Need help with the following problem. (a) Consider an assembly of n weakly interacting magnetic atoms per unit volume at atemperature T and describe the situation classically. Then each magnetic moment ucan 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+d0is simply proportional to the solid angle 27 sin Ode enclosed in this range. In the presenceof a magnetic field H in the z-direction, this probability must further be proportional tothe Boltzmann factor e-BE where E is the magnetic energy of the moment u making thisangle 0 with the z-axis. Use this result to calculate the classical expression for the meanmagnetic moment M, of the n atoms (per unit volume). [Remember that theclassical magnetic moment u in a magnetic field H is given by E = -µ·H = -µH cos 0.]energy ofa(b) Show that in the high temperature regime (kT > µH), the magnetization M; in part(a) has the form M, = xH, where the susceptibility x is given bynu?X =3kTThus a classical magnetic moment obeys Curie's Law, just like a quantum magneticmoment, but x is a factor of 3 smaller classically than for a quantum spin-1/2 particle.(For those who are interested, one can show that a quantum spin with spin J approachesthe classical result for J → .)

Solution: The magnetic atoms can orient at any angle θ between 0 to π. Here θ is the continuous…

(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.]