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The Force Between Neutral Composite Particles

There are composite particles or structures which are overall neutral with respect to
some force but because of internal separations experience a net force. For example, positronim,
the combination of an electron and positron, might be such a composite structure.
The neutron is overall neutral concerning electrostatic charge, but there is a spatial
variation in positive and negative charge within it. The most important example of this
phenomena is the matter of nucleons (protons and neutrons) being made up of three quarks.
The nucleons are conjectured to be overall neutral with respect to a charge associated
with the strong nuclear force. In close proximity two nucleons experience a net force,
called the strong force because of the interactions of the separate quarks.

To investigate the net force experienced by neutral composite structures consider the
simplest version.

This is a system of two composite particles whose centers of mass are separated by a distance r.
Each composite particle is composed of two charged point particles, one positive and one negative
separated by a distance s.
Thus each composite particle is neutral. For r much greater than s there is no net force
on the composite particles. However for r of a magnitude near that of s there is a net force.

Assume the force between the point particles is inversely proportional to the distance
squared and that the composite particles are static aligned as shown.

Ignoring the force constant the net force is given by:

F = −1/(r-s)² + 1/r² + 1/r² − 1/(r+s)²
or, equivalently
F = −1/(r-s)² + 2/r² − 1/(r+s)²

This expression can be expressed as a ratio; i.e.,

F = (−6r² + 2s²)/(r²(r-s)²(r+s)²)
or, equivalently
F = (−6r² + 2s²)/(r²(r²-s²)²)

The graph of this function is very interesting. It has a positive pole at r=0 and
a negative pole at r=s. Between those two poles and for r large relative s the value
drops to an insignificant level.