## 4 Discussion  
### Research on Solid Wire and the Extension of Hollow Wire  
According to the integral state of [PERSON]'s equations, the current density distribution, \(J(r)\), and the corresponding skin depth, \(\delta\), on the wire section are expressed by the Bessel function [23; 31]. When studying the \(\delta\) and the \(J(r)\) of wire in the electric explosion process, a new expression related to wire structure is needed. Skin depth and current density can be expressed by new synthetic formulas respectively, as shown in Equations (5) and (6). \(r\in[-D/2,D/2]\) is the radius range,\(f\) is the characteristic frequency of an electric explosion discharge loop, \(\sigma\) is the conductivity of the Cu wire, \(\mu_{\mathrm{o}}\) is the vacuum permeability, \(\mu_{r}\) is the relative permeability, \(J_{0}\) is the theoretical total current phasor, \(D\) is the outer diameter of a wire, \(j\) is the complex unit, \(\Psi_{0}\) is the zero order of the first kind of Bessel function, and \(\Psi_{1}\) is the first order of the first kind of Bessel function. \[\delta=\sqrt{\frac{1}{\pi f\sigma\mu_{0}\mu_{r}}} \tag{5}\]  
\[J(r)=J_{0}\frac{\pi D(1-j)}{\delta}\frac{\Psi_{0}[(1-j)r/\delta]}{\Psi_{1}[(1 -j)D/2\delta]} \tag{6}\]  
According to Table 5 and Equations (5) and (6), when the energy pulse is discharged in a copper wire with a diameter of 0.5 mm, the normalized calculation results of the current density distribution in the horizontal section are shown in Figure 7a. The value of the shadow is the normalized current value, which decays inward on the radial normal. The current density at the core of the wire decays to about 0.2 times the size of its surface. The normalized current density distribution and skin depth in the vertical section of wires with different diameters are shown in Figure 7b. Here, \(m\) is the ratio of the current density at the radius, \(r\), to that at the wire's surface, \(m=J(r)/J(D/2)\), and \(\delta_{1}\), \(\delta_{2}\), and \(\delta_{3}\) are the skin depths of solid wire with three diameters. The results show that the current density at the wire core decays to 0.20-0.35 times the size of its surface, with \(\delta\) ranging from 0.136 mm to 0.147 mm. It is indicated that the non-uniform attenuation effect of the current is more pronounced in larger-diameter wires. The attenuation leads to uneven development of the electric explosion of wire. When the core of the wire is hollowed out, the current density still satisfies the integral form of Maxwell's equations. It means that the current density is still distributed as a Bessel function in the non-hollow part of the wire. The proportional relationship of current densities at different \(r\) does not change. What changes is the absolute value of the current density. The deductions of the normalized current density in the hollow wire are shown in Figure 8a,b. Interceptive current density is 0.45-0.53 times the size of its surface when the hollow ratio is 0.5. Moreover, it is 0.66-0.70 times larger when the hollow ratio is 0.7. The \(J_{a}(r)\) is the simplified expression of the current density when the hollow ratio of wire is \(a\). It is defined as \(J_{0}(r)\), \(J_{0.5}(r)\), and \(J_{0.7}(r)\) when \(a\) is 0, 0.5, and 0.7, respectively. The relation between \(J_{a}(r)\) satisfies Equation (7). \[I=\int_{0}^{0.5D}J_{0}(r)2\pi rdr=\int_{0.25D}^{0.5D}J_{0.5}(r)2\pi rdr=\int_{0. 35D}^{0.5D}J_{0.7}(r)2\pi rdr \tag{7}\]  
Due to the integral interval variation, the hollow wire's current density is amplified in proportion to coefficient, \(K_{a}\), when the hollow ratio of wire is \(a\). The current density at this point is rewritten as \(J_{a}(r)=K_{a}V_{0}(r/\delta)\). The integral of current density in Equation (7)  
Figure 8: The equivalent current density distribution on the vertical section of three diameter wires. (**a**) With hollow ratio of 0.5. (**b**) With hollow ratio of 0.7. Figure 7: Normalized current density distribution of solid wire. (**a**) \(D=0.5\) mm Cu wire in horizontal section.