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This question is an outgrowth of regarding voltage and emf where @sb1 mentioned Faraday's law. However, Faraday's law as part of Maxwell's equations cannot account for the voltage measured between the rim and the axis of a Faraday generator because $\frac {\partial B} {\partial t} = 0$. It would've been a different story if the derivative were $\frac {dB} {dt} $ but it isn't. A palliative solution to this problem is given by invoking the Lorentz force. However, Lorentz force cannot be derived from Maxwell's equation while it must be if we are to consider Maxwell's equations truly describing electromagnetic phenomena. As is known, according to the scientific method, one only experimental fact is needed to be at odds with a theory for the whole theory to collapse. How do you reconcile the scientific method with the above problem?

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I am somewhat sympathetic to your situation. It's good you are not discouraged to ask questions which confuse you. Unfortunately, I don't have much time answering this question now. Just wish to mention that there is absolutely nothing in this, which violates Maxwell's equations. One more friendly advice if you allow me to give. Think a thousand times before expressing such audacious statement like "Maxwell's equations fail in this phenomenon". You possibly don't know what you are saying! – user1355 Oct 7 '11 at 5:00
I'll wait for you to find time to answer the above question. I assure you, however that you will not be able to. Maxwell's equations cannot derive the voltage observed in the unipolar generator which is enough to invalidate them. If you claim that I don't know what I'm saying then explain how they derive said voltage. – ganzewoort Oct 7 '11 at 12:09
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I don't know why you expect Maxwell's equations to derive the said voltage in the first place! This is to do with how a charged particle in motion should interact with a magnetic field which is given by Lorenz force as correctly pointed out by David. I shall not repeat him. Lorentz force is the relativistically invariant force law which is essential for a correct treatment of the dynamics of charged particles. – user1355 Oct 7 '11 at 13:29
You see, you admit Maxwell's equations cannot derive the voltage in question. Leave the palliative solution, the Lorentz force, which I first mentioned, alone. By the way, if Maxwell's equations are truly the foundation of electrodynamics then they must derive Lorentz force. They can't, however. Like I said, they can't even derive emf in an open circuit. As to why I should expect them to derive all this, it's not me, it's the scientific method which expects them to derive it. And, by the way, leave relativity out of this. Relativity is a non-theory which I may prove to you at once in chat. – ganzewoort Oct 7 '11 at 13:47
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As already pointed out by David, Maxwell's equations only describe the dynamics of the electromagnetic field. The dynamics of charged particle requires to use the Lorentz force law. In the classical electrodynamics the charged particles and electromagnetic field, both are real. It is a dualistic theory in this sense. If only the particles can be expressed as a local concentration of the field and its movement as a changing field then only you can hope to find the dynamics of charged particles from field equations alone. There were failed attempts in these lines. cont... – user1355 Oct 7 '11 at 16:09
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I'm not sure if this addresses what you're actually asking (if not I'll convert it to a comment), but Maxwell's equations only describe the dynamics of the EM field itself. The Lorentz force law is something separate, which describes the field's effect on charged particles. I've never heard any serious physicist claim that you can, or should be able to, derive the force law from Maxwell's equations.

Classical electrodynamics takes both Maxwell's equations and the Lorentz force law as "postulates."

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Well, the fact is, however, that Maxwell's equations cannot account for the voltage observed in the unipolar generator. So, we have at least one experimental fact which they cannot account for and that is enough, according to the scientific method, to abandon these equations. It's a separate story if we don't want to abide by the scientific method any more. – ganzewoort Oct 6 '11 at 19:23
Also, Maxwell's equations cannot be taken as postulates as, for example Newton's second law cannot be taken as a postulate, because a plethora of experimental facts can be described exactly by these equations. Thus, they need not be postulated. And, yet, there's at least one experiment where they don't hold. I'd be curious to know how Maxwell's equations account for the emf of a Galvanic cell in an open circuit as well. – ganzewoort Oct 6 '11 at 19:27
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No, that's not what the scientific method is. After all, the same logic would tell you that we should abandon Newton's laws because they cannot account for bremsstrahlung radiation. When you have a fact which your theory cannot account for, all it means is that you've exceeded the domain of applicability of your theory. And systems which involve charged particles reacting to an EM field, like the generator (as I understand it), are outside the domain of applicability of Maxwell's equations. – David Zaslavsky♦ Oct 6 '11 at 19:34
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@ganzewoort (2 comments up): that would be why I put "postulates" in quotes. I was using it to mean the foundational equations of the theory, from which other results are generally derived - not postulates in the sense of mathematical axioms. Anyway, if you would like to continue this let's take it to Physics Chat. – David Zaslavsky♦ Oct 6 '11 at 19:36
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Regarding the unipolar generator:

The stationary rod (or leads on a multi-meter) is actually part of the circuit. You will get the same result with a stationary disk and rotating rod.

For short periods of time, it can be approximated by 2 wires going out from the center with the outer ends touching. As long as the wires are moving through the magnetic field at different velocities, the voltage drops across them will be different, resulting in a current through the wires.

The question of whether or not a rotating magnet will induce a voltage in a stationary disk is irrelevant to the functioning of this type of generator, since it would just add the SAME voltage to both parts of the circuit.

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