# Thermodynamic electric and magnetic susceptibility

Are there differences between how the electric and magnetic susceptibilities are defined in thermodynamics and electrodynamics? My confusion is that it seems the techniques from thermodynamics (from my "elementary," undergraduate level) give different answers in the case of magnetic susceptibility, so I'm not sure if there is other physics going on or if it is as simple as a matter of definition. For example, in thermodynamics one would consider some external magnetic field $$\mathbf B$$ acting on the system, and use that to find the partition function. Then the magnetization is given by (using the free energy) $$M = -\left(\frac{\partial F}{\partial B}\right)_N$$ and the magnetic susceptibility is $$\chi = \frac{\partial M}{\partial B}_{B\to 0}$$ From what I know, that would mean $$\mathbf M = \chi\mathbf B$$. Then in terms of electromagnetism one would have $$\mathbf B = \mu_0(\mathbf H + \mathbf M) = \mu_0(\mathbf H + \chi\mathbf B)\implies \mathbf H = \frac{1}{\mu_0} \mathbf B - \chi \mathbf B$$. That is, however, clearly different from electrodynamics where $$\mathbf M = \chi_m\mathbf H$$. Obviously $$\chi_m$$ and $$\chi$$ are related, but I have seen some sources, lecture notes and thermodynamics textbooks, call $$\chi = \chi_m$$. Why is there this difference? A further point of confusion: In that external field used to calculate the free energy, it seems that in a vacuum one can choose $$\mathbf H$$ or $$\mathbf B$$ and get the same answer, up to a scaling constant, however, the answers will not be simply related by a scaling constant when considering $$\mathbf B = \mu_0(\mathbf H + \mathbf M)$$.

Magnetic susceptibility in EM theory is defined as rate of change of magnetization per unit change of magnetic field strength $$\mathbf H$$, not $$\mathbf B$$. This is an age old convention for which there are good practical reasons. It can and preferably should be used in thermodynamics as well.

Your $$\chi$$ based on $$\mathbf B$$ in general isn't equal to $$\chi_m$$, these two quantities have different units. They may have similar values if magnetization is very weak and a system of units where $$\mathbf B$$ and $$\mathbf H$$ have the same units.

• Okay, thanks. However, I’m still left wondering the last point I made. Depending on what I express energy in terms of, the external field only changes by a scaling, so the magnetic susceptibility only scales from one case to another. That leaves me with wondering where my mistake is in the thermodynamic part. There has to be some difference in the definitions or the partition function when using H or B as external field that I’m not getting. Commented May 3, 2021 at 23:47
• I don't understand your description and problem regarding energy, scaling and thermodynamics. What are you trying to do? I recommend posting a new, better phrased question. Commented May 3, 2021 at 23:52
• I mean one can, in vacuum, replace B with $\mu_0 H$ so then that will not affect the analysis in thermodynamics except by scaling the field by $\mu_0$. Commented May 3, 2021 at 23:56
• I put a new question with a specific example. I’m fairly sure I have some fundamental misunderstanding which is why I am running into this problem. Commented May 4, 2021 at 2:32