Consider two spin-1/2 subsystems with identical magnetic moments (μ) in equal fields (B). The first subsystem has a total of NA spins with initially "a" having magnetic moments pointing against the field and (NA - a) pointing along the field, so that its initial energy is UiA = μB(a - (NA - a) = μB(2a - NA). The second subsystem has a total of NB spins with "b" having moments initially pointing against the field so that its initial energy is UiB = μB(2b - NB). Now suppose that the two subsystems are brought together so that they can exchange energy. Assume that B = constant and a, b, NA, NB >> 1. Show that in equilibrium, a0/NA = b0/NB, and this implies that the two subsystems will have the same "magnetization," i.e. total magnetic moment/spin.
I'm not really sure what equations are useful in this case because I'm having trouble understanding what I need to be doing. I think I need to use multiplicity so
Ω(N,n) = N!/(n!(N-n)!)
The Attempt at a Solution
I think that I have to first figure out the most probable macrostate because that is where the systems would be in equilibrium(?). So do I go along the lines of solving Ω(NA, a) and Ω(NB, b)? I don't even know if that makes sense or what a0 and b0 represent in this question.