Consider a solid sphere of radius r and mass m which has a charge q distributed uniformly over its volume. The sphere is rotated about a diameter with an angular speed ω. Show that the magnetic moment μ and the angular momentum l of the sphere are related as μ=q2ml
The magnetic moment acts along the axis of rotation. Consider a volume element dV.
It contains a charge, q′=Q43πR3dV
So, the current constituted by the element can be given as:
I=3Q4πR3ω2πdV
and magnetic moment can be given as:
dμ=Iμr2sin2θ=3Q4πR3ω2πdVπr2sin2θ
⇒μ=∫π0∫R03Q4πR3(2πr2sinθdθdr)(ωrr2sin2θ)
⇒3Q2R3×ω2×R55×43=QR2ω5
∵L=25mR2ω
∴R2ω=L2m
Hence, μ=QL2m