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## Homework Statement

Given a sphere radius R with surface charge density [itex]\rho_s=k\;cos\theta[/itex]. Find the dipole moment [itex]\;\vec p[/itex].

## Homework Equations

[tex]\vec p=\int \vec r'\rho_s \;d a = \int_0^{2\pi}\int_0^{\pi}\vec r' k\;cos\theta\; R^2d\theta\;d\phi [/tex]

## The Attempt at a Solution

To me, [itex]\vec r' = \hat R R[/itex] in spherical coordinates. But the book claimed from the charge density distribution, [itex]\vec p = \hat z p[/itex] which make sense so the book assumed [itex]\;\vec r'=\hat z z = \hat z R\;cos\theta[/itex]. This all make sense.

My real question is what if I don't know the direction of the [itex]\vec p[/itex] by looking at the charge distribution, how am I going to do the integration and find [itex]\vec p[/itex]? If I just use [itex]\vec r' = \hat R R[/itex], the answer won't be correct. Please help.