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?

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    $\begingroup$ Possible duplicate: physics.stackexchange.com/q/20477 $\endgroup$ – David H Sep 11 '13 at 22:34
  • $\begingroup$ See David Zaslavsky's answer to this Phys.SE question physics.stackexchange.com/q/15443/2451 $\endgroup$ – Qmechanic Sep 11 '13 at 23:47
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    $\begingroup$ 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. $\endgroup$ – Michael Sep 12 '13 at 0:35
  • $\begingroup$ 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. $\endgroup$ – Selene Routley Sep 12 '13 at 1:35
  • $\begingroup$ "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. $\endgroup$ – Philip Wood Oct 17 '18 at 18:53

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.

  • $\begingroup$ This is wrong for the reasons given in my answer, Steve B's answer, and Jonathan's answer to the question that this question duplicates. (Although I disagree with Steve B's over-all interpretation in his answer, he clearly discusses in that answer the fact that they are not completely logically independent as you claim in this answer.) $\endgroup$ – user4552 Sep 12 '13 at 1:21
  • $\begingroup$ +1 but I would be happier if you softened "nothing about" a bit. You can't derive as you say (as Michael Brown says, there are other possibilities to Lorentz force, albeit not in keeping with experiment, so that backs your answer), but there are definite links as discussed in the other answer so I wouldn't want to give the OP the impression that there were no links. $\endgroup$ – Selene Routley Sep 12 '13 at 1:56
  • $\begingroup$ PS: you might be amused to know that even though I've thought about Maxwell's equations in one way or another my whole professional life, I seem to have dreadful crises of understanding in electromagnetism about once every ten years (i'm actually due for one about now). This is all very subtle stuff and often highly nuanced. $\endgroup$ – Selene Routley Sep 12 '13 at 2:00

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