Editing Classical field theory (section)

== Potential theory ==
The term "[[potential theory]]" arises from the fact that, in 19th century physics, the fundamental forces of nature were believed to be derived from [[scalar potential]]s which satisfied [[Laplace's equation]]. Poisson addressed the question of the stability of the planetary [[orbit]]s, which had already been settled by Lagrange to the first degree of approximation from the perturbation forces, and derived the [[Poisson's equation]], named after him. The general form of this equation is

<math display="block">\nabla^2 \phi = \sigma </math>

where ''σ'' is a source function (as a density, a quantity per unit volume) and ø the scalar potential to solve for.

In Newtonian gravitation, masses are the sources of the field so that field lines terminate at objects that have mass. Similarly, charges are the sources and sinks of electrostatic fields: positive charges emanate electric field lines, and field lines terminate at negative charges. These field concepts are also illustrated in the general [[divergence theorem]], specifically Gauss's law's for gravity and electricity. For the cases of time-independent gravity and electromagnetism, the fields are gradients of corresponding potentials
<math display="block">\mathbf{g} = - \nabla \phi_g \,,\quad \mathbf{E} = - \nabla \phi_e </math>
so substituting these into Gauss' law for each case obtains
<math display="block">\nabla^2 \phi_g = 4\pi G \rho_g \,, \quad \nabla^2 \phi_e = 4\pi k_e \rho_e = - {\rho_e \over \varepsilon_0}</math>

where ''ρ<sub>g</sub>'' is the [[mass density]], ''ρ<sub>e</sub>'' the [[charge density]], ''G'' the gravitational constant and ''k<sub>e</sub> = 1/4πε<sub>0</sub>'' the electric force constant.

Incidentally, this similarity arises from the similarity between [[Newton's law of gravitation]] and [[Coulomb's law]].

In the case where there is no source term (e.g. vacuum, or paired charges), these potentials obey [[Laplace's equation]]:
<math display="block">\nabla^2 \phi = 0.</math>

For a distribution of mass (or charge), the potential can be expanded in a series of [[spherical harmonics]], and the ''n''th term in the series can be viewed as a potential arising from the 2<sup>''n''</sup>-moments (see [[multipole expansion]]). For many purposes only the monopole, dipole, and quadrupole terms are needed in calculations.