Calculating Quadrupole Moment from Point Dipoles | Quick Question

[Gear.0]

Homework Statement

This isn't directly a homework question, but a response will really help me on my homework.

How does one calculate the quadrupole moment, when given two point dipole moments?
I would know how to proceed had I been given a system of 4 charges, which is essentially the same thing, but I need to know how to make this calculation if we are only given two point dipoles.

So basically, from the equations below, I just need to know how to find at least Q2, when given two point dipoles that are the same, but rotated 180degrees from each other.

Homework Equations

Quadrupoles:
$$V = \frac{1}{4\pi\epsilon_{0}}\left ( \frac{\hat{\textbf{r}}\cdot Q_{2}\cdot \hat{\textbf{r}}}{r^{3}} \right )$$
$$Q_{2} = \sum_{k=1}^{N}\frac{q_{k}}{2}\left ( 3\textbf{x}_{k}\textbf{x}_{k} - r_{k}^{2}\textbf{I} \right )$$

Dipoles:
$$V = \frac{1}{4\pi\epsilon_{0}}\left ( \frac{\hat{\textbf{r}}\cdot \textbf{p}}{r^{2}} \right )$$
$$\sum_{k=1}^{N}q_{k}\textbf{x}_{k}$$

The Attempt at a Solution

I'm just trying to see from the quadrupole equations and the dipole equations if there is a way to write the quadrupole term as a function of dipole terms, but I don't see a way so far.

 

[gabbagabbahey]

I'm just trying to see from the quadrupole equations and the dipole equations if there is a way to write the quadrupole term as a function of dipole terms, but I don't see a way so far.

I'm not aware of any formula for directly calculating multipole moments of a distribution of ideal dipoles. Ideal (point) dipoles are very special limiting cases point charge distributions ( two equal and opposite point charges in the limit that the distance between them goes to zero, but the product of the charge <of the positive point charge> and the displacement between the charges remains constant. I would suggest treating each of your dipoles as pairs of equal and opposite point charges separated by some small displacement $\textbf{d}_i$, calculate the quadrapole moment of this physical charge distribution, and then take the same limit(s) you would to convert your physical dipoles to ideal ones.