The paradox we are about to relate (or a close resemblance) goes back to 1963 (at the very least,) in a Fig. 2c of a paper by Buchta [1], although its true origin may well be much earlier.
Klein [2] describes the curious circumstance in the follow manner.
“Imagine a box black on a table with two electrodes A and B, protruding from opposite sdes. The experimenter comes from one side, and measures with a static voltmeter a voltage between A and B of, e.g., 1V. He then walks around to the opposite side of the box and measures the voltage between the same two points, but this time he measures only 0.1V.”
The circuit within the box, together with the two terminals A and B, and the connected voltmeters, is shown below. Positioned at the centre is a solenoid or electromagnet, connected to an A.C. supply. In this variation, the voltmeters are connected at the same time.
On the face of it, the voltmeters are connected to the same electrodes, and so the measurements should yield the same potential difference between these two points.
Do they yield the same values? Isn't it impossible that they don't register the same potential difference?
Our speculation
Cf.
http://dx.doi.org/10.13140/RG.2.2.26858.70082
Dear Gary Stephens ,
I have had a look at the paper and I do not see that there is a paradox, unless there is a clear argument showing that the voltages have to be different, or an experiment showing that they are.
Jose Gaite
Dear Professor Gaite,
Thank you very much for your comment. You are the first to venture an opinion.
Not sure I understand exactly your meaning-- do you mean the voltages (being measured from the same terminals) should or rather -are- the same?-- that it is impossible for two voltmeters connected to the same pair of terminals to show different values?
Dear Gary Stephens ,
That is exactly what I mean. If the electric potential is a well-defined magnitude, it has one value at every point. Therefore, the voltmeters cannot measure different values, unless one or both are malfunctioning.
Jose Gaite
Dear Professor Gaite,
You are quite right. Ordinarily, this would be the case. And it would be impossible for voltmeters connected to the same pair of terminals to read different values, unless, like you said, they were malfunctioning. That is what is so interesting about this circuit.
This anomaly has been demonstrated experimentally, (I think more than once,) much to the astonishment of those present.
The actual reason why the two voltmeters are held at (widely) different values, is, I believe, because of the strictures of Relative Motion. Relatively moving observers have to agree on the observed reality, and this trumps everything else, even if it means the manifestation of an apparent impossibility.
What happens is the light (electromagnetic waves) organises itself, such that each voltmeter is held at a different value, notwithstanding they are connected to the same pair of terminals.
To see this, one first must realise the circuit is really two Faraday induction circuits stuck together. Cf.
http://dx.doi.org/10.13140/RG.2.2.26858.70082
Vide also,
https://www.researchgate.net/post/Faradays_Law_and_Relative_Motion
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The converse effect is shown in the below circuit.
https://www.researchgate.net/post/New_Paradox_in_Electrodynamics
Here the voltmeter connected to the middle crossbar, is held at zero volts, (for the same reasons) if both left and right crossbars are in motion.
This converse effect has not yet been demonstrated experimentally.
Dear Gary Stephens ,
I may have a further look at the references, but I still think that what ever happens at the solenoid will not produce a multivalued potential. Of course, if the solenoid somehow impairs one or both solenoids, then one or both are malfunctioning.
Dear Professor Gaite,
Well here is one (quite clear experimental description,) --- admittedly he doesn't actually give the experimental results,-- just hints at them, but I think there are more examples.
D. R. Moorcroft, Faraday's Law -“Demonstration of a Teaser, ” Am. J. Phys. 37 (2), 221 (1969).
Jose Gaite
Dear Professor Gaite,
But you are right, the experimental evidence is annoyingly scant, with academicians rather hinting at their results-- saying they have built the apparatus. Here is another example :
Dear Gary Stephens ,
After recalling that voltmeters generally are electromechanical instruments, I think that they are definitely affected by magnetic fields. If the effect has been measured experimentally, it is most likely due to the magnetic field of the solenoid affecting the functioning of the two voltmeters.
Jose Gaite
Dear Professor Gaite,
So essentially you are saying the voltmeters are malfunctioning due to the proximity of the solenoid, directly through the air, and not actually to do with the layout of the circuit at all?
So there is no paradox, just a spurious effect due to the proximity of the solenoid and voltmeters?
I suppose this could be tested by putting a goodly distance between the voltmeters and the solenoid-- perhaps a couple of feet? at which distance we should expect the voltmeters to register the same values?
Jose Gaite
Dear Professor Gaite,
I take your point, but surely it would have been unlikely that the experimenters were fooled in this way. I mean the two valued potential must have been reproducible, and fixed at widely different values.
Dear Gary Stephens ,
Perhaps, you are too confident on experimenters, some of whom have been notoriously fooled in the past. Consider Blondot's N rays, Mars canals, or have a look at the Journal of Irreproducible Results. Are you certain that this one is reproducible? I do not have a lab but tell me if you can attest that someone does it.
Dear Professor Gaite,
Well I certainly hope this is not another hoax. It took me a long time to figure out what was going on-- why the voltmeters were held at different values-- (a seeming impossibility.)
Thank you for your examples. Another one is the Great Moon Hoax, ---the great astronomical discovery of Sir John Herschel, L.L.D. F.R.S. &c., in 1835. "Man-bats" were spied through the telescope.
https://en.wikipedia.org/wiki/Great_Moon_Hoax
Jose Gaite
There was an on-going discussion in Youtube format on this topic a couple of years ago: https://www.youtube.com/watch?v=0TTEFF0D8SA
Dear René Christensen ,
Thank you for the tip, good explanation! I thought that the voltmeters could be affected by the magnetic field, but it is actually the wires that are affected!
Jose Gaite Yes, and so it is just a matter of a lumped model not really including all effects, unless you put some extra effort into it, and there are no paradoxes. What really stands out, is how arrogantly the professor handles this, instead of explaining 'both sides of the coin'. It is not that complicated. All models are limited, but as long as you understand the limitations, you're good.
Dear René Christensen ,
Good philosophical remarks! Models are just models, not the ultimate truth. I have often pondered over the limitatiions of lumped models.
I have seen this question posed as a problem in Purcell's good old "Electricity and Magnetism" (problem 7.4 in 3rd edition, 2013, co-authored with Morin).
Strangely enough, the answer seems to agree with Klein's paradox.
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