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Covariant macroscopic electromagnetism

  1. Aug 29, 2014 #1

    Dale

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    I wondered if anyone had a good online reference on the covariant formulation of Maxwell's macroscopic equations and the other equations of classical electromagnetism?

    The wikipedia article talks about constituitive equations in vacuum, which doesn't make a lot of sense to me since M and P are 0 by definition.
     
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  3. Aug 29, 2014 #2

    ShayanJ

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    I had the same problem but I couldn't find the answer in online sources.
    I only can suggest you "Classical electromagnetic theory" by Jack Vanderlinde.
     
  4. Aug 29, 2014 #3

    Meir Achuz

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    Which Wikipedia article?
     
  5. Aug 30, 2014 #4

    Dale

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  6. Aug 30, 2014 #5

    vanhees71

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    Hm, but it's given just in the next section of the Wikipedia article, referring to Minkowski's model for the linear-response consitutive relations.
     
  7. Aug 30, 2014 #6

    Dale

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    Yes, but I don't want to assume linear constitutive relationships. The Wikipedia treatment is also just too disjointed for me to follow well. I am looking for a more systematic and general covariant presentation.
     
  8. Aug 30, 2014 #7

    ShayanJ

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    OK, this is from the book I mentioned:
    [itex] H_{\mu \nu} [/itex] is the anti-symmetric tensor containing H and D fields. It has the same dimensions as the H field and the D and H components located in it and what are their signs can be derived by analogy with [itex] F_{\mu \nu} [/itex].
    [itex] P_{\mu \nu} [/itex] is the anti-symmetric tensor containing M and P fields. It has the same dimensions as the M field and, again, where the D and H components located in it and what are their signs can be derived by analogy with [itex] F_{\mu \nu} [/itex].
    Now we have the equations below:
    [itex]
    H_{\mu\nu}=\varepsilon_0c^2F_{\mu\nu}+P_{\mu\nu}
    [/itex]
    [itex]
    \partial_\mu H^{\mu\nu}=J^\nu
    [/itex]
    EDIT: Looks like the Wikipedia page contains the things I wrote in this post. So what else do you need DaleSpam?
     
    Last edited: Aug 30, 2014
  9. Aug 30, 2014 #8

    robphy

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    Last edited: Aug 30, 2014
  10. Aug 30, 2014 #9

    atyy

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  11. Aug 31, 2014 #10

    robphy

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    The covariant constitutive equation relates E with D and B with H, even in the vacuum case (e.g., [itex]\vec D=\epsilon_0 \vec E[/itex]).

    From a more abstract viewpoint, that example is actually more than a proportionality relation between two vector fields. The vector E is the metric-dual of a one-form, and (in (3+1)-dimensions) the vector D is the metric-dual of the Hodge-dual of a twisted two-form. (The latter is similar to how a cross-product of two vectors in 3D can be thought of as a [pseudo-]vector.)
     
  12. Aug 31, 2014 #11

    Dale

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    Thanks. I am still going through these.

    Hehl's "gentle introduction" paper made an interesting comment that I had never considered. After introducing equation 8.2 he says "there is no physical exchange between the bound and the external charges". Since each is separately conserved he is correct, but I guess that means that this approach cannot model something like dielectric breakdown where the bound charge becomes so strong that it turns into a free current.
     
  13. Sep 4, 2014 #12

    DrDu

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    A covariant definition of the polarization-magnetization 2-form P is
    dP=*j
    where j is the charge-current form for the microscopic charges in the medium and * is the dual.
    Obviously P is defined only up to the differential dA of a 1-form C as ddC=0. In optics one usually choses C so as to make magnetization vanish while in static situations this leads to a divergenge as ω→0.
    Note the close analogy to the definition of the vector potential A in terms of the electromagentic field strength
    dA=E
    where A is also defined only up to a total derivative df of a gauge function f.
    Macroscopic equations are obtained by considering the components of P with low wavenumber.
    Covariant constituitive relations are obtained when the induced charge-current density j due to E is calculated covariantly.
     
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