Question

Consider a non-conducting plate of radius r and mass m that has a charge q distributed uniformly over it. The plate is rotated about its axis with an angular speed ω. Show that the magnetic moment µ and the angular momentum l of the plate are related as `mu = q/(2 m)l`

Answer 1

Given:

Radius of the ring =  r

Mass of the ring =  m

Total charge of the ring =  q

Angular speed, w = `(2pi)/T ⇒ T = (2pi)/w`

Current in the ring, `i = q/t = (qw)/(2pi)`

For the ring of area A with current i, magnetic moment,

`mu = niA =ia [n = 1]`

= `(qw)/(2pi)xx pir^2 = (qwr^2)/(2)`

Angular momentum, `l =  Iw`, where I is the moment of inertia of the ring about its axis of rotation.

`I = mr^2`

`so, l =mr^2w`

⇒ `wr^2 = 1/m`

Putting this value in equation (i), we get:

`mu = (ql)/(2m)`