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.
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.
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 three-dimensional.
Moments in Cartesian Form
The ith 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 dij 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 can be used to express moments in a more elegant form. The moments are given by qlm :
where Ylm 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.