# Heat capacity of magnetic dipole in magnetic field

1. Dec 18, 2014

### Nikitin

edit: The title is misleading, sorry. Originally I wanted to ask a question about the heat capacity but I figured it out and changed the question while forgetting to change the thread title..

Hi. OK, assume we have a classic magnetic dipole in a magnetic field with $H= - \vec{\mu} \cdot \vec{B}$. Then you can show that the canonical partition function becomes $Z= \frac{4 \pi}{\beta \mu B} \sinh{\beta \mu B}$, where $\beta = 1/(k_B T)$.

Using $Z$, you can show that the mean energy $<E>= \frac{1}{3} \frac{(\mu B)^2}{k_B T}$ in the limit $T \rightarrow \infty$.

I have a question that isn't really specific to the system, but something more general: As temperature approaches infinity, all the energy-states for my magnetic dipole will become equally probable. Why?

According to the ergodic hypothesis, every microstate is equiprobable. So in the limit $T$ being very large, does every energy state become so small compared to $k_B T$ that they all have roughly the same energy, ~$0$? And then since they have roughly the same energy , the system effectively becomes a micro-canonical ensemble where the ergodic hypothesis applies?

Last edited: Dec 18, 2014
2. Dec 18, 2014

### Staff: Mentor

Right. For large temperatures, the energy difference between the states just becomes negligible.