Wednesday, October 31, 2018

homework and exercises - Understanding the integral for the electric dipole moment of a charge distribution


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 (rR).



The dipole moment is by definition


p=rρ(r)dV


But Griffiths uses z=rcosθ 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 hemisphererρ0dVsouthern hemisphererρ0dV


which gives p=0



when evaluated, which is wrong. Where have I setup my integral wrong?




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