Magnetic Quadrupole: Model & Vector Potential

[linford86]

Homework Statement

By analogy with an electric quadrupole, one can devise a simple model for a magnetic quadrupole
that consists of two small parallel loops with currents circulating in opposite senses and that are
separated by a small distance. Consider two magnetic dipoles of equal dipole moments +/-m0 z-hat
located at z = +/-a. In this case, the total dipole moment is zero. Show that, at large distances,
the vector potential is given approximately by $A_{\phi} = 6 \mu_0 m_0 a sin(\theta)cos(\theta)/(4 \pi r^3)$.

Homework Equations

Multipole Expansion + Taylor Expansion?

The Attempt at a Solution

I think that what I need to do is to perform a multipole expansion to get the field for a single dipole. Then, superpose the two. Finally, I think I need to Taylor expand that result. Is this the correct way of going about doing things? What do I do with the radius of each dipole? (I can't get the radius of the dipole to go away!)

 

[Thaakisfox]

Simply expand the general vector potential till the quadrupole term. You will get an integral which has to be evaluated according to the given configuration.