PRAMANA ­c Indian Academy of Sciences Vol. 55, No. 3 — journal of September 2000 physics pp. 393–404 Comparison between Weber’s electrodynamics and classical electrodynamics A K T ASSIS £ and H TORRES SILVA Facultad de Ingenieria, Universidad de Tarapaca, Av. 18 de Septiembre 2222, Arica, Chile £ Permanent address: Instituto de F´ısica ‘Gleb Wataghin’, Universidade Estadual de Campinas - Unicamp, 13083-970 Campinas, S˜ao Paulo, Brasil Email: [email protected]; [email protected] MS received 20 January 2000; revised 25 April 2000 Abstract. We present the main aspects of Weber’s electrodynamics and of Maxwell’s equations. We discuss Maxwell’s point of view related to Weber’s electrodynamics. We compare Weber’s force with Lorentz’s force. We analyse the relation between Weber’s law and Maxwell’s equations. Finally, we discuss some experiments performed and proposed with which we can distinguish Weber’s force from Lorentz’s one. Keywords. Weber’s electrodynamics; Maxwell’s equations; Lorentz’s force; classical electromag- netism. PACS Nos 41.20.Bt; 41.20.-q; 03.50.De 1. Weber’s electrodynamics Wilhelm Weber (1804–1891) developed his electrodynamics during the same period in which James Clerk Maxwell (1831–1879) was putting together what are known as Maxwell’s equations. In this work we compare these two electrodynamics. We begin presenting the main aspects of Weber’s electrodynamics. For complete ref- Õ ~Ö ¾ erences, see [1] and [2]. Suppose we have a point charge ¾ located on relative to the Ë ~Ú ~a ¾ origin of an inertial frame of reference , moving with velocity ¾ and acceleration , Õ ~Ö ~Ú ~a ½ ½ ½ and another point charge ½ located on and moving with velocity and acceleration relative to Ë . Weber’s force (1846) exerted by 2 on 1 is given by ¾ Õ Õ Ö^ Ö_ ÖÖ ½ ¾ ~ F = ½ · ¾½ ¾ ¾ ¾ 4" Ö ¾c c ¼ Õ Õ Ö^ ½ ¿ ½ ¾ ¾ ½· ~Ú ¡~Ú ´^Ö ¡ ~Úµ · ~Ö ¡ ~a : = (1) ¾ ¾ 4" Ö c ¾ ¼ ½¾ ¾ ½ ¾ " 8:85 ¢ ½¼ In this equation ¼ C N m is called the permittivity of free space, Ö j~Ö ~Ö j ~Ö ~Ö ~Ö ¾ ½ ¾ ½ is the distance between the charges, is the vector pointing from 393 A K T Assis and H Torres Silva Ö^ ~Ö=Ö ~Ú ~Ú ~Ú ¾ 2to1, is the unit vector pointing from 2 to 1, ½ is the relative vectorial ~a ~a ~a Ö_ dÖ=dØ =^Ö¡~Ú ¾ velocity, ½ is the relative vectorial acceleration, is the relative ¾ ¾ ¾ Ö d Ö=dØ =[~Ú¡~Ú ´^Ö ¡ ~Úµ · ~Ö ¡ ~a]=Ö radial velocity, is the relative radial acceleration 8 ½ =¿¢½¼ Ñ× and c is the ratio of electromagnetic and electrostatic units of charge. The experimental value of c was first determined by Weber and Kohlrausch in 1856. In 1848, Weber presented a velocity dependent potential energy from which this force might be derived, namely: ¾ Ö_ Õ Õ ½ ½ ½ ½ : = Í (2) ¾ 4" Ö ¾c ¼ ~ ~ Í F F = Ö^dÍ=dÖ ¾½ The relation between and ¾½ is . These two expressions form the basis of Weber’s electrodynamics. 2. Maxwell–Lorentz’s electrodynamics Based mainly on the works of Coulomb, Amp`ere, Gauss, Neumann and Faraday, Maxwell put together between 1855 and 1864 a set of four equations relating the electric and mag- ~ ~ B netic fields, E and , to their sources: the volumetric charge density and the current ~ density  . In vectorial notation, in differential form, supposing the sources and fields in vacuum, these four equations can be written as ~ Ö¡ E = ; (3) " ¼ ~ ½ @E ~ ~ Ö¢B =  · ; ¼ (4) ¾ c @Ø ~ B =¼; Ö¡ (5) ~ @B ~ E = : Ö¢ (6) @Ø ¾ 7 4 ¢ ½¼ kgÑC In these equations ¼ is called the vacuum permeability. ~  These equations describe how the sources and create the electric and magnetic fields, ~Ú but not how these fields act on another test charge Õ moving with velocity relative to an inertial frame of reference. This last equation was given by H A Lorentz (1853–1928) in 1895: ~ ~ ~ = Õ E · Õ~Ú ¢ B: F (7) In this way we have the main aspects of Maxwell–Lorentz electrodynamics. In order to compare it with Weber’s electrodynamics, we need expressions describing the force Õ Õ ½ exerted by a point charge ¾ on another point charge . This was obtained by the works of Li´enard, Wiechert and Schwarzschild between 1898 and 1903. The final expression valid 394 Pramana – J. Phys., Vol. 55, No. 3, September 2000 Weber’s and classical electrodynamics up to second order in Ú=c and taking into account retardation of time, radiation phenomena and relativistic effects is given by (see ([3], vol. 1, pp. 215–223), [4] and [5]): ~Ú ¡~Ú Õ ½ ¾ ¾ ¾ ~ ~ ~ F = Õ E ´~Ö µ·Õ ~Ú ¢B ´~Ö µ=Õ Ö^ ½· ¾½ ½ ¾ ½ ½ ½ ¾ ½ ½ ¾ ¾ 4" Ö ¾c ¼ ¾ ~Ö ¡ ~a Ö~a ¿ ´^Ö ¡ ~Ú µ Õ ½ ~Ú ¢ Ö^ ¾ ¾ ¾ ¾ ¾ · Õ ~Ú ¢ : ½ ½ (8) ¾ ¾ ¾ ¾ ¾ ¾ c ¾c ¾c 4" Ö c ¼ The energy of interaction between 1 and 2 in Maxwell–Lorentz’s electrodynamics is obtained from Darwin’s lagrangian of 1920, namely ([6], pp. 150–151 and [7], Section 12.7, pp. 593–595): Õ Õ ~Ú ¡ ~Ú ·´~Ú ¡ Ö^µ´~Ú ¡ Ö^µ ½ ½ ¾ ½ ¾ ½ ¾ = ½ : Í (9) ¾ 4" Ö ¾c ¼ In the next section we present Maxwell’s points of view as regards Weber’s electrody- namics and then begin the comparison between these two formulations. 3. Maxwell’s points of view on Weber’s electrodynamics In his first work dealing with electromagnetism, published in 1855, Maxwell praised highly Weber’s electrodynamics. After presenting Faraday’s ideas which he was trying to follow, Maxwell said: There exists however a professedly physical theory of electro-dynamics, which is so elegant, so mathematical, and so entirely different from anything in this paper, that I must state its axioms, at the risk of repeating what ought to be well known. It is contained in M W Weber, Electrodynamic Measurements, and may be found in the Transactions of the Leibniz-Society, and of the Royal Society of Sciences in Saxony [8]. The assumptions are (...). From these ax- ioms are deducible Amp`ere’s laws of the attraction of conductors, and those of Neumann and others, for the induction of currents. Here then is a really physical theory, satisfying the required conditions better perhaps than any yet invented, and put forth by a philosopher whose experimental researches form an ample foundation for his mathematical investigations ([9], pp. 207–208). In his famous article of 1864 in which he completed his electromagnetic theory of light, Maxwell presented similar points of view. After saying that the most natural theories of electromagnetism are based on forces acting directly between the charges he said: In these theories the force acting between the two bodies is treated with refer- ence only to the condition of the bodies and their relative position, and without any express consideration of the surrounding medium. These theories assume, more or less explicitly, the existence of substances the particles of which have the property of acting on one another at a distance by attraction or repulsion. Pramana – J. Phys., Vol. 55, No. 3, September 2000 395 A K T Assis and H Torres Silva The most complete development of a theory of this kind is that of M W We- ber, who has made the same theory include electrostatic and electromagnetic phenomena. In doing so, however, he has found it necessary to assume that the force between two electric particles depends on their relative velocity, as well as on their distance. This theory, as developed by M W Weber and C Neumann, is exceedingly ingenious, and wonderfully comprehensive in its application to the phenomena of statical electricity, electromagnetic attrac- tions, induction of currents and diamagnetic phenomena; and it comes to us with more authority, as it served to guide the speculations of one who has made so great an advance in the practical part of electric science, both by introduc- ing a consistent system of units in electrical measurement, and by actually determining electrical quantities with an accuracy hitherto unknown ([10], pp. 526–527). But if Maxwell knew so well Weber’s electrodynamics and praised it so much, why he did not work with it and develop its properties and applications? Weber presented his force law in 1846. It was the first example appearing in physics of a force depending not only on the distance between the interacting bodies but also on their velocity (Lorentz’s force would appear only in 1895). Only one year after this Helmholtz published his famous and influential work dealing with the conservation of energy. The principle of the conservation of energy had been established by J R Meyer in 1842 and by J P Joule in 1843. In his work of 1847, Helmholtz put this principle in a firm theoretical basis by developing the consequences of central forces. Let us explain some terms which he utilized. At that period ¾ the usual name for ÑÚ was Leibniz’s living force, vis viva, but in this article Helmholtz ¾ =¾ says explicitly that he will call ÑÚ (our kinetic energy) by vis viva, as this last quantity appeared more frequently in mechanics and was more useful. What we call nowadays by ¾ kÜ =¾ potential energy (like Ñg h, etc), Helmholtz called by tension. The main results of his work were presented as: The preceding proposition may be collected together as follows: 1. Whenever natural bodies act upon each other by attractive or repulsive forces, which are independent of time and velocity, the sum of theirvires vivae and tensions must be constant; the maximum quantity of work which can be obtained is therefore a limited quantity. 2. If, on the contrary, natural bodies are possessed of forces which depend upon time and velocity, or which act in other directions than the lines which unite each two separate material points, for example, rotatory forces, then combinations of such bodies would be possible in which forces might be either lost or gained ad infinitum ([11], p.