ò.4 No 1

1

On gradient of potential function

On gradient of potential function of dynamic field

S. B. Karavashkin and O.N. Karavashkina

Special Laboratory for Fundamental Elaboration SELF

187 apt., 38 bldg., Prospect Gagarina, Kharkov, 61140, Ukraine

phone +38 (0572) 7370624;

e-mail: selftrans@yandex.ru , selflab@mail.ru

Abstract

We will study the gradient of potential function of dynamic field and show that in dynamic fields the gradient of function divides into coordinate-dependent and time-dependent parts. We will show the standard expression connecting the electric field strength with vector and scalar potentials to be the consequence of this division of gradient in dynamic fields. Due to this, curl of gradient of potential function is not zero.

Keywords: theoretical physics; mathematical physics; wave physics; vector algebra; EM theory; dynamic potential fields; gradient of potential function of dynamic field; curl of dynamic gradient of potential function; dynamic field of pulsing potential source; dynamic field of oscillating potential source

Classification by MSC 2000: 76B47, 78A02, 78A25, 78A40

Classification by PASC 2001: 03.50.-z; 03.50.De; 41.20.Jb; 41.20.-q; 41.60.-m

1. Introduction

Before, in [1] and [2], we showed the theorems of divergence and curl of vector essentially transforming in dynamic fields; this led us to changes in conventional system of Maxwell's equations. In particular, in [2] we showed that Faraday's law is the corollary of conservation law for curl of dynamic field, and in dynamic fields this curl can be created by both vortical and potential vectors being perpendicular to the field propagation.

This feature of dynamic fields causes the necessity of more careful attitude to operations applied to the study of such fields. In particular, in appendix [3] to the paper [1] we showed that when finding the strength of electric field on the basis of vector potential , double application of curl operation distorts the result, due to the fact that the gradient of potential function is filtered in this operation. From this, the value of electric field strength obtained in such way does not satisfy the conservation law of flux of vector of dynamic field that has been proven in [1]. At the same time, the use of known expression

yielded around curl filters brings us to results that are associated with the EM field behaviour in the far field and fully satisfies the divergence theorem in dynamic fields [1].

If we draw more attention to the derivation of (1) given in [3], we will see it not free of contradictions, though (1) gives correct result.

Actually, we yielded (1) of [3] from the condition

But in accordance with [2], the curl of potential vector vanishes not always but only in case when the vector is directed along the field propagation and depends only on radius-vector. But in the problem studied in [3], gradient of scalar potential depended also on polar angle ; this made (2) incorrect. And we can easily show it.