1.Show that for  a uniformly charged sphere, all multipole moments other than  vanishes.

This is shown by orthogonality of spherical harmonics,

2. In the multipole expansion of Coulomb potential, there are three dipole terms corresponding to . Add these three terms to explicitly show that the expression for the potential is given by .

Since we are only interested in the dipole term, the general expression for the potential for this term is

We can calculate the three dipole moments as follows:

  is similarly calculated and gives,

We now calculate

Substituting these into the expression for the potential we get,

3. A charge distribution is given by

Expand the potential due to this charge density  in a multipole expansion and obtain its leading term for large distances.

We can write . The multipole moments are then given by

The integral over r gives . By orthogonality of the spherical harmonics,  only non zero moments are for .  We have,


Thus, the potential is given by