My name is Alexander FufaeV and here I write about:

# What is the Poisson Equation?

## Important Formula

## What do the formula symbols mean?

## Nabla operator

`$$ \nabla $$`Unit

`$$ \frac{1}{\mathrm m} $$`

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

## Electric potential

`$$ \varphi $$`Unit

`$$ \frac{\mathrm{J}}{\mathrm{C}} $$`

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

## Space charge density

`$$ \rho $$`Unit

`$$ \frac{\mathrm{C}}{\mathrm{m}^3} $$`

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

## Vacuum Permittivity

`$$ \varepsilon_0 $$`Unit

`$$ \frac{\mathrm{As}}{\mathrm{Vm}} $$`

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.