# Transient Current Density in a Pair of Long Parallel Conductors
## 3 Results and Discussion
### Discussion  
[PERSON] proposed a method for the calculation of current density in a solitary conductor [15]. Hundreds of works have since been published (more recently, e.g., [16; 17; 18; 19; 20; 21; 22; 23; 24]) that deal with the calculation of current density in one or two conductors, and exceptionally in more conductors. The calculation of current density in one conductor is accurate, but it can only be applied to the solitary conductor connected to the ideal current source, which, however, cannot be implemented [1]. The published methods, with the exception of those in [1; 3; 24], are approximate, with the current density error unknown. The self-inductance of two conductors cannot be calculated without accurate current density [2] and thus approximate formulae continue to be used for the calculation of self-inductance of two coaxial conductors and two circular cross section conductors [2]. According to these formulae, inductance does not depend on conductor resistivity and voltage source frequency, in contrast to the inductance calculated using accurate current density. The subject of the present article is transient current in two solid long parallel conductors (conductors in the following) connected to the voltage source. A definite contribution of the article is that current is determined using transient current density. In the article, a numerical method is proposed and used for solving a system of differential equations for the calculation of transient current density. The present method of calculating transient current, not only in the conductors considered in the present article, consists in calculating transient current in the circuit \(\mathcal{C}\) with lumped elements, which is a series connection of  
Figure 8: Example 4, \(d=16\) mm, \(f=60\) Hz, \(\Omega=1\) V\(\cdot\)m\({}^{-1}\): Approximation of the voltage \(U(t)\) by \(n\) terms of the series (12) for \(n=5\) (solid line) and \(n=20\) (thin line). The current \(I\) in the conductor \(\mathcal{A}_{1}\) when approximating the voltage by ten terms of the series (12). a resistor (with resistance \(R\)), an inductor (with inductance \(L\)) and a voltage source (with voltage \(U\)). For the circuit \(\mathcal{C}\), [PERSON]'s voltage law holds  
\[L\dot{I}(t)+R\,I(t)=U(t). \tag{13}\]  
Dividing both sides in (13) by \(L\) gives the equation  
\[\dot{I}(t)+\frac{R}{L}\,I(t)=\frac{U(t)}{L}, \tag{14}\]  
which is an ordinary linear inhomogeneous differential equation. According to [PERSON] and [PERSON] [10], there exists just one solution \(I(t)\) to Equation (14) in the interval \(0\leq t<\infty\) that satisfies the given initial condition \(I(0)=I_{0}\)  
\[I(t)=\exp\left(-\frac{R}{L}t\right)\left[C+\int\frac{U(t)}{L}\exp\left(\frac{ R}{L}t\right)\mathrm{d}t\right], \tag{15}\]  
where the constant \(C\) is determined by the initial condition. For the source of sinusoidal voltage \(U(t)=\tilde{U}\sin(\omega t+\alpha)\), the transient current is  
\[I(t)=K\left[\frac{R}{L}\sin(\omega t+\alpha)-\omega\cos(\omega t+\alpha) \right]+C\exp\left(-\frac{R}{L}t\right), \tag{16}\]  
\[K=\tilde{U}\left[\frac{R^{2}}{L}+L\omega^{2}\right]^{-1}.\]  
For the initial condition \(I(0)=I_{0}\), it holds that  
\[C=I_{0}-K\left(\frac{R}{L}\sin\alpha-\omega\cos\alpha\right).\]  
Let us consider the Al conductors \(\mathcal{A}_{1}\) and \(\mathcal{A}_{2}\) at a temperature of 300 K, \(\varrho=2.733\times 10^{-8}\Omega\cdot\)m [13]. The cross section of the two conductors is a circle the radius \(r=8\) mm. The cross sections \(\mathcal{A}_{1}\) and \(\mathcal{A}_{2}\) have their centres at points \((0,0)\) and \((d,0)\), respectively, \(d=2r\). At the instant \(t=0\), the conductors are connected to the source of sinusoidal voltage \(U(t)=\tilde{U}\sin\omega t\), \(\tilde{U}=1\) V\(\cdot\)m\({}^{-1}\), \(f=10^{5}\) Hz. The inductance \(L\) of the conductor pair can be established using the formula [2,15]  
\[L_{\text{lit}}=\frac{\mu_{0}}{\pi}\left[\frac{1}{4}+\ln\frac{d-r}{r}\right].\]  
According to this formula, \(L_{\text{lit}}=10^{-7}\) H\(\cdot\)m\({}^{-1}\).