Dipole moment of a sphere?

humanist rho

1. The problem statement, all variables and given/known data
The electric potential of a grounded conducting sphere of radius r in a
uniform electric field [tex]E_{0}\hat{k}[/tex]along the z direction is given by

[tex]V(r,\theta )=-E_{0}r\cos \theta +\frac{E_{0}R^{3}\cos \theta }{r^{2}}[/tex]

where r is the distance from the centre of the sphere and θ is the
angle the radius vector makes with the z axis.

(a)what is the dipole moment acqured by the sphere?


2. Relevant equations



3. The attempt at a solution


Surface charge density,

[tex]\sigma (\theta )=-\varepsilon \frac{\partial V}{\partial r}%
|_{r=R}=3\varepsilon E_{0}\cos \theta [/tex]
Dipole moment,

[tex]P=\int \sigma (\theta ^{\prime })r^{\prime }ds^{\prime }

=\int_{0}^{\pi }3\varepsilon E_{0}\cos \theta ^{\prime }r^{\prime
}r^{\prime 2}\sin \theta ^{\prime }d\theta ^{\prime }d\phi ^{\prime }

=6\pi \varepsilon E_{0}R^{3}\int_{0}^{\pi }\cos \theta ^{\prime
}\sin \theta ^{\prime }d\theta ^{\prime }

=6\pi \varepsilon E_{0}R^{3}\int_{-1}^{1}\cos \theta ^{\prime
}d(\cos \theta )

=6\pi \varepsilon E_{0}R^{3}[/tex]

But this donot match with the real answer. :(