Global Journal of Science Frontier Research, A: Physics and Space Science, Volume 23 Issue 1

with c designating the speed of light and the symbol =∫= indicating a functional transformation that involves both dimensional conversion and an equivalence between two different physical systems of measurement [8]: E = (9.109 x 10 -31 kg) (2.998 x 10 8 m s -1 ) 2 = 8.187 x 10 -14 Joules =∫= (5.486 x 10 -6 m)(2.998 x 10 8 m s -1 ) 2 = = 4.930 x 10 11 m 3 s - 2 (4) For electromagnetic (photon) energy consisting of oscillating electric and magnetic fields, where the maximum wave velocity is restricted to the speed of light, the well-known Compton electron wavelength ( λ ce = 2.426 x 10 -12 m) can be used to determine the Compton electron frequency: υ ce = c / λ ce = (2.998 x 10 8 m s -1 ) / (2.426 x 10 -12 m) = 1.236 x 10 20 s -1 (5) When energy is quantized, it can be calculated by the product of Planck’s constant (h = 6.626 x 10 -34 J s) and frequency. If the frequency happens to be the Compton electron frequency, then the result is equivalent to the mass-energy of the electron or to the low-energy limit of a gamma ray photon. Modern science shows this calculation as E = h υ ce = (6.626 x 10 -34 J s) (1.236 x 10 20 s -1 ) = 8.187 x 10 -14 Joules (6) matching the results of the E = m e c 2 calculation shown above in equation (4). Now, it also becomes apparent that a functionally equivalent Planck’s constant h, in meter and second units of measure, can be determined by rearranging the above equation as h = E / υ ce = (4.930 x 10 11 m 3 s -2 ) / (1.236 x 10 20 s -1 ) = 3.990 x 10 -9 m 3 s -1 (7) Our limited understanding of charge has confused the issue of the fine-structure of the mass-energy of a mass-bearing charge more than necessary. If we go back to James Maxwell’s (1831-79) theory, charge, or as he stated, “a [fundamental] quantity of electricity”, could be represented using three physical quantities and dimensions, as [9] q = M 0.5 L 1.5 T -1 (M = mass, L = length, T = time) (8) This is in agreement with the physical quantities for charge in the Electrostatic System of Units (ESU) [10], and also as proposed in the work of Harold Aspden (1927-2011) [11]. If the mass to length transformation is applied to this definition of charge it can be seen that the property of charge is equivalent to a function of linear momentum that is treated as being massfree - q = M 0.5 L 1.5 T -1 =∫= L 2 T -1 = p e (9) “q = 4.830 x 10 -10 ESU =∫= e = 1.611 x 10 -19 C =∫= p e = 13.970 m 2 s -1 ” [12] (10) In addition to being expressed in joules, per equation (4) above, the electron mass-energy can also be expressed as charge, represented by “e,” multiplied by voltage, found to be equal to 510,998.950 electron volts (eV), per Codata 2018: E = m e c 2 = (e)(511kV) = h υ ce =∫= (p e ) (h / p e )( υ ce ) (11) The Duane-Hunt Law states that the maximum frequency of X-rays emitted from a tube, resulting from electrons that being accelerated by an applied voltage strike a metal anode, is proportional to the applied voltage and is quantized by Planck’s quantum of action h, per the equation: The Toroidal Fine-Structure of the Electron © 2023 Global Journals 1 Year 2023 2 Global Journal of Science Frontier Research Volume XXIII Issue ersion I VI ( A ) - where p e symbolizes charge, with massfree dimensionality, as an electric linear momentum vector function [11]. Charge is a special type of linear momentum function that can be expressed inertially or electromagnetically in mass dependent systems of units (as q or e, see below) or electrically in the meters and seconds massfree system of units as p e [11]. As shown below in equations (15) and (17), this linear momentum charge function is inherent to the mass-energy of the electron and every massbound charge. The functional equivalences of charge in three different systems of units (ESUs, Coulombs, and meters squared per seconds) are