Dipole moment of a spherical shell

by Idoubt
Tags: dipole, moment, shell, spherical
 P: 157 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?
 P: 6 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.
 P: 157 I see now. My problem was that even though I looked at it as a vector, I didn't realize that the unit vector $\hat{r}$ itself was a function of θ and $\phi$ 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?
P: 2

Dipole moment of a spherical shell

i got confused with this problem too, thank you for taking it out.
 P: 157

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