2. The Particle
The purpose of this section is to demonstrate that a particle can consist of nothing more than an electric field. The first derivation shows that an electric field does not have to propagate, if of the proper orientation. The second derivation shows that the field itself can (and actually should) contain mass, this uses the current view of mass, energy and its correlation, which turns out (as we will see later) not to be exact, but sufficient for this point. The concepts of mass and energy are discussed at length in later sections. The third derivation shows an explicit model of such a particle, which does not represent any specific particle, but is probably similar to an electron. One should not dismiss the idea by the existence of chargless particles, since such particles could have electric fields that are confined to a very small area.
Since the first section proposed a space without any particles (foreign objects of unknown constitution), this section was the necessary first step. While most aether theories enter into a problem of how particles interact with the presence of an aether, it is a moot point here. The particles themselves are forms of electromagnetic propagation and are thus free to travel through the aether without interference as we will see in the next section.
We have not yet defined the divergence of the field. Suppose that we arbitrarily establish a field with a non-zero divergence and we adjust it so that, in fact,
(1)
With such a geometry, Eq.(1-6) would become
(2)
What this means physically is that there is no change in the field intensities over time. Though it is natural for the field to decay, with this particular geometry, the decay is prevented. As it turns out, the requirements for such a field given in Eq.(1) are satisfied by any spherically symmetric field. The condition for stability can be given as
(3)
There can exist fields in this space that do not propagate, if the field has divergence. It is normally assumed that the divergence must be caused by a charge, but the above result is true by the geometry of the field and in itself does not require the existence of any object in the space.
Now consider the electric field of an electron which is given by:
(4)
The energy in this field is given by:
(5)
where l is the radius of the electron. (This limitation is required to avoid an infinite energy.) This yields
(6)
By using the mass of the electron to determine the amount of energy in the field, this radius can be determined yielding
(7)
thus l=1.4 fm. This was called the classical radius of the electron and was considered interesting since the size seemed appropriate1. This presents a serious problem, however. The entire energy of the electron has been accounted for, so there can be no energy within the 1.4 fm radius sphere or the energy will no longer balance.
Where is the particle?
Apparently, the mass energy is in the field, not the particle, so why is a particle needed? If the field could exist by itself without decaying or propagating, then a particle would not be required. It has already been shown above that this is possible.
Imagine an electric field defined around an arbitrary point in space (as described in the introduction) being2
(8)
where R is the position vector from the point in space and l is a constant. q is not to be considered a charge, but only a constant which relates to the divergence of the field. This field has several important features: The field is continuous in all space; The field has a finite energy in all space; The field is spherically symmetric, so will not propagate, but remain in place at constant intensity; The field is extremely close to a 1/r2 field at an appreciable distance, 1% error at 68.5 fm and 10-9% error at 1 nm;3 The field starts at 0 in the center and increases in intensity until a maximum at l and then decreases. The energy in the field is given by:
(9)
using the same method as in Eq.(5), but integrating over all space. Therefore l can be determined by:
(10)
which yields 0.69 fm. This would represent a field that had a total energy content equal to the rest energy of an electron. Actually q can be viewed as a charge if it is realized that Coulomb's Law does not exist in this space and the finite distribution will not explode. Furthermore, the distribution is not a collection of particles but a perfectly continuous field. In this view q can represent the total charge which is spread out into a charge density given by:
(11)
Note that 99% of the charge is within a sphere of a 68 fm radius.
From this viewpoint we can say that f(r)=r/e. When dealing with a charge distribution that is perfectly continuous (as opposed to a collection of elementary charges), we will call it a "charge field" to distinguish it from a collection of charges where force interaction does occur (which will be derived later).
1. The classical radius of the electron is well known as it appears in the textbook by Jackson2, pages 681 & 790 (Though Jackson refers to the classical diameter as the radius).
2. We choose a radially symmetric function here simply to imitate an electron like particle. This function is chosen because it is continuous and continuous in the first derivative (to supply a continuous magnetic field). A finite charge field could be selected here as well. This was chosen only because the whole field is then represented with a single (closed) function. I suspect that there are other limiting conditions that exist for successful charge fields. They must be able to exist at all velocities (see next section) and propagation of the longitudinal field may provide further constraints.
3. The experimental matching of the 1/R2 field is actually not particularly significant here, since a radially symmetric function can be produced to satisfy any experimental precision. This one is selected only to represent the concept.
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