Research website of Vyacheslav Gorchilin
2020-06-26
Dynamic skin effect and transverse wavelength
Dear readers! In this work, we open one more effect for General study and application. While he received only theoretically, but as you know, the theory, in many cases, practice is ahead and we hope for further successful implementation and introduction of this unusual phenomenon in the area of alternative energy.
The phenomenon of skin effect [1] been studied obviously insufficiently and electrodynamics, as it turned out, there are many gaps. The classic explanation of this effect does not hold water though, because it is not quite correctly explain its mechanism, which is well lit, for example, in [2]. But there is another nuance, which reason nothing is said. It lies in the fact that in a conductor the electric field and current density in a plane perpendicular to its course, are distributed uniformly with a constant current and moves to the edge — if it is changed. It is evident from the classical explanation, which follows from the Maxwell equations: ${\Bbb{rot}\, \mathbf{E} = - \frac {\partial \mathbf{B} }{\partial t}} \qquad (1.1)$ Its meaning is quite simple: when changing magnetic induction (right part of the equation), around its lines appear closed lines of the electric field (the left part of the equation), and magnetic and electric lines of force are always mutually perpendicular. According to the classical explanation, the part created in this way, the electric lines of force are directed by the movement of the current, a part — against, that redistributes the pattern of the motion of charges and forces them to move on the surface of the conductor. But! If using our guide runs an alternating sinusoidal current, in the moment of transition of the sine using your maximum and minimum (red circles in Fig. 1), the time derivative will be zero and, thus, the right side of equation (1.1), which means the absence of skin effect at these moments. Conversely, when the sine passes through zero (black circles in Fig. 1), its derivative is maximum, and therefore at this point will be maximum and the electric forces that form the skin effect. It is this nuance and bypasses theoretical electrodynamics. Let's deal...
If our conductor carries a steady current, the current density must be distributed over the cross section of the conductor evenly. This distribution is shown in figure (2), which shows a longitudinal section of the conductor, and the purple lines is a conventional image of the vectors of current density or electric field intensity in it.

Hereinafter in the figures, we will depict only part of the longitudinal section of the conductor, where the coordinate of $$y$$ will be directed upwards. The bottom of the figure will represent the surface of the conductor, and the top of the picture — its middle. On the surface of the conductor the current density is always maximum in charts conditionally approved per unit. The length of the vector indicates the magnitude of the current density and its direction and color polarity.

If the Explorer runs an alternating, sinusoidal current, the vectors of current density must be unevenly distributed to the surface as you increase the derivative $$\partial \mathbf{B} / \partial t$$, i.e. to be the most unevenly distributed in the moments of transition of the sine zero, and in the moments of maximum and minimum of the sine wave to be the same as at DC. We expect this dynamic allocation is presented in figure (3).
 Fig.1. Sine wave and its derivative at different points in time Fig.2. Uniform distribution of current density in the conductor at DC Fig.3. The expected picture of the dynamic skin effect when alternating current
The last figure we get the expected picture of the dynamic skin effect. Let's clarify this assumption. For this we follow as it enters classical electrodynamics in order to circumvent the above caveat. This is done so. First, equation (1.1) simplifies to the case of a real conductor is: $\frac{\partial ^{2}E_{x}}{\partial y^{2}} = \mu \gamma \frac{\partial E_{x}}{\partial t} \qquad (1.2)$ is absolutely the right decision, and then instead of a variable $$E_{x}$$, which represents the distribution of the electric field along the conductor (along the axis of $$x$$), is substituted the following expression [1]: $E_{x}(y,t) = E_{0}(y)\, e^{i\omega t} \qquad (1.3)$ where $$E_{0}(y)$$ is the amplitude value of the electric field, and $$\omega$$ angular frequency. Under the coordinate $$y$$ here refers to the current density distribution along the axis perpendicular to the axis of the conductor (in the drawings vertically). This form of entry is fair for electromagnetic waves, but if the changes in the field strength (and hence current, and magnetic induction) in the conductor is relatively slow, and change according to the sinusoidal law, in fact it should be: $E_{x}(y,t) = E_{0}(y)\, \sin(\omega t) \qquad (1.4)$ And if we substitute in further calculations this is correct according to the author, form, then the final formula and the very physical sense, the skin effect will change fundamentally. The expression (1.2) will then become: $\frac{\partial ^{2}E_{0}}{\partial y^{2}} = \mu \gamma \omega \cot(\omega t) E_{0} \qquad (1.5)$ where: $$\cot(\omega t) = \cos(\omega t) / \sin(\omega t)$$. In this formula it is possible to see the dependence of the field strength distribution from its phase at the time, what distinguishes this solution from the classical one. To obtain the final result we need to recall that the current density and the electric field associated specific conductivity: $$j= \sigma E_0$$. Solving the last differential equation, and giving it a physical meaning, we obtain the following distribution of current density on the coordinate $$y$$ and: $j(y,t) = j_0 \sin(\omega t) \begin{cases} \exp ({- \sqrt{\cot(\omega t)}\, y/\Lambda}), & \mbox{if } \cot(\omega t) \ge 0 \\ \cos( \sqrt{- \cot(\omega t)}\, y/\Lambda ), & \mbox{if } \cot(\omega t) \lt 0 \end{cases} \qquad (1.6)$ or, equivalently: $j(y,t) = j_0 \sin(\omega t)\, Re \exp \left( {- \sqrt{\cot(\omega t)}\, y/\Lambda} \right) \qquad (1.7)$ As we can see, this solution is fundamentally different from the classical [1] the fact that the thickness of the skin layer changes in time, and therefore we can now speak of the dynamic skin effect. In this formula: $$j_0$$ is the current density on the surface of the conductor (its maximum value), $$Re$$ is the real part of the subsequent expression. But the next parameter you need more detail.
The transverse wavelength
Here we introduce a new term, the meaning and application of which remains to be explored. $\Lambda = {1 \over \sqrt{\mu \gamma \omega}} \qquad (1.8)$ In this formula, $$\Lambda$$ is the transverse wavelength in the denominator, under the square root is the product of magnetic permeability, conductivity and angular frequency. If you multiply the parameter to the square root of two, then numerically it is the same as the thickness of the skin effect, but the mathematical and physical sense it is fundamentally different. The transverse wave length determines the number of waves that laid across the wires. Here we can draw a parallel with its longitudinal counterpart, but unlike the latter the transverse length is much less, and the oscillation frequency changes along the axis $$y$$. For comparison: for copper wire and a frequency of 10 kHz longitudinal wave length is 30 kilometers, and the cross is just 0.47 mm!
But everything in order. Figure (3), we assume, should look like the picture of the skin effect in the dynamics. Now let's use mathematical editor MathCAD [3], formula (1.6) or (1.7) will build a dynamic vector plot and compare the estimated and actual figures. If the substitution, we will assume that the diameter of our conductor $$d$$ is equal to its transverse wavelength, we get figure (4). If along the diameter of the conductor are placed two transverse wavelength, we get a more complex picture in figure (5). Similarly, if along the diameter of the conductor are stacked three transverse wavelength, we get a more dynamic figure (6). Only do not forget that in the figures we depicted only a portion of a longitudinal section of the conductor: the bottom of the drawing surface, the top of the picture — its middle.
 Fig.4. Dynamic distribution of current density in the wire when d = Λ Fig.5. Dynamic distribution of current density in the wire when d = 2Λ Fig.6. Dynamic distribution of current density in the wire when d = 3Λ
In fact, with alternating current, the current through a conductor, dynamic skin effect is to generate large numbers of frequencies and wavelengths, but after some value it's fading fast. This value was taken as the length of the shear wave. If we look at the dynamic drawings (5-6), then at some point we will be able to see this value. For greater clarity, we have recorded these moments in the following two static images (Fig. 7-8).
 Fig.7. One moment in the dynamic of the current density distribution at d = 2Λ Fig.8. One moment in the dynamic of the current density distribution at d = 3Λ
As we can see in figure (7) between the surface of the conductor and the stacked 1 wave, and means across the diameter at this point, fit the two waves. Similarly, in figure (8) between the surface of the conductor and its center can be seen 1.5 waves, and means across the diameter at this point is placed three waves.
From this work it is clear that some properties of the skin-effect correspond to classical electrodynamics. For example, it concerns the fact that in alternating current, most of it runs in the surface layer of the conductor. However, some phenomena not considered classics. It refers to the complex structure of currents and their frequencies within the wire. In the following work we will continue to develop this theme and show a more complex picture of the dynamic current density distribution in the conductor, and also, get some of the energy ratio, which gives us this wonderful effect.

The materials used
1. Wikipedia. Skin-effect.
2. Rysin A. B., Rysin O. V., Bonaci V. N., Nikiforov I. K. the Paradox of skin effect. [PDF]