# Derivation of Lorentz force law from the Biot-Savart Law [closed]

How to derive the Lorentz force (F=qv X B) law from the definition of the magnetic field and the Biot-Savart Law?

• To me, the force law $\mathbf{F} = q \mathbf{v} \times \mathbf{B}$ is the definition of the magnetic field. What other definition do you have in mind? – d_b Aug 24 '19 at 1:21
• Like we had studied in middle school the magnetic field was defined on the basis of force felt between two magnets just like the electric field was defined on the basis of the force between two electric charges. – Mayank Pande Aug 24 '19 at 1:33
• Biot-Savart worked out the magnitude of the $B$ field at a given distance from a current carrying wire. It was Lenz who worked out out the force between $2$ current carrying loops or wires. In any case, the force between the current loops or wires requires line integrals - there are no line integrals in the Lorentz force. It's possible the opposite might be true - Lenz from Lorentz. – Cinaed Simson Aug 24 '19 at 3:39

## 2 Answers

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.

• @MayankPande And conclude what? – G. Smith Aug 24 '19 at 2:10
• Consider two parallel current carrying conductors. They will experience a force because they are both behaving as magnets . The magnitude of the force will need to be derived from the classical (nineteenth century) expression for the force between two magnets – Mayank Pande Aug 24 '19 at 2:12
• @MayankPande I still don't see your point – BioPhysicist Aug 24 '19 at 2:15
• What expression for the force between two magnets are you referring to? Please be specific. – G. Smith Aug 24 '19 at 2:17
• @MayankPande That's not the same thing. You're asking about physical laws here, not the usefulness of mathematical formalism. – BioPhysicist Aug 24 '19 at 16:22

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.