V.37(2001) No 3, pp. 233 - 243

241 - 243

Transformation of divergence theorem

p. 241

In the most general case of dynamical vector flux, (20) coincides with the result obtained by Levitch in particular case for the vector potential. At the same time, we identified (, t)  with no specified physical quantity, and studied the vector in the most general form. So we can state that we have proved the

Theorem: When the wave flux propagating in the source-free space, the divergence of flux of vector is proportional to the scalar product of derivative of this vector with respect to time into the unit vector of direction of the flux.

4.  Application to the EM field

Basing on the above results, we can obtain the refined values for the divergence of vectors of electrical and magnetic fields in Maxwell system. Substituting the electrical strength (, t)   into (20) instead (, t), we obtain

Similarly, for the magnetic strength (, t)

For the vector potential (, t)

Knowing that the scalar potential (, t) relates with the vector potential (see e.g. [5, p.106]) as

establishing the integrity of relations in the EM field theory. The fact is important that (25) has been obtained for dynamical fields. It follows from it that it is incorrect to equal the scalar potential to zero in general case. More completely it was proved in [10, pp.25-26].

At the same time, for transverse and , (21)-(23) coincide with the known results (3), since the scalar product of perpendicular vectors in the right part vanishes. The more, zero result of divergence of curl will also keep valid. In fact, in accord with item 3,

we obtain for transverse field, where :

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since we know that the curl of transverse vector (, t)  is perpendicular both to the vector and to the field propagation direction . The similar result will be for the longitudinal vector, since the curl of longitudinal vector is zero.

Thus, for any inclined vector of a field, the divergence of curl of vector remains zero, since the curl of vector is perpendicular to the wave propagation. But it does not concern the results obtained for the divergence of longitudinal vector of the field. In (21) and (22) the derivatives of and   with respect to time are the consequence of wave space-delay and cannot vanish if these vectors have a dynamical pattern. The changes in (21) and (22) give them the wave pattern, since, despite these equations are the first-order, expressions like

are their solutions.

As follows from it, the longitudinal component of EM field has the wave properties too, irrespectively to the conventional concept that in the longitudinal field “the motion of energy is absent, there takes place only a periodical exchange of the energy between the electrical and magnetic components of a field” [11, p.99]. The only we should mark is, the longitudinal component of a field has its properties that basically distinguish it from the transverse wave. Conventional symbols of the vortex vector (, t) do not allow us to describe the magnetic field generating around the longitudinal dynamical E-field, because with it, at any point of the space, not vector but some circulation around the electrical vector will correspond to the magnetic field [10, p.42]. The divergence of this circulation will not vanish, as in (28), since its direction coincides with the flux direction. So it also will have the wave properties.

The fact that the longitudinal EM waves (LEMWs) in free space do exist was corroborated experimentally at the laboratory SELF in 1990. The portable device radiating/receiving the directed LEMW at 30 kHz range has been constructed and multiply demonstrated (see the review by Professor Denisov). But this is the subject of a wide consideration being out of frames of this paper.

1. 5.   Conclusions

As a result of investigation that we carried out, basing on the conventional definition of divergence and using the conventional method to find the flux of vector, we have revealed that:

- in dynamical fields, in general case, the flux and divergence of vector are non-zero;

- for the vectors of flux directed along the wave propagation, the divergence is proportional to the scalar product of the particular derivative of this vector with respect to time into the wave propagation direction. Particularly, for this component of the field, the pair of Maxwell equations describing the flux of vector acquires the wave pattern;

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- for the transverse component of wave, the divergence of vector remains zero, and consequently, Maxwell equations remain valid.

1. Ignazius, G.I. Field theory (Mathematical analysis of multivariable functions). Znanie, Moscow, 1971 (Russian)

2. Rossel, J. General physics. Mir, Moscow, 1964 (Russian; from edition: ‘Physique generale’. Par Jean Rossel. Editions du Griffon – Neuchatel diffusion dunod – Paris; edited in Suisse, 1960)

3. Korn, G.A. and Korn, T.M. Mathematical handbook for scientists and engineers. Definitions, theorems, formulas. Nauka, Moscow, 1968 (Russian; from edition by McGraw-hill book company, Inc., New York – Toronto – London, 1961)

4. Landau, L.D. and Lifshiz, E.M. Theoretical physics, vol.II. The field theory. Nauka, Moscow, 1973 (Russian)

5. Sedov, L.I. Mechanics of continuum, vol.1. Nauka, Moscow, 1973 (Russian)

6. Alpert, Ya.L. Ginzburg, V.L. and Feinberg, E.L. Radio waves propagation. The State publishing. of technical and theoretical literature, Moscow, 1953 (Russian)

7. Levitch, V.G. The course of theoretical physics, vol.1. The State publishing of physical and mathematical literature, Moscow, 1962

8. Eichenwald, A.A. Selected works. The State publishing of technical and theoretical literature, Moscow, 1956 (Russian)

9. Purcell, E.M. Electricity and magnetism. Berkeley physics course, vol.2. Nauka, Moscow, 1971, 447 pp. (Russian; from edition by McGraw-hill book company)

10. Karavashkin, S.B. On longitudinal EM waves. Chapter 1. Lifting the bans. In: SELF Transactions, 1 (1994), 15-47, Eney Ltd., Ukraine  (English),

11. Kugushev, A.M. and Golubeva, N.S. Foundations of radioelectronics. Linear electromagnetic processes. Energia, Moscow, 1969, 880 pp. (Russian)