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

υ max = Ve / h = K / h (12) where υ max is the maximum frequency, V is the voltage applied to the tube, e is charge, and K is equal to the kinetic energy of the accelerated electron. Notice that if the terms are rearranged to solve for energy and if the maximum frequency happens to be the Compton electron frequency ( υ ce ), then the kinetic energy would have the same magnitude as the mass-energy of the electron: K = h υ ce = Ve = E = (e)(511kV) =∫= p e (h / p e ) υ ce (13) In the meter and second system of units, the term (h/p e ) has the physical dimensionality of length which is expressed in meters and is a wavelength. Under the above condition, this wavelength may be determined for the electric equivalent of the electron mass-energy as the quantum of action h divided by the quantum of charge p e ; it is, therefore, a constant: λ x = h / p e = (3.990 x 10 -9 m 3 s -1 ) / (13.970 m 2 s -1 ) = 2.856 x 10 -10 m (14) Expanding the electron mass-energy equation to include both inertial and electric linear momentum functions results in E = m e c 2 = p Ae c = h υ ce =∫= λ e c 2 = p e W x = h υ ce (15) where p Ae is the (photo)inertial linear momentum m e c, p e is the electric linear momentum, and W x is the electric wavefunction corresponding to the intrinsic voltage (511kV) of the electron mass-energy. The function W x is called “the voltage equivalent electric wave speed of the electron mass-energy” [13] and it can be directly expressed by its equivalent electromagnetic form as a function of the Compton frequency and the Duane-Hunt wavelength – thus, as a function of two constants: 511kV =∫= W x = λ x υ ce = (2.856 x 10 -10 m) (1.236 x 10 20 s -1 ) = 3.529 x 10 10 m s -1 (16) This confirms that E = 511keV =∫= λ e c 2 = p e W x = 4.930 x 10 11 m 3 s -2 = (13.970 m 2 s -1 ) (3.529 x 10 10 m s -1 ) = 4.930 x 10 11 m 3 s -2 (17) and it expands the electron mass-energy expression to include W x and its constituents. It results in E = m e c 2 = p Ae c = h υ ce =∫= λ e c 2 = p e W x = λ e W k W x = λ e W k ( λ x υ ce ) (18) Note that the Compton electron frequency υ ce of the electromagnetic equivalent of the electron mass- energy, p Ae c = h υ ce , is shared with the actual electric structure of that mass-energy (W x = λ x υ ce ) [14]. Equation (18) requires another wave speed function, namely W k , which is the magnetic wave speed characteristic of the electron charge. As shown by that equation, there is a fundamental equivalence of c 2 = W k W x . The product of these two wavefunctions, W k and W x , represents a superimposition, where the two separate wavefunctions are superimposed nearly perpendicularly to one another. Since c and W x are known, it is a simple matter to solve for W k : W k = c 2 / W x = (2.998 x 10 8 m s -1 ) 2 / (3.529 x 10 10 m s -1 ) = 2.547 x 10 6 m s -1 (19) Besides being intrinsic to the electric equivalent of the electron mass-energy, the wave speed W k is actually the magnetic wave speed “intrinsic to the elementary charge of the electron” [15]: p e = λ e W k (20) Rearranging the above terms permits the confirmation of the value of W k . An interesting sidebar to the main purpose of this paper is an insight into the meaning of the charge to mass ratio. It can be seen below that this ratio results in a wave speed that is exactly equal to W k , the magnetic wave speed intrinsic to charge [6, 15]: W k = p e / λ e = (13.970 m 2 s -1 ) / (5.486 x 10 -6 m) = 2.547 x 10 6 m s -1 (21) 1 Year 2023 3 © 2023 Global Journals Global Journal of Science Frontier Research Volume XXIII Issue ersion I VI ( A ) The Toroidal Fine-Structure of the Electron