Potential energy

When a body is subjected to a force (e.g. a ball falling from a height H above the ground), it can move for a certain distance under the action of this force's work. It can therefore be considered that when the body started moving under the action of the force, it had a reserve of energy, related to the force, which would be used by the work moving the body. This idea is the basis of the concept of potential energy.

If W_AB is the work done by the force of interaction (e.g. gravitational or electrostatic) being exerted on the object and moving it from point A to point B, and assuming, as is often the case, that the work does not depend on the path of the object from A to B (this is called a conservative force), then the variation in the potential energy of interaction U_AB is equal to the opposite of this work; i.e. -W_AB. Then at A the object can be considered to have had a potential energy U(A) and that a part of this energy was used to bring the object from A to B where the potential energy is equal to U(B)

It must be remembered that the force of interaction is not necessarily the force that can be exerted to move an object (itself submitted to a force of interaction). If the applied force is of opposite sign to the force of interaction, this means that in that case W_AB= U_AB. It is a matter of convention.

The potential energy of a body at a point A is usually defined as the work done by the force applied to this body, and therefore opposed to the force of interaction, required to move the body a finite distance from point A in a given reference frame. This, if the origin of a reference frame is placed at the centre of the Earth, the potential energy U(r) of a body of mass m at a distance r from the centre of the Earth will be equal to:

U(r) = - GmM_E/r²

where G is the gravitational constant and ME the mass of the Earth.

In the case of an electron attracted by a proton placed at the origin of the reference frame:

U(r) = -e²/r²

where e is the electrical charge of an electron.

But in the case of two protons, since the force applied to bring one of the protons from infinity must oppose the electrostatic repulsion, we will have a + sign:

U(r) = +e²/r²