# Electrokinetic Properties of 3D-Printed Conductive Lattice Structures
## 4 [PERSON]'s Law
Working as for Equation (15), one is led to:  
\[E_{z}(z)=-\frac{\rho}{4\pi}2\pi\int_{-\infty}^{+\infty}\,\,\mathrm{d}z^{ \prime}j(z^{\prime})\int_{|z^{\prime}-z|}^{+\infty}\,\,\mathrm{d}R\left[\frac{ 1}{R}\frac{d\eta}{dR}-\frac{\eta}{R^{2}}+(z^{\prime}-z)^{2}\left(\frac{1}{R^{2} }\frac{d^{2}\eta}{dR^{2}}-\frac{3}{R^{3}}\frac{d\eta}{dR}+\frac{3\eta}{R^{4}} \right)\right] \tag{22}\]The integration over the variable \(R\) is readily carried out to give the simple analytical result:  
\[E_{z}(z)=\frac{\rho}{2}\int\int_{-\infty}^{+\infty}\eta^{\prime}(|z^{\prime}-z|)j (z^{\prime})\,\mathrm{d}z^{\prime} \tag{23}\]  
where \(\eta^{\prime}\) is the derivative of the function \(\eta\) with respect to its argument. As a simple illustration, the current density is supposed to be a constant \(j\) in the half space \(z>0\) and zero in the half space \(z<0\). This situation would be realized if there were a current source uniformly distributed on the plane \(z=0\) and a current drain localized on a parallel plane at \(z\to\infty\). Taking into account the properties \(\eta(0)=0\) and \(lim_{z\to\infty}\eta(z)=1\), one obtains:  
\[E_{z}(z) = \frac{\rho j}{2}[1+\eta(z)]\text{ for }z>0 \tag{24}\] \[E_{z}(z) = \frac{\rho j}{2}[1-\eta(-z)]\text{ for }z<0\quad. \tag{25}\]  
As shown in Figure 4, the electric field rapidly converges to \(\rho j\) in the half space that sustains the current and rapidly vanishes in the half space where there is no current. Still, the transition at the interface \(z=0\) is not infinitely sharp, but extends over a region of width \(5\alpha^{-1}\). The minima and maxima of the electric field are due to the maximum that the correction function \(\eta\) must have, as discussed above. With the simple analytical representation used here (Equation (12)), the extrema of \(E_{z}(z)\) take place at \(z=\pm(1+\sqrt{3})\alpha^{-1}\). This value can be considered as a penetration depth of the electric field in a region where the current density is zero. It can be useful to recast Equation (21) in the following form:  
\[ \begin{array}{rcl}\vec{E}(\vec{r})&=&-\frac{\rho}{4\pi}\int_{R_{3}}\frac{3 \bar{R}\,\bar{R}-R^{2}\mathbf{I}}{R^{3}}\cdot\vec{j}(\vec{r}^{\prime})\,\,\mathrm{ d}^{3}r^{\prime}-\\ &&\frac{\rho}{4\pi}\int_{R_{3}}\left[\frac{1}{3\bar{R}}\frac{d\mathbf{I}}{R^{2}} \mathbf{I}+\left(\frac{R^{2}}{3}\frac{d^{2}\eta}{d\mathbf{R}^{2}}-R\frac{d\eta}{d\mathbf{ R}}+\eta-1\right)\,\frac{3\bar{R}\,\bar{R}-R^{2}\mathbf{I}}{R^{3}}\right]\cdot\vec{j}( \vec{r}^{\prime})\,\,\mathrm{d}^{3}r^{\prime}\end{array}  \tag{26}\]  
after the first term, which corresponds to a normal 3D resistive medium, has been extracted. If the right-hand side were limited to this sole first term, the equation obtained would be equivalent to \(\vec{\
abla}\times\vec{\
abla}\times\vec{E}=0\) after \(\vec{j}\) were replaced by \(\sigma E\), with \(\sigma=1/\rho\). The second part of Equation (26) is a correction to Ohm's law coming from the discrete structure of the resistive mesh. The correction term is still a convolution product, like Equation (21), now with a kernel that samples a narrow region around the position \(\vec{r}\), with radius of the order of \(1/\alpha\). If the current density varies slowly with \(\vec{r}^{\prime}\) on the scale  
Figure 4: Variation of the electric field predicted by Equation (21) in the effective medium representing a 3D conductive mesh. The upper half space \(z>0\) is crossed by a uniform current of density \(j\) flowing in the \(z\) direction. There is no current in the lower half space \(z<0\). \(1/\alpha\), it can be expended in a power series of \(\vec{r}^{\prime}-\vec{r}\) restricted to its first few terms. Each term can be integrated over \(\vec{R}\). Carrying on this development is easy in Fourier space, starting from Equation (20). Under the condition \(k\ll\alpha\), one gets \(L(\vec{k})\approx(1+2k^{2}/\alpha^{2})/k^{2}\) and \(\rho(\vec{k})\approx\rho\vec{k}\vec{k}/k^{2}+2\rho\vec{k}\vec{k}/\alpha^{2}\).