# Heat capacity of magnetic dipole in magnetic field

• Nikitin
In summary, the conversation discusses the canonical partition function and mean energy of a classic magnetic dipole in a magnetic field at high temperatures. The question is raised about the equal probability of all energy states at infinite temperature, which is explained by the ergodic hypothesis. At high temperatures, the energy difference between states becomes negligible, making the system effectively a micro-canonical ensemble.

#### 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?

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

## 1. What is the heat capacity of a magnetic dipole in a magnetic field?

The heat capacity of a magnetic dipole in a magnetic field is a measure of the amount of heat energy required to raise the temperature of the dipole by a certain amount. It is dependent on the strength of the magnetic field and the properties of the material the dipole is made of.

## 2. How does the heat capacity of a magnetic dipole change with temperature?

The heat capacity of a magnetic dipole typically increases with temperature. This is because as the temperature increases, the thermal energy of the dipole increases, making it more difficult to raise its temperature further.

## 3. What is the role of thermal energy in determining the heat capacity of a magnetic dipole?

Thermal energy plays a crucial role in determining the heat capacity of a magnetic dipole. It is the thermal energy that is responsible for the motion of the particles that make up the dipole, and this motion contributes to the overall heat capacity of the system.

## 4. How does the heat capacity of a magnetic dipole in a magnetic field differ from that of a non-magnetic dipole?

The heat capacity of a magnetic dipole in a magnetic field is typically higher than that of a non-magnetic dipole. This is because the magnetic field interacts with the magnetic dipole, resulting in additional energy being required to raise the temperature of the system.

## 5. Can the heat capacity of a magnetic dipole be manipulated?

Yes, the heat capacity of a magnetic dipole can be manipulated by changing the strength of the magnetic field or by altering the properties of the material the dipole is made of. This can be useful in various applications, such as in the design of magnetic refrigeration systems.