## Summary

James Clerk Maxwell (13 June 1831, Edinburgh, Scotland – 5 November 1879, Cambridge, England) was a British (Scottish) physicist, mathematician and mechanic. Member of the Royal Society of London (1861). Maxwell laid the foundations of modern classical electrodynamics (Maxwell”s equations), introduced into physics the concepts of displacement current and electromagnetic field, received a number of consequences from his theory (prediction of electromagnetic waves, electromagnetic nature of light, light pressure and others). One of the founders of the kinetic theory of gases (he established the distribution of gas molecules by velocities). He was one of the first who introduced statistical ideas into physics, showed the statistical nature of the second principle of thermodynamics (“Maxwell”s daemon”), obtained a number of important results in molecular physics and thermodynamics (Maxwell”s thermodynamic relations, Maxwell”s rule for the phase transition of liquid – gas, etc.). Pioneer of quantitative color theory; author of three-color principle of color photography. Among Maxwell”s other works are studies in mechanics (photoelasticity, Maxwell”s theorem in elasticity theory, works in motion stability theory, analysis of the stability of Saturn”s rings), optics, mathematics. He prepared for publication manuscripts of Henry Cavendish”s works, paid much attention to the popularization of science, and designed a number of scientific instruments.

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## Origins and Youth. First scientific work (1831-1847)

James Clerk Maxwell belonged to the old Scottish Clerk family of Penicuik. His father, John Clerk Maxwell, was the owner of the Middleby family estate in South Scotland (the second surname Maxwell reflects this fact). He graduated from Edinburgh University and was a member of the bar, but had no love of the law, having a passion in his spare time for science and technology (he even published several articles of an applied nature) and regularly attending meetings of the Royal Society of Edinburgh as an audience. In 1826 he married Frances Cay, daughter of a judge of the Admiralty Court, who five years later bore him a son.

Soon after the birth of their son, the family moved from Edinburgh to their abandoned Middleby estate, where a new house was built, named Glenlair (meaning “den in a narrow hollow”). Here James Clerk Maxwell spent his childhood years, overshadowed by the early death of his mother from cancer. Life in nature made him hardy and curious. From an early age he showed interest in the world around him, was surrounded by various “scientific toys” (for example, the “magic disk” – the forerunner of the cinematograph, a model of the celestial sphere, a “devil” volley, etc.), learned much from communication with his father, was interested in poetry and made his first poetic experiments. It was not until he was ten years old that he had a specially hired home teacher, but such instruction proved ineffective, and in November 1841 Maxwell moved with his aunt Isabella, his father”s sister, to Edinburgh. Here he entered a new school, the so-called Edinburgh Academy, which emphasized a classical education – the study of Latin, Greek and English, Roman literature and the Scriptures.

At first Maxwell was not attracted to learning, but gradually he developed a taste for it and became the best student in the class. At this time he became interested in geometry, making polyhedrons out of cardboard. His appreciation of the beauty of geometric images grew after a lecture by the artist David Ramsay Hay on the art of the Etruscans. Reflection on this subject led Maxwell to invent a method of drawing ovals. This method, which went back to the work of René Descartes, consisted of using focal pins, threads, and a pencil to draw circles (one focus), ellipses (two focuses), and more complex oval shapes (more focuses). These results were reported by Professor James Forbes at a meeting of the Royal Society of Edinburgh and then published in his Proceedings. During his studies at the Academy, Maxwell became close friends with a classmate, Lewis Campbell, later a famous classical philologist and Maxwell”s biographer, and the future famous mathematician Peter Guthrie Tate, who was a class below him.

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## University of Edinburgh. Photoelasticity (1847-1850)

In 1847 his term at the academy ended, and in November Maxwell entered the University of Edinburgh, where he attended lectures by the physicist Forbes, the mathematician Philip Kelland, and the philosopher William Hamilton; he studied numerous works in mathematics, physics, and philosophy and made experiments in optics, chemistry, and magnetism. During his studies, Maxwell prepared a paper on rolling curves, but he focused on the study of the mechanical properties of materials using polarized light. The idea for this research goes back to his acquaintance in the spring of 1847 with the famous Scottish physicist William Nicol, who gave him two polarizing devices of his own design (Nicol prisms). Maxwell realized that polarized radiation could be used to determine the internal stresses of loaded solids. He made models of bodies of various shapes out of gelatin and, subjecting them to deformations, observed in polarized light colored patterns corresponding to the curves of directions of compression and stretching. By comparing the results of his experiments with theoretical calculations, Maxwell checked many old and derived new laws of the theory of elasticity, including in those cases that were too difficult to calculate. In all, he solved 14 problems on the stresses inside hollow cylinders, rods, circular disks, hollow spheres, and flat triangles, thus making a significant contribution to the development of the method of photoelasticity. These results were also of considerable interest for structural mechanics. Maxwell reported them in 1850 at a meeting of the Royal Society of Edinburgh, the first serious recognition of his work.

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## Cambridge (1850-1856)

In 1850, despite his father”s desire to keep his son close to him, it was decided that Maxwell would go to Cambridge University (all his friends had already left Scotland for a more prestigious education). In the fall he arrived in Cambridge, where he enrolled at the cheapest college, Peterhouse, with a room in the college building itself. He was not satisfied with the Peterhouse curriculum, however, and there was little chance of staying at the college after graduation. Many of his relatives and acquaintances, including professors James Forbes and William Thomson (some of his Scottish friends also attended. Eventually, after his first semester at Peterhouse, James convinced his father to transfer to Trinity.

In 1852 Maxwell became a fellow of the college and was given a room directly in its building. At that time he did little scientific work but read a great deal, attended lectures by George Stokes and seminars by William Hopkins, who prepared him for his examinations, made new friends, wrote poems for fun (many of them were later published by Lewis Campbell). Maxwell took an active part in the intellectual life of the university. He was elected to the “club of apostles,” which united twelve people with the most original and profound ideas; there he gave papers on a variety of topics. The fellowship with new people enabled him to compensate for the shyness and reticence he had developed during his years of quiet life at home. James”s daily routine also seemed unusual to many: from seven in the morning to five in the evening he worked, then went to bed, got up at half past ten to read, from two to half past three in the morning to run through the corridors of the dormitory as exercise, then slept again until the morning.

By this time his philosophical and religious views were finally formed. The latter were characterized by a considerable eclecticism dating back to his childhood years, when he attended both his father”s Presbyterian church and his aunt Isabella”s Episcopal church. At Cambridge, Maxwell became an adherent of the theory of Christian socialism developed by the theologian Frederick Denison Maurice, an ideologue of the “broad church” (The broad church and one of the founders of the Working Men”s College. Believing that the main way to improve society was education and the development of culture, James participated in the work of this institution, giving popular lectures there in the evenings. However, despite his unquestioning faith in God, he was not overly religious, receiving warnings repeatedly for missing church services. In a letter to his friend Lewis Campbell, who had decided to choose a theological career, Maxwell ranked the sciences as follows:

In every field of knowledge, progress is proportional to the number of facts on which it is built, and thus related to the possibility of obtaining objective data. In mathematics it is simple. <…> Chemistry is far ahead of all the sciences of Natural History; they are all ahead of Medicine, Medicine ahead of Metaphysics, Law and Ethics; and they are all ahead of Theology. …I believe that the more down-to-earth and material sciences are by no means to be despised in comparison with the sublime study of Mind and Spirit.

In another letter, he formulated the principle of his scientific work and life in general:

Here is my great plan, which has been conceived for a long time, and which now and then dies, now returns to life, and gradually becomes more and more obsessive… The basic rule of this plan is to stubbornly leave nothing unexplored. Nothing must be “holy ground,” sacred Immutable Truth, positive or negative.

In January 1854 Maxwell passed his final three-step examination in mathematics (Mathematical Tripos) and, ranked second in the list of students (Second Wrangler), received his bachelor”s degree. In the next test, a written mathematical study for the traditional Smith Prize, he solved a problem proposed by Stokes concerning the proof of a theorem that is now called the Stokes theorem. At the end of this test he shared the prize with his classmate Edward Rouse.

After passing his examinations, Maxwell decided to remain in Cambridge to prepare for a professorship. He tutored students, took examinations at Cheltenham College, made new friends, continued to work with the Working College, began to write a book on optics at the suggestion of editor Macmillan (it was never finished), and in his spare time visited his father in Glenlaire, whose health had deteriorated dramatically. This was also the time of a mock experimental study on “catcalling” that entered Cambridge folklore: its purpose was to determine the minimum height from which a cat would stand on its four paws if it fell.

However, Maxwell”s main scientific interest at this time was his work on color theory. It originated in the work of Isaac Newton, who adhered to the idea of seven primary colors. Maxwell acted as the continuator of Thomas Jung”s theory, who put forward the idea of three primary colors and connected them with physiological processes in the human body. The important information contained testimonies of patients with color blindness, or color blindness. In experiments on color mixing, which in many respects independently repeated experiments of Hermann Helmholtz, Maxwell applied a “color wave”, the disk of which was divided into painted in different colors sectors, and also a “color box”, an optical system developed by him, which allowed mixing of reference colors. Similar devices were used before, but only Maxwell began to receive with their help quantitative results and to predict quite precisely the colors arising as a result of mixing. So, it has shown, that mixing of dark blue and yellow colors gives not green, as it was often believed, and pinkish shade. Maxwell”s experiments have shown that white color cannot be received by mixture of dark blue, red and yellow as David Brewster and some other scientists believed, and the basic colors are red, green and dark blue. For graphic representation of colors Maxwell, following Jung, used a triangle the points inside which designate result of mixture of the basic colors located in tops of a figure.

Maxwell”s first serious interest in the problem of electricity also dates back to his years at Cambridge. Shortly after passing his examination, in February 1854, he asked William Thomson to recommend literature on the subject and how to read it. At the time Maxwell began his study of electricity and magnetism, there were two views on the nature of electrical and magnetic effects. Most continental scientists, such as André Marie Amper, Franz Neumann, and Wilhelm Weber, held the concept of long-range action, treating electromagnetic forces as analogous to gravitational attraction between two masses that interact instantaneously at a distance. Electrodynamics, developed by these physicists, was a well-formed and rigorous science. On the other hand, Michael Faraday, the discoverer of the phenomenon of electromagnetic induction, put forward the idea of force lines that connect positive and negative electric charges or the north and south poles of a magnet. According to Faraday, lines of force fill the entire surrounding space, forming a field, and cause electrical and magnetic interactions. Maxwell could not accept the concept of action at a distance, it contradicted his physical intuition, so he soon switched to Faraday”s position:

When we observe that one body acts on another at a distance, before we accept that this action is direct and direct, we usually examine whether there is any material connection between the bodies… To whom the properties of air are not familiar, the transfer of force through this invisible medium will seem as incomprehensible as any other example of action at a distance… We should not look at these lines as purely mathematical abstractions. They are directions in which the medium experiences a tension similar to the tension of a rope…

Maxwell faced the question of constructing a mathematical theory that would include both Faraday”s notions and the correct results obtained by the proponents of long-range action. Maxwell decided to use the method of analogies successfully applied by William Thomson, who already in 1842 noticed an analogy between electrical interaction and heat transfer processes in a solid body. This allowed him to apply to electricity the results obtained for heat and give the first mathematical justification of the processes of electrical action transfer through some medium. In 1846 Thomson studied the analogy between electricity and elasticity. Maxwell used another analogy: he developed a hydrodynamic model of force lines, likening them to tubes with a perfect incompressible fluid (the vectors of magnetic and electric induction are analogous to the velocity vector of the fluid), and for the first time he expressed the laws of Faraday”s field picture in mathematical language (differential equations). In the figurative expression of Robert Milliken, Maxwell “clothed the plebeian naked body of Faraday”s ideas in the aristocratic garb of mathematics. However, he did not succeed at that time in uncovering the connection between resting charges and “moving electricity” (currents), the lack of which was apparently one of his main motivations in his work.

In September 1855 Maxwell attended a congress of the British Science Association in Glasgow, stopping on the way to visit his sick father, and on his return to Cambridge successfully passed his examination to become a member of the college board (which implied a vow of celibacy). In the new term Maxwell began lecturing on hydrostatics and optics. In the winter of 1856 he returned to Scotland, moved his father to Edinburgh, and returned to England in February. At that time he learned of a vacancy as professor of natural philosophy at Marischal College, Aberdeen, and decided to try for the position, hoping to be closer to his father and seeing no clear prospects at Cambridge. In March Maxwell took his father back to Glenlair, where he seemed to be getting better, but on April 2 his father passed away. At the end of April Maxwell received an appointment as professor at Aberdeen and, after spending the summer on the family estate, arrived at his new workplace in October.

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## Aberdeen (1856-1860)

From his first days in Aberdeen Maxwell set about setting up teaching in the department of natural philosophy, which was in a neglected state. He sought the right method of teaching, tried to train students to work scientifically, but did not succeed. His lectures, spiced with humor and wordplay, often touched on things so complex that they deterred many. They differed from the earlier model with less emphasis on popular presentation and breadth of subject matter, more modest demonstrations and more attention to the mathematical side of things. In addition, Maxwell was one of the first to engage students in practical work and to offer last year”s students extra classes outside the standard course. As astronomer David Gill, one of his Aberdeen students, recalled,

…Maxwell was not a good teacher; only four or five of us, seventy or eighty of us, learned much from him. We used to stay with him for a couple of hours after lectures, until his awful wife came and dragged him to a meager three o”clock dinner. He was a most pleasant and lovable creature in his own right – he would often fall asleep and wake up suddenly – then talk about whatever came into his head.

In Aberdeen there were serious changes in Maxwell”s personal life: in February 1858 he became engaged to Catherine Mary Dewar, the younger daughter of Marischal College principal Daniel Dewar, professor of church history, and in June they were married. Immediately after the wedding Maxwell was expelled from the Trinity College board because he had broken his vow of celibacy. At the same time Maxwell”s philosophical views on science, as expressed in one of his friendly letters, finally came to fruition:

As far as the material sciences are concerned, these seem to me to be the direct path to any scientific truth concerning metaphysics, one”s own thoughts, or society. The sum of knowledge that exists in these subjects takes a large portion of its value from ideas derived by drawing analogies from the material sciences, and the remainder, though important to humanity, is not scientific but aphoristic. The basic philosophical value of physics is that it gives the brain something definite to rely on. If you find yourself somewhere wrong, nature itself will tell you right away.

As for scientific work in Aberdeen, at first Maxwell was engaged in designing a “dynamic wave”, which was created at his request and demonstrated some aspects of the theory of rotation of solids. In 1857, the Proceedings of the Cambridge Philosophical Society published his article On Faraday”s lines of force, which contained the results of his research on electricity over the previous few years. In March, Maxwell sent it to major British physicists, including Faraday himself, with whom he struck up a friendly correspondence. Another issue he addressed at this time was geometric optics. In the article “On the general laws of optical instruments” (On the general laws of optical instruments) he analyzed the conditions that a perfect optical instrument must have. Subsequently, Maxwell returned more than once to the topic of refraction of light in complex systems, applying his results to the operation of specific devices.

However, much more Maxwell”s attention at this time was attracted by the study of the nature of Saturn”s rings, proposed in 1855 by Cambridge University for the Adams Prize (the work had to be completed in two years). The rings were discovered by Galileo Galilei in the early 17th century and had long been a natural mystery: the planet seemed to be surrounded by three continuous concentric rings composed of matter of unknown nature (the third ring had been discovered shortly before by George Bond). William Herschel believed them to be continuous solid objects. Pierre Simon Laplace proved that solid rings must be inhomogeneous, very narrow and must necessarily rotate. After carrying out a mathematical analysis of the different variants of the structure of the rings, Maxwell was convinced that they could be neither solid nor liquid (in the latter case, the ring would quickly collapse, disintegrating into droplets). He concluded that such a structure could only be stable if it consisted of a swarm of unconnected meteorites. Stability of the rings is provided by their attraction to Saturn and mutual movement of the planet and meteorites. Using Fourier analysis, Maxwell studied wave propagation in such a ring and showed that under certain conditions meteorites do not collide with each other. For the case of two rings, he determined at what ratios of their radii comes a state of instability. For this work back in 1857 Maxwell received the Adams Prize, but continued to work on this topic, which resulted in the publication in 1859 of his treatise “On the stability of the motion of Saturn”s rings” (On the stability of the motion of Saturn”s rings). This work was immediately recognized in scientific circles. The royal astronomer George Airy declared it the most brilliant application of mathematics to physics he had ever seen. Later, influenced by the methods of the kinetic theory of gases, Maxwell tried to develop the kinetic theory of rings, but was not successful in this endeavor. The problem turned out to be much more difficult than in the case of gases, because of the inelasticity of meteorite collisions and the essential anisotropy of their velocity distribution. In 1895, James Keeler and Aristarchus Belopolsky measured the Doppler shift of different parts of Saturn”s rings and found that the inner parts moved faster than the outer parts. This confirmed Maxwell”s conclusion that the rings consist of many small bodies obeying Kepler”s laws. Maxwell”s work on the stability of Saturn”s rings is considered “the first work on the theory of collective processes done at the modern level.

Maxwell”s other main scientific activity at this time was the kinetic theory of gases, based on the notion of heat as a kind of motion of gas particles (atoms or molecules). Maxwell continued the ideas of Rudolf Clausius, who introduced the concepts of average free path length and average speed of molecules (it was assumed that in equilibrium all molecules have the same speed). Clausius, on the other hand, introduced elements of probability theory into kinetic theory. Maxwell decided to take up the subject after reading the work of the German scientist in the February 1859 issue of Philosophical Magazine, initially intending to disprove Clausius”s views, but then recognizing them as worthy of attention and development. As early as September 1859, Maxwell gave a paper on his work at a meeting of the British Association in Aberdeen. The results contained in the report were published in the article “Illustrations of the Dynamical Theory of Gases” (Illustrations of the Dynamical Theory of Gases), which appeared in three parts in January and July 1860. Maxwell proceeded from the idea of a gas as an ensemble of a set of perfectly elastic balls moving chaotically in a closed space and colliding with each other. The balls-molecules can be divided into groups according to their velocities, and in the stationary state the number of molecules in each group remains constant, although they can change their velocity after collisions. It followed from such consideration that in equilibrium the particles do not have the same velocity, but are distributed by velocities according to the Gauss curve (Maxwell distribution). Using the resulting distribution function, Maxwell calculated a number of quantities that play an important role in transport phenomena: the number of particles in a certain velocity range, the average velocity and the mean square of the velocity. The total distribution function was calculated as the product of the distribution functions for each of the coordinates. This implied their independence, which at the time seemed unobvious to many and required a proof (it was given later).

Maxwell further refined the numerical coefficient in the expression for the mean free path length and also proved the equality of the mean kinetic energies in an equilibrium mixture of two gases. Having considered the problem of internal friction (viscosity), Maxwell was able to estimate for the first time the value of the mean free path length, obtaining the correct order of magnitude. Another consequence of the theory was a seemingly paradoxical conclusion about independence of the coefficient of internal friction of a gas from its density, which was later confirmed experimentally. In addition, the explanation of Avogadro”s law followed directly from the theory. Thus, in his work of 1860, Maxwell actually constructed the first statistical model of microprocesses in the history of physics, which laid the foundation for the development of statistical mechanics.

In the second part of the paper Maxwell, in addition to internal friction, considered from the same position other transport processes – diffusion and heat conduction. In the third part he turned to the question of rotational motion of colliding particles and for the first time obtained the law of equal distribution of kinetic energy in translational and rotational degrees of freedom. He reported the results of the application of his theory to transport phenomena at the regular convention of the British Association in Oxford in June 1860.

Maxwell was quite satisfied with his workplace, which required his presence only from October to April; the rest of the time he spent in Glenlair. He liked the free atmosphere of the college, the lack of rigid duties, though as one of the four regents he had to attend occasional meetings of the college senate. In addition, once a week at the so-called Aberdeen School of Science he lectured for a fee on practical purposes to craftsmen and mechanics, still, as at Cambridge, interested in the education of workers. Maxwell”s position changed at the end of 1859, when a decree was issued to merge the two Aberdeen colleges, Marischal College and King”s College, into the University of Aberdeen. In this connection, the professorial seat held by Maxwell was abolished as of September 1860 (the united chair was given to the influential King”s College professor, David Thomson). An attempt to win the competition for the post of professor of natural philosophy at the University of Edinburgh, vacated by Forbes” departure, failed: his old friend Peter Tat got the position. In the early summer of 1860 Maxwell was invited to take up the post of Professor of Natural Philosophy at King”s College, London.

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## London (1860-1865)

The summer and early fall of 1860 before moving to London Maxwell spent in his native manor of Glenlair, where he was seriously ill with smallpox and recovered only thanks to the care of his wife. Work at King”s College, where the emphasis was placed on experimental science (there were some of the best equipped physical laboratories) and where there were a large number of students, left him little free time. However, he had time to conduct home experiments with soap bubbles and color box, experiments to measure the viscosity of gases. In 1861 Maxwell joined the Committee on Standards, whose task was to determine the basic electrical units. An alloy of platinum and silver was taken as the material of the electrical resistance standard. The results of his careful measurements were published in 1863 and were the basis for the recommendation of the International Congress of Electrical Engineers (1881) that the ohm, the ampere and the volt were basic units. Maxwell also continued his work in the theory of elasticity and the calculation of structures, considered the methods of graphostatics for stresses in trusses (Maxwell”s theorem), analyzed the equilibrium conditions of spherical shells, and developed methods for constructing diagrams of internal stresses in bodies. For these works of great practical importance he was awarded the Keith Medal of the Royal Society of Edinburgh.

In June 1860, at the British Association convention in Oxford, Maxwell reported on his results in the field of color theory, backing them up with experimental demonstrations using a color box. Later that year the Royal Society of London awarded him the Rumford Medal for his research on color mixing and optics. On May 17, 1861 at a lecture at the Royal Institution on “The theory of three primary colors,” Maxwell presented yet another convincing proof of the correctness of his theory – the world”s first color photography, which he had conceived as early as 1855. Together with the photographer Thomas Sutton he produced three negatives of colored tape on glass coated with photographic emulsion (colloid). The negatives were shot through green, red and blue filters (solutions of salts of various metals). By then illuminating the negatives through the same filters, it was possible to obtain a color image. As it was shown almost a hundred years later by the Kodak staff who recreated the conditions of Maxwell”s experiment, the available photographic material did not allow to demonstrate color photography and, in particular, to obtain red and green images. By happy coincidence, the image obtained by Maxwell was formed as a result of the mixing of quite different colors – waves in the blue range and near-ultraviolet. Nevertheless, Maxwell”s experiment contained the correct principle of obtaining color photography, used many years later, when light-sensitive dyes were discovered.

Influenced by the ideas of Faraday and Thomson, Maxwell came to the conclusion that magnetism has a vortex nature and electric current has a translational nature. To clearly describe electromagnetic effects, he created a new, purely mechanical model, according to which rotating “molecular vortices” produce a magnetic field, while tiny transferring “idle wheels” provide rotation of vortices in one direction. Progressive motion of these transfer wheels (“particles of electricity”, according to Maxwell”s terminology) provides formation of electric current. The magnetic field directed along the axis of vortex rotation is perpendicular to the direction of the current, which is expressed in Maxwell”s grounded “rule of the borax”. Within the framework of this mechanical model it was possible not only to give an adequate visual illustration of the phenomenon of electromagnetic induction and the vortex character of the field generated by the current, but also to introduce an effect symmetrical to Faraday”s: changes in the electric field (the so-called displacement current created by the shift of transmission wheels, or bound molecular charges, under the action of the field) must lead to the appearance of a magnetic field. The bias current led directly to the continuity equation for electric charge, that is, to the notion of open currents (previously, all currents were considered closed). Symmetry considerations of the equations apparently did not play any role in this process. The famous physicist J.J. Thomson called the discovery of bias current “Maxwell”s greatest contribution to physics. These results were summarized in On physical lines of force (On physical lines of force). (On physical lines of force), published in several parts in 1861-1862.

In the same article Maxwell, having passed to consideration of propagation of disturbances in his model, noticed the similarity of properties of his vortex medium and light-carrying Fresnel ether. This found expression in the practical coincidence of the rate of propagation of perturbations (the ratio of the electromagnetic and electrostatic units of electricity as defined by Weber and Rudolf Kohlrausch) and the speed of light as measured by Hippolyte Fizeau. Thus, Maxwell took a decisive step toward the construction of the electromagnetic theory of light:

We can hardly reject the conclusion that light consists of transverse vibrations of the same medium that is the cause of electrical and magnetic phenomena.

However, this medium (ether) and its properties were not of primary interest to Maxwell, although he certainly shared the idea of electromagnetism as a result of applying the laws of mechanics to ether. As Henri Poincaré noted on this subject, “Maxwell does not give a mechanical explanation of electricity and magnetism; he confines himself to proving the possibility of such an explanation.

In 1864, Maxwell”s next article “A dynamical theory of the electromagnetic field” was published. (A dynamical theory of the electromagnetic field), in which a more detailed formulation of his theory was given (the term “electromagnetic field” first appeared here). In this case, he discarded the crude mechanical model (such representations, according to the scientist, were introduced solely “as illustrative, not as explanatory”), leaving a purely mathematical formulation of the field equations (Maxwell”s equations), which were first treated as a physically real system with a certain energy. Apparently, this is related to the first realization of the reality of the delayed charge interaction (and delayed interaction in general) discussed by Maxwell. In the same paper, he actually predicted the existence of electromagnetic waves, although, following Faraday, he wrote only about magnetic waves (electromagnetic waves in the full sense of the word appeared in an 1868 paper). The speed of these transverse waves turned out to be equal to the speed of light, and thus the idea of the electromagnetic nature of light finally took shape. Moreover, in the same paper Maxwell applied his theory to the problem of light propagation in crystals, whose dielectric or magnetic permeability depends on the direction, and in metals, obtaining a wave equation taking into account the conductivity of the material.

In parallel with his studies in electromagnetism, Maxwell set up several experiments in London to verify his results in kinetic theory. He constructed a special apparatus for determining the viscosity of air, and with its help he verified the validity of the conclusion that the coefficient of internal friction is independent of density (these experiments he conducted with his wife). Subsequently Lord Rayleigh wrote that “in the whole field of science there is no more beautiful or more significant discovery than the invariability of the viscosity of a gas at all densities. After 1862, when Clausius criticized several provisions of Maxwell”s theory (especially with regard to questions of heat conduction), he agreed with these remarks and began to correct the results. However, he soon came to the conclusion that the method based on the notion of the mean free path was unsuitable for the consideration of transport processes (as evidenced by the impossibility of explaining the temperature dependence of viscosity).

## Glenlair (1865-1871)

In 1865 Maxwell decided to leave London and return to his native estate. The reason for this was a desire to devote more time to scientific work, as well as pedagogical failures: he could not manage to maintain discipline in his extremely difficult lectures. Shortly after moving to Glenlair, he became seriously ill with head rot as a result of an injury sustained on one of his horse rides. After his recovery, Maxwell actively undertook household chores, rebuilding and expanding his estate. He regularly visited London as well as Cambridge, where he took part in examinations. Under his influence questions and problems of an applied nature began to be introduced into the examinations. Thus, in 1869 he proposed for the examination a study which represented the first theory of dispersion, based on the interaction of the incident wave with molecules possessing a certain frequency of natural oscillations. The dependence of the refractive index on frequency obtained in this model was independently deduced three years later by Werner von Sellmeier. The Maxwell-Sellmeier theory of dispersion was confirmed at the end of the 19th century in the experiments of Heinrich Rubens.

In the spring of 1867 Maxwell spent the spring of 1867 with his frequently ill wife on the advice of a doctor in Italy, seeing the sights of Rome and Florence, meeting with Professor Carlo Matteucci, practicing languages (he knew Greek, Latin, Italian, French and German well). Through Germany, France, and Holland they returned to their homeland. In 1870 Maxwell spoke as president of the mathematics and physics section of the British Association convention in Liverpool.

Maxwell continued to pursue kinetic theory, constructing in On the dynamical theory of gases (1866) a more general theory of transport processes than previously. As a result of his experiments on measuring the viscosity of gases, he decided to abandon the idea of molecules as elastic balls. In his new work he regarded molecules as small bodies repulsing each other with a force depending on the distance between them (from his experiments he deduced that this repulsion is inversely proportional to the distance to the fifth power). Having phenomenologically considered the viscosity of the medium on the basis of the simplest model of molecules for calculations (“Maxwellian molecules”), he first introduced the concept of relaxation time as a time of equilibrium establishment. Further, he mathematically analyzed from unified positions the processes of interaction of two molecules of the same or different species, for the first time introducing into the theory the collision integral, later generalized by Ludwig Boltzmann. Having considered transport processes, he determined values of diffusion and conduction coefficients, relating them with experimental data. Although some of Maxwell”s statements turned out to be incorrect (e.g., the laws of interaction of molecules are more complex), the general approach he developed proved to be very fruitful. In particular, the foundations were laid for the theory of viscoelasticity on the basis of the medium model known as Maxwell”s medium (Maxwell material). In the same work of 1866, he gave a new conclusion of the distribution of molecules by velocities, based on a condition later called the principle of detailed equilibrium.

Maxwell devoted much attention to writing his monographs on the kinetic theory of gases and on electricity. At Glenlair he completed his textbook Theory of Heat, published in 1871 and reprinted several times during his lifetime. Most of this book was devoted to a phenomenological treatment of thermal phenomena. The last chapter contained basic information on molecular-kinetic theory, combined with Maxwell”s statistical ideas. There he also opposed the second principle of thermodynamics as formulated by Thomson and Clausius, which led to the “thermal death of the universe. Disagreeing with this purely mechanical point of view, he was the first to realize the statistical nature of the second principle. According to Maxwell, it can be violated by individual molecules, but it remains valid for large populations of particles. To illustrate this point, he proposed a paradox known as “Maxwell”s demon” (a term suggested by Thomson; Maxwell himself preferred the word “valve”). It consists in the fact that some controlling system (“demon”) is capable of reducing the entropy of the system without the cost of work. Maxwell”s demon paradox was already solved in the 20th century in the works of Marian Smoluchowski, who pointed out the role of fluctuations in the controlling element itself, and Leo Szilard, who showed that getting information about molecules by the “demon” leads to increasing entropy. Thus, the second beginning of thermodynamics is not violated.

In 1868 Maxwell published another paper on electromagnetism. A year earlier there was an occasion to greatly simplify the presentation of the results of the work. He had read Peter Tat”s An elementary treatise on quaternions and decided to apply the quaternion notation to numerous mathematical relations of his theory, which made it possible to reduce and clarify their notation. One of the most useful tools was the Hamiltonian operator nabla, whose name was suggested by William Robertson Smith, a friend of Maxwell”s, by analogy with the ancient Assyrian type of harp with a triangular backbone. Maxwell wrote a mock ode, “To the Chief Musician on Playing the Nabla,” dedicated to Tat. The success of this poem ensured that the new term would gain a foothold in scientific usage. Maxwell was also the first to write down the equations of the electromagnetic field in invariant vector form through the Hamiltonian operator. It is worth noting that he owes his pseudonym dpdt{displaystyle dpdt}, which he used to sign his letters and poems, to Tat. The fact is that in their “Treatise on Natural Philosophy,” Thomson and Tat presented the second principle of thermodynamics in the form JCM=dpdt{displaystyle JCM=dpdt}. Since the left part is the same as Maxwell”s initials, he decided to use the right part for his signature in the future. Among other achievements of the Glener period is an article called “On governors” (On governors, 1868), in which the stability of the centrifugal regulator is analyzed by the methods of the theory of small oscillations.

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## Cavendish Laboratory (1871-1879)

In 1868 Maxwell refused to take the post of rector of the University of St. Andrews, not wishing to part with the secluded life in the estate. However, three years later, after much hesitation, he still accepted the offer to head the newly organized physics laboratory at Cambridge University and to take the corresponding position of professor of experimental physics (before that William Thomson and Hermann Helmholtz declined the invitation). The laboratory was named in honor of the reclusive scientist Henry Cavendish, whose grand-nephew, the Duke of Devonshire, was at the time chancellor of the university and provided the finances for its construction. The formation of the first laboratory in Cambridge was in keeping with the realization of the importance of experimental research for the further progress of science. On March 8, 1871, Maxwell received his appointment and immediately began his duties. He set up the work of building and equipping the laboratory (initially using his personal instruments) and lectured on experimental physics (courses on heat, electricity and magnetism).

In 1873 Maxwell published a major two-volume work A Treatise on Electricity and Magnetism, which contained information on pre-existing theories of electricity, methods of measurement and features of experimental apparatus, but the main attention was paid to the treatment of electromagnetism from a single, Faradayan position. The presentation of the material was even to the detriment of Maxwell”s own ideas. As Edmund Whittaker noted,

Doctrines belonging exclusively to Maxwell-the existence of displacement currents and of electromagnetic oscillations identical with light-were not presented in the first volume or in the first half of the second volume; and their description was hardly more complete, and probably less attractive, than that which he gave in his first scientific writings.

The “Treatise” contained the basic equations of the electromagnetic field, now known as Maxwell”s equations. However, they were not presented in a very convenient form (through scalar and vector potentials, moreover, in quaternionic notation), and there were quite a few of them – twelve. Subsequently, Heinrich Hertz and Oliver Heaviside rewrote them through electric and magnetic field vectors, resulting in four equations in the modern form. Heaviside also first noted the symmetry of Maxwell”s equations. A direct consequence of these equations was the prediction of the existence of electromagnetic waves, experimentally discovered by Hertz in 1887-1888. Other important results set forth in the “Treatise” were the proof of the electromagnetic nature of light and the prediction of the effect of light pressure (as a result of the ponderomotive action of electromagnetic waves), discovered much later in the famous experiments of Peter Lebedev. On the basis of his theory, Maxwell also gave an explanation of the influence of the magnetic field on the propagation of light (the Faraday effect). Another proof of the validity of Maxwell”s theory – the quadratic relationship between the optical (refractive index) and electrical (permittivity) characteristics of the medium – was published by Ludwig Boltzmann soon after the publication of his “Treatise.

Maxwell”s fundamental work was coolly accepted by most of the coryphaei of science at the time – Stokes, Airy, Thomson (he called his friend”s theory “a curious and original, but not too logical hypothesis,” and only after Lebedev”s experiments his conviction was somewhat shaken), Helmholtz, who unsuccessfully tried to reconcile new views with old theories based on long range action. Tat considered the main achievement of the “Treatise” to be only the final debunking of the long-range action. Particularly difficult to understand was the concept of the displacement current, which must exist even in the absence of matter, that is, in the ether. Even Hertz, a student of Helmholtz, avoided references to Maxwell, whose works were extremely unpopular in Germany, and wrote that his experiments on electromagnetic waves were “convincing regardless of any theory.” The peculiarities of style – the lack of notation and the often confused presentation – did not contribute to the understanding of new ideas, as noted, for example, by French scientists Henri Poincaré and Pierre Duhem. The latter wrote: “We thought we were entering the peaceful and orderly dwelling of deductive reason, but instead we found ourselves in some kind of factory. The historian of physics Mario Liozzi summarized the impression Maxwell”s work left as follows

Maxwell builds his theory step by step with “sleight of hand,” as Poincaré aptly put it, referring to the logical strains that scientists sometimes allow themselves when formulating new theories. When, in the course of analytical construction, Maxwell encounters an apparent contradiction, he does not hesitate to overcome it with disconcerting liberties. For example, it costs him nothing to exclude a term, to replace an inappropriate sign of expression with an inverse sign, to substitute the meaning of a letter. For those who admired the infallible logical construction of Ampere”s electrodynamics, Maxwell”s theory must have made an unpleasant impression.

Only a few scientists, mostly young ones, were seriously interested in Maxwell”s theory: Arthur Schuster (Oliver Lodge, who set out to discover electromagnetic waves; George Fitzgerald, who unsuccessfully tried to convince Thomson (Russian scientists Nikolai Umov and Alexander Stoletov. The famous Dutch physicist Hendrik Anton Lorenz, one of the first to apply Maxwell”s theory in his work, wrote many years later:

“Treatise on Electricity and Magnetism” made perhaps one of the strongest impressions of my life: its interpretation of light as an electromagnetic phenomenon surpassed in its audacity anything I had ever known before. But Maxwell”s book was not an easy one!

On June 16, 1874, the three-story Cavendish Laboratory building was inaugurated. On the same day the Duke of Devonshire presented Maxwell with twenty bags of manuscripts of Henry Cavendish. For the next five years Maxwell worked on the legacy of this unsociable scientist who turned out to have made a number of outstanding discoveries: he measured the capacitance and dielectric constants of a number of substances, determined the resistance of electrolytes and anticipated the discovery of Ohm”s law, established the law of interaction of charges (known as Coulomb”s law). Maxwell carefully studied the features and conditions of Cavendish”s experiments, many of them were reproduced in the laboratory. In October 1879 he edited a two-volume collection of essays, The Electrical Researches of the Honourable Henry Cavendish.

In the 1870s Maxwell actively engaged in the popularization of science. He wrote several articles for the Encyclopedia Britannica (“Atom,” “Attraction,” “Aether,” and others). In the same year, 1873, when “A Treatise on Electricity and Magnetism” was published, a small book, “Matter and Motion,” was published. Until the last days of his life he labored on Electricity in an Elementary Statement, published in 1881. In his popular writings he allowed himself to express his ideas more freely, his views on the atomic and molecular structure of bodies (and even the ether) and the role of statistical approaches, to share his doubts with readers (for example, about the indivisibility of atoms or the infinity of the world). It must be said that at that time the idea of the atom itself was by no means considered indisputable. Maxwell, being a supporter of the ideas of atomism, highlighted a number of problems unsolvable at that time: what is a molecule, and how do atoms form it? what is the nature of interatomic forces? how to understand the identity and immutability of all atoms or molecules of a given substance, as it follows from spectroscopy? These questions were not answered until after the advent of quantum theory.

At Cambridge, Maxwell continued to develop specific questions of molecular physics. In 1873, following data from the work of Johannes Loschmidt, he calculated the sizes and masses of molecules of a number of gases and determined the value of the Loschmidt constant. As a result of the discussion of the equilibrium of the vertical column of gas, he gave a simple derivation of the generalized distribution of molecules in the potential force field previously obtained by Boltzmann (the Maxwell-Boltzmann distribution). In 1875, after the appearance of the work of Jan Diderik van der Waals, he proved that on the transition curve between the gaseous and liquid states, the straight line corresponding to the transition region cuts off equal areas (Maxwell”s rule).

In recent years, Maxwell paid much attention to the work of Willard Gibbs, who developed geometrical methods as applied to thermodynamics. These methods were adopted by Maxwell in preparing reprints of the Theory of Heat and were promoted in articles and speeches in every possible way. Based on them, he gave a correct interpretation of the concept of entropy (and even approached to its treatment as a property depending on the knowledge of the system) and obtained four thermodynamic relations (the so-called Maxwell relations). He made several models of thermodynamic surfaces, one of which he sent to Gibbs.

In 1879 Maxwell”s last two works on molecular physics were published. The first of them gave the basics of the theory of inhomogeneous dilute gases. He also considered the interaction of gas with the surface of a solid body in connection with the thermal effect of light in the radiometer invented by William Crooks (originally it was assumed that this device recorded the pressure of light). In his second article, “On Boltzmann”s theorem on the average distribution of energy in a system of material points,” Maxwell introduced the terms “system phase” (for the set of coordinates and momentum) and “degree of freedom of a molecule,” which are still used today; he actually expressed the ergodic hypothesis for mechanical systems with constant energy; he considered gas distribution under the action of centrifugal forces, that is, he laid the foundation for centrifuge theory. This work was an important step on the way to statistical mechanics, which was later developed in the works of Gibbs.

In Cambridge, Maxwell performed various administrative duties, was a member of the Senate of the University, was a member of the committee to reform the mathematical examination and one of the organizers of the new, natural science examination, was elected president of the Cambridge Philosophical Society (1876-1877). At this time his first students appeared – George Chrystal, Richard Glazebrook (Maxwell studied with him the propagation of waves in biaxial crystals), Arthur Schuster, Ambrose Fleming, John Henry Poynting. As a rule, Maxwell left the choice of research topics up to his students, but he was willing to offer helpful advice when necessary. His colleagues noted his simplicity, his focus on his research, his ability to penetrate deeply into the essence of a problem, his insight, his sensitivity to criticism, his lack of desire for fame, but at the same time his capacity for subtle sarcasm.

The first symptoms of the disease appeared in Maxwell in early 1877. Gradually he began to have difficulty breathing, difficulty swallowing food, and pain. In the spring of 1879 he struggled to lecture and became quickly fatigued. In June, he returned to Glenlair with his wife, his condition steadily worsening. Doctors diagnosed him with abdominal cancer. In early October, the finally weakened Maxwell returned to Cambridge under the care of the famous Dr. James Paget. Soon, on November 5, 1879, the scientist died. The coffin containing Maxwell”s body was transported to his estate and he was buried next to his parents in a small cemetery in the village of Parton.

Although Maxwell”s contribution to the development of physics (especially electrodynamics) was not properly appreciated during his lifetime, in the following years there was a growing awareness of the true place of his work in the history of science. Many major scientists noted this in their assessments. For example, Max Planck drew attention to Maxwell”s universalism as a scientist:

Maxwell”s great thoughts were no accident: they naturally flowed from the richness of his genius; this is best proved by the fact that he was a pioneer in the most varied branches of physics, and in all its sections he was an expert and teacher.

However, according to Planck, it is Maxwell”s work on electromagnetism that is the pinnacle of his work:

…in the doctrine of electricity his genius appears to us in its full glory. It is in this field, after many years of quiet research, that Maxwell has had a success which we must rank among the most astonishing deeds of the human spirit. He succeeded in coaxing out from nature, by pure thought alone, such secrets as only a generation later and only partially could be shown in witty and laborious experiments.

As Rudolf Peierls noted, Maxwell”s work on electromagnetic field theory contributed to the acceptance of the idea of the field as such, which found wide application in twentieth-century physics:

It is good that after assimilating Maxwell”s ideas, physicists got used to perceiving as a basic physical fact the statement that there is some field of a certain kind in a certain point of space, since it has long been impossible to be limited to the electromagnetic field. Many other fields have appeared in physics and, of course, we do not wish or expect to explain them through models of various kinds.

The importance of the field concept in Maxwell”s work was pointed out by Albert Einstein and Leopold Infeld in their popular book The Evolution of Physics:

The formulation of these equations is the most important event since Newton, not only because of the value of their content, but also because they provide an example of a new type of laws. The characteristic feature of Maxwell”s equations, which appears in all other equations of modern physics, can be expressed in one sentence: Maxwell”s equations are laws expressing the field structure… The theoretical discovery of an electromagnetic wave propagating at the speed of light is one of the greatest achievements in the history of science.

Einstein also acknowledged that “the theory of relativity owes its origin to Maxwell”s equations for the electromagnetic field. It is also worth noting that Maxwell”s theory was the first gauge-invariant theory. It gave impetus to the further development of the principle of gauge symmetry, which is the basis of the modern Standard Model. Finally, numerous practical applications of Maxwell”s electrodynamics, augmented by the concept of the Maxwell stress tensor, are worth mentioning. These include the calculation and construction of industrial plants, the use of radio waves, and modern numerical modeling of the electromagnetic field in complex systems.

Niels Bohr, in his speech at the celebration of Maxwell”s centennial, pointed out that the development of quantum theory has by no means diminished the significance of the British scientist”s achievements:

The development of atomic theory, as we know, soon took us beyond the direct and consistent application of Maxwell”s theory. However, I must emphasize that it was the possibility of analyzing radiation phenomena thanks to the electromagnetic theory of light that led to the recognition of essentially new features in the laws of nature… And yet in this position Maxwell”s theory continued to be the leading theory… We must not forget that only the classical ideas of material particles and electromagnetic waves have an unambiguous application, while the concepts of photon and electronic waves have none… In fact, we must realize that an unambiguous interpretation of any measurement

At the time of his death, Maxwell was known primarily for his contributions to molecular-kinetic theory, in the development of which he was a recognized leader. Of great importance in the development of science, in addition to the many concrete results in this field, was Maxwell”s development of statistical methods, which eventually led to the development of statistical mechanics. The term “statistical mechanics” itself was introduced by Maxwell in 1878. A vivid example of understanding the importance of this approach is the statistical interpretation of the second principle of thermodynamics and the paradox of Maxwell”s “demon”, which influenced the formulation of information theory already in the 20th century. Maxwell”s methods in the theory of transport processes have also found fruitful development and application in modern physics in the works of Paul Langevin, Sidney Chapman, David Enskog, John Lennard-Jones, and others.

Maxwell”s writings on color theory laid the foundation for methods of accurately quantifying colors resulting from mixing. These results were used by the International Commission on Illumination to develop color charts, taking into account both the spectral characteristics of colors and their saturation levels. Maxwell”s analysis of the stability of Saturn”s rings and his work on kinetic theory find their continuation not only in modern approaches to the description of the ring structure features, many of which have not yet been explained, but also in the description of similar astrophysical structures (such as accretion disks). Moreover, Maxwell”s ideas about the stability of particle systems have found application and development in completely different fields – the analysis of the dynamics of waves and charged particles in ring gas pedals, plasma, nonlinear optical media and so on (systems of Vlasov-Maxwell equations, Schrödinger-Maxwell, Wigner-Maxwell).

As a final assessment of Maxwell”s contribution to science, it is appropriate to quote Lord Rayleigh (1890):

There can be no doubt that later generations will consider as the highest achievement in this field his electromagnetic theory of light, thanks to which optics becomes a section of electricity. …only slightly less important, if at all, than his work on electricity was Maxwell”s participation in the development of the dynamic theory of gases…

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## Translations into Russian

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