This one is a **BIG DEAL!**

## Calculating The Electron Charge From *Theory*

In 1997, Williamson and van der Mark published their model of the substructure of the electron. (For more on the electron’s substructure, see understanding electrons.)

Based upon their model, a charge of the order of 10^{-19} Coulombs can be *calculated* for the electron, based purely upon the energy and topology of its internal photon.

The following is a short summary of this calculation. More detail can be found in their 1997 paper.

When an electron and positron annihilate, they yield photons with a Compton wavelength of 𝜆_{C} = *h/m _{e}c* = 2.43 x 10

^{-12}m. Williamson and van der Mark considered the possibility that these were the photons that were confined to form the particles, and as such, these are also the Compton wavelengths of the electron and the positron.

In the diagram below of the internal photon as a twisted strip:

a) the top image depicts a single wavelength for a circularly-polarised photon in flat space. The **B**-field is in the plane of the strip and the **E**-field is perpendicular to it.

b) the bottom image depicts a similar photon in a closed path in curved space with periodic boundary conditions of length 𝜆_{C} . The **E**-field vector is radial and directed inwards, and the **B**-field is vertical. (The magnetic moment μ, angular momentum L, and direction of propagation with velocity *c* are also indicated.)

Assuming that the fields can indeed be folded in such a way, so that we have two equal and opposite charges, we can estimate the charge magnitude from the field magnitude close to the object.

Given the toroidal topology of the Williamson/van der Mark electron (*shown in fig. 2 above*), it is clear that there is a range of possible torus thicknesses — a range of possible loop sizes — for the two loops of the double-loop rotation. On one hand, the two paths could be coincident, with identical loop sizes; on the other hand, one loop could be large and the other small — as long as their combined paths add up to the Compton wavelength 𝜆_{C}. These two cases would yield different values for the electron’s charge (*e*), which increases as the loop size difference increases. (*See below*.)

For their idealized calculation of the electron charge (*e*), Williamson and van der Mark consider a toroidal structure where the loops are coincident. In such a case, the radius of the loops is a consequence of the Compton wavelength (𝜆_{C}) being wrapped into a 720° rotational double-loop, yielding an electron topology with a toroidal radius (the “mean radius of energy transport”) of *r* = 𝜆_{C}/4𝜋. (This is* labeled in fig. 2A above.*)

They further assume in their idealized calculation that the photon is contained within a spherical cavity of diameter exactly equal to 𝜆, which means a sphere of radius *r* = 𝜆/2. The reasoning is as follows.

An upper limit on the size of the object is obtained by considering the extremal paths which also fulfil the condition that they have length 𝜆_{C}. These paths, which correspond to the limiting case where the confined photon travels radially outwards and inwards, have maximal radius 𝜆_{C}/2. The limitation of the speed of light means that only paths within this radius can provide a contribution to the circulation of energy inside the object. This radius constitutes a “rotation horizon” and sets an upper limit on the size of the object equal to a Compton wavelength, 𝜆_{C}.

At the same time, we can estimate a lower limit on the size from the fact that it is impossible to confine an arbitrary wave into a box which is smaller than one half of its wavelength. Moreover, the minimum diameter of, for example, a spherically symmetric dielectric cavity is equal to a full wavelength. These conditions constrain the effective size of our self-confined object to be close to or equal to 𝜆_{C} .

In order to calculate the charge, they thus consider their photon of wavelength 𝜆_{C} confined into a spherical volume of:Under such conditions (and with the details of the derivation omitted from this summary), the average magnitude of the electric field inside the model electron is given by:

To estimate the charge in our model we need to compare the magnitude of the inward-directed electric field to that for a point charge at the origin. Making the plausible assumptions that (a) the relevant length scale from where the electric field is effectively inward-directed is the mean radius of energy transport *r*=𝜆_{C}/4𝜋, and (b) that the average electric field of the confined photon, ⟨*E*⟩, is a good estimate of the field at this radius, we then obtain the effective charge, *q*, by comparing this to the Coulomb field of a point charge at this distance.If we equate the two E values, this yields an elementary charge of:In S.I. units, this works out to *q* = 1.458 x 10^{-19} C.

Since the elementary charge, *e* = 1.609 x 10^{-19} C, this equates to a calculated electron charge of:*q* = **0.906 e**.

The calculation of the electron charge will vary as the above toroidal variables are manipulated. In the idealized case discussed in this paper, the elementary charge (*e*) is found to be close to the accepted value. With minimal perturbation of the variables — such as a variation in the relative sizes of the loops in the topology — the charge increases from 0.9*e*, through 1*e*, and up to as much as an order of magnitude larger in the extremal case, in which one loop is small and the other is large.

It seems self-evident that the extremal, high-charge state would not represent the most stable or lowest energy equilibrium state for an electron. From an angular momentum standpoint alone, it seems reasonable that the rotational topology will be most stable when the two half-wavelength loops are of a very similar (though not necessarily equal) radius. The toroidal structure shown in figure 2B (*above*) is intended to portray this.

We would like to emphasize that the photon remains uncharged. It is the confinement — the *topology* — and the commensurability of the field components with the orbital path which are responsible for the resulting charged state.

(For a more detailed derivation, see the Williamson/van der Mark 1997 paper.)

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