We have seen that an electric current is surrounded by a magnetic field; and also that, if a wire carrying a current is situated in an external magnetic field, it experiences a force at right angles to the current. It is therefore not surprising that two current-carrying wires exert forces upon each other.
More precisely, if there are two parallel wires each carrying a current in the same direction, the two wires will attract each other with a force that depends on the strength of the current in each, and the distance between the wires.
Definition. One amp (also called an ampère) is that steady current which, flowing in each of two parallel wires of negligible cross-section one metre apart in vacuo, gives rise to a force between them of 2 × 10-7 newtons per metre of their length.
At last! We now know what an amp is, and consequently we know what a coulomb, a volt and an ohm are. We have been left in a state of uncertainty until now. No longer!
But you may ask: Why the factor 2 × 10-7? Why not define an amp in such a manner that the force is 1 N m-1? This is a good question, and its answer is tied to the long and tortuous history of units in electromagnetism. I shall probably discuss this history, and the various “CGS” units, in a later chapter. In brief, it took a long time to understand that electrostatics, magnetism and current electricity were all aspects of the same basic phenomena, and different systems of units developed within each topic. In particular a so-called “practical” unit, the amp (defined in terms of the rate of deposition of silver from an electrolytic solution) became so entrenched that it was felt impractical to abandon it. Consequently when all the various systems of electromagnetic units became unified in the twentieth century (starting with proposals by Giorgi based on the metre, kilogram and second (MKS) as long ago as 1895) in the “Système International” (SI), it was determined that the fundamental unit of current should be identical with what had always been known as the ampère. (The factor 2, by the way, is not related to their being two wires in the definition.) The amp is the only SI unit in which any number other than “one” is incorporated into its definition, and the exception was forced by the desire to maintain the amp.
A proposal to be considered (and probably passed) by the Conférence Générale des Poids et Mesures in 2018 would re-define the coulomb in such a manner that the magnitude of the charge on a single electron is exactly 1.60217 x 10-19 C.
One last point before leaving this section. In the opening paragraph I wrote that “It is therefore not surprising that two current-carrying wires exert forces upon each other.” Yet when I first learned, as a student, of the mutual attraction of two parallel electric currents, I was very astonished indeed. The reason why this is astonishing is discussed in Chapter 15 (Special Relativity) of the Classical Mechanics section of these notes.