(a) Consider an assembly of n weakly interacting magnetic atoms per unit volume at a temperature T and describe the situation classically. Then each magnetic moment u can make an arbitrary angle 0 with respect to a given direction (call it the z direction). In the absence of a magnetic field, the probability that this angle lies between 0 and 0+d0 is simply proportional to the solid angle 27 sin Od0 enclosed in this range. In the presence of a magnetic field H in the z-direction, this probability must further be proportional to the Boltzmann factor e-BE, where E is the magnetic energy of the moment µ making this angle 0 with the z-axis. Use this result to calculate the classical expression for the mean magnetic moment M, of the n atoms (per unit volume). [Remember that the energy of a classical magnetic moment µ in a magnetic field H is given by E = –µ·H = -µH cos 0.]
(a) Consider an assembly of n weakly interacting magnetic atoms per unit volume at a temperature T and describe the situation classically. Then each magnetic moment u can make an arbitrary angle 0 with respect to a given direction (call it the z direction). In the absence of a magnetic field, the probability that this angle lies between 0 and 0+d0 is simply proportional to the solid angle 27 sin Od0 enclosed in this range. In the presence of a magnetic field H in the z-direction, this probability must further be proportional to the Boltzmann factor e-BE, where E is the magnetic energy of the moment µ making this angle 0 with the z-axis. Use this result to calculate the classical expression for the mean magnetic moment M, of the n atoms (per unit volume). [Remember that the energy of a classical magnetic moment µ in a magnetic field H is given by E = –µ·H = -µH cos 0.]
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