# Replacing an equipotential surface with a conductor

If i replace an equipotential surface generated by a charge distribution with a conductor (i already know that the conductor has an equipotential surface), what assures me that the original charge distribution's field is not modified by the conductor?

• Have a look at uniqueness theorems. – sbp Nov 17 '17 at 12:48

Since the field is uniquely determined from the potential: $$\vec E=-\vec\nabla V$$ there would be no difference in electric fields between the "before" and "after" situations since the equipotentials outside the conductor are the same (assuming there is no charge in the region where you locate the conductor). Since the field is the same before and after, there effect on the charge distribution will not change.