Linking Maxwell, Helmholtz and Gauss through the Linking Integral

We take the Gauss’ linking integral of two curves as a starting point to discuss the connection between the equation of continuity and the inhomogeneous Maxwell equations. Gauss’ formula has been discussed before, as being derivable from the line integral of a magnetic field generated by a steady current flowing through a loop. We argue that a purely geometrical result such as Gauss’ formula cannot be claimed to be derivable from a law of Nature, i.e., from one of Maxwell’s equations, which is the departing point for the calculation of the magnetic field. We thus discuss anew the derivation of Gauss’ formula, this time resting on Helmholtz’s theorem for vector fields. Such a derivation, in turn, serves to shed light into the connection existing between a conservation law like charge conservation and the Maxwell equations. The key role played by the constitutive equations in the construction of Maxwell’s electromagnetism is briefly discussed, as well.