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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?

user2122387
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Yes, they're independent of one another.

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.

Larry Harson
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