jd7h/texify 🖼️✓ → 📝

The potential $V_i$ of cell $\mathcal{C}_i$ centred at position $\mathbf{r}i$ is related to the surface charge densities $\sigma_j$ of cells $\mathcal{C}j$ $j\in[1,N]$ through the superposition principle as: $$V_i,=,\sum{j=0}^{N},\frac{\sigma_j}{4\pi\varepsilon_0},\int{\mathcal{C}j}\frac{1}{|\mathbf{r}i-\mathbf{r}^{\prime}|},\mathrm{d}^2\mathbf{r}^{\prime},=,\sum{j=0}^{N},Q{ij},\sigma_j,$$ 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 $\mathcal{C}_j$.