1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Energy of electromagnetic field

  1. May 14, 2015 #1

    VVS

    User Avatar

    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]div(A)=0[/itex]

    and

    [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?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Energy of electromagnetic field
  1. Electromagnetic field (Replies: 4)

Loading...