What is the Poisson Equation?

The Poisson equation looks like this: $$ \nabla^2 \, \varphi ~=~ - \frac{\rho}{\varepsilon_0} $$

The Poisson equation is a differential equation that relates the electrostatic potential \(\varphi\) to the charge density \(\rho\). With a known charge density \(\rho\), the electric field \(\boldsymbol{E}\) can be determined by integrating the Poisson equation, using \( \boldsymbol{E} = - \nabla \, \varphi \). The boundary conditions of the respective problem determine the integration constants.

Or: If the potential \(\varphi\) is known, the charge density is calculated by differentiating it twice.