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Maxwell's eqn in invariant form

  1. Dec 17, 2003 #1
    Maxwell's eqn, in invariant form reads:

    [tex]F^{\mu \nu}{}_{;\nu} = J^{\mu}[/tex]

    and

    [tex]F_{\alpha \beta ;\gamma} + F_{\beta \gamma ;\alpha}+F_{\gamma \alpha; \beta} = 0 [/tex]

    Can someone give Maxwell's eqn if there is magnetic charge and current? I do not believe the form (matrix element) of F change, however, if it does, please state that as well.
     
  2. jcsd
  3. Dec 17, 2003 #2

    DW

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    The second can be written in terms of the electromagnetic duel tensor [tex]D^{\mu \nu}[/tex] as
    [tex]D^{\mu \nu}{}_{;\nu} = 0[/tex]
    Instead of setting that equal to zero try setting it proportional to your hypothetical magnetic four current [tex]M^\mu[/tex] like:
    [tex]D^{\mu \nu}{}_{;\nu} = kM^{\mu}[/tex]
    (Normally I would explicitely put in the constants determied by your system of units for both sets of equations)
    I haven't checked into this, but off the top of my head I think this would work. Of course your next job will be to go out and find a magnetic monopole in order to justify having done this.
     
  4. Dec 17, 2003 #3
    How do you define the dual tensor?
     
  5. Dec 18, 2003 #4

    DW

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    The electromagnetic duel tensor [tex]D_{\mu\nu}[/tex] is related to the electromagnetic tensor [tex]F^{\mu\nu}[/tex] and the rank 4 Levi-Civita tensor [tex]\epsilon_{\alpha\beta\mu\nu}[/tex] by
    [tex]D_{\mu\nu} = \frac{1}{2}F^{\alpha\beta}\epsilon_{\alpha\beta\mu\nu}[/tex].
     
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