If I connected my voltmeter leads across the left-hand crossbar in the top figure, the voltmeter would presumably read an e.m.f of "E"?
But then, what would my voltmeter read for the left-hand crossbar in middle and bottom figures?
Also what is the Relativistic expression for the emf in the moving crossbar?
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Below we have a straight-wire "crossbar" and a U-shaped wire. A uniform magnetic field B is directed out of the paper.
Here, in the top figure, with the crossbar moving to the left, touching the U-shaped wire, we have, by the relation (v x B), an emf appearing across the left-hand crossbar, as indicated.
In the middle figure, with an observer moving in concert with the moving crossbar, we have, by the relation (v x B), an emf appearing across the right-hand crossbar.
In the lower figure, with an observer moving to the left, at half the rate of the left-hand crossbar, we have, by (v x B), an emf appearing across both right and left hand crossbars.
Faraday's law involves both a time-varying aspect and a convective aspect. The former is about the force induced on a charged particle that is placed at a point in a magnetic field that is changing in time. The latter is about the force that is induced on a charged particle that is in motion in a magnetic field. The latter aspect of Faraday's law can be isolated as the equation F = qvxB. It's the latter aspect that is relevant for the EMF induced in a conducting wire that is in motion in a magnetic field.
You don't need to involve relativity.
Frederick David Tombe
Dear Professor Tombe,
Thank you very much for your reply. You are the first to dare a response. I know from the literature this can be a very confusing and difficult subject.
Would the integral form of Faraday's Law be the best suited for the above problem?
Do you have any opinion on what the voltmeter would read? If you recall the question was :
"What would my voltmeter read for the left-hand crossbar in middle and bottom figures? "
Gary Stephens , You don't need to use the integral form of Faraday's law. The equation E = vxB is sufficient.
The answer is the same in all three cases as far as I am concerned because I don't believe that the relative speed of the observer affects the situation in the slightest.
Gary Stephens , And even if we were to believe in special relativity, the EMF readings would still be the same, because both cross bars are in parallel.
Frederick David Tombe
Dear Professor Tombe,
The relativistic expression is also interesting, in that the "relativistically-modified" emf in the moving crossbar (let's call it gamma.E) is fed back round, and so is also the emf in the stationary crossbar. In terms of the Dingle-clock case, it would be as if the "moving clock runs slow" but also (fed back around) the "stationary clock also runs slows". In the emf case, this might actually work (be free from contradictions,) as contra-distinct to the well know Dingle case where we have the clock running both faster and slower.
Gary Stephens , But it's only a very particular and simple case where it would work. There are many other cases where it is absurd.
Frederick David Tombe
Dear Professor Tombe,
That's true. I believe Einstein was well aware of this, but, the way out of these difficulties (ie. the ones Dingle highlighted,) was opaque to him, just as it is opaque to everyone else.
In my view the Minkowski space-time framework is valid, (especially as arrived at by A A Robb,) but it's more a description of the odd behaviour of light, in the presence of mankind, than anything else.
Frederick David Tombe
Dear Professor Tombe,
Related to the above post is the one below.
https://www.researchgate.net/post/New_Paradox_in_Electrodynamics
I'm guessing that since you think the emf is not located anywhere in particular in the circuit,-- that both inductions could, for example, be located in the middle crossbar, then you would agree that no induction can take place in this circuit, if both crossbars are in motion?
Gary Stephens , Einstein's singular, but fatal, error was abandoning the luminiferous medium. That medium served as a wave-carrying medium for EM waves, as well as a standard of rest for inertial motion, and the Lorentz transformations are only correctly understood physically in connection with the tiny aethereal vortices that comprise the luminiferous medium. The time-terms apply to the angular frequency of the tiny vortices and not to astronomical time.
There is no time dilation as per Einstein's theories.
As for 4D space-time, it is an over-sensational terminology for the luminiferous medium, reflecting the 4D interaction between frequency and wavelength in the optical Doppler effect, which is the parent theory of the Lorentz transformations, and which requires the use of four-vector algebra when extrapolated to the electric and magnetic fields at a deeper level in EM radiation.
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