Question

Why must electric field lines be perpendicular to equipotential surfaces?

Answer 1

Because if they weren’t, then there would be some component of the electric field along the equipotential surface, and some work would have to be done against that field-component if we wanted to move a charge around on the surface— and that work done would give rise to a potential difference between any two points on the surface, because [math]W = q \Delta V.[/math]