Consider a general charge distribution
It is very difficult, in general, to find the electric and magnetic fields a point _{} due to a charge distribution _{}
Why is this problem so difficult ?
So physicists ask if there are situations that make the problem more tractable.
If _{} is large compared to _{} [very localized charge distribution at a significant distance from the location where the fields are needed ] one might expect that the details as to how the charge is distributed are unimportant. The dominant feature will be the total charge.
This point of view is mathematically justified by expanding the complete solution in terms of _{} the distance between a point in the charge distribution and the location where the fields need to be evaluated. The problem then separates into a set of terms where each term is characterized by a specific d dependence and a characterization of the structure of the charge distribution as a multipole.
The details of this procedure can be found in most E&M text books and on the web. For example the following sites describe this expansion nicely.
http://en.wikipedia.org/wiki/Spherical_multipole_moments |
http://en.wikipedia.org/wiki/Multipole_expansion |
multipole moments are defined in
the above references:
let the charge distribution be
defined by primed cooridantes
charge distribution |
_{} |
field point |
_{} |
_{}
_{}
Usually the student is introduced to this procedure based on the problem expressed in Cartesian coordinates. The more revealing expansion is done in terms of spherical coordinates because the moments can be classified by their transformation properties under rotations.
How would one describe the monopole moment in terms of rotational properties?
{independent è a rotation will not change the monopole moment}
How about the dipole?
{vector è a rotation will mix the three components}
Originally a vector was introduced as a quantity with magnitude and direction. This type of entity was important because so many physics concepts force, position … possess this structure. However another natural way to introduce a vector is to characterize by its transformation characteristics under rotations. There are objects that transform as scalers, vectors, tensors under rotation.
To return to the problem of the charge distribution we discovered that we could break a general charge distribution into parts:
Under rotations the form of any element sill change but not mix.
Consider the monopole as a point it doesn’t change under rotation.
Consider the dipole as a vector. It rotates by pointing in a different direction. The new dipole is simply a transformation from the old dipole only.
This is not a proof but a statement of how these objects transform. The convenient separation can be illustrated by the following:
_{}
Above we have simply written all the elements of the charge distribution as a column matrix. The length of course goes to infinity.
Because of our choice of elements the rotation matrix takes on a special form.
_{}
The general matrix reduces to a block diagonal form where much smaller matrices can be used 1, r, z, _{}. If you can find those ways to break up a general structure such that the transformations on a general structure can be reduced because the elements in the structure don’t mix. Then you call the general structure reducible. So the next step is to separate the elements.
_{}
If the resulting matrices 1, r, z, _{}can not be further reduces then we say that each of the above equations represents a way for the group to behave. On the monopole all the group elements (ie all possible rotations) just behave as the identity element. For the dipole there are a set of 3x3 matrices and each rotation in the group has an associated matrix. For the quadrupole we need to specifically state that we choose the spherical coordinate expansion where the quadrupole has 5 elements. So now each rotation can be reprepresented as a 5x5 matrix. In theoretical terms we have found the irreducible representations of the rotation group. Along with finding the matrices we have found a set of structures (monopole, dipole, quad….).
I turns out that these transformations have a convenient mathematical form.
_{}
For transformations on a real space we can drop the imaginary number.
_{}
[http://en.wikipedia.org/wiki/Euler's_rotation_theorem]
_{}
Start with a rotation in 2-d. Imagine that it is a rotation about the z axis by _{}.
_{}
Choose the matrix J_{z}_{
}as shown below. We can then test to see if indeed the above matrix can be
obtained using the exponential form and the
_{}
_{}
_{}
_{}
So for a simple rotation about the z-axis we see that one can define a matrix J that can be used build a rotation about the z axis.
That is there is an operator _{} and a set of
parameters _{}that can be used to carry out any rotation. To make sense of the above equation we simply
use a
The interesting fact is that using the operator J which can be used to generate a rotation has separate the rotation into two convenient parts. A set of parameters and an independent operation.
The structures that we have uncovered can be labeled in several different ways. For our example we could state the size of the matrix needed to perform the rotation. Monopole=1, dipole=3… One can also use the generator J to label the state. In QM this operator is the angular momentum operator. The structures then possess and intrinsic angular momentum. In particle physics we build elementary particles from the irreducible elements of the rotation group and label them by the their angular momentum which is defined as J the generator of the rotations.