Is there a theory of electrostatics in the presence of conductors with finite conductivity? A substantial part of my university EM course was devoted to the study of conductors in the presence of static electric fields. Poisson and Laplace equations were covered, along with the uniqueness theorems and the like.
However, as far as I understand, the whole theory applied to perfect conductors, i.e. those with conductivity $\sigma =+\infty$, since the electric field ${\bf  E}=\frac{{\bf J}}{\sigma}$ need not be null when $\sigma$ is finite. 
For example, let's say we have a spherical conductors such that $\sigma < +\infty$, and a fixed point charge $q$ at a certain distance from it. Assume for simplicity that the conductor is uncharged, $Q=0$. How does one go about determining the EM fields in this scenario? Is there a whole different theory for conductors with finite conductivity or am I misunderstanding something?
 A: In electrostatics, it doesn't matter.
A finite conductivity just means that it will take a non-zero amount of time for the charges to rearrange themselves. However, the charges will eventually get to the same place as the infinite conductivity case because that's still the lowest energy state. Think about a capacitor discharging thru a wire. The end state is the same whether the wire has zero or non-zero resistance.
Since electrostatics is about the steady state, we don't care about the transient where it takes charges a little while to rearrange themselves.
The case is very different for electrodynamics, where we very much care about the transients.
A: You do not need infinite conductivity to describe conductors in electrostatic problems. It suffices to have a finite conductivity $\sigma$ large enough so that the dielectric relaxation time $\tau=\epsilon/\sigma$ of the involved metals is short compared to the characteristic times of the experiments. Then the surface charges always produce a zero electric field inside the metal and along the surfaces, corresponding to a constant electrostatic potential of a metal. 
