Evaluation of forces in magnetic materials by means of energy and co-energy methods

The evaluation of the total force of magnetic origin acting upon a body in a stationary magnetic field is often carried out using the so-called magnetic energy (or co-energy) method, which is based on the derivation of the magnetic energy (or co-energy) with respect to a virtual rigid displacement of the considered body. The application of this method is usually justified by resorting to the energy conservation principle written in terms both of electrical and of mechanical quantities. In this paper we shall re-examine the whole matter in the context of classical thermodynamics, in order to obtain a more comprehensive and general proof of the validity of the energy (or co-energy) approach and to point out its limitations. Two typical configurations will be discussed; in the first one, the field sources are represented by conducting bodies carrying free currents, whereas in the second one a permanent magnet creates the driving field. All magnetic materials are assumed to be non-hysteretic and permanent magnets are represented by means of the well-known linear model in the second quadrant of the (B,H) plane.