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 _{}
The potential at _{}for a point charge located at _{}is
_{}
For a charge distribution defined
by primed coordinates
charge distribution |
_{} |
field point |
_{} |
_{}
_{}
Usually the student is introduced to this procedure in E&M based on the above problem but 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 will 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….).