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

Question

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 ?

Answer 1

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.

Answer 2

If we are considering electrostatics,then yes the time varying component of the Electric field is zero and thus no magnetic field.

Answer 3

You have to differentiate between stationary, divergence-free currents and currents due to a locally changing charge density. In a stationary loop current there is no accumulation or depletion of charge anywhere. This is typically the case in all dc electrical circuits. In general, from the Maxwell law you mentioned follows the general law of conservation of charge $$ \nabla\cdot(\nabla\times{\vec B})= \mu_0\nabla\cdot(\vec J +\epsilon_0 \epsilon_r\frac{\partial \vec E}{\partial t})=0 $$ $\implies$ $$ \nabla\cdot{\vec J}=-\frac {\partial \rho}{\partial t}$$ Thus you can have divergence-free currents without any charges changing anywhere. You have just a charge flow constant in time without any time varying charge anywhere.