Mastering Physics- Induced EMF and Current in a Shrinking Loop?
A circular loop made of flexible iron wire initially has a circumference of 168 cm. Due to a tangential pull on the wire, the circumference is decreasing at a constant rate of 15.0 cm/s. The loop is placed in a uniform magnetic field with a constant magnitude of 1.00 T, oriented perpendicular to the plane of the loop. Assume you are facing the loop, and the magnetic field is directed into the loop.
Determine the magnitude of the electromotive force (emf) induced in the loop exactly 8.00 seconds after the circumference starts to decrease.
1 Answers
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
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