Poisson Equation

If the potential is given, the electric field can be calculated by reversing the integral according to $$\vec{E} = -\vec\nabla \varphi$$ which is the gradient of the potential. The divergence of $\vec{E}$ is then given as $$\mathrm{div}\vec{E} = -\vec\nabla\cdot \vec\nabla \varphi = -\Delta\varphi = \frac{\varrho}{\varepsilon_0}$$ according to the previously found relation between the divergence of the electric field and the charge density. Here it was taken into account that the dot product of two nabla operators $\vec{\nabla}$ equals the Laplace operator $\Delta$. This equation is called the Poisson equation. Integrating this equation allows us to calculate the potential and electric field of any charge distribution. If an area is chosen, where no charge is existing, then $\varrho$ equals 0 and this equation simplifies to the so-called Laplace equation: $$\Delta \varphi = 0$$