# Potential Energy Formula

What is the potential energy formula for a gravitational field, a spring or an electric field? How is potential energy defined? Answer to all these questions and more are provided here, along with ready-to-use potential energy calculators.

Omkar Phatak

Last Updated: Mar 6, 2018

*electromagnetic, gravitational, strong or weak*), every change that occurs in the universe is a conversion from

*potential*to

*kinetic*energy. From merely being in a position in space, in presence of forces, a particle has an inherent energy associated with it, that's termed as potential energy. Depending on the force involved, the associated potential energy formulas vary. For a detailed explanation of potential energy, scroll down further. In what follows, the most commonly used formulas are presented, along with calculators.

Potential Energy Formulas

The formula for potential energy of a system depends on the

*forces*at work and the

*constituents*of a system. All macroscopic systems outside the atomic nucleus are either governed by the electromagnetic or gravitational forces. Potential energy is measured in SI unit of '

**Joule**'.

Gravitational potential energy is a function of the position of the object in a gravitational field, force of gravity at that point and mass of the object. The formula is as follows:

Gravitational Potential Energy = mgh

*where m is the mass, g is the acceleration due to gravity (9.8 m/s*

^{2}) and h is the height above Earth surface.The formula for elastic potential energy of a spring with a constant k is as follows:

Elastic Potential Energy = ½ kx

^{2}*where x is the displacement parameter for a spring.*

The formula for a test charge 'q' that has been placed in the presence of a source charge 'Q', is as follows:

Electric Potential Energy = q/4πε

_{o}∑^{N}_{i = 1}[Q_{i}/R_{i}]*where q is the test charge, ε*

_{o}is the permittivity of free space, Q is the field charge and R is the distance between the two point charges.What is Potential Energy?

Potential energy is the dormant stored energy in any system. It is a type of energy that a system possesses on account of its configuration and position of various constituents of that system in presence of a force. It is the energy stored in an object when work is done against a force. Depending on the configuration of a system and the forces operational in it, there are various forms of potential energy. Motion occurs when this stored energy is converted into kinetic energy.

The formula for potential energy of an object in a force field depends on its position and other inherent factors like mass and charge. For example, when an object is lifted up, work is done against the gravitational force (

*that pulls it down*). This work done on the object is stored as potential energy. When such an object is released from the height to which it has been lifted and it falls freely, gravitational potential energy gets converted into kinetic energy. So one can say that objects at higher altitudes on Earth have a higher potential energy.

The knowledge of potential energy of a system at a point is a useful result, but what we ultimately want is an equation that predicts the position of a particle at any moment of time, along with its energy. To get that, is to derive the equation of motion of a system. There are various ways of deriving it. One of them is through solution of Lagrangian equations of motion. The Lagrangian is the difference of potential and kinetic energy of a system. You need to know the potential energy formulas for particular systems along with the kinetic energy expressions, to set up the Lagrangian. Ergo, to understand potential energy and its computation is just the first step in your journey into classical mechanics.