Potential Energy Explained: Formulas, Units & Derivations

Potential energy is the negative of the work done by a conservative force on an object, representing mechanical energy stored due to position within a force field. It is a scalar quantity with the same dimensions as energy, ML²/T², measured in joules (J). Common forms include gravitational, elastic (spring), electric, and magnetic potential energy, each defined relative to a chosen reference point.

Key Takeaways

• Potential energy (PE) has units of joules (J) and dimensions ML²/T², identical to all other forms of energy.
• The gravitational potential energy formula PE = -mMG/r applies to inverse-square gravitational fields, while PE = mgh is a near-Earth approximation valid when height h is much smaller than Earth’s radius.
• Elastic potential energy in a spring is given by PE = ½kx², where k is the spring constant and x is displacement from equilibrium.
• Electric potential energy equals qV, the charge q multiplied by the electric potential V at that point.
• Only changes in potential energy affect physical motion; adding an arbitrary constant to a potential energy function does not change any measurable prediction.
• When both conservative and non-conservative forces act on a system, the quantity E + PE minus the work done by non-conservative forces remains constant.

What Is Potential Energy?

Potential energy is defined as the negative of the work done by a conservative force. A conservative force is one for which the work done moving between two points does not depend on the path taken, only on the start and end positions. Gravity near Earth’s surface is a conservative force; friction is not.

Potential energy is always defined relative to an arbitrary reference level, such as a starting height or a distance of infinity. This reference point can be chosen for convenience without changing any physical outcome, since only differences in potential energy affect motion.

Worked Example: Gravity and Friction Together

Consider an object of mass m moving upward by a vertical height h along a curved path of total length s. The object loses mechanical energy equal to the integral of mg over dh due to gravity, plus the integral of the friction force dotted with ds due to friction. Because gravity is conservative, the gravity integral simplifies to mgh when the gravitational acceleration g is treated as constant. The friction term cannot be simplified this way because friction is non-conservative and depends on the path taken.

What Equations Govern Potential Energy?

Work–Energy Theorem

The work-energy theorem states that kinetic energy E minus the work W done by the total force equals a constant:

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

Energy Conservation for Conservative Forces

When only a conservative force acts on a system, kinetic energy E plus potential energy PE equals a constant:

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

Mixed Conservative and Non-Conservative Forces

When both conservative and non-conservative forces act, kinetic energy E plus potential energy PE minus the work done by non-conservative forces equals a constant:

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

Common Forms of Potential Energy

Standard potential energy formulas by force type

Force type	Formula	Notes
Gravitational (inverse-square field)	$$PE = -\dfrac{mMG}{r}$$	Exact form for gravity between two masses separated by distance r
Gravitational (uniform-field approximation)	$$PE = mgh$$	Valid near Earth’s surface where g is treated as constant
Elastic (spring)	$$PE = \tfrac{1}{2} k x^2$$	k is the spring constant, x is displacement from equilibrium
Electric (charge in a potential)	$$PE = qV = -\int \mathbf{E}\cdot d\mathbf{s}$$	q is charge, V is electric potential
Magnetic (magnetic moment)	$$PE = -\mathbf{\mu}\cdot\mathbf{B}$$	μ is magnetic moment, B is magnetic field

Is Potential Energy Really “Energy”?

Potential energy’s status as “energy” depends on which formulation is used. In the work-energy theorem, potential energy enters through the work done by conservative forces rather than standing alone. In the energy conservation statement, where only conservative forces act, potential energy is explicitly part of the total mechanical energy alongside kinetic energy.

How Does Potential Energy Differ from Potential?

Electric potential, commonly called voltage, is potential energy per unit charge, while gravitational potential is potential energy per unit mass. These are distinct from potential energy itself. For example, the gravitational potential of a point mass is -GM/r, while the gravitational potential energy of a test mass m at that same location is -mGM/r, the potential multiplied by the test mass.

The same symbol is sometimes used loosely for both “field” and “force.” For an inline field, the force on a particle equals the field vector multiplied by the particle’s coupling constant, such as its charge or mass. If a vector field is written as F_field and there exists a scalar potential U such that F_field equals the negative gradient of U, then the potential energy for a particle with coupling q is PE = qU. This can be shown as:

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

Why Is Potential Energy Always Relative to a Reference Point?

Only changes in potential energy affect motion and forces, so an arbitrary constant can be added to any potential energy expression without changing physical predictions. This flexibility is why physicists choose a convenient reference point, such as r equals infinity for inverse-square forces or the equilibrium position for a spring.

For an attractive inverse-square force, potential energy with a reference point at r₀ can be written as PE = -mC/r + mC/r₀. Choosing r₀ equal to infinity simplifies this expression to the standard form PE = -mC/r.

How Is mgh Derived from the Inverse-Square Gravitational Potential?

The near-Earth approximation PE = mgh can be derived directly from the general point-mass gravitational potential energy formula. Starting with the change in potential energy between two heights:

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

For heights h that are much smaller than Earth’s radius, this expression simplifies to:

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

This recovers the familiar near-Earth result, with gravitational acceleration g equal to MG divided by the square of Earth’s radius.

Frequently Asked Questions

What are the units of potential energy?

Potential energy is measured in joules (J), the standard SI unit of energy. Its dimensions are ML²/T², identical to kinetic energy and all other forms of mechanical energy, since potential energy is simply a form of stored energy.

Why can potential energy be negative?

Potential energy can be negative because it depends on an arbitrarily chosen reference point. For example, gravitational potential energy given by PE = -mMG/r is negative when the reference point is set at r equals infinity, reflecting that work must be done against gravity to separate the masses to infinite distance.

Is gravitational potential energy the same as gravitational potential?

No. Gravitational potential is potential energy per unit mass, given by -GM/r for a point mass. Gravitational potential energy is that potential multiplied by the test mass m, giving -mGM/r. The two quantities have different units and represent different physical concepts.

When can I use PE = mgh instead of PE = -mMG/r?

PE = mgh is valid as an approximation when the height h is much smaller than Earth’s radius, such as in most everyday and laboratory settings near Earth’s surface. It is derived directly from the general inverse-square formula PE = -mMG/r by taking the limit where h is small relative to Earth’s radius.

Does friction have a potential energy?

No. Friction is a non-conservative force, meaning the work it does depends on the path taken, not just the start and end points. Because potential energy can only be defined for conservative forces, friction cannot be assigned a potential energy function.

What is the potential energy formula for a spring?

The potential energy stored in an ideal spring is PE = ½kx², where k is the spring constant and x is the displacement from the spring’s equilibrium position. This formula assumes the spring obeys Hooke’s law.

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