V.4 No 1

77

Study of electromotive force induced by heterogeneous magnetic field

To clear up, which of two formulations does not reflect the reality - or rather, which is not enough general - we can consider another scheme, in which the emf inductance is present in the wire moving in the magnetic field. We can use for this purpose a no less known model of unipolar generator, see Fig. 3.

Fig. 3. Standard unipolar generator

In Fig. 3 we see that the standard unipolar generator, the same as the loop with a movable side, has the secondary loop, but the area of this loop does not change with the disk rotation. This area is perpendicular both to the rotation axis of disk and to the direction of magnetic field. Given the magnetic field inductance is here constant, in accordance with Maxwell formulation we cannot state that the flux of vector through the cross-section of secondary loop changes. The more that the vector of plane of the secondary loop is perpendicular to the direction of magnetic field. Due to this, according to Maxwell formulation, even under condition of homogeneous magnetic field, we yield

It is understandable that with an inhomogeneous but time-constant magnetic field the result will be also zero.

As opposite to this, if we use the Lorenz formula to find the induction emf, we yield

where R is the radius of disk. With it the left-hand part of (26) will remain nonzero both in homogeneous and inhomogeneous magnetic field. And this result is corroborated experimentally.

It is interesting that (26) also cannot be transformed to the following integral over the area:

where determines the vector of angular shift of disk, and  d₁   = r [d d] determines an elementary area on the disk interacting with the magnetic field. Thus, we see that if we determine the inductance in Maxwell formulation not through the area which the taps cover but through the cross-section which interacts with magnetic field immediately, the Maxwell formulation describes the processes in unipolar generator the same well as Faraday formalism. So we can lift the duality in description of induction existing now. True, this fully changes the today meaning of Maxwell formulation. In reality, the phenomenology of induction process in a moving wire is based on the charges of wire crossing the force lines of magnetic field. The vector of flux of magnetic induction in general case is irrelevant to the process of this kind of induction. The emf is induced exceptionally in that part of secondary loop which is located in the region of nonzero magnetic field and moves in this field. And the main, emf is excited in the immediate interaction of outer field in the region of wire, with charges of this wire moving in this field with the wire. It also follows from this, inequal emf is induced along the wire. The distribution of this emf depends on the magnetic field distribution and charges velocity in this field. In particular, in case of unipolar generator at the axis of disk there will be induced a minimal emf, as in this region the charges velocity is minimal. While we measure the emf consisting of induced emfs in elementary parts of secondary loop, and we have to take this feature into account in study of electromagnetic induction.

Similarly, "if any rigid closed conducting loop moved onward in a homogeneous magnetic field, no emf (total emf is meant - authors) arises in this loop, as the magnetic flux bonded with it will remain constant (due to the filed homogeneity). With it in separate parts of loop there can arise the emfs, but their algebraic sum will be zero, as the values of oppositely acting emfs will be equal" [2, p. 424]. And this revelation of local emfs at zero total emf over the loop is important to understand the induction in moving wire in magnetic field, because, should the emf be induced in the loop as the whole, local emfs in the loop sides would not arise. They can arise only in case if Faraday formulation was true. And we can easily detect them, terminating the taps to the opposite sides of closed loop. This also shows that Maxwell's integral formulation describes the induction only mathematically, even noting the change of idea of secondary loop which we made above, while Faraday formulation exactly describes the induction both phenomenologically and mathematically.

Having cleared up the meaning of induction in a moving wire and seeing that the phenomenology of induction is determined by Faraday formulation, we can return and to study the analogy between the induction in permanent and time-variable magnetic fields. We will see, now this analogy is established on the basis of Maxwell, not Faraday formulation - just the formulation that does not reflect the essence of process. Actually, "When two loops with the current (1 and 2) are in the magnetic fields of each other (Fig. 4), with every change of intensity of current in one of loops, the flux bonded with another loop changes, and in this last the emf is induced. Such phenomenon is called the inter-induction" [2, p. 450].

Fig. 4. Explanation of inter-induction of loops. Copied from [2, p. 450]

Noting the above study, we can considerably improve the interpretation of inter-induction. Now on the basis of Faraday formulation we have to expect that the magnetic field created by each loop directly interacts with the charges of wire of second loop, exciting the emf along the loop as the sum of emfs of local interactions of magnetic field with the parts of second loop. The features introduced into the induction due to time-dependence of magnetic field are added to this.

The first thing which we have to note is that the analogy between the movable wire in permanent field and immovable wire in time-variable field is broken by the known fact that magnetic field does not supply work, as it affects the charged body perpendicularly to its motion. In case when the wire moved, "the positive work of forces affecting the wire (ponderomotive forces) is equal to the negative work of emfs induced in the wire, so the full work of magnetic forces is zero" [2, p. 423]. In other words, in the process of induction the magnetic field is only the connection between the affecting force doing the work and the wire in which this work is done. And in case of stationary wire in time-variable field, the charges in wire can be thought stationary in average, as the direction of their motion is absent and the average velocity of their chaotic motion is zero, only the amplitude of magnetic field varies. But in accordance with standard theory, this field itself does not do the work. So, if we grounded on the standard formulation, the essence will not change - no matter, is the field permanent or variable. However, the experience shows that the variable magnetic field excites the emf in secondary loop, which means that the very variable field does supply work, exciting the directed motion of charges in the secondary loop. There is no external affection in this case, except the magnetic field. But if it is so and the very variable field is able to accelerate the charges in a definite direction, this changes the very idea of magnetic field and its basic properties.

One more difficulty follows from this. Basically, so different properties of permanent and variable magnetic fields could be interpreted as an existence of two independent fields, each of which has its own properties, - and so we suggested in [1]. But in reality we cannot speak of two different (orientation and induction) fields, as these virtual fields have both individual features and identical properties. We can easily make sure of it, if we put the above unipolar generator (Fig. 3) to the variable magnetic field. In this case, with the stationary disk in the secondary circuit, the emf will be induced in full accordance with Faraday law of induction. And we will detect the maximal induction emf, when the inducing field is directed in parallel to the plane of disk, and that will be minimal, when it is perpendicular to the plane of disk. If we choose some intermediate direction of magnetic field and rotate the disk, we will see the induction emf growing. The maximal addition will take place, when the magnetic field is directed normally to the plane of disk, i.e. at the position when the induction addition is minimal! While in the intermediate positions of inductor we will observe the simultaneous affection of two fields, which will increase the induction emf.

This evidences that the magnetic field is integral in its structure but different in its revelations, dependently on its space distribution and time dependence of its amplitude.