Finding Dipole Moment: Solve Integral & Calculate \vec p

[yungman]

Homework Statement

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

Homework Equations

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

The Attempt at a Solution

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

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

 

[ehild]

Resolve the vector r into x, y, and z components, expressed in polar coordinates and calculate these components by separate integrals. You will see that both the x and y components cancel.

ehild

 

[yungman]

Resolve the vector r into x, y, and z components, expressed in polar coordinates and calculate these components by separate integrals. You will see that both the x and y components cancel.

ehild

I got it, thanks for your help.