Equipotential Surfaces

Think of a hiking map. Contour lines connect points at the same elevation, and walking along one means you neither climb nor descend. Equipotential surfaces do the same job for electric potential: every point on one surface has the identical value of V. Since work done by the electric force equals -q times the change in potential, and delta V is zero along the surface, sliding a charge anywhere on it costs zero work. Gravity never does work on you walking along a contour line; the electric field never does work on a charge sliding along an equipotential.

This fact forces a geometric rule. The electric field points in the direction potential drops fastest, like water flowing straight downhill, perpendicular to the contours. If the field had any component running along the surface, that component would push the charge and do work, which would change V where V is supposed to stay fixed. That contradiction is impossible, so the field must meet every equipotential surface head-on, at ninety degrees, with no exceptions.

Sketching follows directly from the field lines you already know. Near an isolated point charge, equipotentials are concentric spheres, since V depends only on distance. Around a dipole, draw the field lines curving from positive to negative charge, then draw closed, egg-shaped surfaces crossing each field line perpendicularly, bunching tightly near each charge where the field is strong and spreading out between them. Between two oppositely charged parallel plates, the field is uniform and horizontal, so the equipotentials become evenly spaced flat planes running parallel to the plates, each one a step of constant voltage between the two terminals.