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Delta function help

  1. Feb 23, 2005 #1
    I'm having trouble with the following problem:
    The spherically symmetrical electrostatic potential of a particular object is given (in spherical coordinates) by:
    [tex]V(\vec{r})=V(r)=c\frac{exp{(\frac{-2r}{a})}}{4\pi\varepsilon r} (1+\frac{r}{a})[/tex]
    I found the electrostatic field in spherical coords (I think it's right),
    [tex]\vec{E}(\ver{r})=\frac{c}{4 \pi \varepsilon} (\frac{2}{a r} +\frac{1}{r^2} +\frac{2}{a^2}) exp(\frac{-2 r}{a})\hat{r}[/tex]
    Now I'm looking for the charge density [itex]\rho(\vec{r})[/itex] in spherical coords.
    My problem is with representing the singularities with the Dirac Delta function.
    I'm not sure how to do this.
    Any help would be greatly appreciated.
  2. jcsd
  3. Feb 23, 2005 #2


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    Compute [itex] \rho [/itex] first and then we shall see whether a delta-Dirac is necessary.

  4. Feb 23, 2005 #3
    I got
    [tex]\rho=\frac{-c}{r \pi a^2} (r-a+1) exp(\frac{-2 r}{a})[/tex]
    Will this need a Delta Function?
  5. Feb 23, 2005 #4
    This looks like a hydrogen atom type charge density?
    An easy way to find a delta function is to guess at where it is, integrate the E field over a sphere of radius r around it.. that will tell you the total charge inside the sphere as a function of r.. then take the limit as r goes to zero.. if that doesn't go zero, then you must have a point charge at the center of the sphere.
  6. Feb 23, 2005 #5


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    Go back to basics. A 1/r potential, being generated by a point charge of unit magnitude, is a green's function, and its source is represented by a delta function -- in this case delta(r). As r->0 your potential goes like 1/r, so there's a delta function. for practical purposes, del squared(1/r) = - delta(r) -- there could be a few 2pi s I've missed). With the chain rule, that's all you need. (Also see Jackson, or any E&M or Boundary Values or Potential Theory or..........)

    Reilly Atkinson
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