Mechanical Forces between Electric Currents and Saturated Magnetic Fields

The general case considered is that of N independent electric circuits placed in a medium of variable permeability and subject to saturation, in parts or as a whole. The problem is to determine the component (in a given direction) of the mechanical force acting upon one of the electric circuits, upon a group of circuits, or upon a group of circuits with part of the magnetic medium rigidly attached to them. It is believed that the problem has not been solved in this general form, heretofore. Use is made of the expression for the stored electromagnetic energy, W, of the system, assuming all the electric circuits to be originally open and then closed one by one. Such a treatment necessitates a number of partial saturation curves, giving g the linkages with each individual electric circuit when some of the remaining circuits are closed and the rest are open. A virtual displacement, ?? s, is then given to the part of the system under consideration, keeping either the linkages or the currents constant, and the mechanical force, F, is determined from a comparison of the work done, F. ?? s, with the change in the stored energy, ?? W. It is shown that the familiar reciprocal relationship for the mutual inductance, M12 = M21, which holds true in a medium without saturation, can be generalized to a more involved integral expression for a saturated medium.