# One Hundred and Fifty Years of Skin Effect
## 2 Solitary Conductor Concept
The amplitude, \(\hat{J}_{\text{sol}}\), is the largest on the conductor surface, \(r=r_{1}\), and it decreases with deceasing \(r\); for \(r=r_{1}-\delta\), its value for high frequencies is e-times smaller than on the conductor surface, where \(\delta\) is the so-called penetration depth [30]:  
\[\delta=\sqrt{\frac{2\varrho}{\omega\mu_{0}}}. \tag{8}\]  
Related to sinusoidal current density in a solitary conductor is the technical term skin effect, which should characterize current density in conductors. According to ref. [31], item 121-13-18, \"Skin effect is a phenomenon in which the current density is greater near the surface than in the interior of the conductor for an alternating electric current in the conductor\". In the literature are further definitions, which are either similar to the definition in ref. [31] or they define the skin effect as a phenomenon in which most of the current in the conductor flows through a thin layer near the conductor surface. In all the definitions, only the word 'conductor' is given, although all the definitions were proposed in analyses of the current density in a'solitary cylindrical conductor'. Examples are given below that raise doubts as to the given skin effect definitions. Figure 2 gives the current density \(J_{\text{sol}}=J_{\text{sol}}(r,t)\) in Example 1 at five instants, \(t=\theta T\), where \(\theta=0,0.3,0.4,0.5\), and \(T=1/f,f=10^{4}\,\text{Hz}\). It follows from Figure 2 that for a solitary cylindrical conductor the conductor surface exhibits the maximum \(\hat{J}_{\text{sol}}\) but not always the maximum current density. For a solitary conductor of rectangular shape, the situation is similar. **Example 2**. : _Solitary Al conductor \((\varrho=2.650\times 10^{-8}\,\Omega\cdot m\) at 20 \({}^{\circ}\)C [29]) with the rectangle cross section \([0,10]\,mm\ \times\ [0,16]\,mm\). The current through the conductor is_  
\[I_{\text{rec}}(t)=I_{\text{rec}}\sin\omega t,\ \text{where}\ \hat{J}_{\text{rec}}=1 \,\text{A},f=10^{4}\,\text{Hz}.\]  
\[i(t)=i^{(0)}(t),\quad i^{(k+1)}(t)=\frac{{\rm d}^{k+1}}{{\rm d}t^{k+1}}\,i(t). \tag{4}\]0 Dependence of instantaneous current density on \(r\) for \(t=\theta T\); \(\theta=0,0.3,0.4\), and \(0.5\), in Example 1. The amplitude, \(\hat{J}_{\text{sol}}\), is indicated by the red line and the quantity, \(\delta\), is determined by Formula (8). For \(r\in[0,3)\,\text{mm}\), the current density magnitude is smaller than \(6.943\,\text{A}\cdot\text{m}^{-2}\). This example is given, inter alia, because the current density,  
\[i(t)=i^{(0)}(t),\quad i^{(k+1)}(t)=\frac{{\rm d}^{k+1}}{{\rm d}t^{k+1}}\,i(t). \tag{4}\]1  
in a solitary rectangular conductor has not been calculated using the method proposed by [PERSON] nor by methods derived from his method. The reason for this is ignorance of the initial and boundary conditions necessary for the solution of the respective differential equation. Figure 3 provides the dependence of the amplitude, \(\hat{J}_{\rm rec}\), on \(x\) for several values of \(y\) in Example 2. The values of \(\hat{J}_{\rm rec}\) were obtained by a method published in refs. [9; 10; 11; 12], and in Figure 3 they are only given in one quarter of the conductor cross section because not only the amplitude but also the initial current density phase are symmetrical along the symmetry axes of the cross section \(x=5\,\)mm and \(y=8\,\)mm. Figure 4 gives the current density in a quarter of the rectangular cross section of a solitary conductor at the instant \(t=0.39\,T\) in Example 2. For a solitary tubular conductor with internal non-zero radius and constant resistivity, the skin effect obviously does not occur. \(\hat{J}_{\rm sol}\) is maximal and minimal near the outer and the inner surface, respectively; see Figures 2, 4, and 6 in ref. [16] and Figure 3 in ref. [17]. The skin effect is associated in practice not only with solitary conductors but also with two conductors, although in its definition (item 121-13-18 in ref. [31]) there is only 'conductor'.