The Green's function

In two dimensional problems, determining the scalar potential from the charge density is relatively straightforward. The Green's function is essentially equal to the logarithm of the distance from the charge.

$$G _{ \rho } \left( \mathbf{r _{1}},\mathbf{r _{2}} \right) =... ...}} ln \left\vert \mathbf{r _{1}} - \mathbf{r _{2}} \right\vert$$ (1)

The scalar potential at a point is then determined from the sum of all the charges multiplied by their respective values of the Green's function.