Shrinking Loop. A circular loop of flexible iron wire has an initial circumferen

Question

Shrinking Loop. A circular loop of flexible iron wire has an initial circumference of 163cm , but its circumference is decreasing at a constant rate of 12.0cm/s due to a tangential pull on the wire. The loop is in a constant uniform magnetic field of magnitude 0.500T , 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.

Part A

Find the magnitude of the emf E induced in the loop after exactly time 2.00s has passed since the circumference of the loop started to decrease.

Express your answer numerically in volts to three significant figures.

Explanation / Answer

circumference, C = 2*pi*r

rate of change of C = dC/dt = 2*pi*(dr/dt) = -12 cm/s = -0.12 m/s <----- negative sign for decrease

So, dr/dt = -0.12/(2*pi) = -0.019 m/s

B = 0.5 T

area of the loop, A = pi*r^2

So, flux , Q = B*A = B*(pi*r^2)

Now, rate of decrease of circumference, dC/dt = -0.12 m/s

So, in 1s, C decreases by 0.12 m

in 2s, it will decreases by 0.12*2 = 0.24 m = 24 cm

So,after 2s, its circumference will be, 163 - 24 = 139 cm

when C= 2*pi*r = 139 cm = 1.29m

So, r = 0.205 m

magnitude of induced emf, E = dQ/dt = B*pi*(2r)*dr/dt

So, E = 0.5*pi*(2*0.205)*0.019 = 0.012 V <------------answer

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