We introduced multipole expansions and we worked out a few problems using spherical harmonics.

We started our discussion from a typical problem in which one was asked to identify the first non vanishing multipoles of some distribution of charge, and we then moved on to define things more formally. In particular we pointed out how the definition of multipoles depends on the origin of axes, and this is why one can always choose the system of axes that brings to the most simple solution. We highlighted how to obtain the electric field cartesian components starting from the potential in spherical coordinates. We also went over the homework solutions in class. Again we had animated discussions (please do not hesitate to let me know, possibly during my office hours, how you feel about spending time for discussions and questions in class). We are going to cover a few math concepts, tensors, distribution functions etc.

We covered topics in Ch.4 of Jackson.

1) A long cylindrical rod is placed in a uniform field E. The rod has radius "a" and dielectric constant epsilon. The rod lies perpendicular to the electric field. Find the potential inside and outside the rod.
(Repeat using cylindrical coordinates)

2) Find the potential at any point in space for the distribution:
&rho = q/a (cos &phi - sin 2 &phi)
up to order 1/r³.
What geometrical shape does the given density correspond to?

3) A sphere of dielectric material has radius R and a constant polarization P directed along the z-axis.
a) What are the volume and surface charge densities?
b) By direct integration over the surface charge density, find an expression for the potential at a point "z" (on the z axis) with z>R is: