© Harold Aspden, 1998
Research Note: 6/98: June 18, 1998
I am writing this as an item supplementary to the Essay: Question No. 5, which is entitled What is a Neutrino?
In presenting that Essay I stated that I would show how an in-line combination or 'string' element comprising an electron and a positron in close surface charge contact would have an energy of 1.25 times that of the individual electron. A larger 'string' or 'chain' of four such charges has an energy 2.25 times that of an isolated electron and a six-long composition has an energy of 3.225 times that of the electron.
The analysis is quite simple. One just needs to formulate the energy E given by the J J Thomson formula for the electron as:
E = 2e2/3a ........... (1)
where a is the radius of a sphere bounding the electron charge e, our analysis here being in the classical c.g.s system of units, where we take the dielectric constant of the vacuum as unity.
From this it is evident that two electron-sized charge spheres of opposite polarity will, when in surface contact, have two units of energy E, offset by the energy:
e2/2a = 3E/4 ......... (2)
That gives the result 1.25E for the energy of the two-charge electron-positron string.
With four such charges in line the offset as applied to 4E is three of the quantites stated in equation (2) plus the offset of one third of the equation (2) value, but there is then the addition of positive energy owing to two interactions of half the value given by equation (2).
That is (2.25 + 0.25)E as an offset with 0.75E as an addition to the 4E value, giving, overall, 2.25E as the energy of the four-charge string.
To derive the energy of six charges in a string, the procedure is the same. Take 6E, offset 5(3E/4), offset 3(E/4), offset (3E/4)/5, add 4(3E/4)/2, and add 2(3E/4)/4. The result is 3.225E.
Had we worked with other charged lepton forms, muons or taons, the result is the same, E becoming the energy of such a single basic charge form. The question one must ask, however, is how composite charge forms move, and whether they travel as a single spherical charge form, dividing into separate charge spheres and regrouping as they proceed.
The above completes the simple analysis needed to support what is said in
What is a Neutrino? but, of course, anyone reading this without that introduction will wonder how electrons and positrons forming strings or chains can possibly survive without annihilating one another. Well, of course, they do annihilate one another in their pairs, but I ask you then how that the energy shed is then deployed. Surely it goes back into creating electrons and positrons in the same string configuration, because energy has to assume a particle form somehow and, in a sense, electrons and positrons are the lowest form of life in the charged particle spectrum!
I suspect that it is the ability of charge to engage in such transformations that allows electric currents to migrate through forms of matter linked by such electron-positron chains. The movement of isolated charges at different speeds and the consequent problem of wondering how kinetic energy is represented in Nature's storehouse is otherwise a very problematic topic. I will try to address that in these Web pages, particularly by reference to proton motion. There is now some evidence experimentally that tells us that an antiproton and a proton have identical mass to within less than a part in one billion. The way that is measured opens the question discussed in the earlier Research Note Research Note No. 3/98 and that concerns the issue of whether Coulomb interaction energy, such as that of equation (2) above, exhibits the mass property in motion. That is indeed a very interesting topic for the scientific minds of the 21st Century to contemplate.
June 18, 1998