Alexander Fufaev
My name is Alexander FufaeV and here I write about:

What is the Poisson Equation?

Important Formula

Formula: Poisson equation for electric potential (3d)
What do the formula symbols mean?

Nabla operator

Nabla operator is a differential operator whose components are partial derivatives of the spatial coordinates.

Electric potential

Electric potential that can be used to calculate the electric field.

Space charge density

The electric charge density indicates how close together electric charges are. It indicates the charge per volume.

Vacuum Permittivity

The vacuum permittivity is a physical constant that appears in equations involving electromagnetic fields. It has the following experimentally determined value: $$ \varepsilon_0 ~\approx~ 8.854 \, 187 \, 8128 ~\cdot~ 10^{-12} \, \frac{\mathrm{As}}{\mathrm{Vm}} $$

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.