Calculating Quadrupole Moment of Sphere w/ Surface Charge

[KleZMeR]

1.

Looking for quadrupole moment of spherical shell with surface charge (sigma)=(sigma_o)*cos(theta) and (sigma_o) => constant
Sphere is at the origin with radius=a.

2.

Well, I think I am using the right equation, by integration of the quadrapole moment taken from the quadrupole term, but I am questioning my [(r')^2] factor. 3.

I use vector product I get a zero. I guess a vector crossed with itself is always zero because of no span. I have already found the monopole and dipole (monopole =0), and I think I take the vector product of r', and not the dot product, because the dot product is a scalar... and I found by another theorem that whether it is shell or sphere, the quadrapole and higher terms are zero. So is this where the quadrapole zeroes out?? from the [(r')^2] factor?

 

[olgranpappy]

i'm pretty sure only the dipole moment should be non-zero. you can write all the moments in terms of the charge density integrated against spherical harmonics. E.g., Y_{0,0} for monopole term, Y_{1,m} for dipole term, Y_{2,m} for quadrupole term, etc. But since the charge density itself is proportional to Y_{1,0} only the l=1 multipole moments should be non-zero due to orthogonality of spherical harmonics.