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

distinguished scientists, drawn from all areas of science, engineering, and medicine. The Society was founded in 1660 to recognize, promote, and support excellence in science, and to encourage the development and use of science for the benefi t of humanity. The Society has played a part in some of the most fundamental, signifi cant, and life-chang-ing discoveries in scientifi c history. Royal Society scientists continue to make outstanding contribu-tions to science in many research areas. The Royal Society is the national academy of science in the UK, and its core are its Fellowship and Foreign Membership, supported by a dedicated staff in London and elsewhere. The Fellowship comprises the most eminent scientists of the UK, Ireland, and the Commonwealth.

6. A photo of the Old Cavendish Laboratory, the Department of Physics at the University of Cam-bridge, part of the University’s School of Physical Sciences. It was opened in 1874 as a teaching labo ratory. It was named to commemorate British chemist and physicist Henry Cavendish, for contri butions to science, and his relative, William

Cavendish, seventh Duke of Devonshire, who served as Chancellor of the University and donated money for the construction of the laboratory. Prof. James Clerk Maxwell was a founder of the lab, and became the fi rst Cavendish Professor of Physics in 1871. The Duke of Devonshire had given to Maxwell, as Head of the Laboratory, the manu-scripts of Henry Cavendish’s unpublished “Electri-cal Works.” The editing and publishing of these was Maxwell’s main scientifi c work while he was at the laboratory. Cavendish’s work aroused Maxwell’s intense admiration, and he decided to call the laboratory (formerly known as the Devon shire Laboratory) the Cavendish Laboratory, and to thus commemorate both the Duke and Henry Cavendish.

7. James Clerk Maxwell is buried, with his parents and his wife, within the ruins of the Old Kirk (1592), which lies in the graveyard of Parton Parish Church (1834). The Old Kirk, roofl ess, with just front and side walls, is shown in the photo. Parton is about seven miles by road from Maxwell House at Glenlair.

From Electromagnetism to the ElectromagneticField:

TheGenesisofMaxwell’sEquations

Ovidio Mario Bucci

University of Naples Federico II, Naples, ItalyE-mail: bucci@unina.it

Abstract

This contribution outlines the main stages of the path that in ten years led James Clerk Maxwell to the introduction of the concept of the electromagnetic fi eld, to the formulation of the electromagnetic theory of light, and to the development of the equations we still adopt for the description of electromagnetic phenomena.

1. Introduction

Exactly 150 years ago, on December 8, 1864, James Clerk Maxwell (1831-1879) read to the Royal Society of Lon-

don his third and last fundamental memoir on electromagnet-ism, entitled “A Dynamical Theory of the Electromagnetic Field,” the abstract of which had been submitted on October 27. In this memoir, published in the Transactions of the Soci ety the following year [1], the equations bearing his name appeared for the fi rst time. This introduced a way of looking at electromagnetic phe-nomena that opened completely new conceptual (and practi-cal) horizons. Together with his other seminal contribution, the kinetic theory of gases, Maxwell modifi ed the body of accepted theories and physical conceptions, namely what Kuhn [2] 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. It must be stressed that the development of both theories was not motivated by new experimental fi ndings that did not fi t in the existing scientifi c body of knowledge. In particular, in the case of electromagnetism, all the facts known in Maxwell’s time had been satisfactorily interpreted within the Newtonian paradigm, and incorporated in a theoretical frame work of excellent predictive power. This was indeed exploited and further developed until Hertz’s experimental verifi cation of the most striking physical consequence of Maxwell’s the ory, i.e., the existence of electromagnetic waves.

As for Newton before him and Einstein after, the main motivation behind Maxwell’s effort was metaphysical, i.e., his adherence to a “world view” alternative to the dominant one, a vision introduced by Faraday in connection with his studies on electromagnetic induction and polarization. The construction of a coherent and satisfactory theory based on this alternative conception required ten years, and an exceptional intellectual effort. Following [3, 4], this paper outlines the main stages of this enterprise, in order to follow and attempt to clarify the evolution of Maxwell’s thoughts, which led to the introduction of the concept of the electromagnetic fi eld, to the electromag-netic theory of light, and to the formulation of those equations we still adopt for the description of electromagnetic phenom-ena.

2. “Mathematizing” Faraday

Maxwell was introduced to the study of magnetism by William Thomson (1824-1907, Figure 1), the future Lord Kelvin, when he was still an undergraduate student at the Trinity College in Cambridge. However, Maxwell’s explicit interest in electromagnetism started just after his successful graduation at the fi nal honors examination for the BA in Mathematics (the so called “Tripos”) in January 1854, when he was nearly 23 years old (Figure 2). As a matter of fact, in a letter to Thomson, dated February 20, 1854, we read [5]:

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

Figure 2. James C. Maxwell, about 1854.Figure 1. William Thomson in 1852.

Figure 3. Michael Faraday in his laboratory at the Royal Institution (1850s).

Suppose a man to have a popular knowledge of electrical show experiments and a little antipathy to Murphys Electricity1, how ought he to proceed in reading & working so to get a little insight into the subject wh(ich) may be of use in further reading? If he wished to read Ampère Faraday &c how should they be arranged, and in what order might he read your articles in the Cambridge Journal?

In choosing as a fi eld of enquiry electromagnetism, an area at the forefront of current research, Maxwell was natu rally drawn to the work of Michael Faraday (1791-1867, Fig ure 3), whose extraordinary series of experimental discoveries formed (in Maxwell’s words) “the nucleus of everything elec tric since 1830.” Maxwell was also drawn to the work of the German scientists Franz Newmann, Gustav Kirchhoff, and, above all, Wilhelm Eduard Weber (1804-1891, Figure 4). Weber had developed a comprehensive explanation of all electromagnetic phenomena in the classic Newtonian frame work of a direct action at distance between charges and cur rents. These last were conceived according to a hypothesis fi rst formulated in 1845 by the Leipzig professor Gustav Theodor Fechner, as a streaming of opposite charges, traveling with equal velocities in opposite directions.

In modern vector notation and units, which will be adopted henceforth, Weber’s expression for the force exerted in vacuo by a point charge, 1q , on another point charge, 2q , reads [6]

1A popular textbook of the day.

Figure 4. Wilhelm Eduard Weber.

( ) ( )23 2 2 2 21 2 04 1 1 2 1q q r c dr dt c r d r dtπε = − +

F r

(1)

with 0ε 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 electro-static 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 electromag-netic 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

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) wh:(ich) it passed that something might be done by considering “magnetic polarization” as a property of a “magnetic fi eld” or space and developing the geometric ideas according to this view.

This clearly shows that since the beginning, Maxwell was strongly infl uenced by Faraday’s conception that the transmission of forces is mediated by the action of contiguous particles of matter in the space between charged or magnetized bodies, i.e., through the action of “lines of force” in space. This commitment to Faraday’s ideas underlies all Maxwell’s subsequent work, and was central to the development of his fi eld theory of electromagnetic phenomena.

Maxwell immediately realized that the fi rst necessary step to make Faraday’s conceptions acceptable was to show that contrary to what had been generally thought, they were compatible with a mathematical theory of electromagnetic phenomena. In fact, in a letter to Thomson, dated May 15, 1855 [8], we read,

I am trying to construct two theories, mathematically identical, in one of which the elementary conceptions shall be about fl uid particles attracting at a distance while in the other nothing (mathematical) is considered but various states of polarization, tension, etc., existing at various part of space.

The construction of the envisaged theory was carried out dur-ing the summer and autumn of the same year, and led to the memoir, “On Faraday’s Lines of Forces.” This was presented to the Cambridge Philosophical Society in two parts, Decem ber

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

Figure 2. James C. Maxwell, about 1854.Figure 1. William Thomson in 1852.

Figure 3. Michael Faraday in his laboratory at the Royal Institution (1850s).

Suppose a man to have a popular knowledge of electrical show experiments and a little antipathy to Murphys Electricity1, how ought he to proceed in reading & working so to get a little insight into the subject wh(ich) may be of use in further reading? If he wished to read Ampère Faraday &c how should they be arranged, and in what order might he read your articles in the Cambridge Journal?

In choosing as a fi eld of enquiry electromagnetism, an area at the forefront of current research, Maxwell was natu rally drawn to the work of Michael Faraday (1791-1867, Fig ure 3), whose extraordinary series of experimental discoveries formed (in Maxwell’s words) “the nucleus of everything elec tric since 1830.” Maxwell was also drawn to the work of the German scientists Franz Newmann, Gustav Kirchhoff, and, above all, Wilhelm Eduard Weber (1804-1891, Figure 4). Weber had developed a comprehensive explanation of all electromagnetic phenomena in the classic Newtonian frame work of a direct action at distance between charges and cur rents. These last were conceived according to a hypothesis fi rst formulated in 1845 by the Leipzig professor Gustav Theodor Fechner, as a streaming of opposite charges, traveling with equal velocities in opposite directions.

In modern vector notation and units, which will be adopted henceforth, Weber’s expression for the force exerted in vacuo by a point charge, 1q , on another point charge, 2q , reads [6]

1A popular textbook of the day.

Figure 4. Wilhelm Eduard Weber.

( ) ( )23 2 2 2 21 2 04 1 1 2 1q q r c dr dt c r d r dtπε = − +

F r

(1)

with 0ε 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 electro-static 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 electromag-netic 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

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) wh:(ich) it passed that something might be done by considering “magnetic polarization” as a property of a “magnetic fi eld” or space and developing the geometric ideas according to this view.

This clearly shows that since the beginning, Maxwell was strongly infl uenced by Faraday’s conception that the transmission of forces is mediated by the action of contiguous particles of matter in the space between charged or magnetized bodies, i.e., through the action of “lines of force” in space. This commitment to Faraday’s ideas underlies all Maxwell’s subsequent work, and was central to the development of his fi eld theory of electromagnetic phenomena.

Maxwell immediately realized that the fi rst necessary step to make Faraday’s conceptions acceptable was to show that contrary to what had been generally thought, they were compatible with a mathematical theory of electromagnetic phenomena. In fact, in a letter to Thomson, dated May 15, 1855 [8], we read,

I am trying to construct two theories, mathematically identical, in one of which the elementary conceptions shall be about fl uid particles attracting at a distance while in the other nothing (mathematical) is considered but various states of polarization, tension, etc., existing at various part of space.

The construction of the envisaged theory was carried out dur-ing the summer and autumn of the same year, and led to the memoir, “On Faraday’s Lines of Forces.” This was presented to the Cambridge Philosophical Society in two parts, Decem ber

AP_Mag_Dec_2014_Final.indd 301 12/27/2014 2:04:57 PM

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

10, 1855, and February 11, 1856, and published in extenso the same year in the Transactions of the Society [9].

The fi rst sentence of the paper was trenchant, almost arrogant, for a 24-year-old neo-graduate:

The present state of electrical science seems particularly unfavourable to speculation.

At the very beginning, Maxwell then stated the purpose of his work:

...to show how, by a strict application of the ideas and methods of Faraday, the connection of the very different orders of phenomena which he has discovered can be clearly placed before the mathematical mind.

The method Maxwell adopted to deal with electrostatics, magnetostatics, and electric conduction was that of the physi-cal analogy, i.e., in his own words, “that partial similarity between the laws of one science and those of another which makes each of them illustrate the other.” To this end, he exploited a mechanical model, that of an imponderable and incompressible fl uid moving through a resisting medium, which exerts on it a retarding force proportional to its velocity. The fl uid can be supplied or swallowed by sources and sinks within the considered region of space, or from outside through its boundaries. In this framework, Faraday’s lines of force and tubes of fl ux correspond to lines and tubes of (steady) fl uid motion, respectively, the geometrical and dynamical properties of which are examined in detail in the fi rst two sections of the fi rst part of the memoir.

By properly reinterpreting the mechanical quantities of the model, in the third section, Maxwell obtained the laws of magnetostatics (in absence of currents), electrostatics, and electric conduction. In the same section, he also started to address the phenomena of electrodynamics and electromag-netic induction.

Concerning the mutual actions between currents, Maxwell made a crucial observation:

We must recollect however that no experiments have been made on these elements of currents except under the form of closed currents....Hence if Ampère’s formulae applied to closed currents give true results, their truth is not proved for elements of currents unless we assume that the action between two such elements must be along the line which joins them.

He then put down his main argument:

Although this assumption is most warrantable and philosophical in the present state of science, it will be more conducive to freedom of investigation if we endeavour to do without it, and to assume the laws of closed currents as the ultimate datum

of experiment....In the following investigation, therefore, the laws established by Faraday will be assumed as true,....

However, no physical analogy was now available, so that Maxwell wrote at the end of the fi rst part:

...I can do no more than simply state the mathematical methods by which I believe that electrical phenomena can be best comprehended and reduced to calculation.

This was not an easy task, as was testifi ed by the much higher mathematical level of the second part of the memoir. In con-nection with electromagnetic induction, Maxwell made refer-ence to a somewhat vague concept introduced by Faraday, that of “Electro-tonic state.” For its mathematical description, he adopted what today we call the vector potential, defi ned, with the usual meaning of the symbols, by

= ∇ × A , (2) 0∇ =A .

Note that in analogy to the model adopted in the fi rst part, to make the vector potential unique (which is obviously neces-sary if it must represent a physical property), Maxwell enforced what is today called the Coulomb gauge.

By a full exploitation of the results of the vector analysis available at his time, he then could state the laws of electrody-namics and electromagnetic induction in explicit mathematical form:

= ∇ ×J H , (3)

i= −∇Φ +E E , (4) i t= − ∂ ∂E A .

Maxwell also provided the expression of the potential energy of a closed current in a magnetic fi eld in terms of the current intensity and the circulation of A, from which all the dynami cal actions can be derived. It is noteworthy that Maxwell explicitly stressed that the so-called “Ampère” law, Equa tion (3), can only be valid for closed currents, and said:

Our investigation are therefore for the present limited to closed currents; and we know little on the magnetic effects of any currents which are not closed.

Maxwell had achieved his goal. However, he was well aware of the purely mathematical and somehow artifi cial character of his construction, and wrote:

....I do not think that it contains even the shadow of a true physical theory; in fact, its chief merit as a temporary instrument of research is that it does not, even in appearance, account for anything.

After this – even excessive – understatement of the relevance of his work, and a great praise of Weber’s electrodynamics, as he wanted to anticipate the objections of continental physi cists, he asked:

what is the use then of imaging an electrotonic state of which we have no distinctly physical conception, instead of a formula of attraction which we can readily understand?

Maxwell provided two answers to this question. One was related to the dependence on velocity of Weber’s force. Just one year after its presentation, Hermann L. von Helmholtz (1821-1894) had published his famous and infl uential mono-graph [10], which put on a fi rm theoretical basis the principle of conservation of energy. Based on Helmholtz’s results, Maxwell and others thought that Weber’s electrodynamics did not comply with this principle. Although in 1848 Weber had shown that his force could be derived by a potential, it was only in 1869 and 1871 that he proved in detail that it satisfi es the principle of conservation of energy. After that, Maxwell obviously corrected himself, but in the meantime the electro-magnetic fi eld theory had been fully developed.

The other answer was, again, of methodological and phi-losophical nature:

I would answer, that it is a good thing to have two ways of looking at a subject, and to admit that there are two ways of looking at it. Besides, I do not think that we have any right at present to understand the action of electricity, and I hold that the chief merit of temporary theory is, that it shall guide experiments, without impeding the progress of the true theory when it appears.

The key words here are “temporary” and “true.” They clearly unveiled Maxwell’s deep conviction that Weber’s theory was unsatisfactory (hence “temporary”), not so much because of its weak points but because of its being an action at distance the ory, while the way toward a “true theory” was that paved by Faraday, through his conception of an action mediated by the medium. However, to develop a “true physical theory,” mathematics was not enough: such a theory must also rely on sound physical, i.e., (for Maxwell’s times) mechanical bases. This explains why Maxwell closed the fi rst part of the memoir with a hope, which was also a program:

By a careful study of the laws of elastic solids and of the motions of viscous fl uid fl uids, I hope to discover a method of forming a mechanical conception of the electro-tonic state adapted to general reasoning.

3. Electromagnetic Clockwork

A relatively long time (for Maxwell standards) had to pass before the hope expressed in the fi rst memoir could be realized. In the meantime, Maxwell published, among the oth ers, fi ve

memoirs about the color vision, his outstanding memoir about the stability of Saturn’s rings, and the fi rst of his epochal memoirs on the kinetic theory of gases.

However, he didn’t stop to refl ect on electromagnetism. His letters to C. J. Monro (May 20, 1857), Faraday (November 9, 1857), and Thomson (January 30, 1858) [11-13] showed an increasing interest in the theory of molecular vortices, pro posed by Thomson to explain the rotation of the plane of polarization of linearly polarized light by a magnetic fi eld. Thomson supposed that this phenomenon, discovered by Faraday in 1845, was caused by the rotation of molecular vor tices in an Aether, having their axis of rotation along the lines of forces of the magnetic fi eld.

No wonder, then, that the theory of molecular vortices was the cornerstone of Maxwell’s second memoir, “On Physi-cal Lines of Force,” which was published in the Philosophical Magazine in March, April, and May, 1861 (Parts I and II), and January and February, 1862 (Parts III and IV) [14], when he was 30 years old (Figure 5). Again, his goal was clearly stated since the beginning:

My object in this paper is to clear the way for speculation in this direction (i.e., Faraday’s point of view) by investigating the mechanical results of certain states of tension and motions in a medium, and comparing these with the observed phenomena of magnetism and electricity. By pointing out the

Figure 5. J. C. Maxwell (circa 1862).

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

10, 1855, and February 11, 1856, and published in extenso the same year in the Transactions of the Society [9].

The fi rst sentence of the paper was trenchant, almost arrogant, for a 24-year-old neo-graduate:

The present state of electrical science seems particularly unfavourable to speculation.

At the very beginning, Maxwell then stated the purpose of his work:

...to show how, by a strict application of the ideas and methods of Faraday, the connection of the very different orders of phenomena which he has discovered can be clearly placed before the mathematical mind.

The method Maxwell adopted to deal with electrostatics, magnetostatics, and electric conduction was that of the physi-cal analogy, i.e., in his own words, “that partial similarity between the laws of one science and those of another which makes each of them illustrate the other.” To this end, he exploited a mechanical model, that of an imponderable and incompressible fl uid moving through a resisting medium, which exerts on it a retarding force proportional to its velocity. The fl uid can be supplied or swallowed by sources and sinks within the considered region of space, or from outside through its boundaries. In this framework, Faraday’s lines of force and tubes of fl ux correspond to lines and tubes of (steady) fl uid motion, respectively, the geometrical and dynamical properties of which are examined in detail in the fi rst two sections of the fi rst part of the memoir.

By properly reinterpreting the mechanical quantities of the model, in the third section, Maxwell obtained the laws of magnetostatics (in absence of currents), electrostatics, and electric conduction. In the same section, he also started to address the phenomena of electrodynamics and electromag-netic induction.

Concerning the mutual actions between currents, Maxwell made a crucial observation:

We must recollect however that no experiments have been made on these elements of currents except under the form of closed currents....Hence if Ampère’s formulae applied to closed currents give true results, their truth is not proved for elements of currents unless we assume that the action between two such elements must be along the line which joins them.

He then put down his main argument:

Although this assumption is most warrantable and philosophical in the present state of science, it will be more conducive to freedom of investigation if we endeavour to do without it, and to assume the laws of closed currents as the ultimate datum

of experiment....In the following investigation, therefore, the laws established by Faraday will be assumed as true,....

However, no physical analogy was now available, so that Maxwell wrote at the end of the fi rst part:

...I can do no more than simply state the mathematical methods by which I believe that electrical phenomena can be best comprehended and reduced to calculation.

This was not an easy task, as was testifi ed by the much higher mathematical level of the second part of the memoir. In con-nection with electromagnetic induction, Maxwell made refer-ence to a somewhat vague concept introduced by Faraday, that of “Electro-tonic state.” For its mathematical description, he adopted what today we call the vector potential, defi ned, with the usual meaning of the symbols, by

= ∇ × A , (2) 0∇ =A .

Note that in analogy to the model adopted in the fi rst part, to make the vector potential unique (which is obviously neces-sary if it must represent a physical property), Maxwell enforced what is today called the Coulomb gauge.

By a full exploitation of the results of the vector analysis available at his time, he then could state the laws of electrody-namics and electromagnetic induction in explicit mathematical form:

= ∇ ×J H , (3)

i= −∇Φ +E E , (4) i t= − ∂ ∂E A .

Maxwell also provided the expression of the potential energy of a closed current in a magnetic fi eld in terms of the current intensity and the circulation of A, from which all the dynami cal actions can be derived. It is noteworthy that Maxwell explicitly stressed that the so-called “Ampère” law, Equa tion (3), can only be valid for closed currents, and said:

Our investigation are therefore for the present limited to closed currents; and we know little on the magnetic effects of any currents which are not closed.

Maxwell had achieved his goal. However, he was well aware of the purely mathematical and somehow artifi cial character of his construction, and wrote:

....I do not think that it contains even the shadow of a true physical theory; in fact, its chief merit as a temporary instrument of research is that it does not, even in appearance, account for anything.

After this – even excessive – understatement of the relevance of his work, and a great praise of Weber’s electrodynamics, as he wanted to anticipate the objections of continental physi cists, he asked:

what is the use then of imaging an electrotonic state of which we have no distinctly physical conception, instead of a formula of attraction which we can readily understand?

Maxwell provided two answers to this question. One was related to the dependence on velocity of Weber’s force. Just one year after its presentation, Hermann L. von Helmholtz (1821-1894) had published his famous and infl uential mono-graph [10], which put on a fi rm theoretical basis the principle of conservation of energy. Based on Helmholtz’s results, Maxwell and others thought that Weber’s electrodynamics did not comply with this principle. Although in 1848 Weber had shown that his force could be derived by a potential, it was only in 1869 and 1871 that he proved in detail that it satisfi es the principle of conservation of energy. After that, Maxwell obviously corrected himself, but in the meantime the electro-magnetic fi eld theory had been fully developed.

The other answer was, again, of methodological and phi-losophical nature:

I would answer, that it is a good thing to have two ways of looking at a subject, and to admit that there are two ways of looking at it. Besides, I do not think that we have any right at present to understand the action of electricity, and I hold that the chief merit of temporary theory is, that it shall guide experiments, without impeding the progress of the true theory when it appears.

The key words here are “temporary” and “true.” They clearly unveiled Maxwell’s deep conviction that Weber’s theory was unsatisfactory (hence “temporary”), not so much because of its weak points but because of its being an action at distance the ory, while the way toward a “true theory” was that paved by Faraday, through his conception of an action mediated by the medium. However, to develop a “true physical theory,” mathematics was not enough: such a theory must also rely on sound physical, i.e., (for Maxwell’s times) mechanical bases. This explains why Maxwell closed the fi rst part of the memoir with a hope, which was also a program:

By a careful study of the laws of elastic solids and of the motions of viscous fl uid fl uids, I hope to discover a method of forming a mechanical conception of the electro-tonic state adapted to general reasoning.

3. Electromagnetic Clockwork

A relatively long time (for Maxwell standards) had to pass before the hope expressed in the fi rst memoir could be realized. In the meantime, Maxwell published, among the oth ers, fi ve

memoirs about the color vision, his outstanding memoir about the stability of Saturn’s rings, and the fi rst of his epochal memoirs on the kinetic theory of gases.

However, he didn’t stop to refl ect on electromagnetism. His letters to C. J. Monro (May 20, 1857), Faraday (November 9, 1857), and Thomson (January 30, 1858) [11-13] showed an increasing interest in the theory of molecular vortices, pro posed by Thomson to explain the rotation of the plane of polarization of linearly polarized light by a magnetic fi eld. Thomson supposed that this phenomenon, discovered by Faraday in 1845, was caused by the rotation of molecular vor tices in an Aether, having their axis of rotation along the lines of forces of the magnetic fi eld.

No wonder, then, that the theory of molecular vortices was the cornerstone of Maxwell’s second memoir, “On Physi-cal Lines of Force,” which was published in the Philosophical Magazine in March, April, and May, 1861 (Parts I and II), and January and February, 1862 (Parts III and IV) [14], when he was 30 years old (Figure 5). Again, his goal was clearly stated since the beginning:

My object in this paper is to clear the way for speculation in this direction (i.e., Faraday’s point of view) by investigating the mechanical results of certain states of tension and motions in a medium, and comparing these with the observed phenomena of magnetism and electricity. By pointing out the

Figure 5. J. C. Maxwell (circa 1862).

AP_Mag_Dec_2014_Final.indd 303 12/27/2014 2:04:58 PM

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

mechanical consequences of such hypothesis, I hope to be of some use to those who consider the phenomena as due to the action of a medium, but are in doubt as to the relation of this hypothesis to the experimental laws already established, which have been generally expressed in the language of other hypotheses (action at distance).

As it is sketched in Figure 6, which appeared in the memoir, the hypothetic medium consisted of molecular vortices, with the axes directed along the magnetic lines of forces, revolving with a peripheral velocity proportional to the intensity of the magnetic fi eld. The density of the vortices was proportional to the magnetic permeability.

To overcome the obvious diffi culty that the periphery of contiguous vortices must move in opposite directions, Maxwell made recourse to the interposition of a layer of round particles between contiguous vortices, playing the role of “idle wheels” (see Figure 6). These particles, which were in rolling contact with the vortices but did not rub against each other, played the role of electricity. Their motion of translation con-stituted an electric current, while their rotation transmitted the motion of the vortices from one part of the fi eld to another. The corresponding tangential stresses thus called into play constituted electromotive force.

By applying to this model the laws of continuum mechan-ics, in the fi rst two parts of the memoir Maxwell derived the Ampère law, Equation (3), the (local) law of the magnetic force on a current,

= ×f J B , (5)

and that of electromagnetic induction, in the form we now call the fi rst Maxwell equation:

/ /t tµ∇ × = −∂ ∂ = −∂ ∂E H B . (6)

He showed that it was equivalent to Equation (4), which was generalized to the case of moving bodies by adding the convec-tive term ×v B .

Of course, Maxwell was well aware of the awkwardness of his mechanical model, as he clearly stated at the end of the second part. However, he had achieved his goal, and it is likely that he originally envisaged his paper as consisting of only those two parts. However, during the summer of 1861, in his country house of Glenlair, in Scotland, he developed his mechanical theory along lines which led to revolutionary and apparently unexpected results. His excitement was clearly testifi ed to by all extant scientifi c letters between the summer of 1861 and January 1862, just before the publication of the last two parts of the memoir. To account for the phenomena of electrostatics, Maxwell extended his model by providing the medium with elastic properties. As the vortices’ rotation was no more of interest, he now spoke of “elastic cells,” surrounded by the layer of

Figure 6. Maxwell’s model of molecular vortices.

particles that played the role of electricity. When the electric particles were displaced from their equilibrium positions, they distorted the cells and called into play a force arising from their elasticity, equal and opposite to that which urged the par ticles away from their equilibrium positions. This state of par ticles displaced from their equilibrium positions and distorted cells was assumed to represent an electrostatic fi eld. However, what happens during the displacement?

In the light of his model, the answer was apparently obvi-ous to Maxwell, who said:

This displacement does not amount to a current because when it has attained a certain value it remains constant, but it is a commencement of a current, and its variations constitute currents, in the positive or negative direction, according as the displacement is increasing or diminishing.

He then performed the crucial step, i.e., the inclusion of the displacement currents in the Ampère law, Equation (3). Assuming a linear relationship between electromotive force (i.e., electric fi eld) and displacement,

2E= −E h , (7)

h being the displacement and E a constant depending on the elastic constants of the medium, he obtained

d= ∇ × +J H J

t= ∇ × + ∂ ∂H h (8)

( )21 E t= ∇ × − ∂ ∂H E .

Note that in coherence with the model, the displacement cur rent, dJ , was added to that due to the molecular vortices, i.e., ∇ × H

. The correct result in terms of E (i.e., what we call the second Maxwell equation) was obtained because of the minus sign in Equation (7), which implied that in Equation (7), E must be identifi ed with the elastic reaction exerted by the cells on the electric particles and not vice versa, as it should be.

This ambiguity shows that while Maxwell was by then sure that the displacement current must be taken fully into account, he was still not aware of the fact that the “total” cur rent must be closed (soleinodal), as well as of the relevance of the step he performed. As a matter of fact, he used Equa tion (6) only to derive, through the equation of continuity, the Gauss equation (and that is why he needed the minus sign in Equation (5)), which he exploited to fi nd the force acting between two charged bodies, in order to relate the mechanical constant 2E to the dielectric constant.

Instead, he came back to the mechanical model, and deter-mined the velocity of propagation of transverse vibrations through the medium. This turned out to be equal (in air) to the same ratio between the measures of an electric charge in the electrostatic and electromagnetic systems that appears in Equa tion (1). This ratio, determined by Weber and Kohlrausch in 1857, agreed so nearly with the velocity of light in air, as determined by Fizeau in 1849, that he concluded:

….we can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.”

Maxwell had unexpectedly established the basis for the elec-tromagnetic theory of light! Since the beginning, he was fully persuaded of the correct-ness of his hypothesis. This is clearly shown by his interest for a precise determination of the “electromagnetic” value of the light velocity, and by his fi rst attempts to explain the refl ection and refraction of light in the framework of the theory.

It must be stressed that the almost perfect coincidence between the constant c in Equation (1) and the velocity of light in vacuo had been noted by Weber and Kohlrausch in their 1856 paper, but they did not consider this to be physically signifi cant. Even more impressive is the fact that in 1857, Gustav Kirchhoff (1824-1887), applying Weber’s electrody namics to the study of the propagation of electrical signals along metallic wires [15], had shown that they propagate with fi nite velocity, which in the case of vanishingly small resis tance is equal to the velocity of light. Again, he did not develop the implications of this result.

The contrast with Maxwell’s attitude is a striking illustra-tion of the infl uence of paradigms on the development of sci-ence. If we look at electromagnetic and optical phenomena under different paradigms, the Newtonian (action at distance between particles) and Cartesian (action by contact, through an interposed medium), respectively, a numerical coincidence

between electromagnetic and optical properties appears for-tuitous. On the other side, if we look at them under the same paradigm, as Maxwell did, the coincidence becomes physi cally relevant, and discloses their possible substantial unity, opening the way to a scientifi c revolution.

However, at least two points had to be addressed before the theory could be considered satisfactory. First of all, the equations for the electromagnetic fi eld and the fundamental property of the total current needed to be clearly and explicitly stated. The properties of the electromagnetic (and optical) waves then had to be derived from the equations, and not exploiting a model so tricky to be considered “imaginary” by Maxwell himself.

The diffi culty of this goal is quite evident: it required about three years to be achieved, and during this period Maxwell published just one minor geometrical paper.

4. Electromagnetic Field

The task was apparently completed in the summer of 1864, as witnessed by a letter to C. Hockin dated September 7, 1864.

On October 27, Maxwell presented to the Royal Society the abstract of his third memoir, which, as already said, was read on December 8, and published in extenso the following year. In the introduction, which summarized in detail the motivations and the content of the memoir, he stated the essence of his theory:

The theory I propose may therefore be called a theory of the Electromagnetic Field, because it has to do with the space in the neighbourhood of the electric and magnetic bodies, and it may be called a Dynamical theory, because it assumes that in that space there is matter in motion, by which the observed electromagnetic phenomena are produced.

It is clear from this statement that Maxwell ascribed a physical, material, reality to the electromagnetic medium, and that, in his view, the electromagnetic phenomena were just the expression of the mechanical properties of this medium. This was by no means surprising. On the contrary, it was in complete agree ment with the then widely accepted view that the propagation of light and radiant heat consisted of undulations of an omnipervasive Aether. In fact, he wrote:

We may therefore receive, as a datum from a branch of science, independent of that with which we have to deal, the existence of a pervading medium, of small but real density, capable of being set in motion, and of transmitting motion from one part to another with great, but not infi nite velocity.

However, this time only very general assumptions concerning the properties of the medium were made, namely the capacity of

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

mechanical consequences of such hypothesis, I hope to be of some use to those who consider the phenomena as due to the action of a medium, but are in doubt as to the relation of this hypothesis to the experimental laws already established, which have been generally expressed in the language of other hypotheses (action at distance).

As it is sketched in Figure 6, which appeared in the memoir, the hypothetic medium consisted of molecular vortices, with the axes directed along the magnetic lines of forces, revolving with a peripheral velocity proportional to the intensity of the magnetic fi eld. The density of the vortices was proportional to the magnetic permeability.

To overcome the obvious diffi culty that the periphery of contiguous vortices must move in opposite directions, Maxwell made recourse to the interposition of a layer of round particles between contiguous vortices, playing the role of “idle wheels” (see Figure 6). These particles, which were in rolling contact with the vortices but did not rub against each other, played the role of electricity. Their motion of translation con-stituted an electric current, while their rotation transmitted the motion of the vortices from one part of the fi eld to another. The corresponding tangential stresses thus called into play constituted electromotive force.

By applying to this model the laws of continuum mechan-ics, in the fi rst two parts of the memoir Maxwell derived the Ampère law, Equation (3), the (local) law of the magnetic force on a current,

= ×f J B , (5)

and that of electromagnetic induction, in the form we now call the fi rst Maxwell equation:

/ /t tµ∇ × = −∂ ∂ = −∂ ∂E H B . (6)

He showed that it was equivalent to Equation (4), which was generalized to the case of moving bodies by adding the convec-tive term ×v B .

Of course, Maxwell was well aware of the awkwardness of his mechanical model, as he clearly stated at the end of the second part. However, he had achieved his goal, and it is likely that he originally envisaged his paper as consisting of only those two parts. However, during the summer of 1861, in his country house of Glenlair, in Scotland, he developed his mechanical theory along lines which led to revolutionary and apparently unexpected results. His excitement was clearly testifi ed to by all extant scientifi c letters between the summer of 1861 and January 1862, just before the publication of the last two parts of the memoir. To account for the phenomena of electrostatics, Maxwell extended his model by providing the medium with elastic properties. As the vortices’ rotation was no more of interest, he now spoke of “elastic cells,” surrounded by the layer of

Figure 6. Maxwell’s model of molecular vortices.

particles that played the role of electricity. When the electric particles were displaced from their equilibrium positions, they distorted the cells and called into play a force arising from their elasticity, equal and opposite to that which urged the par ticles away from their equilibrium positions. This state of par ticles displaced from their equilibrium positions and distorted cells was assumed to represent an electrostatic fi eld. However, what happens during the displacement?

In the light of his model, the answer was apparently obvi-ous to Maxwell, who said:

This displacement does not amount to a current because when it has attained a certain value it remains constant, but it is a commencement of a current, and its variations constitute currents, in the positive or negative direction, according as the displacement is increasing or diminishing.

He then performed the crucial step, i.e., the inclusion of the displacement currents in the Ampère law, Equation (3). Assuming a linear relationship between electromotive force (i.e., electric fi eld) and displacement,

2E= −E h , (7)

h being the displacement and E a constant depending on the elastic constants of the medium, he obtained

d= ∇ × +J H J

t= ∇ × + ∂ ∂H h (8)

( )21 E t= ∇ × − ∂ ∂H E .

Note that in coherence with the model, the displacement cur rent, dJ , was added to that due to the molecular vortices, i.e., ∇ × H

. The correct result in terms of E (i.e., what we call the second Maxwell equation) was obtained because of the minus sign in Equation (7), which implied that in Equation (7), E must be identifi ed with the elastic reaction exerted by the cells on the electric particles and not vice versa, as it should be.

This ambiguity shows that while Maxwell was by then sure that the displacement current must be taken fully into account, he was still not aware of the fact that the “total” cur rent must be closed (soleinodal), as well as of the relevance of the step he performed. As a matter of fact, he used Equa tion (6) only to derive, through the equation of continuity, the Gauss equation (and that is why he needed the minus sign in Equation (5)), which he exploited to fi nd the force acting between two charged bodies, in order to relate the mechanical constant 2E to the dielectric constant.

Instead, he came back to the mechanical model, and deter-mined the velocity of propagation of transverse vibrations through the medium. This turned out to be equal (in air) to the same ratio between the measures of an electric charge in the electrostatic and electromagnetic systems that appears in Equa tion (1). This ratio, determined by Weber and Kohlrausch in 1857, agreed so nearly with the velocity of light in air, as determined by Fizeau in 1849, that he concluded:

….we can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.”

Maxwell had unexpectedly established the basis for the elec-tromagnetic theory of light! Since the beginning, he was fully persuaded of the correct-ness of his hypothesis. This is clearly shown by his interest for a precise determination of the “electromagnetic” value of the light velocity, and by his fi rst attempts to explain the refl ection and refraction of light in the framework of the theory.

It must be stressed that the almost perfect coincidence between the constant c in Equation (1) and the velocity of light in vacuo had been noted by Weber and Kohlrausch in their 1856 paper, but they did not consider this to be physically signifi cant. Even more impressive is the fact that in 1857, Gustav Kirchhoff (1824-1887), applying Weber’s electrody namics to the study of the propagation of electrical signals along metallic wires [15], had shown that they propagate with fi nite velocity, which in the case of vanishingly small resis tance is equal to the velocity of light. Again, he did not develop the implications of this result.

The contrast with Maxwell’s attitude is a striking illustra-tion of the infl uence of paradigms on the development of sci-ence. If we look at electromagnetic and optical phenomena under different paradigms, the Newtonian (action at distance between particles) and Cartesian (action by contact, through an interposed medium), respectively, a numerical coincidence

between electromagnetic and optical properties appears for-tuitous. On the other side, if we look at them under the same paradigm, as Maxwell did, the coincidence becomes physi cally relevant, and discloses their possible substantial unity, opening the way to a scientifi c revolution.

However, at least two points had to be addressed before the theory could be considered satisfactory. First of all, the equations for the electromagnetic fi eld and the fundamental property of the total current needed to be clearly and explicitly stated. The properties of the electromagnetic (and optical) waves then had to be derived from the equations, and not exploiting a model so tricky to be considered “imaginary” by Maxwell himself.

The diffi culty of this goal is quite evident: it required about three years to be achieved, and during this period Maxwell published just one minor geometrical paper.

4. Electromagnetic Field

The task was apparently completed in the summer of 1864, as witnessed by a letter to C. Hockin dated September 7, 1864.

On October 27, Maxwell presented to the Royal Society the abstract of his third memoir, which, as already said, was read on December 8, and published in extenso the following year. In the introduction, which summarized in detail the motivations and the content of the memoir, he stated the essence of his theory:

The theory I propose may therefore be called a theory of the Electromagnetic Field, because it has to do with the space in the neighbourhood of the electric and magnetic bodies, and it may be called a Dynamical theory, because it assumes that in that space there is matter in motion, by which the observed electromagnetic phenomena are produced.

It is clear from this statement that Maxwell ascribed a physical, material, reality to the electromagnetic medium, and that, in his view, the electromagnetic phenomena were just the expression of the mechanical properties of this medium. This was by no means surprising. On the contrary, it was in complete agree ment with the then widely accepted view that the propagation of light and radiant heat consisted of undulations of an omnipervasive Aether. In fact, he wrote:

We may therefore receive, as a datum from a branch of science, independent of that with which we have to deal, the existence of a pervading medium, of small but real density, capable of being set in motion, and of transmitting motion from one part to another with great, but not infi nite velocity.

However, this time only very general assumptions concerning the properties of the medium were made, namely the capacity of

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

receiving and storing both “actual” (i.e., kinetic) and “potential” energy (through some kind of elasticity), and the fact that it must be subject to the general laws of dynamics.

In the following two sections, expressing the laws of elec tromagnetic induction in the language of Lagrangian dynamics, Maxwell identifi ed the vector potential as the gen-eralized momentum conjugate to the current density, which he explicitly requires to be the total current, namely the sum of the conduction current and the displacement current. This allowed him to get the following equations:

A) t= + ∂ ∂C K D eq. of total currents

B) µ = ∇ ×H A eq. of magnetic force

C) ∇ × =H C eq. of currents

D) tµ= × − ∂ ∂ − ∇ΦE V H A eq. of electromotive force

to which he added the following equations. He thus obtained a set of 20 scalar equations in 20 unknowns:

E) k=E D eq. of electric elasticity

F) ρ= −E K eq. of electric resistance

G) 0e∇ + =D eq. of free electricity

H) 0e t∇ + ∂ ∂ =K eq. of continuity

He then derived the expressions of the magnetic (i.e., kinetic) and electric (i.e., elastic) energy densities, which was exploited in the following section to fi nd the forces on cur rents and magnets.

Putting 0=V (stationary media) and µ=B H , we imme-diately recognize that Equations A to D are equivalent to the fi rst two Maxwell’s equations, as we write them today, plus the fourth one (that expressing that the magnetic induc tion is solenoidal). On the other side, Equation G is not Gauss’ equation, as it should be, because of the wrong sign. This is exactly the opposite of what happened in the second memoir, wherein Gauss’ law was correct, whereas the relation between the electric fi eld and electric displacement had the wrong sign: see Equation (7). Moreover, this time the relation Equation F between the fi eld and conduction current also had the wrong sign.

The presence of these errors, which would be corrected in the Treatise [16], was quite surprising, in the light of Maxwell’s deep physical and mathematical attitude. In fact, Equation G is mathematically inconsistent with Equations C and H, while Equation F clearly contrasts with the dissipative nature of conduction currents. We will come back to this point in a while.

Anyway, the errors did not have a direct impact on Maxwell’s main goal: the electromagnetic theory of light, which was addressed in Section VI, wherein the properties of electromagnetic waves were deduced directly from the fi eld equations (even if in a somewhat involved way), and their propagation in isotropic and anisotropic media was examined in detail and compared with that of optical waves. This allows him to state:

Hence electromagnetic science leads to exactly the same conclusions as optical science with respect to the direction of the disturbances which can propagate through the fi eld; both affi rm the propagation of transverse vibrations and both give the same velocity of propagation....The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the fi eld according to electromagnetic laws.

The electromagnetic theory of light had been established.

Maxwell then proceeded to analyze the propagation of electromagnetic waves in conductive media. It is interesting to note that he avoided the consequence of the sign error in Ohm’s law, Equation F, by means of a further change of sign, assuming for a sinusoidal plane wave along x the expression

( ) ( ) ( ), exp cosA x t px qx nt= − + , (9)

i.e., a wave attenuating in the forward direction, but propa-gating in the backward direction!

The root of this kind of plus-minus dyslexia lies in Maxwell’s attitude toward the nature of electric charges and currents. In accordance with his emphasis on the role of the medium, Maxwell (and the British Maxwellians after him) considered charges and currents not as the sources of the fi eld, but, vice versa, as a product or a property of the fi eld itself. In other words, the Aether was the only fundamental physical entity, and the description of the phenomena must be obtained by a proper characterization of its dynamical properties (i.e., its Lagrangian or Hamiltonian).

Apart from being doomed to failure – because charged matter possesses its own degrees of freedom, so that it consti-tutes a dynamical system distinct from (even if coupled to) the electromagnetic fi eld – this attitude explains the fact that nei ther Maxwell nor the Maxwellians considered the question of generating electromagnetic waves distinct from light. As is well known, this crucial validation of Maxwell’s equations was performed by Heinrich R. Hertz (1857-1894) only in October 1886, twenty-two years after their formulation, whereas the defi nitive abandonment of mechanical Aether theories had to wait for Einstein’s epochal paper of 1905.

In the meantime, the atomic theory of matter had been developed and the quantum revolution had started. However, the equations stood unchanged, and today are still those envis-aged by Maxwell’s genius 150 years ago.

5. References

1. J. C. Maxwell, “A Dynamical Theory of the Electromag netic Field,” Phil. Trans. Roy. Soc., 155, pp. 459-512, 1865.

2. T. S. Kuhn, The Structure of Scientifi c Revolutions, Chi cago, The University of Chicago Press, 1962.

3. O. M. Bucci, “The Genesis of Maxwell’s Equations,” in T. K. Sarkar et al. (eds.), History of Wireless, New York, John Wiley & Sons, 2006, Chapter 5, pp. 189-214.

4. O. M. Bucci, “The Birth of Maxwell’s Equations,” 44th European Microwave Conference, Rome, Italy, October 2014.

5. P. M. Harmon (ed.), The Scientifi c Letters and Papers of James Clerk Maxwell, Volume 1, Cambridge, Cambridge Uni-versity Press, 1990, pp. 237-238.

6. W. Weber, “Elektrodynamische Maassbestimmungen, uber ein allgemeines Grundgesetz der Elektrischen Wirkung,” Leipzig Abhandl., 1846, pp. 211-378.

7. P. M. Harmon (ed.), The Scientifi c Letters and Papers of James Clerk Maxwell, Volume 1, Cambridge, Cambridge Uni-versity Press, 1990, pp.254-263.

8. The Scientifi c Letters and Papers of James Clerk Maxwell, Volume 1, Cambridge, Cambridge University Press, 1990, p. 294.

9. J. C. Maxwell, “On Faraday’s Lines of Force,” Trans. Camb. Phil. Soc., X, 1856, pp. 27-83.

10. H. L. Helmholtz, Uber die Erhaltung der Kraft, Berlin, G. A. Reimer, 1847.

11. P. M. Harmon (ed.), The Scientifi c Letters and Papers of James Clerk Maxwell, Volume 1, Cambridge, Cambridge Uni-versity Press, 1990, pp.505-507.

12. The Scientifi c Letters and Papers of James Clerk Maxwell, Volume 1, Cambridge, Cambridge University Press, 1990, pp. 548-552.

13. The Scientifi c Letters and Papers of James Clerk Maxwell, Volume 1, Cambridge, Cambridge University Press, 1990, pp. 578-581.

14. J. C. Maxwell, “On Physical Lines of Force,” Phil. Mag., XXI, pp. 161-175, 281-291, 338-348, 1861 (Parts I and II); Phil. Mag., XXIII, pp. 12-25, 85-95, 1862 (Parts III and IV).

15. G. Kirchhoff, “Uber die Bewegung der Electricitat in Drahten,” Ann. Phys., 100, pp. 193-217, 1857.

16. J. C. Maxwell, A Treatise on Electricity and Magnetism, Oxford, Clarendon Press, 1873.

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

receiving and storing both “actual” (i.e., kinetic) and “potential” energy (through some kind of elasticity), and the fact that it must be subject to the general laws of dynamics.

In the following two sections, expressing the laws of elec tromagnetic induction in the language of Lagrangian dynamics, Maxwell identifi ed the vector potential as the gen-eralized momentum conjugate to the current density, which he explicitly requires to be the total current, namely the sum of the conduction current and the displacement current. This allowed him to get the following equations:

A) t= + ∂ ∂C K D eq. of total currents

B) µ = ∇ ×H A eq. of magnetic force

C) ∇ × =H C eq. of currents

D) tµ= × − ∂ ∂ − ∇ΦE V H A eq. of electromotive force

to which he added the following equations. He thus obtained a set of 20 scalar equations in 20 unknowns:

E) k=E D eq. of electric elasticity

F) ρ= −E K eq. of electric resistance

G) 0e∇ + =D eq. of free electricity

H) 0e t∇ + ∂ ∂ =K eq. of continuity

He then derived the expressions of the magnetic (i.e., kinetic) and electric (i.e., elastic) energy densities, which was exploited in the following section to fi nd the forces on cur rents and magnets.

Putting 0=V (stationary media) and µ=B H , we imme-diately recognize that Equations A to D are equivalent to the fi rst two Maxwell’s equations, as we write them today, plus the fourth one (that expressing that the magnetic induc tion is solenoidal). On the other side, Equation G is not Gauss’ equation, as it should be, because of the wrong sign. This is exactly the opposite of what happened in the second memoir, wherein Gauss’ law was correct, whereas the relation between the electric fi eld and electric displacement had the wrong sign: see Equation (7). Moreover, this time the relation Equation F between the fi eld and conduction current also had the wrong sign.

The presence of these errors, which would be corrected in the Treatise [16], was quite surprising, in the light of Maxwell’s deep physical and mathematical attitude. In fact, Equation G is mathematically inconsistent with Equations C and H, while Equation F clearly contrasts with the dissipative nature of conduction currents. We will come back to this point in a while.

Anyway, the errors did not have a direct impact on Maxwell’s main goal: the electromagnetic theory of light, which was addressed in Section VI, wherein the properties of electromagnetic waves were deduced directly from the fi eld equations (even if in a somewhat involved way), and their propagation in isotropic and anisotropic media was examined in detail and compared with that of optical waves. This allows him to state:

Hence electromagnetic science leads to exactly the same conclusions as optical science with respect to the direction of the disturbances which can propagate through the fi eld; both affi rm the propagation of transverse vibrations and both give the same velocity of propagation....The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the fi eld according to electromagnetic laws.

The electromagnetic theory of light had been established.

Maxwell then proceeded to analyze the propagation of electromagnetic waves in conductive media. It is interesting to note that he avoided the consequence of the sign error in Ohm’s law, Equation F, by means of a further change of sign, assuming for a sinusoidal plane wave along x the expression

( ) ( ) ( ), exp cosA x t px qx nt= − + , (9)

i.e., a wave attenuating in the forward direction, but propa-gating in the backward direction!

The root of this kind of plus-minus dyslexia lies in Maxwell’s attitude toward the nature of electric charges and currents. In accordance with his emphasis on the role of the medium, Maxwell (and the British Maxwellians after him) considered charges and currents not as the sources of the fi eld, but, vice versa, as a product or a property of the fi eld itself. In other words, the Aether was the only fundamental physical entity, and the description of the phenomena must be obtained by a proper characterization of its dynamical properties (i.e., its Lagrangian or Hamiltonian).

Apart from being doomed to failure – because charged matter possesses its own degrees of freedom, so that it consti-tutes a dynamical system distinct from (even if coupled to) the electromagnetic fi eld – this attitude explains the fact that nei ther Maxwell nor the Maxwellians considered the question of generating electromagnetic waves distinct from light. As is well known, this crucial validation of Maxwell’s equations was performed by Heinrich R. Hertz (1857-1894) only in October 1886, twenty-two years after their formulation, whereas the defi nitive abandonment of mechanical Aether theories had to wait for Einstein’s epochal paper of 1905.

In the meantime, the atomic theory of matter had been developed and the quantum revolution had started. However, the equations stood unchanged, and today are still those envis-aged by Maxwell’s genius 150 years ago.

5. References

1. J. C. Maxwell, “A Dynamical Theory of the Electromag netic Field,” Phil. Trans. Roy. Soc., 155, pp. 459-512, 1865.

2. T. S. Kuhn, The Structure of Scientifi c Revolutions, Chi cago, The University of Chicago Press, 1962.

3. O. M. Bucci, “The Genesis of Maxwell’s Equations,” in T. K. Sarkar et al. (eds.), History of Wireless, New York, John Wiley & Sons, 2006, Chapter 5, pp. 189-214.

4. O. M. Bucci, “The Birth of Maxwell’s Equations,” 44th European Microwave Conference, Rome, Italy, October 2014.

5. P. M. Harmon (ed.), The Scientifi c Letters and Papers of James Clerk Maxwell, Volume 1, Cambridge, Cambridge Uni-versity Press, 1990, pp. 237-238.

6. W. Weber, “Elektrodynamische Maassbestimmungen, uber ein allgemeines Grundgesetz der Elektrischen Wirkung,” Leipzig Abhandl., 1846, pp. 211-378.

7. P. M. Harmon (ed.), The Scientifi c Letters and Papers of James Clerk Maxwell, Volume 1, Cambridge, Cambridge Uni-versity Press, 1990, pp.254-263.

8. The Scientifi c Letters and Papers of James Clerk Maxwell, Volume 1, Cambridge, Cambridge University Press, 1990, p. 294.

9. J. C. Maxwell, “On Faraday’s Lines of Force,” Trans. Camb. Phil. Soc., X, 1856, pp. 27-83.

10. H. L. Helmholtz, Uber die Erhaltung der Kraft, Berlin, G. A. Reimer, 1847.

11. P. M. Harmon (ed.), The Scientifi c Letters and Papers of James Clerk Maxwell, Volume 1, Cambridge, Cambridge Uni-versity Press, 1990, pp.505-507.

12. The Scientifi c Letters and Papers of James Clerk Maxwell, Volume 1, Cambridge, Cambridge University Press, 1990, pp. 548-552.

13. The Scientifi c Letters and Papers of James Clerk Maxwell, Volume 1, Cambridge, Cambridge University Press, 1990, pp. 578-581.

14. J. C. Maxwell, “On Physical Lines of Force,” Phil. Mag., XXI, pp. 161-175, 281-291, 338-348, 1861 (Parts I and II); Phil. Mag., XXIII, pp. 12-25, 85-95, 1862 (Parts III and IV).

15. G. Kirchhoff, “Uber die Bewegung der Electricitat in Drahten,” Ann. Phys., 100, pp. 193-217, 1857.

16. J. C. Maxwell, A Treatise on Electricity and Magnetism, Oxford, Clarendon Press, 1873.

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