How to calculate the dipole moment of the spherical shell?

[lrcarr]

Homework Statement

A spherical shell of radius R has a surface charge distribution σ = k sinφ.
Calculate the dipole moment of the spherical shell.

Homework Equations

P[/B]' = ∫r' σ(r') da'

The Attempt at a Solution

So I believe my dipole will be directed along the y axis, as the function sinφ is positive in the region 0-π and negative in π-2π. Dipoles point from negative to positive, so the right hand side of my sphere will be positive and left negative.

I also know that da' in spherical co-ordinates is r² sinθdθdφ.

I also thought that maybe because I know my dipole is in y direction, I can do r' → y → rsinθsinφ. I saw another similar problem where the dipole was orientated in z, and they did r' → z → rcosθ.

However no matter what I do, zero seems to fall out at the end. I feel like my understanding of the whole prime system, or spherical co-ordinates, is maybe confused or I am misunderstanding the affect that the direction of the dipole has on this.

What am I doing wrong?

 

[kuruman]

Remember that dipole moment is a vector. So p_x = ∫x' σ(r') dA' and so on for y' and z'. There are three integrals to do.