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Potential energy is energy stored within a physical system as a result of the position or configuration of the different parts of that system. It is called potential energy because it has the potential to be converted into other forms of energy, such as kinetic energy, and to do work in the process. The SI unit of measure for energy (including potential energy) and work is the joule (symbol J).

The term "potential energy" was coined by the 19th century Scottish engineer and physicist William Rankine.

## Overview

Potential energy is energy that is stored within a system. It exists when there is a force that tends to pull an object back towards some lower energy position. This force is often called a restoring force. For example, when a spring is stretched to the left, it exerts a force to the right so as to return to its original, unstretched position. Similarly, when a mass is lifted up, the force of gravity will act so as to bring it back down. The initial action of stretching the spring or lifting the mass both require energy to perform. The energy that went into lifting up the mass is stored in its position in the gravitational field, while similarly, the energy it took to deform the spring is stored in the metal. According to the principle of conservation of energy, energy cannot be created or destroyed; hence this energy cannot disappear. Instead, it is stored as potential energy. If the spring is released or the mass is dropped, this stored energy will be converted into kinetic energy by the restoring force, which is elasticity in the case of the spring, and gravity in the case of the mass. Think of a roller coaster. When the coaster climbs a hill it has potential energy. At the very top of the hill is its maximum potential energy. When the car speeds down the hill potential energy turns into kinetic. Kinetic energy is greatest at the bottom.

The more formal definition is that potential energy is the energy difference between the energy of object in a given position and its energy at a reference position.

There are various types of potential energy, each associated with a particular type of force. More specifically, every conservative force gives rise to potential energy. For example, the work of elastic force is called elastic potential energy; work of gravitational force is called gravitational potential energy, work of the Coulomb force is called electric potential energy; work of strong nuclear force or weak nuclear force acting on the baryon charge is called nuclear potential energy; work of intermolecular forces is called intermolecular potential energy. Chemical potential energy, such as the energy stored in fossil fuels, is the work of the Coulomb force during rearrangement of mutual positions of electrons and nuclei in atoms and molecules. Thermal energy usually has two components: the kinetic energy of random motion of particles and potential energy of their mutual positions.

As a general rule, the work done by a conservative force F will be
\,W = -\Delta U
where \Delta U is the change in the potential energy associated with that particular force. The most common notations for potential energy are PE and U.

## Reference level

The potential energy is a function of the state a system is in, defined relative to an arbitrary reference energy. This energy can be chosen for convenience, and/or such that for a particular state the potential energy is zero. Typically the reference is chosen such that the potential energy depends on the relative positions of its components only.

In the case of inverse-square law forces, a common choice is to define the potential energy as tending to zero when the distances between all bodies tend to infinity.

## Gravitational potential energy

Gravitational energy is the potential energy associated with gravitational force. If an object falls from one point to another inside a gravitational field, the force of gravity will do positive work on the object and the gravitational potential energy will decrease by the same amount.

For example, consider a book, placed on top of a table. When the book is raised from the floor to the table, some external force works against the gravitational force. If the book falls back to the floor, the same work will be done by the gravitational force. Thus, if the book falls off the table, this called potential energy goes to accelerate the mass of the book (and is converted into kinetic energy). When the book hits the floor this kinetic energy is converted into heat and sound by the impact.

The factors that affect an object's gravitational potential energy are its height relative to some reference point, its mass, and the strength of the gravitational field it is in. Thus, a book lying on a table has less gravitational potential energy than the same book on top of a taller cupboard, and less gravitational potential energy than a heavier book lying on the same table. An object at a certain height above the Moon's surface has less gravitational potential energy than at the same height above the Earth's surface because the Moon's gravity is weaker. (This follows from Newton's law of gravitation because the mass of the moon is much smaller than that of the Earth.) It is important to note that "height" in the common sense of the term cannot be used for gravitational potential energy calculations when gravity is not assumed to be a constant. The following sections provide more detail.

The strength of a gravitational field varies with location. However, when the change of distance is small in relation to the distances from the center of the source of the gravitational field, this variation in field strength is negligible and we can assume that the force of gravity on a particular object is constant. Near the surface of the Earth, for example, we assume that the acceleration due to gravity is a constant ("standard gravity"). In this case, a simple expression for gravitational potential energy can be derived using the W = Fd equation for work, and the equation
W_F = -\Delta U_F.\!

When accounting only for mass, gravity, and altitude, the equation is:
U = mgh,\!
where U is the potential energy of the object relative to its being on the Earth's surface, m is the mass of the object, g is the acceleration due to gravity, and h is the altitude of the object.. If m is expressed in kilograms, g in meters per second squared and h in meters then U will be calculated in Joules.

Hence, the potential difference is
\,\Delta U = mg \Delta h.

However, if the force of gravity varies too much for this approximation to be valid, then we have to use the general, integral definition of work to determine gravitational potential energy. Now taking the arbitrary reference point where U = 0 to be when the two objects are infinite distance apart:

The (now negative) gravitational potential energy of a system of masses m1 and m2 at a distance R using gravitational constant G is

U = -G \frac{m_1 m_2}{R}.

for the computation of the potential energy we can integrate the gravitational force (whose magnitude is given by Newton's law of gravitation) with respect to the distance r between the two bodies from to .

The total potential energy of a system of n bodies is found by summing, for all \frac{n ( n - 1 )}{2} pairs of two bodies, the potential energy of the system of those two bodies.

Considering the system of bodies as the combined set of small particles the bodies consist of, and applying the previous on the particle level we get the negative gravitational binding energy. This potential energy is more strongly negative than the total potential energy of the system of bodies as such since it also includes the negative gravitational binding energy of each body. The potential energy of the system of bodies as such is the negative of the energy needed to separate the bodies from each other to infinity, while the gravitational binding energy is the energy needed to separate all particles from each other to infinity.

### Uses

Gravitational potential energy has a number of practical uses, notably the generation of hydroelectricity. For example in Dinorwig, Wales, there are two lakes, one at a higher elevation than the other. At times when surplus electricity is not required (and so is comparatively cheap), water is pumped up to the higher lake, thus converting the electrical energy (running the pump) to gravitational potential energy. At times of peak demand for electricity, the water flows back down through electrical generator turbines, converting the potential energy into kinetic energy and then back into electricity. (The process is not completely efficient and much of the original energy from the surplus electricity is in fact lost to friction.) See also pumped storage.

Gravitational potential energy is also used to power clocks in which falling weights operate the mechanism.

## Elastic potential energy

Elastic potential energy is the potential energy of an elastic object (for example a bow or a catapult) that is deformed under tension or compression (often termed under the word stress by physicists). It arises as a consequence of a force that tries to restore the object to its original shape, which is most often the electromagnetic force between the atoms and molecules that constitute the object. If the stretch is released, it is transformed into mechanical energy.

### Calculation of elastic potential energy

In the case of a spring of natural length l and modulus of elasticity λ under an extension of x, elastic potential energy can be calculated using the formula:

E = \frac{\lambda x^2}{2l}

This formula is obtained from the integral of Hooke's Law:

U_e = -\int\vec{F}\cdot d\vec{x}=-\int {-k x}\, dx = \frac {1} {2} k x^2

The equation is often used in calculations of positions of mechanical equilibrium.

In the general case, elastic energy is given by the Helmholtz potential per unit of volume f as a function of the strain tensor components εij:

f(\epsilon_{ij}) = \lambda \left ( \sum_{i=1}^{3} \epsilon_{ii}\right)^2+2\mu \sum_{i=1}^{3} \sum_{j=1}^{3} \epsilon_{ij}^2

Where λ and μ are the Lamé elastical coefficients. The connection between stress tensor components and strain tensor components is:

\sigma_{ij} = \left ( \frac{\partial f}{\partial \epsilon_{ij}} \right)_S

For a material of Young's modulus, Y (same as modulus of elasticity λ), cross sectional area, A0, initial length, l0, which is stretched by a length, \Delta l:

U_e = \int {\frac{Y A_0 \Delta l} {l_0}}\, dl = \frac {Y A_0 {\Delta l}^2} {2 l_0}

where Ue is the elastic potential energy.

The elastic potential energy per unit volume is given by:
\frac{U_e} {A_0 l_0} = \frac {Y {\Delta l}^2} {2 l_0^2} = \frac {1} {2} Y {\varepsilon}^2

where \varepsilon = \frac {\Delta l} {l_0} is the strain in the material.

## Chemical potential energy

Chemical potential energy is a form of potential energy related to the structural arrangement of atoms or molecules. This arrangement may be the result of chemical bonds within a molecule or otherwise. Chemical energy of a chemical substance can be transformed to other forms of energy by a chemical reaction. As an example, when a fuel is burned the chemical energy is converted to heat, same is the case with digestion of food metabolized in a biological organism. Green plants transform solar energy to chemical energy through the process known as photosynthesis, and electrical energy can be converted to chemical energy through electrochemical reactions.

The similar term chemical potential is used by chemists to indicate the potential of a substance to undergo a chemical reaction.

## Electrical potential energy

An object can have potential energy by virtue of its electric charge and several forces related to their presence. There are two main types of this kind of potential energy: electrostatic potential energy, electrodynamic potential energy (also sometimes called magnetic potential energy).

### Electrostatic potential energy

In case the electric charge of an object can be assumed to be at rest, it has potential energy due to its position relative to other charged objects.

The electrostatic potential energy is the energy of an electrically charged particle (at rest) in an electric field. It is defined as the work that must be done to move it from an infinite distance away to its present location, in the absence of any non-electrical forces on the object. This energy is non-zero if there is another electrically charged object nearby.

The simplest example is the case of two point-like objects A1 and A2 with electrical charges q1 and q2. The work W required to move A1 from an infinite distance to a distance r away from A2 is given by:
W=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r},
where ε0 is the electric constant.

This equation is obtained by integrating the Coulomb force between the limits of infinity and r.

A related quantity called electric potential (commonly denoted with a V for voltage) is equal to the electric potential energy per unit charge.

### Electrodynamic potential energy

In case a charged object or its constituent charged particles are not at rest, it generates a magnetic field giving rise to yet another form of potential energy, often termed as magnetic potential energy. This kind of potential energy is a result of the phenomenon magnetism, whereby an object that is magnetic has the potential to move other similar objects. Magnetic objects are said to have some magnetic moment. Magnetic fields and their effects are best studied under electrodynamics.

## Nuclear potential energy

Nuclear potential energy is the potential energy of the particles inside an atomic nucleus. The nuclear particles are bound together by the strong nuclear force. Weak nuclear forces provide the potential energy for certain kinds of radioactive decay, such as beta decay.

Nuclear particles like protons and neutrons are not destroyed in fission and fusion processes, but collections of them have less mass than if they were individually free, and this mass difference is liberated as heat and radiation in nuclear reactions (the heat and radiation have the missing mass, but it often escapes from the system, where it is not measured). The energy from the Sun is an example of this form of energy conversion. In the Sun, the process of hydrogen fusion converts about 4 million tonnes of solar matter per second into light, which is radiated into space.

## Relation between potential energy, potential and force

Potential energy is closely linked with forces. If the work done moving along a path which starts and ends in the same location is zero, then the force is said to be conservative and it is possible to define a numerical value of potential associated with every point in space. A force field can be re-obtained by taking the vector gradient of the potential field.

For example, gravity is a conservative force. The associated potential is the gravitational potential, often denoted by \Phi or V, corresponding to the energy of a unit mass as a function of position. As the potential energy of a particle of mass m at a distance R from a body of mass M is
U = -\frac{G M m}{R},
the gravitational potential at a distance R from a body of mass M is given by
\Phi = -\frac{G M}{R}.

The work done against gravity by moving a unit mass from point A with U = a to point B with U = b is (b - a) and the work done going back the other way is (a - b) so that the total work done going from A to B is
U_{A \to B \to A} = (b - a) + (a - b) = 0. \,
If the potential is redefined at A to be a + c and the potential at B to be b + c, where c is a constant (i.e. c can be any number, positive or negative, but it must be the same at A as it is at B) then the work done going from A to B is
U_{A \to B} = (b + c) - (a + c) = b - a \,
as before.

In practical terms, this means that one can set the zero of U and \Phi anywhere one likes. One may set it to be zero at the surface of the Earth, or may find it more convenient to set zero at infinity (as in the expressions given earlier in this section).

A thing to note about conservative forces is that the work done going from A to B does not depend on the route taken. If it did then it would be pointless to define a potential at each point in space. An example of a non-conservative force is friction. With friction, the route taken does affect the amount of work done, and it makes little sense to define a potential associated with friction.

All the examples above are actually force field stored energy (sometimes in disguise). For example in elastic potential energy, stretching an elastic material forces the atoms very slightly further apart. The equilibrium between electromagnetic forces and Pauli repulsion of electrons (they are fermions obeying Fermi statistics) is slightly violated resulting in a small returning force. Scientists rarely discuss forces on an atomic scale. Often interactions are described in terms of energy rather than force. One may think of potential energy as being derived from force or think of force as being derived from potential energy (though the latter approach requires a definition of energy that is independent from force which does not currently exist).

A conservative force can be expressed in the language of differential geometry as a closed form. As Euclidean space is contractible, its de Rham cohomology vanishes, so every closed form is exact, for example, is the gradient of a scalar field. This gives a mathematical justification of the fact that all conservative forces are gradients of a potential field.

## References

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