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:

$$V_ {i}\,=\,\sum_ {j=0}^{N}\,\frac{\sigma_ {j}}{4\pi\varepsilon_ {0}}\,\int_{\mathcal{E}_ {j}}\frac{1}{\left|\mathbf{r}_ {i}-\mathbf{r}^{\prime}\right|}\,\mathrm{d}^{2}\mathbf{r}^{\prime}\,=\,\sum_{j=0}^{N}\,Q_ {ij}\,\sigma_{j},$$

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 $\mathcal{C}_ {j}$.