A circular loop of flexible iron wire has an initial circumference of 163 cm, bu

Question

A circular loop of flexible iron wire has an initial circumference of 163 cm, but its circumference is decreasing at a constant rate of 11.0 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 epsilon induced in the loop after exactly time 3.00 s has passed since the circumference of the loop started to decrease.

Explanation / Answer

EMF= magnetic field * area
The derivative of this equation with respect to time is the change in EMF with respect to time

Eq.#1. = dEMF / dt = B * dA / dt
As the area changes, the EMF changes.

We need an equation that expresses the area in terms of the circumference.
C = 2 * * r and Area = * r^2
r = C ÷ (2 * )

Area = * (C ÷ 2 * )^2
Eq. #2 = Area = C^2 ÷ (4 * )
The derivative of this function with respect to time is the change in area with respect to time

dA/dt = d(C^2 ÷ 4 )/dt = (2 C/4 )* dC / dt
dA/dt =(C/2 )* dC / dt
Now substitute this expression for dA / dt into Eq.#1

Eq.#1. = dEMF / dt = B * dA / dt
Eq.#1. = dEMF / dt = B * (C/2 )* dC / dt
C = 1.63 m
B = 1.00 T
dC / dt = 0.11 m/s
At 3 seconds, C = 1.63 – (0.11 * 3) = 1.30 m

dEMF / dt = B * (C/2 )* dC / dt
dEMF / dt = 1.00 * (1.30/2 ) * 0.11 = 0.02275

Express your answer numerically in volts to three significant figures
Volts = 2.275 * 10^-2 volts

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