The Maths of Electrostatic Interaction
for two like-charged particles

The starting point for all electrostatic interactions is that between two point electric charges such as that between a positron and an electron. In the 19th century scientists did not understand the interaction, and invented the concept of “potential energy” to provide the force that drove these particles together. It was only when Einstein developed the equivalence of mass and energy that the true source of the energy became apparent.

You are probably familiar with the concept that electric fields contain energy and it is the interaction between the fields of the two charges that leads to changes in their energy and hence to the forces between them, energy being simply the integral of force over distance. This paper develops the equations for the interaction at any point in space near the charges, without recourse to “potential energy”.

As you can see from this paper the concept of electrostatic potential energy is a Newtonian approximation that holds true only so long as the energy involved is a minuscule part of the rest energy/mass of the point charges. Changes in potential energy are equivalent to changes in the rest mass/energy of the system, which can become significant at high energies.

Electrostatic potential energy

The most basic model to use for analysing electrostatic potential energy is that of an electron and a positron in a closed system - one through whose boundary no energy flows. The electron and positron are initially at infinite separation and then are allowed to fall towards each other in free fall. The observer and the origin of the system are at the midpoint between the two, as shown in Figure 1.

The origin of the system is at the median point on the line joining their centers. The electron is considered to be at [x,y] = [0,-a] and the positron at [0,+a] so they are separated by ‘2.a’. The potential energy available in bringing them together from infinite separation to the separation of ‘2.a’ is...

where q is the positron charge and -q is that of the electron in coulombs. This is somewhat unsatisfactory in that potential energy starts off at zero for the isolated particles at infinite separation. It then goes negative as they approach. So we have a source of energy that seems without limit. What we really need to know is what this potential energy is, where it comes from and what effect it has on the rest of the system.

Consider just the electron. It is surrounded by an electrostatic field. The electrostatic field strength X_e is given by...

This is a vector directed inwards towards the centre of the electron. The electrostatic field energy density over space dE/dS at any point in this field is...

Where does the electrostatic field energy come from? It can come only from the creation energy of the particle. An electron is created with 8.2E^-14 Joules of energy and a portion of this goes into the electrostatic field. However, from E=m.c² the rest mass of the electron is based on its whole creation energy, so it is equivalent to say that the electric field contains a portion of the creation energy, or that same portion of the rest mass.

When the electron and positron approach their electrostatic fields overlap. The field strengths are added vectorially and unless the field vectors are orthogonal to each other the energy density will be changed. As an example, if at a given point in space the electrostatic field vectors from the electron and positron lie on the same axis but oppose each other the energy density will be given by...

This is an fall from the energy of the isolated particles proportional to (2.|X_p.X_e|). So the energy densities change as electrostatic fields merge. Hence if we consider the electrostatic potential energy in this system to be zero when the total field energy is at its isolated value, we can think of potential energy becoming negative as the field energy drops below that value. The latter approach is conceptually more satisfying, since we do not end up with negative values of energy. I will term this concept the field potential energy, and refer to the classic form as the classical potential energy. It is now necessary to demonstrate that the field energy lost is equivalent to the potential energy.

We can analyse this in detail by looking at a point [x,y] in the plane of Figure 2, separating the electric field vectors into orthogonal x-components and y-components in order to sum the fields from both particles. We need the change in energy from the isolated-particle case, rather than the total creation energy. Figure 2 has cylindrical symmetry (around the y-axis) so it is necessary to analyse only the plane shown.

The field strengths at [x,y] from the electron at [0,-a] and the positron at [0,+a] are first converted to orthogonal coordinates.

They can now be converted into orthogonal field strengths in 'x' (X_px) and in 'y' (X_py) and summed.

Since the electric field vectors have been split into orthogonal components we can simply add the energy densities.

Consider only the "change" in energy dE_c(x,y)/dS from the isolated- particle values (the constant components do not give rise to energy changes or forces):-

Observation shows that the numerator term (a²-x²-y²) describes a circle of radius ‘a’ centered on the origin and passing through the points [0,a] and [0,-a] when the term is zero. When rotated around the y-axis into the entire volume this circle describes a sphere. On the surface of the sphere the value of this term is zero and there is no change in energy density from the sum of the isolated values of each particle. Inside the sphere the energy density is greater than the sum of the isolated values, while on the outside the energy density is lower. The loss of energy over the whole of the outside is greater than the gain on the inside, so the net effect is a drop in the total energy over all space as the particles approach.

To work out the total energy over all space, use the cylindrical symmetry and multiply the energy density for each point [x,y] by the circumference at each radius ‘x’ and integrate over positive x. Multiply by two and integrate only over positive ‘y’. So the total energy change E_c at a separation of (2.a) is (integrating over all positive x,y):-

This is a pretty evil integration. Add the limits of the integration and you get a really serious problem, making it easier to use numerical integration.

Worse still, to get the forces you have to look at the change in energy with separation of the particles. Worse and worse! Numerical computation, although less accurate, gives a way around this.