Translational Motion of a Conductor

Consider a C-shaped conducting frame ABCD of wires, with a straight conducting rod BC of length l free to slide along the frame (parallel to AD) in a uniform magnetic field $\vec{B}$ perpendicular to the plane of the frame. As BC slides outward with velocity $\vec{v}$, increasing the enclosed length x, the loop's area (and hence the flux $\Phi = Blx$ linked with it) grows with time. By the flux rule, the magnitude of the induced emf is $|e| = \left|\frac{d\Phi}{dt}\right| = Bl\frac{dx}{dt} = Blv$.

This same result follows independently from the microscopic Lorentz force on the charge carriers inside the moving rod BC. A charge q carried along with the rod experiences a force $\vec{F} = q(\vec{v}\times\vec{B})$, which -- since $\vec{v}$ and $\vec{B}$ are both constant along BC -- is constant in magnitude and directed along the length of the rod (parallel to BC), and is zero everywhere else in the stationary part of the frame (where v = 0). As charge q traverses the rod's length l under this constant force, the work done is $W = Fl = qvB\sin\theta \cdot l$, where $\theta$ is the angle between $\vec{B}$ and $\vec{v}$. Since emf is work done per unit charge, $e = \frac{W}{q} = Blv\sin\theta$, which for the usual case of $\vec{v}\perp\vec{B}$ (i.e. $\theta=90°$, so $\sin\theta=1$) gives the maximum value $e_{max} = Blv$ -- exactly matching the flux-rule result. …