equipotential

An equipotential is a surface or region on which the electric potential has a single, constant value. In electrostatics an equipotential surface is the locus of points where the electric potential V is constant, with potential defined relative to a chosen reference.

The electric field E is the negative gradient of the potential, E = -∇V. As a consequence, E is everywhere perpendicular to equipotential surfaces, and a test charge moving along an equipotential undergoes no net work by the field. The potential difference between two points is the integral of E along any path, which vanishes if both points lie on the same equipotential.

Examples: For a single point charge, the equipotential surfaces are spheres centered on the charge. In a uniform external field, the equipotential surfaces are planes perpendicular to the field. In complex charge configurations, the surfaces can be highly nonplanar but retain the property of being everywhere perpendicular to the field lines.

Mathematics: Equipotentials are level sets of the scalar potential, V(x) = c. In regions without charges, V satisfies Laplace's equation ∇^2V = 0. In general, equipotential surfaces are three-dimensional regions in space where V is constant.

Applications: Mapping equipotential surfaces helps visualize electric fields and potential distributions. They are relevant in the design of capacitors, insulation, and sensor systems; voltmeters measure potential differences between points on different equipotentials.