Page:Elementary Text-book of Physics (Anthony, 1897).djvu/368

354 and any current whatever through $$B,$$ it is found that $$B$$ does not move. The opposing forces due to the actions of $$A$$ and $$C$$ upon $$B$$ are in equilibrium. From this fourth case of equilibrium is deduced the law that the force between two current elements is inversely as the square of the distance between them.

Ampère made the assumption that the action between two current elements is in the line joining them. Prom the four cases of equilibrium he then deduced an expression for the attraction between two current elements. It is In this formula $$ds$$ and $$ds'$$ represent the elements of the two circuits, $$i$$ and $$i'$$ the strength of current in those circuits measured in electromagnetic units, $$r$$ the distance between the current elements, $$\epsilon$$ the angles made by the two elements with one another, $$\theta$$ and $$\theta '$$ the angles made by $$ds$$ and $$ds'$$ with $$r$$ or $$r'$$ produced, the direction of the two elements being taken in the sense of their respective currents.

298. Solenoids and Electromagnets.—Ampère also showed that the action between two small plane circuits is the same as that between two small magnetic shells, and that a circuit, or system of circuits, may be constructed which is the complete equivalent of any magnet. A long bar magnet may be looked on as made up of a great number of equal and similar magnetic shells arranged perpendicularly to the axis of the magnet, with their similar faces all in one direction. In order to produce the equivalent of this arrangement with the circuit, a long insulated wire is wound into a close spiral, straight and of uniform cross-section. The end of the wire is passed back through the spiral. When the current passes, the action of each turn of the spiral may be resolved into two parts—that due to the projection of the spiral on the plane normal to the axis, and that due to its projection on the axis. This latter component, for every turn, is neutralized by the current in the returning wire, and the action of the spiral is reduced to that of a number of similar plane circuits perpendicular to its axis.