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There's couple of related question before. See Reference:

  1. Ampère's law vs Biot Savart law

  2. What is the difference between Biot-Savart law and Ampere's law?

  3. Is Biot-Savart law obtained empirically or can it be derived?

However, it didn't exactly answer all my doubts, and I have some contradiction information as well.

There's several major insights given in the previous posts:

  1. user103515: Biot Savart Law was experimental observation law.

  2. user26872: Biot-Savart law is a consequence of Maxwell's equations.

  3. Emilio Pisanty: both Ampère's law and the Biot-Savart law always hold.

  4. Contradict Information:

    a. Ján Lalinský and Self-teaching worker: A magnetic field around the capacitor does not obey Ampère law.

    b. Lelouch: The law is not incorrect except in capacitor type cases when the second term in maxwell's eqn. needs to be taken into account.

  5. My Instructor: There was a paper some where claim that Ampere's law was more general... not sure...

Recently, I've got in touch with lagragian density and action principle a lot. And, this might be a bit suspicious, but Ampere's law looked very, or say exactly, like the boundary term added to the action. This made it act like a gauge, or things of sort, which seemed to be more convient to be understood than Biot Savart. It's coincide with what peanut_butter had observed, that Ampere's law was hard to use unless there's some symmetry.

  1. Could you help me to clarify the point 4, and make a comparative argument between Ampere's law and Biot Savart Law?

  2. Especially, what's the current status on the view of the subject, and weather there's any connection between action principle?

  3. If there's does, then wasn't Biot Savart a special case under Lorentz symmetry? while Ampere's law was a gauge theory in general?(Well, it's gauge of conserved current, so I suppose that's an underling statement that they indicate to the same symmetry?)

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    $\begingroup$ Why are you focused on these rather than on Maxwell’s equations? I would argue that Maxwell’s equations are “more fundamental” than either Ampere’s Law or the Biot-Savart Law. Differential equations explain physics using local concepts. Integral laws are inelegant (although useful) and somewhat mysterious by comparison with the simplicity of differential relationships at a point. $\endgroup$ – G. Smith Dec 12 '19 at 5:35
  • $\begingroup$ @G.Smith Maxwell are basically SR, and relativity principles. So I guess they some what arise to the sameting. But Maxwell was esentially F tensor in the expression, but Ampere and Biot Savart are of current and boundary terms. Basically they are stating the related things, but Ampere and Biot Savart involves more applied calculation, thus more interesting? $\endgroup$ – ShoutOutAndCalculate Dec 12 '19 at 5:39
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    $\begingroup$ There is no contradictory information in the posts of Ján Lalinský and Self-teaching worker and Lelouch rather all of them say that the "simplified" version of Amperes law with only currents considered needs to have an extra term in it relating to the displacement current to be true for all situations. $\endgroup$ – Farcher Dec 12 '19 at 5:45
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    $\begingroup$ If you read the first paragraph of the Wikipedia article Biot-Savart Law you will find a clear statement of when this law can be used. $\endgroup$ – Farcher Dec 12 '19 at 5:48
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    $\begingroup$ en.wikipedia.org/wiki/… $\endgroup$ – anna v Dec 12 '19 at 6:01
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All laws in physics are a distillate of a large number or experimental measurements and observations. They are used in order to pick from the plethora of algebraic or differential equations, those equations that are relevant for modeling the observations. They are the "axioms" of the physics model.

In this case, the laws used by Maxwell to model electromagnetism, picking up the relevant solutions of his differential equations do not include the Biot Savart Law. The reason is found here in wikipedia

In a magnetostatic situation, the magnetic field B as calculated from the Biot–Savart law will always satisfy Gauss's law for magnetism and Ampère's law. In a non-magnetostatic situation, the Biot–Savart law ceases to be true (it is superseded by Jefimenko's equations), while Gauss's law for magnetism and the Maxwell–Ampère law are still true.

As Maxwell's equations managed to unify electricity and magnetism in one mathematical model, using the laws as "axioms" the Biot Savart law in not fundamental to electromagnetism.

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Ampere's law is one of the Maxwell equations, which is very useful in cases with high symmetry, whereas biot savart law can be derived from ampere's law, it is something similar to the relationship between Gauss's Law and Coulumb's law.

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