Motivation for Maxwell-Ampere equations

Question

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?

Answer 1

Adding a solenoidal field to (1) does not change the underlying problem namely that $\mathbf{J}$ is not solenoidal unless the charge distribution is static and hence the current is stationary. You still have the unresolved problem of a vector equation being solenoidal on the left side and non-solenoidal on the right side manifesting itself with the same problem of the flux between the plates of a capacitor. Since we (Coulomb. Ampere, Maxwell...) understand the physics of a dielectric-filled capacitor reasonably well it is reasonable to modify the equation so that some of the contradiction is removed by including the polarization current after which the real puzzlement that remains is what happens in a vacuum.

You may always add a lamellar field to $\mathbf{B}$ since the curl of the the sum does not change (curlgrad=0) but that lamellar field is not observable.