My name is Alexander FufaeV and here I write about:
What is the Poisson Equation?
Important Formula
$$\nabla^2 \, \varphi ~=~ - \frac{\rho}{\varepsilon_0}$$
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.