Potential (physics, mathematics, and general use)

Common types

Potential energy is energy associated with position or configuration, such as gravitational or elastic potential energy. The force on a particle in a conservative field is the negative gradient of the potential energy (F = -∇U).

Electric (electrostatic) potential is a scalar whose difference between two points equals the work per unit charge required to move a test charge. Related is the vector potential, a field from which magnetic fields are obtained; vector potentials are not unique and are subject to gauge freedom.

Chemical potential governs the tendency of species to diffuse, react, or partition between phases; it appears in thermodynamics and statistical mechanics.

Mathematical potential theory

Potential theory studies scalar potentials and related functions, especially harmonic functions that satisfy Laplace's equation. It provides tools such as Green's functions and boundary value methods used to solve problems in electrostatics, gravitation and fluid flow. A scalar potential is determined only up to an additive constant; boundary conditions and physical conventions fix that freedom.

Properties and significance

• Conservative fields: If a force is conservative, it can be written as the gradient of a potential.
• Nonuniqueness: Scalar potentials are defined up to a constant; vector potentials admit gauge transformations.
• Physical differences: Often only differences of potential are physically meaningful (for example, potential differences drive currents).
• Quantum mechanics: Potentials enter the equations of motion for wavefunctions and can affect observable phenomena even when local field values vanish.

Applications

Potentials are central in engineering and electronics (voltage and circuit behavior), geophysics (gravitational and magnetic surveys), chemistry and materials science (reaction equilibria and diffusion), and in mathematical modeling of boundary value problems. The idea of potential also extends metaphorically to human, economic or social contexts to describe latent capacity for development.