As a result of studying this chapter, the student should:

know

• empirical and theoretical foundations of the electromagnetic field theory;
• the history of the creation of the theory of the electromagnetic field, the history of the discovery of light pressure and electromagnetic waves;
• physical essence of Maxwell's equations (in integral and differential forms);
• the main stages of the biography of J. K. Maxwell;
• the main directions in the development of electrodynamics after J.K. Maxwell;
• achievements of J.K. Maxwell in molecular physics and thermodynamics;

be able to

• evaluate the role of Maxwell in the development of the theory of electricity and magnetism, the fundamental significance of Maxwell's equations, the place of the book "Treatise on Electricity and Magnetism" in the history of science, the historical experiments of G. Hertz and P. N. Lebedev;
• discuss the biographies of the leading scientists working in the field of electromagnetism;

own

Skills of operating with the basic concepts of the theory of the electromagnetic field.

Key terms: electromagnetic field, Maxwell's equations, electromagnetic waves, light pressure.

Faraday's discoveries revolutionized the science of electricity. With his light hand, electricity began to gain new positions in technology. Earned an electromagnetic telegraph. In the early 70s. In the 19th century, it already connected Europe with the USA, India and South America, the first electric current generators and electric motors appeared, electricity began to be widely used in chemistry. Electromagnetic processes invaded science ever deeper. An era has come when the electromagnetic picture of the world was ready to replace the mechanical one. We needed a man of genius who could, like Newton in his time, combine the facts and knowledge accumulated by that time and, on their basis, create a new theory describing the foundations of the new world. J.K. Maxwell became such a person.

James Clerk Maxwell(Fig. 10.1) was born in 1831. His father, John Clerk Maxwell, was clearly an outstanding man. A lawyer by profession, he, nevertheless, devoted considerable time to other things that were more interesting to him: he traveled, designed cars, set up physical experiments, and even published several scientific articles. When Maxwell was 10 years old, his father sent him to study at the Edinburgh Academy, where he stayed for six years - until entering the university. At the age of 14, Maxwell wrote the first scientific paper on the geometry of oval curves. A summary of it was published in Proceedings of the Royal Society of Edinburgh, 1846.

In 1847, Maxwell entered the University of Edinburgh, where he began to study mathematics in depth. At this time, two more scientific work gifted student were published in Proceedings of the Royal Society of Edinburgh. The content of one of them (about rolling curves) was introduced to the society by Professor Kelland, the other (about the elastic properties of solids) was first presented by the author himself.

In 1850, Maxwell continued his education at Peterhouse - St. Peter's College, Cambridge University, and from there moved to the Holy Trinity College - Trinity College, which gave the world I. Newton, and later V. V. Nabokov, B. Russell and others. In 1854 Mr. Maxwell passes the exam and receives a bachelor's degree. Then he was left at Trinity College as a teacher. However, he was more concerned scientific problems. At Cambridge, Maxwell began to study color and color vision. In 1852, he came to the conclusion that the mixing of spectral colors does not coincide with the mixing of colors. Maxwell develops the theory of color vision, designs a color top (Fig. 10.2).

Rice. 10.1.

Rice. 10.2.

In addition to his old hobbies - geometry and color problems, Maxwell became interested in electricity. In 1854, on February 20, he wrote a letter from Cambridge to W. Thomson in Glasgow. Here is the beginning of that famous letter:

"Dear Thomson! Now that I've entered the unholy undergraduate class, I've begun to think about reading. It is very pleasant sometimes to be among deservedly recognized books that I have not yet read, but must read. But we have a strong desire to return to physical things, and some of us here want to attack electricity.”

After completing the course, Maxwell became a member of Trinity College, Cambridge University, and in 1855 became a member of the Royal Society of Edinburgh. However, he soon left Cambridge and returned to his native Scotland. Professor Forbes informed him that a vacancy for a professor of physics had opened up in Aberdeen, at Marishall College, and he had every chance of filling it. Maxwell accepted the offer and in April 1856 (at the age of 24!) took up a new position. In Aberdeen, Maxwell continued to work on the problems of electrodynamics. In 1857, he sent M. Faraday his work "On Faraday's lines of force."

Of Maxwell's other work at Aberdeen, his work on the stability of Saturn's rings was widely known. From the study of the mechanics of the rings of Saturn, it was quite natural to move on to the consideration of the motions of gas molecules. In 1859, Maxwell spoke at a meeting of the British Association for the Advancement of Sciences with a report "On the dynamic theory of gases." This report marked the beginning of his fruitful research in the field of the kinetic theory of gases and statistical physics.

In 1860, Maxwell accepted an invitation from King's College London and worked there as a professor for five years. He was not a brilliant lecturer and did not particularly enjoy lecturing. Therefore, the ensuing break in teaching was more desirable than annoying for him, and allowed him to completely immerse himself in solving fascinating problems of theoretical physics.

According to A. Einstein, Faraday and Maxwell played the same roles in the science of electricity that Galileo and Newton played in mechanics. Just as Newton gave the mechanical effects discovered by Galileo a mathematical form and physical justification, so Maxwell did the same with respect to Faraday's discoveries. Maxwell gave Faraday's ideas a rigorous mathematical form, introduced the term "electromagnetic field", and formulated mathematical laws describing this field. Galileo and Newton laid the foundations for the mechanical picture of the world, Faraday and Maxwell for the electromagnetic one.

Maxwell began to think about his ideas about electromagnetism in 1857, when the already mentioned article "On Faraday's lines of force" was written. Here he makes extensive use of hydrodynamic and mechanical analogies. This allowed Maxwell to apply the mathematical apparatus of the Irish mathematician W. Hamilton and thus express the electrodynamic relations in mathematical language. In the future, hydrodynamic analogies are replaced by methods of the theory of elasticity: the concepts of deformation, pressure, vortices, etc. Proceeding from this, Maxwell comes to the field equations, which at this stage have not yet been reduced to a single system. Investigating dielectrics, Maxwell expresses the idea of ​​"displacement current", as well as, as yet vaguely, the idea of ​​the connection between light and the electromagnetic field ("electrotonic state") in the Faraday formulation that Maxwell then used.

These ideas are set forth in the articles "On the physical lines of forces" (1861-1862). They were written during the most prolific London period (1860-1865). At the same time, Maxwell's famous articles "Dynamical Theory of the Electromagnetic Field" (1864-1865) were published, where thoughts were expressed about the unified nature of electromagnetic waves.

From 1866 to 1871 Maxwell lived at his family estate, Middleby, leaving occasionally for exams in Cambridge. Being engaged in economic affairs, Maxwell did not leave scientific studies. He worked hard on the main work of his life, "Treatise on Electricity and Magnetism", wrote the book "Theory of Heat", a number of articles on the kinetic theory of gases.

In 1871 there was significant event. At the expense of the descendants of G. Cavendish, the Department of Experimental Physics was established in Cambridge and the construction of the experimental laboratory building began, which in the history of physics is known as the Cavendish Laboratory (Fig. 10.3). Maxwell was invited to become the first professor of the department and head of the laboratory. In October 1871 he delivered an inaugural lecture on the trends and significance of experimental research in university education. This lecture became a program for teaching experimental physics for many years to come. On June 16, 1874, the Cavendish Laboratory was opened.

Since then, the laboratory has become the center of world physical science for many decades, and it is the same now. For more than a hundred years, thousands of scientists have passed through it, among them many of those who have made the glory of world physical science. After Maxwell, the Cavendish Laboratory was headed by many outstanding scientists: J. J. Thomson, E. Rutherford, L. Bragg, N. F. Mott, A. B. Pippard, and others.

Rice. 10.3.

After the release of the "Treatise on Electricity and Magnetism", in which the theory of the electromagnetic field was formulated, Maxwell decides to write the book "Electricity in an Elementary Presentation" in order to popularize and disseminate his ideas. Maxwell worked on the book, but his health was getting worse. He died on November 5, 1879, without witnessing the triumph of his theory.

Let us dwell on the creative heritage of the scientist. Maxwell left a deep mark in all areas of physical science. No wonder a number of physical theories bear his name. He proposed a thermodynamic paradox that haunted physicists for many years - "Maxwell's demon". AT kinetic theory he introduced the concepts known as: "Maxwell distribution" and "Maxwell-Boltzmann statistics". He also wrote an elegant study of the stability of Saturn's rings. In addition, Maxwell created many small scientific masterpieces in a wide variety of fields - from the implementation of the world's first color photograph to the development of a method for radically removing grease stains from clothes.

Let's move on to the discussion electromagnetic field theory- the quintessence of Maxwell's scientific creativity.

It is noteworthy that James Clerk Maxwell was born in the same year that Michael Faraday discovered the phenomenon of electromagnetic induction. Maxwell was particularly impressed by Faraday's book Experimental studies on electricity."

In Maxwell's time, there were two alternative theories of electricity: Faraday's theory of "lines of force" and the theory developed by the French scientists Coulomb, Ampère, Biot, Savart, Arago and Laplace. The initial position of the latter is the idea of ​​long-range action - the instantaneous transfer of interaction from one body to another without the help of any intermediate medium. Realistically thinking Faraday could not reconcile himself with such a theory. He was absolutely convinced that "matter cannot act where it does not exist." The medium through which the action is transmitted, Faraday called the "field". The field, he believed, was permeated with magnetic and electrical "lines of force."

In 1857 Maxwell's article "On Faraday's lines of force" appeared in the Proceedings of the Cambridge Philosophical Society. It contained the entire program of research on electricity. Note that Maxwell's equations have already been written in this article, but so far without a bias current. The article "On Faraday's lines of force" required continuation. Electrohydraulic analogies have given a lot. With their help, useful differential equations were written. But not everything could be subordinated to electrohydraulic analogies. The most important law of electromagnetic induction did not fit into their framework. It was necessary to come up with a new auxiliary mechanism that would facilitate the understanding of the process, reflecting both the translational movement of currents and the rotational, vortex nature of the magnetic field.

Maxwell suggested special environment, in which the vortices are so small that they fit inside the molecules. Rotating "molecular vortices" produce a magnetic field. The direction of the axes of the vortices of molecules coincides with their lines of force, and they themselves can be represented as thin rotating cylinders. But the external, touching parts of the vortices must move in opposite directions, i.e. prevent mutual movement. How can two adjacent gears rotate in the same direction? Maxwell suggested that between the rows of molecular vortices a layer of tiny spherical particles ("idle wheels") capable of rotation is placed. Now the vortices could rotate in the same direction and interact with each other.

Maxwell also began to study the behavior of his mechanical model in the case of conductors and dielectrics and came to the conclusion that electrical phenomena can also occur in a medium that prevents the passage of current - in a dielectric. Suppose that the “idle wheels” could not move forward in these media under the action of an electric field, but when the electric field is applied and removed, they are displaced from their positions. It took Maxwell great scientific courage to identify this displacement of bound charges with electric current. After all, this current - bias current- no one has watched yet. After that, Maxwell inevitably had to take the next step - to recognize behind this current the ability to create its own magnetic field.

Thus, Maxwell's mechanical model made it possible to draw the following conclusion: a change in the electric field leads to the appearance of a magnetic field, i.e. to the phenomenon opposite to Faraday, when a change in the magnetic field leads to the appearance of an electric field.

Maxwell's next article on electricity and magnetism is "On Physical Lines of Force". Electrical phenomena demanded an ether as hard as steel for their explanation. Maxwell unexpectedly found himself in the role of O. Fresnel, forced to "invent" his own "optical" ether to explain polarization phenomena, as hard as steel and as permeable as air. Maxwell notes the similarity of two media: "luminiferous" and "electric". He is gradually approaching his great discovery of the "single nature" of light and electromagnetic waves.

In the next article - "Dynamical theory of the electromagnetic field" - Maxwell used the term "electromagnetic field" for the first time. “The theory that I propose may be called the theory of the electromagnetic field, because it deals with the space surrounding electric or magnetic bodies, and it may also be called the dynamical theory, since it assumes that there is matter in this space, which is in movement, through which the observed electro magnetic phenomena».

When Maxwell deduced his equations in the Dynamic Theory of the Electromagnetic Field, one of them seemed to indicate exactly what Faraday was talking about: magnetic influences really propagated in the form of transverse waves. Maxwell did not notice then that more follows from his equations: along with the magnetic action, an electrical disturbance propagates in all directions. An electromagnetic wave in the full sense of the word, including both electrical and magnetic perturbations, appeared in Maxwell later, already in Middleby, in 1868, in the article “On the method of direct comparison of the electrostatic force with the electromagnetic force with a note on the electromagnetic theory of light” .

In Middleby, Maxwell completed the main work of his life - "A Treatise on Electricity and Magnetism", first published in 1873 and subsequently reprinted several times. The content of this book, of course, was primarily articles on electromagnetism. In the "Treatise" the basics of vector calculus are systematically given. Then there are four parts: electrostatics, electrokinematics, magnetism, electromagnetism.

Note that Maxwell's research method differs sharply from the methods of other researchers. Not only every mathematical quantity, but also every mathematical operation is endowed with a deep physical meaning. At the same time, each physical quantity corresponds to a clear mathematical characteristic. One of the chapters of the "Treatise" is called "Basic Equations of the Electromagnetic Field". Here are the basic equations of the electromagnetic field from this Treatise. Thus, with the help of vector calculus, Maxwell did more simply what he had done earlier with the help of mechanical models - he derived the equations of the electromagnetic field.

Let us consider the physical meaning of Maxwell's equations. The first equation says that the sources of the magnetic field are currents and an electric field that changes with time. Maxwell's brilliant conjecture was his introduction of a fundamentally new concept - displacement current - as a separate term in the generalized Ampère - Maxwell law:

where H- vector of magnetic field strength; j is the electric current density vector, to which the displacement current was added by Maxwell; D- electric induction vector; c is some constant.

This equation expresses the magnetoelectric induction, discovered by Maxwell and based on the concepts of displacement currents.

Another idea that immediately won Maxwell's recognition was Faraday's idea of ​​the nature of electromagnetic induction - the occurrence of an inductive current in a circuit, the number of magnetic lines of force in which changes either due to the relative motion of the circuit and the magnet, or due to a change in the magnetic field. Maxwell wrote the following equation:

where Yo- electric field strength vector; AT- century-

torus of magnetic field strength and, respectively: - -

change of the magnetic field in time, s - some constant.

This equation reflects Faraday's law of electromagnetic induction.

It is necessary to take into account one more important property of the vectors of electric and magnetic induction Yo and B. While electric lines of force begin and end on the charges that are the sources of the field, the lines of force of the magnetic field are closed on themselves.

In mathematics, to denote the characteristics of a vector field, the operator of "divergence" (differentiation of the field flow) - div is used. Using this, Maxwell adds to the two existing equations two more:

where p is the density of electric charges.

Maxwell's third equation expresses the law of conservation of the amount of electricity, the fourth - the vortex nature of the magnetic field (or the absence of magnetic charges in nature).

The vectors of electric and magnetic induction and the vectors of electric and magnetic fields included in the considered equations are connected by simple relations and can be written in the form of the following equations:

where e is the dielectric constant; p is the magnetic permeability of the medium.

In addition, one more relation can be written that relates the intensity vector Yo and specific conductivity at:

To represent the complete system of Maxwell's equations, it is also necessary to write down the boundary conditions. These conditions must be satisfied by the electromagnetic field at the interface between two media.

where about- surface density of electric charges; i is the surface conduction current density at the considered interface. In the particular case when there are no surface currents, the last condition turns into:

Thus, J. Maxwell comes to the definition of the electromagnetic field as a type of matter, expressing all its manifestations in the form of a system of equations. Note that Maxwell did not use vector notation and wrote his equations in rather cumbersome component form. The modern form of Maxwell's equations appeared around 1884 after the work of O. Heaviside and G. Hertz.

Maxwell's equations are one of the greatest achievements not only of physics, but of civilization in general. They combine the strict logic of the natural sciences with the beauty and proportion that characterize the arts and the humanities. Equations with the maximum possible accuracy reflect the essence of natural phenomena. The potential of Maxwell's equations is far from being exhausted; on their basis, more and more new works appear, explanations of the latest discoveries in various fields of physics - from superconductivity to astrophysics. Maxwell's system of equations is the basis of modern physics, and so far there is not a single experimental fact that would contradict these equations. Knowledge of Maxwell's equations, at least their physical essence, is mandatory for any educated person, not only a physicist.

Maxwell's equations were the forerunner of a new non-classical physics. Although Maxwell himself, according to his scientific convictions, was a “classical” person to the marrow of his bones, the equations he wrote belonged to a different science, different from the one that was known and close to the scientist. This is evidenced at least by the fact that Maxwell's equations are not invariant under the Galilean transformations, but they are invariant under the Lorentz transformations, which, in turn, underlie relativistic physics.

Based on the equations obtained, Maxwell solved specific problems: he determined the electrical permeability coefficients of a number of dielectrics, calculated the coefficients of self-induction, mutual induction of coils, etc.

Maxwell's equations allow us to draw a number of important conclusions. Maybe the main one is the existence of transverse electromagnetic waves propagating at a speed c.

Maxwell found that the unknown number c turned out to be approximately equal to the ratio of electromagnetic and electrostatic units of charge, which is approximately 300,000 kilometers per second. Convinced of the universality of his equations, he shows that "light is an electromagnetic disturbance." Recognition of the finite, albeit very high, speed of propagation of the electromagnetic field of stone on stone did not leave the supporters of "instantaneous long-range action" from theories.

The most important consequence of the electromagnetic theory of light was the prediction by Maxwell light pressure. He managed to calculate that in the case when in clear weather sunlight, absorbed by a plane of one square meter, gives 123.1 kilogram meters of energy per second. This means that it presses on this surface in the direction of its fall with a force of 0.41 milligrams. Thus, Maxwell's theory was strengthened or collapsed depending on the results of experiments not yet carried out. Are there electromagnetic waves in nature with properties similar to light? Is there light pressure? Already after the death of Maxwell, Heinrich Hertz answered the first question, and Pyotr Nikolaevich Lebedev answered the second.

J.K. Maxwell is a giant figure in physical science and as a person. Maxwell will live in people's memory for as long as humanity exists. Maxwell's name is immortalized in the name of a crater on the Moon. The highest mountains on Venus are named after the great scientist (Maxwell's mountains). They rise 11.5 km above the average surface level. Also, his name is the world's largest telescope that can operate in the submillimeter range (0.3-2 mm) - the telescope named after. J.C. Maxwell (JCMT). It is located in the Hawaiian Islands (USA), in the highlands of Mauna Kea (4200 m). The JCMT's 15-meter main mirror is made from 276 individual pieces of aluminum, tightly butted together. The Maxwell telescope is used to study the solar system, interstellar dust and gas, and distant galaxies.

After Maxwell, electrodynamics became fundamentally different. How did she develop? We note the most important direction of development - experimental confirmation of the main provisions of the theory. But the theory itself also required some interpretation. In this regard, it is necessary to note the merits of the Russian scientist Nikolai Alekseevich Umov, who headed the Department of Physics at Moscow University from 1896 to 1911.

Nikolai Alekseevich Umov (1846-1915) - Russian physicist, born in Simbirsk (now Ulyanovsk), graduated from Moscow University. He taught at Novorossiysk University (Odessa), and then at Moscow University, where from 1896, after the death of A. G. Stoletov, he headed the Department of Physics.

Umov's works are devoted to various problems of physics. The main one was the creation of the doctrine of the movement of energy (the Umov vector), which he outlined in 1874 in his doctoral dissertation. Umov was endowed with high civic responsibility. Together with other professors (V. I. Vernadsky, K. A. Timiryazev,

N. D. Zelinsky, P. N. Lebedev), he left Moscow University in 1911 in protest against the actions of the reactionary Minister of Education L. A. Kasso.

Umov was an active propagandist of science, popularizer of scientific knowledge. Almost the first of the physicists, he realized the need for serious and targeted research on the methods of teaching physics. Most of the Methodist scholars of the older generation are his students and followers.

The main merit of Umov - development of the doctrine of the movement of energy. In 1874, he obtained a general expression for the energy flux density vector as applied to elastic media and viscous fluids (the Umov vector). After 11 years, an English scientist John Henry Poynting(1852-1914) did the same for the flow of electromagnetic energy. Thus, in the theory of electromagnetism, the well-known Umov vector - Pointing.

Poynting was one of those scientists who immediately accepted Maxwell's theory. It cannot be said that there were many such scientists, which Maxwell himself understood. Maxwell's theory was not immediately understood even in the Cavendish Laboratory he created. Nevertheless, with the advent of the theory of electromagnetism, the knowledge of nature has risen to a qualitatively different level, which, as always happens, increasingly removes us from direct sensory representations. This is a normal natural process that accompanies the entire development of physics. The history of physics provides many such examples. It suffices to recall the provisions quantum mechanics, special relativity, other modern theories. So the electromagnetic field at the time of Maxwell was hardly accessible to the understanding of people, including the scientific community, and even more so not accessible to their sensory perception. Nevertheless, after the experimental work of Hertz, ideas arose about creating wireless communications using electromagnetic waves, culminating in the invention of radio. Thus, the emergence and development of radio communication technology has turned the electromagnetic field into a well-known and familiar concept for everyone.

The German physicist played a decisive role in the victory of Maxwell's theory of electromagnetic field Heinrich Rudolf Hertz. Hertz's interest in electrodynamics was stimulated by G. L. Helmholtz, who, considering it necessary to "order" this area of ​​physics, suggested that Hertz should study processes in open electrical circuits. At first, Hertz abandoned the topic, but then, while working in Karlsruhe, he discovered devices there that could be used for such studies. This predetermined his choice, especially since Hertz himself, knowing Maxwell's theory well, was fully prepared for such studies.

Heinrich Rudolf Hertz (1857-1894) - German physicist, was born in 1857 in Hamburg in the family of a lawyer. He studied at the University of Munich, and then - in Berlin with G. Helmholtz. Since 1885, Hertz has been working at the Technische Hochschule in Karlsruhe, where he began his research, which led to the discovery of electromagnetic waves. They were continued in 1890 in Bonn, where Hertz moved, replacing R. Clausius as professor of experimental physics. Here he continues to study electrodynamics, but gradually his interests shift to mechanics. Hertz died on January 1, 1894 in the prime of his talent at the age of 36.

By the beginning of Hertz's work, electrical oscillations had already been studied in some detail. William Thomson (Lord Kelvin) received an expression that is now known to every schoolchild:

where T- period electrical oscillations; BUT- inductance, which Thomson called the "electrodynamic capacitance" of the conductor; C is the capacitance of the capacitor. The formula has been confirmed in experiments Berend Wilhelm Feddersen(1832-1918), who studied the oscillations of the spark discharge of a Leyden jar.

In the article "On very fast electrical oscillations" (1887), Hertz gives a description of his experiments. Figure 10.4 explains their essence. In its final form, the oscillatory circuit used by Hertz consisted of two conductors CuC ", located at a distance of about 3 m from each other and connected by a copper wire, in the middle of which there was a spark gap AT induction coil. The receiver was a circuit acdb with dimensions 80 x 120 cm, with spark gap M on one of the short sides. Detection was determined by the presence of a weak spark in the spark gap M. The conductors with which Hertz experimented are, saying modern language, antenna with detector. They are now named vibrator and Hertz resonator.

Rice. 10.4.

The essence of the results obtained was that the electric spark in the spark gap AT caused a spark in the discharger M. At first, Hertz, in explaining the experiments, does not speak of Maxwellian waves. He speaks only of the "interaction of conductors" and tries to find an explanation in the theory of long-range interaction. While conducting experiments, Hertz discovered that at short distances the nature of propagation " electrical force» is similar to the dipole field, and then it decreases more slowly and has an angular dependence. We would now say that the spark gap has an anisotropic radiation pattern. This, of course, fundamentally contradicts the theory of long-range action.

After analyzing the results of experiments and conducting his own theoretical research, Hertz accepts Maxwell's theory. He comes to the conclusion about the existence of electromagnetic waves propagating with a finite speed. Now Maxwell's equations are no longer an abstract mathematical system and they should be brought to such a form that they are convenient to use.

Hertz received electromagnetic waves experimentally predicted by Maxwell's theory and, no less important, proved their identity with light. To do this, it was necessary to prove that with the help of electromagnetic waves one can observe the known effects of optics: refraction and reflection, polarization, etc. Hertz carried out these studies, which required virtuoso experimental skill: he conducted experiments on the propagation, reflection, refraction, and polarization of electromagnetic waves discovered by him. He built mirrors for experiments with these waves (Hertz mirrors), an asphalt prism, and so on. Hertz mirrors are shown in fig. 10.5. The experiments showed the complete identity of the observed effects with those that were well known for light waves.

Rice. 10.5.

In 1887, in his work “On the Influence of Ultraviolet Light on an Electric Discharge,” Hertz describes a phenomenon that later became known as external photoelectric effect. He found that when high-voltage electrodes are irradiated with ultraviolet rays, the discharge occurs at a greater distance between the electrodes than without irradiation.

This effect was then comprehensively investigated by the Russian scientist Alexander Grigorievich Stoletov (1839-1896).

In 1889, at a congress of German natural scientists and physicians, Hertz delivered a report "On the relationship between light and electricity", in which he expressed his opinion on the great importance of Maxwell's theory, now confirmed by experiments.

Hertz's experiments made a splash in scientific world. They have been repeated and modified many times. One of those who did this was Petr Nikolaevich Lebedev. He received the shortest electromagnetic waves at that time and in 1895 made experiments with them on birefringence. In his work, Lebedev set the task of gradually reducing the wavelength of electromagnetic radiation in order to finally connect them with long infrared waves. Lebedev himself failed to do this, but it was carried out in the 20s of the XX century by Russian scientists Alexandra Andreevna Glagoleva-Arkadieva(1884-1945) and Maria Afanasyevna Levitskaya (1883-1963).

Petr Nikolaevich Lebedev (1866-1912) - Russian physicist, born in 1866 in Moscow, graduated from the University of Strasbourg and in 1891 began working at Moscow University. Lebedev remained in the history of physics as a virtuoso experimenter, the author of research carried out with modest means on the verge of the technical capabilities of that time, and also as the founder of the generally recognized scientific school in Moscow, where famous Russian scientists P. P. Lazarev, S. I. Vavilov, A. R. Kolli and others came from.

Lebedev died in 1912 shortly after he, along with other professors, left Moscow University in protest against the actions of the reactionary Minister of Education L. A. Kasso.

However, Lebedev's main contribution to physics is that he experimentally measured the light pressure predicted by Maxwell's theory. Lebedev devoted his whole life to studying this effect: in 1899 an experiment was set up that proved the presence of light pressure on solids (Fig. 10.6), and in 1907 on gases. Lebedev's works on light pressure have become classics; they are one of the pinnacles of experiment in the late 19th and early 20th centuries.

Lebedev's experiments on light pressure brought him worldwide fame. On this occasion, W. Thomson said, "All my life I fought with Maxwell, not recognizing his light motion, but ... Lebedev forced me to surrender to his experiments."

Rice. 10.6.

The experiments of Hertz and Lebedev finally confirmed the priority of Maxwell's theory. As for practice, i.e. practical application laws of electromagnetism, then by the beginning of the 20th century. humanity already lived in a world in which electricity began to play a huge role. This was facilitated by vigorous inventive activity in the field of application of electrical and magnetic phenomena discovered by physicists. Let's take a look at some of these inventions.

One of the first applications of electromagnetism found in communications technology. The telegraph had already existed since 1831. In 1876, an American physicist, inventor and businessman Alexander Bell(1847-1922) invented the telephone, which was further improved by the famous American inventor Thomas Alva Edison (1847-1931).

In 1892 an English physicist William Crookes(1832-1912) formulated the principles of radio communication. Russian physicist Alexander Stepanovich Popov(1859-1906) and Italian scientist Guglielmo Marconi(1874-1937) actually put them into practice at the same time. The question usually arises as to the priority of the present invention. Popov demonstrated the capabilities of the device he created a little earlier, but did not patent it, as Marconi did. The latter determined the tradition prevailing in the West to consider Marconi the "father" of radio. This was facilitated by the award of the Nobel Prize to him in 1909. Popov, apparently, would also have been among the laureates, but by that time he was no longer alive, and Nobel Prize awarded only to living scientists. More about the history of the invention of the radio will be told in part VI of the book.

They tried to use electrical phenomena for lighting as early as the 18th century. (voltaic arc), later this device was improved Pavel Nikolaevich Yablochkov(1847-1894), who in 1876 invented the first practical electric light source (Yablochkov's candle). However, it did not find wide application, primarily because in 1879 T. Edison created an incandescent lamp of a sufficiently durable design and convenient for industrial production. Note that the incandescent lamp was invented back in 1872 by a Russian electrical engineer Alexander Nikolaevich Lodygin (1847- 1923).

test questions

• 1. What research did Maxwell do while working at Marischal College? What role did Maxwell play in the development of the theory of electricity and magnetism?
• 2. When was the Cavendish Laboratory organized? Who became its first director?
• 3. What law could not be described using electrohydraulic analogies?
• 4. With what model did Maxwell come to the conclusion about the existence of a displacement current and the phenomenon of magnetoelectric induction?
• 5. In which article did Maxwell first use the term "electromagnetic field"?
• 6. How is the system of equations compiled by Maxwell written?
• 7. Why are Maxwell's equations considered one of the triumphant achievements of human civilization?
• 8. What conclusions did Maxwell draw from the theory of the electromagnetic field?
• 9. How did electrodynamics develop after Maxwell?
• 10. How did Hertz come to the conclusion about the existence of electromagnetic waves?
• 11. What is Lebedev's main contribution to physics?
• 12. How is the electromagnetic field theory used in engineering?

Tasks for independent work

• 1. J. K. Maxwell. Biography and scientific achievements in electrodynamics and other areas of physics.
• 2. Empirical and theoretical foundations of Maxwell's electromagnetic field theory.
• 3. The history of the creation of Maxwell's equations.
• 4. Physical essence of Maxwell's equations.
• 5. J. K. Maxwell - first director of the Cavendish Laboratory.
• 6. How is Maxwell's system of equations currently written: a) in integral form; b) in differential form?
• 7. G. Hertz. Biography and scientific achievements.
• 8. History of detection of electromagnetic waves and their identification with light.
• 9. P. N. Lebedev’s Experiments on the Detection of Light Pressure: Scheme, Problems, Difficulties, and Significance.
• 10. Works by A. A. Glagoleva-Arkadyeva and M. A. Levitskaya on the generation of short electromagnetic waves.
• 11. History of the discovery and study of the photoelectric effect.
• 12. Development of Maxwell's electromagnetic theory. Works by J. G. Poynting, N. A. Umov, O. Heaviside.
• 13. How was the electric telegraph invented and improved?
• 14. Historical stages in the development of electrical and radio engineering.
• 15. The history of the creation of lighting devices.

• 1. Kudryavtsev, P. S. Course in the history of physics. - 2nd ed. - M.: Enlightenment, 1982.
• 2. Kudryavtsev, P. S. History of physics: in 3 volumes - M.: Education, 1956-1971.
• 3. Spassky, B. I. History of physics: in 2 volumes - M .: graduate School, 1977.
• 4. Dorfman, Ya. G. The World History physics: in 2 volumes - M .: Nauka, 1974-1979.
• 5. Golin, G. M. Classics of physical science (from ancient times to the beginning of the 20th century) / G. M. Golin, S. R. Filonovich. - M.: Higher school, 1989.
• 6. Khramov, Yu. A. Physicists: a biographical guide. - M.: Nauka, 1983.
• 7. Virginsky, V. S. Essays on the history of science and technology in 1870-1917. / V. S. Virginsky, V. F. Khoteenkov. - M.: Enlightenment, 1988.
• 8. Witkowski, N. A sentimental history of science. - M.: Hummingbird, 2007.
• 9. Maxwell, J.K. Selected works on the theory of the electromagnetic field. - M.: GITTL, 1952.
• 10. Kuznetsova, O. V. Maxwell and the Development of Physics in the 19th-20th Centuries: Sat. articles / resp. ed. L. S. POLAK. - M.: Nauka, 1985.
• 11. Maxwell, J.K. Treatise on electricity and magnetism: in 2 volumes - M .: Nauka, 1989.
• 12. Kartsev, V.P. Maxwell. - M.: Young Guard, 1974.
• 13. Niven, W. Life and scientific activity of J. K. Maxwell: a brief essay (1890) // J. K. Maxwell. Matter and motion. - M.: Izhevsk: RHD, 2001.
• 14. Harman, R. M. The natural philosophy of James Clerk Maxwell. - Cambridge: University Press, 2001.
• 15. Bolotovsky, B. M. Oliver Heaviside. - M.: Nauka, 1985.
• 16. Gorokhov, V. G. The formation of radio engineering theory: from theory to practice on the example of technical consequences from the discovery of G. Hertz // VIET. - 2006. - No. 2.
• 17. Book series "ZhZL": "People of Science", "Creators of Science and Technology".

By about 1860, thanks to the work of Neumann, Weber, Helmholtz and Felici (see § 11), electrodynamics was considered to be a finally systematized science, with clearly defined boundaries. The main research now seemed to have to follow the path of finding and deriving all the consequences from the established principles and their practical application, which the inventive techniques had already embarked on.

However, the prospect of such a quiet work was violated by the young Scottish physicist James Clark Maxwell (1831-1879), pointing to a much wider area of ​​application of electrodynamics. With good reason, Duhem wrote:

“No logical necessity pushed Maxwell to invent a new electrodynamics; he was guided only by some analogies and the desire to complete the work of Faraday in the same spirit as the works of Coulomb and Poisson were completed by Ampère's electrodynamics, and also, perhaps, by an intuitive sense of the electromagnetic nature of light " (P. Duhem, Les theories electriques de J. Clerk Maxwell, Paris, 1902, p. eight).

Perhaps the main motivation that made Maxwell take up work that was not at all required by the science of those years was admiration for Faraday's new ideas, so original that the scientists of that time were not able to perceive and assimilate them. To a generation of theoretical physicists, brought up on the concepts and mathematical elegance of the works of Laplace, Poisson and Ampère, Faraday's thoughts seemed too vague, and to experimental physicists - too sophisticated and abstract. A strange thing happened: Faraday, who was not a mathematician by training (he started his career as a peddler in a bookshop and then joined Davy's laboratory as a half-assistant, half-service), felt an urgent need to develop some theoretical method as effective as and mathematical equations. Maxwell guessed it.

“Having begun to study the work of Faraday,” Maxwell wrote in the preface to his famous Treatise, “I found that his method of understanding phenomena was also mathematical, although not presented in the form of ordinary mathematical symbols. I have also found that this method1 can be expressed in the usual mathematical form and thus compared with the methods of professional mathematicians. So, for example, Faraday saw lines of force penetrating all space, where mathematicians saw centers of forces attracting at a distance; Faraday saw the medium where they saw nothing but distance; Faraday assumed the source and cause of phenomena in real actions occurring in a medium, but they were satisfied that they found them in the force of action at a distance attributed to electric fluids.

When I translated what I considered Faraday's ideas into mathematical form, I found that in most cases the results of both methods coincided, so that they explained the same phenomena and deduced the same laws of action, but that Faraday's methods were like those in which we start from the whole and arrive at the particular by analysis, while the usual mathematical methods are based on the principle of moving from particulars and building the whole by synthesis.

I also found that many of the fruitful methods of investigation discovered by mathematicians could be expressed much better with the help of ideas arising from the works of Faraday than in their original form ”( J. Clerk Maxwell, A Treatise on Electricity and Magnetism, London, 1873; 2nd ed., Oxford, 1881.).

As for Faraday's mathematical method, Maxwell notes elsewhere that mathematicians who considered Faraday's method devoid of scientific accuracy did not themselves come up with anything better than using hypotheses about the interaction of things that do not have physical reality, such as current elements, " which arise from nothing, pass through a section of wire and then turn back into nothing.”

To give Faraday's ideas a mathematical form, Maxwell began by creating the electrodynamics of dielectrics. Maxwell's theory is directly related to Mossotti's theory. While Faraday, in his theory of dielectric polarization, deliberately left open the question of the nature of electricity, Mossotti, a supporter of Franklin's ideas, imagines electricity as a single fluid, which he calls ether and which, in his opinion, is present with a certain degree of density in all molecules. . When a molecule is under the action of an inductive force, the ether is concentrated at one end of the molecule and rarefied at the other; because of this, a positive force arises at the first end and an equal negative force at the second. Maxwell fully accepts this concept. In his Treatise, he writes:

“The electric polarization of a dielectric is a state of deformation into which a body enters under the influence of an electromotive force and which disappears simultaneously with the termination of this force. We can think of it as something that can be called an electrical displacement produced by an electromotive force. When an electromotive force acts in a conducting medium, it induces a current there, but if the medium is non-conductive or dielectric, then the current cannot pass through this medium. Electricity, however, is displaced in it in the direction of the electromotive force, and the magnitude of this displacement depends on the magnitude of the electromotive force. If the electromotive force increases or decreases, then the electric displacement increases or decreases correspondingly in the same proportion.

The amount of displacement is measured by the amount of electricity that crosses a unit area as the displacement increases from zero to a maximum value. Such, therefore, is the measure of electric polarization.

If a polarized dielectric consists of a collection of conducting particles scattered in an insulating medium, on which electricity is distributed in a certain way, then any change in the state of polarization must be accompanied by a change in the distribution of electricity in each particle, i.e., a real electric current, though limited only by the volume of the conducting particle. In other words, each change in the state of polarization is accompanied by a bias current. In the same Treatise, Maxwell says:

“Changes in electrical displacement obviously cause electrical currents. But these currents can only exist during a change in displacement, and since the displacement cannot exceed a certain amount without causing a destructive discharge, these currents cannot continue indefinitely in the same direction, like currents in conductors..

After Maxwell introduces the concept of field strength, which is a mathematical interpretation of the Faraday concept of the field of forces, he writes down the mathematical relationship for the mentioned concepts of electric displacement and displacement current. He concludes that the so-called charge of a conductor is the surface charge of the surrounding dielectric, that energy is stored in the dielectric in the form of a state of voltage, that the movement of electricity is subject to the same conditions as the movement of an incompressible fluid. Maxwell himself summarizes his theory thus:

“Electrization energy is concentrated in a dielectric medium, whether it be a solid, liquid or gas, a dense medium, or rarefied, or completely devoid of weighty matter, so long as it is able to transmit electrical action.

Energy is contained in each point of the medium in the form of a state of deformation, called electric polarization, the magnitude of which depends on the electromotive force acting at that point ...

In dielectric liquids, electric polarization is accompanied by tension in the direction of the induction lines and an equal pressure in all directions perpendicular to the induction lines; the magnitude of this tension or pressure per unit area is numerically equal to the energy per unit volume at that point.”

It is difficult to express more clearly the main idea of ​​this approach, which is the idea of ​​Faraday: the place in which electrical phenomena occur is the environment. As if to emphasize that this is the main thing in his treatise, Maxwell ends it with the following words:

“If we accept this environment as a hypothesis, I believe that it should occupy a prominent place in our studies and that we should try to construct a rational idea of ​​all the details of its operation, which was my constant goal in this treatise”.

Having substantiated the theory of dielectrics, Maxwell transferred its concepts with the necessary corrections to magnetism and created the theory of electromagnetic induction. He summarizes his entire theoretical construction in several equations that have now become famous: in Maxwell's six equations.

These equations are very different from the usual equations of mechanics - they determine the structure of the electromagnetic field. While the laws of mechanics apply to areas of space in which matter is present, Maxwell's equations apply to all of space whether or not bodies or electric charges are present. They determine the changes in the field, while the laws of mechanics determine the changes in material particles. In addition, Newtonian mechanics refused, as we said in Chap. 6, from the continuity of action in space and time, while Maxwell's equations establish the continuity of phenomena. They connect events that are adjacent in space and time: given the state of the field "here" and "now" we can deduce the state of the field in close proximity at close times. Such an understanding of the field is absolutely consistent with Faraday's idea. but is in insurmountable contradiction with the two-century tradition. Therefore, it is not surprising that it met with resistance.

The objections that were put forward against Maxwell's theory of electricity were numerous and related both to the fundamental concepts underlying the theory and, perhaps even more so, to the too free manner that Maxwell uses in deriving consequences from it. Maxwell builds his theory step by step with the help of "sleight of fingers", as Poincaré aptly put it, referring to the theological stretches that scientists sometimes allow themselves to formulate new theories. When, in the course of an analytic construction, Maxwell encounters an obvious contradiction, he does not hesitate to overcome the era with the help of discouraging liberties. For example, it doesn't cost him anything to exclude a member, replace an inappropriate sign of an expression with a reverse, change the meaning of a letter. For those who admired the infallible logic of Ampère's electrodynamics, Maxwell's theory must have made an unpleasant impression. Physicists failed to bring it into order, that is, to free it from logical errors and inconsistencies. But. on the other hand, they could not abandon the theory, which, as we shall see later, organically connected optics with electricity. Therefore, at the end of the last century, the leading physicists adhered to the thesis put forward in 1890 by Hertz: since the reasoning and calculations by which Maxwell arrived at his theory of electromagnetism are full of errors that we cannot correct, let us accept Maxwell's six equations as the initial hypothesis, as postulates on which the whole theory of electromagnetism will be based. "The main thing in Maxwell's theory is Maxwell's equations," Hertz says.

21. ELECTROMAGNETIC THEORY OF LIGHT

The formula found by Weber for the force of interaction of two electric charges moving relative to each other includes a coefficient that has the meaning of a certain speed. Weber himself and Kohlrausch determined the value of this speed experimentally in the work of 1856, which became a classic; this value turned out to be somewhat greater than the speed of light. The following year, Kirchhoff "from Weber's theory derived the law of propagation of electrodynamic induction along a wire: if the resistance is zero, then the speed of propagation of an electric wave does not depend on the cross section of the wire, on its nature and the density of electricity and is almost equal to the speed of propagation of light in a vacuum. Weber, in one of his theoretical and experimental works in 1864, confirmed the results of Kirchhoff, showing that the Kirchhoff constant is quantitatively equal to the number of electrostatic units contained in an electromagnetic unit, and noticed that the coincidence of the propagation velocity of electric waves and the speed of light can be considered as an indication of there is a close connection between the two phenomena. However, before talking about this, one should first find out exactly what is the true meaning of the concept of the speed of propagation of electricity: "and this meaning," Weber concludes melancholy, "is not at all such as to arouse great hopes."

Maxwell did not have any doubts, perhaps because he found support in Faraday's ideas regarding the nature of light (see § 17).

“In various places of this treatise,” writes Maxwell, starting in Chapter XX of the fourth part to present the electromagnetic theory of light, “an attempt was made to explain electromagnetic phenomena with the help of a mechanical action transmitted from one body to another through a medium that occupies the space between these bodies. The wave theory of light also allows for the existence of some kind of medium. We must now show that the properties of the electromagnetic medium are identical with those of the luminiferous medium...

We can obtain a numerical value for certain properties of a medium, such as the speed with which a disturbance propagates through it, which can be calculated from electromagnetic experiments and also observed directly in the case of light. If it were found that the speed of propagation of electromagnetic disturbances is the same as the speed of light, not only in air, but also in other transparent media, we would get a good reason to consider light as an electromagnetic phenomenon, and then the combination of optical and electrical evidence will give the same proof of the reality of the environment, which we receive in the case of other forms of matter on the basis of the totality of evidence from our senses" ( Ibid. pp. 550-551 of the Russian edition).

As in the first work of 1864, Maxwell proceeds from his equations and, after a series of transformations, comes to the conclusion that in vacuum, transverse displacement currents propagate at the same speed as light, which "represents a confirmation of the electromagnetic theory of light" - Maxwell confidently states.

Then Maxwell studies the properties of electromagnetic disturbances in more detail and comes to conclusions that are already well known today: an oscillating electric charge creates an alternating electric field, inextricably linked with an alternating magnetic field; this is a generalization of Oersted's experience. Maxwell's equations make it possible to trace the changes in the field over time at any point in space. The result of such a study shows that electric and magnetic oscillations arise at each point in space, i.e., the intensity of the electric and magnetic fields changes periodically; these fields are inseparable from each other and polarized mutually perpendicular. These oscillations propagate in space at a certain speed and form a transverse electromagnetic wave: electric and magnetic oscillations at each point occur perpendicular to the direction of wave propagation.

Among the many particular consequences that follow from Maxwell's theory, we mention the following: the assertion that the dielectric constant is equal to the square of the refractive index of optical rays in a given medium, which was especially often criticized; the presence of light pressure in the direction of light propagation; orthogonality of two polarized waves - electric and magnetic.

22. ELECTROMAGNETIC WAVES

In § 11 we have already said that the oscillatory nature of the discharge of the Leyden jar has been established. This phenomenon from 1858 to 1862 was again subjected to careful analysis by Wilhelm Feddersen (1832-1918). He noticed that if two capacitor plates are connected by a small resistance, then the discharge is oscillatory in nature and the duration of the oscillation period is proportional to the square root of the capacitance of the capacitor. In 1855, Thomson deduced from potential theory that the period of oscillation of an oscillating discharge is proportional to the square root of the product of the capacitance of a capacitor and its coefficient of self-induction. Finally, in 1864, Kirchhoff gave the theory of an oscillatory discharge, and in 1869, Helmholtz showed that similar oscillations can also be obtained in an induction coil, the ends of which are connected to the capacitor plates.

In 1884, Heinrich Hertz (1857-1894), a former student and assistant of Helmholtz, began to study Maxwell's theory (see Ch. 12). In 1887 he repeated Helmholtz's experiments with two induction coils. After several attempts, he managed to stage his classical experiments, which are now well known. With the help of a “generator” and a “resonator”, Hertz experimentally proved (in a way that is described in all textbooks today) that an oscillatory discharge causes waves in space, consisting of two oscillations - electric and magnetic, polarized perpendicular to each other. Hertz also established the reflection, refraction and interference of these waves, showing that all his experiments are fully explainable by Maxwell's theory.

Many experimenters rushed along the path discovered by Hertz, but they did not manage to add much to understanding the similarity of light and electric waves, because, using the same wavelength that Hertz took (about 66 cm), they came across diffraction phenomena that obscured all others. effects. To avoid this, installations of such large sizes were needed, which were practically unrealizable at that time. A big step forward was made by Augusto Righi (1850-1920), who, with the help of a new type of generator he created, managed to excite waves several centimeters long (most often he worked with waves 10.6 cm long). Thus, Rigi managed to reproduce all optical phenomena with the help of devices that are basically analogues of the corresponding optical instruments. In particular, Rigi was the first to obtain double refraction of electromagnetic waves. Riga's work, begun in 1893 and described from time to time in notes and articles published in scientific journals, was then combined and supplemented in the now classic book "Ottica delle oscillazioni elettriche" ("Optics of electrical oscillations"), published in 1897, whose name alone expresses the content of an entire era in the history of physics.

The ability of a metal powder placed in a tube to become conductive under the action of a discharge from a nearby electrostatic machine was studied by Snez (1853-1922) in 1884, and ten years later this ability was used by Dodge a.d. and many others to indicate electromagnetic waves. Combining the Riga generator and the Demolish indicator with the ingenious ideas of "antenna" and "grounding", at the end of 1895 Guglielmo Marconi (1874-1937) successfully carried out the first practical experiments ( As you know, the priority in the invention of radio belongs to the Russian scientist A.S. Popov, who read his report on May 7, 1895 at a meeting of the Physics Department of the Russian Physical) in the field of radiotelegraphy, the rapid development and amazing results of which truly border on a miracle.

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What does the light tell Suvorov Sergey Georgievich

Maxwell's electromagnetic field theory

Maxwell's merit lies in the fact that he found the mathematical form of equations in which the values ​​​​of electric and magnetic intensities that create electromagnetic waves are connected together with the speed of their propagation in media with certain electrical and magnetic characteristics. In short, Maxwell's merit lies in the creation of the theory electromagnetic fields.

The creation of this theory allowed Maxwell to express another remarkable idea.

In the specific case of the interaction of currents and charges, he measured electrical and magnetic voltages, took into account the quantities characterizing the electrical and magnetic properties of a space devoid of a material medium (“emptiness”). Substituting all this data into his equations, he calculated the speed of propagation of an electromagnetic wave. According to his calculations, it turned out to be equal to 300 thousand kilometers per second, i.e. equal to the speed of light! But at one time the speed of light was determined purely optically: the distance traveled by the light signal from the source to the receiver was divided by the time of its movement; no one could even think about electrical and magnetic intensities, or about the electrical and magnetic properties of the medium.

Is this coincidence of speeds accidental?

Maxwell made a bold assumption: the speed of light and the speed of electromagnetic waves are the same because light has the same nature - electromagnetic.

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String field theory Since Faraday's pioneering work, all physical theories have been written as fields. Maxwell's theory of light is based on field theory, as is Einstein's theory. In fact, all particle physics is based on field theory. Not based on it

In 1860-1865, D. Maxwell developed the theory of the electromagnetic field, the apex of which is the system of Maxwell's equations. Maxwell's theory was the greatest contribution to the development of classical physics and allowed from a general point of view to cover a huge range of phenomena, ranging from the electrostatic field of stationary charges to the electromagnetic nature of light.

Maxwell's theory is a phenomenological theory of the electromagnetic field. This means that the internal mechanism of the phenomena occurring in the medium and causing the appearance of electric and magnetic fields is not considered in the theory.

Maxwell's theory is a macroscopic theory of the electromagnetic field. It deals with electric and magnetic fields created by macroscopic charges and currents.

Maxwell's first equation is a generalization of the law of electromagnetic induction. Maxwell suggested that an alternating magnetic field at any point in space creates vortex electric field, regardless of whether the conductor is at this point or not.

The circulation of the electric field strength vector E along an arbitrary closed loop L is equal to the rate of change of the magnetic flux through the surface S, bounded by the loop L, taken with the opposite sign:

The electric field (vortex) is generated by an alternating magnetic field.

But if we proceed from the unity of electric and magnetic fields, then we can assume the existence of the reverse process: the magnetic field must be generated by an alternating electric field. Developing this idea, Maxwell introduced the concept of displacement current.

If a circuit containing a capacitor is connected to an alternating voltage source, then a current will appear in it. However, how does this current close through the capacitor plates? The current I in the external circuit is associated with a change in the charge q of the capacitor plate

the mark "> S and the mark" > D - the electrical displacement between the plates (D = the mark "> I lost the physical meaning of the conduction current, it began to describe the rate of change in the electrical displacement between the plates of the capacitor, and therefore is called the bias current selection"> Displacement current is a special current, which is created not by the directed movement of charges, but by an alternating electric field, but in the same way as the conduction current, it generates a magnetic field.

Displacement current density as follows from (14..gif" border="0" align="absmiddle" alt="(!LANG:

The generalized total current law has the form:

the mark ">L is equal to the total current penetrating the surface bounded by this circuit.

Equation (14.3) can be written for the circulation of the vector B:

define-e "> the second Maxwell equation in integral form.

Maxwell's third equation is the Ostrogradsky-Gauss theorem for the flow of the electric field strength vector through an arbitrary closed surface S, covering the total charge mark "> V, enclosed inside the closed surface S:

define "> The fourth Maxwell equation is the Ostrogradsky-Gauss theorem for the magnetic flux through an arbitrary closed surface S: the vector flux through the closed surface is equal to zero.

selection"> This equation is a consequence of the fact that free magnetic charges do not exist in nature.

Maxwell's equations (14.1), (14.4), (14.5), (14.6) show that the sources of the electric field can be either electric charges or time-varying magnetic fields. Magnetic fields can be excited either by moving electric charges (electric currents) or by alternating electric fields. The equations are not symmetrical with respect to electric and magnetic fields. This is due to the fact that in nature there are electric charges, but there are no magnetic charges. The desire to achieve the symmetry of the equations of electrodynamics made Dirac put forward a hypothesis about the existence of magnetic charges - monopoles. Numerous attempts to experimentally detect monopoles have not yet yielded a positive result.

Maxwell's equations are axioms of electrodynamics obtained by generalizing experimental facts.

Maxwell's fundamental equations do not contain any constants that characterize the properties of the medium in which the electromagnetic field is excited. The relations (material equations), with the help of which the electrical and magnetic characteristics of substances are introduced formula !lang::

formula" src="http://hi-edu.ru/e-books/xbook785/files/epsilon.gif" border="0" align="absmiddle" alt="(!LANG:.gif" border="0" align="absmiddle" alt="(!LANG:- electrical conductivity of the medium.

The physical essence of Maxwell's equations lies in the fact that the electromagnetic field can be divided into electric and magnetic only relatively. A changing magnetic field generates an electric field, and a changing electric field excites a magnetic field, and these fields are interconnected - there is a single whole - an electromagnetic field.

In some reference systems B = const or E = const and then the equations (14.1) and (14.4) take a simpler form. In these special cases, the electric and magnetic fields can be considered independently of each other..gif" border="0" align="absmiddle" alt="(!LANG:

2) mark "> q, the sources of the magnetic field are only conduction currents I.

Using mathematical operations (based on the Stokes theorem), Maxwell's equations (14.1), (14.4) can also be represented in differential form:

define "> Maxwell's equations in differential form:

define "e"> an electromagnetic wave.

Electromagnetic waves can be generated either by antennas due to alternating current oscillations (low frequencies), or due to processes performed in lamp or semiconductor devices by rapidly moving charges (higher frequencies), or due to intra-atomic processes or deceleration of electrons in metals (light and x-rays) and etc.

The electromagnetic field is a special form of matter and is completely determined by the electric strength vector E and the induction vector B (or strength vector H) of the magnetic fields.

Vectors E and B are perpendicular to each other, as well as to the direction of wave propagation, Fig. 61
. For this reason, an electromagnetic wave in an unlimited space is transverse wave.

The wave equations for the vectors E and H have the form:

formula" src="http://hi-edu.ru/e-books/xbook785/files/130-2.gif" border="0" align="absmiddle" alt="(!LANG:

where the formula is" src="http://hi-edu.ru/e-books/xbook785/files/epsilon0.gif" border="0" align="absmiddle" alt="(!LANG:and the ">v" mark.

The energy transfer of the electromagnetic field is characterized by the Umov-Poynting vector

mark ">S - electromagnetic energy flux density.

The direction of the energy flow coincides with the direction of the phase velocity v, which, together with the vectors E and H, makes up a right-handed three of vectors.

There is no fundamental difference between electromagnetic waves of different lengths (frequency). It manifests itself only when interacting with matter, when the ratio of the wavelength and some size characteristic of the observed effect is important.

It is customary to classify electromagnetic waves according to wavelength intervals. We give a brief description of the main intervals.

1. Range of long radio waves (definition "> 2. Range of medium and short radio waves (definition"> 3. Meter range (10 - 1 m).

This radio wave band is used for television and radar. In television, it is necessary to transmit not only sound, but also images over long distances. Therefore, for high-quality work, a much larger frequency band is needed than for a radio station. The disadvantage of wave propagation in this range is that they pass well through the ionosphere and therefore do not go around the Earth's surface. In this regard, it is necessary to build many relay stations, or satellites are launched into specially selected near-Earth orbits, constantly located above the given area, which serve as relays.

Radar requires waves to be focused in a certain direction. For this, the dimensions of the antenna reflector were of the same order of magnitude as the wavelength or greater.

4. Centimeter range (defined "> 5. Millimeter (microwave) range (10 - 1 mm)

This range is used for precise radar, as well as for scientific purposes, since this range includes the rotational spectra of polyatomic molecules, and the structure of molecules can be judged from the absorption of millimeter waves.

6. Infrared range (definition "> 7. Visible light (0.7 - 0.45 microns).

This range represents all the information that we perceive with our eyes.

8. Ultraviolet range (0.4 - 0.1 microns).

Waves of this range are able to actively influence the substance. Ultraviolet tanning from the sun or a quartz lamp can have a strong physiological effect and even lead to burns. Due to the strong interaction with matter, the ultraviolet radiation of the Sun is almost completely absorbed by the atmosphere (up to 99%), and only because of this, the conditions for the existence of life on Earth arose.

9. X-ray and gamma radiation range (less than 0.1 µm).

This radiation is widely used in medicine, as well as in flaw detection technology. X-ray lasers are used to destroy enemy targets, gamma rays are one of the factors in the destruction of nuclear weapons.

Control questions and tasks

1. What is Maxwell's generalization of the law of electromagnetic induction?
2. What is bias current and how can it be detected?
3. What is a vortex electric field? What properties does it have?
4. Write the complete system of Maxwell's equations. What is the physical meaning of these equations?
5. Write down the material equations.
6. Write Maxwell's equations for stationary electric and magnetic fields.
7. What is an electromagnetic wave? What characteristics does she have?
8. Write down the wave equations of an electromagnetic wave.
9. What determines the propagation velocity of the phase surface of an electromagnetic wave?
10. What ranges of electromagnetic waves do you know?

In the 60s of the last century (around 1860), Maxwell, based on the ideas of Faraday, generalized the laws of electrostatics and electromagnetism: the Gauss-Ostrogradsky theorem for an electrostatic field and for a magnetic field; total current law ; the law of electromagnetic induction, and as a result developed a complete theory of the electromagnetic field.

Maxwell's theory was the greatest contribution to the development of classical physics. It made it possible to understand a wide range of phenomena from a unified point of view, ranging from the electrostatic field of stationary charges to the electromagnetic nature of light.

mathematical expression Maxwell's theory is served by four Maxwell equations. which are usually written in two forms: integral and differential. Differential equations are obtained from integral equations with the help of two theorems of vector analysis - the Gauss theorem and the Stokes theorem. Gauss theorem:

(1)

(2)

- vector projections on the axes; V- volume bounded by a surface S.

Stokes' theorem: . (3)

here rot- vector rotor , which is a vector and is expressed in Cartesian coordinates as follows: rot , (4)

S- area bounded by a contour L.

Maxwell's equations in integral form express relationships that are valid for immobile closed contours and surfaces mentally drawn in an electromagnetic field.

Maxwell's equations in differential form show how the characteristics of the electromagnetic field and the density of charges and currents at each point of this field are related.

12.1. Maxwell's first equation

It is a generalization of the law of electromagnetic induction ,

and in integral form has the following form (5)

and claims that a vortex electric field is inextricably linked with an alternating magnetic field, which does not depend on whether there are conductors in it or not. From (3) it follows that . (6)

From comparison (5) and (6) we find that (7)

This is the first Maxwell equation in differential form.

12.2. mixing current. Maxwell's second equation

Maxwell generalized the total current law assuming that an alternating electric field, as well as an electric current, is a source of a magnetic field. To quantitatively characterize the "magnetic action" of an alternating electric field, Maxwell introduced the concept bias current.

According to the Gauss-Ostrogradsky theorem, the flow of electrical mixing through a closed surface

Differentiating this expression with respect to time, we obtain for a fixed and non-deformable surface S (8)

The left side of this formula has the dimension of current, which, as is known, is expressed in terms of the current density vector . (9)

From comparison (8) and (9) it follows that it has the dimension of current density: A /m 2 . Maxwell proposed to call the displacement current density:

. (10)

Bias current . (11)

Of all the physical properties inherent in the real current (conduction current) associated with the transfer of charges, the mixing current gives only one: the ability to create a magnetic field. When the bias current "flows" in a vacuum or dielectric, no heat is generated. An example of a bias current is an alternating current through a capacitor. In the general case, conduction and displacement currents are not separated in space, and we can speak of a total current equal to the sum of conduction and displacement currents: (12)

With this in mind, Maxwell generalized the total current law by adding mixing current to its right side. (13)

So, the second Maxwell equation in integral form has the form:

. (14)

From (3) it follows that . (15)

From comparison (14) and (15) we find that . (16)

This is the second Maxwell equation in differential form.

12.3. Third and fourth Maxwell equations

Maxwell generalized the Gauss-Ostrogradsky theorem for the electrostatic field. He suggested that this theorem is valid for any electric field, both stationary and variable. Accordingly, the third Maxwell equation in integral form has the form: . (I7) or . (18)

where - bulk density of free charges, \u003d C / m 3

From (1) it follows that . (19)

From comparison (18) and (19) we find that . (20)

Maxwell's fourth equation in integral and differential forms has

the following form: , (21) . (22)

12.4. Complete system of Maxwell's equations in differential form

. (23)

This system of equations needs to be supplemented material equations characterizing the electrical and magnetic properties of the medium:

, , . (24)

So, after the discovery of the relationship between electric and magnetic fields, it became clear that these fields do not exist in isolation, independently of each other. It is impossible to create an alternating magnetic field without simultaneously generating an electric field in space.

Note that an electric charge at rest in a certain frame of reference creates only an electrostatic field in this frame of reference, but it will create a magnetic field in the frames of reference with respect to which it moves. The same applies to a fixed magnet. Note also that Maxwell's equations are invariant to Lorentz transformations: moreover, for inertial frames of reference To and TO' the following relations hold: , . (25)

Based on the foregoing, we can conclude that electric and magnetic fields are a manifestation of a single field, which is called the electromagnetic field. It propagates in the form of electromagnetic waves.

8) Boundary conditions on the media interface. An ideal conductor in an electrostatic field. surface charges. Electric field near the tip.

Boundary conditions at the media interface

At the interface between two dielectrics with different absolute permittivities e 1 and e 2, the tangential components of the field strength are equal to each other

Here index 1 refers to the first dielectric, and index 2 to the second.

Conditions can also be presented in this form

From these boundary conditions, one more condition can be obtained - the condition for the refraction of field lines when they pass from one dielectric to another:

q 1 and q 2 are the angles between the intensity (or displacement) vector and the normals to the interface between the media.

In this case, if the intensity vector is perpendicular to the interface, the field strength changes abruptly.

When passing through the interface between two dielectrics, the electric potential does not undergo jumps.

Ideal conductor in an electrostatic field

Near the surface of a charged conductor, the lines of force are perpendicular to its surface, and therefore the work to move the charge along any line on the surface of the conductor .

For electrostatic phenomena, the field inside the conductor is zero

Surface charges

charge density is the amount of charge per unit length, area, or volume.

If the conductor is given an excess charge, then this charge spread over the surface of the conductor.

The field strength on the surface of the conductor must be directed at each point along the normal to the surface, otherwise the component appears directed along the surface, which will lead to the movement of charges until the component disappears. Therefore, in the case of equilibrium of charges, the surface of the conductor will be equipotential. If a conducting body is given a certain charge q, then it will be distributed so that the equilibrium conditions are met. Imagine an arbitrary closed surface completely enclosed within the body. Since there is no field at any point inside the conductor when the charges are in equilibrium, the flux of the electric displacement vector through the surface is zero. According to the Gauss theorem, the algebraic sum of charges inside the surface will also be equal to zero.

Electric field near the tip

The lines of tension near the tip thicken, and they are discharged in the troughs.

9) Coefficients of capacitance and mutual capacitance of conductors. Capacitors. The capacitance of the capacitors.

Coefficients of capacitance and mutual capacitance of conductors. Capacitors

Capacitor(from lat. condensare- “compact”, “thicken”) - a two-terminal network with a certain value of capacitance and low ohmic conductivity; electric field charge and energy storage device

Capacitor capacitance

The main characteristic of a capacitor is its capacity characterizing the ability of a capacitor to store an electric charge.

The capacitance of a flat capacitor, consisting of two parallel metal plates with an area of ​​\u200b\u200beach, located at a distance from each other, in the SI system is expressed by the formula: constant numerically equal to f/m

10) Energy of interaction of electric charges. Energy of a system of charged conductors. The energy of a charged capacitor. Electrostatic field energy density

Energy of interaction of electric charges

Two point charges in vacuum act on each other with forces that are proportional to the product of the moduli of these charges, inversely proportional to the square of the distance between them, and directed along the straight line connecting these charges. These forces are called electrostatic (Coulomb).

Energy of a system of charged conductors

A charged conductor can be represented as a set of interacting point charges. It has one feature that is characteristic of conductors - the entire volume of the conductor is equipotential, i.e. for all the charges included in the conductor there is the same potential. Therefore, to find the energy of a charged conductor, you can use the formula (5.10)

, (5.11)

where is the charge of the conductor; is the potential of the conductor. Using the definition of the capacitance of a solitary conductor, formula (5.11) can be rewritten as:

.(5.12)

From formula (5.12) it follows that the energy of a charged conductor (regardless of the sign of the charge) is always positive.

The scope of formula (5.10), taking into account expression (5.11), can be changed: instead of determining the interaction energy of point charges, it can be used to calculate the interaction energy of charged conductors. In this case, instead of the parameters of point charges in (5.10), the parameters of charged conductors will appear.

Based on the results obtained above, we can consider common task- definition energy of a system of charged conductors.

The simplest example of a system of charged conductors is a capacitor. In a capacitor, one conductor (plate), on which the charge is located, has a potential, and the potential of the plate, on which the charge is located, is equal to. According to formula (5.10), the energy of such a system of charges is defined as

where is the potential difference between the capacitor plates. Using the definition of the capacitance of a capacitor (5.3), the formula for the energy of a charged capacitor can be represented as:

Energy of a charged capacitor

If on the capacitor plates with electrical capacity FROM electric charges are +q and - q, then, according to formula (20.1), the voltage between the capacitor plates is

Electrostatic field energy density

This is a physical quantity, numerically equal to the ratio of the potential energy of the field contained in the volume element to this volume. For a uniform field, the volume energy density is . For a flat capacitor, the volume of which is Sd, where S is the area of ​​the plates, d is the distance between the plates, we have

Considering that and

11) Dielectrics in an electric field. Dielectric polarization. Vectors of polarization and electric induction (electrical mixing). Dielectric constant and susceptibility

Dielectrics in an electric field

Unlike conductors, dielectrics do not have free charges. All charges are bound: electrons belong to their atoms, and ions of solid dielectrics oscillate

near the nodes of the crystal lattice.

Accordingly, when a dielectric is placed in an electric field, there is no directed movement of charges. Therefore, for dielectrics, our proofs of the properties of conductors do not pass - after all, all these arguments were based on the possibility of the appearance of a current. Indeed, none of the four properties of conductors formulated in the previous article apply to dielectrics.

2. The volume charge density in a dielectric can be different from zero.

3. Tension lines may not be perpendicular to the surface of the dielectric.

4. Different points of the dielectric may have different potentials. Therefore, talking about

"dielectric potential" is not necessary.

But nevertheless, dielectrics have one most important common property, and you know it

(Remember the formula for the field strength of a point charge in a dielectric!). tension

The field decreases inside the dielectric by a certain number of times compared with vacuum.

The value " is given in the tables and is called the permittivity of the dielectric.

Dielectric polarization

Polarization of dielectrics- a phenomenon associated with a limited displacement of bound charges in a dielectric or rotation of electric dipoles, usually under the influence of an external electric field, sometimes under the influence of other external forces or spontaneously.

The polarization of dielectrics is characterized by electric polarization vector. physical meaning of the electric polarization vector is the dipole moment per unit volume of the dielectric. Sometimes the polarization vector is briefly referred to as simply the polarization.

The polarization vector is applicable to describe the macroscopic state of polarization not only of ordinary dielectrics, but also of ferroelectrics, and, in principle, of any media with similar properties. It is applicable not only to describe induced polarization, but also spontaneous polarization (for ferroelectrics).

Polarization is the state of a dielectric, which is characterized by the presence of an electric dipole moment in any (or almost any) element of its volume.

A distinction is made between polarization induced in a dielectric under the action of an external electric field and spontaneous (spontaneous) polarization, which occurs in ferroelectrics in the absence of an external field. In some cases, the polarization of a dielectric (ferroelectric) occurs under the action of mechanical stresses, friction forces, or due to temperature changes.

Polarization does not change the total charge in any macroscopic volume inside a homogeneous dielectric. However, it is accompanied by the appearance on its surface of bound electric charges with a certain surface density σ. These bound charges create in the dielectric an additional macroscopic field with a strength of E 1 directed against an external field with a strength of E 0 . The resulting field strength E inside the dielectric E=E 0 -E 1 .

Vectors of polarization and electric induction (electric mixing)

Polarization vector- vector physical quantity, the dipole moment reduced by an external electric field to a unit volume of a substance, quantitatively the characteristics of the dielectric polarization.

Denoted by the letter, in SI it is measured in V / m.

electrical induction (electrical displacement) is a vector quantity equal to the sum of the electric field strength vector and the polarization vector.

Dielectric constant and susceptibility

Absolute permittivity- a physical quantity showing the dependence of electric induction on the strength of the electric field. AT foreign literature denoted by the letter ε, in domestic (where it usually denotes the relative permittivity), the combination is predominantly used, where is the electrical constant. This article uses .

Relative permittivity environment ε is a dimensionless physical quantity that characterizes the properties of an insulating (dielectric) medium. It is connected with the effect of polarization of dielectrics under the action of an electric field (and with the value of the dielectric susceptibility of the medium characterizing this effect). The value of ε shows how many times the force of interaction of two electric charges in a medium is less than in vacuum. The relative permittivity of air and most other gases under normal conditions is close to unity (because of their low density). For most solid or liquid dielectrics, the relative permittivity ranges from 2 to 8 (for a static field). The dielectric constant of water in a static field is quite high - about 80. Its values ​​are large for substances with molecules that have a large electric dipole. The relative permittivity of ferroelectrics is tens and hundreds of thousands.

Relative permittivity of a substance εr can be determined by comparing the capacitance of a test capacitor with a given dielectric (C x) and the capacitance of the same capacitor in vacuum (C o):

Dielectric susceptibility(or polarizability) substances - a physical quantity, a measure of the ability of a substance to polarize under the influence of an electric field. Dielectric susceptibility χ e- coefficient of linear connection between the polarization of the dielectric P and external electric field E in sufficiently small fields:

In the SI system:

where ε 0 - electrical constant; product ε 0 χ e called in SI system absolute dielectric susceptibility.

In case of vacuum

In dielectrics, as a rule, the dielectric susceptibility is positive. The dielectric susceptibility is a dimensionless quantity.

Polarizability is related to permittivity ε by the relation:

ε = 1 + 4πχ (CGS)

ε = 1 + χ (SI)

12) Constant electric current. Conditions for the existence of a current. Current strength. current density. Resistance. Conductivity. Ohm's and Joule-Lenz's laws in integral and differential form

Constant electric current.

Electricity- ordered uncompensated movement of free electrically charged particles, for example, under the influence of an electric field. Such particles can be: in conductors - electrons, in electrolytes - ions (cations and anions), in gases - ions and electrons, in vacuum under certain conditions - electrons, in semiconductors - electrons and holes (electron-hole conductivity). Historically, it is accepted that the direction of the current coincides with the direction of movement of positive charges in the conductor. D.C- current, the direction and magnitude of which varies slightly with time.

Conditions for the existence of a current.

For the occurrence and maintenance of current in any medium, two conditions must be met:
-the presence of free electric charges in the environment
- creation of an electric field in the environment. ( the presence of a power source. in which the conversion of any type of energy into the energy of an electric field is carried out.)
In different media, the carriers of electric current are different charged particles.

To maintain the current in the electric circuit, the charges, in addition to the Coulomb forces, must be affected by non-electrical forces (external forces).
A device that creates external forces, maintains a potential difference in a circuit and converts various types of energy into electrical energy, is called a current source.
For the existence of electric current in a closed circuit, it is necessary to include a current source in it.

Current strength. current density. Resistance. Conductivity.

1. Current strength - I, unit of measurement - 1 A (Ampere).
The current strength is a value equal to the charge flowing through the cross section of the conductor per unit time.
I = ∆q/∆t.
Formula (1) is valid for direct current, at which the current strength and its direction do not change with time. If the strength of the current and its direction change with time, then such a current is called variable.
For AC:
I = lim Δq/Δt , (*)
∆t -> 0
those. I = q’, where q’ is the derivative of the charge with respect to time.

2. Current density - j, unit of measurement - 1 A/m2.
The current density is the quantity equal to strength current flowing through a single cross section of the conductor:
j = I/S .

3. The electromotive force of the current source - emf. (ε), unit of measurement - 1 V (Volt). E.m.f. is a physical quantity equal to the work done by external forces when moving along an electric circuit of a single positive charge:
ε = Ast./q.

4. Conductor resistance - R, unit of measurement - 1 Ohm.
Under the action of an electric field in a vacuum, free charges would move at an accelerated rate. In matter, they move uniformly on average, because part of the energy is given to particles of matter in collisions.

The theory states that the energy of the ordered movement of charges is dissipated by the distortions of the crystal lattice. Based on the nature of electrical resistance, it follows that
R = ρ*l/S,
where
l - conductor length,
S - cross-sectional area,
ρ is a proportionality factor, called the resistivity of the material.
This formula is well confirmed by experience.
The interaction of conductor particles with charges moving in the current depends on the chaotic motion of particles, i.e. on the temperature of the conductor. It is known that
ρ = ρ0(1 + ∆t) ,
R = R0(1 + ∆t)

The coefficient k is called the temperature coefficient of resistance:
k = (R - R0)/R0*t .

For chemically pure metals K > 0 and equal to 1/273 K-1. For alloys, temperature coefficients are less important. The dependence r(t) for metals is linear:

In 1911, the phenomenon of superconductivity was discovered, which consists in the fact that at a temperature close to absolute zero, the resistance of some metals drops abruptly to zero.

For some substances (for example, electrolytes and semiconductors), the resistivity decreases with increasing temperature, which is explained by an increase in the concentration of free charges.
The reciprocal of the resistivity is called the electrical conductivity G
G = 1/ρ .

Ohm's and Joule-Lenz's laws in integral and differential form

Homogeneous section of the chain (e = 0):

Observations show that the current strength in the circuit section is directly proportional to the voltage (I ~ U) and inversely proportional to the resistance (I ~ 1/R). Consequently,

Formula (10) is Ohm's law for a homogeneous section of the chain.

The current-voltage characteristic has the form shown on the graph:

From formula (10) it follows that U = I*R. The product I*R is called the voltage drop.

When writing equations for direct current in metals, it follows that all time derivatives in Maxwell's equations are set equal to zero. Thus, the following equations are accepted as basic equations for direct current in metals:

Joule-Lenz law- a physical law that quantifies the thermal effect of an electric current. Established in 1841 by James Joule and independently in 1842 by Emil Lenz.

Mathematically it can be expressed in the following form:

where w- power of heat release per unit volume, - electric current density, - electric field strength, σ - conductivity of the medium.

The law can also be formulated in integral form for the case of current flow in thin wires:

The amount of heat released per unit time in the considered section of the circuit is proportional to the product of the square of the current strength in this section and the resistance of the section

In mathematical form, this law has the form

where dQ- the amount of heat released over a period of time dt, I- current strength, R- resistance, Q is the total amount of heat released during the time interval from t1 before t2. In the case of constant current and resistance.