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Multipole expansion

  1. Oct 29, 2012 #1
    1. The problem statement, all variables and given/known data

    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.


    2. Relevant equations


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


    3. The attempt at a solution

    So far I have worked that the monople and dipole moment is zero. So i realise 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.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Oct 29, 2012 #2

    mfb

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    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:
    641f2fa691959ec8094e7d055345c686.png
     
  4. Oct 29, 2012 #3
    I still don't properly understand? What do you mean every pair (i,j)?
     
  5. Oct 29, 2012 #4

    mfb

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    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.
     
  6. Feb 27, 2013 #5
    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.
     
  7. Feb 27, 2013 #6

    mfb

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    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).
     
  8. Feb 27, 2013 #7
    Perfect, thank you VERY MUCH!
     
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