The Genesis of Maxwell's Equations

Home , James Clerk Maxwell, Wilhelm Eduard Weber

distinguished scientists, drawn from all areas of Cavendish, seventh Duke of Devonshire, who science, engineering, and medicine. The Society served as Chancellor of the University and donated From Electromagnetism to the was founded in 1660 to recognize, promote, and money for the construction of the laboratory. Prof. support excellence in science, and to encourage the James Clerk Maxwell was a founder of the lab, and development and use of science for the benefi t of became the fi rst Cavendish Professor of Physics Electromagnetic Field: humanity. The Society has played a part in some of in 1871. The Duke of Devonshire had given to the most fundamental, signifi cant, and life-chang- Maxwell, as Head of the Laboratory, the manu- The Genesis of Maxwell’s Equations ing discoveries in scientifi c history. Royal Society scripts of Henry Cavendish’s unpublished “Electri- scientists continue to make outstanding contribu- cal Works.” The editing and publishing of these was tions to science in many research areas. The Royal Maxwell’s main scientifi c work while he was at the Ovidio Mario Bucci Society is the national academy of science in the laboratory. Cavendish’s work aroused Maxwell’s UK, and its core are its Fellowship and Foreign intense admiration, and he decided to call the University of Naples Federico II, Naples, Italy Membership, supported by a dedicated staff in laboratory (formerly known as the Devon shire E-mail: [email protected] London and elsewhere. The Fellowship comprises Laboratory) the Cavendish Laboratory, and to thus the most eminent scientists of the UK, Ireland, and commemorate both the Duke and Henry Cavendish. the Commonwealth. 7. James Clerk Maxwell is buried, with his parents and 6. A photo of the Old Cavendish Laboratory, the his wife, within the ruins of the Old Kirk (1592), Department of Physics at the University of Cam- which lies in the graveyard of Parton Parish Church Abstract bridge, part of the University’s School of Physical (1834). The Old Kirk, roofl ess, with just front and Sciences. It was opened in 1874 as a teaching side walls, is shown in the photo. Parton is about labo ratory. It was named to commemorate British seven miles by road from Maxwell House at This contribution outlines the main stages of the path that in ten years led James Clerk Maxwell to the introduction of the chemist and physicist Henry Cavendish, for Glenlair. concept of the electromagnetic ﬁ eld, to the formulation of the electromagnetic theory of light, and to the development of contributions to science, and his relative, William the equations we still adopt for the description of electromagnetic phenomena.

1. Introduction

xactly 150 years ago, on December 8, 1864, James Clerk As for Newton before him and Einstein after, the main EMaxwell (1831-1879) read to the Royal Society of Lon- motivation behind Maxwell’s effort was metaphysical, i.e., his don his third and last fundamental memoir on electromagnet- adherence to a “world view” alternative to the dominant one, a ism, entitled “A Dynamical Theory of the Electromagnetic vision introduced by Faraday in connection with his studies on Field,” the abstract of which had been submitted on October electromagnetic induction and polarization. The construction 27. In this memoir, published in the Transactions of the Society of a coherent and satisfactory theory based on this alternative the following year [1], the equations bearing his name appeared conception required ten years, and an exceptional intellectual for the fi rst time. effort. Following [3, 4], this paper outlines the main stages of this enterprise, in order to follow and attempt to clarify the This introduced a way of looking at electromagnetic phe- evolution of Maxwell’s thoughts, which led to the introduction nomena that opened completely new conceptual (and practi- of the concept of the electromagnetic fi eld, to the electromag- cal) horizons. Together with his other seminal contribution, the netic theory of light, and to the formulation of those equations kinetic theory of gases, Maxwell modifi ed the body of accepted we still adopt for the description of electromagnetic phenom- theories and physical conceptions, namely what Kuhn [2] ena. called the scientifi c “paradigms” of “normal” sci ence, laying the foundations of today’s vision of a physical world made of particles interacting through fi elds. 2. “Mathematizing” Faraday

It must be stressed that the development of both theories Maxwell was introduced to the study of magnetism by was not motivated by new experimental fi ndings that did not William Thomson (1824-1907, Figure 1), the future Lord fi t in the existing scientifi c body of knowledge. In particular, in Kelvin, when he was still an undergraduate student at the Trinity the case of electromagnetism, all the facts known in Maxwell’s College in Cambridge. However, Maxwell’s explicit interest in time had been satisfactorily interpreted within the Newtonian electromagnetism started just after his successful graduation at paradigm, and incorporated in a theoretical frame work of the fi nal honors examination for the BA in Mathematics (the so excellent predictive power. This was indeed exploited and called “Tripos”) in January 1854, when he was nearly 23 years further developed until Hertz’s experimental verifi cation of the old (Figure 2). As a matter of fact, in a letter to Thomson, dated most striking physical consequence of Maxwell’s theory, i.e., February 20, 1854, we read [5]: the existence of electromagnetic waves.

IEEE Antennas and Propagation Magazine, Vol. 56, No. 6, December 2014 299

AP_Mag_Dec_2014_Final.indd 299 12/27/2014 2:04:56 PM 2 Fr=q q4πε r3 1 −+ 12 c 2() dr dt1 c222 r d r dt 12 0 () (1)

with ε0 being the vacuum permittivity, r being the vector pointing from 1 to 2, r being its length, and c being the ratio between the units of charge in the electromagnetic and electrostatic systems. The fi rst term on the right-hand side of Equa- tion (1) clearly corresponds to the Coulomb interaction, the second is equivalent to Ampère’s law for the force between current elements, while the third accounts for the electromagnetic induction.

Maxwell made rapid progress. In a letter to Thomson, dated November 13, 1854 [7], he wrote,

....Then I tried to make out the theory of attractions of currents but tho’ I could see how the effects could be determined I was not satisfi ed with the form of the theory which treats of elementary currents & their reciprocal action....Now I have heard you speak of “magnetic lines of force” & Faradays seems

Figure 1. William Thomson in 1852. Figure 2. James C. Maxwell, about 1854. Figure 4. Wilhelm Eduard Weber. to make great use of them....Now I thought that as every current generated magnetic lines & was acted on in a manner determined by the lines thro(ough) Suppose a man to have a popular knowledge of wh:(ich) it passed that something might be done by electrical show experiments and a little antipathy to considering “magnetic polarization” as a property Murphys Electricity1, how ought he to proceed in of a “magnetic fi eld” or space and developing the reading & working so to get a little insight into the geometric ideas according to this view. subject wh(ich) may be of use in further reading? If he wished to read Ampère Faraday &c how should This clearly shows that since the beginning, Maxwell they be arranged, and in what order might he read was strongly infl uenced by Faraday’s conception that the your articles in the Cambridge Journal? transmission of forces is mediated by the action of contiguous particles of matter in the space between charged or magnetized In choosing as a fi eld of enquiry electromagnetism, an bodies, i.e., through the action of “lines of force” in space. area at the forefront of current research, Maxwell was naturally This commitment to Faraday’s ideas underlies all Maxwell’s drawn to the work of Michael Faraday (1791-1867, Fig ure 3), subsequent work, and was central to the development of his whose extraordinary series of experimental discoveries formed fi eld theory of electromagnetic phenomena. (in Maxwell’s words) “the nucleus of everything electric since 1830.” Maxwell was also drawn to the work of the German Maxwell immediately realized that the fi rst necessary scientists Franz Newmann, Gustav Kirchhoff, and, above all, step to make Faraday’s conceptions acceptable was to show Wilhelm Eduard Weber (1804-1891, Figure 4). Weber had that contrary to what had been generally thought, they were developed a comprehensive explanation of all electromagnetic compatible with a mathematical theory of electromagnetic phenomena in the classic Newtonian framework of a direct phenomena. In fact, in a letter to Thomson, dated May 15, 1855 action at distance between charges and cur rents. These last [8], we read, were conceived according to a hypothesis fi rst formulated in 1845 by the Leipzig professor Gustav Theodor Fechner, as a I am trying to construct two theories, mathematically streaming of opposite charges, traveling with equal velocities identical, in one of which the elementary conceptions in opposite directions. shall be about fl uid particles attracting at a distance while in the other nothing (mathematical) is In modern vector notation and units, which will be adopted considered but various states of polarization, henceforth, Weber’s expression for the force exerted in vacuo tension, etc., existing at various part of space. by a point charge, q1 , on another point charge, q2 , reads [6] The construction of the envisaged theory was carried out during the summer and autumn of the same year, and led to the Figure 3. Michael Faraday in his laboratory at the Royal Institution (1850s). memoir, “On Faraday’s Lines of Forces.” This was presented 1A popular textbook of the day. to the Cambridge Philosophical Society in two parts, Decem ber

300 IEEE Antennas and Propagation Magazine, Vol. 56, No. 6, December 2014

AP_Mag_Dec_2014_Final.indd 300 12/27/2014 2:04:57 PM 2 Fr=q q4πε r3 1 −+ 12 c 2() dr dt1 c222 r d r dt 12 0 () (1)

Maxwell made rapid progress. In a letter to Thomson, dated November 13, 1854 [7], he wrote,

Figure 1. William Thomson in 1852. Figure 2. James C. Maxwell, about 1854. Figure 4. Wilhelm Eduard Weber. to make great use of them....Now I thought that as every current generated magnetic lines & was acted on in a manner determined by the lines thro(ough) Suppose a man to have a popular knowledge of wh:(ich) it passed that something might be done by electrical show experiments and a little antipathy to considering “magnetic polarization” as a property Murphys Electricity1, how ought he to proceed in of a “magnetic fi eld” or space and developing the reading & working so to get a little insight into the geometric ideas according to this view. subject wh(ich) may be of use in further reading? If he wished to read Ampère Faraday &c how should This clearly shows that since the beginning, Maxwell they be arranged, and in what order might he read was strongly infl uenced by Faraday’s conception that the your articles in the Cambridge Journal? transmission of forces is mediated by the action of contiguous particles of matter in the space between charged or magnetized In choosing as a fi eld of enquiry electromagnetism, an bodies, i.e., through the action of “lines of force” in space. area at the forefront of current research, Maxwell was naturally This commitment to Faraday’s ideas underlies all Maxwell’s drawn to the work of Michael Faraday (1791-1867, Fig ure 3), subsequent work, and was central to the development of his whose extraordinary series of experimental discoveries formed fi eld theory of electromagnetic phenomena. (in Maxwell’s words) “the nucleus of everything electric since 1830.” Maxwell was also drawn to the work of the German Maxwell immediately realized that the fi rst necessary scientists Franz Newmann, Gustav Kirchhoff, and, above all, step to make Faraday’s conceptions acceptable was to show Wilhelm Eduard Weber (1804-1891, Figure 4). Weber had that contrary to what had been generally thought, they were developed a comprehensive explanation of all electromagnetic compatible with a mathematical theory of electromagnetic phenomena in the classic Newtonian frame work of a direct phenomena. In fact, in a letter to Thomson, dated May 15, 1855 action at distance between charges and cur rents. These last [8], we read, were conceived according to a hypothesis fi rst formulated in 1845 by the Leipzig professor Gustav Theodor Fechner, as a I am trying to construct two theories, mathematically streaming of opposite charges, traveling with equal velocities identical, in one of which the elementary conceptions in opposite directions. shall be about fl uid particles attracting at a distance while in the other nothing (mathematical) is In modern vector notation and units, which will be adopted considered but various states of polarization, henceforth, Weber’s expression for the force exerted in vacuo tension, etc., existing at various part of space. by a point charge, q1 , on another point charge, q2 , reads [6] The construction of the envisaged theory was carried out during the summer and autumn of the same year, and led to the Figure 3. Michael Faraday in his laboratory at the Royal Institution (1850s). memoir, “On Faraday’s Lines of Forces.” This was presented 1A popular textbook of the day. to the Cambridge Philosophical Society in two parts, Decem ber

IEEE Antennas and Propagation Magazine, Vol. 56, No. 6, December 2014 301

AP_Mag_Dec_2014_Final.indd 301 12/27/2014 2:04:57 PM 10, 1855, and February 11, 1856, and published in extenso the of experiment....In the following investigation, After this – even excessive – understatement of the relevance memoirs about the color vision, his outstanding memoir about same year in the Transactions of the Society [9]. therefore, the laws established by Faraday will be of his work, and a great praise of Weber’s electrodynamics, as the stability of Saturn’s rings, and the fi rst of his epochal assumed as true,.... he wanted to anticipate the objections of continental physicists, memoirs on the kinetic theory of gases. The fi rst sentence of the paper was trenchant, almost he asked: arrogant, for a 24-year-old neo-graduate: However, no physical analogy was now available, so that However, he didn’t stop to refl ect on electromagnetism. Maxwell wrote at the end of the fi rst part: what is the use then of imaging an electrotonic state His letters to C. J. Monro (May 20, 1857), Faraday (November The present state of electrical science seems of which we have no distinctly physical conception, 9, 1857), and Thomson (January 30, 1858) [11-13] showed an particularly unfavourable to speculation. ...I can do no more than simply state the instead of a formula of attraction which we can increasing interest in the theory of molecular vortices, pro posed mathematical methods by which I believe that readily understand? by Thomson to explain the rotation of the plane of polarization At the very beginning, Maxwell then stated the purpose of his electrical phenomena can be best comprehended of linearly polarized light by a magnetic fi eld. Thomson work: and reduced to calculation. Maxwell provided two answers to this question. One was supposed that this phenomenon, discovered by Faraday in 1845, related to the dependence on velocity of Weber’s force. Just was caused by the rotation of molecular vortices in an Aether, This was not an easy task, as was testifi ed by the much higher ...to show how, by a strict application of the ideas one year after its presentation, Hermann L. von Helmholtz having their axis of rotation along the lines of forces of the mathematical level of the second part of the memoir. In con- and methods of Faraday, the connection of the (1821-1894) had published his famous and infl uential mono- magnetic fi eld. nection with electromagnetic induction, Maxwell made refer- very different orders of phenomena which he graph [10], which put on a fi rm theoretical basis the principle ence to a somewhat vague concept introduced by Faraday, that has discovered can be clearly placed before the of conservation of energy. Based on Helmholtz’s results, No wonder, then, that the theory of molecular vortices of “Electro-tonic state.” For its mathematical description, he mathematical mind. Maxwell and others thought that Weber’s electrodynamics did was the cornerstone of Maxwell’s second memoir, “On Physi- adopted what today we call the vector potential, defi ned, with not comply with this principle. Although in 1848 Weber had cal Lines of Force,” which was published in the Philosophical the usual meaning of the symbols, by The method Maxwell adopted to deal with electrostatics, shown that his force could be derived by a potential, it was Magazine in March, April, and May, 1861 (Parts I and II), and magnetostatics, and electric conduction was that of the physi- only in 1869 and 1871 that he proved in detail that it satisfi es January and February, 1862 (Parts III and IV) [14], when he was ÂA=∇× , cal analogy, i.e., in his own words, “that partial similarity the principle of conservation of energy. After that, Maxwell 30 years old (Figure 5). Again, his goal was clearly stated since (2) between the laws of one science and those of another which obviously corrected himself, but in the meantime the electro- the beginning: ∇= A 0 . makes each of them illustrate the other.” To this end, he magnetic fi eld theory had been fully developed. exploited a mechanical model, that of an imponderable and My object in this paper is to clear the way for incompressible fl uid moving through a resisting medium, Note that in analogy to the model adopted in the fi rst part, to The other answer was, again, of methodological and phi- speculation in this direction (i.e., Faraday’s point which exerts on it a retarding force proportional to its velocity. make the vector potential unique (which is obviously neces- losophical nature: of view) by investigating the mechanical results of The fl uid can be supplied or swallowed by sources and sinks sary if it must represent a physical property), Maxwell enforced certain states of tension and motions in a medium, within the considered region of space, or from outside through what is today called the Coulomb gauge. I would answer, that it is a good thing to have two and comparing these with the observed phenomena its boundaries. In this framework, Faraday’s lines of force and ways of looking at a subject, and to admit that there of magnetism and electricity. By pointing out the By a full exploitation of the results of the vector analysis tubes of fl ux correspond to lines and tubes of (steady) fluid are two ways of looking at it. Besides, I do not think available at his time, he then could state the laws of electrody- motion, respectively, the geometrical and dynamical properties that we have any right at present to understand the namics and electromagnetic induction in explicit mathematical of which are examined in detail in the fi rst two sections of the action of electricity, and I hold that the chief merit of form: fi rst part of the memoir. temporary theory is, that it shall guide experiments, without impeding the progress of the true theory JH=∇× , (3) By properly reinterpreting the mechanical quantities of when it appears. the model, in the third section, Maxwell obtained the laws of EE= −∇Φ + , magnetostatics (in absence of currents), electrostatics, and i The key words here are “temporary” and “true.” They clearly electric conduction. In the same section, he also started to (4) unveiled Maxwell’s deep conviction that Weber’s theory was address the phenomena of electrodynamics and electromag- EAi =−∂ ∂t . unsatisfactory (hence “temporary”), not so much because of netic induction. its weak points but because of its being an action at distance Maxwell also provided the expression of the potential energy the ory, while the way toward a “true theory” was that paved Concerning the mutual actions between currents, Maxwell of a closed current in a magnetic fi eld in terms of the current by Faraday, through his conception of an action mediated by made a crucial observation: intensity and the circulation of A, from which all the dynamical the medium. However, to develop a “true physical theory,” actions can be derived. It is noteworthy that Maxwell explicitly mathematics was not enough: such a theory must also rely on We must recollect however that no experiments stressed that the so-called “Ampère” law, Equa tion (3), can only sound physical, i.e., (for Maxwell’s times) mechanical bases. have been made on these elements of currents be valid for closed currents, and said: This explains why Maxwell closed the fi rst part of the memoir except under the form of closed currents....Hence if with a hope, which was also a program: Ampère’s formulae applied to closed currents give Our investigation are therefore for the present true results, their truth is not proved for elements of limited to closed currents; and we know little on By a careful study of the laws of elastic solids and of currents unless we assume that the action between the magnetic effects of any currents which are not the motions of viscous fl uid fl uids, I hope to discover two such elements must be along the line which closed. a method of forming a mechanical conception of the joins them. electro-tonic state adapted to general reasoning. Maxwell had achieved his goal. However, he was well aware of He then put down his main argument: the purely mathematical and somehow artifi cial character of his construction, and wrote: Although this assumption is most warrantable and 3. Electromagnetic Clockwork philosophical in the present state of science, it will ....I do not think that it contains even the shadow of be more conducive to freedom of investigation a true physical theory; in fact, its chief merit as a A relatively long time (for Maxwell standards) had to pass if we endeavour to do without it, and to assume temporary instrument of research is that it does not, before the hope expressed in the fi rst memoir could be realized. the laws of closed currents as the ultimate datum even in appearance, account for anything. In the meantime, Maxwell published, among the oth ers, fi ve Figure 5. J. C. Maxwell (circa 1862).

302 IEEE Antennas and Propagation Magazine, Vol. 56, No. 6, December 2014

AP_Mag_Dec_2014_Final.indd 302 12/27/2014 2:04:57 PM 10, 1855, and February 11, 1856, and published in extenso the of experiment....In the following investigation, After this – even excessive – understatement of the relevance memoirs about the color vision, his outstanding memoir about same year in the Transactions of the Society [9]. therefore, the laws established by Faraday will be of his work, and a great praise of Weber’s electrodynamics, as the stability of Saturn’s rings, and the fi rst of his epochal assumed as true,.... he wanted to anticipate the objections of continental physicists, memoirs on the kinetic theory of gases. The fi rst sentence of the paper was trenchant, almost he asked: arrogant, for a 24-year-old neo-graduate: However, no physical analogy was now available, so that However, he didn’t stop to refl ect on electromagnetism. Maxwell wrote at the end of the fi rst part: what is the use then of imaging an electrotonic state His letters to C. J. Monro (May 20, 1857), Faraday (November The present state of electrical science seems of which we have no distinctly physical conception, 9, 1857), and Thomson (January 30, 1858) [11-13] showed an particularly unfavourable to speculation. ...I can do no more than simply state the instead of a formula of attraction which we can increasing interest in the theory of molecular vortices, pro posed mathematical methods by which I believe that readily understand? by Thomson to explain the rotation of the plane of polarization At the very beginning, Maxwell then stated the purpose of his electrical phenomena can be best comprehended of linearly polarized light by a magnetic fi eld. Thomson work: and reduced to calculation. Maxwell provided two answers to this question. One was supposed that this phenomenon, discovered by Faraday in 1845, related to the dependence on velocity of Weber’s force. Just was caused by the rotation of molecular vortices in an Aether, This was not an easy task, as was testifi ed by the much higher ...to show how, by a strict application of the ideas one year after its presentation, Hermann L. von Helmholtz having their axis of rotation along the lines of forces of the mathematical level of the second part of the memoir. In con- and methods of Faraday, the connection of the (1821-1894) had published his famous and infl uential mono- magnetic fi eld. nection with electromagnetic induction, Maxwell made refer- very different orders of phenomena which he graph [10], which put on a fi rm theoretical basis the principle ence to a somewhat vague concept introduced by Faraday, that has discovered can be clearly placed before the of conservation of energy. Based on Helmholtz’s results, No wonder, then, that the theory of molecular vortices of “Electro-tonic state.” For its mathematical description, he mathematical mind. Maxwell and others thought that Weber’s electrodynamics did was the cornerstone of Maxwell’s second memoir, “On Physi- adopted what today we call the vector potential, defi ned, with not comply with this principle. Although in 1848 Weber had cal Lines of Force,” which was published in the Philosophical the usual meaning of the symbols, by The method Maxwell adopted to deal with electrostatics, shown that his force could be derived by a potential, it was Magazine in March, April, and May, 1861 (Parts I and II), and magnetostatics, and electric conduction was that of the physi- only in 1869 and 1871 that he proved in detail that it satisfi es January and February, 1862 (Parts III and IV) [14], when he was ÂA=∇× , cal analogy, i.e., in his own words, “that partial similarity the principle of conservation of energy. After that, Maxwell 30 years old (Figure 5). Again, his goal was clearly stated since (2) between the laws of one science and those of another which obviously corrected himself, but in the meantime the electro- the beginning: ∇= A 0 . makes each of them illustrate the other.” To this end, he magnetic fi eld theory had been fully developed. exploited a mechanical model, that of an imponderable and My object in this paper is to clear the way for incompressible fl uid moving through a resisting medium, Note that in analogy to the model adopted in the fi rst part, to The other answer was, again, of methodological and phi- speculation in this direction (i.e., Faraday’s point which exerts on it a retarding force proportional to its velocity. make the vector potential unique (which is obviously neces- losophical nature: of view) by investigating the mechanical results of The fl uid can be supplied or swallowed by sources and sinks sary if it must represent a physical property), Maxwell enforced certain states of tension and motions in a medium, within the considered region of space, or from outside through what is today called the Coulomb gauge. I would answer, that it is a good thing to have two and comparing these with the observed phenomena its boundaries. In this framework, Faraday’s lines of force and ways of looking at a subject, and to admit that there of magnetism and electricity. By pointing out the By a full exploitation of the results of the vector analysis tubes of fl ux correspond to lines and tubes of (steady) fluid are two ways of looking at it. Besides, I do not think available at his time, he then could state the laws of electrody- motion, respectively, the geometrical and dynamical properties that we have any right at present to understand the namics and electromagnetic induction in explicit mathematical of which are examined in detail in the fi rst two sections of the action of electricity, and I hold that the chief merit of form: fi rst part of the memoir. temporary theory is, that it shall guide experiments, without impeding the progress of the true theory JH=∇× , (3) By properly reinterpreting the mechanical quantities of when it appears. the model, in the third section, Maxwell obtained the laws of EE= −∇Φ + , magnetostatics (in absence of currents), electrostatics, and i The key words here are “temporary” and “true.” They clearly electric conduction. In the same section, he also started to (4) unveiled Maxwell’s deep conviction that Weber’s theory was address the phenomena of electrodynamics and electromag- EAi =−∂ ∂t . unsatisfactory (hence “temporary”), not so much because of netic induction. its weak points but because of its being an action at distance Maxwell also provided the expression of the potential energy the ory, while the way toward a “true theory” was that paved Concerning the mutual actions between currents, Maxwell of a closed current in a magnetic fi eld in terms of the current by Faraday, through his conception of an action mediated by made a crucial observation: intensity and the circulation of A, from which all the dynamical the medium. However, to develop a “true physical theory,” actions can be derived. It is noteworthy that Maxwell explicitly mathematics was not enough: such a theory must also rely on We must recollect however that no experiments stressed that the so-called “Ampère” law, Equa tion (3), can only sound physical, i.e., (for Maxwell’s times) mechanical bases. have been made on these elements of currents be valid for closed currents, and said: This explains why Maxwell closed the fi rst part of the memoir except under the form of closed currents....Hence if with a hope, which was also a program: Ampère’s formulae applied to closed currents give Our investigation are therefore for the present true results, their truth is not proved for elements of limited to closed currents; and we know little on By a careful study of the laws of elastic solids and of currents unless we assume that the action between the magnetic effects of any currents which are not the motions of viscous fl uid fl uids, I hope to discover two such elements must be along the line which closed. a method of forming a mechanical conception of the joins them. electro-tonic state adapted to general reasoning. Maxwell had achieved his goal. However, he was well aware of He then put down his main argument: the purely mathematical and somehow artifi cial character of his construction, and wrote: Although this assumption is most warrantable and 3. Electromagnetic Clockwork philosophical in the present state of science, it will ....I do not think that it contains even the shadow of be more conducive to freedom of investigation a true physical theory; in fact, its chief merit as a A relatively long time (for Maxwell standards) had to pass if we endeavour to do without it, and to assume temporary instrument of research is that it does not, before the hope expressed in the fi rst memoir could be realized. the laws of closed currents as the ultimate datum even in appearance, account for anything. In the meantime, Maxwell published, among the oth ers, fi ve Figure 5. J. C. Maxwell (circa 1862).

IEEE Antennas and Propagation Magazine, Vol. 56, No. 6, December 2014 303

AP_Mag_Dec_2014_Final.indd 303 12/27/2014 2:04:58 PM mechanical consequences of such hypothesis, I Note that in coherence with the model, the displacement current, between electromagnetic and optical properties appears for- hope to be of some use to those who consider the Jd , was added to that due to the molecular vortices, i.e., ∇×H tuitous. On the other side, if we look at them under the same phenomena as due to the action of a medium, but are . The correct result in terms of E (i.e., what we call the second paradigm, as Maxwell did, the coincidence becomes physically in doubt as to the relation of this hypothesis to the Maxwell equation) was obtained because of the minus sign in relevant, and discloses their possible substantial unity, opening experimental laws already established, which have Equation (7), which implied that in Equation (7), E must be the way to a scientifi c revolution. been generally expressed in the language of other identifi ed with the elastic reaction exerted by the cells on the hypotheses (action at distance). electric particles and not vice versa, as it should be. However, at least two points had to be addressed before the theory could be considered satisfactory. First of all, the As it is sketched in Figure 6, which appeared in the memoir, This ambiguity shows that while Maxwell was by then equations for the electromagnetic fi eld and the fundamental the hypothetic medium consisted of molecular vortices, with sure that the displacement current must be taken fully into property of the total current needed to be clearly and explicitly the axes directed along the magnetic lines of forces, revolving account, he was still not aware of the fact that the “total” cur rent stated. The properties of the electromagnetic (and optical) waves with a peripheral velocity proportional to the intensity of the must be closed (soleinodal), as well as of the relevance of the then had to be derived from the equations, and not exploiting magnetic fi eld. The density of the vortices was proportional to step he performed. As a matter of fact, he used Equation (6) a model so tricky to be considered “imaginary” by Maxwell the magnetic permeability. only to derive, through the equation of continuity, the Gauss himself. equation (and that is why he needed the minus sign in To overcome the obvious diffi culty that the periphery Equation (5)), which he exploited to fi nd the force acting The diffi culty of this goal is quiteevident: it required about of contiguous vortices must move in opposite directions, between two charged bodies, in order to relate the mechanical three years to be achieved, and during this period Maxwell Maxwell made recourse to the interposition of a layer of round constant E2 to the dielectric constant. published just one minor geometrical paper. particles between contiguous vortices, playing the role of “idle wheels” (see Figure 6). These particles, which were in rolling Figure 6. Maxwell’s model of molecular vortices. Instead, he came back to the mechanical model, and deter- contact with the vortices but did not rub against each other, mined the velocity of propagation of transverse vibrations 4. Electromagnetic Field played the role of electricity. Their motion of translation con- through the medium. This turned out to be equal (in air) to stituted an electric current, while their rotation transmitted the the same ratio between the measures of an electric charge in The task was apparently completed in the summer of motion of the vortices from one part of the fi eld to another. the electrostatic and electromagnetic systems that appears in 1864, as witnessed by a letter to C. Hockin dated September 7, The corresponding tangential stresses thus called into play particles that played the role of electricity. When the electric Equa tion (1). This ratio, determined by Weber and Kohlrausch 1864. constituted electromotive force. particles were displaced from their equilibrium positions, they in 1857, agreed so nearly with the velocity of light in air, as distorted the cells and called into play a force arising from their determined by Fizeau in 1849, that he concluded: On October 27, Maxwell presented to the Royal Society By applying to this model the laws of continuum mechan- elasticity, equal and opposite to that which urged the particles the abstract of his third memoir, which, as already said, was ics, in the fi rst two parts of the memoir Maxwell derived the away from their equilibrium positions. This state of par ticles ….we can scarcely avoid the inference that light read on December 8, and published in extenso the following Ampère law, Equation (3), the (local) law of the magnetic force displaced from their equilibrium positions and distorted cells consists in the transverse undulations of the same year. In the introduction, which summarized in detail the on a current, was assumed to represent an electrostatic fi eld. However, what medium which is the cause of electric and magnetic motivations and the content of the memoir, he stated the essence happens during the displacement? phenomena.” of his theory: f= JB × , (5) In the light of his model, the answer was apparently obvi- Maxwell had unexpectedly established the basis for the elec- The theory I propose may therefore be called a and that of electromagnetic induction, in the form we now call ous to Maxwell, who said: tromagnetic theory of light! theory of the Electromagnetic Field, because it has the fi rst Maxwell equation: to do with the space in the neighbourhood of the This displacement does not amount to a current Since the beginning, he was fully persuaded of the correct- electric and magnetic bodies, and it may be called ∇ × = −∂µ ∂ = −∂ ∂ . (6) EH//tt B because when it has attained a certain value it ness of his hypothesis. This is clearly shown by his interest for a Dynamical theory, because it assumes that in remains constant, but it is a commencement of a a precise determination of the “electromagnetic” value of the that space there is matter in motion, by which the He showed that it was equivalent to Equation (4), which was current, and its variations constitute currents, in light velocity, and by his fi rst attempts to explain the refl ection observed electromagnetic phenomena are produced. generalized to the case of moving bodies by adding the convec- the positive or negative direction, according as the and refraction of light in the framework of the theory. tive term vB× . displacement is increasing or diminishing. It is clear from this statement that Maxwell ascribed a physical, It must be stressed that the almost perfect coincidence material, reality to the electromagnetic medium, and that, in his Of course, Maxwell was well aware of the awkwardness He then performed the crucial step, i.e., the inclusion of between the constant c in Equation (1) and the velocity of light view, the electromagnetic phenomena were just the expression of his mechanical model, as he clearly stated at the end of the the displacement currents in the Ampère law, Equation (3). in vacuo had been noted by Weber and Kohlrausch in their 1856 of the mechanical properties of this medium. This was by no second part. However, he had achieved his goal, and it is likely Assuming a linear relationship between electromotive force paper, but they did not consider this to be physically signifi cant. means surprising. On the contrary, it was in complete agree ment that he originally envisaged his paper as consisting of only those (i.e., electric fi eld) and displacement, Even more impressive is the fact that in 1857, Gustav Kirchhoff with the then widely accepted view that the propagation of light two parts. However, during the summer of 1861, in his country (1824-1887), applying Weber’s electrodynamics to the study of and radiant heat consisted of undulations of an omnipervasive house of Glenlair, in Scotland, he developed his mechanical Eh= −E2 , (7) the propagation of electrical signals along metallic wires [15], Aether. In fact, he wrote: theory along lines which led to revolutionary and apparently had shown that they propagate with ﬁ nite velocity, which in the unexpected results. His excitement was clearly testifi ed to by h being the displacement and E a constant depending on the case of vanishingly small resistance is equal to the velocity of We may therefore receive, as a datum from a branch all extant scientifi c letters between the summer of 1861 and elastic constants of the medium, he obtained light. Again, he did not develop the implications of this result. of science, independent of that with which we January 1862, just before the publication of the last two parts of have to deal, the existence of a pervading medium, the memoir. J=∇× HJ + d The contrast with Maxwell’s attitude is a striking illustra- of small but real density, capable of being set in tion of the infl uence of paradigms on the development of sci- motion, and of transmitting motion from one part to To account for the phenomena of electrostatics, Maxwell =∇×Hh +∂ ∂t (8) ence. If we look at electromagnetic and optical phenomena another with great, but not infi nite velocity. extended his model by providing the medium with elastic under different paradigms, the Newtonian (action at distance properties. As the vortices’ rotation was no more of interest, However, this time only very general assumptions concerning 2 between particles) and Cartesian (action by contact, through he now spoke of “elastic cells,” surrounded by the layer of =∇×HE −()1 Et ∂ ∂ . an interposed medium), respectively, a numerical coincidence the properties of the medium were made, namely the capacity of

304 IEEE Antennas and Propagation Magazine, Vol. 56, No. 6, December 2014

AP_Mag_Dec_2014_Final.indd 304 12/27/2014 2:04:58 PM mechanical consequences of such hypothesis, I Note that in coherence with the model, the displacement current, between electromagnetic and optical properties appears for- hope to be of some use to those who consider the Jd , was added to that due to the molecular vortices, i.e., ∇×H tuitous. On the other side, if we look at them under the same phenomena as due to the action of a medium, but are . The correct result in terms of E (i.e., what we call the second paradigm, as Maxwell did, the coincidence becomes physically in doubt as to the relation of this hypothesis to the Maxwell equation) was obtained because of the minus sign in relevant, and discloses their possible substantial unity, opening experimental laws already established, which have Equation (7), which implied that in Equation (7), E must be the way to a scientifi c revolution. been generally expressed in the language of other identifi ed with the elastic reaction exerted by the cells on the hypotheses (action at distance). electric particles and not vice versa, as it should be. However, at least two points had to be addressed before the theory could be considered satisfactory. First of all, the As it is sketched in Figure 6, which appeared in the memoir, This ambiguity shows that while Maxwell was by then equations for the electromagnetic fi eld and the fundamental the hypothetic medium consisted of molecular vortices, with sure that the displacement current must be taken fully into property of the total current needed to be clearly and explicitly the axes directed along the magnetic lines of forces, revolving account, he was still not aware of the fact that the “total” cur rent stated. The properties of the electromagnetic (and optical) waves with a peripheral velocity proportional to the intensity of the must be closed (soleinodal), as well as of the relevance of the then had to be derived from the equations, and not exploiting magnetic fi eld. The density of the vortices was proportional to step he performed. As a matter of fact, he used Equation (6) a model so tricky to be considered “imaginary” by Maxwell the magnetic permeability. only to derive, through the equation of continuity, the Gauss himself. equation (and that is why he needed the minus sign in To overcome the obvious diffi culty that the periphery Equation (5)), which he exploited to fi nd the force acting The diffi culty of this goal is quiteevident: it required about of contiguous vortices must move in opposite directions, between two charged bodies, in order to relate the mechanical three years to be achieved, and during this period Maxwell Maxwell made recourse to the interposition of a layer of round constant E2 to the dielectric constant. published just one minor geometrical paper. particles between contiguous vortices, playing the role of “idle wheels” (see Figure 6). These particles, which were in rolling Figure 6. Maxwell’s model of molecular vortices. Instead, he came back to the mechanical model, and deter- contact with the vortices but did not rub against each other, mined the velocity of propagation of transverse vibrations 4. Electromagnetic Field played the role of electricity. Their motion of translation con- through the medium. This turned out to be equal (in air) to stituted an electric current, while their rotation transmitted the the same ratio between the measures of an electric charge in The task was apparently completed in the summer of motion of the vortices from one part of the fi eld to another. the electrostatic and electromagnetic systems that appears in 1864, as witnessed by a letter to C. Hockin dated September 7, The corresponding tangential stresses thus called into play particles that played the role of electricity. When the electric Equa tion (1). This ratio, determined by Weber and Kohlrausch 1864. constituted electromotive force. particles were displaced from their equilibrium positions, they in 1857, agreed so nearly with the velocity of light in air, as distorted the cells and called into play a force arising from their determined by Fizeau in 1849, that he concluded: On October 27, Maxwell presented to the Royal Society By applying to this model the laws of continuum mechan- elasticity, equal and opposite to that which urged the particles the abstract of his third memoir, which, as already said, was ics, in the fi rst two parts of the memoir Maxwell derived the away from their equilibrium positions. This state of par ticles ….we can scarcely avoid the inference that light read on December 8, and published in extenso the following Ampère law, Equation (3), the (local) law of the magnetic force displaced from their equilibrium positions and distorted cells consists in the transverse undulations of the same year. In the introduction, which summarized in detail the on a current, was assumed to represent an electrostatic fi eld. However, what medium which is the cause of electric and magnetic motivations and the content of the memoir, he stated the essence happens during the displacement? phenomena.” of his theory: f= JB × , (5) In the light of his model, the answer was apparently obvi- Maxwell had unexpectedly established the basis for the elec- The theory I propose may therefore be called a and that of electromagnetic induction, in the form we now call ous to Maxwell, who said: tromagnetic theory of light! theory of the Electromagnetic Field, because it has the fi rst Maxwell equation: to do with the space in the neighbourhood of the This displacement does not amount to a current Since the beginning, he was fully persuaded of the correct- electric and magnetic bodies, and it may be called ∇ × = −∂µ ∂ = −∂ ∂ . (6) EH//tt B because when it has attained a certain value it ness of his hypothesis. This is clearly shown by his interest for a Dynamical theory, because it assumes that in remains constant, but it is a commencement of a a precise determination of the “electromagnetic” value of the that space there is matter in motion, by which the He showed that it was equivalent to Equation (4), which was current, and its variations constitute currents, in light velocity, and by his fi rst attempts to explain the refl ection observed electromagnetic phenomena are produced. generalized to the case of moving bodies by adding the convec- the positive or negative direction, according as the and refraction of light in the framework of the theory. tive term vB× . displacement is increasing or diminishing. It is clear from this statement that Maxwell ascribed a physical, It must be stressed that the almost perfect coincidence material, reality to the electromagnetic medium, and that, in his Of course, Maxwell was well aware of the awkwardness He then performed the crucial step, i.e., the inclusion of between the constant c in Equation (1) and the velocity of light view, the electromagnetic phenomena were just the expression of his mechanical model, as he clearly stated at the end of the the displacement currents in the Ampère law, Equation (3). in vacuo had been noted by Weber and Kohlrausch in their 1856 of the mechanical properties of this medium. This was by no second part. However, he had achieved his goal, and it is likely Assuming a linear relationship between electromotive force paper, but they did not consider this to be physically signifi cant. means surprising. On the contrary, it was in complete agree ment that he originally envisaged his paper as consisting of only those (i.e., electric fi eld) and displacement, Even more impressive is the fact that in 1857, Gustav Kirchhoff with the then widely accepted view that the propagation of light two parts. However, during the summer of 1861, in his country (1824-1887), applying Weber’s electrodynamics to the study of and radiant heat consisted of undulations of an omnipervasive house of Glenlair, in Scotland, he developed his mechanical Eh= −E2 , (7) the propagation of electrical signals along metallic wires [15], Aether. In fact, he wrote: theory along lines which led to revolutionary and apparently had shown that they propagate with ﬁ nite velocity, which in the unexpected results. His excitement was clearly testifi ed to by h being the displacement and E a constant depending on the case of vanishingly small resistance is equal to the velocity of We may therefore receive, as a datum from a branch all extant scientifi c letters between the summer of 1861 and elastic constants of the medium, he obtained light. Again, he did not develop the implications of this result. of science, independent of that with which we January 1862, just before the publication of the last two parts of have to deal, the existence of a pervading medium, the memoir. J=∇× HJ + d The contrast with Maxwell’s attitude is a striking illustra- of small but real density, capable of being set in tion of the infl uence of paradigms on the development of sci- motion, and of transmitting motion from one part to To account for the phenomena of electrostatics, Maxwell =∇×Hh +∂ ∂t (8) ence. If we look at electromagnetic and optical phenomena another with great, but not infi nite velocity. extended his model by providing the medium with elastic under different paradigms, the Newtonian (action at distance properties. As the vortices’ rotation was no more of interest, However, this time only very general assumptions concerning 2 between particles) and Cartesian (action by contact, through he now spoke of “elastic cells,” surrounded by the layer of =∇×HE −()1 Et ∂ ∂ . an interposed medium), respectively, a numerical coincidence the properties of the medium were made, namely the capacity of

IEEE Antennas and Propagation Magazine, Vol. 56, No. 6, December 2014 305

AP_Mag_Dec_2014_Final.indd 305 12/27/2014 2:04:58 PM receiving and storing both “actual” (i.e., kinetic) and “potential” Anyway, the errors did not have a direct impact on In the meantime, the atomic theory of matter had been 8. The Scientifi c Letters and Papers of James Clerk Maxwell, energy (through some kind of elasticity), and the fact that it must Maxwell’s main goal: the electromagnetic theory of light, developed and the quantum revolution had started. However, Volume 1, Cambridge, Cambridge University Press, 1990, p. be subject to the general laws of dynamics. which was addressed in Section VI, wherein the properties of the equations stood unchanged, and today are still those envis- 294. electromagnetic waves were deduced directly from the fi eld aged by Maxwell’s genius 150 years ago. In the following two sections, expressing the laws of equations (even if in a somewhat involved way), and their 9. J. C. Maxwell, “On Faraday’s Lines of Force,” Trans. Camb. electromagnetic induction in the language of Lagrangian propagation in isotropic and anisotropic media was examined in Phil. Soc., X, 1856, pp. 27-83. dynamics, Maxwell identifi ed the vector potential as the gen- detail and compared with that of optical waves. This allows him eralized momentum conjugate to the current density, which he to state: 5. References 10. H. L. Helmholtz, Uber die Erhaltung der Kraft, Berlin, G. explicitly requires to be the total current, namely the sum of the A. Reimer, 1847. conduction current and the displacement current. This allowed Hence electromagnetic science leads to exactly 1. J. C. Maxwell, “A Dynamical Theory of the Electromagnetic him to get the following equations: the same conclusions as optical science with Field,” Phil. Trans. Roy. Soc., 155, pp. 459-512, 1865. 11. P. M. Harmon (ed.), The Scientifi c Letters and Papers of respect to the direction of the disturbances which James Clerk Maxwell, Volume 1, Cambridge, Cambridge Uni- A) CK= +∂ D ∂t eq. of total currents can propagate through the fi eld; both affi rm the 2. T. S. Kuhn, The Structure of Scientifi c Revolutions, Chi cago, versity Press, 1990, pp.505-507. propagation of transverse vibrations and both give The University of Chicago Press, 1962. B) µHA=∇× eq. of magnetic force the same velocity of propagation....The agreement of 12. The Scientifi c Letters and Papers of James Clerk Maxwell, the results seems to show that light and magnetism 3. O. M. Bucci, “The Genesis of Maxwell’s Equations,” in T. K. Volume 1, Cambridge, Cambridge University Press, 1990, pp. Sarkar et al. (eds.), History of Wireless, New York, John Wiley C) ∇×HC = eq. of currents are affections of the same substance, and that light is 548-552. an electromagnetic disturbance propagated through & Sons, 2006, Chapter 5, pp. 189-214. D) EV= ×µ H −∂ A ∂t −∇Φ eq. of electromotive force the fi eld according to electromagnetic laws. 13. The Scientifi c Letters and Papers of James Clerk Maxwell, 4. O. M. Bucci, “The Birth of Maxwell’s Equations,” 44th Volume 1, Cambridge, Cambridge University Press, 1990, pp. The electromagnetic theory of light had been established. European Microwave Conference, Rome, Italy, October 2014. 578-581. to which he added the following equations. He thus obtained a set of 20 scalar equations in 20 unknowns: Maxwell then proceeded to analyze the propagation of 5. P. M. Harmon (ed.), The Scientifi c Letters and Papers of 14. J. C. Maxwell, “On Physical Lines of Force,” Phil. Mag., electromagnetic waves in conductive media. It is interesting to James Clerk Maxwell, Volume 1, Cambridge, Cambridge Uni- XXI, pp. 161-175, 281-291, 338-348, 1861 (Parts I and II); E) ED= k eq. of electric elasticity note that he avoided the consequence of the sign error in Ohm’s versity Press, 1990, pp. 237-238. Phil. Mag., XXIII, pp. 12-25, 85-95, 1862 (Parts III and IV). law, Equation F, by means of a further change of sign, assuming F) EK= −ρ eq. of electric resistance for a sinusoidal plane wave along x the expression 6. W. Weber, “Elektrodynamische Maassbestimmungen, uber 15. G. Kirchhoff, “Uber die Bewegung der Electricitat in ein allgemeines Grundgesetz der Elektrischen Wirkung,” Drahten,” Ann. Phys., 100, pp. 193-217, 1857. G) ∇ += eq. of free electricity  D e 0 A()()() x, t=−+ exp px cos qx nt , (9) Leipzig Abhandl., 1846, pp. 211-378. 16. J. C. Maxwell, A Treatise on Electricity and Magnetism, H) ∇ K +∂et ∂ =0 eq. of continuity 7. P. M. Harmon (ed.), The Scientifi c Letters and Papers of i.e., a wave attenuating in the forward direction, but propa- Oxford, Clarendon Press, 1873. James Clerk Maxwell, Volume 1, Cambridge, Cambridge Uni- gating in the backward direction! He then derived the expressions of the magnetic (i.e., kinetic) versity Press, 1990, pp.254-263. and electric (i.e., elastic) energy densities, which was exploited The root of this kind of plus-minus dyslexia lies in in the following section to fi nd the forces on cur rents and Maxwell’s attitude toward the nature of electric charges and magnets. currents. In accordance with his emphasis on the role of the medium, Maxwell (and the British Maxwellians after him) Putting V = 0 (stationary media) and BH= µ , we imme- considered charges and currents not as the sources of the fi eld, diately recognize that Equations A to D are equivalent to the but, vice versa, as a product or a property of the fi eld itself. fi rst two Maxwell’s equations, as we write them today, plus the In other words, the Aether was the only fundamental physical fourth one (that expressing that the magnetic induction is entity, and the description of the phenomena must be obtained solenoidal). On the other side, Equation G is not Gauss’ by a proper characterization of its dynamical properties (i.e., its equation, as it should be, because of the wrong sign. This is Lagrangian or Hamiltonian). exactly the opposite of what happened in the second memoir, wherein Gauss’ law was correct, whereas the relation between Apart from being doomed to failure – because charged the electric fi eld and electric displacement had the wrong sign: matter possesses its own degrees of freedom, so that it consti- see Equation (7). Moreover, this time the relation Equation F tutes a dynamical system distinct from (even if coupled to) between the fi eld and conduction current also had the wrong the electromagnetic fi eld – this attitude explains the fact that sign. neither Maxwell nor the Maxwellians considered the question of generating electromagnetic waves distinct from light. As is The presence of these errors, which would be corrected in well known, this crucial validation of Maxwell’s equations was the Treatise [16], was quite surprising, in the light of Maxwell’s performed by Heinrich R. Hertz (1857-1894) only in October deep physical and mathematical attitude. In fact, Equation G 1886, twenty-two years after their formulation, whereas the is mathematically inconsistent with Equations C and H, while defi nitive abandonment of mechanical Aether theories had to Equation F clearly contrasts with the dissipative nature of wait for Einstein’s epochal paper of 1905. conduction currents. We will come back to this point in a while.

306 IEEE Antennas and Propagation Magazine, Vol. 56, No. 6, December 2014

AP_Mag_Dec_2014_Final.indd 306 12/27/2014 2:04:58 PM receiving and storing both “actual” (i.e., kinetic) and “potential” Anyway, the errors did not have a direct impact on In the meantime, the atomic theory of matter had been 8. The Scientifi c Letters and Papers of James Clerk Maxwell, energy (through some kind of elasticity), and the fact that it must Maxwell’s main goal: the electromagnetic theory of light, developed and the quantum revolution had started. However, Volume 1, Cambridge, Cambridge University Press, 1990, p. be subject to the general laws of dynamics. which was addressed in Section VI, wherein the properties of the equations stood unchanged, and today are still those envis- 294. electromagnetic waves were deduced directly from the fi eld aged by Maxwell’s genius 150 years ago. In the following two sections, expressing the laws of equations (even if in a somewhat involved way), and their 9. J. C. Maxwell, “On Faraday’s Lines of Force,” Trans. Camb. electromagnetic induction in the language of Lagrangian propagation in isotropic and anisotropic media was examined in Phil. Soc., X, 1856, pp. 27-83. dynamics, Maxwell identifi ed the vector potential as the gen- detail and compared with that of optical waves. This allows him eralized momentum conjugate to the current density, which he to state: 5. References 10. H. L. Helmholtz, Uber die Erhaltung der Kraft, Berlin, G. explicitly requires to be the total current, namely the sum of the A. Reimer, 1847. conduction current and the displacement current. This allowed Hence electromagnetic science leads to exactly 1. J. C. Maxwell, “A Dynamical Theory of the Electromagnetic him to get the following equations: the same conclusions as optical science with Field,” Phil. Trans. Roy. Soc., 155, pp. 459-512, 1865. 11. P. M. Harmon (ed.), The Scientifi c Letters and Papers of respect to the direction of the disturbances which James Clerk Maxwell, Volume 1, Cambridge, Cambridge Uni- A) CK= +∂ D ∂t eq. of total currents can propagate through the fi eld; both affi rm the 2. T. S. Kuhn, The Structure of Scientifi c Revolutions, Chi cago, versity Press, 1990, pp.505-507. propagation of transverse vibrations and both give The University of Chicago Press, 1962. B) µHA=∇× eq. of magnetic force the same velocity of propagation....The agreement of 12. The Scientifi c Letters and Papers of James Clerk Maxwell, the results seems to show that light and magnetism 3. O. M. Bucci, “The Genesis of Maxwell’s Equations,” in T. K. Volume 1, Cambridge, Cambridge University Press, 1990, pp. Sarkar et al. (eds.), History of Wireless, New York, John Wiley C) ∇×HC = eq. of currents are affections of the same substance, and that light is 548-552. an electromagnetic disturbance propagated through & Sons, 2006, Chapter 5, pp. 189-214. D) EV= ×µ H −∂ A ∂t −∇Φ eq. of electromotive force the fi eld according to electromagnetic laws. 13. The Scientifi c Letters and Papers of James Clerk Maxwell, 4. O. M. Bucci, “The Birth of Maxwell’s Equations,” 44th Volume 1, Cambridge, Cambridge University Press, 1990, pp. The electromagnetic theory of light had been established. European Microwave Conference, Rome, Italy, October 2014. 578-581. to which he added the following equations. He thus obtained a set of 20 scalar equations in 20 unknowns: Maxwell then proceeded to analyze the propagation of 5. P. M. Harmon (ed.), The Scientifi c Letters and Papers of 14. J. C. Maxwell, “On Physical Lines of Force,” Phil. Mag., electromagnetic waves in conductive media. It is interesting to James Clerk Maxwell, Volume 1, Cambridge, Cambridge Uni- XXI, pp. 161-175, 281-291, 338-348, 1861 (Parts I and II); E) ED= k eq. of electric elasticity note that he avoided the consequence of the sign error in Ohm’s versity Press, 1990, pp. 237-238. Phil. Mag., XXIII, pp. 12-25, 85-95, 1862 (Parts III and IV). law, Equation F, by means of a further change of sign, assuming F) EK= −ρ eq. of electric resistance for a sinusoidal plane wave along x the expression 6. W. Weber, “Elektrodynamische Maassbestimmungen, uber 15. G. Kirchhoff, “Uber die Bewegung der Electricitat in ein allgemeines Grundgesetz der Elektrischen Wirkung,” Drahten,” Ann. Phys., 100, pp. 193-217, 1857. G) ∇ += eq. of free electricity  D e 0 A()()() x, t=−+ exp px cos qx nt , (9) Leipzig Abhandl., 1846, pp. 211-378. 16. J. C. Maxwell, A Treatise on Electricity and Magnetism, H) ∇ K +∂et ∂ =0 eq. of continuity 7. P. M. Harmon (ed.), The Scientifi c Letters and Papers of i.e., a wave attenuating in the forward direction, but propa- Oxford, Clarendon Press, 1873. James Clerk Maxwell, Volume 1, Cambridge, Cambridge Uni- gating in the backward direction! He then derived the expressions of the magnetic (i.e., kinetic) versity Press, 1990, pp.254-263. and electric (i.e., elastic) energy densities, which was exploited The root of this kind of plus-minus dyslexia lies in in the following section to fi nd the forces on cur rents and Maxwell’s attitude toward the nature of electric charges and magnets. currents. In accordance with his emphasis on the role of the medium, Maxwell (and the British Maxwellians after him) Putting V = 0 (stationary media) and BH= µ , we imme- considered charges and currents not as the sources of the fi eld, diately recognize that Equations A to D are equivalent to the but, vice versa, as a product or a property of the fi eld itself. fi rst two Maxwell’s equations, as we write them today, plus the In other words, the Aether was the only fundamental physical fourth one (that expressing that the magnetic induction is entity, and the description of the phenomena must be obtained solenoidal). On the other side, Equation G is not Gauss’ by a proper characterization of its dynamical properties (i.e., its equation, as it should be, because of the wrong sign. This is Lagrangian or Hamiltonian). exactly the opposite of what happened in the second memoir, wherein Gauss’ law was correct, whereas the relation between Apart from being doomed to failure – because charged the electric fi eld and electric displacement had the wrong sign: matter possesses its own degrees of freedom, so that it consti- see Equation (7). Moreover, this time the relation Equation F tutes a dynamical system distinct from (even if coupled to) between the fi eld and conduction current also had the wrong the electromagnetic fi eld – this attitude explains the fact that sign. neither Maxwell nor the Maxwellians considered the question of generating electromagnetic waves distinct from light. As is The presence of these errors, which would be corrected in well known, this crucial validation of Maxwell’s equations was the Treatise [16], was quite surprising, in the light of Maxwell’s performed by Heinrich R. Hertz (1857-1894) only in October deep physical and mathematical attitude. In fact, Equation G 1886, twenty-two years after their formulation, whereas the is mathematically inconsistent with Equations C and H, while defi nitive abandonment of mechanical Aether theories had to Equation F clearly contrasts with the dissipative nature of wait for Einstein’s epochal paper of 1905. conduction currents. We will come back to this point in a while.

IEEE Antennas and Propagation Magazine, Vol. 56, No. 6, December 2014 307

AP_Mag_Dec_2014_Final.indd 307 12/27/2014 2:04:58 PM