Multipole Expansion: Quadrupole Moment Calculation

[Josephk1508]

Homework Statement

Four point charges: q at a^z; q at -a^z; -q at a^y and -q at -a^y
where ^z and ^y are the unit vectors along the z and y axes.

Homework Equations

Find the approximate expression (i.e. calculate the first non-zero term in the multipole expansion) for the electrostatic potential at large distances.

The Attempt at a Solution

So far I have worked that the monople and dipole moment is zero. So i realize I need to move onto the quadrupole moment. Which is what I'm struggling with.

The formula is Qij= Ʃ(i) qi (3RiRj-R^2(δij))

Any help would be appreciated.

 

[mfb]

You can simply calculate the sum for every pair (i,j). The result is a matrix with 9 components.

However, you should not use the index i for a coordinate and the charge index at the same time. Wikipedia has a better version:

 

[Josephk1508]

I still don't properly understand? What do you mean every pair (i,j)?

 

[mfb]

Calculate Q₁₁: simply insert i=1 and j=1 everywhere.
When you are done, calculate Q₁₂, and Q₁₃, and so on.

Q₂₃ as example:
$$Q_{23}=\sum_l q_l (3r_{2l}r_{3l}-r_l^2 \delta_{23})$$
$\delta_{23}=0$, therefore
$$Q_{23}=q(3*0*a-a^2*0) + q(3*0*(-a)-a^2*0) - q(3*a*0-a^2*0) - q(3*(-a)*0-a^2*0)=0$$

It is not necessary/useful to write PMs, I see this thread in the subscribed threads.

 

[sindarintech]

You can simply calculate the sum for every pair (i,j). The result is a matrix with 9 components.

Can you speak more about what the r_il and r_jl elements map to, using the coordinates used in this problem? Specifically, what is 0 and what is a/-a? I'm having a brain-cramp about it... and I know it's something straightforward.

 

[mfb]

Those are values taken from the initial problem. a is part of coordinates, and 0 is just zero.
l is the index of the charges, i and j are numbers for coordinates (x=1, y=2, z=3).

 

[sindarintech]

Perfect, thank you VERY MUCH!