On the Definition of “Hidden” Momentum

In 1822 [1, 2], Ampère argued that the magnetic interaction between two electrical circuits could be written as a double integral over the current elements in the two circuits, with the vector integrand being along the line of centers of pairs of current elements. This insured that the total force on each circuit obeyed Newton’s third law, i.e., that mechanical momentum is conserved. Ampère hesitated to interpret the integrand as the force law between isolated current elements, perhaps because the integrand did not factorize. In 1845, Grassmann [3] gave an alternate form of the integrand which did factorize (into what is now called the Biot-Savart form), such that the force on each circuit had the same value as in Ampère’s calculation, but the forces on a pair of isolated current elements did not obey Newton’s third law, i.e., that mechanical momentum would not be conserved in the interaction of isolated current elements. Maxwell (sec. 527 of [4]) favored Ampère over Grassmann (and the school of Neumann and Weber) in this matter, and was skeptical of the notion of free moving charges. The concept of free moving charges was revived by J.J. Thomson beginning with speculations as to electromagnetic mass/momentum (1881) [8], and continuing with a concept of momentum stored in the electromagnetic field (1891) [9, 10], with the fieldmomentum density being the Poynting vector [11] divided by c, pEM = S/c , where c is the speed of light in vacuum. Thomson was also the first to consider (1904) [15, 16, 17] an example in which an electromechanical system initially “at rest” leads to motion of part of the system if the magnetic field later drops to zero. This motion was attributed to a conversion of the initial electromagnetic field momentum into final mechanical momentum. However, Thomson did not notice that it was odd that a system initially “at rest” could end with nonzero total momentum. Thomson’s 1904 paradox seems to have gone unnoticed until 1949, when Slepian published a delightful version of it involving a (quasistatic) AC circuit [19], arguing that once electromagnetic field momentum is taken into consideration, total momentum is conserved and the device cannot be used for electromagnetic rocket propulsion. A variant closer to