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The Meaning of the Constant C in Weber's Electrodynamics

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The Meaning of the Constant C in Weber's Electrodynamics Proe. Int. conf. "Galileo Back in Italy - II" (Soc. Ed. Androme-:ia, Bologna, 2000), pp. 23-36, R. Monti (editor) The Meaning of the Constant c in Weber's Electrodynamics A. K. T. Assis' Instituto de Fisica 'Gleb Wat.aghin' Universidade Estadual de Campinas - Unicamp 13083-970 Campinas, Sao Paulo, Brasil Abstract In this work it is analysed three basic electromagnetic systems of units utilized during last century by Ampere, Gauss, Weber, Maxwell and all the others: The electrostatic, electrodynamic and electromagnetic ones. It is presented how the basic equations of electromagnetism are written in these systems (and also in the present day international system of units MKSA). Then it is shown how the constant c was introduced in physics by Weber's force. It is shown that it has the unit of velocity and is the ratio of the electromagnetic and electrostatic Wlits of charge. Weber and Kohlrausch '5 experiment to determine c is presented, emphasizing that they were 'the first to measure this quantity and obtained the same value as that of light velocity in vacuum. It is shown how Kirchhoff and Weber obtained independently of one another, both working in the framev.-ork of \Veber's electrodynamics, the fact that an electromagnetic signal (of current or potential) propagate at light velocity along a thin wire of negligible resistivity. Key Words: Electromagnetic units, light velocity, wave equation. PACS: O1.55.+b (General physics), 01.65.+g (History of science), 41.20.·q (Electric, magnetic, and electromagnetic fields) ·E-mail: assisOiCLunicamp.br, homepage: www.lCi.unicamp.brrassis AA. VV•. Alli del Couvegno lnltmalionale "'G.mco back in ltaly- U" • p" iI ritonw della nrxioMIiIQ IItlLl ~ IfIOdtma - 23· 1 Introduction One of the most fundamental constants in physics is the ratio of electromagnetic and electrostatic units 9f charge, c. Its origin in Weber's electrodynamics is traced in this work, along with its first measurement and the first time the wave equation describing an electromagnetic disturbance was obtained. Initially the basic systems of units utilized during last century for describing electromagnetic quantities is presented, along with a short review of Weber's electrodynamics and the measurement of c. How the equation of telegraphy was first derived is also considered. These are the main goals of this work. 2 Systems of Units According to Newton's first and second definitions in the Principia, [New34J, the quantity of motion is the mass of a body times its velocity. His second law of motion states; "The change of motion is proporcional to the motive force impressed; and is made in the direction of the right line in which that force is impressed." Representing this force vectorially by F, the mass by m and the velocity by V, the second law can be written as (1) where Kl is a constant of proportionality. Considering this constant to be the unitless number 1 and supposing a constant mass during the motion yields Newton's second law in the usual form F=mii, (2) where ii = diJjdt is the acceleration of the body. H the force F is constant during the time t, this equations yields ii = F jm = constant and v = Va + at, where Va is the initial velocity of the body. The unit force is then that constant force which when it acts upon the unit of mass imparts to this mass a unit of velocity in unit of ;'ime, [Web72, see p. 2J. Usually the basic units o( mechanics are chosen to be the mass, length and time. Weber used to Consider milligrams, millimeters and seconds as his basic units. Newton's law of universal gravitation of 1687 states that this force is pro­ portional to the product of the masses m and m'. and inversely proportional to the square of their distance r. If r is the unit vector pointing from m' to m, the force on m due to m' is given by _ mm'_ F= -K2 (3) -,-r.r AA. VY. Ani del Convegno lotemazionaJe "GalUeo back in Ualy • II" . Per il rilomD dtUa ra:imwJ.itil nella Sdm:a modtma ·24- This force is along the straight line connecting the masses and is al ways at­ tractive. The constant K2 allows for different systems of units. H we take this constant K2 = 1 unitiess, then combining Eq. (3) with Eq. (2) yields the unit of mass as deduced (or derived) from the units of time and length, see [Max:54, Article 5, pp. 3-5J. The force on a particle of mass m can be written as - _ (" K,m' _) F=mg=m -~--,::2r . (4) Here 9 is called the gravitational field acting on m due to all the masses m'. It is the force per unit mass. According to Coulomb's law of 1785, the force between two point charges e and e' is proportional to their product and inversely proportional to the square of their distance r: (5) where the force is along the straight line connecting the charges and is repulsive (attractive) if ee' > 0 (ee' < 0). Two equal charges e = e' are said to have unit magnitude when they exert upon one another a unit force when separated by a unit distance. The first system of units applicable to electric quantities to be considered here is the electrostatic system of units. In this system K3 = 1 unitless and the unit (or dimension) of the charges e and e' is called electrostatic unit, esu. The force on a Charge e can be written as F=eE=e- - ("~~f K,e' ) . (6) Here E is called the electric field acting on e due to all the charges e'. It is the force per unit charge. Coulomb also obtained in 1785 the force between two magnetic poles p and p' separated by a distance r as given by a similar expression. In the case of long thin bar magnets. the poles are located at the extremeties. Usually a north pole of a bar magnet (Which points towards the geographic north of the earth) is considered positive and a south pole negative. Coulomb's expression for the force between magnetic poles is given by: (7) Once more two equal magnetic poles p = p' are said to have unit magnitude when exert a unit of force when separated by a unit distance. There will be a force of repulsion (attraction) when pp' > 0 (pp' < 0). It is also along the straight line connecting the poles. AA. VV.· AlIi dd COQvegno Intenwionalc "Galileo hack in Italy' If'· Peril rilqnw della I'IJliOMJild tWIa XYIIlJ/ flWliema The second system of units utilized during last century is the electromagnetic system of units. In it K( = 1 unltIess and the unit (or dimension) of p and p' is called electromagnetic unit, emu. The force on a magnetic pole p can be written as F=PB=P(L,~'t"f) . (8) Here B is called the magnetic field acting on p due to all the poles p'. It is the force per unit magnetic pole. Between 1820 and 1826 Ampere obtained the force between two current elements. He was led to his researches after Oersted great discovery of 1820 that a current carrying wire afects a magnet in its vicinity. Following Oersted discovery Ampere decided to consider the direct action between currents. From his experiments and theoretical considerations he was led to his force expression. If the circuits carry currents i and e and the current elements separated by a distance r have lengths ds and ds', respectively, Ampere's force is given by - ii'dsds' r:PF=Ks r!! f(3cosBcos8' -2cose) : .. , K"~ [3(f-ds"')(f-dS')-2(ds·dS')]_ (9) = r The constant Ks allows for different systems of units. In this expression e and ()' are the angles b~tween the positive directions of the currents in the elements and the connecting right line between them, e is the angle between the positive directions of the currents in the elements, f is the unit vector connecting them, ciS and ciS' are the vectors pointing along the direction of the currents and having magnitude equal to the length of the elements. After integrating this expression Ampere obtained the force exerted by a closed circuit G1 where flows a current i' on a current element id$ of another circuit as given by: (10) A simple example is given here. Integrating this expression to obtain the force dF exerted by a long straight ",ire where flows a current i ' acting on a current element ids at a distance l to the wire and parallel to it is given by (11) pointing from the current element to the wire (supposing that both currents flow in the same direction). AA.VV.· Atli del Dlnvegno lmem.uionale ''Galilco back in Ita1y -lI". PerU ritonwtkl41lidonaliltl MI41 $timw1lWdt11lll ·26- The physical connection between magnetic pole and current was given by Oersted experiment. That is, he observed that a galvanic current orients a small magnet in the same way as others magnets (or the earth) do. From Ampere's force law we can obtain a mathematical connection between these two concepts. This is done writing Eq. (10) as dF = idS x B, (12) where B is called the magnetic field generated by the closed circuit C'. It is only possible to call it a magnetic field by Oersted's experiment. That is, the force exerted on a unit magnetic pole located at the same place as idi' by the current carrying circuit C' is given by this magnetic field. This means that p and ids have the same units.
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