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# What is the Poisson Equation?

## Important Formula

What do the formula symbols mean?

## Nabla operator

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

## Electric potential

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

## Space charge density

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

## Vacuum Permittivity

Unit
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.