An insulating thin rod of length ℓ has linear charge density ρ(x)=ρ₀x/ℓ. The rod is rotated about an axis passing through the origin (x=0) and perpendicular to the rod. If the rod makes n rotations per second, then its time-averaged magnetic moment is:
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Question

An insulating thin rod of length ℓ has linear charge density ρ(x)=ρ₀x/ℓ. The rod is rotated about an axis passing through the origin (x=0) and perpendicular to the rod. If the rod makes n rotations per second, then its time-averaged magnetic moment is:

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G1 Physics · Accepted Answer

For an element dq=(ρ₀x/ℓ)dx rotating with angular speed ω=2πn, dμ=(ω/2)x²dq. Hence μ=(ωρ₀/2ℓ)∫₀^ℓx³dx=(π/4)nρ₀ℓ³. Hence A.