Deriving ε = −N·dΦ/dt from the Lorentz force and energy conservation

Step 1 of 4: Magnetic force on a charge carrier in the sliding rod

Intuition

Like dust kicked up when you swing a plank through the air: the charge carriers in the rod get 'swept' across the field lines and the field flings them toward the two ends of the rod.

F_m = q\,\vec v \times \vec B

Analysis

A conducting rod of length $\ell$ slides at speed $v$ on conducting rails in a uniform magnetic field $B$ perpendicular to the rail plane. Every charge carrier $q$ inside the rod is dragged along at speed $v$ , so it feels a magnetic force $F_m = qvB$ directed along the rod — this force separates the charges and produces an EMF.