# Transient Current Density in a Pair of Long Parallel Conductors
## 2 Calculation of Current Density and Current in Conductors  
The essence of the method for calculating current density as published in [1] consists in the conductors being replaced by the partial conductors \(A_{i},\forall i\). The symbol \(\mathbb{V}\) means that the quantity after it takes the values \(1,2,\ldots,N\) successively. The cross section of each partial conductor \(A_{i}\) is also denoted by the symbol \(A_{i}\). Each cross section \(\mathcal{A}_{\ell},\ell=1,2\), can be replaced with arbitrary precision by the union \(a(\mathcal{A}_{\ell})\) of identical disjunct rectangles that have been formed using the net of parallel lines  
\[x=k_{x}\Delta_{x},\quad y=k_{y}\Delta_{y},\]  
where \(k_{x},k_{y}\) are integers and \(\Delta_{x}\) and \(\Delta_{y}\) are the chosen lengths of the rectangle sides. The rectangles of the net, which are subsets in \(a(\mathcal{A}_{\ell}),\ell=1,2\), are the cross sections of the partial conductors \(A_{i},\forall i\). \(A_{i}\subset a(\mathcal{A}_{\ell})\) if \((X_{i},Y_{i})\in\mathcal{A}_{\ell}\), where \((X_{i},Y_{i})\) is the centre of the rectangle \(A_{i}\). The partial conductors \(A_{i}\), which form \(a(\mathcal{A}_{1})\) and \(a(\mathcal{A}_{2})\), must be numbered such that the graph with \(N\) vertices \((X_{i},Y_{i})\) and \(N-1\) edges is a path graph [1,8]. It is assumed that, in the partial conductor \(A_{i},\forall i\), there is a constant resistivity \(\varrho_{i}=\varrho(X_{i},Y_{i})\) and that the current density \(J_{i}(t)\) does not depend on \(x\) and \(y\). According to [PERSON] [1], the current densities \(J_{i}(t),\forall i\), in conductors \(\mathcal{A}_{1}\) and \(\mathcal{A}_{2}\) are a solution to the system of \(N-1\) ordinary differential equations  
\[\varrho_{i}\,J_{i}(t)-\varrho_{i+1}\,J_{i+1}(t)+\frac{\mu_{0}}{4\pi}\sum_{k=1 }^{N}\phi_{ik}\,J_{k}(t)=\delta_{i}\,U(t),\quad i=1,2,\ldots,N-1, \tag{1}\]  
and one algebraic equation  
\[\sum_{k=1}^{N}J_{k}(t)=0. \tag{2}\]  
The differentiation of a function with respect to time \(t\) is denoted by a dot over the symbol of the function. The coefficients \(\phi_{ik}\) do not depend on \(t\), and their calculation is described in the Appendix of [9]. \(J_{k}(t)(z_{2}-z_{1})\mu_{0}\,\phi_{ik}/(4\pi)\) is the contribution of the \(k\)th partial conductor to the magnetic flux through the segment (between the planes \(z=z_{1}\) and  
Figure 1: The cross sections of the conductors \(\mathcal{A}_{1}\) and \(\mathcal{A}_{2}\). \(z=z_{2}\)) of the loop formed by partial conductors \(A_{i}\) and \(A_{i+1}\). The value of the coefficient \(\delta_{i}\) on the right-hand side of Equation (1) is given by the relation  
\[\delta_{i}=\left\{ \begin{array}{rl}1&\text{if }(A_{i}\subset a(\mathscr{A}_{1})) \wedge(A_{i+1}\subset a(\mathscr{A}_{2})),\\ -1&\text{if }(A_{i}\subset a(\mathscr{A}_{2}))\wedge(A_{i+1}\subset a(\mathscr{A }_{1})),\\ 0&\text{otherwise}.\end{array} \right. \tag{3}\]  
Prior to the numerical solution to the system formed by Equations (1) and (2), Equation (2) is substituted by the equation  
\[\sum_{k=1}^{N}J_{k}(t)=0. \tag{4}\]  
The system of Equations (1) and (4) can after rearrangement be replaced by a single equation  
\[\phi\,\dot{\mathrm{J}}(t)=10^{7}\mathrm{D}+10^{7}U(t)\,\delta, \tag{5}\]  
where the matrix \(\phi\) is of the type \((N,N)\), on whose first \(N-1\) rows are the fluxes \(\phi_{ik},\forall k,\ i=1,2,\ldots,N-1\), while on the \(N\)th row are ones (coefficients on the left-hand side of Equation (4)); \(10^{7}=4\pi/\mu_{0}\); the matrices \(\mathrm{J}\), \(\mathrm{D}\) and \(\delta\) are of the type \((N,1)\). \(\mathrm{J}=[J_{i}(t)]\); \(\mathrm{D}=[D_{i}],D_{i}=\phi_{i+1}I_{i+1}-\phi_{i}I_{i}\) for \(i=1,2,\ldots,N-1,D_{N}=0\). \(\delta=[\delta_{i}]\), \(\delta_{i}\) is defined by relation (3) for \(i=1,2,\ldots,N-1\), \(\delta_{N}=0\).