Finding charge density from spherical potential with delta function singularities

[upsidedown314]

Hello,
I'm having trouble with the following problem:
The spherically symmetrical electrostatic potential of a particular object is given (in spherical coordinates) by:
$$V(\vec{r})=V(r)=c\frac{exp{(\frac{-2r}{a})}}{4\pi\varepsilon r} (1+\frac{r}{a})$$
I found the electrostatic field in spherical coords (I think it's right),
$$\vec{E}(\ver{r})=\frac{c}{4 \pi \varepsilon} (\frac{2}{a r} +\frac{1}{r^2} +\frac{2}{a^2}) exp(\frac{-2 r}{a})\hat{r}$$
Now I'm looking for the charge density $\rho(\vec{r})$ 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.
Thanks

 

[dextercioby]

Compute $\rho$ first and then we shall see whether a delta-Dirac is necessary.

Daniel.

 

[upsidedown314]

I got
$$\rho=\frac{-c}{r \pi a^2} (r-a+1) exp(\frac{-2 r}{a})$$
Will this need a Delta Function?

 

[kanato]

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.

 

[reilly]

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...)

Regards,
Reilly Atkinson