For a charge distribution located near the origin we have
with
.
Here Q = total charge, p = dipole moment, Qij = quadrupole moment tensor of the charge distribution. If the problem has rotational symmetry about the z-axis, such that Qxx=Qyy=-½Qzz, then Qzz is called the quadrupole moment.
In spherical coordinates we have
,
with
.
ql-m=(-1)mqlm*
The qlm are called the multipole moments.
Relationship between Cartesian and spherical multipoles:
The energy of a charge distribution near the origin in an external field is given by
.
This expansion shows how the various multipoles interact with the external field.
 | Consider a charge q at  .
| ||||
| Consider a line charge l parallel to the x-axis
at  .
| ||||
| Assume the z = 0 plane is a plane interface between
two dielectrics.
|
Let 
in spherical
coordinates. Then
.
Let 
in cylindrical coordinates. Then
.
When solving electrostatic problems, we often rely on the uniqueness theorem. If f or its normal derivatives are specified at the boundaries of a volume V, then a unique solution exists for f inside V.
.
.
on the z-axis makes the sphere an equipotential with f=0. Add q’’ at the center of the sphere to
change the potential or the total charge on the sphere.
parallel to the x-axis makes the sphere an equipotential with
. Add V0
everywhere to change the potential of the cylinder.
in e1. Then
placing an image charge 
at
d’=-d on the z-axis in a medium with e1
gives the potential and field in dielectric 1. And placing an image charge 
at
d’’=d on the z-axis
in a medium with e2 gives the
potential and field in dielectric 2.