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Consider a solid sphere of radius r and mass m which has charge q distributed uniformly over its volume.The sphere is rotated about a diameter with an angular speed x . Show that the magnetic moment y and the angular momentum l of the sphere are related as :- qL/2m pls show the steps

Consider a solid sphere of radius r and mass m which has charge q distributed uniformly over its volume.The sphere is rotated about a diameter with an angular speed x . Show that the magnetic moment y and the angular momentum l of the sphere are related as :-


qL/2m


pls show the steps

Grade:12

1 Answers

ROSHAN MUJEEB
askIITians Faculty 829 Points
one year ago

The magnetic moment acts along the axis of rotation. Consider a volume element dV.
It contains a charge,q′=34​πR3Q​dV
So, the current constituted by the element can be given as:
I=4πR33Q​2πω​dV
and magnetic moment can be given as:
dμ=Iμr2sin2θ=4πR33Q​2πω​dVπr2sin2θ
⇒μ=∫0π​∫0R​4πR33Q​(2πr2sinθdθdr)(rω​r2sin2θ)
⇒2R33Q​×2ω​×5R5​×34​=5QR2ω​
∵L=52​mR2ω
∴R2ω=2mL​
Hence,μ=2mQL​

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