A rod has a total charge Q uniformly distributed along its length L. If the rod rotates with angular velocity `omega` about its end, compute its magnetic moment.

Explanation

Charge on small length `dx` at distance `x` from origin
`dq=(Q)/(L)dx`
This charge is rotating with angular velocity `omega`,
Current `i = (dq)/(T) = (Qdx//L)/(2pi//omega)`
`= (Q omega)/(2pi L)`
Small magnetic moment
`dM = i.pi x^(2)`
`= (Q omega)/(2L) x^(2) dx`
`M = (Q omega)/(2L) int_(0)^(L) x^(2) dx`
`= (Q omega)/(2L)|(x^(3))/(3)|_(0)^(L)`
`= (Q omega)/(2L)((L^(3))/(3) - 0)`
`= (Q omega L^(2))/(6)`