# Are Lorentz force and maxwell's equations independent? [duplicate]

The Lorentz force and Maxwell's Equations gives answers to many physics problems, and the answers given by both methods are consistent.

For example, consider the problem of a conducting rod of length $\ell$ sliding at speed $v$ on two rails, with a $\mathbf{B}$ field normal to the plane of the rod/rails. Lenz's law, which is derived from Maxwell's equations can be used to find $V=-B\frac{dA}{dt}=-B\ell{v}$. On the other hand, consider a charge carrier $q$ in the conducting rod. $q(E+vB)=0$, so $E=-vB$ and integrating over the length of the rod gives $V=-B\ell{v}$. Both Lorentz force and Maxwell's Equations have given the same answer.

However, it appears that Maxwell's equations and Lorentz force appear self contained; Maxwell's equations is not concerned with particles at all. Can one be derived from the other, or is there an overarching structure from which Maxwell's Equations and Lorentz force are corollaries?

• Possible duplicate: physics.stackexchange.com/q/20477 – David H Sep 11 '13 at 22:34
• See David Zaslavsky's answer to this Phys.SE question physics.stackexchange.com/q/15443/2451 – Qmechanic Sep 11 '13 at 23:47
• You can derive the continuity equation from Maxwell's equations, but not the force law unless you bring in some additional assumptions (e.g. arxiv.org/abs/physics/0206022). Though note that the Lorentz force is the only thing you can write down for point particles that is consistent with relativity. – Michael Sep 12 '13 at 0:35
• Further to @MichaelBrown 's comment: an interesting "historical" point is that Maxwell included the Lorentz force amongst his equations in his "A Dynamical Theory of the Electromagnetic Field" of 1865 and was aware of it by 1861 at the latest in his On Physical Lines of Force. – Selene Routley Sep 12 '13 at 1:35
• "Lenz's law, which is derived from Maxwell's equations can be used to find V=−BdA/dt=−Bℓv" I don't think this is right. Lenz's law deals only with the direction of the induced emf, while the Faraday-Maxwell law (integral form) deals with the electric field (and hence the emf) induced by a changing magnetic field around a loop stationary in the frame of reference to which the equation is being applied. – Philip Wood Oct 17 '18 at 18:53

Maxwell's equations only tell use how electric and magnetic field evolve over a time $dt$ given their values at $(\vec r, t)$ and the currents densities at time t. They tell us nothing about the effects of electric and magnetic fields on charge, and for that you need the Lorentz force law.