Proof Part 2: Photon Geometries

For an ideal capacitor and an ideal inductor, the general forms for calculating C and L are given by the equations,

1) C = Kε ε0 d'

2) L = Kμ μ0 N**2 d'

where Kε is the dielectric constant, Kμ is a permeability constant representing the ferromagnetic effects of the coils core, N is the number of turns in the coil and d' is determined by geometrical considerations and has units of length. For an ideal parallel plate capacitor, d' is the area divided by the spacing between the plates. For an ideal coil, d' is the area divided by the length of the coil. In both cases, the parameter is similarly related to the volume enclosing the capacitor or inductor. The more formal definitions of capacitance and inductance are functions of electric and magnetic fields as defined by Maxwell's equations.

We can reasonably say that any dimensions with a justifiable physical basis must be proportional to the wavelength of the photon. If we set the number of turns to be 1 and constrain d' to reflect that the capacitor and inductor occupy the same volume of space, we can set d' to be a constant times the wavelength. We will set this constant to 1/2π and generate these new equations for L and C.

        λ
3) C = --- Kε ε0
       2π

        λ
4) L = --- Kμ μ0
       2π

where λ is the wavelength. While the distance constant, 1/2π, seems arbitrary, Kε and Kμ are dimensionless constants that will be derived later and will indicate any wrong guesses about the distance constant. An additional consideration is that Kμ*Kε must be 1 or else the speed of light is violated and another is that the capacitor and inductor occupy the same volume of space. The value of 1/2π turns out to be the only value that can work. The geometrical justification is for an equivalent area of λ**2/2π acting over a distance of λ. The actual area is π/4 times λ**2, or the area of a circle whose diameter is λ. The equivalent area is smaller by a factor of π**2/2 which accounts for the sinusoidal distribution of curvature and anticurvature outside of the SOE. We can also treat the geometric constant (1/2π) as an unknown and there are enough equations that it can be solved for.

We can then compute the impedance at resonance and the resonant frequencies as sqrt(L/C) and 1/sqrt(LC) respectively, resulting in,

             Kμ μ0
5) Ζr = sqrt(-----)
             Kε ε0

        2π           1
6)  ω = --- sqrt(-----------)
         λ       ε0 μ0 Kε Kμ

By recognizing that sqrt(μ0/ε0) is the impedance of free space, that sqrt(1/(ε0 μ0)) is the speed of light, and that (2π c)/λ is the radian frequency, ω, we can rewrite these as,

7) Ζr = Ζ0 sqrt(Kμ/Kε)

8)  ω = ω sqrt(1/(Kμ Kε))

To solve for the constants, we combine equation 10, from the photon Energy analysis and equation 7 and dividing both sides of equation 8 by ω, resulting in these new expressions,

9)  α = sqrt(Kε/Kμ)

10) sqrt(Kε Kμ) = 1

Solving for Kε and Kμ, we get,

11) Kε = α

12) Kμ = 1/α

Again, the fine structure constant shows up as these arbitrary constants which quantify the space comprising the L and C relative to free space.

Part 5 will describe how these values can only arise if the space comprising the capacitor and inductor are curved and anticurved respectively. But first, we can quantify the resistance of the Universe to curvature as described by the First Law of CTE starting with Part 3 which shows how an intrinsic resistance to curvature is indistinguishable from gravity.

(C) 1997-2004 George White, All Rights Reserved