Derivation of Lorentz force law from the Biot-Savart Law

Question

How to derive the Lorentz force ($\vec{F}=q\vec{v} × \vec{B}$) law from the definition of the magnetic field and the Biot-Savart Law?

Answer 1

This is not possible. The Biot-Savart Law tells how the moving charges that make up steady currents produce the magnetic field ${\bf B}$. The Lorentz Force Law tells how the magnetic field, in turn, produces a force on a moving charge. The two phenomena are, at this level, independent, and one cannot be derived from the other.

Answer 2

To me it is natural to think of the magnetic field as being defined (implicitly) by Maxwell's equations. If you go by this definition, then there is no way of deriving the expression for the Lorentz force due to the magnetic field from the Biot-Savart Law. The evidence for the Lorentz force law is historically experimental, but they can also be theoretically motivated: see the answers to this question.

The Biot-Savart law can be derived from Maxwell's equations for static (or slowly-varying) fields and sources. If you think of the magnetic field as being defined by Maxwell's equations, since the Biot-Savart law is also a consequence of Maxwell's equations, the derivation you are seeking would only be possible if the magnetic Lorentz force can be derived from Maxwell's equations. However, this cannot be done (at least not without some additional assumptions: again, see the answers to this question). Classical electrodynamics, in addition to Maxwell's equations, ordinarily requires a prescription for the force applied on charges due to fields, i.e. the Lorentz force law.