To find the dipole moment in a dielectric sphere, I normally use the bound charges density which helps me to find the potential differences and then negative gradient of that potential can give me the Electric field.
I want to explore how I can deal this kind of problem when there are different charge density exist.
For example: We have a spherical shell with radius R which is nonconducting. Upper half of the shell has charge density $+\sigma$ and the lower half of the charge density is $-\sigma$. If I want to know the dipole moment of the entire shell/sphere, how should I proceed? Any suggested reading or details answer will be highly appreciated.