If I were rotating a disc of radius $r$ with an angular velocity $w$ completely enclosed by a uniform magnetic field $B$, perpendicular to the plane, will there be any emf produced? And how will the current flow?

I am aware that the motional emf produced is similar to that produced between two ends of a rod with length $r$ rotating around a circular conducting ring (radius $r$) with one end at the centre and the other touching the circumference of the ring. But how does the current flow? And the emf is between what two points?

The formula of the emf is $$\frac{wBR^{2}}{2}$$