Multipole Expansion Homework: Calculate Approx. Electrostatic Potential

AI Thread Summary
The discussion focuses on calculating the approximate electrostatic potential from three charges using multipole expansion. The user expresses confusion about the necessity of multipole expansion since the charges appear as a point charge when measured far from the origin. They note that the total charge is not zero, which complicates the use of multipole expansion. The user attempts to calculate the potential using distances from the charges but finds that two charges cancel out, leaving only the negative charge's contribution. They seek guidance on rewriting the potential as a sum of individual charge potentials for a general point in space and how to proceed with the multipole expansion.
Ruddiger27
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Homework Statement



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.

Homework 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'


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.
 
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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.
 
Thanks for that.
 
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