Question

Can someone please explain to me why an electric field cannot exist in a conductor for electrostatics, but when the charge moves, such as when there is a current, an electric field has to exist in a conductor (such as a wire) in order to move those charges?

Answer 1

One characteristic of a conductor at electrostatic equilibrium is that the electric field anywhere beneath the surface of a charged conductor is zero. If an electric field did exist beneath the surface of a conductor (and inside of it), then the electric field would exert a force on all electrons that were present there. This net force would begin to accelerate and move these electrons. But objects at electrostatic equilibrium have no further motion of charge about the surface. So if this were to occur, then the original claim that the object was at electrostatic equilibrium would be a false claim. If the electrons within a conductor have assumed an equilibrium state, then the net force upon those electrons is zero. The electric field lines either begin or end upon a charge and in the case of a conductor, the charge exists solely upon its outer surface. The lines extend from this surface outward, not inward. This of course presumes that our conductor does not surround a region of space where there was another charge.
while if charge moves ( current flows through wire) that means they experience a force F
but F=e*E as e=electronic charge constant so E =electric field has to exist:)

Answer 2

Because electrodynamics does not describe an equilibrium situation. Transient and time-dependent electric fields certainly can exist within conductors, and do so when they are conducting currents. Electrostatics describes the situation after all currents have stopped flowing, and all charge has reached its position of lowest potential energy. Only under those circumstances can you say there are no electric fields in a conductor.