# Transient Current Density in a Pair of Long Parallel Conductors
## 1 Introduction  
The article deals with the calculation of transient current density and transient current in a long loop formed by two long parallel solid conductors of arbitrary cross section and a voltage source. The source voltage is assumed to be dependent on time \(t\). Due to induction, the current density in the conductors is a function of \(t\) too. It is only recently that a method has been published for accurately calculating current density in a group of long parallel conductors [1]. The method has thus far been applied to the calculation of steady-state current density in a loop connected to a sinusoidal voltage source [2].  
The original contributions of the present article are as follows:  
* adjustment of the method of current density calculation proposed in [1] to a form suitable for numerical calculation of transient current density;
* presentation of results of the calculation of transient current density and transient current;
* analysis of transient current when connecting on and short-circuiting a voltage source;
* results of the calculation of the dependences of maximum current density, maximum current and the time of achieving steady state on the distance of circular cross section conductors, the frequency of sinusoidal voltage source and the resistivity of conductors.  
In the existing methods for the calculation of transient current in a pair of conductors of circular cross section, the current is not determined by transient current density. Either there is a constant current density in the conductor cross section [3] or the loop is replaced by a circuit with lumped parameters. The circuit is a series connection of a resistor with the resistance \(R\) and an inductor with the inductance \(L\). The quantities \(R\) and \(L\) are specified prior to the calculation of transient current, which is problematic [2]. Using the proposed method, it is therefore possible to solve problems that have been solved in a steady state (e.g., [4, 5, 6, 7]).  
In the calculation of current density, a quasi-stationary process is assumed and the displacement current and leakage current between conductors are neglected. The conductors do not move. The permeability of the conductors and their surroundings equals the vacuum permeability \(\mu_{0}\). In the chosen coordinate system \(xyz\), the conductors are parallel to the axis \(z\) and their cross sections do not depend on \(z\). Figure 1 gives an example of the cross sections of the conductors \(\mathcal{A}_{1}\) and \(\mathcal{A}_{2}\) in the plane \(x\), the cross sections are denoted by the same symbols as the conductors. The resistivity of the conductors is a function of \(x\) and \(y\), \(\varrho=\varrho(x,y)\). The voltage source the conductors are connected to produces a potential \(V(z,t)\) between the conductors, while the cross section of each conductor is an equipotential area. When calculating the current density, the voltage \(U(t)=V(z_{1},t)-V(z_{2},t)/(z_{2}-z_{1})\), where \(z_{2}>z_{1}\), is assumed to be specified. The current \(I(t)\) in the loop is the flux of current density vector \(\mathbf{J}\) through the cross section of one of the loop conductors. \(J_{z}(x,y)=J(x,y)\) is the only non-zero component of \(\mathbf{J}\).