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A Different versions of covariant Maxwell's equations

  1. Jan 12, 2017 #1

    ShayanJ

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    The standard way of writing Maxwell's equations is by assuming a vector potential ## A^\mu ## and then defining ## F_{\mu \nu}=\partial_\mu A_\nu-\partial_\nu A_\mu ##. Then by considering the action ## \displaystyle \mathcal S=-\int d^4 x \left[ \frac 1 {16 \pi} F_{\mu \nu}F^{\mu\nu}+\frac 1 c J_\mu A^\mu \right] ##, the Maxwell's equations will be given by the e.o.m. of the action, which is ## \partial_\mu F^{\mu \nu}=\frac {4\pi} c J^\nu ##, and the Bianchi identity, ## \partial_\lambda F_{\mu \nu}+\partial_\nu F_{\lambda\mu}+\partial_\mu F_{\nu \lambda}=0 ##.


    But Maxwell's equations can also be written in the language of forms ## \mathbf{d\star F=4\pi \star J} \ , \ \mathbf{d F}=0 ## Where ## \mathbf F ## is the E.M. two-form and ## \star \mathbf S ## is the Hodge dual of the p-form ## \mathbf S ##.


    Yet another way of writing Maxwell's equations, is by considering two actions ## \displaystyle \mathcal S=-\int d^4 x \left[ \frac 1 {16 \pi} F_{\mu \nu}F^{\mu\nu}+\frac 1 c J_\mu A^\mu \right] ## and ## \displaystyle \mathcal S=\int d^4 x F_{\mu \nu}\mathcal F^{\mu\nu} ## where ## \mathcal F^{\mu \nu}=\frac 1 2 \varepsilon^{\mu \nu \lambda \sigma}F_{\lambda \sigma} ## represents the components of ## \star \mathbf F ##. These two actions give the equations ## \partial_\mu F^{\mu \nu}=\frac {4\pi} c J^\nu ## and ## \partial_\mu \mathcal F^{\mu \nu}=0 ##.


    1) Is there any other way of writing Maxwell's equations in covariant form?
    2) Is there any relationship between all the above ways or are they just different independent ways of doing it?

    Thanks
     
  2. jcsd
  3. Jan 12, 2017 #2

    phyzguy

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    In the language of Geometric Algebra, you can write [itex] DF = \mu_0 J [/itex], where F is the EM Field bi-vector, J is the current vector, and D is the geometric derivative. This combines the above two separate relations into a single relation, since [itex] D\cdot F = \mu_0 J [/itex] and [itex] D \wedge F = 0 [/itex]. Of course, these are all just different notational ways of packaging the same physics.
     
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