SELF

18

S.B. Karavashkin and O.N. Karavashkina

We can easily show it, deriving the regularities for Lienard- Wiehert potentials which Landau presented in [5, p. 210- 213]. To derive this regularity for delaying potentials, he introduced the following substitution of variables:

and on the grounds of this substitution wrote so: "In the reference frame where at the moment of time t'   the particle is in rest (? - authors), the field at the observation point at the moment t  is given simply as the Coulomb potential, i.e.

[5, p. 211]. At the same time, by just his definition, "according to the formulas of delaying potentials, the field at the observation point P(x, y, z)  at the moment of time t  is determined by the state of charge motion at the  previous moment of time t', for which the time of light signal propagation from the point of charge location r₀(t')   to the observation point P  just coincides with the difference t - t' " [5, p. 210]. Proceeding from this and from the fact that we observe the field not at the moment t', when the signal was radiated by the source, but at the moment t common for the studied region of field (or we would may neither differentiate with respect to spatial coordinates nor determine the strength of electric and magnetic fields), the denominator of the first expression in (15) becomes time-dependent for different points of field. It follows straight from (14). Substituting to (14) different values of R , we for one and the same instant of time t yield different values of t', which in their turn will correct the initial value of R   which we choose, because different locations of the source in time will correspond to these values t'. So the potential of neighbouring points of field will depend not only on the spatial distribution, as it takes place for stationary fields, but also on the location of source in time, which is tantamount to the time dependence. And this will be not Coulomb field already. This will be fully dynamic field which has to be described by the corresponding function of dynamic field, not by Coulomb law which is true only in frames of long-range formalism.

Thus, we see that with all outward simplicity and elegance of expressions (12) and (13), they are too approximative, and this approximation will have the same problems which we showed in Fig. 4. In the near field it will be possible, but we already will not have a necessity in it.

As opposite to the described, when specifying the model of radiator, taking into account the features of particular radiator, we automatically account the feature of near field of dynamic process (just as we did it in calculation for Fig. 4). When plotting "in frontal attack", we take into account the delay phase at different moments of time and dependently on different position of sources, as it stands to reason. So, lest to encounter in future the necessity to introduce the correspondence, in this investigation we will confine ourselves to the particular model of dipole whose charges vary in time harmonically in anti-phase and remain mutually immovable. As we showed in [1], this model is realisable when the standing wave forms in the radiator and in case of pulsing potential sources of field. In frames of this model, it will be enough to determine the dynamic distribution of scalar potential for each of pulsing sources and then to study their additive sum, taking into account the delay phases.

To determine the dynamic distribution of scalar potential excited in space by the pulsing source, we can use the results of study conducted in [6, p. 40- 42]. There we showed that in the view of wave physics, for material continuum, the wave process excited by a point source satisfies the differential equation

where is the density of continuum, is its viscosity, is the strength module in the medium, r is the distance from the field source, and y is the momentary shift of the element of medium. In case of non-viscous medium, (16) simplifies and takes the form

or

where is the wave propagation velocity is space.

We can easily transform this equation for EM field, applying to it the dynamic electromagnetic analogy DEMA whose fundamentals have been stated in [7]. In accordance with this analogy, the difference of potentials between the studied point of EM analogue and the point of zero potential corresponds to the momentary shift of elementary volumes of mechanical continuum. Since as the zero potential we can surely choose the space free of field sources, in our case y will straight correspond to the sought scalar potential . With it (18) will take the form