Volt Circular
A circular loop of flexible iron wire has an initial circumference of 170 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 0.700 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 EMF induced in the loop after exactly time 5.00 s has passed since the circumference of the loop started to decrease.
Express your answer numerically in volts to three significant figures.
From the wording of the problem, it’s not the average rate of change of area over the 0 – 5 s interval that you want, it’s the current rate of change of area at t = 5 s.
First calculate r, dr/dt and dA/dt.
dr/dt = -0.15/(2pi) = -2.38732414637843E-02 m/s
circumference C = C0-t*dC/dt = 1.7-5*0.15 = 0.95 m
dA/dt = dr/dt*C = -2.26795793905951E-02 m^2/s.
Finally V = d(BA)/dt = B*dA/dt = 1.58757053030549E-02 V. (answer)
EDIT: I decided to work this out, just being curious, so numbers are added. Also removed a bunch of unnecessary stuff (computing r0, dr/dt and r to find C).
To visualize why dA/dt = dr/dt*C, think of unrolling C to a straight line and adding another line a distance dr directly above it. The area enclosed, dr*C, is dA.
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