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In this chapter, we evaluate Charles Coulomb's electronic-force equation, which possesses spherical symmetry (as does Newton's gravitational-force equation) and contains the permittivity constant, ε0.
This FUPCON does not exist in nature. The reason for its existence is to prevent electronic equations from using SI units by forcing the use of the quantum attributes of the electron as units of measure and to express electronic force in terms of inertial force.
In this section, we recreate Coulomb's electronic-force proportion.
Note: The normal proportion symbol is unavailable in HTML. In this presentation of this book, the Icelandic thorn, Þ, is used to represent the "is proportional to" symbol.
First, let us briefly analyze the environment of a single point charge: We notice that the charge creates radiating force vectors in three dimensions. They diverge as the distance from the charge increases. Therefore, the influence of the point charge at a particular location declines as the location's distance from the charge increases. However, as Michael Faraday demonstrated, the total number of force vectors remain the same, regardless of the distance. We can say, then, that the total potential force, TFe, of the point charge, q1, is proportional to the value of the charge, as follows:
TFe1 Þ q1 (18
and is distributed over the surface of a charge's sphere of influence. At the distance, r, from the point charge, the area of the surface of its sphere of influence is (4 π r2). The potential force per unit area (pressure) at the surface of the sphere of influence is represented by the following proportion:
Fe1 Þ q1 (4 π r2)-1 (19
Two point charges, q1 and q2, create a force between them that is proportional, as follows:
Fe Þ q1 q2 (4 π r2)-1 (20
which is Coulomb's electronic-force proportion.
To form this proportion, the force, Fe, the two electrical charges, q1 and q2, and the distance, r, between them can be measured by using any arbitrary value for the unit of each of these factors. However, to convert the proportion into an equation, a constant of proportionality can and must be inserted and assume whatever value satisfies the equation. The more arbitrary, the choice of units is; the more irrational, the constant of proportionality becomes.
Coulomb's electronic-force equation using SI units of measure takes the historical form:
Fem = q1 q2 (ε0 4 π r2)-1 (21
For the equation to work, the value of the constant of proportionality, ε0, in SI units must be:
ε0 = 8.9 × 10-12 s2·C2·kg-1·m-3 (22
which, historically, is called the permittivity constant. This FUPCON was not created by trying to measure it, because you cannot measure nonexistent entities. It is simply the SI value that is necessary to convert the SI-based Coulomb proportion into an equation.
We determine the SE value of the permittivity constant, ε0, by converting its SI units of measure to SE units, as follows:
ε0 = 8.9 × 10-12 s2·C2·kg-1·m-3 (23
ε0 = (8.9 × 10-12) (1.2 × 1020 te)2 (6.2 × 1018 qe)2 ×
(1.1 × 1030 me)-1 (4.1 × 1011 λe)-3 (24
ε0 = 68.5 te2·qe2·me-1·λe-3 (25
The value, 68.5, is one half the value of the reciprocal of the fine-structure constant, α, which is dimensionless. Its value is:
α = (137.036 ...)-1 (26
For information about the fine-structure constant, see Chapter 11.
For the remainder of this book, we do not use the α form of the fine-structure constant. We use a new β form, which is the reciprocal of α, as follows:
β = α-1 = 137.036 ... (27
Using β, we express the SE value of the permittivity constant as:
ε0 = β 2-1 te2·qe2·me-1·λe-3 (28
Rearranging the factors in preparation for removing them from the permittivity constant (making it disappear by converting its value to dimensionless unity):
ε0 = (2-1) (β me-1·λe-1·te2) (qe2·λe-2) (29
To see these numbers in context, let us put them in place of the historical value of the permittivity constant, ε0, in the SE version of Coulomb's force equation as follows:
Fee = [2 (β-1 me·λe·te-2) (qe-2·λe2)] ×
[q1 q2 (4 π r2)-1] (30
where the first set of square brackets encloses the reciprocal of the SE value of the permittivity constant, ε0, and the second set, the rest of the right side of the equation. Of course, the variables, q1, q2, and r, use SE units of measure. Separating these variables' SE values from their SE units of measure gives:
Fee = [2 (β-1 me·λe·te-2)(qe-2·λe2)] ×
[|q1 q2|e (4 π |r|e2)-1 (qe2·λe-2)] (31
In Equation 31, the reciprocal of the SE-based permittivity constant, ε0, consists of three groups of factors: (2), (β-1 me·λe·te-2), and (qe-2·λe2). We eliminate each group in turn.
The number, 2, belongs with the q1 and q2 factors. In essence, the q1 charge contributes (q1 q2) to the force, and the q2 charge does the same for a total of (2 q1 q2). Placing the number, 2, with the two charges gives:
Fee = [(β-1 me·λe·te-2)(qe-2·λe2)] ×
[2 |q1 q2|e (4 π |r|e2)-1 (qe2·λe-2)] (32
Each of the two remaining groups of factors in the permittivity constant, ε0, represents the SE unit of electronic force, but is expressed in terms of the action that created the force reaction and not the unit of force, itself, as it should be. The SE unit of inertial force, fie, replaces (me·λe·te-2), and the SE unit of electronic force, fee(λ), replaces (qe2·λe-2) to give:
Fee = [(β-1 fie) (fee(λ)-1)] ×
[2 |q1 q2|e (4 π |r|e2)-1 (fee(λ))] (33
which shows more-easily that [(β-1 fie) (fee(λ)-1)] exists to convert the unit of force that is used in the equation from electronic to inertial; therefore:
(β-1 fie) = (fee(λ)) (34
and
[(β-1 fie)(fee(λ)-1)] = 1 (35
and the permittivity constant, ε0, no longer exists as such.
Coulomb's SE electronic-force equation is now:
Fee = 2 |q1 q2|e (4 π |r|e2)-1 fee(λ) (36
or, by converting the reaction, fee(λ), back into the action that caused it, qe2·λe-2, and reinserting these units back into |q1 q2|e and |r|e2, the final form becomes,
Fee = 2 q1 q2 (4 π r2)-1 (37
We compare the SI- and SE-based electronic-force equations using the SI newton, N, as the comparison unit of force, where q1 and q2 each equals qe, and r equals λe. The four steps of this procedure are as follows:
1) Determine the force Fem (in newtons), using the historic form of Coulomb's force equation:
Fem = qe2 (ε0 4 π λe2)-1 = (1.6 × 10-19 C)2 ×
[(8.9 × 10-12 s2·C2·kg-1·m-3) 4 π (2.4 × 10-12 m)2]-1 =
3.9 × 10-5 N (38
2) Determine the force Fee, (in fee(λ)), using the new SE-based form of Coulomb's force equation (without a constant of proportionality):
Fee = 2 qe2 (4 π λe2)-1 = 2 (1 qe2) [4 π (λe2)]-1 =
0.16 qe2·λe-2 (39
or, converting from the action to reaction form,
Fee = 0.16 fee(λ) (40
3) Convert the SE value of fee(λ) to the SI value:
fee(λ) = (4 π) 2-1 Fem = [2 π (3.9 × 10-5 N)] =
2.5 × 10-4 N (41
4) Convert the SE value of Fee to the SI value:
Fem = fee(λ) Fee = [(2.5 × 10-4 N) (fee(λ)-1)] (0.16 fee(λ)) =
3.9 × 10-5 N (42
and Equations 38 and 42 equal each other.
The presentation in this chapter proves mathematically that the two quantum actions,
Although, we now know the origin of the fine-structure constant, its significance still eludes us. However, in Chapters 8 through 11, we examine Newton's gravitational equation and discover, to our amazement, that the fine-structure constant, β, enters into it in the same manner as it does with Coulomb's electromagnetic equation. This opens a whole new area of speculation as to its significance, which I pursue in the subjective and metaphysical Part Two.
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