Equipotential surface

The CED calls these isolines of electric potential, and the contour-map analogy is exact. Contour lines join points of equal height; equipotentials join points of equal potential.

Four rules, all of them statements from Topic 10.5.

• Equipotential lines represent lines of equal electric potential in space.
• Isolines are perpendicular to electric field vectors, and either map can be constructed from the other.
• An electric field vector points in the direction of decreasing potential.
• There is no component of the electric field along an isoline.

The last one is where the zero-work result comes from. Move a charge along an equipotential and $\Delta V = 0$, so $\Delta U_E = q \Delta V = 0$, so the electric force does no work on any path that stays on the surface. Getting anywhere on it costs the same.

Reading a map: closely spaced isolines mean a strong field, because the same $\Delta V$ falls across a shorter distance. Evenly spaced parallel isolines mean a uniform field. Two isolines can never cross, since one point cannot hold two potentials.

Shapes worth recognizing at a glance are concentric spheres around an isolated point charge, flat parallel planes between capacitor plates, and the surface of a conductor in electrostatic equilibrium, which is an equipotential precisely because the field just outside it is perpendicular to it.