Exercise 4.1.17: Rotating charged ring

The whole charge passes any point of the ring once per revolution, $T=2\pi /\omega$:

(Equivalently ${\rho }_L=Q/2\pi b$ moving at $v=\omega b$ gives $I={\rho }_{L}v$.)

The ring is a current loop, so on its axis (Problem 4.1.2)

and at the centre $B(0)={\mu }_{0}I/2b=4\pi \times 10^{-7}\times 5\times 10^{-7}/0.10=6.3\times 10^{-12}$ T.

$m=I\pi b^2=\frac{Q\omega b^2}{2}=0.5\times 10^{-6}\times \pi \times 0.0025=3.9\times 10^{-9}$ A m${}^2$.

Units: C/s $=$ A, and ${\mu }_{0}I/b$ is T. The disk result of Problem 4.1.3 is recovered by integrating $dI={\rho }_S\omega {\rho }^{\prime }d{\rho }^{\prime }$ over the rings. Note the ratio $m/L$ with the angular momentum $L=M{b}^2\omega$ of a ring of mass $M$: $m/L=Q/2M$, the classical gyromagnetic ratio, the same for any rigid body with charge and mass distributed alike.