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Potnetial of a spherical Shell

  1. Oct 27, 2006 #1
    Griffith's EM problem 3.28
    A spherical shell of radius R has a surface charge [itex] \sigma = k \cos \theta [/itex]

    a) Calculate the dipole moment of this charge distribution.
    i know that
    [tex] p = r' \sigma(r') da' [/tex]

    but here sigma depends on theta
    would the dipole moment p then turn into
    [tex] p = \theta' \sigma(theta') da' [/tex]

    and the radius of the sphere is constant theta and phi are constant
    so that
    [tex] p = \int_{0}^{\pi} \int_{0}^{2 pi} \theta' \sigma(\theta') R^2 \sin\theta' d \theta' d \phi [/tex]
    i get a negative dipole moemnt as a result of this though... which amkes no sense
    what am i doing wrong??

    please help!!!

    thanks :)
     
    Last edited: Oct 28, 2006
  2. jcsd
  3. Oct 27, 2006 #2

    quasar987

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    Look at equ. 3.98. p and r' are VECTORS in there.
     
  4. Oct 28, 2006 #3
    I take it you are refering to Griffith's textbook?
     
  5. Oct 28, 2006 #4
    right they are vectors...

    so then i cant use theta the way i used it

    so
    [tex] \vec{p} = \int \vec{r'} \simga(\theta') d\vec{a'} [/tex]
    [tex] p = \int_{0}^{\pi} \int_{0}^{2\pi} r' k \cos\theta' r'^2 \sin\theta d\theta d\phi [/tex]

    but the integral
    [tex] \int_{0}^{2\pi} \cos\theta' \sin\theta' d\theta = 0 [/tex]!
    cant have zero dipole moment...
     
  6. Oct 28, 2006 #5
    problem 3.28
    page 151
     
  7. Oct 28, 2006 #6

    quasar987

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    [tex]\vec{r}=r\hat{r}=r(\hat{x}\sin\theta \cos \phi+\hat{y}\sin\theta\sin\phi+\hat{z}\cos\theta)[/tex]
     
  8. Oct 28, 2006 #7

    quasar987

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    [itex]\hat{r}[/itex] is not a vector like [itex]\hat{x},\hat{y},\hat{z}[/itex]. The latest are constants vectors while [itex]\hat{r}[/itex] points towards the point that you're integrating (if I may say so). So it changes as you "sum" each [itex]d\theta[/itex] and [itex]d\phi[/itex] (if I may be so ruthless). So you can't pull it out of the integral as opposed to "inert" vectors like [itex]\hat{x},\hat{y},\hat{z}[/itex].
     
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