Poisson's equation

Poisson’s equation is the PDE that links the Electric potential $V$ to the Charge density ${\rho }_v$ in a region:

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

The Laplacian ${\nabla }^2$ is applied to the scalar potential $V$; the right side is the local volume charge density divided by the Permittivity of the medium.

In a region with no charge (${\rho }_v=0$), Poisson’s equation reduces to Laplace’s equation ${\nabla }^{2}V=0$.

Derivation

Combine three relations:

1. Differential Gauss’s law: $\nabla \cdot \mathbf{D}={\rho }_v$.
2. Constitutive relation: $\mathbf{D}=ϵ\mathbf{E}$.
3. Potential definition: $\mathbf{E}=-\nabla V$.

Substitute:

$\nabla \cdot \mathbf{D}=\nabla \cdot (ϵ\mathbf{E})=-ϵ\nabla \cdot \nabla V=-ϵ{\nabla }^{2}V={\rho }_v,$

so ${\nabla }^{2}V=-{\rho }_v/ϵ$.

Why it’s the natural object to solve

Three reasons to compute $V$ first and then recover $\mathbf{E}=-\nabla V$, rather than compute $\mathbf{E}$ directly:

1. $V$ is a scalar. One equation ${\nabla }^{2}V=-{\rho }_v/ϵ$ replaces three coupled equations for the components of $\mathbf{E}$.
2. Boundary conditions are simpler. $V$ is constant on the surface of a conductor — a Dirichlet boundary condition that determines the problem cleanly. By contrast, $\mathbf{E}$ has only its normal component pinned at a conductor surface ($E_n={\rho }_s/ϵ$, where ${\rho }_s$ is initially unknown).
3. Lots of solution methods exist. Separation of variables, method of images, Green’s functions, conformal mapping in 2D, finite-element methods, etc. — all developed for the scalar Poisson equation.

Typical use cases

Charged region with known ${\rho }_v$: solve ${\nabla }^{2}V=-{\rho }_v/ϵ$ in the region, match boundary conditions, recover $V$ and $\mathbf{E}$.

Boundary-value problem with charge-free interior (e.g., capacitor): Poisson reduces to Laplace’s equation; specify $V$ on the conductors and solve.

Semiconductor device physics: charge density depends nonlinearly on potential (the Boltzmann factor for carrier concentrations); the result is a nonlinear Poisson equation that’s solved numerically.

Solutions in symmetric cases

In Cartesian, cylindrical, or spherical coordinates with appropriate symmetry, Poisson’s equation reduces to an ODE and can be integrated directly. For example, in 1D with ${\rho }_v$ depending only on $x$:

$\frac{d^{2}V}{d{x}^2}=-\frac{{\rho }_v(x)}{ϵ}.$

Integrate twice; the two constants of integration are fixed by boundary conditions.

For a uniformly charged sphere of radius $a$, charge density ${\rho }_v$ constant inside, zero outside — spherical symmetry reduces Poisson’s equation to a 1D problem in $r$ and gives the textbook result: $V$ parabolic inside the sphere, $1/r$ outside.

Integral form

The “particular solution” of Poisson’s equation in free space (with boundary at infinity, where $V\to 0$) is

$V(\mathbf{R})=\frac{1}{4\pi ϵ}\int \frac{{\rho }_v({\mathbf{R}}^{\prime })}{|\mathbf{R}-{\mathbf{R}}^{\prime }|}d{v}^{\prime }.$

This is the same expression you’d get from superposing point-charge potentials $q/(4\pi ϵR)$ over the distribution. Poisson’s equation and the Coulomb integral are two routes to the same answer; which one is easier depends on whether the geometry is dominated by the source distribution or by the boundary.

In context

Poisson’s equation extends well beyond electrostatics — same PDE shows up in Newtonian gravity (${\nabla }^2\Phi =4\pi G\rho$ for gravitational potential), in steady-state diffusion with a source term, and in stress analysis of elastic solids. The harmonic-function machinery (maximum principle, mean value property, conformal mapping) carries over directly from Laplace’s equation to the inhomogeneous Poisson case.