Exercise:
A rectangular coil with sides LO and WO is pushed o a homogeneous magnetic field with strength BO. The velocity vector has a magnitude vO. The area of the coil is perpicular to the magnetic field lines. abcliste abc For the two orientations shown in the figure below make a quantitative graph for the induced emf vs. time. abc Discuss the properties of the induced emf for other orientations of the coil still perpicular to the magnetic field. abcliste center includegraphicswidthtextwidth#image_path:rectangular-coil# center

Solution:
abcliste abc While the coil is entering the magnetic field the induced emf is mathcalEt -dotPhi_mt -B dotAt -B ydot xt where y is the vertical dimension of the coil and xt is the distance by which the coil has already entered the field. medskip In the first situation we find mathcalEt Va -BtimesLtimesv Va approx resultVaP- The time for the coil to enter the magnetic field is given by Delta t taF fracWv ta approxtaP- For the second situation the same calculations apply with the roles of width and length switched. We find a voltage of VbP- during a time of tbP-. center includegraphicswidthcm#image_path:rectangular-coil-# center abc For an arbitrary angle there are three different phases: itemize item Phase I: The area in the field as a function of the horizontal position is given by a quadratic function area of a triangle with a linearly increasing height. This corresponds to a linear rate of change of the area in the field and therefore to a linear increase of the induced emf. item Phase II: Between the positions A and B in the figure below the area increases at a constant rate. This is equivalent to a constant induced emf. item Phase III: The area decreases in the same way that it increases in phase I. As a consequence the induced emf decreases linearly. itemize abcliste center includegraphicswidth.mm#image_path:rectangular-coil-rotated-# center For a coil rotated by an angle alpha with respect to the first situation in a the times at which the different phases start and are given by t_fracWcosalphav t_fracLsinalphav t_t_+t_fracWcosalphav+fracLsinalphav The area in the magnetic field can be expessed as follows: At cases fracv^sinalphacosalpha t^ & leq tleq t_ fracW^tanalpha+fracv Wsinalphat-t_ & t_leq tleq t_ L W-fracv^sinalphacosalpha t_-t^ & t_leq tleq t_ cases The induced emf then follows from Faraday's law: mathcalE-Bdot Atcases -Bfracv^sinalphacosalpha t & leq tleq t_ -Bfracv Wsinalpha & t_leq tleq t_ -Bfracv Wsinalpha-fracv^sinalhpacosalpha t & t_leq tleq t_ cases The corresponding graph for alphadegree has been included in the diagram below. The area between the red graph and the time axis is -DeltaPhi -Bfracv Wsinalpha t_ -Bfracv Wsinalpha fracLsinalphav -B L W which corresponds once again to the total negative flux change during this process. center includegraphicswidthcm#image_path:rectangular-coil-rotated# center