Potential Energy Explained — Definitions & Formulas

Definition / Summary

Potential energy is the negative of the work done by a conservative force. It represents stored mechanical energy associated with position in a force field.

The work–energy theorem relates work and kinetic energy. For conservative forces this leads to an energy conservation statement in which mechanical energy plus potential energy is constant.

For example: an object of mass $m$ that moves upward by a vertical height $h$ along a curved path and a distance $s$ loses mechanical energy equal to $\int m g\, dh$ because of gravity and $\int \mathbf{F}\cdot d\mathbf{s}$ because of friction. Gravity is a conservative force (friction is not), so the first integral reduces to $mgh$ when $g$ is effectively constant.

Potential energy is defined relative to an arbitrary reference level (for example, the starting level or infinity). Potential energy (commonly shortened to PE) is a scalar with the same dimensions and units as energy: $ML^2/T^2$, measured in joules ($J$).

Equations

Work–energy theorem

$$E – W = E – \int \mathbf{F}_{\text{total}}\cdot d\mathbf{s} = \text{constant}$$

Conservative force (energy conservation)

$$E + PE = \text{constant}$$

Mixed conservative and non-conservative forces

$$E – W_{\text{conservative}} – W_{\text{non-conservative}} = E + PE – W_{\text{non-conservative}} = \text{constant}$$

Common forms of potential energy

• Gravitational (inverse-square field): $$PE = -\dfrac{mMG}{r}$$
• Gravitational (uniform-field approximation): $$PE = mgh \quad (\text{for near-Earth, constant } g)$$
• Elastic (spring): $$PE = \tfrac{1}{2} k x^2$$
• Electric (charge in an electric potential): $$PE = qV = -\int \mathbf{E}\cdot d\mathbf{s}$$
• Magnetic (magnetic moment): $$PE = -\mathbf{\mu}\cdot\mathbf{B}$$

Extended explanation

Is potential energy “energy”?

Some confusion arises about whether potential energy is part of “energy.” In the work–energy formulation, potential energy appears through the work done by conservative forces. In the conservation-of-energy statement (when only conservative forces are involved), potential energy is part of the total mechanical energy.

Potential vs. potential energy

Careful: “potential” and “potential energy” are different quantities. Electric potential (voltage) is potential energy per unit charge; gravitational potential is potential energy per unit mass. For example, the gravitational potential of a point mass is $-GM/r$, while the gravitational potential energy of a test mass $m$ at that location is $-mGM/r$.

The same symbol is sometimes used for “field” and “force.” For an inline field, the force on a particle equals the field vector times the particle’s coupling (for example, charge or mass). If a vector field is written as $\mathbf{F}_{\text{field}}$ and there exists a scalar potential $U$ such that $\mathbf{F}_{\text{field}} = -\nabla U$, then the potential energy for a particle with coupling $q$ is $$PE = qU$$. Equivalently:

$$-\int \mathbf{F}_{\text{force}}\cdot d\mathbf{s} = -q\int \mathbf{F}_{\text{field}}\cdot d\mathbf{s} = q\int (\nabla U)\cdot d\mathbf{s} = qU$$

Potential energy is relative

Only changes in potential energy affect motion and forces. An arbitrary constant can be added to a potential without changing physical predictions. That is why one picks a convenient reference point (e.g., $r=\infty$ for inverse-square forces or the equilibrium position for a spring).

For an attractive inverse-square force, the potential energy with reference $r=r_0$ can be written as $PE = -mC/r + mC/r_0$. Choosing $r_0=\infty$ simplifies this to $PE = -mC/r$.

Derivation of $mgh$ from the general gravitational potential

Start with the change in potential energy using the point-mass gravitational potential:

$$\Delta PE = \Delta\!\left(-\dfrac{mMG}{r}\right) = -\dfrac{mMG}{r_{\text{earth}}+H+h} + \dfrac{mMG}{r_{\text{earth}}+H}$$

For heights $h \ll r_{\text{earth}}$

$$\Delta PE \approx \dfrac{mMG\,h}{r_{\text{earth}}^2} = mgh$$

which recovers the familiar near-Earth result with $g = MG/r_{\text{earth}}^2$.

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