# Current in a rolling ring

In the given figure, the ring with charge "$$Q$$", mass "$$M$$", and radius "$$R$$" is pushed in the magnetic field "$$B$$" with translational speed "$$V$$" and no angular velocity. After entering the field, the ring starts pure rolling. How can we define the current flowing the ring?

Given: $$Q=\sqrt{2}\,\frac{MV}{BR}$$

Will it be a function of polar coordinate from centre of ring or a function of time, or a constant? If so, what would be the expression? I am completely blank on how to proceed here?

We can see that after it passes through the line $$A$$ the Lorentz force applies the torque that results in the rolling.

When you say "Will it be a function of polar coordinate from centre of ring", this actually raises a good question about relativity. If we look at it from the center of the ring's perspective (whose center is moving at $$V=0$$ from its perspective) we get a different result than if we look at it from a rest observer's perspective (someone who sees the ring moving at $$V$$).

From the rest observer's perspective we see this (press the play button next to $$T$$ to see the animation).

From the center of the ring's perspective we see this (press the play button next to $$T$$ to see the animation).

The velocity vectors, $$\vec{v}$$, for each point look different in each frame. This difference can be viewed as the freedom to find a convient frame in a more classical picture but has interesting implications in special relativity.

For your problem you can pick either frame, as long as you specify which one, and solve the problem in that frame (one is more convinient). If you want it in the other frame this just ammounts to a Galilean transformation for the classical current (i.e., not the special relativity version).

You can proceed from here and comment any further questions.