# Potential energy

Potential energy (U, or Ep), a kind of scalar potential, is energy by virtue of matter being able to move to a lower-energy state, releasing energy in some form. For example a mass released above the Earth has energy resulting from the gravitational attraction of the Earth which is transferred in to kinetic energy.

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## Types

### Gravitational potential energy

This energy is stored as a result of the elevated position of an object such as a rock on top of a hill or water behind a dam. It is written as

[itex]U_g = m g h \,[itex]

where [itex]m[itex] is the mass of the object, [itex]g[itex] the acceleration due to gravity and [itex]h[itex] the height above a chosen reference level (typical units would be kilograms for [itex]m[itex], metres/second2 for [itex]g[itex], and metres for [itex]h[itex]).

In relation to spacecraft and astronomy g is not constant and the formula becomes an integral. In the case of a sphere of uniform mass (such as a planet), with h measured above the surface, the integral takes the form:

[itex]U_g = \int_{h_0}^{h + h_0} {GmM \over r^2} dr[itex]

Where [itex]h_0[itex] is the radius of the sphere, M is the mass of the sphere, and G is the gravitational constant.

If h is instead taken to be the distance from the center of the sphere, then outside the sphere the potential energy relative to that at the center has two terms:

[itex]U_g = \int_{h_0}^h {GmM \over r^2} dr + \int_0^{h_0} {GmM \over h_0^2} {r \over h_0} dr[itex]

which evaluates to:

[itex]U_g = GmM \left[{1 \over h_0} - {1 \over h}\right] + {1 \over 2} {GmM \over h_0} = GmM \left[{3 \over 2h_0} - {1 \over h}\right][itex]

[We may also want to link to an explanation of that second term (the gravitational forces created by hollow spherical shells)]

A frequently adopted convention is that an object infinitely far away from an attracting source has zero potential energy. Relative to this, an object at a finite distance r from a source of gravitation has negative potential energy. If the source is approximated as a point mass, the potential energy simplifies to:

[itex]U_g = - {GmM \over r}[itex]

### Elastic potential energy

This energy is stored as the result of a deformed solid such as a stretched spring. As a result of Hooke's law, it is given by:

[itex]U_e = {1\over2}kx^2[itex]

where [itex]k[itex] is the spring constant (a measure of the stiffness of the spring), expressed in N/m, and [itex]x[itex] is the displacement from the equilibrium position, expressed in metres (see Main Article: Elastic potential energy).

### Chemical energy

Chemical energy is a form of potential energy related to the breaking and forming of chemical bonds.

### Rest mass energy

Albert Einstein's famous equation, derived in his special theory of relativity, can be written:

[itex]E_0 = m c^2 \,[itex]

where E0 is the rest mass energy, m is mass of the body, and c is the speed of light in a vacuum. (The subscript zero is used here to distinguish this form of energy from the others that follow. In most other contexts, the equation is written with no subscript.)

The rest mass energy is the amount of energy inherent in the mass when it is at rest. This equation quantifies the equivalence of mass and energy: A small amount of mass is equivalent to a very large amount of energy. (i.e., 90 petajoule/kg ≈ 21 megatons/kg)

### Electrical potential energy

The electrical potential energy per unit charge is called electrical potential. It is expressed in volts. The fact that a potential is always relative to a reference point is often made explicit by using the term potential difference. The term voltage is also common.

The electrical potential energy between two charges [itex]q_1[itex] and [itex]q_2[itex] is:

[itex]

U = - \frac{q_1 q_2} {4 \pi \epsilon_o r} [itex]

The electric potential generated by charges [itex]q_1[itex] (denoted [itex]V_1[itex]) and [itex]q_2[itex] (denoted [itex]V_2[itex]) is:

[itex]

V_1 = - \frac{q_1} {4 \pi \epsilon_o r} [itex]

[itex]

V_2 = - \frac{q_2} {4 \pi \epsilon_o r} [itex]

## Relation between potential energy and force

Potential energy is closely linked with forces. If the work done going around a loop 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 work done by a unit mass going from point A with [itex]U = a[itex] to point B with [itex]U = b[itex] by gravity is [itex](b - a)[itex] and the work done going back the other way is [itex](a - b)[itex] so that the total work done from

[itex]U_{A \to B \to A} = (b - a) + (a - b) = 0 \,[itex]

The nice thing about potential energy is that you can add any number to all points in space and it doesn't affect the physics. If we redefine the potential at A to be [itex]a + c[itex] and the potential at B to be [itex]b + c[itex] [where [itex]c[itex] can be any number, positive or negative, but it must be the same number for all points] then the work done going from

[itex]U_{A \to B} = (b + c) - (a + c) = b - a \,[itex]

as before.

In practical terms, this means that you can set the zero of [itex]U[itex] anywhere you like. You might set it to be zero at the surface of the Earth or you might find it more convenient to set it zero at infinity.

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 you take does affect the amount of work done, and it makes no sense at all 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. Powerful electromagnetic forces try to keep the atoms at their optimal distance and so elastic potential is actually electromagnetic potential. Having said that, scientists rarely talk about forces on an atomic scale. Everything is phrased in terms of energy rather than force. You can think of potential energy as being derived from force or you can think of force as being derived from potential energy.

A conservative force can be expressed in the language of differential geometry as an exact form. Because Euclidean space is contractible, its de Rham cohomology vanishes, so every exact form is closed, i.e., 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.

## Graphical representation

A graph of a 1D or 2D potential function with the function value scale increasing upward is useful to visualize the potential field: a ball rolling to the lowest part corresponds to a mass or charge, etc. being attracted.

E.g. a mass, being an area of attraction, may be called a gravitational well. See also potential well.

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