Savchenko Problem 11.2.14
Physics · mechanics
Problem
On a nonconducting ring of mass m and radius r, the charge q is uniformly distributed. The ring can rotate freely around its axis. At the initial moment, the ring is at rest. In the central region of the ring of radius l < r, there is a magnetic field perpendicular to the ring plane, the induction of which is uniformly reduced to zero. What angular velocity will the ring acquire by the time the field disappears? Will the result change if the induction of B decreases to zero unevenly? The magnetic field induction generated by the rotating ring can be ignored.Intuitive explanation
This problem hinges on Faraday's law and angular momentum conservation: as the magnetic flux through the ring decreases, an induced electric field circulates around the axis, exerting a tangential force on the distributed charge. Since the ring is free to rotate, this force spins it up. The key insight is that the total impulse delivered to the charge depends only on the total change in flux, not on how fast that change happens—so the final angular velocity is independent of the time profile of .
Intuitive explanation
This problem hinges on Faraday's law and angular momentum conservation: as the magnetic flux through the ring decreases, an induced electric field circulates around the axis, exerting a tangential force on the distributed charge. Since the ring is free to rotate, this force spins it up. The key insight is that the total impulse delivered to the charge depends only on the total change in flux, not on how fast that change happens—so the final angular velocity is independent of the time profile of .