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Poisson's Equation

Describes the relationship between the electric potential and the electric field: ∇ ⋅ E = ρ/ε₀, where ρ is charge density and ε₀ is the permittivity of free space.
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The statement of the theorem

In a domain Ω⊂R3\Omega \subset \mathbb{R}^3 where the electric potential ϕ(r⃗)\phi(\vec{r}) is defined, and assuming the medium is linear and isotropic, the governing relationship between the potential and the charge density ρ(r⃗)\rho(\vec{r}) is established by the Poisson equation. This equation is derived from Gauss's Law in differential form (∇⋅E⃗=ρ/ϵ0\nabla \cdot \vec{E} = \rho/\epsilon_0) and the definition of the electric field (E⃗=−∇ϕ\vec{E} = -\nabla \phi). The resulting partial differential equation is:\n\n∇2ϕ(r⃗)=−1ϵ0ρ(r⃗)\nabla^2 \phi(\vec{r}) = -\frac{1}{\epsilon_0} \rho(\vec{r})\n\nwhere ∇2\nabla^2 is the Laplacian operator, defined in Cartesian coordinates as ∇2=∂2∂x2+∂2∂y2+∂2∂z2\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}. The equation holds subject to appropriate boundary conditions on the boundary ∂Ω\partial \Omega, such as Dirichlet (ϕ∣∂Ω=f\phi|\partial \Omega = f) or Neumann (∂ϕ∂n∣∂Ω=g\frac{\partial \phi}{\partial n}|\partial \Omega = g). Here, ϵ0\epsilon_0 is the permittivity of free space, and ρ(r⃗)\rho(\vec{r}) is the volume charge density.
Source: Wikipedia