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Maxwell Equations and Fourier Expansions

  1. Dec 13, 2012 #1
    1. The problem statement, all variables and given/known data

    The field E(r,t) can be written as a Fourier expansion of plane waves [tex]E(r,t)=∫E(k,w)e^{i(kr-wt)}d^{3}kdw[/tex] with similar expansions for other fields.

    Need to show the derivation of [tex]kXE(k,w)=wB(k,w)[/tex] from Faraday's law [tex]∇XE(r,t)=-∂B(r,t)/∂t[/tex] and also the derivation of [tex]kXH(k,w)=-wD(k,w)[/tex] from Ampere's law [tex]∇XH(r,t)=∂D(r,t)/∂t[/tex]

    2. Relevant equations

    [tex]∫e^{ax}=(1/a)e^{ax}[/tex]

    3. The attempt at a solution

    I thought the Fourier expansion expression for E meant to integrate with once with respect to w and three times with respect to k, so get:

    [tex]∫E(k,w)e^{i(kr-wt)}d^{3}kdw = (1/r)x(1/r)x(1/r)x(-1/t)e^{i(kr-wt)}=(-1/r^{3}t)e^{i(kr-wt)}[/tex]

    But that clearly doesn't give the result, no k or w there at all... what am I doing wrong? I get this isn't tricky but can't figure it out.
     
    Last edited: Dec 13, 2012
  2. jcsd
  3. Dec 13, 2012 #2
    Just try to evaluate curlE : (curlE)i = εijkjEk and compare the result with the definition of cross product : (A×B)i = εijkΑjBk. You don't have to try to evaluate the Fourier integral; actually you can't, since E(k,w) is unknown (if known, then the field E(r,t) would also be known). Make the partial derivations of the Fourier integral by derivating directly the integrand.
     
  4. Dec 13, 2012 #3
    I still don't get how to do this :frown:

    So I should take [tex](∇E)i = ε_{ijk}∂_{j}E_{k}[/tex] and compare the result with the definition of cross product [tex](A×B)i = ε_{ijk}Α_{j}B_{k}[/tex]?

    So [tex](∇E) = ∂_{j}E_{k} = A_{j}B_{k}=AXB=kXE[/tex]?
     
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