Electric potential for an axial quadrupole

[Telemachus]

Find the electric potential for an axial quadrupole: point charges q, -2q, q over the z axis at distances l,0,l from the origin. Find the electric potential only for distances r>>l and demonstrate that the potential is proportional to one of the zonal armonics.

Well, I found at wikipedia that an axial multipole has an electric potential given by:
$$\Phi(r)=\frac{1}{4\pi \epsilon_0 r}\sum_{k=0}^{\infty}qa^k \left ( \frac{1}{r^{k+1}} \right ) P_k(\cos\theta)$$

But I don't know how to apply this to my problem. I don't know neither what the zonal armonics are.

 

[kuruman]

Zonal harmonics are functions of the form

φ_n=rⁿP_n(cosθ) or φ_n=r^-(n+1)P_n(cosθ) where P_n(cosθ) are Legendre polynomials.

In relation to your problem, first find the potential using simple superposition of three point charges, then expand in Taylor series for r>>l. You should get something that matches the functional dependence of one of the two forms. Can you predict which one?

 

[Telemachus]

Thanks.