This question is an outgrowth of What is the difference between electric potential, potential difference (PD), voltage and electromotive force (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?
4 Answers
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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There are three separate possibilities:
- A theory correctly predicts an experiment result
- A theory predicts something but an experiment contradicts it.
- A theory makes no prediction whatsoever regarding some experimental result.
You're treating the second and third possibility as if they were the same thing, but they're entirely different.
For example, let's take Newton's laws (of motion and of gravity), as originally understood (i.e. as exact laws, not just as an approximate theory in a certain limit which is how we think of these laws today)...
An example of the first is that Newton's laws correctly predict the moon's motion orbiting earth.
An example of the second is that Newton's laws predict that there can be no perihelion precession of Mercury, whereas measurements showed that there is actually a perihelion precession of Mercury. So this contradicts Newton's laws. [Correction 2015-05: I mean, additional precession beyond the precession caused by the gravitational pull of other planets. Thanks David Hammen.]
An example of the third is the fact that uranium undergoes radioactive decay. Newton's laws make no prediction that uranium does or does not undergo radioactive decay. It's outside the scope of the laws.
Every theory (except a "theory of everything" like string theory) has things in the third category. There's nothing wrong with that. The stuff in the third category is not evidence against the theory.
So let's talk about Maxwell's equations (as they are normally written today, which is a bit different than how Maxwell originally defined them). Maxwell's equations by themselves make no predictions whatsoever about any kind of electromagnetic forces, because there are no electromagnetic forces in the equations, just electric fields and magnetic fields.
EMF is in the category of electromagnetic forces (it's not exactly a force, but it's an integral of force over distance divided by charge). Therefore anything you say about electromagnetic forces or EMFs is in the third category---something that Maxwell's equations by themselves make no prediction about.
That is why we have the Lorentz Force Law! Maxwell's equations plus the Lorentz force law DO in fact make all kinds of predictions about electromagnetic forces and EMFs. These predictions, yes including the EMF generated by a homopolar (Faraday) generator, are correct (first category).
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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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I'll take a slightly different take on the question than the other answers already posted, because I want to address a different side of the question.
Just because a theory makes incorrect predictions doesn't make it useless. Take Newtonian gravity, for instance. It doesn't correctly predict the bending of light or the precession of Mercury, both of which were triumphs of general relativity. Yet we still learn Newtonian gravity in high school. Why? Because it is more convenient to do calculations using $F=G \frac{m_1m_2}{r^2}$ than using $R_{ab}-\frac{1}{2}Rg_{ab}+ \Lambda g_{ab}=\frac{8 \pi G}{c^4}T_{ab}$.
I don't know much about Maxwell's equations (and hence can't verify or dispute your statement), but I can say that if a theory makes an accurate prediction in one scenario, it is useful. It may not be right in all circumstances, but if it is easier to calculate with, I think most of us will be happy to use it.
By the way, this isn't an attempt at the bounty. As I've already said, I believe another user deserves it.
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