I am trying to understand the motivation behind one of the Maxwell equations (say in vaccum rather than matter for simplicity).
As far as I understand in magnetostatics there is a theorem which says $$rot \vec B=\frac{4\pi}{c}\vec j\tag{1}$$ where $\vec B$ is a constant magnetic field, $\vec j$ is a current density.
This result cannot be true for time dependend magnetic fields as it contradicts to the continuity equation $\frac{\partial \rho}{\partial t} +div (\vec j)=0$. Hence Maxwell observed that if one modifies the equation as $$rot \vec B=\frac{4\pi}{c}\vec j +\frac{1}{c}\frac{\partial \vec E}{\partial t}\tag{2}$$ the contradiction is removed. Moreover for time independent fields (2) becomes (1).
This procedure is highly non-unique. In fact one can add to the right hand side of (1) any divergency free field which vanishes for time independent fields, e.g. $\frac{\partial \vec B}{\partial t}$. Are there any other a priori reasons leading to equation (2)?
I realize that a posteriori this equation was confirmed by all the experiments. Was Maxwell just lucky?