Production of Induced EMF by Changing the Area of the Coil

What this figure shows. A conducting rod of length l lies across a U-shaped, fixed rectangular metal framework ABCD, immersed in a uniform field $\vec B$ directed into the page, with the rod free to slide along the frame's rails and shown moving with velocity v towards the left. As the rod slides from position AB towards DC over a small time interval dt, the enclosed area ABCD (and hence the flux through it) steadily shrinks, since the loop between the rod's current position and the fixed end DC becomes progressively smaller. The figure supports the derivation that the small change in area swept out in time dt is $l\times(v\,dt)$, so the flux change is $d\Phi_B = Bl\,v\,dt$, giving exactly the motional emf $\varepsilon = Blv$ already met via the Lorentz-force route in section 4.1.6, no …

Worked out. A circular metal disc of area 0.03 m$^2$ rotates about an axis through its centre, perpendicular to its plane and parallel to a uniform field of 0.4 T, completing 20 revolutions per second, with the disc's own resistance being 4 $\Omega$; the induced emf between the axis and rim, and the induced current, are required. The area swept out per unit time by the rotating disc is $dA/dt = (\text{area})\times(\text{frequency}) = 0.03\times20 = 0.6\ \text{m}^2/\text{s}$, so the induced emf is $\varepsilon = B\,dA/dt = 0.4\times0.6=0.24$ V. The induced current then follows directly from Ohm's law, $i = \varepsilon/R = 0.24/4 = 0.06$ A. This example applies the changing-area mechanism to a spinning disc rather than a sliding rod, illustrating that the same $\varepsilon=B\,dA/dt$ logic covers …