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Dipole moment of a spherical shell

  1. Dec 21, 2011 #1
    1. The problem statement, all variables and given/known data

    I'm trying to do problem 3.28 in griffith's electrodynamics. The problem statement is, to find the dipole moment of a spherical shell with charge distribution σ = kcosθ

    The way I tried to do it was to use the definition of dipole moment, which griffith defines as

    P= ∫r σ dζ

    where r = position of charge w.r.t origin ( in this case R ) and dζ is volume element.

    The above integral gives 0 ( unless i did something stupid)

    I looked up the solution manual and the way it does it is to use Rcosθ ie z instead of r. Can some1 explain why this is?
     
  2. jcsd
  3. Dec 23, 2011 #2
    You should consider r in the integral as a vector. and since the charge distribution has symmetry with respect to x and y axes, we only consider z component of r which is Rcosθ. You can check x and y and make sure that they are zero.
     
  4. Dec 24, 2011 #3
    I see now. My problem was that even though I looked at it as a vector, I didn't realize that the unit vector [itex]\hat{r}[/itex] itself was a function of θ and [itex]\phi[/itex] and I took it out of the integral. When i rewrite it in cartesian co-ordinates and do the integral for each component ( cartesian unit vectors are constant so I can take it out of the integral ) it comes out fine. But when I look at this, vector integration with spherical co-ordinates seems very complicated, is there an easier way than rewriting in cartesian co-ords and integrating?
     
  5. Feb 22, 2012 #4

    mpg

    User Avatar

    i got confused with this problem too, thank you for taking it out. :smile:
     
  6. Feb 23, 2012 #5
    :smile:
     
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