# Free energy of a magnetic system

## Homework Statement

Please see P4, P5, and P6 in http://panda.unm.edu/pandaweb/graduate/prelims/SM_S09.pdf

"...What is the minimum work that must be performed to completely withdraw the core at constant I and T?"

## The Attempt at a Solution

I have no idea how to do this type of problems. But if I look at P5, using $$S=-\left(\frac{\partial F}{\partial T}\right)_V$$, given the free energy in P4, I get the answer for the case when the core is in the solenoid. So I guess that the free energy when the core is withdrawn is $$F=-\sigma TV\ln \frac{T}{T_0}$$. Is it correct?

To increase free energy, one has to put in work. Since free energy limits the maximum amount of work that can be extracted from a system, it should also limit the minimum amount of work that can be done on the system to do something. Since the magnetic field tends to pull the core into the solenoid and decrease the core's energy by the term $$\frac{1}{2}\mu \eta^2 I^2 V$$ in the free energy, that is the minimum amount of work we have to put in. This is my interpretation of my guess.

Any hint or useful equations for P6?

## Answers and Replies

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