equipotentialforbinding

Equipotential lines and surfaces are loci of constant electric potential in a field. In electrostatics, the potential V is a scalar field such that E = -∇V, where E is the electric field. An equipotential is a surface (or line in two dimensions) on which V takes the same value. Because the gradient is perpendicular to level surfaces, the electric field is everywhere perpendicular to equipotential surfaces. The potential difference between two points equals the negative line integral of E along any path connecting them: ΔV = -∫ E · dl.

For a single point charge q, equipotential surfaces are spheres centered on the charge. For multiple charges, the surfaces become distorted, reflecting the superposition of potentials. The magnitude of E is larger where equipotentials crowd together, and smaller where they are spaced farther apart.

Within a conductor at electrostatic equilibrium, E = 0 inside, so V is constant there; the interior is an equipotential region and the conducting surface itself is an equipotential surface.

Equipotentials extend to gravitational fields as well, where the gravitational field g = -∇Φ and surfaces of constant Φ are the gravitational equipotentials.

Applications include visualizing fields, solving boundary-value problems, shielding design, and computing capacitances. Equipotential maps also aid intuition in other areas of physics and engineering, where analogous potential functions exist.