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Using divB as a condition for wave

  1. Nov 9, 2009 #1
    1. The problem statement, all variables and given/known data

    By considering divergence, show whether the expression

    B=kBzei(kzz-wt)

    is a valid function for an electromagnetic plane wave.

    2. Relevant equations

    divB=0


    3. The attempt at a solution

    I have found divB=ikzBzei(kzz-wt).

    Does this satisfy divB=0 because it is imaginary?
     
  2. jcsd
  3. Nov 9, 2009 #2

    gabbagabbahey

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    The field you give is usually denoted as

    [tex]\tilde{\textbf{B}}=\hat{\mathbf{k}}\tilde{B}_ze^{i(k_z z- \omega t)}[/tex]

    and is itself complex-valued... The actual magnetic field [itex]\textbf{B}[/itex] is taken to be the real part of [tex]\tilde{\textbf{B}}[/itex] and so

    [tex]\mathbf{\nabla}\cdot\textbf{B}=\mathbf{\nabla}\cdot\text{Re}\left[\tilde{\textbf{B}}\right]=0[/tex]

    So, only if the divergence of the real part of the field you gave vanishes, can it be a valid magnetic field (The fact that the magnetic field is polarized in the same direction as the propagation should give you a little bit of hesitation here).
     
    Last edited: Nov 9, 2009
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