Faraday's disk and "absolute" magnetic fields

[PAllen]

I'm struggling with this one. It seems the $v\times B$ gives the correct answer. The problem is what is $v$? In the video $v$ is the velocity relative to the lab frame and $B$ is the field in the lab frame. What I only partially get is that spinning the magnet doesn't change the magnetic field or induce an electric field in the lab (this is what I think is shown by the experiment). The one case not done is spinning magnet+disk+stator. My guess is this would produce no EMF which I find, well, confusing.

No, see the second video, which corotates the stator as well and still shows electric polarization.

 

[Paul Colby]

No, see the second video,

Thanks, I gave up too early. The experiment shown removes one of my major issues with the previous experiment which uses a stationary electroscope. Having the co-rotating data logger really removes changes of the circuit/current path while rotating. I haven't worked out the math but I would expect an $E$ field in the rotating frame since from the lab frame the rotating observer is after all rotating. Why is the claim made that this violates relativity? It now seems consistent to me.

 

[PAllen]

Thanks, I gave up too early. The experiment shown removes one of my major issues with the previous experiment which uses a stationary electroscope. Having the co-rotating data logger really removes changes of the circuit/current path while rotating. I haven't worked out the math but I would expect an $E$ field in the rotating frame since from the lab frame the rotating observer is after all rotating. Why is the claim made that this violates relativity? It now seems consistent to me.

I think it clearly does not violate relativity. Whether, historically, some relativity experts including possibly Einstein had an incomplete or incorrect understanding of the problem is not something I can give informed comment on.

 

[Dale]

I came across this which demonstrates the capacitor experiment I mentioned earlier. Both videos are well done and worth]

I just watched the video. I am not sure why he thinks his results contradict Einstein in any way. I think that the author has some misunderstanding about relativity.

 

[rrogers]

I would start here: http://www.doiserbia.nb.rs/img/doi/0353-3670/2011/0353-36701102221B.pdf

Nice paper :) Thanks, I think I can do better at the mathematics but I think this is concrete enough to write the equations.

 

[vanhees71]

Okay, I understand.
Now if somebody could show me the mathematical derivation in terms of Maxwell's equations and/or the electromagnetic two-form, I would appreciate it. It would save me some trouble and thinking. Preferably with very few words.

I don't know, what the movies are about, and I've not the time to watch them. I gave a derivation for the most simple example of a homopolar generator, i.e., a homogeneously magnetized rotating sphere, here:

https://th.physik.uni-frankfurt.de/~hees/pf-faq/

Something seems to be wrong with our web server, but it should get online again very soon (so I hope ;-)).

 

[rrogers]

I don't know, what the movies are about, and I've not the time to watch them. I gave a derivation for the most simple example of a homopolar generator, i.e., a homogeneously magnetized rotating sphere, here:

https://th.physik.uni-frankfurt.de/~hees/pf-faq/

Something seems to be wrong with our web server, but it should get online again very soon (so I hope ;-)).

Worked fine for me. I do have one complaint about the comments beyond Eq: 32. I think it is misleading since a loop of wire in a time-varying magnetic field certainly does demonstrate a curl-type electric field; i.e. a transformer. It's true that the original source might be electric but the local field doesn't care. All it needs is the boundary conditions and controlling PDE; which can be made around a very small region. That's the reason PDE's rule FEA's

 

[vanhees71]

Maybe, the statement is a bit misleading in the way I expressed it, but for sure one should write the Maxwell equations in the following form, grouping them in terms of homogeneous
$$\vec{\nabla} \times \vec{E}+\frac{1}{c} \partial_t \vec{B}=0, \quad \vec{\nabla} \cdot \vec{B}=0$$
and inhomogeneous equations (from the point of view of the electromagnetic field),
$$\vec{\nabla} \times \vec{B} -\frac{1}{c} \partial_t \vec{E}=\frac{1}{c} \vec{j}, \quad \vec{\nabla} \cdot \vec{E}=\rho.$$
When solving the equations for the full time-dependent problem, you can easily derive inhomogeneous wave equations for the electromagnetic field components, and you solve these for the usual situation that you have given charge-current distributions at sources of outgoing electromagnetic waves, using the retarded Green's function, which leads to the socalled Jefimenko equations. These are of a form, to be expected from a relativistic local field theory: They lead to local expressions, taking into account the finite speed of propagation (in this case of a massless field it's the speed of light). This solution has also the great advantage of being manifestly Lorentz covariant (when written in the appropriate way as integrals over Minkowski-space tensor fields).

You can, of course, write solutions of the time-dependent Maxwell equations in a way, only using the 3D Helmholtz fundamental theorem of vector calculus at a fixed time. Then you get a kind of "instantaneous" form of integral solutions. However they look very ugly and unintuitive by just looking at them ;-)). They are also completely useless to solve the problem. You find a nice discussion with a lot of equations in

https://arxiv.org/abs/1609.08149

 

[rrogers]

Maybe, the statement is a bit misleading in the way I expressed it, but for sure one should write the Maxwell equations in the following form, grouping them in terms of homogeneous
$$\vec{\nabla} \times \vec{E}+\frac{1}{c} \partial_t \vec{B}=0, \quad \vec{\nabla} \cdot \vec{B}=0$$
and inhomogeneous equations (from the point of view of the electromagnetic field),
$$\vec{\nabla} \times \vec{B} -\frac{1}{c} \partial_t \vec{E}=\frac{1}{c} \vec{j}, \quad \vec{\nabla} \cdot \vec{E}=\rho.$$
When solving the equations for the full time-dependent problem, you can easily derive inhomogeneous wave equations for the electromagnetic field components, and you solve these for the usual situation that you have given charge-current distributions at sources of outgoing electromagnetic waves, using the retarded Green's function, which leads to the socalled Jefimenko equations. These are of a form, to be expected from a relativistic local field theory: They lead to local expressions, taking into account the finite speed of propagation (in this case of a massless field it's the speed of light). This solution has also the great advantage of being manifestly Lorentz covariant (when written in the appropriate way as integrals over Minkowski-space tensor fields).

You can, of course, write solutions of the time-dependent Maxwell equations in a way, only using the 3D Helmholtz fundamental theorem of vector calculus at a fixed time. Then you get a kind of "instantaneous" form of integral solutions. However they look very ugly and unintuitive by just looking at them ;-)). They are also completely useless to solve the problem. You find a nice discussion with a lot of equations in

https://arxiv.org/abs/1609.08149

Actually, the using the differential notation/antisymmetric tensor form reduces the symbolic clutter a lot. You can fit rotations and velocity changes on one line. Actually, the physics fit's on four lines of crisp geometric objects and then everything else is Minkowski geometry. Simple things for simple people like me. I always try to keep in mind that our mathematics is just a language describing an underlying reality (I am a die-hard Neo-Platonist) and there are probably other better languages possible. Maxwell's equations versus the geometric differential forms is an example.