Potential of an inverse-radius spherical charge density
= Potential of an inverse-radius spherical charge density
A charge cloud $\rho=A/r$ for $0<r<a$ encloses $Q(r)=2\pi Ar^2$. With no central point charge and potential zero at infinity, <Gauss's law> gives $\phi=A(2a-r)/(2\epsilon_0)$ inside and $Aa^2/(2\epsilon_0r)$ outside. The <density> singularity is integrable, and the potential is finite at the origin.