Where can I find a proof for the Ampere Circuital law? Wherever I look, I just find a proof for an infinitely long current carrying conductor.
There is no proof for it, it is an empirically measured law.
So is it an axiom? I always thought they declared a law only after some mathematical proof. I mean, for all we know, the experiment might be erroneous.
Edit:Were all Maxwell's equations experimentally determined? Gauss's law seems pretty intuitive. I can't understand the others.
There is no way to mathematically prove how the universe works. It simply isn't possible. All of Maxwells equations were obtained through experiment, as were all other physical theories.
Ampère's circuital law is the magnetic version of Gauss's law, if Gauss's law is intuitive, why isn't Ampère's?
I mean, for all we know, the experiment might be erroneous.
Gauss's law seems pretty intuitive. I can't understand the others.
We are really talking about two mathematical theorems here (as WannabeNewton mentioned). The divergence theorem and the Kelvin-Stokes theorem. Now you say they are completely different, but actually, they are both special cases of the same theorem: the generalised Stokes' theorem http://en.wikipedia.org/wiki/Stokes'_theoremGauss's law is simply based on the fact that if any curve enters/exits a closed Gaussian surface, it must exit/enter it as well, as long as it does not have an end/origin bounded by the closed surface. Ampere's circuital law is completely different.