Electromagnetic
Fields and Matter
A force is
exerted on an electric charge by an electric field. Positive charges
accelerate in the direction of the applied field while negative charges
accelerate opposite to the direction of the electric field. Charge
currents, on the other hand, deflect perpendicular to both the
direction of the magnetic field and perpendicular to the flow of current.
Since electric forces tend to be larger than magnetic
forces, we will ignore magnetic fields for now.
Dipole Moment
As shown below (Figure (a)),
a uniform electric field causes the positive and negative charges to
separate. The equilibrium distance, d, between the charges is
determined by the competition between the applied electric field, which
pulls the two charges apart, and the active force between the positive
and negative charge.
As shown in Figure (b), the
dipole moment, p, is a vector defined as the product
of the magnitude of each charge and distance between the two charges.
The dipole moment is a vector that points from positive to negative
charge and is represented by the black arrow.
Quadrupole Moment
The electric field of a plane wave at one snapshot in
time is illustrated in the left hand portion of the figure below. Due
to the sinusoidal spatial variation of the electric field, the dipole
moment induced at one point in space is oriented in the opposite
direction to the adjacent dipole moment, as shown in the right part of
the figure.
Recall that the electric dipole moment is a
property of two opposite charges that are spatially separated.
Analogously, a quadrupole moment is the property of a pair of dipoles
that are spatially separated and point in opposite directions.
Quadrupole moments are induced when the
electric field is different in two points within a material. We call
such fields inhomogeneous. Thus, dipoles are induced in a uniform
electric field while quadrupole are formed in a field gradient.
Higher Order Moments
One possible arrangement for an octupole and dodecapole
moment are shown in the figure below. To create the next higher moment,
one continues the pattern of adding another dipole and arranging them
with equal angles. There are other patterns that give higher order
moments and many of these are threedimensional.

Moments in Cartesian Form
The i^{th} Cartesian component of
the dipole moment is expressed as an integral over the charge density, r:
Given the point charges below, the above
integral can be evaluated to yield the expected dipole moment.
Three parameters can be used to describe a
dipole, namely its three cartesian components.
The quadrupole moment is given by
where d_{ij}
is the Kronecker delta. Below is an example of the calculation of the
quadrupole moment for four point charges:
While it may not be obvious, a total of 5
parameters describes a quadrupole.
Spherical Tensors
Spherical tensors can be used to express
moments in a more elegant form. The moments are given by q_{lm}
:
where Y_{lm} are the spherical
harmonics and r the radial variable. l =1 for
the dipole moment, and m can have the values (1,0,1). The quadrupole
moment is given by l =2, with m having values (2,1,0,1,2).
For the moment given by l, there are 2l+1
parameters describing that moment.
For the dipole, the cartesian form is
related to the spherical matrix form by,
Arbitrary Charge Distribution
An arbitrary charge distribution can be
described by its moments. Thus, it is equivalent to describe a
distribution of charge by either its charge density at all points in
space, or by the collection of all moments.
There are many situations in which a charge
distribution is well approximated by a small number of moments. In
these cases, a broad range of phenomena can be described in terms of a
few parameters. For example, light scattering from molecules and atoms
are well described by the dipole approximation.
