Inmathematics andphysics, anequipotential orisopotential refers to aregion in space where every point is at the samepotential.[1][2][3] This usually refers to ascalar potential (in that case it is alevel set of the potential), although it can also be applied tovector potentials. An equipotential of a scalar potentialfunction inn-dimensional space is typically an (n − 1)-dimensional space. Thedel operator illustrates the relationship between a vector field and its associated scalar potential field. An equipotential region might be referred as being 'of equipotential' or simply be called 'an equipotential'.
An equipotential region of a scalar potential in three-dimensional space is often anequipotential surface (orpotentialisosurface), but it can also be a three-dimensionalmathematical solid in space. Thegradient of the scalar potential (and hence also its opposite, as in the case of a vector field with an associated potential field) is everywhere perpendicular to the equipotential surface, and zero inside a three-dimensional equipotential region.
Electrical conductors offer an intuitive example. Ifa andb are any two points within or at the surface of a given conductor, and given there is no flow of charge being exchanged between the two points, then thepotential difference is zero between the two points. Thus, an equipotential would contain both pointsa andb as they have the samepotential. Extending this definition, an isopotential is the locus of all points that are of the same potential.
Gravity is perpendicular to the equipotential surfaces of thegravity potential, and inelectrostatics and steadyelectric currents, theelectric field (and hence the current, if any) is perpendicular to the equipotential surfaces of theelectric potential (voltage).
In gravity, a hollow sphere has a three-dimensional equipotential region inside, with no gravity from the sphere (seeshell theorem). In electrostatics, a conductor is a three-dimensional equipotential region. In the case of a hollow conductor (Faraday cage[4]), the equipotential region includes the space inside.
A ball will not be accelerated left or right by the force of gravity if it is resting on a flat,horizontal surface, because it is an equipotential surface.For thegravity of Earth, the correspondinggeopotentialisosurface (theequigeopotential) that best fitsmean sea level is called thegeoid.