Consider a solid containing N atoms per unit volume, each atom having a magnetic dipole

Chapter 32, Problem 45

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Consider a solid containing N atoms per unit volume, each atom having a magnetic dipole moment . Suppose the direction of can be only parallel or antiparallel to an externally applied magnetic field (this will be the case if is due to the spin of a single electron). According to statistical mechanics, the probability of an atom being in a state with energy U is proportional to eU/kT, where T is the temperature and k is Boltzmanns constant. Thus, because energy U is , the fraction of atoms whose dipole moment is parallel to is proportional to emB/kT and the fraction of atoms whose dipole moment is antiparallel to is proportional to emB/kT. (a) Show that the magnitude of the magnetization of this solid is M Nm tanh(mB/kT). Here tanh is the hyperbolic tangent function: tanh(x) (ex ex )/(ex ex ). (b) Show that the result given in (a) reduces to M Nm2 B/kT for . (c) Show that the result of (a) reduces to M Nm for . (d) Show that both (b) and (c) agree qualitatively with Fig. 32-14.