# Paradox: electric current in a coil on a disc - will this disc spin if the circuit is opened?

I encountered this problem in a book, but there were no solution written there.

The setup: there is a plastic (insulator) circular disc, that is suspended in a way, that can very easily rotate (so the friction is negligible). On the disc, a coil is placed exactly around the axis of rotation. The ends of the coil are connected through a battery. The battery is assumed to be small in size. There is I current flowing in the circuit. At the circumference of the disc, small metal spheres are placed at equal distances. These metal spheres are seperated from each other. Each sphere has the same electric charge Q. The metal spheres are fixed to the place. (They can't roll away.)

Now, the disc is not in motion. The circuit is then broken.

The paradox: two outcomes are possible (but of course, only one could be right):

1. The magnetic flux decreases, an electric field is induced, that will exert a force on the spheres, torque is applied to the disc, the disc will rotate.

2. The angular momentum is conserved. The system had 0 angular momentum at the beginning, so it will have 0 later. The disc will not rotate.

Where does the momentum originally come from in this case? Imagine setting it up. You have to have the current on before you place the charges, otherwise the disc begins spinning when you start the current. That means you have to bring the charges in from infinity. For simplicity, imagine that the charges initially form a ring of infinite radius in the xy plane, and we contract that ring until the charges rest on the disc. Each charge will be moving through a magnetic field from the wire, and therefore will feel a force $\mathbf{F}=q\mathbf{v}\times \mathbf{B}$ which will point around the rim of the ring. The charges will try to spin, and you will have to apply a torque to prevent this. It's this torque which produces the angular momentum stored in the disc configuration.