Shrinking Loop. A circular loop of flexible iron wire has an initial circumference of 168 cm, but its circumference is decreasing at a constant rate of 15.0 cm/s due to a tangential pull on the wire. The loop is in a constant uniform magnetic field of magnitude 1.00 T, which is oriented perpendicular to the plane of the loop. Assume that you are facing the loop and that the magnetic field points into the loop.
Find the magnitude of the emf induced in the loop after exactly time 8.00 s has passed since the circumference of the loop started to decrease.
1 Answer

induced emf V = d(B*A)/dt = B*dA/dt
B = 1.00 T
A = pi*r^2
circumference c = 2pi*r
r = c / 2pi
so A = pi*(c / 2pi)^2
A = c^2 / 4pi
so dA/dc = c / 2pi = r
dA/dt = dA/dc * dc/dt
dc/dt = 15 cm/s = 0.15 m/s
At t = 8.00s, c = 168  (8*15)m = 48 cm = 0.48m
so dA/dc = r = 0.48m / 2pi = 0.0764 m
so dA/dt = 0.0764m * 0.15 m/s = 0.01146 m^2 /s
so E = 1.00T * 0.01146 m^2/s = 0.01146V = 11.5 mV <= ANS