What Is the Minimum Work to Withdraw a Magnetic Core?

[Truecrimson]

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?"

Homework Equations

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 So I guess that the free energy when the core is withdrawn is $$F=-\sigma TV\ln \frac{T}{T_0}$$.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?

 

[mark726]

Hi there,

Thank you for posting your question on the forum. The problem you are working on is related to the thermodynamics of a magnetic system, and it involves calculating the minimum work required to withdraw a core from a solenoid at constant temperature and current.

In order to solve this problem, we can start by considering the free energy expression given in P4: F = -\sigma TV\ln \frac{T}{T_0}. This expression is valid when the core is inside the solenoid. However, when the core is withdrawn, the magnetic field inside the solenoid will decrease, leading to a change in the free energy of the system. This change in free energy can be calculated using the Maxwell's equations for a magnetic system, which relate the magnetic field, current, and magnetic flux.

Using these equations, we can calculate the change in free energy as the core is withdrawn from the solenoid. This change in free energy corresponds to the minimum work that must be performed in order to withdraw the core at constant temperature and current. Therefore, the minimum work required is given by the change in free energy, which can be calculated using the Maxwell's equations.

As for P6, it involves calculating the change in entropy of the system as the core is withdrawn from the solenoid. This can be done by considering the change in the magnetic field and current, and using the expression for entropy given in P5: S=-\left(\frac{\partial F}{\partial T}\right)_V. By calculating the change in entropy, we can determine how the entropy of the system changes as the core is withdrawn, which can give us insights into the thermodynamics of the system.

I hope this helps. Good luck with your problem!