Fhd expression bothering me for weeks

[da_willem]

Suppose the magnetization of some substance depends on the field H and temperature, i.e. M(H,T) and we have the mathematical identity

$$\nabla \int _0 ^H M dH = M \nabla H + \int _0 ^H \nabla M dH$$

then it is derived in Ferrohydrodynamics (fhd) by Rosensweig that

$$\nabla \int _0 ^H M dH = M \nabla H + \int _0 ^H \frac{\partial M}{\partial T} dH$$

but what happened with the dependence of M on H?! Shouldn't the above include a term

$$\int _0 ^H \frac{\partial M}{\partial H} \nabla H dH$$

or is there some reason this should vanish? Can someone help me out of my misery?

 

[eljose79]

Hello there,

Thank you for bringing this interesting topic to our attention. I understand your confusion and I will try my best to provide some insight into this matter.

Firstly, it is important to note that the mathematical identity you have mentioned is known as the Helmholtz free energy equation, which is a fundamental equation in thermodynamics. It relates the internal energy of a system to its entropy and volume, and can also be used to derive other thermodynamic equations.

Now, in the case of ferrohydrodynamics, we are dealing with a magnetic substance that also exhibits fluid-like behavior. This means that the substance is not only affected by the magnetic field H, but also by temperature T. Therefore, the magnetization M is a function of both H and T, i.e. M(H,T).

When we apply the Helmholtz free energy equation to this system, we need to take into account the variations of M with both H and T. This is why the term \frac{\partial M}{\partial T} appears in the equation derived by Rosensweig. This term represents the change in magnetization with respect to temperature, which is a crucial factor in ferrohydrodynamics.

As for the missing term \int _0 ^H \frac{\partial M}{\partial H} \nabla H dH, it is important to note that this term is not included in the Helmholtz free energy equation. This is because the equation assumes that the magnetic field H is constant and does not change with respect to itself. In other words, the equation only considers the change in H with respect to other variables, such as temperature.

I hope this helps clarify your doubts. If you need further assistance, please do not hesitate to ask. Keep up the good work in your research!

 

[charng]

I can understand your frustration with this expression and the potential discrepancies in the mathematical identity. It is important to note that in scientific research, there is always room for further exploration and clarification. It is possible that there may be additional factors or assumptions that need to be considered in this equation.

One potential explanation could be that the dependence of M on H is negligible in this particular scenario, and therefore the term \int _0 ^H \frac{\partial M}{\partial H} \nabla H dH can be omitted. However, it is always important to thoroughly examine and test all variables in order to ensure the accuracy and validity of the results.

In order to fully understand the implications of this equation, it may be helpful to consult with other experts in the field or conduct further research to gain a deeper understanding of the underlying principles. As scientists, it is our responsibility to continue to question and seek answers in order to advance our understanding of the natural world. I hope this response has provided some insight and guidance in your quest for clarification.