I am trying to derive the formula for E.m.f across the ends of a rod rotating in a magnetic field when the field is perpendicular to the plane of rotation: [tex] |\epsilon| = B \omega r^2 [/tex] using the Lorentz Formula only. Suppose, there is a rod of length 'R'. Let, 'n' be the no. of electrons per unit length. Let, at a distance 'r' from the center of rotation, there be a small length element dr. The no. of electrons in this length element is [itex]ndr[/itex]. Also, the velocity, [itex]v = r\omega[/itex] is perpendicular to the field always as it lies in the plane of rotation of the rod. Hence, the Lorentz force here is given as: [tex] dF = en r\omega B dr [/tex] What do i do after this? Do integrate it? This integral only talks about a 'net force' on all the electrons. How do I use it to compute the e.m.f across the two ends? Do I first compute work using [itex]\int W = \int F.dr[/itex] and the differentiate w.r.t charge. But, what charge is the Work a function of?