Ampère formulated an elegant and powerful mathematical theory in which he treated the force between two current elements as an action at a distance having an inverse-square law of decay with distance of separation. The formulation of this theory, and a closely related one due to contemporaries Biot and Savart, are for now beyond the scope of what we are doing. In Newtonian fashion, Ampère avoided any hypothesis or model for the transmission of the force from one portion of the wire to the other. His theory was very successful in accounting for all effects that were observable at that time. It is now recognized that his basic equation is applicable only to conditions of steady (or slowly varying) current, but with this restriction it is a valid description of the force between current-carrying conductors.

Coupled with the great unification and simplification afforded by the overall view of magnetism as fundamentally associated with electric current, Ampère's theory dealt a final blow to the still-lingering hypothesis of imponderable austral (southerly) and boreal (northerly) fluids. Mention of these soon vanished from the literature.

Modern research in magnetism, still a highly active and rapidly progressing branch of solid state physics, has gone far beyond Ampère and has yielded many deep insights into the atomic-molecular mechanisms that govern the magnetic behavior of various materials. Materials such as iron, nickel, and cobalt are capable of very strong permanent magnetization and are called ferromagnetic. Certain other materials, such as manganese compounds and the so-called rare-earth elements, do not acquire permanent magnetization at ordinary temperatures and are only weakly magnetized and attracted in the vicinity of a strong magnet; these materials are called paramagnetic. A third class of materials, exemplified by bismuth, are very weakly repelled by a strong magnet, and are called diamagnetic. There is now some understanding of what characteristics of atomic structure make some substances ferromagnetic and other not, but Ampère's basic qualitative idea of the role of electricity in the magnetism of a bar magnet has been vindicated.

During the weeks while Ampère was performing his experiments with parallel wires, his friend and colleague Arago showed that iron could be weakly magnetized near a current-carrying conductor (Humphry Davy made essentially the same discoveries independently at the Royal Institution at about the same time). Ampère suggested that the effect might be greatly enhanced by placing an iron bar or steel needle within a solenoid where the magnetic influence would be stronger than near a single wire. This prediction was quickly confirmed. Since soft iron does not become permanently magnetized but readily shows induced magnetism when placed in a magnetic field, a soft iron bar inserted in a solenoid is magnetized when current flows in the solenoid, and not magnetized when the circuit is broken. This device is called an electromagnet, and still is used in everything from doorbells, through automatic valves on washing machines, to control elements in elaborate machinery.

Almost as soon as these various discoveries were made, Laplace, Ampère, and others pointed out that if wires were run between two widely separated points, the closing of a circuit at the first point, and the consequent flow of current in the wire, could be used to produce motion at the second point by virtue of the effect on a compass needle, or still better, by energizing an electromagnet. After adoption of a code, information might thus be transmitted between widely separated places by means of electrical signals. In subsequent years many individuals attempted the construction of such a telegraph, the first fully successful long-distance operation being carried out between Baltimore and Washington, D. C., in 1844 by Samuel F. B. Morse. The researches of Oersted, Ampère, and Arago, however, had been motivated by sheer curiosity concerning the phenomena of electricity and magnetism, and not by a plan to invent the electric telegraph. These men did not disdain the useful application of their discoveries, but the possibility of useful application played little or no direct role in their research.

From the 1820's and 1830's, applied and basic research in electricity and magnetism proceeded in closely related channels. Powerful electromagnets were being constructed by 1825. Electric motors soon followed. More stable and reproducible voltaic batters were manufactured. Generators and dynamos made their appearance after Faraday's discovery of electromagnetic induction in 1831. Engineering applications of the basic discoveries not only accelerated technological development, they also fed back into the laboratory, providing steadier and more powerful sources of electricity, leading to more precise measurements and to the construction of experiments not possible with earlier devices. This in turn led to new empirical discoveries and to further efforts aimed at theoretical synthesis of the accumulating experimental knowledge.

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