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Magnetic fields are wondrous things, bound by geometric relationships to the moving currents that generate them. Learn these links and the things they govern, from transformers to electric motors.

A closed loop of wire that consists of a semicircle of radius \(3.2\text{ cm}\) and a straight line, as shown in the above figure, is lying in a uniform magnetic field \(\vec{B}\) of magnitude \(74\text{ mT}.\) The magnetic field is perpendicular to the straight line of the loop and makes an angle of \(45^\circ\) with the plane of the semicircle. If the magnetic field is reduced to zero at a uniform rate during a time interval of \(6.0\text{ ms},\) what is the approximate magnitude of the electromotive force induced in the loop during this interval?

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Consider a conducting loop of a half-circle of radius \(r=0.30\text{ m},\) as shown in the above figure. The loop lies in a uniform magnetic field \(\vec{B}\) that is directed out of the screen. If the field magnitude is given by \(B=4.0t^2+4.0t+3.0,\) where \(B\) is in teslas and \(t\) is in seconds, what is the approximate magnitude of the electromotive force induced around the loop by that field \(\vec{B}\) at \(t=5\text{ s}?\)

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A rectangular loop of wire is immersed in a nonuniform and varying magnetic field \(\vec{B}\) that is perpendicular to and directed into the screen, as shown in the above figure. The loop has width \(W=3.0\text{ m}\) and height \(H=2.0\text{ m}.\) If the magnitude of the magnetic field is given by \(B=5t^2x^2,\) where \(B\) is in teslas, \(t\) is in seconds, and \(x\) is in meters, what is the magnitude of the induced electromotive force around the loop at \(t=0.30\text{ s}?\)

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