Can the Lorentz force expression be derived from Maxwell's equations?

Question

The electromagnetic force on a charge $e$ is

$$F=e(E+v\times B),$$

the Lorentz force. But, is this a separate assumption added to the full Maxwell's equations? (the result of some empirical evidence?) Or is it somewhere hidden in Maxwell's equations?

Answer 1

Maxwell's equations do not contain any information about the effect of fields on charges. One can imagine an alternate universe where electric and magnetic fields create no forces on any charges, yet Maxwell's equations still hold. (E and B would be unobservable and totally pointless to calculate in this universe, but you could still calculate them!) So you can't derive the Lorentz force law from Maxwell's equations alone. It is a separate law.

However...

--> Some people count a broad version of "Faraday's law" as part of "Maxwell's equations". The broad version of Faraday's law is "EMF = derivative of flux" (as opposed to the narrow version "curl E = derivative of B"). EMF is defined as the energy gain of charges traveling through a circuit, so this law gives information about forces on charges, and I think you can derive the Lorentz force starting from here. (By comparison, "curl E = dB/dt" talks about electric and magnetic fields, but doesn't explicitly say how or whether those fields affect charges.)

--> Some people take the Lorentz force law to be essentially the definition of electric and magnetic fields, in which case it's part of the foundation on which Maxwell's equations are built.

--> If you assume the electric force part of the Lorentz force law (F=qE), AND you assume special relativity, you can derive the magnetic force part (F=qv x B) from Maxwell's equations, because an electric force in one frame is magnetic in other frames. The reverse is also true: If you assume the magnetic force formula and you assume special relativity, then you can derive the electric force formula.

--> If you assume the formulas for the energy and/or momentum of electromagnetic fields, then conservation of energy and/or momentum implies that the fields have to generate forces on charges, and presumably you can derive the exact Lorentz force law.

Answer 2

Steve B gives a very, very good answer, but I have one thing to add to his third point. He says if you assume the electric part of the force, you can derive the magnetic part from relativity. I have a different derivation for the magnetic part that doesn't exactly use relativity in an obvious way. I take a freely propagating e-m wave travelling between two metal plates. From Maxwell's equations we can get the induced charges in the plates, and also the induced currents. If we know the electrostatic force due to the charges, then the two plates must be attracted to each other. It turns out that the magnetic force is exactly equal and opposite to the electric force, so there is no net force between the plates. It's a nice calculation, and I'd like to say it allows me to derive the magnetic force, but I was never able to think of a physical reason why I would be entitled to assume that the total force between the plates must be zero.

I talk about this problem on my physics blog .

Answer 3

I haven't seen this mentioned in the answers so I thought I should at least mention it. If you take the perspective that Maxwell's equations are the equations describing a $U(1)$ gauge field, then minimal coupling (which is, in a sense, the only gauge invariant way of coupling matter to a gauge field) ensures than any charged particle obeys the Lorentz force law, with the only freedom being the value $e$ of its charge. So while Maxwell's equations themselves, without some additional assumptions, may not necessarily imply the Lorentz force law, $U(1)$ gauge invariance does imply the Lorentz force law. In fact, if you take $U(1)$ gauge invariance as being the fundamental starting point, then it implies both Maxwell's equations and the Lorentz force law. Again, this is a matter of perspective, so I am not disagreeing with the other answers, but I think that this is the modern point of view.