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Multipole Expansion

  1. Mar 11, 2007 #1
    1. The problem statement, all variables and given/known data

    I have to calculate the approximate electrostatic potential far from the origin for the following arrangement of three charges: +q at (0,0,a), -q at (0,a,0) and (0,-a,0). I have to give the final answer in spherical coordinates and keep the first two non-zero terms in the multipole expansion.

    2. Relevant equations

    So the equation for the multipole expansion is:

    V(r) = (1/4piEo)SUM(1/(r^(n+1)))*int(r')^n*Pn*cos(theta)*pho(r')dr'

    3. The attempt at a solution

    My main problem here is that I don't see the need for a multipole expansion, since we're taking the measurement far from the origin the charges will appear as a point charge. Also I'm pretty sure the total charge has to be zero to use the multipole expansion, but the total charge isn't zero here.
    What I've got is a triangle on the z-y axes, with a point on the x-axis where I'm measuring the potential. I've got

    V(x,y,z)= (1/4*piEo)(q/L - q/D - q/S), where L, D and S are the distances from the charges to the point of observation, and I've drawn them such that L=D=S= sqrt(a^2 + x^2)

    This doesn't really work, since two of the charges cancel out and leave one of the negative charges as the sole contributor to the potential. Also I think I've missed the point of the question by placing the point of observation on the x-axis.
    I think I should rewrite the integral for the potential for the multipole expansion as a point charge distribution, but I'm not sure how to go about that.
  2. jcsd
  3. Mar 11, 2007 #2

    Dr Transport

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    Science Advisor
    Gold Member

    First of all, write the poential as a sum of potentials due to the individual charges for a general point in space, then write the resulting potential terms in an expansion like you alluded to.
  4. Mar 13, 2007 #3
    Thanks for that.
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