# One Hundred and Fifty Years of Skin Effect
## 3 Solitary Conductor versus Reality  
Significant for practice are the theoretical models that are the most faithful image of reality. An infinitely long cylindrical conductor through which steady sinusoidal current flows is the basis of the solitary conductor concept. A conductor connected to the ideal current source can be considered solitary [32]. The ideal current source cannot be realized because, on a finite segment of the conductor, it produces an infinitely large voltage, but in spite of this a finite voltage is assumed in ref. [1], Art. 690. [PERSON]. [PERSON], who prepared the third edition [3] of [1], added in the form of a footnote a derivation of the formula for the calculation of inductance of a solitary conductor and the dependence of effective resistance on the frequency of the sinusoidal source the examined conductor is connected to. The skin effect is defined for the solitary conductor. Questioning the existence of solitary conductor also means questioning the skin effect. Closest to the cylindrical conductor is in fact a pair of coaxial conductors connected to a source. The relation between current density in a solitary cylindrical conductor and in a pair of coaxial conductors is demonstrated in the following example. **Example 3**. : _A pair of coaxial conductors at a temperature of \(25\,^{\circ}C\), whose axis is the axis \(z\), is formed by an inner cylindrical \(Cu\) conductor: cross section radius \(r_{1}=10\,nm\), resistivity \(\varrho_{1}=1.712\times 10^{-8}\,\Omega\cdot m\)[29], and by an outer tubular \(Al\) conductor: inner radius \(r_{2}=11\,mm\), resistivity \(\varrho_{1}=2.709\times 10^{-8}\,\Omega\cdot m\)[29], wall thickness \(\Delta r\). A source of steady sinusoidal voltage with \(f=10^{3}\,Hz\) causes the drop \(\hat{V}\sin\omega t\), where \(\hat{V}=1\,V\cdot m^{-1}\)._  
The steady current density in coaxial conductors,  
\[J_{\rm coa}(r,t)=\hat{f}_{\rm coa}(r)\sin[\omega t+\varepsilon_{\rm coa}(r)],\]  
is symmetrical with respect to the axis \(z\); for the given parameters \(r_{1},r_{2},\,\Delta r,\varrho_{1},\varrho_{2},f\), it only depends on the distance, \(r\), from the axis \(z\) and on \(t\), and is directly proportional to the voltage drop on a segment of both conductors. According to refs. [13; 18; 22], it holds for a given voltage drop:  
\[\lim_{r_{2}\to\infty}\hat{f}_{\rm coa}(r)=0,\text{ for all }r. \tag{9}\]  
Figure 4: Instantaneous current density, \(J_{\rm rec}(x,y,t)\), for \(t=0.39\,T\) in Example 2. It should be noted that (9) holds for the quasi-stationary state, which is assumed in the present review ( see Section 1), but assuming the quasi-stationary state is not adequate for a large \(r_{2}\). According to ref. [33], the assumption of the quasi-stationary state is only adequate when the maximum dimension \(\ell_{\text{max}}\) of the system in the plane \(xy\) is much smaller than the free space wavelength, i.e.,  
\[\ell_{\text{max}}\ll c/f,\text{ where }c\text{ is the speed of light.} \tag{10}\]  
If the maximum magnitudes of \(r_{1}\) and \(\Delta r\) are several centimetres, then the inequality, (10) corresponds to the inequality \(r_{2}\ll c/f\). In Figure 5, the function \(\hat{J}_{\text{coa}}(r)\) in Example 3 is illustrated for three values of \(\Delta r\). The method for calculating current density in two coaxial conductors was described in refs. [9; 12; 13; 14; 15]. In Example 3, \(r_{2}=0.02c/f\) was chosen, i.e., \(r_{2}=6000\,\text{m}\), in agreement with (10). It was established by calculation that for \(\Delta r=5\,\text{mm}\) and for all \(r\), \(\hat{J}_{\text{coa}}\) is smaller than \(6.73\times 10^{5}\,\text{A}\cdot\text{m}^{-2}\) and \(1.13\times 10^{3}\,\text{A}\cdot\text{m}^{-2}\) in the inner and outer conductor, respectively. The maximum value of \(\hat{J}_{\text{coa}}\) for \(r_{2}=6000\,\text{m}\) is only \(3\,\%\) of the maximum value of \(\hat{J}_{\text{coa}}\) for \(r_{2}=11\,\text{mm}\). From the above, it follows that a solitary conductor connected to the ideal current source cannot be implemented.