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

237 - 238

Transformation of divergence theorem

Plot the diagram of (x, t)  space-time variation, remembering the progressive pattern of wave (7). Basing on the diagrams of F(x, t)  shown in Fig. 1, centre, we can easy determine _0i , since the integration amounts to a simple multiplying of  (x, t)  parameters at a picked out moment of time t  at a studied surface into the size of this surface. The obtained results calculated for the moments fixed in Fig. 1, left are shown in Fig. 1, right. As we can see from calculation, despite we used the conventional definition and method, the obtained value _0i  in general case does not vanish at the boundaries of the picked out regions. It is different both for all the surfaces  a₁, a₂, a₃   and all the moments of time, though _0i  was calculated relatively to the surface a₀ common for all the regions and simultaneously for all the picked out regions. The regularity _0i(t) shown in Fig. 2 reflects this peculiarity; it is plotted on the basis of calculation of Fig.1. As we see from Fig. 2, both amplitude and phase of _0i(t)  are different for the picked out regions, the same as these parameters are different for G_i(t)   whose regularity is shown in Fig. 3. The more, when diminishing the size of picked out region, the amplitude of G_i  increases. It confirms that the flux through a picked out region is time-inconstant in dynamical fields, and the fact of _0i   time-variation is conditioned not by the space parameters of flux, but namely by the progressive pattern of wave, which we can easy prove mathematically.

Fig. 2. The time dependencies of the flux through the picked out space on the size of given region

Actually, for any region picked out at a moment t  we have

(where x_i = x_i - x₀  is the distance between the surfaces  a_i  and  a₀). Considering that in the studied case V_i = x_iS, we obtain (8) for G_i as

Because the regularity G_i(x_i) is conditioned by the finite velocity of the wave space-propagation, we can express x_i  through the time characteristic of the wave delay t_i:

p. 238

Substituting (11) into (10), we obtain

(where c is the wave propagation velocity). As we see from (12), both amplitude and phase of G_i depend on t_i; it completely corresponds to the plot shown in Fig. 3. With it the specific flux amplitude depends on the ratio of sine of argument t_i / 2 to this argument, determining the first significant limit. It backgrounds that the inequality of flux to zero in dynamic fields is the objective fact.

Fig. 3. The time dependencies of the specific flux through the picked out space on the size of given region

To determine the divergence on the basis of (12) in accord to (2), it is sufficient to find the limit of G_i at t_i 0. Taking it for any of picked out regions at the moment t, we obtain:

Thus, similarly to the flux, the divergence of (x, t) does not vanish too, and Levitch’s expression (5) in the case of 1D flux and harmonic time-dependence of longitudinal vector (, t)  will completely correspond to (13). On one hand it corroborates the result obtained by Levitch for the vector potential of dipole radiator, and on the other hand, it shows more general pattern of the result.

So we can state that at least for any 1D flux in dynamical fields the divergence of its vector does not vanish, being inconsistent with the conventional concepts basing on Poisson theorem.

We should note again, as we mentioned in the introduction, the difference between the obtained results and conventional concept is conditioned by the fact that in all the theorems proved before on the basis of divergence definition, in fact only stationary fields and stable fluxes were considered. While in dynamical fields, not only the density of space-distribution of the force lines, but also the wave propagation velocity, being finite and causing the phase delay, have effect on the value of divergence.