Theory and system analysis of field-dependent thermodynamic variables and Maxwell relations.

Abstract

Using previously developed principles of thermodynamics in the presence of fields a generalized set of field-dependent, extensive and intensive, thermodynamic variables is formulated. It is shown that extensive variables can depend on field constraints in the same way as the intensive ones do. Using these generalized variables, Maxwell relations are formulated and then tested under different field constraints. Conventional magnetic field systems are used to evaluate directly thermodynamic variables, which are then compared to those predicted by the theory. This provides a direct test and the means to illustrate the validity and significance of the theory in magnetic circuits. These tests involve uniform fields, continua and discrete systems that are subject to different field constraints. In this way the dependence of the thermodynamic variables on the field constraints is verified both directly and by the theory. The case of a sphere in a uniform field is used to show how thermodynamic variables can be defined in discrete systems. To this end an effective thermodynamic permeability of the sphere is defined. This facilitates the use of a model where the field is considered to be entirely within the boundary of the sphere, as if it were an ordinary thermodynamic system. It is shown that for this discrete system, there are more ways to define the magnetic chemical potential as compared to the case of a continuum. This is due to the fact that field constraints can be imposed independently on the sphere and on the uniform field. The significance of the volume of a subsystem that contains the sphere is considered. It is shown that the magnetic energy of this subsystem is a function of the volume of the sphere but not of that of the subsystem. This provides further insight concerning the significance of variables, such as volume, when the thermodynamics of a system is considered in the presence of fields. Finally, it is shown that, for the sphere, the ratio of the work delivered by a current source to build the field to the one that is delivered to an external mechanical work source, that balances quasistatically the magnetic pull, is one-third or less.