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Transforming dipole moment

  1. Jan 7, 2014 #1
    I just wondered how I transform electromagnetic dipole moments.
    For example assuming I have magnetic dipole moment μ in a frame without E dipole moment. Then I boost orthogonal to μ. Now I would like to determine the electric dipole moment.

    I could use the magnetic potential and transform it to the new frame. But this quiet tedious to me.
    My second guess was to argue that there is a linear dependence of the magnetic fields on μ.
    And the E field in the new frame will also have a linear dependence on B and thus on μ. By looking at the boost component I could also find a relation. But this would assume that I have an electric dipole moment in the new frame.

    I just wondered whether there is any easier and more elegant method.
    I would be very glad about any advise.
    Thank you :D
     
  2. jcsd
  3. Jan 7, 2014 #2

    maajdl

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    Gold Member

    Just a thought: writing the Lagrangian in terms of moments.
     
  4. Jan 7, 2014 #3

    vanhees71

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    Science Advisor
    2016 Award

    It's just better to start from covariant expressions. As the electromagnetic field itself the electric and magnetic dipole moment densities build an antisymmetric 2nd-rank tensor in Minkowski space-time:

    [tex]M^{\alpha \beta}=\begin{pmatrix}
    0 & P_1 & P_2 & P_3 \\
    -P_1 & 0 & -M_3 & M_2 \\
    -P_2 & M_3 & 0 & -M_1 \\
    -P_3 & -M_2 & M_1 & 0
    \end{pmatrix},[/tex]
    where [itex]\vec{P}[/itex] and [itex]\vec{M}[/itex] are the electric and magnetic dipole density vector fields of the 1+3-dimensional formalism.

    They transform under Lorentz transformations as any other 2nd rank tensor components
    [tex]M'^{\gamma \delta} = {\Lambda^{\gamma}}_{\alpha} {\Lambda^{\delta}}_{\beta} M^{\alpha \beta}.[/tex]
     
  5. Jan 7, 2014 #4
    oh really? I was looking for sth like this. thats good thank you!
     
  6. Jan 7, 2014 #5
    Magnetization-polarization tensor I found it now
     
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