Consider a paramagnet; i.e. a system of magnetic dipoles in an external magnetic field, B~ , where interactions between dipoles can be ignored. Any dipole can be considered the ”system”, S and all others the ”reservoir” at constant temperature, T. Recall that the energy for a dipole that is aligned with B~ is ↑ = −µB and that the energy for a dipole anti-aligned with B~ is ↓ = µB. (Recall that µ is the magnetic moment of the dipole and B is the strength of the magnetic field.) It might be helpful for you to use e^(x) + e^(-x) = 2 cosh x and if you do use this you may need (d)/(dx) cosh x = sinh x and tanh x = (sinh x)/(cosh x).
(a) (3 pts) Calculate the partition function for one dipole.
(b) (3 pts) Calculate the probability of finding the dipole in the ”up” state. Also calculate the probability of finding the dipole in the ”down” state.
(c) (5 pts) Now consider the whole system of N dipoles. What is the average number of dipoles in the ”up” state, ¯n↑?
(d) (15 pts) Calculate the average of the ratio of the number of dipoles in the ”up” state to the number of dipoles in the ”down” state, i.e. n↓/n↑
(e) (10 pts) Make a sketch of this quantity n↓/n↑ versus temperature and describe in detail what happens T = 0 and T = ∞. Make sure you say ”why” the system behaves the way it does at these limits!
(f) (15 pts) Assume that the particles are distinguishable and write down an expression for the partition function for the whole system of N dipoles.
(g) (10 pts) Calculate the entropy of this system (N distinguishable, non-interacting dipoles) in terms of N, kB, T, µ, and B