## Dipole moment of a sphere?

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 $$E_{0}\hat{k}$$along the z direction is given by

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

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,

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

$$P=\int \sigma (\theta ^{\prime })r^{\prime }ds^{\prime } =\int_{0}^{\pi }3\varepsilon E_{0}\cos \theta ^{\prime }r^{\prime }r^{\prime 2}\sin \theta ^{\prime }d\theta ^{\prime }d\phi ^{\prime } =6\pi \varepsilon E_{0}R^{3}\int_{0}^{\pi }\cos \theta ^{\prime }\sin \theta ^{\prime }d\theta ^{\prime } =6\pi \varepsilon E_{0}R^{3}\int_{-1}^{1}\cos \theta ^{\prime }d(\cos \theta ) =6\pi \varepsilon E_{0}R^{3}$$

But this donot match with the real answer. :(

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