SELF

78

S.B. Karavashkin and O.N. Karavashkina

One more important feature of induction in time-variable magnetic field follows from this. As is known, in accordance with Biot- Savart- Laplace law, "the strength at each point of magnetic field created by a closed loop of current is such as if it were vectorially combined of strengths created by its separate sections, if we admit (Fig. 5) that the magnetic field strength dH created by each element of loop is in proportion to the current I, length of element dl, sine of angle between the direction dl and radius-vector r drawn from the beginning of considered element to the point at which we determine the magnetic field, and in inverse proportion to the squared distance r" [2, p. 275].

Fig. 5. Explanation of Biot- Savart- Laplace law

In other words, if in Fig. 3 we choose the first loop as the primary, in case of permanent magnetic field

Given the nature of orientation and induction magnetic field is the same, with the time-variable current going in the primary loop we will yield

With it the magnetic field will be already determined by the delaying potentials, as the elements of current of primary loop are in general case unequally distanced from the point at which we are observing the field.

Given the vector of magnetic induction is formed just by the magnetic field strength described by (29), we can write the induction emf E_ind excited in the element of wire dl₂ of the second loop as

where is the angle between the inducing current and radius-vector from the element of this current to the observation point. We would like to draw your attention that in (30) we intentionally passed from the vector form of record to that scalar. Because, as we said above, the process of excitation of current in stationary secondary wire by the time-variable magnetic field still is not determined finally. So, should we now try to write (30) in standard vectorial form, we would have to note that the magnetic field perpendicular to the secondary wire excites the strength of induction field along the secondary wire. In the Maxwellian formulation this difficulty was avoided through using the idea of vector of secondary loop cross-section and finding the scalar product of vectors of magnetic induction and the loop square. This all the same did not solve the problem, how to describe the process of excitation of current in stationary secondary loop by magnetic field, as all the derivation was reduced to the scalar equality

in the left-hand part of which we had the scalar product of vectors parallel to the plane of secondary loop, and in the right-hand part - the scalar product of vectors nonparallel to this plane. It remains unclear from (31), in which way the time-variable field affected the stationary charges. While the analogue of (31) for Faraday formulation is absent and the formula for Lorenz force in its original representation

is unable to describe the induction with stationary charges. So we still will confine ourselves to the record for induction emf in non-vectorial form, hoping to improve this expression in the following works, as we will clear the very process of induction in time-variable magnetic field.

Basing on (30), we can easily write the expression for the emf of induction excited along the secondary loop as the whole:

The structure of (33) shows that, if we do not abstract from the source of magnetic field, as it is done in the standard mathematical record of electromagnetic induction law, it will appear that the induction is determined by the immediate interaction of the elements of current of the primary and secondary loops. This is in full accordance with the conclusions of [1] obtained on the basis of experimental study of the induction with the help of single probe.

To show it visually, consider the model of two quadratic large-size loops shown in Fig. 6.

Fig. 6. The circuit explaining the induction process with the large-size loops; the green loop is primary, and the blue loop is secondary

If we choose the size of loops so that

the emf induced in each selected element of the secondary loop will be determined by that part of integral (33) for which r is comparable with h. In this connection, we can neglect the affection of the side CD on the value of emf excited in A'B' of secondary loop, as the condition (34) provides the small affection of this side, because it is far from the side of secondary loop. Thus, with enough accuracy we can write (33) so:

The expression (35) shows that the induction process is caused by the direct interaction of elements of current of primary and secondary loop, which corroborates the said.

The features of induction process revealed in this study, with common physical nature of magnetic field, make the study of nature of this process quite difficult. Not occasionally, speaking of Biot- Savart- Laplace law, Schtrauff underlines: "in reality we deal always with the magnetic field of closed loop, so direct experimental check of Biot- Savart- Laplace law is a hard task" [2, p. 274]. None the less, even at this level of knowledge we can distinguish from masking effects the essence of inductive interaction of time-variable magnetic field with a secondary loop and to clear, which of interpretations - Maxwell's either Faraday's - describes the induction process correctly. And we did it practically.