William Gilbert established the force formula K.x₁.x₂ / R² (for magnetism)
Where K is a constant, x₁ & x₂ are charges (electrical or magnetic) and R is the distance between the charges.
Newton used this formula to define the gravitational force between two masses; F = G.m₁.m₂/R²
and
Coulomb used this formula to define electrical force bewtween two charges; F = k.e₁.e₂/R²
And like Newton's G; Coulomb's constant was originally given units of convenience: N.m²/C²
The formula for Coulomb's constant 'k' is;
k = μₒ.c² / 4π = μ.c² = 8.98755184732667E+09 {kg.m/C² . (m/s)²}
If we break open; μ = Rₙ.mₑ/e² = 1E-07 (exact) {kg.m³ / C².s²}
we find that Rₙ.mₑ.c²/e² = 8.98755184732667E+09 {kg.m³ / s²C²}
It isn't difficult to see the equivalency of the two constants when one considers their units
k; kg.m³ / s²C²
G: m³ / s².kg
however, their units show that both constants are inconsistent. An explanation is provided in our Magnetism page.
Given that;
G = aₒ.c² / mₙ
where;
Rydberg's radius; aₒ = Rₙ.(ξᵥ/4π)²
unit mass of ultimate density; mₙ = mₑ.√[ξₘ/Σ]
G = Rₙ.(ξᵥ/4π)² . c² . √Σ / √[mₑ.mₚ]
and
k = μ.c² = mₑ.Rₙ/e² . c² {kg.m³ / s².C²}
G/k = Rₙ.(ξᵥ/4π)² . c² . √Σ / √[mₑ.mₚ] ÷ (Rₙ.mₑ/e² . c²)
G/k = (ξᵥ/4π)² . √Σ / √ξₘ.mₑ² . e²
G/k = (ξᵥ/4π)² . √Σ.√ξₘ . e²/mₑ.mₚ
because; (ξᵥ/4π)² . √[Σ.ξₘ] = φ
G/k = φ . e²/mₑ.mₚ
which is the definition of the coupling ratio:
φ = G.mₑ.mₚ / k.e²
Note: refer to our Mathematical Constants page for definitions of all of the above constants.
You will find further reading on this subject in reference publications(68, 69, & 70)