What is the size of the electic field inside a charged conductor?
The electric field in the bulk of a conductor, charged or otherwise, is zero (at least in the static case) .
Note that there is a non-zero electric field in a conductor when a current is flowing through it.
A conductor has mobile charge carriers - this, after all, is what makes it a conductor. As a result, even if a electric field is set up inside a conductor, the charge carriers will move in response. If, as in most cases, the carriers are electrons, they will move against the field. This will cause a charge separation, giving rise to a counter field. As long as the original field is larger than this opposing field, the electrons will continue to move, increasing the counter field further.
The process will stop only when the two fields balance out - leaving no net electric field inside the conductor.
All this takes a very short time to happen, and once things settle down, the electric field will vanish.
Note that when a current is flowing in a conductor electrons which move to one end are carried back to the other end by the external power source (battery). As a result, electrons do not accumulate at one end. As a result, there is no opposing electric field. A current carrying conductor does have an electric field inside. This electric field is the potential difference divided by the length of the conductor, leading to
Thus the electric field in a conductor carrying a current is proportional to the current density and the specific resistance.