potentials

In physics and mathematics, a potential is a scalar field that encodes stored energy and from which forces can be derived. A potential function assigns to each point a scalar value. If a force is conservative, the force F is the negative gradient of a potential energy U, F = -∇U. In electrostatics, the relevant scalar is the electric potential V, with E = -∇V. The potential difference between two points equals the work done per unit charge in moving between them.

Examples include gravitational potential energy U = mφ near a planet, with φ(r) = -GM/r for a point mass; and electric potential V(r) = kQ/r for a point charge. Potential energy is not the same as the potential itself; it is obtained by multiplying the potential by the relevant quantity (U = mφ for gravity, U = qV for electricity).

Key properties: conservative forces yield path-independent work; equipotential surfaces are where the potential is constant. In electrostatics and gravity, potentials satisfy differential equations: ∇·E = ρ/ε0 leads to Poisson's equation ∇^2V = -ρ/ε0; in charge-free regions, ∇^2V = 0 (Laplace equation).

In mathematics, potential theory studies harmonic functions, Green's and Poisson's equations, and boundary value problems. It provides tools such as Green's identities and conformal mapping, with applications in physics, engineering, and probability.

Other uses of the term include chemical potential in thermodynamics, and potential energy surfaces in chemistry and molecular dynamics. In physics, potentials extend beyond scalar fields to include vector potentials for magnetic fields, where B = ∇×A and, in quantum mechanics, the scalar and vector potentials enter the Schrödinger equation.