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Homework Help: Fourier Tranform of electric dipole charge density

  1. Jul 5, 2011 #1
    1. The problem statement, all variables and given/known data

    This is supposed to be simple, so I guess I miss something..
    We have charge q at x1=d*cos (w*t), y=0, z=0. and charge -q at
    x2=-d*cos (w*t), y=0, z=0. I need to do fourier transform to the charge density.

    2. Relevant equations
    The fourier transform is : [itex]\rho_{\omega} = \int \rho (r,t)e^{i\omega t} dt [/itex]

    3. The attempt at a solution
    I first try only for the first charge. We have [itex]\rho (r,t) = \delta(x-d\cdot cos (\omega t))\cdot\delta(y)\cdot\delta(z) [/itex]
    But I don't know how to do the intergral :
    [itex]\rho_{\omega} = \int \delta(x-d\cdot cos (\omega t))\cdot\delta(y)\cdot\delta(z)e^{i\omega t} dt [/itex]
    Any ideas? Do I really need to do this integral?
  2. jcsd
  3. Jul 5, 2011 #2


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    Homework Helper

    Generally, if you have an integral over a delta function of another function, you can use the relation

    [tex]\delta(f(t)) = \sum_{j}\frac{\delta(t-t^\ast_j)}{|f'(t^\ast_j)|},[/tex]
    where the [itex]t^\ast_j[/itex] are the solutions of [itex]f(t^\ast) = 0 [/itex]. Note that this relation doesn't work if the derivative is zero at a solution.
  4. Jul 6, 2011 #3

    Thanks for your answer.
    Actually the question is to find the potential of this charges far from the origin,so I guess I should transform the dipole moment vector only..
    Using the identity for the delta function I get weird things like exp(arccos (x/d))...
    It is just interesting if the transform of the charge density could be useful.
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