Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/282

of two magnetic shells, of strengths $\phi$ and $\phi ^{\prime }$ and bounded by the closed curves $s$ and $s^{\prime }$ respectively, is

where $\epsilon$ is the angle between the directions of $ds$ and $ds^{\prime }$, and $r$ is the distance between them. We also found in Art. 521 that the mutual energy of two circuits $s$ and $s^{\prime }$, in which currents $i$ and $i^{\prime }$ flow, is

If $i$, $i^{\prime }$ are equal to $\phi$, $\phi ^{\prime }$ respectively, the mechanical action between the magnetic shells is equal to that between the corresponding electric circuits, and in the same direction. In the case of the magnetic shells, the force tends to diminish, their mutual potential energy, in the case of the circuits it tends to increase their mutual energy, because this energy is kinetic.

It is impossible, by any arrangement of magnetized matter, to produce a system corresponding in all respects to an electric circuit, for the potential of the magnetic system is single valued at every point of space, whereas that of the electric system is many- valued.

But it is always possible, by a proper arrangement of infinitely small electric circuits, to produce a system corresponding in all respects to any magnetic system, provided the line of integration which we follow in calculating the potential is prevented from passing through any of these small circuits. This will be more fully explained in Art. 833.

The action of magnets at a distance is perfectly identical with that of electric currents. We therefore endeavour to trace both to the same cause, and since we cannot explain electric currents by means of magnets, we must adopt the other alternative, and explain magnets by means of molecular electric currents.

638.] In our investigation of magnetic phenomena, in Part III of this treatise, we made no attempt to account for magnetic action at a distance, but treated this action as a fundamental fact of experience. We therefore assumed that the energy of a magnetic system is potential energy, and that this energy is diminished when the parts of the system yield to the magnetic forces which act on them.

If, however, we regard magnets as deriving their properties from electric currents circulating within their molecules, their energy