An electric multipole expansion is a determination of the voltage V due to a collection of charges obtained by performing a multipole expansion. This corresponds to a series expansion of the charge density in terms of its moments, normalized by the distance to a point R far from the charge distribution. In cgs, the electric multipole expansion is given by

where is a Legendre polynomial and is the polar angle, defined such that

In MKS, the corresponding formula has the following normalization

where is the permittivity of free space.
For example, for an electric monopole (i.e., a point charge) in cgs,, where is the 3-dimensional delta function and , so




The first term arises from , while all further terms vanish as a a result of being a polynomial in x for , giving for all .
For an electric dipole in cgs, , so the n = 0 terms vanishes. Set up the coordinate system so that measures the angle from the charge-charge line with the midpoint of this line being the origin. Then the n = 1 term is given by



Defining the dipole moment by

then gives

where

While this is the dominant term for a dipole, there are also higher-order terms in the multipole expansion that become smaller as R becomes large.
The electric quadrupole term in cgs is given by