- #1

coaxmetal

- 4

- 0

## Homework Statement

show that the charge distribution ([tex]|\vec{r}|\equiv r [/tex])

[tex]\rho(r) = Z\delta^3(\vec{r})-\frac{Ze^{-r/R}}{4\pi R^2r}[/tex]

has zero net charge for any Z and R. Explain the meaning of Z.

## Homework Equations

none given, but divergence (gauss) theorem and poisson's equation may be necessary.

## The Attempt at a Solution

integrate to get net charge:

[tex]Q = \displaystyle\int_V \rho(\vec{r})d\tau[/tex]

I can split this into 2 integrals and use the volume element for spherical polar coordinates ([tex]d\tau = r^2dr \sin{\theta} d\theta d\varphi[/tex]), so the first integral is

[tex]\displaystyle\int_V Z\delta^3(\vec{r})r^2dr \sin{\theta} d\theta d\varphi[/tex]

This is what I get stuck on. I don't know how to handle the delta function (I assume the Dirac delta, I am given no other information) raised to a power. The fact that the volume integral is a triple integral seems like it may be important here, but I don't recall any tricks or anything about volume integrals.