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the same direction. In general we write

                                                  H 0 (r, t) − H i (r, t) = H d (r, t)
                        where H d is the demagnetizing field produced by the magnetic dipole moments of the
                        material. Each electron responds to the internal field by precessing as described above
                        until the precession damps out and the electron moments align with the magnetic field.
                        At this point the ferrite is saturated. Because the demagnetizing field depends strongly
                        on the shape of the material we choose to ignore it as a first approximation, and this
                        allows us to concentrate our study on the fundamental atomic properties of the ferrite.
                          For purposes of understanding its magnetic properties, we view the ferrite as a dense
                        collection of electrons and write

                                                      M(r, t) = Nm(r, t)
                        where N is the number density of electrons. Since we are assuming the ferrite is homoge-
                        neous, we take N to be independent of time and position. Multiplying (4.108) by −Nγ ,
                        we obtain an equation describing the evolution of M:

                                                dM(r, t)
                                                        =−γ M(r, t) × B i (r, t).             (4.109)
                                                   dt
                        To determine the temporal response of the ferrite we must include a time-dependent
                        component of the applied field. We now let

                                               H 0 (r, t) = H i (r, t) = H T (r, t) + H dc
                        where H T is the time-dependent component superimposed with the uniform static com-
                        ponent H dc . Using B = µ 0 (H + M) we have from (4.109)
                                       dM(r, t)
                                               =−γµ 0 M(r, t) × [H T (r, t) + H dc + M(r, t)].
                                          dt
                        With M = M T (r, t) + M dc and M × M = 0 this becomes
                                 dM T (r, t)  dM dc
                                          +       =−γµ 0 [M T (r, t) × H T (r, t) + M T (r, t) × H dc +
                                    dt        dt
                                                  + M dc × H T (r, t) + M dc × H dc .         (4.110)

                        Let us assume that the ferrite is saturated. Then M dc is aligned with H dc and their cross
                        product vanishes. Let us further assume that the spectrum of H T is small compared
                                                ˜
                        to H dc at all frequencies: |H T (r,ω)|  H dc . This small-signal assumption allows us to
                        neglect M T × H T . Using these and noting that the time derivative of M dc is zero, we see
                        that (4.110) reduces to
                                       dM T (r, t)
                                                =−γµ 0 [M T (r, t) × H dc + M dc × H T (r, t)].  (4.111)
                                          dt
                          To determine the frequency response we write (4.111) in terms of inverse Fourier
                        transforms and invoke the Fourier integral theorem to find that
                                                         ˜
                                                                              ˜
                                         ˜
                                      jωM T (r,ω) =−γµ 0 [M T (r,ω) × H dc + M dc × H T (r,ω)].
                        Defining
                                                        γµ 0 M dc = ω M ,


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