Equipotential Surfaces and Their Link to the Field

Work to move a charge equals charge times the potential difference between start and end. If start and end sit on the same equipotential surface, that difference is zero, so the work is zero no matter how far or in what path you move, as long as you stay on the surface. Around a point charge the equipotentials are spheres centered on the charge, so sliding sideways along one of those spheres costs nothing.

The field always points in the direction potential drops fastest, and that direction is always perpendicular to the equipotential surface at that point. If the field had any component along the surface, moving along the surface would take work, contradicting what an equipotential surface is by definition. So field lines and equipotential surfaces always meet at right angles. Where surfaces are drawn close together, the potential is changing fast over a short distance, meaning the field is strong there; where they spread out, the field is weak. For a dipole, the equipotentials bunch up tightly near each charge and in the region between the charges, showing where the field is most intense, and open out far away where the field fades.

This connects directly to the field as the local steepness of the potential hill. Potential is like elevation; the field is like the slope. A place can sit at zero elevation while the hill is still steep there, just as potential can be zero at a point while the field is strong, because the field cares about the rate of change, not the value itself.