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For a magnetic system held at constant T and H (see 1),

An Introduction to Thermal Physics | 1st Edition | ISBN: 9780201380279 | Authors: Daniel V. Schroeder ISBN: 9780201380279 40

Solution for problem 47P Chapter 5

An Introduction to Thermal Physics | 1st Edition

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An Introduction to Thermal Physics | 1st Edition | ISBN: 9780201380279 | Authors: Daniel V. Schroeder

An Introduction to Thermal Physics | 1st Edition

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Problem 47P

For a magnetic system held at constant T and H (see Problem 1), the quantity that is minimized is the magnetic analogue of the Gibbs free energy, which obeys the thermodynamic identity

Phase diagrams for two magnetic systems are shown in Figure 5.14; the vertical axis on each of these figures is μ0H.

(a) Derive an analogue of the Clausius-Clapeyron relation for the slope of a phase boundary in the H-T plane. Write your equation in terms of the difference in entropy between the two phases.

(b) Discuss the application of your equation to the ferromagnet phase diagram in Figure 5.14.

(c) In a type-I superconductor, surface currents flow in such a way as to completely cancel the magnetic field (B, not H) inside. Assuming that M is negligible when the material is in its normal (non-superconducting) state, discuss the application of your equation to the superconductor phase diagram in Figure 5.14. Which phase has the greater entropy? What happens to the difference in entropy between the phases at each end of the phase boundary?

Problem 1:

The enthalpy and Gibbs free energy, as defined in this section, give Special treatment to mechanical (compression-expansion) work, – P dV. Analogous quantities can be defined for other kinds of work, for instance, magnetic work. Consider the situation shown in below Figure, where a long solenoid [N turns, total length L) surrounds a magnetic specimen (perhaps a paramagnetic solid). If the magnetic field inside the specimen is  and its total magnetic moment is , then we define an auxiliary field  (often called simply the magnetic field) by the relation

where μo is the “permeability of free space,” 4π × 10–7 N/A2. Assuming cylindrical symmetry, all vectors must point either left of right, so we can drop the symbols and agree that rightward is positive, leftward negative. From Ampere’s law, one can also show that when the current in the wire is I, the H field inside the solenoidis NI/L, whether or not the specimen is present.

(a) Imagine making an infinitesimal change in the current in the wire, resulting in infinitesimal changes in B, M, and H. Use Faraday’s law to show that the work required (from the power supply) to accomplish this change is Wtotal = VHdB. (Neglect the resistance of the wire.)

(b) Rewrite the result of part (a) in terms of H and M, then subtract off the work that would be required even if the specimen wore not present. If we define W, the work done on the system J to be what’s left, show that W = μoHdM.

(c) What is the thermodynamic identity for this system? (Include magnetic work but not, mechanical work or particle flow.)

(d) How would you define analogues of the enthalpy and Gibbs free energy for a magnetic system? (The Helmholtz free energy is defined in the same way as for a mechanical system.) Derive the thermodynamic identities for each of these quantities, and discuss their interpretations.

Figure 1: A long solenoid, surrounding a magnetic specimen, connected to a power supply that can change the current, performing magnetic work.

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Textbook: An Introduction to Thermal Physics
Edition: 1
Author: Daniel V. Schroeder
ISBN: 9780201380279

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For a magnetic system held at constant T and H (see 1),