# If DRDO falls under power supplies

## A new perspective on current density

ORIGINAL ITEM

LUCENA, Lucas Ruas de 

LUCENA, Lucas Ruas de. A new perspective on current density. Revista Científica Multidisciplinar Núcleo do Conhecimento. Volume 05, Ed. 08, Vol. 03, pp. 151-167. August 2020. ISSN: 2448-0959, access link: https://www.nucleodoconhecimento.com.br/fisica-de/kettendichte, DOI: 10.32749 / nucleodoconhecimento.com.br / fisica-de / kettendichte

Contents

### SUMMARY

This article deals with alternative and supplementary proposals to J.C. Maxwell's laws of classical electromagnetism, based on certain hypotheses, hypothetical examples and calculations, with results that can derive new interpretations about the physical phenomenon conduction current density. These new interpretations bring a new understanding of the dynamics of Gauss's law and make Ampère-Maxwell's law, verily, completely symmetrical to Faraday-Lenz-Maxwell's law, with no mathematical or physical inconsistency. These understandings inevitably bring implications and points of view that complement the classical theory of electromagnetism.

Keywords: electromagnetism, current density, continuity equation.

### 1. INTRODUCTION

First, the solution given by J.C. Maxwell is represented in such a way that the Ampere equation (see formula 1) becomes mathematically consistent, whereby the vector identity applied in (see formula 2) is respected and agrees with the continuity equation (see formula 3).

Below is an example of applying Gauss (see Formula 6) to a closed area around one of the plates of a capacitor, Figure 1.

It is proposed as a hypothesis that the temporal variation of the total electrical flux, which crosses the Gaussian surface in a completely closed manner, is always equal to zero. Hence, Gauss law (see formula 6) applied to dynamic situations would become the equation (see formula 9). In order for this hypothesis to build, it is deemed necessary that there be a temporal variation in the electric field density, the same modulus, the same direction and direction of the current density, in the area of ​​intercession between the cylindrical volume of the conductor and the Gaussian surface that is on it shows Figure 1.

To check the consistency of this hypothesis, consider an ideal example where there is a continuous and homogeneous flow in an infinite rectilinear cylindrical wire along the z-axis. Then the vector time variation of the electric field (see formula 12) at point P (0,0,0) is calculated due to the simultaneous and immediate shift of all loads, before and after P (0,0,0). This calculation leads to equality (see formula 16).

Based on the physical truthfulness of equality (see Formula 16), there are inevitable implications, formatting, and theoretical additions to classical equations of electromagnetism by Faraday, Lenz, Biot-Savart, and Maxwell. In the end, a laboratory experiment was proposed to confirm or disprove the theory developed from the analysis of the proposed hypothesis.

### 2. THE DISPLACEMENT AND DRIVING CHAINS

Ampere's switching law in its punctual form (see Formula 1) had mathematical inadequacies (JACKSON et. al., 1998). The result of the equation (see Formula 2) should be zero because the divergence of the rotation value equal to zero represents a vector identity. The continuity equation (HAYT; BUCK et. al., 2013; SHADIKU, 2004), is incompatible with the equation (see formula 2). This correction was brilliantly made by J.C. Maxwell as follows (JACKSON et. al., 1998): So the ampere equation (see formula 1) became completely consistent (see formula 5). Both physically, taking into account the generation of the magnetic field from the temporal variation of the density of the electric field, and mathematically with regard to the vector identity cited in the equation (see formula 2).

The in Term added to equation (see Formula 4) has the same unit of current density, amps per square meter ter, denoted by J.C. Maxwell of the displacement current density, represented by the calibration (see formula 5). Identifying this term was fundamental to understanding electromagnetic wave propagation. The term in the equation (see formula 5) refers to the density of the conduction current.

### 3. HYPOTHESIS OF A DYNAMIC APPLICATION BY GAUSS LAW

Gauss ’Law, equation (see Formula 6), determines that the total electric field density flux that crosses a closed surface is equal to the total electric charge that is contained in that surface. Maxwell observed it in the dot form, equation (see Formula 7) known as maxwell's first equation (HAYT; BUCK et. al., 2013; SHADIKU, 2004). Let's consider the first example, which is a closed area around one of the plates of a capacitor charged by a variable voltage power supply, as shown in Figure 1.

Figure 1. Gaussian surface enclosed on a capacitor plate

Observation of Figure 1, with a line current density in the driver , the equations (see formula 6) or (see formula 7): for a completely closed Gaussian surface; and for one point on the capacitor plate they become equations (see formula 8) and (see formula 3). (HAYT; BUCK et. al., 2013; SHADIKU, 2004). The equation (see formula 8) determines that the temporal variation of the total density flux of the electric field which crosses the Gaussian surface is equal to the temporal variation of the electric charge contained therein.

The continuity equation (see formula 3), which is applied to every infinitely large volume of the positively charged capacitor plate, determines that the conduction current density that the given volume leaves is equal to the rate of time that the volumetric load density falls on it.

The hypothesis proposed in this article assumes that the equations (see Formula 8) and (see Formula 3) are physically complementary. Therefore, it is assumed that with a completely closed Gaussian surface around a capacitor plate under dynamic conditions, the equations (see formula 8) and (see formula 3) would have the following format. Where is t is the temporal variation of the vector electric field density, which is caused by the density of the conduction current, of the same module, of the same direction and direction is generated on the cross-sectional area of ​​the intersection between the cylindrical volume of the conductor and the Gaussian pointing thereon. And it is the temporal variation of the vector density of the electric field, in the closed Gaussian surface that points to it, produced by the displacement current density (temporal variation of the number of electric field lines crossing the closed Gaussian due to the temporal variation of the total electrical load of the plate from the internal capacitor to the Gaussian); (HAYT; BUCK et. Al., 2013; SHADIKU, 2004). Illustration 1.

The equations (see formula 9) and (see formula 10) determine that the temporal variation of the total flux of the electric field density on a completely closed Gaussian surface is equal to zero. The equation (see formula 10) does not contradict the equation of continuity (see formula 3), since it relates the flow of the electrical load, an infinitely simal volume, to the temporal variation of the volumetric density of the electrical charge in it; already that is only related to the dynamics of the electric field.

It is then assumed that the physical phenomenon of conduction current density, creates a variation in the electric field density at the Crossing of the cylindrical conductor with the Gaussian surface, the same module, direction and direction of the vector. One of the advantages of testing this hypothesis would be to understand the Ampere-Maxwell equation (see Formula 4) in the following format (see Formula 11). So it would be intuited that the interactions between fields, electric or magnetic, and charged particles would primarily be interactions only between fields.

In the search for the correctness of the proposed hypothesis, an example of an ideal hypothetical situation is created for the calculation of the vector time variation of the 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.

### 4. EXAMPLE OF AN IDEAL HYPOTHETICAL SITUATION

Let us assume the following ideal configuration: in a cylindrical, rectilinear, uniform, homogeneous conductor, of infinite length, passing 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) at point P (0,0,0) of the Cartesian plane in Figure 2 is generated due to the volumetric displacement, simultaneous and instantaneous, of all positive loads, before and after this Point, in the positive direction of the z-axis. Although the Jennisch is in a real conductor with a different potential in its extremities, the displacement of positive charges in the positive direction of the Z-axis was chosen for the calculation according to the classical pattern.

It is known by symmetry that the resulting vector electric field E., which is created by the sum of all loads present along the conductor, positive and negative, at point P (0.0.0), is zero.

However, let us first consider the calculation of the static electric field (see formula 14) that is generated by a cylindrical volumetric differential element in cylindrical coordinates with a positive charge volume density ρv is generated in an initial position z ’ is centered, where dQ which is differential load element, r ’ is the constant value assigned to the radius of the cylinder conductor, R. the vector distance between point P (0,0,0) and the volumetric difference element dV, azis the supply in the positive direction of z, and ε is the electrical permittivity of the conductor, as shown in FIG.

Please note that since we are taking the measurement point of the electric field fixed in P (0,0,0) as a reference, the direction of dE always that aR.versor will be opposite. Hence the negative sign in the equation (see formula 13). Figure 2. Traction current

Then when you consider that this loaded differential volume disk has constant speed. The electric field is only a function of the variable time t. Taking into account the starting position z ’and der speed of the v-Scheibe Since one wants to calculate the temporal variation of the electric field (see formula 12) at the origin, caused by the displacement of every equally added differential cylinder Elements, it is only taken into account for this calculation that the function of the equation (see formula 15) only depends on the variable position z. Since the contribution of each cylindrical differential element along the entire Z-axis that moves to form the vector of the total time variation of the electric field at point P (0.0.0) because it is instant, does not depend on the variable time t ab. Replace z ’= z and include the variable t = t0 = 0, The equation (see formula 16) then shows the vector equivalence between the time variation of the electric field density at the point P(0,0,0), generated by the immediate and simultaneous displacement of all positive charges along the infinite cylindrical wire, and the conduction current density at the same point.

### 5. Impact

If the hypothesis presented has some physical truthfulness, the following implications are observed:

1. Interpretation of the application of a dynamics of the Gaussian;
2. New conception of the laws of Ampere-Maxwell and Faraday-Lenz-Maxwell;
3. Suggested experiment to prove the theory.

### 5.1 INTERPRETATION OF THE APPLICATION OF A DYNAMICS BY GAUSS LAW

Equality (see Formula 16) is considered consistent with Equations (see Formula 9) and (see Formula 10). It is therefore understood as appropriate that the following physical interpretation of the Gaussian applied to the situation shown in Figure 1 is appropriate: The temporal variation of the total flux of the electric field density on a completely closed Gaussian surface is exactly zero (see formula 17) and (see formula 18). Due to the classic choice of the current direction, which is the displacement of positive loads, in order to make the equations (see formula 24) and (see formula 29) symmetrical with one another, the negative sign (-) was inserted into the equality (see formula 19) . The following interpretation of the physical phenomena considered in equality is proposed here (see formula 19).

The vector time variation of the electric field density , a result of conduction current density, appears to be proportional to the longitudinal speed with which the electric field lines cross a surface element of a Gaussian surface. Theorising can be theorized: in whichvL.is the velocity vector of the electric field lines that cross a surface element of a Gaussian surface, aS.the provider of the vector area element of this surface and K1 a constant, Figure 3.

Figure 3. Variation in time of the density of the electric field caused by

The vector time variation of the electric field density, which results from the displacement current density, is related to the temporal variation of the quantitative number of electric field lines that cross a surface element of a Gaussian surface (HAYT; BUCK et. al., 2013; SHADIKU, 2004).

The theory proposed from the initial hypothesis that the variation in time of the total electrical flux that crosses a Gaussian surface is due both to the variation in time of the number of electric field lines, per unit area they cross, and are formed by the longitudinal speed of the electric field lines that cross a surface element can (see formula 20); it is intuitive through symmetry that the magnetic flux flowing through a Gaussian surface behaves in the same way (see formula 21).

In one way, that it could be generated by varying the number of magnetic field lines in a surface element over time (HAYT; BUCK et. al., 2013; SHADIKU, 2004), ; as well as the longitudinal speed with which magnetic field lines cross a surface element (see formula 22). By symmetry, it is proposed analogously to the equation (see formula 20), the equation (see formula 22). In whichvmLis the velocity vector of magnetic field lines that cross a surface element of a Gaussian surface, aS. is the supplier of the surface element vector of this surface and K2 a constant. ### 5.2 NEW DESIGN OF AMPERE-MAXWELL AND FARADAY-LENZ-MAXWELL LAWS

In view of the possibility of truthfulness in the presented hypothesis, one could imagine the Ampere-Maxwell law (see formula 5) and the Faraday-Lenz-Maxwell law (see formula 23). (see formula 24) and (see formula 25). ### 5.3 SUGGESTED EXPERIMENT TO PROVE THE THEORY.

The aim of this article, in a nutshell, is to propose at the end a theory based on hypothetical situations, with no physical experiments in the laboratory to prove it.

However, what follows is a proposal for a laboratory experiment to prove or disprove the correctness of the proposed theory.

For the calculation of a magnetic field Hthat consists exclusively of a conduction current density using the Equality is generated (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 check the correctness of the theory that the electric and magnetic fields can be generated by the speed of movement of the fields (magnetic and electric), the following experiment is proposed.

It is an electrical circuit formed by an insulated conductive wire that is wound in a distributed and continuous manner around a ferromagnetic material of toroidal topology gewarea A constant of the cross-section, fed by a direct current source, with a current which is set so that it does not magnetically saturate the ferromagnetic material. It becomes a magnetic field density B. give, the is restricted to all toroidal ferromagnetic materials (HALLIDAY; ​​RESNICK; WALKER et. al., 2013), Figure 4. Whereby the magnetic permeability of the ferromagnetic material, N the number of revolutions and l the circumference traversed by the cross-section of the toroid during its total exploration.

Then an isolated conductive wire gauge twist is used around the cross section of the toroid connected to a VDC voltmeter so it is possible to slide the sensing coil along the toroidal circumference, Figure 4.

It is known that the magnetic field generated by an ideal toroidal electrical circuit is zero outside of it (HALLIDAY; ​​RESNICK; WALKER et. al., 2013).

According to the presented theory, when the measuring coil moves at a speed V. along the toroidal circumference, even if the number of magnetic field density lines internal to the turn is not changed, these lines are formed by the Gaussian surface through the circumference of the measuring coil, with -V. Speed.

Applying the proposed equation (see Formula 22) to the Faraday-Lenz-Maxwell Law (see Formula 23) the voltage measured in the voltmeter should be like: Where is the area of ​​the cross section of the toroid.

It is therefore proposed to use a curve of the measurements VDC x V. to increase, where VDC is the voltage measured in the voltmeter und V is the velocity vector of the measurement rotation along the toroidal circumference, Figure 4. According to the proposed theory, this curve should be a straight line with a AK2, Be incline, Figure 4.

Figure 4. Proposed experiment to prove the proposed theory

If the experiment is carried out with a result confirming the proposed theory, it is also suggested the existence of the following conductive magnetic Current density to be taken into account. ### 6. CONCLUSION

In order to better understand the nature of the displacement and conduction currents, the hypothesis was set up that in a completely closed Gaussian surface the temporal variation of the total electric field flux in it could always be zero (see formula 9) and (see formula 10), which does not contradict the continuity equation (see formula 3).

An example of an ideal hypothetical situation was created that would allow a mathematical analysis of the hypothesis created.

The result of this analysis, equality (see formula 16), confirms the idea that the vector time variation of the total density of the electric field (see formula 19) can consist of the following two different physical phenomena.

1 – Temporal variation of the vector density of the electric field in a surface element, depending on the temporal variation of the number of electric field lines that cross it. This is the classical understanding of the displacement current density phenomenon first recognized by J.C. Maxwell. (HAYT; BUCK et. al. 2013; SHADIKU, 2004)

2 – - Temporal variation of the vector density of the electric field in a surface element, depending on the longitudinal speed of the electric field lines that cross it (see formula 20), due to a conduction current density (see formula 16);

Under the premise of symmetry between the behavior of the electric and magnetic fields, similar to the equations (see formula 19) and (see formula 20), the possibility of the magnetic field was considered in the same way (see formula 21) and (see formula 22) considered.

Thus a theory is proposed that all interactions between fields, electric or magnetic, and electrically charged particles are primarily interactions only between fields. If you z. For example, to apply a potential difference in an electrical circuit, the electric field created by the potential difference interacts with the electric field of the free loads, forcing them to move. The shift of the loads implies the shift of their electric field lines. The longitudinal speed with which these lines traverse an area difference element would be proportional to the amount contained therein Vector.

Taking into account the possibility of the truthfulness of equations (see formula 9) and (see formula 10), derived from the dynamics of Gauss law, the Ampere-Maxwell law (see formula 5) in the format (see formula 24) is larger Symmetry to the Faraday-Lenz-Maxwell law (see formula 25).

Finally, a laboratory experiment is proposed to confirm or discredit the proposed theory and its equations. ### CREDENTIALS

FARADAY, M. Experimental Researches in Electricity. B. Quaritch, Londres, 1939, p.1855.

FARADAY, M. Great Books of the Western World. Vol. 45, encyclopedia Britannica Inc., Chicago, 1952, p. 217-866.

HALLIDAY, D; RESNICK, R; WALKER, J. Fundamentos de Física. V.3, ED.9ª, LTC, Rio de Janeiro 2013.

HAYT, W.H; BUCK, J.A.Eletromagnetismo. ED.8ª, AMGH, Porto Alegre, 2013, p. 48-290.

JACKSON, J.D. Classical Electronics. ED.3ª, University of California, Berkeley, 1998, p. 237-239.

MAXWELL, J.C. A Treatise on Electricity and Magnetism. Oxford Univ. Press, Oxford, 2002.

MAXWELL, J.C. The Scientific Papers of James Clerk Maxwell. Dover Publ., New York, 1965, v. 1, p. 451

PLONSEY, R; COLLIN, R.E. Principles and Applications of Electromagnetic Fields. McGraw-Hill, New York, 1961, Topic 2.

SHADIKU, M.N.O. Elementos de Eletromagnetismo. ED.3ª, Bookman, Porto Alegre, 2004, p. 107-350.

 Postgraduate in Electrical Engineering (Instrumentation) from PUC-MG (2008), Degree in Electrical Engineering from CEFET-MG (2007).

Submitted: July 2020.

Approved: August 2020. Degree in Electrical Engineering (Instrumentation) from PUC-MG (2008), Degree in Industrial Electrical Engineering from CEFET-MG (2007).