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Magnetic Flux, Induction, and Ampere's Circuital Law

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.

Using Magnetic Flux/Faraday's Law

         

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?

Suppose that an elastic conducting material, which is placed in a uniform \(0.800\text{ T}\) magnetic field, is stretched into a circle loop with radius \(15.0\text{ cm}.\) The plane of the loop is perpendicular to the magnetic field, and the stretched loop is released to shrink at an instantaneous rate of \(75.0\text{ cm/s}.\) What is the approximate magnitude of the electromotive force induced in the shrinking loop?

A circular wire loop with radius \(0.30\text{ m}\) is lying in a magnetic field with magnitude \(0.50\text{ T}.\) The plane of the loop is perpendicular to the magnetic field. If the direction of the magnetic field is reversed and its magnitude is reduced to \(0.20\text{ T}\) in \(2.0\text{ s},\) what is the approximate magnitude of the average induced electromotive force in the loop during this time interval?

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 \(t=5\text{ s}?\)

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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