Question

Consider a solid sphere of radius r and mass m that has a charge q distributed uniformly over its volume. The sphere is rotated about its diameter with an angular speed ω. Show that the magnetic moment µ and the angular momentum l of the sphere are related as $\mathrm{\mu }=\frac{q}{2m}l$.

Answer 1

Considering the strip of width dx at a distance x from the centre of the sphere.
Small area of the strip is given as,
da = 4π x dx
$$\begin{gathered} i=\frac{dq}{dt}=\frac{q\omega }{2\mathrm{\pi }} \\ \mathrm{\mu }=ia=\frac{q\omega }{2\mathrm{\pi }}4\mathrm{\pi }\int x\mathit{}dx \\ =q.\omega r^2=\frac{q}{m}l \end{gathered}$$