Appendix A The Origins and Meanings of the Magnetic Force F = qv x B In Chapter 2 we showed that the electric component of Lorentz's force (FE = qE, with E = -\l4> - 8XIOt) had been utilized by Kirchhoff in 1857, although Kirchhoff did not speak in terms of electric fields. The magnetic vector potential X had been introduced by Franz Neumann in 1845. We also saw that B = \l x X, where B is the magnetic field. On the other hand the expression for the magnetic force had a later and more tortuous origin, and this is the subject of this appendix. We first discussed this topic in (Assis and Peixoto, 1992). The first relevant information to be stressed is that the expression for the magnetic force appeared after Maxwell's death (1879). According to Whittaker the first to arrive at the magnetic force were J. J. Thomson (1856-1940) and O. Heaviside (1850-1925), in 1881 and 1889, respectively (Whittaker, 1973, Vol. 1, pp. 306 to 310). One of the goals of Thomson's theoretical paper (Thomson, 1881) was to know how an electrified body is affected bya magnet. Thomson followed Maxwell's theory and in particular he utilized the idea that a displacement current (c8EIOt) produces the same effect as an ordinary conduction current 1, namely, a magnetic field B. He supposed a uniformly charged sphere moving in a certain medium with dielectric constant c and magnetic permeability Il, and calculated the displacement current at an external point Q. Then he calculated at another external point P the value of the vector potential X due to this displacement current in Q, and integrated over all points Q in space. But he then observed that the value of \l . X at this point P was different from zero. Maxwell, however, always assumed \l . X = O. Then to satisfy this condition Thomson supposed the existence of another component in X, adding this component to what he had obtained for X (he did not justify the physical origin of this extra component of X). Through B = \l x X he obtained the value of B at P. He then calculated the value of ii in this medium, Bill. Next he calculated the force exerted by a magnet (which generates B) on The Origins and Meanings 01 the Magnetic Force F = qv x B 239 a charged body which moves in this medium. To this end he calculated the interacting energy E = J J J(B. ii)2)dV, and utilized Lagrange's equations to get the force. His final result was ~ v x B F=q-- (Al) 2 This is half the present day value of classical electromagnetism. The most important aspect which we want to emphasize here is the meaning of the velocity which appears in (Al). Here Thomson was careful. He called this the "actual velocity" of the charge. On page 248 of his article he says: "It must be remarked that what we have for convenience called the actual velocity of the particle is, in fact, the velocity of the particle relative to the medium through which it is moving " ... , "medium whose magnetic permeability is /-t." This shows that to Thomson the velocity v in (Al) was not the velocity of the charge relative to the ether (as this medium and the earth might be moving relative to the ether without dragging it), nor relative to the magnet, and not either the velocity relative to the observer. Thomson extended his researches in this direction with a paper published in 1889 (Thomson, 1889). In 1889, in another theoretical paper, Heaviside obtained (Heaviside, 1889): F = qv x B . (A2) The main difference of his work from Thomson's 1S that he included, following Fitzgerald in 1881, the convection current as a source of magnetic field. In the other aspects he followed Thomson's work. He did not make any additional comment on the velocity v in (A2). So it can be assumed that for hirn as weIl this is the velocity of the charge q relative to the medium of magnetic permeability /-t and dielectric constant c. This is even more evident from the title of his paper: On the electromagnetic effects due to the motion of electrification through a dielectric. A discussion of the works of Thomson and Heaviside can be found in (Buchwald, 1985, Appendix I: Maxwellian Analysis of Charge Convection, pp. 269 - 277). 240 Appendix A In 1895, the theoretical physicist Lorentz presented the known expression (Lorentz, 1895; Pais, 1982, p. 125; and Pais, 1986, p. 76): F = qE + qv x B . (A3) Lorentz did not comment on the works of Thomson and Heaviside, and arrived at the magnetic part of this expression from Grassmann's force, substituting qv for I d~ although he did not mention Grassmann's work as weH. This can be seen in Lorentz most famous book, The Theory 01 Electrons (Lorentz, 1915, pp. 14 and 15). This book is based on a course he delivered in 1906 at Columbia University, which was edited for the first time in 1909. Unfortunately Lorentz did not specify in (A3) what is the object, medium or system relative to which the velocity v of the charge q is to be understood. As Lorentz still accepted MaxweH's ether (that is, a medium in astate of absolute rest relative to the frame of the fixed stars; see (Pais, 1982, p. 111)), it is natural to suppose that for him this was the velocity of the charge q relative to this ether, and not relative to any other medium or observer. In support of this statement we have Lorentz's own words on this same page 14: "Now, in accordance with the general principles of MaxweH's theory, we shall consider this force as caused by the state of the ether, and even, since this medium pervades the electrons, as exerted by the ether on all internal points of these particles where there is acharge" (Lorentz, 1915, p. 14). A conclusive proof of this interpretation can be found in another work of Lorentz: Lectures on Theoretical Physics (Lorentz, 1931, Vol. 3, p. 306; se also O'Rahilly, 1965, Vol. 2, p. 566). Here Lorentz says that if a wire carrying an electric current (and thus generating B), and a charge are at rest relative to the ether, then there will be no magnetic force. On the other hand if both share a common translation with velocity v relative to the ether (while the observer and the laboratory also translate with this same velocity v, because he gives as an example of this velocity that of the whole earth relative to the ether), then the says that there will be a magnetic force on the test charge. In this example there is no relative motion between the test charge and the current carrying wire, the observer, nor the laboratory, but only relative to the ether. On the other hand nowadays we utilize expression (A3) with v being the velocity of the charge q relative to an observer. This change of meaning happened after Einstein's The Origins and Meanings of the Magnetic Force P = qiJ x jj 241 work of 1905 on the special theory of relativity (Einstein, Lorentz, Minkowski and Weyl, 1952). In this work, after obtaining Lorentz's transformations of coordinates, Einstein applies them for the force (A3) and begins to utilize the velocity as being relative to the observer. For instance, on page 54 he gives (between square brackets are our words) the difference between the old paradigm of electromagnetism and the new one based on his theory of relativity: "Consequently the first three equations above [for the transformation of the field components in different inertial systems] allow themselves to be clothed in words in the two following ways: 1. If a unit electric point charge [q = 1] is in motion in an electromotive field, there acts upon it, in addition to the electric force [PE = qEJ, an "electromotive force" which, if we neglect the terms multiplied by the second and higher powers of v / c, is equal to the vector-product of the velocity of the charge and the magnetic force [.8], divided by the velocity of light [that is, PM = qv x .8 / c, so that the resultantforce is P = qE + qv x .8/ cl. (Old manner of expression). 2. If a unit electric point charge is in motion in an electromagnetic field, the force acting upon it is equal to the electric force [P' = qE'] which is present at the locality of the charge, and which we ascertain by transformation of the field [0 --t 0'] to a system of co­ ordinates [0'] at rest relatively to the electrical charge [iJ I = 0, so that qE+qiJx.8 /c = qE', where all magnitudes with ' refer to the fields in the system 0' which moves with velocity iJ relative to 0]. (New manner of expression)." It is instructive to see this conceptual change in one of the most utilized expressions of physics. It should be stressed that all of these works (Thomson, Heaviside, Lorentz, Einstein) were of a theoretical nature and realized after Maxwell's death. This change in the meaning of v in (A3) is very strange, confusing and unusual in physics. Appendix B Alternative Formulations of Electrodynamics In this book we dealt with mainly Weber's theory and classical electrodynamics (Maxwell's equations, Lorentz's force, retarded and Lienard-Wiechert potentials, etc.) In this Appendix we present some other approaches which have been proposed by some important scientists like Gauss, Riemann, Clausius and Ritz.