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If you take two electrons and perform a piece-wise numerical integration over their interacting electromagnetic fields, it is no surprise you come up with Coulomb’s Law. However, the surprise is that there is a region between the two electrons that provides an attractive force. Its strength is only about 22% of the repulsive force generated by the rest of the interacting fields, so the net effect is the repulsion we see in Coulomb’s Law. (I will go over the field structure later). This comes about because midway between the two electrons the fields are equal but in opposite directions, cancelling each other out, reducing the energy density, and hence generating attractive forces. Far from the electrons the fields are nearly parallel, increasing the energy density as the square of the sum, and hence creating repulsive forces.

What if we postulate a particle that has an electric field with an outer limit to it, so the field stops at some arbitrary radius “r”. This is not too surprising - after all there must be some sort of inner limit to any particle’s electric field or the energy in the field would be infinite (you can search for equations for the Classical Radius of the Electron” on the internet to find the details), so why can’t there be an outer boundary? Let us also assume that within this limit the field follows an inverse-square law.

If two particles of the same field polarisation, with this limited electric field, are brought together, then the characteristics must be:-

• Beyond a separation of 2.r the electric fields do not overlap and there are no interacting forces.
• As the separation drops from 2.r to r the attractive part of the fields increasingly overlaps, and attractive forces result.
• At a separation of r the attractive and repulsive forces are in balance .
• Below a separation of r the field’s repulsion climbs dramatically.

This has the following consequences:-

• The natural separation of these particles is either r - the particles will place themselves at the balance point between repulsion and attraction - or at separations greater than 2.r where the fields do not interact.
• Coulomb’s Law does not apply to this field geometry. Even though the field follows an inverse-square law within its limit, the interaction is nothing like an inverse-square law, except for separations many times less than r, when coulomb’s Law would again apply.

The force/separation picture computed from the numerical integration looks like this:-

Now the last surprise- see here for an example of the force/separation of the strong nuclear force. It has the same shape. Also, the existence of a short-range electric field around the neutron has been reported, although the repulsive part of the field has sometimes been associated with a positive field and the attractive part with a negative field.

Now let us put some numbers in here. If this is an active component of a neutron:-

• The field radius r must be less than 10^-15 meters, and from some reports of force/separation would seem to be about 0.35 E^-15 meters.
• The simple electric field means that the neutron can have no electric dipole moment.
• The presence of the electric field means that a spinning neutron will have a magnetic dipole.
• Neutrons would appear to pack quite easily, since with three in a row there would be no field overlap between the two outer ones but they would be bonded by the central one. The neutrons might build a diamond-like lattice structure (face centred cubic). Under appropriate conditions there would be no practical limit to the size of such a structure, as shown by the existence of neutron stars.
• The peak attractive force is only about 80% of what Coulomb’s Law would give for an electron at that separation (this is still greater than the 22% of the energy in the attractive field, since it is enhanced by the suddenness of the limit). For a strength quoted as 135 times as great as the electron’s field, the neutron field strength (or unit charge) is required to be 13 times as strong as the electron’s (given by the square root of 135/80%).
• If the proton is a simple structure like an electron, it will have negligible interaction with the neutron at any distance, but as it closes within a separation of 2.r the repulsive forces build up dramatically. So for the proton to bind in the neutron it needs a core similar to that of the neutron, even though its far field is similar to an electron’s. Such a core field could readily overcome the proton’s far field repulsion of other protons in the same nucleus.

Want to check the reasoning? Work through it below...

First, the basics of electrostatic interaction...

1. A point in space with an electric field has a vector describing the direction of the field, and the length of the vector (or “modulus” of the vector) is the strength of the field.
2. The energy density at a point an electric field is proportional to the square of the modulus of the electric field strength.
3. If two electric fields overlap in the same space, the field vectors are added as vectors to get the new composite vector, and this usually has a different field strength and a correspondingly different energy density. For example, if two fields “a” and “b” overlap perfectly, they will add to give a field strength of “a+b”, but the energy density will rise from “a²+b²” to “(a+b)²” = “a²+b²+2ab”, an increase of 2ab in energy density.
4. If, when two fields are added and the combined result has a lower energy density than the two individual fields, attractive forces are produced that draw the sources of these fields closer together. Conversely, if the combined result has a higher energy density then repulsive forces will push their sources apart. Hence forces work to reduce the energy in the system.

Next, the basics of electron-electron interaction...

The electron’s electric field is polar, radiating from the centre. If two electrons are placed near each other the field lines that point more-or-less towards each other will oppose and tend to cancel each other out, causing attractive forces; this applies to vectors which intersect at angles between 180 and 90 degrees. Field lines that cross at 90 degrees to each other will be orthogonal, and will have no effect on each other. Field lines that intersect at angles under 90 degrees will reinforce each other and create repulsive forces.

The inside of the sphere shown, with the electrons on the circumference, is a region where the two fields partially cancel, leading to a reduced energy density and hence contribute an attractive force. The outside of the sphere is a region where the two fields reinforce, leading to higher energy densities and hence contribute a repulsive force. The attractive force inside the sphere is only about 22% of the repulsive force outside this sphere, so the net effect is one of repulsion. This is exactly what we see, that two electrons brought together will repel each other.

So what if the neutron had a bounded positive electric field?...

That is, what if the neutron had a positive electric field that stopped at some minuscule radius? How would two neutrons interact? At distances where the two fields did not overlap there would be no effect. At a separation of twice the radius the fields would just touch, then as they came even closer they would attract because only the fields inside the attraction sphere would be interacting.

As they were brought further together the attraction would dramatically increase until the separation drops to about 1.05 times the radius, then as the separation drops further to 1.0 times the radius the attractive force dramatically drops to zero and at this distance, with each neutron on the circumference of the other’s bounded field, the attractive and repulsive forces are in balance. Below this separation the repulsion increases dramatically.

Although the attractive energy involved is only about 22% of the conventional electrostatic field structure, force is the rate of change of energy with change in distance, and because the onset is so rapid compared with the conventional field structure, the attractive forces involved are about 80% of the conventional repulsive value.

This ties in well with the attractive / repulsive behaviour of the neutron. See here for some background info.

But there are two objections:-

1. If we take the field strength as equivalent to an electron, then the forces are too weak.
2. The Proton-neutron interaction is two orders weaker again.

But if....

1. If the neutron’s electric field strength were not only bounded, but about 13 times as strong as the electron’s electrostatic field, then neutron-neutron forces would be of the correct order and behaviour for the strong force.
2. If the proton had a similar core at its centre then neutron-proton and proton-proton short-range interaction would be similar.

...and it would all work out.

This gives a field around each proton and around the nucleus that repels electrons at close quarters, preventing them falling into the nucleus as they would with a positron. The picture then becomes one where the proton is similar to a composite of a positron and a neutron (note that to have a finite amount of energy in the electric field, there must be an inner boundary to every electric field in every charged particle)...

Now being quite speculative, we might imagine that a proton would behave as if it was built from a neutron and a positron (it is unlikely to be this simple)...

• Neutrons would attract each other at close range, but would repel at extreme close range at separations less than the neutron charge radius.
• Protons would repel each other at medium-to-long range, but attract neutrons at close range, and again repel at separations below the neutron radius.