Q:

In electrostatics, if there is a current (i.e. time varying charge) how can there be a time invariant Electric field?

In electrostatics we consider the curl of B to be equal to conductivity multiplied with current density and the time varying Electric field component to be 0. but Electric fields are created by charges. And if there is a current i.e. i = dq/dt this means that the charge is varying with time. So how can the Electric field be independent of time ? 

A:

A current doesn't necessarily mean charge is changing over time. A wire has a net charge density of zero throughout, and the electrons moving through it doesn't change that.
Basically, as an electron leaves a particular location, a new one takes its place. So there's no net change in charge density.

