Can anyone help me to understand why equipotential surfaces must always be perpendicular to the electric field lines that pass through them? My textbook gives what seems to be a simple mathematical explanation, but the logic is escaping me. It states that dV= -E.ds = 0, (where the "." appears to be dot product of the two vectors -E and ds and so E must be perpendicular to the displacement of the equipotential surface. I can understand that the dot product=0 means two vectors are perpendicular. I am looking for a more intuitive explanation to really understand what is going on.