Poisson's equation

Poisson’s equation is a partial differential equation that describes a broad set of phenomena in physics. This includes in describing the electric potential of a medium:

$${\nabla }^{2}V=-\frac{{\rho }_v}{ϵ}$$

and the magnetic vector potential:

$${\nabla }^2\vec{A}=-{\mu }_0\vec{J}$$

Some key definitions:

• ${\nabla }^2$ describes the Laplacian operator.
• For the electrostatic case:
  • ${\rho }_v$ is the charge density of the medium.
  • $ϵ$ is the permittivity of the medium.
• For the magnetostatic case:
  • ${\mu }_0$ is the permeability of the medium.
  • $\vec{J}$ is the current density of the medium.

Note that if there are no charges in the medium (${\rho }_v=0$) and the medium is uniform ($ϵ$ constant), then Poisson’s equation reduces down to Laplace’s equation.

Derivation

The differential form of Gauss’ law is given by:

$$\nabla \cdot \vec{E}=\frac{{\rho }_v}{ϵ}$$

From the formula for electric potential:

$$dV=-\vec{E}\cdot d\vec{l}$$ $$dV=\nabla V\cdot d\vec{l}$$ $$\vec{E}=-\nabla V$$

Then substituting the equation for electric field in:

$$\nabla \cdot (\nabla V)=-\frac{{\rho }_v}{ϵ}$$

As given.