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Homework Help: Energy of electromagnetic field

  1. May 14, 2015 #1


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    Hey everyone!

    I am supposed to calculate the energy contribution of the magnetic field term of an electromagnetic field.

    Basically the term is the following:

    [itex]\int_\Omega dx^3 (curl(\vec{A}))^2[/itex]

    And we can use the following two equations for simplifying:



    [itex]\Box A=0[/itex]

    So basically what I did was express the curl in terms of the levi civita tensor.
    Then you can simplify the volume integral to:

    [itex]\int_\Omega dx^3 \sum_{m,l}\frac{\partial A_m}{\partial x_l} (\frac{\partial A_m}{\partial x_l}-\frac{\partial A_l}{\partial x_m})[/itex]

    Then I can use a trick and take out the first partial derivative with respect to [itex]x_l[/itex]:

    [itex]\int_\Omega dx^3 \sum_{m,l}\frac{\partial}{\partial x_l} \left(A_m(\frac{\partial A_m}{\partial x_l}-\frac{\partial A_l}{\partial x_m})\right)-\left(A_m(\frac{\partial^2 A_m}{\partial x_l^2}-\frac{\partial^2 A_l}{\partial^2 x_mx_l})\right)[/itex]

    The term [itex]\frac{\partial^2 A_l}{\partial^2 x_mx_l}[/itex] vanishes since it contains the divergence of [itex]\vec{A}[/itex] and the term [itex]A_m\frac{\partial^2 A_m}{\partial x_l^2}[/itex]is exactly the result we want: [itex]\frac{1}{c^2}\vec{A}\cdot \ddot{\vec{A}}[/itex] because of the d'Alembertian relation.
    So what is left to be shown is that the first term [itex]\int_\Omega dx^3 \sum_{m,l}\frac{\partial}{\partial x_l} \left(A_m(\frac{\partial A_m}{\partial x_l}-\frac{\partial A_l}{\partial x_m})\right)[/itex] vanishes, which I have no clue how to show. Hope someone can help me.
  2. jcsd
  3. May 19, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
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