Quadrupole moment being tensor in electromagnetism

Question

I was reading "Lectures on Electromagnetism" by Ashok Das and he says that because the moment of a monopole is a scalar, the moment of a dipole is a vector then the quadrupole moment is a 2nd Rank, 3-Dim, symmetric traceless tensor. And that the n-th term of the multipole expansion is a n-th rank tensor, and it doesn´t make sense to me. Con someone explain why the need to use n-th range tensors?

Answer 1

You are expanding the electric potential, a scalar, so you need to end up with a scalar.

In this expansion, the vector term (dipole) is linear in position: the contribution $\mathbf{E}$ will appear coupled to the position $\mathbf{r}$ like $\mathbf{E} \cdot \mathbf{r} = E_i r_i$.

Likewise, the quadrupole moment is quadratic (like the next term in a Taylor expansion). In order to end up with a net scalar term, you have to "couple" it with two vectors now. If the terms is $\mathbf{Q}$, then the contribution will go as $\mathbf{r}^T \cdot \mathbf{Q} \cdot \mathbf{r} = r_i Q_{ij} r_j$. Which tells you $\mathbf{Q}$ is a 3-by-3 matrix. Or more generally, a rank-2 tensor.

And so on for higher terms.