Consider a straight wire AB resting on a pair of conducting rails separated by a distance l lying wholly in a plane perpendicular to a uniform magnetic field \(\vec B\). \(\vec B\) points into the page and the rails are stationary relative to the field and are connected to a stationary resistor R.
Suppose an external agent moves the rod to the right with a constant speed v, perpendicular to its length and to \(\vec B\). As the rod moves through a distance dx = vdt in time dt, the area of the loop ABCD increases by dA = ldx = lv dt.
A conducting rod is moved to the right on conducting rails in a uniform magnetic field
dΦm = BdA = Blvdt
By Faraday’s law of electromagnetic induction, the magnitude of the induced emf
e = \(\cfrac{dΦ_m}{dt}\) = \(\cfrac{Blvdt}{dt}\) = Blv
