Announcements:

New homework set due next Thursday

• I am willing to have a problem session on Wednesday to allow students to ask questions.
• Most of the parts to the problems can be found online or in text books.  Feel free to use whatever resources you wish, however, make sure you understand.  Do not just copy.

Legendre polynomials

• I place a couple of Legendre comments on my web site.  They try to review these functions and their properties.  Because they are widely used they have been studied in great length so there are recursion relationships, differential equations, proofs of orthonormality …..  Our goal is to examine the expansion of a charge distribution in terms of the Spherical Harmonics.  These functions are usually defined by identifying the Legendre polynomials along the way as a familiar set of functions.  Your goal is to believe that I can indeed write down the potential as a sum over functions.

Context:

Einstein began by realizing that the speed of light is constant.

This is a symmetry.

What does it mean that the speed of light is constant?  Can I make the same statement about the speed of a train?

The distinction between the two is that both c and vtrain are invariant if we rotate but only c is invariant if we boost.

The symmetry needs to be defined with respect to a set of transformations.

[A sphere rotated about its center remains unchanged. A sphere translated in some direction has changed.  I will be able to tell that you moved the baseball but unable to tell if you rotate it (without watching).  Sphere possesses symmetry wrt rotation but not translation.]

What is a set of transformations?

Consider rotations.  There is no need to use mathematics to understand that there are such things as rotations.  We understand that I can rotate a book.  We can take these abstract elements and describe some general properties. For example,

• any combination of rotations can be accomplished by a single rotation
• every combination of rotations remains a rotation
• there is a way of rotating back to your original orientation
• we have a notion that we can identify all the members or ways that one can rotate an object
• we can parameterize rotations by continuous parameters [rotations is an plane can be parameterized using the angle q.

We can represent all of the members of the rotation group as 3x3 matrices.  Get this clear the matrices have all of the properties of the abstract rotations.  We will want to examine how to represent physical states and the associated transformations on these states.

Einstein then went a step further with General Relativity to describe transformations of a local nature.

Weyl extended Einsteins notion by reducing the requirements and imagining that lengths could vary in a restricted but more general way than Einstein suggested.

Weyl was wrong

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

1. monopole term: charge is point like in nature
2. dipole term: charge is vector like in nature  [note three components]
3. HO terms: structure is more complicated and tensors are needed to describe the charge structure that is important.   [5 parts] …

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}

{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:

• monopole
• dipole