Multipole expansion of polarized cylinder

[phys-student]

Homework Statement

I need to calculate the electric field on the midplane of a uniformly polarized cylinder at a large distance from the center of the cylinder. The question also says that because the distance is large compared to the radius the dipole dominates the multipole expansion.

Homework Equations

V_dip=(1/4πε₀)(1/r²)∫r'cosαρ(r')dτ'
V(r)=(1/4πε₀)∑(1/rⁿ⁺¹)∫(r')ⁿP_n(cosα)ρ(r')dτ'

The Attempt at a Solution

The polarized cylinder only has charge bound on the top and bottom surfaces and I tried to do the multipole expansion for each disc separately using the 2nd equation and then add the resulting potentials together to get the total potential so I could find the electric field by taking the gradient. However the 2 discs have the same geometry and opposite charges so I ended up getting 0 total potential and then I can't find the electric field. What should I do?. I also tried using the first equation for the dipole potential but ended up with 0 again.

 

[marcusl]

Do you know that you're required to perform a multipole expansion? Since you are told that the dipole term dominates, I would think not. Just calculate the effective dipole moment, and put it into the equation for the electric field from a dipole.