# Electrostatic potential energy

The electrostatic potential energy or electric potential energy is the energy that results from the electrostatic force given by Coulomb’s law.

Coulomb´s law describes the electrostatic force between point charges, which is a conservative force like the gravitational force described by the Newton’s law of universal gravitation. Therefore it has an associated scalar function whose gradient is equal to the negative of the force between the charges. Said scalar function is the electrostatic potential energy U.

In what follows we will focus on studying the electrostatic potential energy for point charges.

Let’s consider the two point charges shown in the next figure:

We will assume that q1 is the source charge (the one that creates the electric field) and that q2 is the test charge (the one that undergoes the force). The force undergone by q2 due to the presence of q1 is given by Coulomb’s law:

The potential electrostatic energy U between the two point charges q1 and q2 is given by:

Where k is the Coulomb constant and its value in SI units in vacuum is:

Or, if we express it as a product of the vacuum permittivity (permittivity of free space) ε0, it is equal to:

ε0 is one the fundamental physical constants and its value in SI units is:

Next we will verify that the force is minus the gradient of U:

The SI unit for the electrostatic potential energy is the joule (J).

When a test charge q is brought near N source point charges, its electrostatic potential energy will be the sum of the potential energies due to each individual source point charges.

The electrostatic potential energy can be written as a function of the electrostatic potential. When the test charge is brought into a region of space where an electrostatic potential V exists, its electrostatic potential energy is:

If the potential V is due to a source point charge q1, the previous expression is equal to the electrostatic potential energy given at the beginning of this page.

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##### Work of the electrostatic force

A source charge (in green) and a test charge (in blue) are represented in the next figure. When the test charge moves from point A to point B, the electrostatic force that q1 exerts on it does a work.

The work done by a force is:

As for any conservative force, the work done by the electrostatic force to move a test charge q2 from point A to point B is the negative change in the potential energy:

If the work is positive, it means that the work is done by the force; if it is negative, a work against the force has to be done to move the test charge from point A to point B.

In the previous figure, the charges have opposite signs because the force is attractive, and as a consequence the product of the charges is negative. Furthermore:

Therefore the work is negative. That means that the electric force itself cannot move the test charge q2 from point A to point B, but that some additional energy is needed (kinetic energy for instance) to do so.

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For convenience, the zero of the potential energy is chosen at infinity r = ∞. Therefore when the distance between two charges is infinite, the potential energy of the test charge will be zero:

The potential electrostatic energy is the work done by the electrostatic force to move the test charge from a distance r of the source charge to infinity:

When a charge moves due to the electrostatic force, its mechanical energy is conserved.

The work done by the electrostatic force to move a test charge q2 from point A to point B can also be written in terms of the potential difference due to the source charge q1 between these two points of space.

Let’s consider for instance the source charge to be positive. The equipotential surfaces of the potential due to the charge are represented in the next figure.

The work done by the electric force to move a test charge q2 from a point A to point B will be:

And between points A and C it would be:

The work done by the electric force to move a charge between two points lying on the same equipotential surface is zero.

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