data/examples/0.md

The potential $V_{i}$ of cell $\mathcal{C}_ {j}$ centred at position $\mathbf{r}_ {i}$ is related to the surface charge densities $\sigma_ {j}$ of cells $\mathcal{E}_ {j}$ $j\in[1,N]$ through the superposition principle as:

$$V_i=\sum _{j=0}^N\frac{{\sigma }_j}{4\pi {\epsilon }_0}{\int }_{{\mathcal{E}}_j}\frac{1}{\mid {\mathbf{r}}_i-{\mathbf{r}}^{\mathrm{\prime }}\mid }{\mathrm{d}}^2{\mathbf{r}}^{\mathrm{\prime }}=\sum _{j=0}^N{Q}_{ij}{\sigma }_j,V\_\; \{i\}\backslash ,=\backslash ,\backslash sum\_\; \{j=0\}^\{N\}\backslash ,\backslash frac\{\backslash sigma\_\; \{j\}\}\{4\backslash pi\backslash varepsilon\_\; \{0\}\}\backslash ,\backslash int\_\{\backslash mathcal\{E\}\_\; \{j\}\}\backslash frac\{1\}\{\backslash left|\backslash mathbf\{r\}\_\; \{i\}-\backslash mathbf\{r\}^\{\backslash prime\}\backslash right|\}\backslash ,\backslash mathrm\{d\}^\{2\}\backslash mathbf\{r\}^\{\backslash prime\}\backslash ,=\backslash ,\backslash sum\_\{j=0\}^\{N\}\backslash ,Q\_\; \{ij\}\backslash ,\backslash sigma\_\{j\},$$

where the integral over the surface of cell $\mathcal{C}_ {j}$ only depends on $\mathcal{C}{j}$ shape and on the relative position of the target point $\mathbf{r}_ {i}$ with respect to $\mathcal{C}_ {j}$ location, as $\sigma_ {j}$ is assumed constant over the whole surface of cell $\mathcal{C}_ {j}$.