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The Fine Structure Constants of Force Formulas 

One of the most famous dimensionless constants is the fine structure constant. This is the ratio of the constant in
the formula for electrostatic force to hc where h is Planck's constant divided by 2π and
c is the speed of light in a vacuum.
Coulomb's Law of Electrostatics is that the force between two charges q_{1} and q_{2} separated by a distance r is given by
where (1/(4πε_{0})) is a constant equal to 9×10^{9} kg*m^{3}/s^{2}. The quantity ε_{0} is known as the permittivity of free space.
The charge of any body is essentially equal to the net number of elementary charges it contains times the value of the elementary charge; q_{i}=q_{e}n_{i}. Thus the force formula could be represented as
Thus the force constant in units of kg*m^{3}/s^{2} is
(1/(4πε))q_{e}²=(9×10^{9})(1.60218x10^{19})²
= 2.3103×10^{28} kg*m^{3}/s^{2}.
The ratio of this constant to hc=3.1616×10^{26} kg*m^{3}/s^{2}
is 7.34844×10^{3} or approximately 1/137.06.
The electrostatic case suggests that the structure constant for the general case would be kμ²/ (hc)
where k is the force constant and μ is the unit of charge. But the more general force formula would be
where f(z) is a function of the dimensionless ratio r/r_{0} and r_{0} is a length parameter for the force. The generic charges q and Q are multiples n and N of the unit of charge μ so the force formula is more properly
Now it can be said that the proper formula for the structure constant for a force is kμ²/ (hc).
Newton's Law of Gravitation is that the force between masses m_{1} and m_{2} separated by a distance r is given by
where G is a constant equal to 6.67259×10^{11} m^{3}/kg.
The mass of any body is essentially equal to the number of nucleons it contains times the mass of a nucleon; m_{i}=m_{n}n_{i}. Thus the force formula could be represented as
Thus the force constant in units of kg*m^{3}/s^{2} is Gm_{n}² = (6.67259×10^{11})(1.6749×10^{27})^{2} =1.871855×10^{64} kg*m^{3}/s^{2}.
The ratio of this constant to the product of hbar and the speed of light
in a vacuum,
hc=3.1616×10^{26} kg*m^{3}/s^{2},
is 3.76915×10^{39}. This is 5.166×10^{37} times the strucure constant
for the electrostatic force.
The force between two nucleons may be given by the formula
The justification for this formula is that the nuclear force is carried by particles subject to decay; i.e., the π mesons. The population of remaining particles is a negative exponential function the time since emission which translates into a negative exponential function of distance. These remaining particles are spread over an area of 4πr². The intensity is thus proportional to e^{−λr}/r². For more on this model see Nuclear Force.
A value of r_{0} may be found such that 1/r_{0} is equal to &lambda..
An estimate of H based upon the separation distance of the nucleons in a deuteron
being 3.2 fermi is 3.392372×10^{26}
kg*m^{3}/s^{2}. This makes H equal to 1.073105hc=(1/0.931875)hc.
This is 147.08 times the value for the electrostatic force.
If the fine structure constant for the electrostatic force is denoted as α then the structure constant for gravitation is 5.166×10^{37}α and that of the nuclear force is 147.08α
Planck's constant is a dimensioned quantity and so its magnitude can literally be any positive value. Nothing of physical significance can depend upon its magnitude. It is proportional to the fine structure constant α and the constant of proportionality depends upon the dimensions used.
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