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?
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9$\begingroup$ 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. $\endgroup$– user1355Commented Oct 7, 2011 at 13:29
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3$\begingroup$ Could you supply a reference or a clarifying article to expound on your claim that "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"? I have never heard this claim before and I am curious where it comes from. $\endgroup$– Scott GriffithsCommented Sep 23, 2014 at 18:12
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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$\begingroup$ 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. $\endgroup$ Commented Oct 6, 2011 at 19:23
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1$\begingroup$ 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. $\endgroup$ Commented Oct 6, 2011 at 19:27
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34$\begingroup$ 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. $\endgroup$– David ZCommented Oct 6, 2011 at 19:34
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6$\begingroup$ @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. $\endgroup$– David ZCommented Oct 6, 2011 at 19:36
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4$\begingroup$ @JerrySchirmer That's circular reasoning, the electric field is defined as the force per unit charge a motionless test particle feels. The text linked uses that fact (on page 60) to say $\rho\vec{E}\cdot\vec{v}$ is a power density and then derives the expressions for field energy and momentum from that fact. Maxwell's equations are really evolution equations for fields, they can't make a single experimental prediction until you couple them to matter via the Lorentz Force Law (or something equivalent like the field energy and field momentum neither of which follow from Maxwell alone). $\endgroup$– TimaeusCommented Mar 23, 2015 at 0:49
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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3$\begingroup$ I'm not downvoting, but please fix this: An example of the second is that Newton's laws predict that there can be no perihelion precession of Mercury, whereas measurements showed that is actually a perihelion precession of Mercury. That statement is incorrect. Newton's laws predict a significant perihelion precession of Mercury. The predicted value of 527 arcseconds per century is less than the observed value by about 43 arcseconds per century. The difference is significant, so it still does stand as a failure of Newtonian mechanics. $\endgroup$ Commented May 4, 2015 at 9:44
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$\begingroup$ Thank you, and now plus one (we're apparently not supposed to do +1 or -1 anymore, at least not on some parts of the SE network). If it's just a minor typo, incorrect grammar, a bad link, I'll edit. If an edit changes the meaning, I don't do it. I really don't like it when some noobie chasing a badge changes the meaning of one of my answers. So I don't do that to others. $\endgroup$ Commented May 4, 2015 at 12:55
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.
