Why Use z Instead of r for Dipole Moment Calculation?

[Idoubt]

Homework Statement

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?

 

[N2O2]

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.

 

[Idoubt]

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?

 

[mpg]

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

 

[Idoubt]