# What is the volume charge density (in spherical coordinates)?

What is the volume charge density (in spherical coordinates) of a uniform, in finitesimally thin spherical shell of radius $$R$$ and total charge $$Q$$, centered at the origin? Give your answer in terms of the Dirac delta function.

My attempt:

I think the answer is $$\rho(\mathbf{r})=Q\delta^3(\mathbf{r}-R\hat{\mathbf{r}})$$ because the Dirac delta function will blow up only if the radial component of $$\mathbf{r}$$ is $$R$$. But I just wanted to confirm.

The delta function you wrote down is tricky to interpret (what’s the definition of $$\hat{r}$$ inside a delta function?), and while it could be made rigorous, there’s a more systematic approach that is widely generalizable.

You want to find a distribution $$\rho$$ which only has support on the spherical shell $$r=R$$, and has spherical symmetry. As an ansatz, we may write

$$\rho=A\delta(r-R).$$

The next requirement is that the total charge is $$Q$$. This gives

$$\int\mathrm{d}^3\textbf{x}\,\rho(\textbf{x})=4\pi\int_0^{\infty}\mathrm{d}r\,r^2\rho(r)=4\pi R^2A=Q.$$

Thus, $$A=Q/4\pi R^2$$, and

$$\rho=\frac{Q}{4\pi R^2}\delta(r-R).$$