A new perspective on driving current density

This article discusses alternative and complementary proposals to j.C.Maxwell’s laws of classical electromagnetism, based on certain hypotheses, hypothetical examples and calculations, with results that may infer new interpretations about the physical phenomenon conduction current density. These new interpretations bring a new understanding to the dynamics of Gauss’s Law, and, being true, make the Amperre-Maxwell Law totally symmetrical to the faraday-lenz-maxwell law, without any mathematical or physical inconsistency. These understandings inevitably bring implications and points of view complementary to the classical theory of electromagnetism.


INTRODUCTION
First, the solution given by J.C.Maxwell is presented so that the Ampere equation (see formula 1) becomes mathematically consistent, respecting the vector identity applied in (see formula 2), and consistent with the continuity equation (see formula 3).
Below is an example of the application of Gauss's Law (see formula 6) to a closed surface around one of the plates of a capacitor, Figure 1.
By hypothesis, it is proposed that the temporal variation of the total electrical flow, which crosses the gaussian surface totally closed, is always equal to zero. Therefore, Gauss's Law (see formula 6), applied to dynamic situations, would become the equation (see formula 9).
For this hypothesis to be based, it will be considered necessary that there is a temporal variation of electric field density, of the same modulus, direction and direction of current density, in the area of intercession between the cylindrical volume of the conductor and the Gaussian surface, pointing into it, Figure 1.
In order to verify the consistency of this hypothesis, an ideal example will be considered in which there is a continuous and homogeneous current in an infinite rectilinear cylindrical wire along the z axis. Then the vector temporal variation of the electric field is calculated (see formula 12) at point P(0,0,0) due to the simultaneous and instantaneous displacement of all loads, upstream and downstream of P(0,0.0). This calculation results in equality (see formula 16).
Finally, starting from the physical veracity of equality (see formula 16), there are inevitable implications, formatting and theoretical complements for classical equations of electromagnetism, by Faraday, Lenz, Biot-Savart and Maxwell. At the end, a laboratory experiment was suggested to confirm or disprove the theory developed from the analysis of the proposed hypothesis. that is being loaded by a variable voltage power supply, as shown in Figure 1. Observing Figure 1, with a conduction current density in the driver, the equations (see formula 6) and (see formula 7), respectively: for a fully closed Gaussian surface; and for a point on the capacitor plate, they become equations (see formula 8) and (see formula 3). (HAYT;BUCK et. al., 2013;SHADIKU, 2004  Thus, it would be intuited that the interactions between fields, electrical or magnetic, and charged particles, would be, first of all, interactions between fields only. In the search to verify the veracity of the proposed hypothesis, an example of an ideal hypothetical situation is created for the calculation of the vector temporal variation of electric field density (see formula 12), at the origin, generated by a continuous and homogeneous current in a cylindrical conductor of infinite length along the z axis.

EXAMPLE OF AN IDEAL HYPOTHETICAL SITUATION
Suppose the following ideal configuration: in a cylindrical, rectilinear, uniform, homogeneous conductor, of infinite length, which runs through a direct current, of positive, uniform and homogeneous loads in the positive direction of the z-axis, Figure 2.
In cylindrical coordinates, the temporal variation of the electric field vector (see formula 12) generated at point P(0,0,0) of the Cartesian plane in Figure  Although, in a real conductor with a difference in potential applied in its extremities, the electron is the one, by classical pattern, the displacement of positive charges in the positive direction of the z-axis was chosen for the calculation.
www.nucleodoconhecimento.com.br It is known, by symmetry, that the resulting vector electric field E generated by the sum of all loads existing along the conductor, positive and negative, at point P(0.0.0), is null.
However, let us consider, first, the calculation of the static electric field (see formula 14), generated by a cylindrical volumetric differential element, in cylindrical coordinates, of volumetric density of positive load ρ v , centered in an i nitial position z', where dQ is the differential element of load, r' is the constant value assigned to the radius of the cylinder conductor, R the vector distance between the point P (0.0,0) and the volumetric differential element dV , a z is the versor in the positive direction of z, and ε the electrical permissiveness of the conductor, as shown in Figure 2.
Please point out that, because we take as reference the measuring point of the electric field fixed in P(0,0,0), the direction of dE will always be opposite to the a R versor. Therefore, the negative sign in the equation (see formula 13).  As what one wants to calculate is the temporal variation of the electric field (see formula 12), at the origin, caused by the displacement of each differential cylindrical element added along instantly, it will be considered exclusively for this calculation that the function of the equation Suggested experiment to prove the theory. 6. www.nucleodoconhecimento.com.br

INTERPRETATION OF THE APPLICATION OF A DYNAMICS OF GAUSS LAW
Equality (see formula 16) is considered to be consistent with equations (see formula 9) and (see formula 10). Thus, it is understood to be reasonable the following physical interpretation of gauss's law applied to the situation presented in Figure 1: the temporal variation of the total flow of electric field density on any fully closed Gaussian surface is exactly zero, (see The following interpretation about the physical phenomena contemplated in equality is proposed here (see formula 19).
The vector temporal variation of electric field density , a result of conduction current density, appears to be proportional to the longitudinal velocity at which the electric field lines cross an area element of a Gaussian surface. Theorizing can be theorized: where v L is the velocity vector of the electric field lines that cross an area element of a Gaussian surface, a S theversor of the vector area element of that surface, and K 1 a constant,  The vector temporal variation of electric field density , resulting from the displacement current density , is related to the temporal variation of the quantitative number of electric field lines that cross an area element of a Gaussian surface (HAYT;BUCK et. al., 2013;SHADIKU, 2004).
Being true the theory proposed, from the initial hypothesis, that the temporal variation of the total electrical flow that crosses a Gaussian surface, can be formed both by the temporal variation of the number of electric field lines, per unit of area, which cross it , and by the longitudinal velocity of the electric field lines that cross an area element, (see formula 20); it is intuited, by symmetry to (see formula 19), that the magnetic flux passing through a Gaussian surface behaves in an equivalent way (see formula 21).
In such a way that it could be generated both by temporal variation of the number of magnetic field lines in an area element (HAYT;BUCK et. al., 2013;SHADIKU, 2004 The present article aims, succinctly, to propose a theory, based on hypothetical situations, without physical experimentation in the laboratory for proof, at the end of it.
However, a proposal for a laboratory experiment will be presented below, for those of interest, to prove the veracity of the proposed theory, or to disprove it.
For the calculation of a magnetic field H, generated exclusively from a conduction current density , using equality (see formula 16), the Biot-Savart Law (see formula 27) could be described as in (see formula 28).
Then, it is proposed to write the modified Biot-Savart Law (see formula 28), symmetrically for the calculation of the electric field, as in (see formula 29).
In order to verify the veracity of the theory that the fields, electrical and magnetic, can be generated, respectively, by the speed of displacement of the lines of the fields (magnetic and electrical), the following experiment is proposed.
It is an electrical circuit formed by an isolated conductive wire, coiled in a distributed and continuous way around a ferromagnetic material of toroidal topology, area A constant of the cross section, fed by a direct current source, with a current adjusted such that it does not A new perspective on driving current density www.nucleodoconhecimento.com.br magnetically saturate the ferromagnetic material. There will be a magnetic field density B confined to all toroidal ferromagnetic material (HALLIDAY; RESNICK;WALKER et. al., 2013), Where μ is the magnetic permeability of the ferromagnetic material, N the number of turns, and l the perimeter traversed by the cross section of the toroide throughout its revolution.
Then, an isolated conductive wire measuring turn is used around the cross section of the toroide, connected to a VDC voltmeter, in such a way that it is possible to shift the measuring coil along the toroidal perimeter, Figure 4.
It is known that the magnetic field generated by any ideal toroidal electrical circuit, external to it, is zero (HALLIDAY; RESNICK;WALKER et. al., 2013).
By the theory presented, when the measuring coil moves with a velocity V along the toroidal perimeter, even if the number of magnetic field density lines internal to the turn is not changed, these lines will pass through the Gaussian surface formed by the circumference of the measuring coil, with -V velocity.  If the experiment is carried out with a result that corroborates the proposed theory, it is also proposed to consider the existence of the following conduction magnetic current density.

CONCLUSION
In order to better understand the nature of the displacement and conduction currents, the hypothesis was raised that, in a fully closed Gaussian surface, the temporal variation of total electric field flow, in it, could always be equal to zero (see formula 9) and (see formula 10), not contradicting the continuity equation (see formula 3).
An example of an ideal hypothetical situation was created, which would allow mathematical analysis of the hypothesis created.
The result of this analysis, equality (see formula 16), corroborates the idea that the vector temporal variation of total density of electric field (see formula 19) can exist from the following two distinct physical phenomena. Thus, it is proposed a theory that all interactions between fields, electrical or magnetic, and electrically charged particles, are, first of all, interactions between fields only. For example, when applying a potential difference in an electrical circuit, the electric field generated by the potential difference will interact with the electric field of the free loads, forcing them to move.
The displacement of the loads implies the displacement of their electric field lines. The longitudinal velocity at which these lines traverse an area differential element would be Finally, a laboratory experiment is proposed to confirm or discredit the proposed theory and its equations.
The following are the equations proposed in this article.