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Magnetic dipole moment of a sphere

  1. Dec 25, 2006 #1
    1. The problem statement, all variables and given/known data
    Find the magnetic dipole moment of a spherical shell of radiu R carrying a uniform surface charge sigma, set spinning at angular velocity omega.

    2. Relevant equations
    [tex] \vec{m} = \frac{1}{2} \int_{S} \vec{r'} \times \vec{K} (\vec{r'}) da' [/tex]

    3. The attempt at a solution
    So we gotta figure out the surface charge density (since it is a spherical shell)

    [tex] K = \sigma v [/tex]

    and [tex] v = \omega times R [/tex]
    this is where i am doubtful...
    the angle between v and R varies from 0 to 2 pi

    so this cross product is not unique...
    or am i thinking about this the wrong way??

    please help!
  2. jcsd
  3. Dec 26, 2006 #2


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    Science Advisor
    Gold Member

    You are right that the angle changes but v will always point in the correct direction. The changing angle ([tex]sin(\theta)[/tex]) accounts for the "azimuthal radius", that is, distance from the spin axis z to the shell measured parallel to the xy plane, that changes with polar angle. That distance is also the r you need to use in your integral.

    BTW, polar angle only varies from 0 to [tex]\pi[/tex]
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