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

Given

radius of solid sphere = r

mass of sphere = m

charge on the sphere =q

angular speed of the sphere = ω

magnetic moment and the angular momentum ℓ of the sphere are related –

Lets consider a differential strip of width dx at a distance x from the centre of the sphere.

Area of the strip is given as,

now, current i is given by-

Angular speed,

Now frequency f

current i becomes

Magnetic moment is given by-

where,

A= area of cross section

i=current flowing through it

n = number of turns

for number of turns n = 1

Integrating over the sphere-