Halliday, Resnick & Walker — Maxwell's Equations; Magnetism of Matter: Problem 37

37 Consider a solid containing N atoms per unit volume, each atom having a magnetic dipole moment \(\vec{\mu}\). Suppose the direction of \(\vec{\mu}\) can be only parallel or antiparallel to an externally applied magnetic field \(\vec{B}\) (this will be the case if \(\vec{\mu}\) 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 \(e^{-U/kT}\), where T is the temperature and k is Boltzmann’s constant. Thus, because energy U is \(-\vec{\mu} \cdot \vec{B}\), the fraction of atoms whose dipole moment is parallel to \(\vec{B}\) is proportional to \(e^{\mu B/kT}\) and the fraction of atoms whose dipole moment is antiparallel to \(\vec{B}\) is proportional to \(e^{-\mu B/kT}\). (a) Show that the magnitude of the magnetization of this solid is \(M = N\mu \tanh(\mu B/kT)\). Here \(\tanh\) is the hyperbolic tangent function: \(\tanh(x) = (e^x - e^{-x})/(e^x + e^{-x})\). (b) Show that the result given in (a) reduces to \(M \approx N\mu^2 B/kT\) for \(\mu B \ll kT\). (c) Show that the result of (a) reduces to \(M \approx N\mu\) for \(\mu B \gg kT\). (d) Show that both (b) and (c) agree qualitatively with Fig. 32-14.