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Curl of the curl?

  1. Aug 27, 2010 #1
    given curl E = -1/c*([tex]\partial[/tex]H/[tex]\partial[/tex]t)
    div E = 0
    div H = 0
    curl H = 1/c*([tex]\partial[/tex]E/[tex]\partial[/tex]t), find

    [tex]\nabla[/tex] x ([tex]\nabla[/tex] x E)

    how do i take [tex]\nabla[/tex] x curl E? i tried to do it by determinants, but i'm not sure which values correspond to the i, j, and k. so my next assumption is that there is some property that i can take advantage of to solve the problem. please help. thanks.
     
  2. jcsd
  3. Aug 27, 2010 #2

    lanedance

    User Avatar
    Homework Helper

    [tex]
    \nabla \times E = -\frac{1}{c} \frac{\partial H}{\partial t}
    [/tex]


    you could do it directly
    [tex]
    \nabla \times (\nabla \times E)
    =\nabla \times ( -\frac{1}{c} \frac{\partial H}{\partial t})
    =-\frac{1}{c}(\nabla \times \frac{\partial H}{\partial t})
    [/tex]

    and i think it should be ok to take the time derivative outside the curl, though you may want to confirm that...
    [tex]
    -\frac{1}{c}(\nabla \times \frac{\partial H}{\partial t})=-\frac{1}{c}\frac{\partial}{\partial t}(\nabla \times H)
    [/tex]

    and it should follow, otherwise, if you know the original field you could make use of the equality
    [tex]
    \nabla \times (\nabla \times E)
    =\nabla (\nabla \cdot E) - \nabla^2 E
    [/tex]

    which should simplify with some of the other info
     
    Last edited: Aug 27, 2010
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