In problem 3.35 of Griffiths' Introduction to electrodynamics, he states:
A solid sphere, radius R, is centered at the origin. The “northern” hemisphere carries a uniform charge density ρ0, and the “southern” hemisphere a uniform charge density −ρ0. Find the approximate field E(r,θ) for points far from the sphere (r≫R).
The dipole moment is by definition
p=∭r′ρ(r′)dV
But Griffiths uses z=r′cosθ and says
p=∭zρ(r′)dV
How does this work? Aren't you supposed to use r′ in the integral?
In my calculations I get
p=∭northern hemispherer′ρ0dV−∭southern hemispherer′ρ0dV
which gives p=0
when evaluated, which is wrong. Where have I setup my integral wrong?
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