# One Hundred and Fifty Years of Skin Effect
## 1 Introduction
The basic task is the calculation of the current density \(\mathbf{f}_{\text{las}}(x,y,z)\), in a group of conductors that do not move and have the following properties;  
* Each of them has a long part parallel to the axis \(z\) in the coordinate system \(xyz\);
* their cross section is arbitrary but does not change along the long part;
* some are connected at the end to a voltage or current source;
* some are connected at the end to other conductors in the group;
* the permeability of the conductors and their surroundings equals the vacuum permeability, \(\mu_{0}\);
* they are placed in a magnetic field produced by a number of conductors that belong to the group or by an external source. Solving the basic task is trivial if the magnetic field in which the conductors are placed does not change with time. In the case of a magnetic field varying with time, the solving of the basic task is problematic if some parts of the conductors are not parallel to \(z\). In the physical model of the basic task it is assumed that  
* the conductors in the group have only a long part, which is infinitely long, and each conductor is therefore determined by its cross section;
* the conductor resistivity, \(\varrho\), is given and only depends on \(x\) and \(y\);
* the phenomena are quasi-stationary, i.e., the propagation speed of current and magnetic field is infinitely high;
* the displacement current and leakage current between conductors are neglected. The physical model is based, in the first place, on [PERSON]'s law of electromagnetic induction and is an application of the [PERSON] and Savart law, loop current method, [PERSON]'s voltage law, and [PERSON]'s law. Applying these laws to a finite segment of conductors yields a system of equations whose solution is a current density whose nonzero component is only the \(z\)-component:  
\[\mathbf{J}(x,y)=\langle 0,0,J(x,y)\rangle.\]  
In the mathematical model, the conductor cross sections are approximated by the \(a(N,d)\) aggregate of disjunct rectangles \(A_{i}\),  
\[a(N,d)=\bigcup_{i=1}^{N}A_{i},\]in which only the resistivity, \(\varrho_{i}\), and the current density, \(J_{i}\), have constant values, and  
\[d=\max_{i=1,2,\ldots,N}d_{i},\]  
where \(d_{i}\) is the length of the diagonal of rectangle \(A_{i}\). The mathematical model is a system of \(N\) ordinary differential equations of the first order [9; 10; 11; 12]. A detailed description of the physical and mathematical models is given in [9; 10; 11; 12], together with the solution to specific cases. An exact solution to the mathematical model is the set  
\[J_{\rm mat}=\{J_{1},J_{2},\ldots,J_{N}\}. \tag{1}\]  
\(J_{\rm mat}\) can also be considered a piecewise constant function, \(J_{\rm mat}=J_{\rm mat}(x,y,d)\), defined in the plane \(xy\):  
\[J_{\rm mat}(x,y,d)=\left\{ \begin{array}{ll}0&\mbox{for }(x,y)\
otin a(N,d)\\ J_{i}&\mbox{for }(x,y)\in A_{i}.\end{array} \right. \tag{2}\]  
It holds that  
\[\lim_{d\to 0}J_{\rm mat}(x,y,d)=J(x,y).\]  
This means that \(J_{\rm mat}\) can approximate the solution \(J(x,y)\) of the physical model, theoretically with an arbitrary accuracy. In practice, the accuracy is limited by the level of the calculation means used. For a sufficiently small \(d\), \(J_{\rm mat}(x,y,d)\) is an accurate solution to the physical model and can be considered a solution to the basic task. The method for the calculation of the current density is numerical. In the calculation, the number of rectangles that approximate the cross sections of conductors is approximately 10,000. The accuracy of the calculation is assessed via comparing the results for different values of \(d\). Part of the calculation of current density in conductors is the calculation of independent magnetic fluxes. The determination of independent fluxes is described in [9; 10; 11; 12].