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Homework Help: Dipole moment of a sphere?

  1. May 6, 2012 #1
    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. :(
  2. jcsd
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