Ivan Nikiforov said:
View attachment 356917
Good afternoon! I would like to understand how the Faraday disk works and get answers to two questions. The working conditions are as follows: 1) we rotate the external circuit with a voltmeter relative to a stationary disk; 2) the external circuit is not in a magnetic field. Questions: 1) will an electric current flow in such a system? 2) what forces will act on the external circuit with the voltmeter? I would like to consider these processes precisely from the point of view of the theory of relativity, since the Faraday disk is based on the principle of relativity of simultaneity. Thank you in advance!
I am a bit confused - isn't the whole point that the rotating part of the disk is in a magnetic field? I'll assume that this was a typo.
One way of describing the relativistic point of view is that the electric and magnetic fields are part of the Faraday tensor, which is a rank 2 anti-symmetric tensor. This tensor has six non-zero components (there are 16 compoonents in a rank 2 tensor, 4 of them are zero, and because it is anti-symmetric 6 of the 12 components are unique. We identify 3 of the components as the electric field, the other three as the magnetic field.
The wiki article on the Faraday tensor is here and describes the arrangement of the components, i.e. how you pack the 6 different components into a 4x4 matrix. There are some factors of c, the speed of light, when conventional units are used, but it is easier and common practice to use units where the speed of light is equal to 1 to avoid having to worry about this factor. The wiki articles will also illustrate the mathematical simplicity of the Lorentz force law and Maxwell's equations when the formulation using the Faraday tensor is viewed.
One can also describe the relativistic point of view without tensors, using the traditional E and B fields, though some of the transformation laws appear complicated. A key point of the relativistic formulation is that knowing E and B in one frame of reference, one can compute them in any frame of reference. This actually makes problems simpler to understand than the typical undergraduate formulation in many respects, as in the undergraduate approach, one does all the work to derive the solution in each difrent frame of reference, while the relativistic formulation one only needs to solve the problem in one frame of reference, and the solution can be transformed to any frame of reference. To do so successfully requires that one also transform the source fields relativistically, which is why the undergraduate solutions don't take advantage of the fact that knowing the solution in one frame of reference allows one to transform the solution to any frame of reference.
From a physical point of view, it is important that charges and curend densities be transformed in a relativistic manner. Charge density (also number density) depends on the frame of reference due to length contraction, for instance. There are other effects as well. The appropriate relativistic transformation laws for number density and charge density are the number-flux four vector and the charge-current 4-vector. This is discussed in Wikipedia at
https://en.wikipedia.org/wiki/Four-vector.
Introductory treatments of electromagnetism can use the more familiar 3-vector formulations, but that's not the way I think of it. The 3-vector form of the transformation laws appears more complicated, see for instance
https://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity#E_and_B_fields.
I believe Griffiths texbtook on electromagnetism does some motivation for why these rather complex appearing laws are the correct way to transform electric and magnetic field without using 4-vectors and tensors.
Basically, the relativistic explanation of the homopolar genrator is that in the non-rotating frame, there is only a magnetic field, and the relativistic transformation laws give rise to an electric field in the rotating frame of the disk.
You may prefer the non-tensor exposition of relativity - unfortunately, I've forgotten much of that. But do let us know which formulation works best for you, it's pointless to spend a lot of time discussing the 4-vector formalism if you're not interested in it and want to see a 3-vector treatment.