Motional emf in a Rotating Bar

A rotating bar is a genuinely different case from a sliding bar, because different points along its length move at different speeds. Consider a conducting rod pivoted at one end and rotating with angular velocity $\omega$ in a magnetic field $\vec{B}$ perpendicular to the plane of rotation. Take a small element of the rod of length dr, at a distance r from the pivot: this element moves with (tangential) speed $v = \omega r$, and -- exactly as for a sliding conductor -- has a small motional emf induced in it, $de = Bv\,dr = B\omega r\,dr$.

Since all such elements along the rod's length can be imagined as tiny emf sources connected in SERIES, their individual emfs simply add. Integrating from the pivot (r = 0) to the free end (r = l) gives the total emf induced across the rotating rod: $e = \int_0^l B\omega r\,dr = \frac{1}{2}B\omega l^2$. …