Page:Elementary Text-book of Physics (Anthony, 1897).djvu/358

344 work is done in this displacement. The system then returns to its original condition, and work equal to $$4 \pi i$$ is done upon the pole. This is expressed by saying that the potential of a closed current is multiply-valued. The work done during any movement depends not only on the position of the initial and, final points in the path, as in the case of the ordinary single-valued gravitational, electrical, and magnetic potentials, but also on the path traversed by the moving magnet pole. Every time the path encloses the current, work equal to $$4 \pi i$$ is done. The work done in moving by a path which does not enclose the current, from a point where the solid angle subtended by the circuit is $$\Omega '$$ to one where it is $$\Omega,$$ is, as in the case of the magnetic shell, equal to $$i (\Omega ' - \Omega).$$ If the path further enclose the current n times, the work done is $$4 \pi n i,$$ so that the total work done, or the total difEerence of potential between the two points, is where $$n$$ may have any value from 0 to infinity.

The fact that the potential of a current is multiply-valued is well illustrated by any one of a series of experiments due to Faraday. If we imagine a wire frame forming three sides of a rectangle to be mounted on a support so as to turn freely about one of its sides as a vertical axis, while the free end of the opposite side dips in mercury contained in a circular trough of which the axis of rotation passes through the centre, and if we suppose a current to be sent through the axis and the frame, passing out through the mercury; then if a magnet be placed vertically with its centre on the level of the trough, and with either pole confronting the frame, the frame will rotate continuously about the axis.

Other arrangements are made by which more complicated rotations of circuits can be effected. If the circuit be fixed and the magnet movable, similar arrangements will give rise to motions of the magnet or to rotations about its own axis.

291. Electromagnetic Unit of Current.—The relation which has been discussed between a circuit and the equivalent magnetic shell affords a means of defining a unit of current different from that