The Fine-Structure constant is a dimensionless number expressing the ratio of electromagnetic to plank energy. It would seem that the value of (α)-1 being equal 137.035999... is responsible for the existence of all Baryonic (ordinary) matter. I have become interested in the Fine-Structure_constant because of 'an other' project that I embarked upon [a dastardly cinema science-fiction plot]. Before completing that project I wanted to become knowledgeably comfortable with the 'Fine-Structure_constant', thus I researched CRC, Wikipedia, Richtmyer, Miles Mathis, et-al.
In perusing literature I encountered dilemma involving unclear and "apparently" inconsistent mathematical expressions used to compute the numerical value of alpha (α). This will be discussed at greater length in later paragraphs. My starting point, the CRC Handbook of Chemistry and Physics 76th Edition correctly lists the following expression for the Fine-Structure constant:
Since each of the four 'constant parameters' have non-zero values and also have units, old fashion rigor suggests that all four constants need be included in an [understandable] expression used to compute alpha. The CRC Handbook of Chemistry & Physics expression for (α) yields results that are both numerically and dimensionally correct. The following image shows the CRC expression to be, as claimed, a dimensionless ratio.
Additionally, the CRC equation can be used to compute alpha (α) in CGS units equally as easy and more importantly, in understandable terms. The task is outlined in the image below and amounts to simply converting all MKS units to appropriate CGS units; I so doing there is no need or purpose for an 'esu' value for electronic charge manipulation as shown in the Wiki 'CGS' version of (α) that will be discussed later.
Wikipedia lists two expressions for the Fine-Structure constant; The 1st one in (MKS) units is the same as CRC and I have no problem with it. A 2nd version is reported where the electronic charge is said to be in (cgs) 'esu' stat-coulombs and the equation for alpha is called an 'abbreviated' expression as follows:
The 'esu' (cgs) version is where I had a problem. After detailed deliberation I am of the opinion that this 'simplified' version of alpha has been convolved to obfuscate understanding. I would welcome an argument that details where my conclusion is incorrect; any such argument ought include units leading to dimensionless (α) as I have presented above for the CRC Chemistry & Physics handbook equation. Here I detail the reason I think obfuscation was afoot some decades ago.
Now I have to admit that the (cgs) 'esu' stat-coulomb alpha equation looks deceptively Simple and Cool: α =
hc). Only three terms on the right; an (e2) topside and a ( hc)
denominator. The CRC expression for alpha looks a bit more involved since it necessarily includes a permeability
of vacuum (μo) term that is buried (and all but ignored: think units) in the 'esu' expression.
As can be seen above, the value of the 'esu' stat-coulomb is the 'standard' electronic charge value of 1.602E-19 coulomb multiplied by the speed of light and divided by 10. Since 'e' in the numerator is squared an unseen 'c2' term shows up in the numerator; And to PREVENT ERROR, the denominator must have a c2 inserted next to plank's constant. The mysterious 'e' in the numerator retains the same 'e' symbol ('c' is hidden & 'e' is therefore obfuscated). The original 'c' in the numerator gets cancelled by one of the two 'c's inserted in the denominator. This makes the original 'c' term appear to move from the numerator to the denominator. The other strange 'esu' 'e' manipulation that takes place; the standard electron charge (1.602E-19) coulomb is divided by 10. If I had done all this, the coulomb police would probably charge me and lock me in a Faraday cage.
The permeability of vacuum 4πE-7 (Newton/ampere2) also has to be accounted for. (2π) of the
(4π) can be divided into planks constant (h/2π) to become (
h) the reduced planks constant. The
lay problem here is that the (μo) constant also has units of Newton-Amp-2 that tend to
mysteriously 'disappear' in the (cgs) version of Alpha.
Another Reference#1 that I have on hand shows the same expression as the 'Wiki' (cgs) version of alpha.
The above Referenced text was printed in 1955 and is from my undergraduate years at VPI. This would reinforce the
Wiki comment for an "abbreviated expression: α = e2/(
hc) commonly appearing
in physics literature"
The above image shows my version of 'abbreviated' expressions for the Fine-Structure_
constant. Notice that 'c', '
h', and 'μo' are adjusted to (MKS) or (cgs) units per
appropriate conversion factors for the two system of units. Notice also that electronic charge is measured in coulombs
in either system of units.
Also Notice: the (4π) portion of the Permeability of Vacuum (μo) is separated from it's
exponent; (2π) of it is divided (as a unitlesss number) into plank (h) to convert into reduced plank
h). The other (2)-portion cancels the (2) in the Denominator of the original (α) expression.
The Permeability of Vacuum units (Newton/Amp2 & (Dyne/Amp2) are, however, carried with the
original (μo) exponent(s) as indicated.
Derivation of these two expressions for Alpha do not use arbitrary setting of constants to one [unity], fudge units required to show alpha to be dimensionless, or 'sport' an esu stat-coulomb electronic charge with spooky units. Perhaps I am the only bofin on the 3rd rock that had a problem following explanations in Wiki and elsewhere? I hope this work is helpful in de-mystifying computation of the fine-structure-constant (α).