New understanding of the skin-effect

A.D. Rudnev

 The essence of the skin-effect in the fact that the conductivity of the conductor isn't uniform on cross-section and  is defined,  in practice, as a thin surface layer of the section.

 This is perhaps the oldest, most famous of electrical effects. It studied many times, but only in the sense of quantitative expressions for approximated description [1]. To its credit, science didn't assignedd to this effect hypothetical mechanism of its creation, despite the presence of multiple versions [2] of it. We assume the effect as a benign tumor, that not leading to the disease of other "organs". The study of the effect is needed more for ourselves for demonstration the abilities of real physics.

Such an investigation  must be preceded by a summarizing of features of the toolkit. The main tool will be our model of the electron [3], as well as the real physics of magnetism.  We would emphasize one of the properties of an electron, the ability to orient the axis of symmetry (Fig. 1) along the line of motion.


Fig. 1. Scheme of inner cinematics of electron

 

 Energy-mass (EM) performs central-angle oscillation "through the meridians" an imaginary sphere with frequency ω1 = c / re. An imaginary sphere, and with it -axis of symmetry - make reactive opposite vibrations that creates the effect of spherically symmetric field of energy in space. Simultaneously, the EM does orbital rotation with angular frequency ω 0 = αω1, i.e.  with a constant linear velocity υ 0 = αc. The action of an external electric fieldcauses the electron to orient the axis of symmetry along the line of tension. at the same timeposition of the plane of the orbit is also stabilizing, therefore in the space formsdirected vortex of magnetic field. This is due to the CoulombEM interaction with sptial charges [6]. In the motion electron also orients its axis of symmetry along the line of motion - thanks to the same interaction. In this case, the magnetic field is proportional to speed.  To analyze the physics of the processes in this effect, we need to consider the action of vortex magnetic field on the electrons, that create this field. In modernconception there is no bridge linking the magnetic moment with a structureless electron. In our scheme (Fig. 1) an electron at rest has the property of the oscillationsНам

На рис.2 изображен проводник, в котором протекает ток, первоначально

предполагаемый нами как равномерный по сечению.

of orbit on ± π / 2, making the projection of the magnetic moment (MM) on an arbitrary

plane equal to zero.

The narrowing of the angle of oscillation, which occurs at the symmetry axis orientation, violates the condition that the MM is zero at the conductor and around and creates a magnetic field vortex
This means that the electron can not be indifferent to the magnetic field and that own magnetic field also affects the trajectory of the electrons. 
Remain to us to take into account the full range of possible events simultaneously.

Figure 2 shows a conductor in which current flows initially assumed as a uniform by section.


Fig. 2. Vortex magnetic field of conductor with current.

 

Since the current density is assumed constant γ = Const, then the total current in conductor section

 I = γS
 ( 1)

 

выражается диаграммой на рис.2 (справа).

Соответственно, в теле проводника формируется вихревое магнитное поле с

напряженностью

 expressed by the diagram in Figure 2 (right).

Accordingly, in the body of the conductor is formed a rotational magnetic field with tension

 

( 2).

 

 However, this is only a description of the magnetic field generated by current covered by the section dS in the section of the conductor (Fig. 2) induction for each elementary ring formed in the covered circle is external. The Lorentz force FN, defined by the rule of the left hand, is directed radially to the center of conductor. And there comes the most subtle point related to the behaviorof electrons. First, the volume density of free charges σ (FC) becomes heterogeneous, higher in the center of conductor. Second, the path of the electrons became curvilinear,along withelectric tension in the conductor, there is operation of magnetic induction.

He = U / L
 (3)

 

But the geometric combination of forces shows only the deviation of velocity of an electron from the vector of tension of electric field. 

 

Fig. 3. Compression of the structure of free electrons by Lorentz force.

 

And as what is reaction of electron - it should change the orientation of the axis of symmetry by the new velocity vector? It is easy to see that there is a single possibility to compromise - movement of romb formed by the electric and magnetic forces (Fig. 3).

 

Fig. 4. The diagram of vector forces, operating on electrons in the conductor with current.

 The axis of symmetry of the electron deviates from the velocity vector at an angle α. Since induction -is  the result of collective interaction of the electrons, the radialmagnetic force is greater than the electrical force acting on the individual electron. That is, the radial displacement of electrons significantly reduces the value of the current. A measure of replacement energy of the electrons movement is the work of compressionthe structure.

Ранее нами определено [8], что в воздушной среде расстояние между свободными

We have previously determined [8], that in the air distance between the free electrons , and for specific medium it can be determined with use of the ratio of bulk densities. For example, for copper

 ( 4).

 

According to the force of Lorentz, compressing the flow of electrons

 Fт = Вmυqe

 ( 5)

 

 -it is impossible to calculate without knowing the velocity of the electrons. Modern physics does not know number of free electrons in the conductor, and the speed of their movement. the flow of electrons

Традиционным способом можно вычислить постоянный ток в проводнике заданного

 We can calculate the direct current of given section of conductor with given tensity in the traditional way/

For example, in a copper conductor with diameter of 0.2 mm (at a voltage of 0.1 V at 1 m long), a current flows with a value of  0,1848 А.

Lets calculate.

The volume of conductor v = 3,14159E - 08 m .

 

The atomic mass of copper ma = 1,05E - 25 kg,

the quantityof copper atoms х = M / ma = ρv / ma = 2,66287E + 21 ,

 

the volume of copper atom va = 8,78453E - 30 m^3,

the free volume in conductor

v0 = v − хva = 8,02383E - 09 m^3,

 the number of free electrons in it

n = v0 / z М^3 = 5,19893E + 18 

 total moving charge

 Q = nqe = 8,32E - 01 C.

 ( 6).

 

By the way, we obtain a useful constant - the specific charge of the masses

 θ = Q / ρv = 2975,049656 (Кул/кг)

 ( 7).

 

В таком случае, скорость движения электронов (для выбранного примера) определится

величиной

 In this case, the velocity of the electrons (for selected example) will be determined by value

υ = Iz / Q = 2,57E - 10 (m/s)

( 8).

 

Стартовое включение проводника вызывает пару взаимно-перпендикулярных сил:

The starting turning on of the conductor causing a pair of mutually perpendicular forces:

axial  F=qEVe and radial - force of Lorentz. Lets find the radial shift of electrons in conductor

 

For this we will define the initial value of the Coulomb force of interaction of free electrons of the structure

( 9).

 

Comparing it with the electric force F=qEVe= 1,60E - 20 N, заключаем, что она

не способна изменять решетку пространственных свободных электронов. Магнитная

conclude that it can not change the spatial lattice of free electrons. Magnetic force, whose graph is similar to the current function (Fig. 2), even smaller. But there is a nuance:each electron producesits strength of interaction with the magnetic field, so towards the center forces are added. Deformation structures are different radial guide and they are also summarized. But, the summation  is in the opposite direction.

Деформацию ячеек структуры на радиусе r оценим относительно силы кулоновского

сжатия

 Deformation of the cell structure at a radius r we will calculate relative to the strength of the Coulomb compression


 ( 10).

 

 Just substitute not the value of magnetic force (5), but the sum of forces acting on section from the surface of the conductor to the current radius


Воспользовавшись линейностью зависимости силы от радиуса, введем среднее значение

силы на заданном отрезке и заменим суммирование умножением

 Using the linear dependence of force on the radius, we introduce the mean value of  force on a given segment and replace the summation by multiplying

( 11)

 

D = γµ0υ0qe /4zM = 6,43E - 32 ( N / m^ 2 ) ,

Substituting the values ​​of the permanent members obtain a simple expression


( 12).

 

On the fig 5 shown the graphs for this equation

a)

 

b)

Fig.5. Dependence of Lorentz force on radius of conductor (а) and sum of radial forces (b).

 

 Like a column of liquid, the total force rises the deformation while approaches to the center of conductor(4) of the electrons structure. Not only  total deformation of column, but also particular values ​​have essentially nonlinear dependence on the radius (Fig. 6).

a)

b)

 Fig. 6 Dependence of the deformation of structure of the electrons on radius of the conductor (a) and total deformation of "beam" of electrons (b).

A characteristic feature of the functions is that the total force is maximal near the center section, and the total deformation, on the periphery. So energy of deformation (compression work) has pronounced extremum(Fig. 7). 

 

Fig. 7. Nature of work change of structure compression along the length of the electron beam.

 

Here we saw the work of compression of single electrons beam.

 

dE(r ) = F∑dz 
( 13).

 

It is clear that in the section of the conductor  the number of such rays is equal to πR / z, and the number of electronic layers (sections) along the length of the conductor is equal to L / z, ie, the scale of the curve in Figure 7 should be increased by πRL / z^2 that in our example  is equal to 2,35E + 14 .
Summary diagram of the energy costs on electrons structure compression  is shown in Figure 8

Fig. 8. Total energy costs diagram

  Lets recall that the primary driving force was the force created by the tension (3) of the electric field. The movement of electrons created a current, and it created the magnetic field. So that the effect of compressing the structure of electrons occurs in the final due to inhibition of the electrons, ie, by reducing the current. If we will summarize the energy consumption E Σ (r), we would obtain the maximal work necessary to recieve DC in conductor. In this example this value is Е Σ = 7,31E - 25 J. How to compare the energy with electric power? We need parameter of time. But this parameter need to affect only real processes - movement of electrons. So if the power of electric loss UI = 1,85E - 02Wt we need to consider only the power of movement 

N = nFυ = 2,14E - 11 Wt
 (14).

 Everything else - is the energy loss by the interaction of electrons with atoms, and this is tthe reason why uniform motion of electrons does not go uniformly accelerated. Interesting - by how many times the total loss of energy is greater than the effective loss, it turned out that by 8,65 E +08 times.


So, even with such low efficiency it seems to us, that there is enough time

τ = E Σ / N = 3,42E - 14c

 ( 15)

 

to complete the transition processes. But no, we should emphasize the nature of structure compression process: the pressure comes from the surface of the conductor to the center, and the deformation grows in the opposite direction. Consequently, for each coordinate of elementary ring, the cycle of  "deformation - the force appearance" and vice versa is repeated. That is, the deformation occurrs alternately, causing the compression time to be stretched. Assumed mean value of coordinate

r = R/2

we find the average number of cycles x = R / 2zm

деформаций в одной координате. Тогда полный опрос электронного «луча» произойдет за

время

 deformations in one coordinate. Then, complete progress of electrons "beam" over all the cycles will occur in time

(16)

 

In our examle R = 0,0001 м х = 1,87 E + 09 and time Т = хτ = 6,40E - 05s  means that after 64 ms after the conductor is turned on, current in it will become uniform over the cross section. The frequency dependence of the process is in simple stabilization of the operating point - in the extremum of the curve in Figure 8. In this case the amplitude value of the maximum in the graph will fall in proportion of the period to parameter Т = х τ .


 

Fig. 9. Recieving graph «remaining energy».


Fig.10. Diagram of steady-state current (dashed line) and limiting transitional process.

 

Чтобы перейти к закономерности распределения тока по сечению проводника,

необходимо отметить следующее: если работа магнитных сил происходит за счет тока,

он не может стать нулевым, т.к. иначе исчезнет и магнитное поле. Словом, существование

магнитного поля во время установления режима тока должно происходить в паритете

энергии. Иначе говоря, максимум энергии на диаграмме рис.8, соответствует максимуму

возможностей системы в целом. Это значит, что энергию полезного тока можно

отсчитывать от этого пика, как от предела вычитанием энергии сжатия структуры (рис.9).

To proceed to low of current distribution over the cross section of the conductor, should be noted following:  if the work of the magnetic forces is due to the current, this current cannot be zero, since otherwise the magnetic field would dessapear as well. In short, the existence of magnetic field during  the current mode shoud be set with balance of energy. In other words, the maximum energy in Figure 8, corresponds to the maximum capacity of the system as a whole. This means that the energy efficiency of current can be measured from this peak, as from the limit by subtracting the compression energy of structure (Figure 9). 

And since it is clear that 100 percent expression of elementary ring cross section energy capacity is its ability to provide current γπr^2 , so we correlate  remaining energy Е = 1,11Е − 26 J with the nominal value of constant current I (r ) .

 

Consequently, the transition to the current I (r) of transition process is sufficient to normalize the steady current I (r ) = γπr^2 with  similarity coefficient k = E remaining / E max

 I (r ) = γπr^ 2E remaining (r ) / E max

 (17).

 

 Diagram of limiting transition process current  in the conductor is shown in Fig. 10. The width of the δ zone of conductivity, defined as the decreasing in current density by a factor e, covers about 65% of the total current of the conductor. The process of transition by the limiting mode is known in science as "self-inductance effect of the conductor."

 The perfect equality of the magnetic energy and the energy corresponding to the electrical conductivity, in practice violated. This occurs due to diamagnetism or paramagnetism of the material of the conductor. In the first case, the magnetic properties are weakened, causing compression of the structure to be smaller. In the second case, all on the contrary. And the result of the manifestation of these properties is illustrated by the diagrams in Fig. 11.


Рис. 11. Influence of  diamagnetism and paramagnetism of materials on limiting mode.

 

As can be seen from the figure, it is demonstrated here another new feature of the phenomenon negative current of paramagnetics in some elementary rings of cross section near the  0.6 radius of the conductor.Manifestation of the skin effect on alternating current does not reach the limit curve. Working curve with increasing the frequency shifts from the dashed line in Figure 10 to the dashed line in Figure 11. Therefore, in practice it has a form characteristic for the limit diamagnetic mode.

Bibliography:

1.G. Ebert  "Kratkii spravochnik po fizike" (Quick Reference Guide to Physics). M, F-M, 1963, p.444

2. T.I. Trofimova  "Kurs fiziki" (physics Course). M. The high school, 1997

3. A.D. Rudnev  The new conception of physics. http://www.sciteclibrary.ru/rus/catalog/pages/6910.html

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