There are several explanations of the principle of operation of the transformer, for example, in [1,2]. All of them are based on the operation of Faraday's law of electromagnetic induction [3], which states that the EMF (modulo) is equal to the rate of change of the magnetic flux passing through the inductor. We have been using this principle for almost 200 years, and it really agrees well with many known phenomena, and the calculation of electrical machines reflects reality quite accurately. And everything would have been great if generators had not appeared in due time, to explain the work of which this law alone was not enough. One of them is a unipolar generator [4], whose operation we will consider here. In this note, we will temporarily forget about the existence of the law of electromagnetic induction, and explore the unipolar generator using completely different tools.

Generally speaking, if you, as an engineer, only know Lorentz's law [5], and the accompanying left-hand rule, then you can easily understand many electric machines and even invent new ones :) One of them will be discussed in this work.

Faraday came to this paradox when he studied the unipolar generator invented by him [4]. He first made a version of this generator with a fixed magnet, where his work was satisfactorily explained. But when the magnet was made movable, nuances appeared that gave rise to the well-known paradox. It is this version of the generator that we will consider further. Its device is very simple: a round magnet M is located on the same axis with a copper disk D, where they can rotate independently of each other. Current collectors are connected to the center and edge of the disk, closed through an ammeter A, together representing an external circuit (Fig. 1).

The paradox was that the rotation of the magnet together with the disk led to the appearance of an emf in a stationary external circuit. The science of that time rushed to explain this paradox, but disagreements were observed in it even after the discovery of the electron. Scientists even proposed a hypothesis that the lines of force of a magnet are motionless and independent of its rotation, and if so, then unipolar induction was placed in the category of relativistic effects. When, nevertheless, a discrepancy came up with the special theory of relativity [6], indefatigable science has tried to explain this paradox by using the dependence of the scalar potential of the charge on its relative velocity [7]. You will be surprised, but such misconceptions in science are still present.

The operation of the Faraday disk can be fully explained using the Lorentz force [5] or Ampère's law [8], which in its form and essence is one and the same, if we consider only the magnetic component (Fig. 2). Visualizes these two laws the rule of the left hand, which states that if the lines of magnetic induction B enter the palm of the left hand, and the direction of their movement is indicated by four fingers, then the direction of the Lorentz force F acting on the charge will show the thumb. In the case of a conductor, nothing changes, because it also has free electrons, which are affected by the same force, only here is the charge moving relative to the magnetic lines. And if so, then we automatically get Ampère's law, where the product of the charge q speed \(v\) is replaced by the product of the current I and the length of the conductor l. If there is an angle other than 90° between the induction lines and the direction of their movement (or current), then the sine of this angle (angle α) must be added to the force formula.

Returning to paFaraday's radius, we will correctly display the magnetic field lines passing through the disk and through the circuit (Fig. 3). There are only two types of them. The first type of magnetic lines enters the disk D and exits below it (points 1, 2, 3). If we sum up the currents they create, then they will be equal to zero, because. the induction decreases with distance from the center, while the linear velocity \(v\) increases. The magnetic lines of force enter the disk with a high density (with a large B), but with a low speed (point 2), and exit with a low density (with a low B), but with a high speed (point 3). Since the entry angle changes from plus (point 2) to minus (point 3), according to the following formula, the total force must be compensated: \[ F = B\, q\, v\, \sin(\alpha) \tag{1}\] The Lorentz force F shifts the electrons in the disc D to one side or the other, depending on the angle of entry of the magnetic line α. The charge q is determined by the number of free electrons in the disk material, through which the magnetic lines pass with the speed \(v\). Such a compensation of forces (and hence currents) is possible, of course, only with a symmetrically magnetized cylindrical magnet located in the center of the disk. This is clearly shown in the following experiment [13], although there the author of the video does not draw a completely correct conclusion from the result obtained. In fact, the magnetic lines move with the magnet, as Faraday originally intended, which is also intuitive and logical. The fact of rotation of magnetic lines of force is also confirmed by this experiment [14].

From here we immediately conclude that the first type of magnetic lines does not add current to the Cr circuit, and therefore these lines can be disregarded further. The second view remains, and these are the lines that enter the disk and the chain at the same time (points 4,5). If the disk and the chain rotate or are at rest together, then the forces, and hence the currents generated by these magnetic lines, will also be compensated, as in the first case. But if the disk is rotating and the chain is at rest, or the disk is at rest and the chain is rotating, then an uncompensated force arises, which generates a current, but since the common circuit is closed, this current is recorded by an ammeter A. This is the whole "secret" Faraday paradox!

Now it becomes obvious why it doesn't matter for the experiment whether the magnet rotates or is at rest. In more detail, all cases are summarized in the following table, where ω means rotation, and zero - its absence. In the column "Current" zero indicates the absence of current in the ammeter, and the number in brackets conditionally indicates the point at which the current is formed (according to Fig. 3). If this current is uncompensated, then the ammeter will show it.

Faraday's paradox is completely resolved using Lorentz's law and the left hand rule. In contrast to the explanations of the transformer principle of operation, here it was not even necessary to use a second magnetic field (there are also such attempts), although it still exists to some extent. We can safely say that a unipolar generator is an electric machine with one winding turn. The coil is formed by a line: the center of the disk is its edge, if the magnet rotates relative to it, and is at rest relative to the external circuit, or the turn is formed by the line of the external circuit, ifand the magnet rotates relative to it, and rests relative to the disk. If the magnet rotates simultaneously both relative to the disk and relative to the external circuit, then the currents compensate each other, and their total value turns out to be zero.