Average Internal Energy of 2 Paramagnets

[Poop-Loops]

Homework Statement

The problem is that there are two , two-state paramagnets with Na = Nb (number of dipoles in each). uB = |kT|, energies of each dipole are + or - uB. Different internal energies to start, but they are brought together at thermal equilibrium and I need to find the average internal energy of each after that happens.

Homework Equations

$$U = -N \frac{(\mu B)^2}{kT}$$

The Attempt at a Solution

Since N=N, my only thought is Ua + Ub divided by 2. If Ua = Ub, then that makes sense. If Ua > Ub, but both < 0, then Ua would give energy to Ub, since the temperature is higher, yes? If Ua < Ub but both Ub > 0, Ua <0, then it depends on the exact values, but they would both tend towards having infinite temperature, since they would both want to have higher entropy, right? It just seems like I'm missing something really big here.

My book is Schroeder's Thermal Physics, which doesn't have any answers in the back, so I can't even tell if I'm doing the problems correctly.

EDIT:

My other idea is that both temperatures have to be equal to have thermal equilibrium, so I should solve for temperature first. Then solve it so that Ta = Tb and see what happens then?

EDIT2: That doesn't seem to work, either, since I get Ua/Ub = Ta/Tb, but have no idea what to do with that...

 

[Poop-Loops]

Hate to bump, but I have to. I have no idea whether I'm right or not, or whether I'm even on the right track.