There's already a question like this here so that my question could be considered duplicate, but I'll try to make my point clear that this is a different question.
Is there a way to derive Biot-Savart law from the Lorentz' Force law or just from Maxwell's Equations?
The point is that we usually define, based on experiments, that the force felt by a moving charge on the presence of a magnetic field is $\mathbf {F} = q\mathbf{v}\times \mathbf{B}$, but in that case the magnetic field is usually left to be defined later.
Now can that force law be used in some way to obtain Biot-Savart law like we obtain the equation for the electric field directly from Coulomb's Force law?
I wanted to know that because as pointed out in the question I've mentioned, although Maxwell's Equations can be considered more fundamental, those equations are obtained after we know Coulomb's and Biot-Savart's laws, so if we start with Maxwell's Equations to obtain Biot-Savart's having use it to find Maxwell's Equations then I think we'll fall into a circular argument.
In that case, without recoursing to Maxwell's Equations the only way to obtain Biot-Savart's law is through observations or can it be derived somehow?
Best Answer
$\def\VA{{\bf A}} \def\VB{{\bf B}} \def\VJ{{\bf J}} \def\VE{{\bf E}} \def\vr{{\bf r}}$The Biot-Savart law is a consequence of Maxwell's equations.
We assume Maxwell's equations and choose the Coulomb gauge, $\nabla\cdot\VA = 0$. Then $$\nabla\times\VB = \nabla\times(\nabla\times\VA) = \nabla(\nabla\cdot\VA) - \nabla^2\VA = -\nabla^2\VA.$$ But $$\nabla\times\VB - \frac{1}{c^2}\frac{\partial\VE}{\partial t} = \mu_0 \VJ.$$ In the steady state this implies $$\nabla^2\VA = -\mu_0 \VJ.$$ Thus, we have Poisson's equation for each component of the above equation. The solution is $$\VA(\vr) = \frac{\mu_0}{4\pi}\int \frac{\VJ(\vr')}{|\vr-\vr'|}d^3 r'.$$ Now we need only calculate $\VB = \nabla\times\VA$. But $$\nabla\times\frac{\VJ(\vr')}{|\vr-\vr'|} = \frac{\VJ(\vr')\times(\vr-\vr')}{|\vr-\vr'|^3}$$ and so $$\VB(\vr) = \frac{\mu_0}{4\pi}\int \frac{\VJ(\vr')\times(\vr-\vr')}{|\vr-\vr'|^3} d^3 r'.$$ This is the Biot-Savart law for a wire of finite thickness. For a thin wire this reduces to $$\VB(\vr) = \frac{\mu_0}{4\pi}\int \frac{I d{\bf l}\times(\vr-\vr')}{|\vr-\vr'|^3}.$$
Addendum: In mathematics and science it is important to keep in mind the distinction between the historical and the logical development of a subject. Knowing the history of a subject can be useful to get a sense of the personalities involved and sometimes to develop an intuition about the subject. The logical presentation of the subject is the way practitioners think about it. It encapsulates the main ideas in the most complete and simple fashion. From this standpoint, electromagnetism is the study of Maxwell's equations and the Lorentz force law. Everything else is secondary, including the Biot-Savart law.