# Electrokinetic Properties of 3D-Printed Conductive Lattice Structures
## 4 [PERSON]'s Law  
The continuous-medium description of a lattice mesh obtained thus far reproduces some important electrokinetic properties of the latter. Even though the electrokinetics of a network of resistors is governed by topology much more than by geometry, it is tempting to explore the continuous-medium approach slightly further. Forgetting the discrete nature of the interconnected rods, a microscopic Ohm's law will be derived for the continuous medium that would respond to a current source distribution according to Equation (13). [PERSON]'s law is first derived in Fourier space. A non-local form of [PERSON]'s law in real space will be obtained afterward by back Fourier transform of the results obtained. Taking the Fourier transform of Equation (13) versus \(\vec{r}\) yields:  
\[V(\vec{k})=\rho L(\vec{k})S(\vec{k})\quad\text{with}\quad L(\vec{k})=\frac{1}{ 4\pi}\,\int_{IR_{3}}\frac{\eta(r)}{r}\,e^{i\vec{k}\cdot\vec{r}}\,\mathrm{d}^{3}r \tag{17}\]  
where \(V(\vec{k})\) and \(S(\vec{k})\) are the Fourier transforms of \(V(\vec{r})\) and \(s(\vec{r})\), respectively. In the stationary case, \(\vec{E}(\vec{r})=-\vec{\
abla}V(\vec{r})\) and \(s(\vec{r})=\vec{\
abla}\cdot\vec{j}(\vec{r})\). In Fourier space, \(\vec{E}(\vec{k})=-i\vec{k}V(\vec{k})\) and \(S(\vec{k})=i\vec{k}\cdot\vec{j}(\vec{k})\) with \(\vec{E}(\vec{k})\) and \(\vec{j}(\vec{k})\) the Fourier transforms of the electric field and the current density, respectively. Inserting these expressions into Equation (17) yields:  
\[\vec{E}(\vec{k})=\rho\vec{k}L(\vec{k})\vec{k}\cdot\vec{j}(\vec{k})\quad. \tag{18}\]  
The set of factors on the left-hand side of the dot product symbol represents the resistivity tensor:  
\[\mathbf{\rho}(\vec{k})=\rho L(\vec{k})\,\vec{k}\,\vec{k} \tag{19}\]  
Using the definition of \(L(\vec{k})\) (Equation (17)) and assuming as here above that \(\eta(\vec{r})\) is a function of the distance \(r\) irrespective of the direction of \(\vec{r}\) leads one to:  
\[L(\vec{k})=\frac{1}{k}\,\int_{0}^{\infty}\eta(r)\sin(kr)\,\,\mathrm{d}r=\frac{ 1}{k^{2}}\,\frac{1+5k^{2}/\alpha^{2}}{(1+k^{2}/\alpha^{2})^{3}} \tag{20}\]  
The last term follows from Equation (9) along with the mathematical model provided by Equation (12). A back-Fourier transform of Equation (18) allows one to write the electric field in real space as a convolution of the current density. The result is:  
\[\vec{E}(\vec{r})=-\frac{\rho}{4\pi}\,\int_{IR_{3}}\left[\frac{1}{3R}\frac{d^ {2}\eta}{dR^{2}}\,\mathbf{I}+\left(\frac{R^{2}}{3}\frac{d^{2}\eta}{dR^{2}}-R\frac {d\eta}{dR}+\eta\right)\,\frac{3\vec{R}\,\vec{R}-R^{2}\mathbf{I}}{R^{5}}\right] \cdot\vec{j}(\vec{r}^{\prime})\,\,\mathrm{d}^{3}r^{\prime} \tag{21}\]  
with \(\vec{R}=\vec{r}^{\prime}-\vec{r}\), \(\mathbf{I}\) being the identity tensor. This expression is the main result of the paper. It gives the macroscopic electric field at location \(\vec{r}\) due to the actual distribution of current density. In order to catch the content of the obtained non-local Ohm's law for an infinite medium, let us consider a uniform flow of \(\vec{j}(\vec{r})\) in one direction, taken as the \(z\) axis, with an intensity \(j(z)\) depending on the \(z\) coordinate only. The integral over \(\vec{r}^{\prime}\) in Equation (21) can be performed in cylindrical coordinates around the axis parallel to \(z\) passing through the local position \(\vec{r}\). It is easy to show that the electric field will be oriented along \(z\) with a component that depends on \(z\) only.