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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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