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Gauss' Law as a derivative of the electromagnetic field tensor

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

    Prove [tex]\nabla \bullet E =4 \pi \rho[/tex] from [tex]\partial_{\beta}F^{\alpha \beta}=4 \pi J^{\alpha}[/tex] where [tex]J^{\alpha}=(\rho, J^{1}, J^{2}, J^{3})[/tex].

    2. Relevant equations

    We are given that [tex]F_{\alpha \beta}[/tex] is

    0~~~~E_x~~~E_y~~~E_z
    -E_x~~~0~~~~-B_z~~B_y
    -E_y~~B_z~~~~0~~~-B_x
    -E_z~~-B_y~~~B_x~~~0

    (Sorry, don't know how to do matrices.)

    Raising the indices I should get

    0~~~-E_x~~-E_y~~-E_z
    E_x~~~0~~~~-B_z~~B_y
    E_y~~B_z~~~~0~~~-B_x
    E_z~~-B_y~~~B_x~~~0

    3. The attempt at a solution

    [tex]\partial_{\beta}F^{\alpha \beta}=4 \pi J^{\alpha}=>-\partial_{i}F^{0i}=\rho=-\partial_{i}E_{i}[/tex]. I don't know why I keep getting that pesky negative sign! Can anyone point me in the right direction?
     
  2. jcsd
  3. Nov 1, 2009 #2
    For whatever reason I can't edit the latex, but it should say it as follows:

    (interestingly enough after this post the op corrected itself)

    1. The problem statement, all variables and given/known data

    Prove [tex]\nabla \bullet E =4 \pi \rho[/tex] from [tex]\partial_{\beta}F^{\alpha \beta}=4 \pi J^{\alpha}[/tex] where [tex]J^{\alpha}=(\rho, J^{1}, J^{2}, J^{3})[/tex].

    2. Relevant equations

    We are given that [tex]F_{\alpha \beta}[/tex] is

    0~~~~E_x~~~E_y~~~E_z
    -E_x~~~0~~~~-B_z~~B_y
    -E_y~~B_z~~~~0~~~-B_x
    -E_z~~-B_y~~~B_x~~~0

    (Sorry, don't know how to do matrices.)

    Raising the indices I should get

    0~~~-E_x~~-E_y~~-E_z
    E_x~~~0~~~~-B_z~~B_y
    E_y~~B_z~~~~0~~~-B_x
    E_z~~-B_y~~~B_x~~~0

    3. The attempt at a solution

    [tex]\partial_{\beta}F^{\alpha \beta}=4 \pi J^{\alpha}=>-\partial_{i}F^{0i}=\rho=-\partial_{i}E^{i}[/tex]. I don't know why I keep getting that pesky negative sign! Can anyone point me in the right direction?
     
  4. Nov 1, 2009 #3
    Bleh, never mind, I got it. Just got the rows and columns confused somehow... Thanks anyway, sorry for wasting your time.
     
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