# Calculating Quadrupole Moment from Point Dipoles | Quick Question

• Gear.0
In summary, the question is about calculating the quadrupole moment for two point dipoles, which are essentially the same as a system of 4 charges. The equations provided for quadrupoles and dipoles do not have a direct relationship, so a different approach is needed. Treating each dipole as pairs of point charges and taking the appropriate limits may be a possible solution.
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

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

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.

Gear.0 said:
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.

## What is a quick quadrupole question?

A quick quadrupole question refers to a scientific inquiry about quadrupoles, which are four-pole magnets used in particle accelerators to focus and steer charged particles.

## What are the applications of quadrupoles?

Quadrupoles have various applications in the field of particle physics, including focusing and steering charged particles in accelerators, creating deflection fields in mass spectrometers, and controlling the beam size and shape in synchrotron radiation facilities.

Quadrupoles work by producing an electric field that alternates in direction along the length of the magnet, resulting in a focusing effect on charged particles passing through it. This is achieved by using four poles, two of which have a positive charge and two with a negative charge, arranged in a specific configuration.

## What are the different types of quadrupoles?

There are two main types of quadrupoles: electrostatic and electromagnetic. Electrostatic quadrupoles use static electric fields, while electromagnetic quadrupoles use alternating electric fields produced by oscillating currents.

## Why are quadrupoles important in particle accelerators?

Quadrupoles play a crucial role in particle accelerators by focusing and steering charged particles, allowing them to reach high energies and collide with other particles. They also help to maintain the stability of the particle beam, ensuring accurate and precise results from experiments.

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