hese paradoxes are generally resolved by the fact that an EMF may be created by a changing flux in a circuit as explained in Faraday's law or by the movement of a conductor in a magnetic field. This is explained by Feynman as noted below. See also A. Sommerfeld, Vol III Electrodynamics Academic Press, page 362.

The equipment
See also: electrical generator

Figure 1: Faraday's disc electric generator. The disc rotates with angular rate ω, sweeping the conducting disc circularly in the static magnetic field B due to a permanent magnet. The magnetic Lorentz force v × B drives the current radially across the conducting disc to the conducting rim, and from there the circuit path completes through the lower brush and the axle supporting the disc. Thus, current is generated from mechanical motion.
The experiment requires a few simple components (see Figure 1): a cylindrical magnet, a conducting disc with a conducting rim, a conducting axle, some wiring, and a galvanometer. The disc and the magnet are fitted a short distance apart on the axle, on which they are free to rotate about their own axes of symmetry. An electrical circuit is formed by connecting sliding contacts: one to the axle of the disc, the other to its rim. A galvanometer can be inserted in the circuit to measure the current.

The procedure
The experiment proceeds in three steps:

The magnet is held to prevent it from rotating, while the disc is spun on its axis. The result is that the galvanometer registers a direct current. The apparatus therefore acts as a generator, variously called the Faraday generator, the Faraday disc, or the homopolar (or unipolar) generator.
The disc is held stationary while the magnet is spun on its axis. The result is that the galvanometer registers no current.
The disc and magnet are spun together. The galvanometer registers a current, as it did in step 1.
Why is this paradoxical?
The experiment is described by some as a "paradox" as it seems, at first sight, to violate Faraday's law of electromagnetic induction, because the flux through the disc appears to be the same no matter what is rotating. Hence, the EMF is predicted to be zero in all three cases of rotation. The discussion below shows this viewpoint stems from an incorrect choice of surface over which to calculate the flux.

The paradox appears a bit different from the lines of flux viewpoint: in Faraday's model of electromagnetic induction, a magnetic field consisted of imaginary lines of magnetic flux, similar to the lines that appear when iron filings are sprinkled on paper and held near a magnet. The EMF is proposed to be proportional to the rate of cutting lines of flux. If the lines of flux are imagined to originate in the magnet, then they would be stationary in the frame of the magnet, and rotating the disc relative to the magnet, whether by rotating the magnet or the disc, should produce an EMF, but rotating both of them together should not.

Faraday's explanation
In Faraday's model of electromagnetic induction, a circuit received an induced current when it cut lines of magnetic flux. According to this model, the Faraday disc should have worked when either the disc or the magnet was rotated, but not both. Faraday attempted to explain the disagreement with observation by assuming that the magnet's field, complete with its lines of flux, remained stationary as the magnet rotated (a completely accurate picture, but maybe not intuitive in the lines-of-flux model). In other words, the lines of flux have their own frame of reference. As we shall see in the next section, modern physics (since the discovery of the electron) does not need the lines-of-flux picture and dispels the paradox.

Modern explanations
Taking the return path into account
In step 2, since there is no current observed, one might conclude that the magnetic field did not rotate with the rotating magnet. (Whether it does or does not effectively or relatively, the Lorentz force is zero since v is zero relative to the laboratory frame. So there is no current measuring from laboratory frame.) The use of the Lorentz equation to explain this paradox has led to a debate in the literature as to whether or not a magnetic field rotates with a magnet. Since the force on charges expressed by the Lorentz equation depends upon the relative motion of the magnetic field (i.e. the laboratory frame) to the conductor where the EMF is located it was speculated that in the case when the magnet rotates with the disk but a voltage still develops, the magnetic field (i.e. the laboratory frame) must therefore not rotate with the magnetic material (of course since it is the laboratory frame), while the effective definition of magnetic field frame or the "effective/relative rotation of the field" turns with no relative motion with respect to the conductive disk.

Careful thought showed that, if the magnetic field was assumed to rotate with the magnet and the magnet rotated with the disk, a current should still be produced, not by EMF in the disk (there is no relative motion between the disk and the magnet) but in the external circuit linking the brushes,[9] which is in fact in relative motion with respect to the rotating magnet. (The brushes are in the laboratory frame.)

This mechanism agrees with the observations involving return paths: an EMF is generated whenever the disc moves relative to the return path, regardless of the rotation of the magnet. In fact it was shown that so long as a current loop is used to measure induced EMFs from the motion of the disk and magnet it is not possible to tell if the magnetic field does or does not rotate with the magnet. (This depends on the definition, the motion of a field can be only defined effectively/relatively. If you hold the view that the field flux is a physical entity, it does rotate or depends on how it is generated. But this does not alter what is used in the Lorentz formula, especially the v, the velocity of the charge carrier relative to the frame where measurement takes place and field strength varies according to relativity at any spacetime point.)

Several experiments have been proposed using electrostatic measurements or electron beams to resolve the issue, but apparently none have been successfully performed to date.