# Heat capacity of magnetic dipole in magnetic field

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?

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## Answers and Replies

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