Poisson's Equation - Electrostatics

Electrostatics

One of the cornerstones of electrostatics is setting-up and solving problems described by the Poisson equation. Finding φ for some given f is an important practical problem, since this is the usual way to find the electric potential for a given charge distribution described by the density function.

The mathematical details behind Poisson's equation in electrostatics are as follows (SI units are used rather than Gaussian units, which are also frequently used in electromagnetism).

Starting with Gauss' law for electricity (also one of Maxwell's equations) in differential form, we have:

where is the divergence operator, D = electric displacement field, and ρ_f = free charge density (describing charges brought from outside). Assuming the medium is linear, isotropic, and homogeneous (see polarization density), we have the constitutive equation:

where ε = permittivity of the medium and E = electric field. Substituting this into Gauss' law and assuming ε is spatially constant in the region of interest obtains:

In the absence of a changing magnetic field, B, Faraday's law of induction gives:

where is the curl operator and t is time. Since the curl of the electric field is zero, it is defined by a scalar electric potential field, (see Helmholtz decomposition).

The derivation of Poisson's equation under these circumstances is straightforward. Substituting the potential gradient for the electric field

directly obtains Poisson's equation for electrostatics, which is:

Solving Poisson's equation for the potential requires knowing the charge density distribution. If the charge density is zero, then Laplace's equation results. If the charge density follows a Boltzmann distribution, then the Poisson-Boltzmann equation results. The Poisson–Boltzmann equation plays a role in the development of the Debye–Hückel theory of dilute electrolyte solutions.

The above discussion assumes that the magnetic field is not varying in time. The same Poisson equation arises even if it does vary in time, as long as the Coulomb gauge is used. In this more general context, computing φ is no longer sufficient to calculate E, since E also depends on the magnetic vector potential A, which must be independently computed. See Maxwell's equation in potential formulation for more on φ and A in Maxwell's equations and how Poisson's equation is obtained in this case.