Potential energy

Potential energy is the energy held by an object or system because of its position in relation to other objects, or because of how its own particles are arranged. This energy equals the work needed to overcome restoring forces, like gravity or the force inside a stretched spring. The Scottish engineer and physicist William Rankine coined the term "potential energy" in the 19th century, drawing on Aristotle's older idea of potentiality.

Several common types exist. Gravitational potential energy involves the force of gravity, elastic potential energy involves a deformed spring, and electric potential energy involves an electric charge in an electric field. The SI unit for all energy, including potential energy, is the joule (symbol J). Potential energy arises from forces whose total work on a body depends only on where the body starts and ends, not on the path it takes. These are called conservative forces. When such a force varies across space, it creates a force field—a vector field. A conservative vector field can be expressed as the gradient of a scalar function called a scalar potential, and potential energy is derived from this function.

Different forces give rise to different types of potential energy: elastic force yields elastic potential energy, gravitational force yields gravitational potential energy, and the Coulomb force yields electric potential energy. Nuclear forces acting on baryon charge produce nuclear potential energy, and intermolecular forces produce intermolecular potential energy. Chemical potential energy, like that in fossil fuels, comes from the Coulomb force during rearrangements of electrons and nuclei in atoms and molecules. Thermal energy has two parts: the kinetic energy of random particle motion and the potential energy of their configuration.

For a conservative force, the work done is given by \( W = -\Delta U \), where \( \Delta U \) is the change in potential energy. The negative sign means that doing work against the force field increases potential energy, while work done by the force field decreases it. Common symbols for potential energy include PE, U, V, and Ep. Potential energy is stored in the force field when an external force works against it, such as lifting a mass or stretching a spring. If the external force is removed, the force field does work on the body, moving it back to its original position and reducing potential energy. For example, a ball of mass \( m \) dropped from height \( h \) has gravitational potential energy \( U_g = mgh \), assuming constant gravitational acceleration \( g \). More formally, potential energy is the difference between an object's energy at a given position and its energy at a reference position.

Around 1840, scientists began defining and understanding energy and work. Rankine introduced "potential energy" in 1853 as part of a deliberate effort to create terminology, using the pair "actual" versus "potential," which traced back to Aristotle. In 1867, he described potential energy as "energy of configuration," contrasting it with "energy of activity." That same year, William Thomson introduced "kinetic energy" as the opposite of potential energy, proposing that all actual energy took the form \( \frac{1}{2}mv^2 \). Once this idea became widely accepted, the term "actual energy" gradually fell out of use.

Potential energy is tightly linked to forces in conservative fields. If the work done by a force moving a body from point A to point B does not depend on the path, the force is conservative. This work from A assigns a scalar value to every other point in space, defining a scalar potential field. The force can then be expressed as the negative of the vector gradient of that potential field. When work is path-independent, a function \( U(x) \) exists that can be evaluated at points \( x_A \) and \( x_B \) to give the work over any trajectory between them. By convention, this function is defined with a negative sign so that positive work reduces the potential, expressed as \( W = \int_C \mathbf{F} \cdot d\mathbf{x} \).