Charge Q is uniformly distributed on a thin insulating ring of mass m which is initially at rest. To what angular velocity will the ring be accelerated when a magnetic field B, perpendicular to the plane of the ring, is switched on ?

In accordance with Faraday's law of electromagnetic induction, the changing magnetic field induces an electric field in| the ring. Let us imagine the ring to be divided into differential elements of length ds and denote the tangential component of the induced electric field by E_t. The charge on element ds of the ring is $\mathrm{dQ}=\mathrm{Q}\frac{\mathrm{ds}}{2\mathrm{\pi r}}$, where r is the radius of the ring. The force exerted on it is ${\mathrm{dF}}_{\mathrm{t}}={\mathrm{dQE}}_{\mathrm{t}}$ and the resultant torque is $\mathrm{d\tau }={\mathrm{rdF}}_{\mathrm{t}}$.

Thus, the total torque experienced by the ring is

  $\mathrm{\tau }=\int \mathrm{d\tau }=\int \mathrm{rQ}\frac{\mathrm{ds}}{2\mathrm{\pi r}}{\mathrm{E}}_{\mathrm{t}}=\frac{\mathrm{Q}}{2\mathrm{\pi }}\int {\mathrm{E}}_{\mathrm{t}}\mathrm{ds}$

The induced electromotive force along the ring is directly proportional to the rate of change in the magnetic flux, we have

$\int {\mathrm{E}}_{\mathrm{t}}\mathrm{ds}=\frac{\mathrm{d\Phi }}{\mathrm{dt}}={\mathrm{\pi r}}^2\frac{\mathrm{dB}}{\mathrm{dt}}$

As a result of the torque, the ring, which has a moment of inertia I = mr², starts to spin with angular acceleration $\alpha$. During a time interval dt its angular velocity changes by

$\mathrm{d\omega }=\mathrm{\alpha dt}=\frac{\mathrm{\tau }}{\mathrm{I}}\mathrm{dt}=\frac{\mathrm{Q}}{2\mathrm{\pi }}\left({\mathrm{\pi r}}^2\frac{\mathrm{dB}}{\mathrm{dt}}\right)\frac{1}{{\mathrm{mr}}^2}\mathrm{dt}=\frac{\mathrm{Q}}{2\mathrm{m}}\mathrm{dB}$

Since the magnetic field strength increases from zero to B, the final angular velocity of the ring will be

 $\mathrm{\omega }=(\mathrm{QB}/2\mathrm{m})$

$\Rightarrow$The final angular velocity does not depend on the radius of the ring, the time over which the magnetic flux changes, or even on how the magnetic flux increases with time.

$\Rightarrow$In our calculation we ignored the magnetic field produced by the rotating ring.

$\Rightarrow$Except in the case of a cylindrical syntmetric uniform field, it is not possible to find the actual value of the induced electric field within the ring because the geometrical structure of the magnetic field is unknown and we do not know the position of the ring in the magnetic field. We can determine the total induced electromotive force, but not the electric field itself.