SELF

76

S.B. Karavashkin and O.N. Karavashkina

Let us put the loop into a magnetic field of the most general type:

where _B is the unit vector of magnetic field direction, and is the vector of loop area which this magnetic field crosses.

To calculate the expression of induction emf after Faraday, we now have to take into account the inhomogeneity of considered field and connected with it inhomogeneity of emf induced along the movable side of loop. So the general expression will be in the integral form and vectorial:

According to the properties of scalar product (see, for example, [3, p. 160], and given (6) and the fact that we change the size of only one side of loop, we can transform the right-hand part of (13) as follows:

Calculating the induction emf after Maxwell, we have to write

With it we come to (14) with the only difference that in (14) the derivative with respect to time is taken only over the loop area, while in (15) this derivative is taken of the integral as the whole. In the view of mathematical analysis the compared expressions are outwardly non-identical. But if we note that the magnetic field itself does not depend on time, only the loop area changes, in this case we have in (15) to put the derivative versus time under the integral sign in some unusual way, differentiating not the integrand but the very differential of the loop area (!). Then we will yield

Though such transformation is unusual, we can easily show that we did no mathematical violation, there really is differentiated not the variable for which we integrate but the very integrand. Let us prove this as the theorem.

THEOREM 1. If in the integral

the integrand f (l)does not explicitly depend on time t and the integration boundary

the equality

is true.

Actually, if the integrand does not depend on time explicitly, and we integrate with respect to the variable l, the integral I also does not depend on time explicitly. With it the following chain of transformations

is true, as was to be shown.

For our case of flux of vector of magnetic induction   B (S) , when we integrate over the cross-section of loop, we can write (15) as

With it

is the integrand for the integral with respect to h. So in accordance with the theorems of mathematical analysis and given the boundary of integrating for h is time-independent, we have a right to introduce the derivative with respect to time under the sign of integral. With it we will yield

As we can see, according to the statement of problem, the integrand in (23) fully satisfies the Theorem 1, due to which, according to (19), we may substitute in (23) as we did. Then we yield

which coincides with (14) describing the process after Faraday.

This evidences that if we approach to the induction process in the narrow sense, taking as the basis the motion of the side of loop in magnetic field, Faraday and Maxwell interpretations really are mathematically indistinguishable. But in the view of phenomenology there is an essential difference.

If we consider the induction process in the view of Faraday interpretation, the essence of induction process is determined through the immediate interaction of magnetic field with a wire moving in this field. But if we describe this process from the view of Maxwell, it appears that the charges of wire interact with the whole field crossing the loop, in that number with that part where the charges are absent. With it we have to suppose some special "awareness" of charges of wire as to the field out of their location. Not in vain Schtrauff, when speaking of these formulations, specified so: "In practical application of these formulations we have to remember that we have to interpret differently, dependently on, whether we calculate U_ind arising in the wire section or in the closed loop" [2, p.416].