Potential Energy

Potential energy is energy associated with the configuration of a physical system, such as the relative positions of interacting objects or the deformation of an elastic material. Common examples include a raised object interacting with Earth, a compressed spring, and separated electric charges. Unlike kinetic energy, which depends on motion, potential energy describes configuration-dependent interactions. It is usually denoted by $U$, although $V$ is also common. Its unit in the International System of Units is the joule, the same unit used for other forms of energy. (openstax.org)

Definition and reference level

In classical mechanics, potential energy is defined through the work performed by a conservative force. Such a force does work that depends only on the initial and final configurations, not on the path connecting them. For motion from point $A$ to point $B$,

The line integral therefore measures the negative change in potential energy. If the force does positive work, the associated potential energy decreases; if work is done against the force, it increases. Equivalently, a conservative force does zero net work around any closed path. (openstax.org)

The reference value of potential energy is arbitrary in ordinary classical mechanics. Adding a constant to $U$ changes neither potential-energy differences nor the resulting forces. A convenient configuration may therefore be assigned $U=0$. Negative potential energy is not intrinsically abnormal: it indicates a value below the chosen reference. Potential energy properly belongs to an interacting system—for example, an object and Earth—although it is often attributed to one object when the other components are treated as fixed. (openstax.org)

Relation to force and energy conservation

The force can be recovered from the potential-energy function. In one dimension,

where $dU/dx$ is its derivative. In three-dimensional Cartesian coordinates,

with $\nabla U$ the gradient. The negative sign means that the force points toward decreasing potential energy, not necessarily toward decreasing coordinate values. A steeper potential-energy curve corresponds to a larger force magnitude. (openstax.org)

For a system governed by time-independent conservative interactions, with no other energy transfers, the mechanical energy is

A falling object or an oscillating ideal spring illustrates the exchange between kinetic and potential energy. When forces such as friction do work, mechanical energy need not remain constant. In the standard particle description, their work satisfies $W_{\mathrm{nc}}=\Delta(K+U)$. This does not imply destruction of energy: energy may instead enter internal degrees of freedom or be transferred to the surroundings. (openstax.org)

Gravitational potential energy

Near the surface of Earth, where gravitational acceleration $g$ is approximately constant, the change in potential energy of an object of mass $m$ is

Choosing zero at height $h=0$ gives $U=mgh$. Raising the object increases the potential energy of the object–Earth system; lowering it decreases that energy. The approximation becomes less accurate when the change in distance from Earth’s center is large enough for $g$ to vary appreciably. (openstax.org)

For two point masses $M$ and $m$, Newtonian gravity gives

when zero energy is assigned at infinite separation. Here $G$ is the gravitational constant and $r$ is their separation. Bringing the masses closer makes $U$ more negative. The same formula applies to nonoverlapping, spherically symmetric bodies when $r$ is measured between their centers. For an isolated two-body system, negative total energy relative to this reference indicates gravitationally bound motion. (openstax.org)

Elastic and electric potential energy

An ideal spring obeying Hooke’s law exerts the restoring force $F_x=-kx$, where $k$ is the spring constant and $x$ is displacement from its undeformed length. Setting $U=0$ at $x=0$ gives

Both compression and extension increase this energy. The quadratic expression applies within the range where the spring’s response is approximately linear and energy losses are negligible. (openstax.org)

For two stationary point charges in vacuum, Coulomb’s law yields

with zero at infinite separation. Like charges have positive interaction energy under this convention; opposite charges have negative interaction energy. For a charge $q$ in an externally prescribed electrostatic field, $U=q\phi$, where $\phi$ is the electric potential. Potential is energy per unit charge, rather than energy itself. A system of several point charges has interaction energy equal to the sum over distinct pairs. (openstax.org)

Potential-energy diagrams and quantum mechanics

A graph of $U(x)$ reveals possible motion and equilibrium. With fixed mechanical energy $E$, classical motion is allowed where $K=E-U(x)\geq0$. Ordinary turning points occur where $E=U(x)$ and the velocity reverses. Equilibrium occurs where $dU/dx=0$. A strict local minimum is stable, while a local maximum is unstable; where the second derivative vanishes, higher-order behavior must be examined. (openstax.org)

In quantum mechanics, the potential-energy function enters the Schrödinger equation and helps determine the wave function and allowed energies. Unlike a classical particle, a quantum particle can have nonzero probability in a region where $U>E$. Transmission through a finite potential barrier is called quantum tunneling; for a stationary barrier, it does not require a violation of energy conservation. (ocw.mit.edu)