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 Xe 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.|Xp.Xe|). 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' and in
'y' 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 dEc(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 Ec
at a separation of (2.a) is (integrating over all positive x,y):-
This is a pretty evil integration, and difficult to evaluate for anything
but the simple case of all space. A numeric computer integration is more
flexible if less accurate. Appendix A shows the program listing for a
simple finite-element numerical integration to compute the total energy
change values for a range of separations, and compare them with the
theoretical energy values. Some of the results are tabulated in Table 1
below. The error of 2% low in the calculations comes from three sources -
the calculation does not extend to infinity, the singularity at the center
of the electron and positron is avoided, and there is no compensation for
field gradient across each finite element. However, it is clear that
electrostatic potential energy is merely the drop in field energy. A minor
change to the limits for the numerical integration program in Appendix A
shows that the energy increase inside the sphere of radius ‘a’ is
((3.3E-29) / a) Joules for a separation of (2.a).