# One Hundred and Fifty Years of Skin Effect
## 2 Solitary Conductor Concept  
In ref. [1], Art. 689 and 690, [PERSON] proposed a method for the calculation of current density, \(J_{\rm sol}\), in a solitary long cylindrical conductor. In the following analysis of [PERSON]'s result, the quantities are denoted by currently used symbols. In ref. [1], \(J_{\rm sol}\) is the solution to a differential equation; it depends on the distance, \(r\), from the conductor axis and on \(t\),  
\[J_{\rm sol}(r,t)=-\frac{\mu_{0}}{4\pi\rho}\,\sum_{k=0}^{\infty}\left[\frac{ \mu_{0}\,r^{2}}{4\rho}\right]^{k}\frac{i^{(k+1)}(t)}{(k\,!)^{2}}, \tag{3}\]  
where  
\[i(t)=i^{(0)}(t),\quad i^{(k+1)}(t)=\frac{{\rm d}^{k+1}}{{\rm d}t^{k+1}}\,i(t). \tag{4}\]  
\(i(t)\) is the given function and, according to (19) in [22], it holds that  
\[J_{\rm sol}(0,t)=\frac{\mu_{0}}{4\pi\varrho}\,i^{(1)}(t). \tag{5}\]  
If the function \(i(t)\) is sinusoidal, then the function \(J_{\rm sol}\) is also sinusoidal,  
\[J_{\rm sol}(r,t)=\hat{J}_{\rm sol}(r)\sin[\omega t+\varepsilon(r)],\quad \omega=2\pi f,\]and its phasor, \(\underline{J}_{\text{sol}}\), is a solution to the equation  
\[\frac{\text{d}^{2}\underline{J}_{\text{sol}}}{\text{d}r^{2}}+\frac{1}{r}\frac{ \text{d}\underline{J}_{\text{sol}}}{\text{d}r}-\text{j}\kappa^{2}\underline{J} _{\text{sol}}=0,\quad\kappa^{2}=\frac{\omega\mu_{0}}{\varrho}, \tag{6}\]  
which has been derived, for example, in ref. [16]. Using [PERSON]'s equations in differential form, an equation can be derived for the solitary conductor with an arbitrary cross section, which is satisfied by current density over the cross section of a long conductor,  
\[\
abla^{2}\boldsymbol{J}-\mu_{0}\,\gamma\,\frac{\partial\boldsymbol{J}}{ \partial t}=0, \tag{7}\]  
where \(\gamma\) is the electric conductivity of the conductor. Expressing current density in (7) via the vector potential, \(\boldsymbol{A}\), will yield an equation that is similar to (7). The solution to Equation (6) can be expressed using the Bessel or the Kelvin functions [23], Art. 16.4. Since the publication of [1], a number of methods have been published based on the solitary conductor concept; some of the recent ones are, for example, [24; 25; 26; 27; 28]. **Example 1**. : _Cylindrical Al conductor (\(\varrho=2.650\times 10^{-8}\,\Omega\cdot m\) at 20 \({}^{\circ}\)C [29]), whose axis is the axis \(z\) and the cross section radius \(r_{1}=6\,\text{mm}\). The function \(i(t)\) has been chosen such that it holds that \(J_{\text{sol}}(0,t)=\hat{i}\sin\omega t\), where \(\hat{i}=1\,A\cdot m^{-2}\)._  
The solution to Equation (6) in Example 1 is given in Figure 1.  
The curve \(\varepsilon(r)\) for \(f=10^{4}\,\text{Hz}\) in Figure 1 has two parts and is discontinuous. It is only a seeming discontinuity since \(\varepsilon\) is modified such that \(\varepsilon\in(-\pi,\pi]\), which is possible because the period of the function \(\sin\) is \(2\pi\); \(\varepsilon(r)\) is continuous for \(f=10^{3}\,\text{Hz}\). The number of seeming discontinuities of \(\varepsilon(r)\) increases with increasing \(f\). The function \(\tilde{J}_{\text{sol}}(r)\) is increasing (monotonous) in contrast to the function \(J_{\text{sol}}(r)\), which is not monotonous, and for the discontinuous function \(\varepsilon(r)\) it is oscillating. This property of current density is not limited  
Figure 1: Solution to Equation (6) in Example 1. Dependence of amplitude, \(\tilde{J}_{\text{sol}}\), on the distance, \(r\), from the conductor axis for \(f=10^{3},10^{4}\), and \(10^{5}\,\text{Hz}\); dependence of initial phase, \(\varepsilon\), on \(r\) for \(f=10^{3}\,\text{Hz}\) (red line) and for \(f=10^{4}\,\text{Hz}\) (blue lines). to the solitary conductor of circular cross section; it also occurs, for example, with the loop formed by two conductors of rectangular cross section. If the surface of the conductors is parallel with the planes \(xy\) and \(xz\), then the current density, \(J(x,y)\), may oscillate for \(y=\text{const}\), as can be seen in Figure 3 in ref. [21].