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Maxwell's Equations in 4-D Space

  1. Mar 6, 2015 #1

    B3NR4Y

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    Gold Member

    1. The problem statement, all variables and given/known data
    We are using Gaussian units. To begin, the vector Hk is the magnetic field, which relates to a second rank antisymmetric tensor, Fij.
    a.) Prove
    [tex] F_{ij} = \frac{\partial A_{j}}{\partial x_{ i }} - \frac{\partial A_{ i }}{\partial x_{j}} = \partial_{ i } A_{j} - \partial_{j} A_{ i } [/tex]

    where [itex] \nabla \times \vec{A} = \vec{H}[/itex]​
    I'm only posting this first part because I feel like once I get it everything will roll out and I'll be able to answer it all.

    2. Relevant equations
    In the last problem, we proved that [itex] H_{k} = \frac{1}{2} \epsilon_{ijk} F_{ij} \iff F_{ij} = \epsilon_{ijk} H_{k} [/itex]

    3. The attempt at a solution
    The last problem says that the tensor Fij can be written as the following matrix:
    [tex]
    \left(\begin{array}{ccc}
    0 & H_{z} & -H_{y} \\
    -H_{z} & 0 & H_{x} \\
    H_y & -H_x & 0 \end{array}\right) [/tex] and the hint says that
    [tex] (curl \, \vec{A})_{i} = \epsilon_{ijk} \partial_{j} A_{k} = H_{i} [/tex] So, [itex] F_{12} = \partial_{1} A_2 - \partial_2 A_1 [/itex]. Do I just et the partials from the definition of the curl of A being H?
     
    Last edited: Mar 6, 2015
  2. jcsd
  3. Mar 6, 2015 #2

    phyzguy

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    There are several problems with what you've written. First of all, in 4D space-time, the matrix representing Fij is a 4x4 antisymmetric matrix, not 3x3 as you've written it. It contains both the electric field and magnetic field. Secondly, even what you have written for the space part of Fij is wrong, since it should contain Hx, Hy, and Hz. Normally in 4D space-time, time is x0, and the space components are x1,2,3.
    So F12 = ∂1 A2 - ∂2 A1 = ∂x Ay - ∂y Ax = Hz. Try working through it again.
     
  4. Mar 6, 2015 #3

    B3NR4Y

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    Gold Member

    This is the first part of my problem, I should have been more clear, sorry. Later on in the problem it extends the tensor to a 4x4, antisymmetric tensor, but I have trouble proving the first part. I only posted the first part because I felt like I could get the rest if I could get the first part. I mistyped as it was early in the morning, I'll fix my first post, the matrix is:
    [tex]
    \left(\begin{array}{ccc}
    0 & H_{z} & -H_{y} \\
    -H_{z} & 0 & H_{x} \\
    H_y & -H_x & 0 \end{array}\right) [/tex]

    What I noticed, is that [itex] (\nabla \times \vec{A})_{ i } = \epsilon_{ijk} \partial_{j} A_{k} = H_{i} \rightarrow (\nabla \times \vec{A})_{ 1 } = \partial_{2} A_{3} - \partial_{3} A_{2} = H_{1} [/itex], can I use this in my tensor?
     
  5. Mar 6, 2015 #4

    phyzguy

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    Yes, this looks correct now.
     
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