Multipole Expansions

The Dipole:

Let's find the potential at point P due to these two charges.

The Quadrupole:

We now have the following setup..

We will once again assume P is very far away from the charges. It isn't shown, but the bottom line from +Q is r_a, the middle line is r, and the top line is r_b.

The General Case:

We now approach the situation in which we have N number of charges in three dimensional space.  We will use the law of cosines expression just as we did before and the binomial expansion.

Multipole Expansion in Cartesian Coordinates:

Recall that the direction cosines are..

We begin in the exact same manner as before except instead of using a binomial expansion we use a Taylor expansion. (Which is really the same thing!)

The monopole is a scalar, or a Tensor of rank 0. The dipole is a vector or a Tensor of rank 1. And the Quadrupole is a Tensor of Rank 2.