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## Homework Statement

Consider two spin-1/2 subsystems with identical magnetic moments (μ) in equal fields (B). The first subsystem has a total of N

_{A}spins with initially "a" having magnetic moments pointing against the field and (N

_{A}- a) pointing along the field, so that its initial energy is U

_{iA}= μB(a - (N

_{A}- a) = μB(2a - N

_{A}). The second subsystem has a total of N

_{B}spins with "b" having moments initially pointing against the field so that its initial energy is U

_{iB}= μB(2b - N

_{B}). Now suppose that the two subsystems are brought together so that they can exchange energy. Assume that B = constant and a, b, N

_{A}, N

_{B}>> 1. Show that in equilibrium, a

_{0}/N

_{A}= b

_{0}/N

_{B}, and this implies that the two subsystems will have the same "magnetization," i.e. total magnetic moment/spin.

## Homework Equations

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(N

_{A}, a) and Ω(N

_{B}, b)? I don't even know if that makes sense or what a

_{0}and b

_{0}represent in this question.