Is 'Curl of magnetic field in electrostatic is zero' only empirical? I was looking up on the uniqueness of the displacement current.
About the uniqueness of the displacement current this question was exactly what I was looking for, but all the answers seem to go with 'empirically, when the electric field is constant and current density is zero, the curl of the magnetic field is also zero.'
Are there any more explanation other than 'That's what we have found experimentally'? or are we solely relying on the fact the Ampere-Maxwell equation matches the experiments.
 A: No, it's not purely empirical. If the curl of an electrostatic magnetic field was going to be nonzero, it would have to be determined by some new term on the right-hand side of that Maxwell's equation. There is nothing available to serve that purpose that would preserve linearity and have the right symmetry properties under C, P, and T. For instance, $\operatorname{curl}\textbf{B}=\alpha \textbf{B}$, where $\alpha$ is a constant, would violate parity.
At a deeper level, there is very little wiggle room in writing down Maxwell's equations because they're the wave equations of an antisymmetric tensor field. If you try to write down something relativistically valid, you don't have a lot of other options. The main options are either to add magnetic monopoles or to do various trivial redefinitions like flipping the direction of the thing we call the magnetic field.
A: I think that the reason is that the Maxwell equation say so, and the Maxwell equations are empirically right. I can't think of a deeper reason
