. It is unfortunate that Oliver Heaviside's name has been removed. The common public may not have done justice to the man, but wikipedia, an encyclopedia, must not be swayed by Google's page ranks or the absence of it in attributing the credit to the man who deserves it. The article on Oliver Heaviside may please be referred to. YohanN7 has referred to one "Myron Evans". It's not clear how the present edit is related to the person. Discussion is welcome. Bkpsusmitaa (talk) 17:38, 15 September 2014 (UTC)

Your edit was reverted by user YohanN7, most likely because it lacked a reliable source. The revert should have specified the reason and you probably should have been given a little message on your talk page explaining the problem. Anyway, see wp:RS and wp:NOR. Cheers - DVdm (talk) 18:04, 15 September 2014 (UTC)

Here is what I replied on my talk page when you asked me there:

It might be worth a footnote in the history section (if you can source it), not a large (and POV, it didn't read well) chunk in the lead.

And, as a matter of my POV, reorganizing equations is not really the same as discovering them. An undergraduate student can do the former. YohanN7 (talk) 19:46, 15 September 2014 (UTC)

It is clear on my talk page why I refer to Myron Evans. He is the reference for the term "Heaviside-Maxwell equations" usage today in History of Maxwell's equations. And, yes, I believe it is the Myron Evans, the most inappropriate man alive to name equations, see Einstein-Cartan-Evans theory, named so after Myron Evans, by himself. YohanN7 (talk) 19:57, 15 September 2014 (UTC)

I've read Maxwell's papers on E.M. As an undergraduate physics student I'd read the elegant Heaviside equations and wrongly attributed them to Maxwell. Much later, when I read Maxwell's papers I found the two papers were long, difficult and convoluted. The only thing that comes through clearly enough is the speed of light from an algebraic manipulation of permittivity and permeability.

And no, it is beyond the capability of even a talented undergraduate to reinterpret Maxwell's papers to the elegant vectorial form we now know them wrongly as Maxwell's equations. What you say here is very cruel to a man who did so much, including developing the vector calculus notations. Please do read the page on Heaviside along with external references in the wikipedia itself. So either delete that page, or accept my edit in the present wiki page to make the information commensurate with each other.

Bkpsusmitaa 12:02, 1 October 2014 (UTC)

I'll just repeat what I wrote:

It might be worth a footnote in the history section (if you can source it), not a large (and POV, it didn't read well) chunk in the lead.

There are several formulations of the Maxwell equations in use. For this, see the article. I personally think you give Heaviside undue credit. The equations of electromagnetism (whether ugly or pretty) were inconsistent when Maxwell introduced his contributions, and after that they were consistent (whether ugly or pretty). Therein lies the physics. With Maxwell, the classical theory of electromagnetism reached its present form.

Scientific literature older than, say, 50 years is unbearable to read. But just because scientists today are better at presenting science, and better notation is available, it doesn't mean that credit is moved from the original discoverers.

Nobody stops you from making a new attempt, and I certainly will not say in advance that I will revert you. I have nothing against Heaviside, and it appears as the vector calculus formulation is due to him. (I personally think it is messy compared to the tensor and differential form versions.) Just don't give it undue weight.

How about this:

The most common modern formulation, the vector calculus formulation, is due to Oliver Heaviside + citation. YohanN7 (talk) 13:09, 1 October 2014 (UTC)

----------------------------------------

Actually, Heaviside's reformulation revolutionised the development of E.M., as the equations, and with it the theory, became easily comprehensible. And this comprehensibility encouraged further development. A cursory glance at the history of development of E.M. before and after Heaviside's contribution will support this hypothesis.

And no, I won't repost. I would reason with you. If you are convinced, you would revert the earlier page. It is still there.

The POV is that Maxwell's equations, that we use today, are not Maxwell's. It is Heaviside's ;) !

Your logic, "Scientific lit ... older than ... 50 years ... unbearable to read ... better ... presenting...better notation ...doesn't mean ... credit ... moved from the original discoverers" does not apply in this case. Maxwell didn't even frame the simplest one-dimensional E.M. wave equation, or the Poynting Vector. John Henry Poynting, Oliver Heaviside and Nikolay Umov independently co-invented the Poynting vector. So Heaviside is not some fanciful page designer. He is a real scientist who did real work, and so deserves his due credit.

Bkpsusmitaa (talk) 04:53, 2 October 2014 (UTC)

This qualifies as a perennial topic. You are not the first editor to note Heaviside's role in the article. The archives reveal this and the current article still has traces of previous attempts to rename, revise, or otherwise rework the equations' history. Please bear with us, for having seen this before, both in the article, and the talk page. You are seeing the consensus. --Ancheta Wis   (talk | contribs) 06:18, 2 October 2014 (UTC)

@Bkpsusmitaa: I don't think it is giving Heaviside credit for having reformulated Maxwell's equations that is the problem. It is your POV (per above) that is the problem. It — and more — shines through in your edit.

Please let's not get personal, or begin a flame war. I am naturally sympathetic to scientists who are ignored by the highly deterministic, rigid, government-controlled education system and school syllabi. Like Rosalind Franklin, Tycho Brahe, Jagadish Chandra Bose, ... . If these selfless men vanish from our memories science would lose its true history and become a domain of a clan of university-educated scholars, a very distorted view. I maintain that despite many brilliant scientists at the time, the aeroplane owes its invention to bicycle-repairers.

Why forget that science is not about degrees, it is about original queries. I hope I have explained myself. Let's focus on the topic instead.

I'd personally be happy to see a mention, like the one I suggested (+ suitable citation), perhaps in Formulation in terms of electric and magnetic fields, that anyway begins with hailing of the formulation as "powerful". It is after all Maxwell's equations in Heaviside's formulation that is used in this article. Credit goes where credit is due. History is also not undone, yet history does not impinge on this article. (It has its own, as does Heaviside.) Yes, I'd like that. YohanN7 (talk) 09:38, 2 October 2014 (UTC)

I can sympathize with Bkpsusmitaa that this article neglects the modern vector calculus form is due to Heaviside, not Maxwell. Yes Maxwell published the original complete set, but it is important not to neglect Heaviside as this article does. History of Maxwell's equations does not (because I wrote the lead, not the rest of the content). The term "Heaviside-Maxwell equations" is definitely non-standard and should not be used. I'll tweak the article. Along these lines. M∧Ŝc²ħεИ_τlk 09:56, 2 October 2014 (UTC)

If you've read Maxwell's two papers you'd find a remote connection between his equations and the present, elegant formulation. Maxwell included every possible equations available to E.M. till that time. It is our hindsight that allows us to identify Maxwell's set of equations. Bkpsusmitaa (talk) 15:41, 2 October 2014 (UTC)

Hope this change is not too controversial, a source can be added any time. M∧Ŝc²ħεИ_τlk 10:41, 2 October 2014 (UTC)

TBH, I think you overdid it big time, but that is my POV. For me, Maxwell's equations isn't a collection of any particular (unwieldy) collection of mathematical symbols. It is the a synonym for the classical theory of electromagnetism. I oppose to the change. A parenthetical remark (and not in the lead) would suffice IMO. YohanN7 (talk) 11:31, 2 October 2014 (UTC)

Fine, but it read before that Maxwell is responsible for the vector calculus formulation. I'll remove Heaviside out of the lead and not the first vector calculus section. M∧Ŝc²ħεИ_τlk 11:40, 2 October 2014 (UTC)

As a compromise, could we not have something in the lead like, 'the current form of the equations is due to Oliver Heaviside'? Martin Hogbin (talk) 14:11, 2 October 2014 (UTC) I see something like that has already been suggested, I would support that. Anything stronger takes away the credit that is rightly due to Maxwell. Martin Hogbin (talk) 14:15, 2 October 2014 (UTC)

My edit is already on the top of this section. So why not use that instead? :) Bkpsusmitaa (talk) 15:45, 2 October 2014 (UTC)

Why not? (EC: Not referring to the crackpot's of above contribution) Please also remove Heaviside from other mentions (like my last edit) unless there are other reasons to keep him in (like Heaviside-Lorentz units). Keep the article clean, about EM only and not about who did what to develop practical knowledge about EM. We have plenty of articles about such things, and 1000 years from now, there may be millions. This article is about the foundations of it all, and there Coulomb, ..., Ampére, ..., Gauss, ..., and finally Maxwell belong. Not Heaviside, sorry. YohanN7 (talk) 16:01, 2 October 2014 (UTC)

I have seen the edit you have incorporated. It satisfies my first objective. I forgot to say 'thank you'.

But Maxwell's equations are not Maxwell's! That is the moot point. Only one, with the addition of the J vector (original one), is his contribution. And why are you instructing me to keep the article clean? It looks like an allegation. Did I tamper it? And Heaviside did contribute to the foundation, via the Poynting Vector as well. Bkpsusmitaa (talk) 16:34, 2 October 2014 (UTC)

Personally, I think that this would be moot with a short history section just after the lead. Heaviside's name does not need to appear anywhere else IMO. In any case, I strongly object to resurrecting the name Maxwell-Heaviside equations. The purpose of naming equations is to make it easier for scientists to communicate and not to honor the person (people) who developed the equation. There are very few equations, if they are any at all, whose name reflects the people who made the most significant contributions to that law. One note of irony is that one of the main motivations of Heaviside was to purge Maxwell's equations of A and V. Yet, the most valid form of E&M (QED) is not based on E and B but the potentials. TStein (talk) 16:53, 2 October 2014 (UTC)

@Bkpsusmitaa: Yes of course it is unfair to not mention Heaviside at all, but it was a simple tweak to make and it is now made. There is an entire article on the history of the equations, so his work is not neglected in the EM section of wikipedia. Is your concern is solved by now? M∧Ŝc²ħεИ_τlk 18:34, 2 October 2014 (UTC)

All except one: Maxwell's equations are not Maxwell's! So it is better to rename the topic as 'Maxwell's Equation', rather than 'Equations'. People, clans, cults, ghettos, etc., alter history little by little, using apparently innocuous methods. Renaming is one of them. Bkpsusmitaa (talk) 04:05, 3 October 2014 (UTC)

The EM articles here and elsewhere make it clear Maxwell's equations are Guass' laws, Faraday's law, and Ampere's law with modification, and that Maxwell modified Ampere's law like this:

$\nabla \times \mathbf {B} =\mu _{0}\mathbf {J} \quad \rightarrow \quad \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\varepsilon _{0}\mu _{0}{\frac {\partial \mathbf {E} }{\partial t}}$

for time-dependent E fields, and pulled all the equations together to form classical EM as a unified theory, and predict EM waves. The physics literature and community is well aware of this, so no - there is no unfair renaming or loss of knowledge. Which equation is "Maxwell's equation"? The time-dependent generalization of Ampere's law isn't "Maxwell's" equation if that's what you mean. It is absolutely standard to call the complete set of equations "Maxwell's equations" and should stay that way.

In short - no renaming since it is non-standard and wrong. This was a very simple matter that has been resolved, it has dragged on for too long. Can we end it here now? Thanks, M∧Ŝc²ħεИ_τlk 08:11, 3 October 2014 (UTC)

I know! You could stop the discussion whenever you choose, needn't drag the discussion. Please don't get me wrong. I am aware of Maxwell's equations. But I would still say my observations remain valid :). I wrote the following at various times :

(1) The equations that Maxwell arrived at are fairly distant and cumberous from the present form of four elegant equations. These set of four equations owe their present elegance to Oliver Heaviside, who reformulated E.M. field equations in terms of electric and magnetic forces and energy flux.

(2) The common public may not have done justice to the man, but wikipedia, an encyclopedia, must not be swayed by Google's page ranks or the absence of it in attributing the credit to the man who deserves it.

(3) The ... elegant Heaviside equations and wrongly attributed them to Maxwell. Much later, when I read Maxwell's papers I found the two papers were long, difficult and convoluted. The only thing that comes through clearly enough is the speed of light from an algebraic manipulation of permittivity and permeability.

(4) Maxwell didn't even frame the simplest one-dimensional E.M. wave equation, or the Poynting Vector. Only one, with the addition of the J vector (original one), is his contribution.

(5) Some scientists are ignored by the highly deterministic, rigid, government-controlled education system and school syllabi. Like Rosalind Franklin, Tycho Brahe, Jagadish Chandra Bose, ... . If these selfless men vanish from our memories science would lose its true history and become a domain of a clan of university-educated scholars, a very distorted view.

Why forget that science is not about degrees, it is about original queries. I hope I have explained myself. Let's focus on the topic instead.

I add this: (6) The purpose of naming equations is to _is_ to honour the inventor/discoverer, not to make it easier for scientists to communicate. Otherwise, Maxwell's Equations could be better recalled as The Four E.M.Field Equations :) . And scientists are those who work for truth. A man who chooses an easy life of remembering things and not stick to truth is no scientist at all. I maintain that despite many brilliant scientists at the time, the aeroplane owes its invention to bicycle-repairers.

Still, unfortunately, an editor unnecessarily took the time to painstakingly write and post the equation involving the J vector, despite my writing on the matter. Sometime, I find I can't just reach people by writing plainly. To really reach the reader it's also required of the readers' choice to whether read what is written or just ignore it. It makes me believe people are forever imprisoned within their own minds and constructs.

I rest my case, and end this discussion.

Bkpsusmitaa (talk) 04:05, 4 October 2014 (UTC)

Yes many of us oppose the revisionism of attributing credit to discovery. In general it's better to just name equations (laws, theorems, whatever) by their content rather than after people (although people who misplace patriotism will always disagree with that), and attribute credit to people separately, but since the name "Maxwell's equations" have stuck for such a long time, for the purposes of Wikipedia they should be called that, and their history clarified separately. For what it's worth: I have Maxwell's treatise on EM vol 1 and vol 2, so I know what his formulation is like.

In any case you have raised yet again the same points over and over even after they have been fixed - it seems you do not read what others write (YohanN7, DVdm, Ancheta Wis, Martin Hogbin, TStein, possibly others elsewhere, and myself).

I will not post further. M∧Ŝc²ħεИ_τlk 10:42, 4 October 2014 (UTC)

Counter-allegations, false aspersions, Eh!? Good! Bkpsusmitaa (talk) 06:34, 5 October 2014 (UTC)

You have also screwed up this thread by sticking in your cranky comments by intersecting other posts or blocks of posts. Do you mess up on purpose? YohanN7 (talk) 14:29, 4 October 2014 (UTC)

Personal attack? Groupism? Good! Bkpsusmitaa (talk) 06:34, 5 October 2014 (UTC)

I am compelled to post my last comments: No decision is so final in a Human Society that it can't be altered at some time in future. When Maxwell didn't discover the three field equations by himself, why not call this Page "The Four E.M. Field Equations" and redirect search on "Maxwell's Equations" to the renamed page, rather than perpetuate a lie? Wikipedia is a public-edited encyclopedia. Why should it promote a lie popularised by textbooks and vested interests, if any? It should speak the truth. — Preceding unsigned comment added by Bkpsusmitaa (talk • contribs) 06:53, 5 October 2014 (UTC)

Wikipedia speaks the common textbook literature—see also WP:Verifiability, not truth. - DVdm (talk) 09:10, 5 October 2014 (UTC)

Earlier comment, "wikipedia ... should speak the truth", amended: in place of truth in the end, Verifiable truth... LOL
I quote from Wikipedia, "..."Verifiability" was used in this context to mean that material added to Wikipedia must have been published previously by a reliable source...Sources must also be appropriate, and must be used carefully, and must be balanced relative to other sources per Wikipedia's policy on due and undue weight...does not mean Wikipedians have no respect for truth and accuracy...We empower our readers. We don't ask for their blind trust..."
In the light of the above consider these:
(a) It is proven here that Maxwell's equations are not Maxwell's, except the equation involving the J term.
(b) The elegant form that we use today is owing to Oliver Heaviside.
The above facts have been verified in Wikipedia itself.
Why then it negates the proof and calls the said equations Maxwell's equations? Just because the Text Books do?
What is wikipedia doing? In effect, furthering a lie! Call this the Wikipedia Paradox :) LOL
Bkpsusmitaa (talk) 11:43, 5 October 2014 (UTC)

See WP:Common name. JRSpriggs (talk) 11:46, 5 October 2014 (UTC)

Please see History_of_Maxwell's_equations, particularly, "...But it wasn't until 1884 that Oliver Heaviside, concurrently with similar work by Josiah Willard Gibbs and Heinrich Hertz, grouped the twenty equations together into a set of only four, via vector notation. This group of four equations was known variously as the Hertz–Heaviside equations and the Maxwell–Hertz equations,but are now universally known as Maxwell's equations..."
In short, a perpetuation of an incorrect assumption. One could have written, The four Electromagnetic Field Equations, and redirect searches on Maxwell's Equation to this topic to maintain neutrality and accuracy and not perpetuate an incorrect assumption.Bkpsusmitaa (talk) 12:11, 5 October 2014 (UTC)