# Multipole expansion of polarized cylinder

1. Nov 8, 2015

### phys-student

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
Vdip=(1/4πε0)(1/r2)∫r'cosαρ(r')dτ'
V(r)=(1/4πε0)∑(1/rn+1)∫(r')nPn(cosα)ρ(r')dτ'

3. 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.

2. Nov 12, 2015

### 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.