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Delta function representation from EM theory

  1. Jan 18, 2010 #1
    Claim:

    [tex]\nabla \cdot \frac{\hat{e}_r}{r^2}=4\pi\delta^3(\vec{x})[/tex]

    Anyone know of a proof of this? (or a reference which covers it?) We need to show that

    [tex]\frac{1}{4\pi}\int_0^R{(\nabla \cdot \frac{\hat{e}_r}{r^2})f(r)dr=f(0)[/tex].



    The claimed identity can be seen in the solution for the electric field of a point charge in EM theory, where

    [tex]\vec{E}=\frac{q}{r^2}\hat{e}_r[/tex]

    is the solution to

    [tex]\nabla \cdot \vec{E}=4\pi q\delta^3(\vec{x})[/tex]

    It is easy to show in this case that [tex]\nabla \cdot \vec{E}=0[/tex] everywhere but the origin, but I don't know how to show that the delta function relation holds at the origin.
     
    Last edited: Jan 18, 2010
  2. jcsd
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