Error in dipole moment calculation

[Gavroy]

I am calculating the dipole moment of a metal sphere in a uniform electric field [itex]E_0[/itex] in z-direction. From here I know that the charge density [look at page 15][1]


http://www.phy.iitb.ac.in/~dkg/PH-102/conductors.pdf[/PLAIN]

is given by [itex] 3 \epsilon_0 E_0 cos(\theta)[/itex]

Now I wanted to calculate the resulting dipole moment by [itex] \int_0^R \int_0^{2\pi} \int_0^{\pi} r^3 sin(\theta) cos(\theta) 3 \epsilon_0 E_0 d\phi d\theta dr[/itex]

but in this case the integral over [itex]\theta[/itex] is zero and therefore this whole term will be zero which is somewhat strange, since there should be a dipole moment. So what am I doing wrong?

 

[mfb]

One factor of r in your r³ is a vector, not the magnitude of r. It depends on θ.