San Jos� State University

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Thayer Watkins
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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 q1 and q2 separated by a distance r is given by

#### F = (1/(4πε0))q1q2/r²

where (1/(4πε0)) is a constant equal to 9×109 kg*m3/s2. The quantity ε0 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; qi=qeni. Thus the force formula could be represented as

#### F = (1/(4πε))qe²n1n2/r²

Thus the force constant in units of kg*m3/s2 is (1/(4πε))qe²=(9×109)(1.60218x10-19)² = 2.3103×10-28 kg*m3/s2. The ratio of this constant to hc=3.1616×10-26 kg*m3/s2 is 7.34844×10-3 or approximately 1/137.06.

## The General Case

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

#### F = kqQf(r/r0)/r²

where f(z) is a function of the dimensionless ratio r/r0 and r0 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

#### F = kμ²f(r/r0)nN/r² which may be put into the form Fr²/f(r/r0) = kμ²nN

Now it can be said that the proper formula for the structure constant for a force is kμ²/ (hc).

## The Gravitational Force

Newton's Law of Gravitation is that the force between masses m1 and m2 separated by a distance r is given by

#### F = Gm1m2/r²

where G is a constant equal to 6.67259×10-11 m3/kg.

The mass of any body is essentially equal to the number of nucleons it contains times the mass of a nucleon; mi=mnni. Thus the force formula could be represented as

#### F = Gmn²n1n2/r²

Thus the force constant in units of kg*m3/s2 is Gmn² = (6.67259×10-11)(1.6749×10-27)2 =1.871855×10-64 kg*m3/s2.

The ratio of this constant to the product of h-bar and the speed of light in a vacuum, hc=3.1616×10-26 kg*m3/s2, is 3.76915×10-39. This is 5.166×10-37 times the strucure constant for the electrostatic force.

## The Nuclear Force

The force between two nucleons may be given by the formula

#### F = H*e−λr/r²

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 r0 may be found such that 1/r0 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*m3/s2. This makes H equal to 1.073105hc=(1/0.931875)hc. This is 147.08 times the value for the electrostatic force.

## Conclusion

If the fine structure constant for the electrostatic force is denoted as α then the structure constant for gravitation is 5.166×10-37α 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.