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 169cm , but its circumference is decreasing at a constant rate of 10.0cm/s due to a tangential pull on the wire. The loop is in a constant uniform magnetic field of magnitude 1.00T , 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 E induced in the loop after exactly time 1.00s has passed since the circumference of the loop started to decrease.

Explanation / Answer

Circumference C= 2?r
Rate of change of circumference is
dC/dt = 2? dr /dt = - 0.14m/s ( GIVEN)
Minus sign to denote the decrease in value.
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A = ?r^2
The rate of change of area
dA/dt = 2?r dr /dt = - 0.14*r (Replacing 2? dr /dt by -0.14)
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Initial circumference = 1.65 m
In 2 second the C decreases by 2*0.14 = 0.28 m
The radius after 2 s is [1.65 - 0.28] / (2?) = 0.22 m
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e = - d/dt {BA cos ?]
e = - 0.9* dA/dt since cos ? = 1 and B = 0.9
e = - 0.9 *dA/dt
e = 0.9 *[0.14*r] (Substituting for dA/dt as - 0.14*r)
e = 0.9 *0.14*0.22 (Substituting the value of r)
e = 0.028 V

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