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Well, 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. 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 defined based on experiments that the force felt by a moving charge on the presence of a magnetic field is $F = qv\times B$, now can this 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?

And 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?

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Neither Maxwell nor Biot-Savart are fundamental - all such formulas follow from Coulomb and a well-chosen definition of $B$, as mentioned tangentially in this short rant. –  Chris White Jun 8 '13 at 20:13

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$\def\VA{{\bf A}} \def\VB{{\bf B}} \def\VJ{{\bf J}} \def\VE{{\bf E}} \def\vr{{\bf r}}$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.

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But how I've seen it done, Maxwell's equations are derived from the biot-savart law, which would make this circular. –  JLA Jun 8 '13 at 20:24
@JLA: I've added something to address the "circularity" you refer to. –  user26872 Jun 8 '13 at 23:00

It may be true that in days of yore people measured the force resulting from a filamentary current, discovering the Biot-Savart law, and then in turn used that as inspiration to construct Maxwell's equations. If that's how it actually happened historically, fine.

But this is analogous to some alien archaeologist 10 million years from now finding a skeletal hand and foot in the Earth. From the hand, the archaeologist comes to understand what the animal who had that hand liked to do with it: that it could grasp and use tools and so on. From the foot, the archaeologist it comes to understand that the animal it belonged to walked on two legs and that it typically weighed in adulthood around 100-300 pounds.

Only later does the archaeologist that the hand and the foot both belonged to the same animal--a human being. But the nature of the work means that the puzzle of what a human being was has to be broken down into chunks that can be individually understood before the whole picture can come together. That said, it would be backwards to suggest the hand and the foot are more fundamental than the human being itself.

The Maxwell equations have been constructed to be consistent with the Biot-Savart law and other pieces of information, like Coulomb's law. Thus, you can derive Biot-Savart from Maxwell, but not the other way around, for Maxwell is more general and all-encompassing.

If you already know the Lorentz force law, you can infer the strength of the magnetic field from a wire just by shooting charged test particles near the wire and observing their motion. But this calls into question how you already know the Lorentz force law, and so on.

You can go in circles all day over what is or is not fundamental, over what must be based on experimental observation and what is merely constructed to be consistent with those observations, but often there's a preference for "simple" experimental observations being considered fundamental vs. theoretical constructs that incorporate many such observations--see Chris White's comment that Maxwell's equations can be derived from Coulomb's law and some other stuff.

To me, this is silly. Maxwell's equations incorporate the sum total of our observations (those that fit the classical regime, at least). To me, it is what we know about classical electromagnetism. To say that you can derive Maxwell's equation with only one result plus a few assumptions...well, it misses the point that those assumptions also had to be tested and verified in the first place. To me, it is very very backwards to single out special cases (pure electric, pure magnetic, static or dynamic fields) and treat them as "fundamental".

Edit: but really, a physicist needs to work in both directions. To create new theory, we often have special cases that we don't know are connected and must bridge them together. That's building Maxwell's equations from Coulomb's law and Biot-Savart. To analyze a particular problem most easily, one we're not sure there is a special-case formula for, we must resort to the most general description (Maxwell) and try to reduce it down to something simpler and easier to solve (in the case of no current and no time-dependence, you can get back to Coulomb's law). Both approaches are necessary to be as flexible as possible.

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