Fundamentals of Maxwell's theory for electro magnetic field

§ 137. Vortex electric field

From Faraday's law (see (123.2))

ξ = dF/dt follows that any change

the flux of magnetic induction coupled to the circuit leads to the emergence of an electromotive force of induction and, as a result, an induction current appears. Therefore, the occurrence of emf. electromagnetic induction is also possible in a fixed circuit located in an alternating magnetic field. However, emf. in any circuit occurs only when external forces act on current carriers in it - forces of non-electrostatic origin (see § 97). Therefore, the question arises about the nature of extraneous forces in this case.

and is the cause of the induction current in the circuit. According to Maxwell's ideas, the circuit in which the emf appears plays a secondary role, being a kind of only "device" that detects this field.

So, according to Maxwell, a time-varying magnetic field generates an electric field E B, whose circulation, according to (123.3),

where E Bl - vector projection E B to direction d l.

Substituting into formula (137.1) the expression (see (120.2)), we get

If the surface and the contour are fixed, then the operations of differentiation and integration can be interchanged. Hence,

where the partial derivative symbol emphasizes the fact that the integral is

function only of time.

According to (83.3), the circulation of the electrostatic field strength vector (we denote it e q) along any closed contour is equal to zero:

Comparing expressions (137.1) and (137.3), we see that between the considered fields ( E B and e q) there is a fundamental difference: the circulation of the vector E B as opposed to vector circulation e q is not zero. Therefore, the electric field E B , excited by a magnetic field, like the magnetic field itself (see § 118), is vortex.

§ 138. Displacement current

Consider an alternating current circuit containing a capacitor (Fig. 196). There is an alternating electric field between the plates of a charging and discharging capacitor, therefore, according to Maxwell, through the capacitor

Displacement currents “flow”, and in those areas where there are no conductors.

Let's find a quantitative relationship between the changing electric field and the magnetic fields it causes. According to Maxwell, an alternating electric field in a capacitor at each moment of time creates such a magnetic field as if there was a conduction current between the plates of the capacitor, equal to the current in the supply wires. Then it can be argued that the conduction currents ( I) and offsets ( I cm) are equal to: I cm = I. Conduction current near capacitor plates

(surface charge density  on the plates is equal to the electrical displacement D in a condenser (see (92.1)). The integrand in (138.1) can be considered as a special case dot product (dD/d t)d S, when dD/d t and d S mutually parallel. Therefore, for the general case, we can write

Comparing this expression with I=I cm = (see (96.2)), we have

Expression (138.2) was named by Maxwell displacement current density.

Let us consider what is the direction of the vectors of the conduction and displacement current densities j and j see When charging a capacitor (Fig. 197, a) through the conductor connecting the plates, the current flows from the right plate to the left; the field in the capacitor increases, the vector D grows with time;

hence, dD/d t>0, i.e. vector dD/d t

is directed in the same direction as D. It can be seen from the figure that the directions of the vectors

dD/d t and j match. When the capacitor is discharged (Fig. 197, b) through the conductor connecting the plates, the current flows from the left plate to the right; the field in the capacitor is weakened, the vector D decreases with time; hence, dD/d tat

dD/d t is directed opposite to the vector

D. However, the vector dD/d t is directed again like this

same as vector j. From the analyzed examples it follows that the direction of the vector j, and hence the vector j cm matches

with vector direction dD/d t,

as follows from formula (138.2).

We emphasize that of all the physical properties inherent in the conduction current, Maxwell attributed only one to the displacement current - the ability to create a magnetic field in the surrounding space. Thus, the displacement current (in vacuum or matter) creates a magnetic field in the surrounding space (the lines of induction of magnetic fields of displacement currents during charging and discharging of the capacitor are shown in Fig. 197 by a dashed line).

In dielectrics, the displacement current consists of two terms. Since, according to (89.2), D= 0 E+P, where E is the strength of the electrostatic field, and R- polarization (see § 88), then the displacement current density

where  0 dE/d t - displacement current density

in a vacuumdP/d t - polarization current density- current due to ordered movement electric charges in a dielectric (displacement of charges in non-polar molecules or rotation of dipoles in polar molecules). The excitation of a magnetic field by polarization currents is legitimate, since polarization currents by their nature do not differ from conduction currents. However, what the other

( 0 dE/d t),

part of the bias current density ( 0 dE/d t),

not related to the movement of charges, but due to only change in the electric field over time, also excites a magnetic field, is fundamentally new statement Maxwell. Even in a vacuum, any change in time electric field leads to the formation of a magnetic field in the surrounding space.

It should be noted that the name "bias current" is conditional, or rather, historically established, since the bias current is essentially an electric field that changes with time. Displacement current therefore exists not only in vacuum or dielectrics, but also inside conductors through which alternating current flows. However, in this case, it is negligible compared to the conduction current. The presence of displacement currents was confirmed experimentally by the Soviet physicist A. A. Eikhenvald, who studied the magnetic field of the polarization current, which, as follows from (138.3), is part of the displacement current.

Maxwell introduced the concept full current, equal to the sum of conduction currents (as well as convection currents) and displacement. Total current density

j total =j+ dD/d t.

By introducing the concepts of displacement current and total current, Maxwell took a new approach to considering the closure of alternating current circuits. The full current in them is always closed,

i.e., only the conduction current breaks at the ends of the conductor, and in the dielectric (vacuum) between the ends of the conductor there is a displacement current that closes the conduction current.

Maxwell generalized the vector circulation theorem H(see (133.10)) by introducing into its right side the total current I full = through the surface S, stretched over a closed loop L. Then generalized circulation theorem for the vector H will be written in the form

Expression (138.4) is always true, evidence of which is the complete correspondence between theory and experience.

§ 139. Maxwell's equations for the electromagnetic field

The introduction of the concept of displacement current by Maxwell led him to the completion of the unified macroscopic theory of the electromagnetic field he created, which made it possible from a unified point of view not only to explain electrical and magnetic phenomena, but also to predict new ones, the existence of which was subsequently confirmed.

Maxwell's theory is based on the four equations discussed above:

1. The electric field (see § 137) can be both potential ( e q) and vortex ( E B), so the strength of the total field E=E Q+ E b. Since the circulation of the vector e q is equal to zero (see (137.3)), and the circulation of the vector E B is determined by expression (137.2), then the circulation of the total field strength vector

This equation shows that the sources of the electric field can be not only electric charges, but also time-varying magnetic fields.

2. Generalized vector circulation theorem H(see (138.4)):

This equation shows that magnetic fields can be excited either by moving charges (electric currents) or by alternating electric fields.

3. Gauss's theorem for the field D(see (89.3)):

If the charge is distributed continuously inside a closed surface with a bulk density , then formula (139.1) will be written as

4. Gauss's theorem for the field B (see (120.3)):

So, complete system of Maxwell equations in integral form:

The quantities included in the Maxwell equations are not independent and there is the following relationship between them (isotropic non-ferroelectric and non-ferromagnetic media):

D= 0 E,

B= 0 H,

j=E,

where  0 and  0 are respectively electric and magnetic constants,  and  - respectively dielectric and magnetic permeability,  - specific conductivity of the substance.

It follows from Maxwell's equations that the sources of an electric field can be either electric charges or time-varying magnetic fields, and magnetic fields can be excited either by moving electric charges (electric currents) or by alternating electric fields. Maxwell's 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.

For stationary fields (E= const and AT= const) Maxwell's equations take the form

i.e., the sources of the electric field in this case are only electric charges, the sources of the magnetic field are only conduction currents. In this case, the electric and magnetic fields are independent of each other, which makes it possible to study separately permanent electric and magnetic fields.

Using the Stokes and Gauss theorems known from vector analysis

can be imagined the complete system of Maxwell's equations in differential form (characterizing the field at each point in space):

If charges and currents are distributed in space continuously, then both forms of Maxwell's equations are integral

and differential are equivalent. However, when there are fracture surfaces- surfaces on which the properties of the medium or fields change abruptly, then the integral form of the equations is more general.

Maxwell's equations in differential form assume that all quantities in space and time change continuously. To achieve mathematical equivalence of both forms of Maxwell's equations, the differential form is supplemented boundary conditions, which must be satisfied by the electromagnetic field at the interface between two media. The integral form of Maxwell's equations contains these conditions. They have been considered before (see § 90, 134):

D 1 n =D 2 n , E 1  =E 2  , B 1 n =B 2n , H 1  = H 2 

(the first and last equations correspond to cases where there are neither free charges nor conduction currents at the interface).

Maxwell's equations are the most general equations for electric and magnetic fields in resting environments. They play the same role in the theory of electromagnetism as Newton's laws in mechanics. It follows from Maxwell's equations that an alternating magnetic field is always associated with the electric field generated by it, and an alternating electric field is always associated with the magnetic field generated by it, that is, the electric and magnetic fields are inextricably linked with each other - they form a single electromagnetic field.

that the propagation speed of a free electromagnetic field (not related to charges and currents) in vacuum is equal to the speed of light c = 3 10 8 m/s. This conclusion and the theoretical study of the properties of electromagnetic waves led Maxwell to create the electromagnetic theory of light, according to which light is also electromagnetic waves. Electromagnetic waves were experimentally obtained by the German physicist G. Hertz (1857-1894), who proved that the laws of their excitation and propagation are completely described by Maxwell's equations. Thus, Maxwell's theory was experimentally confirmed.

Only Einstein's principle of relativity is applicable to the electromagnetic field, since the fact that electromagnetic waves propagate in vacuum in all frames of reference with the same speed with incompatible with Galileo's principle of relativity.

According to Einstein's principle of relativity mechanical, optical and electromagnetic phenomena in all inertial frames of reference proceed in the same way, that is, they are described by the same equations. Maxwell's equations are invariant under Lorentz transformations: their form does not change when passing

from one inertial frame of reference to another, although the quantities E, V,D,H they are converted according to certain rules.

It follows from the principle of relativity that a separate consideration of electric and magnetic fields has a relative meaning. So, if the electric field is created by a system of fixed charges, then these charges, being fixed with respect to one inertial frame of reference, move relative to another and, therefore, will generate not only an electric, but also a magnetic field. Similarly, a fixed-current conductor with respect to one inertial frame of reference, exciting a constant magnetic field at each point in space, moves relative to other inertial frames, and the alternating magnetic field created by it excites a vortex electric field.

Thus, Maxwell's theory, its experimental confirmation, as well as Einstein's principle of relativity lead to a unified theory of electrical, magnetic and optical phenomena based on the idea of an electromagnetic field.

test questions

What is the cause of the vortex electric field? How is it different from an electrostatic field?

What is the circulation of the vortex electric field?

Why is the concept of displacement current introduced? What does he essentially represent?

Derive and explain the expression for the bias current density.

In what sense is it possible to compare the displacement current and the conduction current?

Write down, explaining the physical meaning, a generalized theorem on the circulation of the magnetic field strength vector.

Write down the complete system of Maxwell's equations in integral and differential forms and explain their physical meaning.

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Ministry of Education of the Russian Federation

St. Petersburg Institute of Mechanical Engineering

Referatin Physics

on the topic:

"The Essence of Maxwell's Electromagnetic Theory"

Performed:

student gr. 2801

Shkeneva Yu.A.

St. Petersburg

Introduction 3

Vortex electric field 6

Bias current 7

Maxwell's equation for an electromagnetic field 9

References 13

Introduction

James Clerk Maxwell was born June 13, 1831. in Edinburgh, in the family of a lawyer - the owner of an estate in Scotland. The boy showed early love for technology and the desire to comprehend the world. His father had a great influence on him - a highly educated person who was deeply interested in the problems of natural science and technology. At school, Maxwell was fascinated by geometry, and his first scientific work, completed at the age of fifteen, was the discovery of a simple but unknown way of drawing oval figures. Maxwell received a good education, first at Edinburgh and then at Cambridge Universities.

In 1856, a young, promising scientist was invited to teach as a professor at a college in the Scottish city of Aberdeen. Here Maxwell enthusiastically works on the problems of theoretical and applied mechanics, optics, physiology of color vision. He brilliantly solves the riddle of Saturn's rings by proving mathematically that they are formed from individual particles. The name of the scientist becomes known, and he is invited to take a chair at King's College in London. The London period (1860-1865) was the most fruitful in the life of a scientist. He resumes and brings to completion theoretical research in electrodynamics, publishes fundamental work on the kinetic theory of gases.

After moving from Aberdeen, Maxwell continued his research with unrelenting intensity, paying particular attention to the kinetic theory of gases. It is said that his wife (the former Katherine Mary Dewar, daughter of the head of Marischal College) made a fire in the basement of their London house in order to enable Maxwell to conduct experiments on the thermal properties of gases in the attic. But the decisive and certainly the greatest achievement of Maxwell was the creation of his electromagnetic theory.

The early nineteenth century was full of exciting discoveries. Shortly after receiving the first stationary currents, Oersted showed that the current flowing through the conductor generates magnetic effects similar to those caused by an ordinary permanent magnet. Therefore, the assumption was made that two conductors with current should behave like two magnets, which, as you know, can either attract or repel. Indeed, the experiments of Ampère and other researchers confirmed the presence of forces of attraction or repulsion between two current-carrying conductors. Soon it was possible to formulate the law of attraction and repulsion with the same accuracy with which Newton formulated the law of gravitational attraction between any two material bodies.

Then Faraday and Henry discovered the remarkable phenomenon of electromagnetic induction and thus demonstrated the close relationship between magnetism and electricity.

However, there was an urgent need to create a unified theory that would meet the necessary requirements, which would make it possible to predict the development of electromagnetic phenomena in time and space in the most general case, under any conceivable specific experimental conditions.

This is exactly what Maxwell's electromagnetic theory turned out to be, formulated by him in the form of a system of several equations that describe the entire variety of properties of electromagnetic fields using two physical quantities - the electric field strength E and the magnetic field strength H. It is remarkable that these Maxwell equations in their final form and to this day, they remain the cornerstone of physics, giving a description of the observed electromagnetic phenomena that corresponds to reality.

When designing a high-voltage line for transmission of electricity over long distances, Maxwell's equations help to create a system that ensures a minimum of losses; when conducting fundamental experiments in the laboratory to study the properties of metals in a high-frequency electric field at very low temperatures, we use Maxwell's equations to determine the nature of the propagation of an electromagnetic field inside a metal; if we are building a new radio telescope capable of capturing the electromagnetic noise of space, then when designing antennas and waveguides that transmit energy from the antenna to the radio receiver, we invariably use Maxwell's equations.

There is a law according to which the force acting on a charge moving in a magnetic field is directly proportional to the product of the magnitude of the charge and the velocity component perpendicular to the direction of the magnetic field; this force is known to us as the "Lorentz force". However, someone calls it the "Laplace force".

With regard to Maxwell's equations, there is no such uncertainty; the honor of this discovery belongs to him alone.

It should be noted that in the last century he was by no means the only physicist who tried to create a comprehensive theory of electromagnetism; others also, not without reason, suspected the existence of a deep connection between light and electrical phenomena.

Maxwell's main merit is that he, in his own way, came to an elegant and simple system of equations that describes all electromagnetic phenomena.

Maxwell's equations not only cover and describe all electromagnetic phenomena known to us; the scope of their application is not limited even by any conceivable electromagnetic phenomena occurring in specific local conditions. Maxwell's theory predicted a completely new effect observed in a space free from material bodies - electromagnetic radiation. This is certainly a unique achievement, crowning the triumph of Maxwell's theory.

Vortex electric field

From Faraday's law e i = - d F / dt it follows that any change in the flux of magnetic induction coupled to the circuit leads to the emergence of an electromotive force of induction and, as a result, an induction current appears. Therefore, the occurrence of emf. electromagnetic induction is also possible in a fixed circuit located in an alternating magnetic field. However, emf. in any circuit occurs only when external forces act on current carriers in it - forces of non-electrostatic origin.

Experience shows that these extraneous forces are not associated with either thermal or chemical processes in the circuit; their occurrence also cannot be explained by Lorentz forces, since they do not act on immovable charges. Maxwell, hypothesized that any alternating magnetic field excites an electric field in the surrounding space, which is the cause of the induction current in the circuit. According to Maxwell's ideas, the circuit in which the emf appears plays a secondary role, being a kind of only “device” that detects this field.

So, according to Maxwell, a time-varying magnetic field generates an electric field E B , whose circulation, according to the formula,

E B dl = E Bl dl = - d F/dt (1)

where, the projection of the vector E Bl is the projection of the vector E onto the direction dl ; the partial derivative ¶Ф/¶t takes into account the dependence of the magnetic induction flux only on time.

Substituting the expression Ф = B dS into this formula (1), we obtain

E B dl = - ¶ / ¶ t B dS

Since the contour and the surface are fixed, the operations of differentiation and integration can be interchanged. Hence,

E B dl = - ¶ B/ ¶ t dS (2)

According to E dl \u003d E l dl \u003d 0, the circulation of the electrostatic field strength vector (let's denote it E Q ) along a closed loop is zero:

E Q dl = E Ql dl = 0 (3)

Comparing expressions (1) and (3), we see that there is a fundamental difference between the considered fields (E B and E Q ): the circulation of the vector E B, unlike the circulation of the vector E Q, is not equal to zero. Therefore, the electric field E B excited by the magnetic field, like the magnetic field itself, is vortex.

Bias current

According to Maxwell, if any alternating magnetic field excites a vortex electric field in the surrounding space, then the opposite phenomenon must also exist: any change in the electric field must cause the appearance of a vortex magnetic field in the surrounding space. Since the magnetic field is always associated with an electric current, Maxwell called the alternating electric field that excites the magnetic field, the displacement current, in contrast to the conduction current due to the ordered movement of charges. For the occurrence of a displacement current, according to Maxwell, only the existence of an alternating electric field is necessary.

Consider an AC circuit containing a capacitor (Fig. 1). There is an alternating electric field between the plates of a charging and discharging capacitor, therefore, according to Maxwell, displacement currents “flow” through the capacitor, and in those areas where there are no conductors. Therefore, since there is an alternating electric field (bias current) between the capacitor plates, a magnetic field is also excited between them.

Let's find a quantitative relationship between the changing electric and magnetic fields caused by it. According to Maxwell, an alternating electric field in a capacitor at each moment of time creates such a magnetic field, as if there were a conduction current between the plates of the capacitor with a force that is equal to the strength of the currents in the supply wires. Then it can be argued that the conduction current densities (j) and displacements (j cm) are equal: j cm = j.

Conduction current density near the capacitor plates j = = = ()= d s /dt , s is the surface charge density, S is the area of the capacitor plates. Therefore, j cm = d s /dt (4). If the electrical displacement in the capacitor is D, then the surface charge density on the plates is s = D. Considering this, expression (4) can be written as: j cm = ¶ D /¶ t , where the sign of the partial derivative indicates that the magnetic field is determined only by the rate of change of the electric displacement in time.

Since the displacement current occurs with any change in the electric field, it exists not only in vacuum or dielectrics, but also inside the conductors through which the alternating current flows. However, in this case, it is negligible compared to the conduction current. The presence of displacement currents was confirmed experimentally by the Soviet physicist A. A. Eikhenvald, who studied the magnetic field of the polarization current, which is part of the displacement current.

In the general case, the conduction currents and displacements in space are not separated, they are in the same volume. Maxwell therefore introduced the concept of total current, which is equal to the sum of conduction currents (as well as convection currents) and displacement. Total current density:

j total = j + ¶ D /¶ t .

Introducing the concept of displacement current and total current, Maxwell took a new approach to considering the closure of alternating current circuits. The total current in them is always closed, that is, only the conduction current breaks at the ends of the conductor, and in the dielectric (vacuum) between the ends of the conductor there is a displacement current that closes the conduction current.

Maxwell generalized the theorem on the circulation of the vector H , introducing into its right side the total current I total = j total dS , covered by a closed loop L . Then the generalized circulation theorem for the vector H can be written as:

H dl = (j + ¶ D/ ¶ t) dS (5)

Expression (5) is always true, which is evidenced by the complete correspondence between theory and experience.

Maxwell's equation for an electromagnetic field

Maxwell's theory is based on the four equations discussed above:

The electric field can be both potential (E Q ) and vortex (E B ), so the total field strength E = E Q + E B . Since the circulation of the vector E Q is equal to zero, and the circulation of the vector E B is determined by expression (2), then the circulation of the total field strength vector

E dl = - ¶B/¶t dS.

This equation shows that the source of the electric field can be not only electric charges, but also time-varying magnetic fields.

Generalized circulation theorem for the vector H:

H dl = (j + ¶D/¶t) dS.

This equation shows that magnetic fields can be excited either by moving charges (electric currents) or by alternating electric fields.

Gauss's theorem for an electrostatic field in a dielectric:

If the charge is distributed continuously inside a closed surface with bulk density ρ, then formula (6) will be written as:

D dS = ρ dV.

Gauss's theorem for the field B :

B dS = 0.

So, the complete system of Maxwell's equations in integral form:

E dl = - ¶ B/ ¶ t dS; D dS = ρ dV;

H dl = (j + ¶D/¶t) dS; B dS = 0.

The quantities included in Maxwell's equations are not independent and there is the following relationship between them:

B = m 0 mH;

J = g E ;

where e 0 and m 0 are the electrical and magnetic constants, respectively, e and m are the dielectric and magnetic permeabilities, respectively, g is the specific conductivity of the substance.

It follows from the Maxwell equation that the sources of an electric field can be either electric charges or time-varying magnetic fields, and magnetic fields can be excited either by moving electric charges (electric currents) or by alternating electric fields. Maxwell's 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.

For stationary fields (E = const and B = const ) Maxwell's equations take the form:

E dl = 0; D dS = Q;

H dl = I; B dS = 0.

In this case, the electric and magnetic fields are independent of each other, which makes it possible to study separately the constant electric and magnetic fields.

Using the Stokes and Gauss theorems known from vector analysis:

A dl = rot A dS;

A dS = div A dV,

it is possible to represent the complete system of Maxwell's equations in differential form:

rot E = - ¶ B/ ¶ t; div D = p;

rot H = j + ¶ D/ ¶ t; div B = 0.

If charges and currents are continuously distributed in space, then both forms of Maxwell's equations - integral and differential - are equivalent. However, when there are discontinuity surfaces - surfaces on which the properties of the medium or fields change abruptly, then the integral form of the equations is more general.

Maxwell's equations are the most general equations for electric and magnetic fields in media at rest. They play the same role in the theory of electromagnetism as Newton's laws in mechanics. It follows from Maxwell's equations that an alternating magnetic field is always associated with the electric field generated by it, and an alternating electric field is always associated with the magnetic field generated by it, i.e. Electric and magnetic fields are inextricably linked with each other - they form a single electromagnetic field.

Maxwell's theory is macroscopic, as it considers electric and magnetic fields created by macroscopic charges and currents. Therefore, this theory could not reveal the internal mechanism of phenomena that occur in the environment and lead to the emergence of electric and magnetic fields. A further development of Maxwell's theory of the electromagnetic field was the electronic theory of Lorentz, and the Maxwell-Lorentz theory was further developed in quantum physics.

Maxwell's theory, being a generalization of the basic laws of electrical and magnetic phenomena, was able to explain not only already known experimental facts, which is also an important consequence of it, but also predicted new phenomena. One of the important conclusions of this theory was the existence of a magnetic field of displacement currents, the existence of electromagnetic waves - an alternating electromagnetic field propagating in space at a finite speed. Later it was proved that the speed of propagation of a free electromagnetic field (not connected by currents) in vacuum is equal to the speed of light c = 3 · 10 8 m/s. This conclusion and the theoretical study of the properties of electromagnetic waves led Maxwell to create the electromagnetic theory of light, according to which light is also electromagnetic waves. Electromagnetic waves were experimentally obtained by G. Hertz (1857 - 1894), who proved that the laws of their excitation and propagation are completely described by Maxwell's equations. Thus, Maxwell's theory received brilliant experimental confirmation.

Later, A. Einstein established that Galileo's principle of relativity for mechanical phenomena extends to all other physical phenomena.

According to Einstein's principle of relativity, mechanical, optical and electromagnetic phenomena proceed in the same way in all inertial frames of reference, i.e. described by the same equations. It follows from this principle that a separate consideration of electric and magnetic fields has a relative meaning. So, if the electric field is created by a system of fixed charges, then these charges, being fixed with respect to one inertial frame of reference, move relative to another and, therefore, will generate not only an electric, but also a magnetic field. Similarly, a conductor with direct current that is motionless with respect to one inertial frame of reference, exciting a constant magnetic field at each point in space, moves relative to other inertial frames, and the alternating magnetic field created by it excites a vortex electric field.

Bibliography

P. S. Kudryavtsev. "Maxwell", M., 1976

D. McDonald. "Faraday", Maxwell and Kelvin", M., 1967

T. I. Trofimova. "Course of Physics", M., 1983

G.M. Golin, S.R. Filonovich. Classics of physical science. " graduate School". M., 1989.

3. Free vibrations in the lc-circuit. Free damped vibrations. The differential equation of damped oscillations and its solution.

4. Forced electrical oscillations. The differential equation of forced oscillations and its solution.

5. Voltage resonance and current resonance.

Fundamentals of Maxwell's theory for the electromagnetic field.

6. General characteristics of Maxwell's theory. Vortex magnetic field. bias current.

7. Maxwell's equations in integral form.

Electromagnetic waves

8.Experimental production of electromagnetic waves. Plane electromagnetic wave. Wave equation for electromagnetic field. Energy of electromagnetic waves. Pressure of electromagnetic waves.

geometric optics

9. Basic laws of geometric optics. Photometric quantities and their units.

10. Refraction of light on spherical surfaces. Thin lenses. The thin lens formula and the construction of images of objects using a thin lens.

11. Light waves

12. Interference of light upon reflection from thin plates. Stripes of equal thickness and equal slope.

13. Newton's rings. Application of the phenomenon of interference. Interferometers. Illumination of optics.

14. Diffraction of light

15. Diffraction of light on a round screen and a round hole.

16. Diffraction of light by one slit. Diffraction grating.

17. 18. Interaction of light with matter. Dispersion and absorption of light. Normal and anomalous dispersion. Bouguer-Lambert law.

19. Polarization of light. Natural and polarized light. The degree of polarization. Small law.

20. Polarization of light during reflection and refraction. Brewster's law. Double refraction. Anisotropy of crystals.

21. Doppler effect for light waves.

22. Thermal radiation. Properties of equilibrium thermal radiation. Completely black body. Distribution of energy in the spectrum of a completely black body. Laws of Kirchhoff, Stefan-Boltzmann, Wien.

23. Elements of the special theory of relativity Postulates of the special theory of relativity. Lorentz transformations.

2. Duration of events in different frames of reference.

24. Basic laws of relativistic dynamics. Law of interrelation of mass and energy.

Fundamentals of Maxwell's theory for the electromagnetic field.

6. General characteristics of Maxwell's theory. Vortex magnetic field. bias current.

7. Maxwell's equations in integral form.

The fundamental equations of classical macroscopic electrodynamics describing electromagnetic phenomena in any medium (including vacuum) were obtained in the 60s. 19th century by J. Maxwell on the basis of a generalization of the empirical laws of electrical and magnetic phenomena and the development of the idea of English. scientist M. Faraday that the interaction between electrically charged bodies is carried out by means of an electromagnetic field.

Maxwell's theory for the electromagnetic field connects the quantities characterizing the electromagnetic field with its sources, i.e. distribution in space of electric charges and currents.

Consider the case of electromagnetic induction. From Faraday's Law

E in = - ∂Ф m /∂t (1)

follows that any a change in the flux of magnetic induction coupled to the circuit leads to the emergence of an electromotive force of induction and the appearance of an inductive current as a result. Maxwell hypothesized that any alternating magnetic field excites an electric field in the surrounding space, which is the cause of the induction current in the circuit. According to Maxwell's ideas, the circuit in which the emf appears plays a secondary role, being a kind of only "device" that detects this field.

Maxwell's first equation in integral form. According to the definition, emf. is equal to the circulation of the electric field strength vector E:

E = ∫E d l , (2)

which is equal to zero for the potential field. In the general case of a changing vortex field for E in we get

∫E· d l = - dФ m /dt = -∫(∂ B/∂t) d S. (3)

(3) – Maxwell's first equation: the circulation of the electric field strength vector along an arbitrary closed contour L is equal to the rate of change of the flux of the magnetic induction vector through the surface bounded by this contour, taken with the opposite sign. The sign "-" corresponds to the Lenz rule for the direction of the induction current. Hence it follows that alternating magnetic field creates in space vortex electric field regardless of whether the conductor is in this field (closed conducting circuit) or not. Equation (3) thus obtained is a generalization of equation (2), which is valid only for a potential field, i.e. electrostatic field.

Displacement current and Maxwell's second equation in integral form. Maxwell hypothesized that the magnetic field is generated not only by electric currents flowing in a conductor, but also by alternating electric fields in dielectrics or vacuum. To establish quantitative relationships between a changing electric field and the magnetic field caused by it, Maxwell introduced the so-called bias current.

Consider an AC circuit containing a capacitor. Between

the plates of the charging and discharging capacitor have an alternating electric field, therefore, according to Maxwell, bias currents “flow” through the capacitor, and in those areas where there are no conductors, and I \u003d I cm \u003d ∫ j cm dS. (*)

The conduction current near the capacitor plates can be written as

I = dq/dt = (d/dt)∫σ dS = ∫(∂σ/∂t)dS = ∫(∂D/∂t)dS (4)

(the surface charge density σ on the capacitor plates is equal to the electrical displacement D in the capacitor). The integrand in (4) can be considered as a special case of the scalar product (∂ D/∂t)dS when (∂ D/∂t) and d S mutually parallel. Therefore, for the general case, we can write

I = ∫(∂ D/∂t)dS.

Comparing this expression with (*), we have

j cm = ∂ D/ ∂t. (5)

Expression (5) Maxwell called bias current density. Direction of the current density vector j and j cm coincides with the direction of the vector ∂ D/∂t. The displacement current excites the magnetic field according to the same law as the conduction current.

In dielectrics, the displacement current consists of two terms. Since in a dielectric D = ε 0 E + P, where E is the electric field strength, and R is the polarization, then the displacement current density

j cm = ε 0 ∂ E/ d∂t + ∂ P/∂t, (6)

where ε 0 ∂ E/ ∂t – displacement current density in vacuum(not related to the movement of charges, but due only to a change in the electric field in time, also excites a magnetic field, is a fundamentally new statement by Maxwell), ∂ P/∂t – polarization current density- current due to the ordered movement of electric charges in the dielectric (displacement of charges in non-polar molecules or rotation of dipoles in polar molecules).

Maxwell introduced the concept full current. full current, equal to the sum displacement current and conduction current, is always closed.

j full = j+ ∂D/∂t. (7)

Maxwell generalized the vector circulation theorem H, introducing into its right side the total current

∫H d l =∫(j + ∂D/d∂t)d S-(8)

Maxwell's second equation: tension vector circulation H magnetic field along any closed loop L is equal to the total conduction current that permeates the surface S stretched over this loop, added with the rate of change in the flow of the electric induction vector D through this surface.

I repeat that alternating magnetic field may be excited moving charges(electric currents) and alternating electric field(bias current).

The third and fourth equations of Maxwell. Maxwell's third equation expresses experimental data on the absence of magnetic charges similar to electric ones (the magnetic field is generated only by electric currents), i.e. the Gauss theorem turned out to be valid not only for electro- and magnetostatic fields, but also for a time-varying vortex electromagnetic field:

∫D d S= q, (9)

∫B d S = 0. (10)

Maxwell's 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 quantities included in Maxwell's equation are not independent and between them exist. following link:

D = D(E), B= B(H), j= j( E). (11)

These equations are called equations of state or material equations, they describe the electromagnetic properties of the medium and for each particular medium have a certain form.

Maxwell's integral equations describe the medium phenomenologically, without considering the complex mechanism of interaction of the electromagnetic field with the charged particles of the medium.

From Maxwell's integral equations (3), (8-10) it is possible to pass to the system of differential equations. Four fundamental ur. Maxwell in integral or differential forms do not form a complete closed system that allows you to calculate electromagnetic processes in the presence of a material environment. They must be supplemented with relations connecting the vectors E, H, D, B and j, which are not independent. The connection between them is determined by the properties of the environment and its state. The electromagnetic properties of the medium are determined by equations, which in the general case are very complex, but in the case of an isotropic homogeneous conducting non-ferromagnetic and non-ferroelectric medium they have the form

D = εε 0 E, B= μμ 0 H, j = γ E. (12)

Equations (3), (8-10) and (12) form a complete system of equations for the electromagnetic field in a medium, the solution of which, under given boundary conditions, allows us to determine the vectors E, H, D, B and j and scalar ρ (distribution density of electric charges in space) at each point of the medium with its given characteristics ε, μ, σ.

Maxwell's equations are the most general equations for electric and magnetic fields in resting environments. It follows from Maxwell's equations that alternating magnetic field is always associated with the electric field generated by it, and the alternating electric field is always associated with its magnetic field, i.e. electric and magnetic fields are inextricably linked with each other - they form a single electromagnetic field. Statics, E = const, B = const. !!!

Maxwell's theory was not only able to explain already known experimental facts, but also predicted new phenomena. One of the important conclusions of this theory was the existence of a magnetic field of displacement currents, which allowed Maxwell to predict the existence electromagnetic waves– an alternating electromagnetic field propagating in space with a finite speed. This led Maxwell to create the electromagnetic theory of light.

Maxwell's equations describe a huge area of phenomena. They form the basis of electrical and radio engineering and play important role in the development of such topical areas of modern physics as plasma physics and the problem of controlled thermonuclear fusion, magnetohydrodynamics, nonlinear optics, astrophysics, etc.

Maxwell's equations are inapplicable only at high frequencies of electromagnetic waves, when quantum effects become significant, i.e. when the energy of individual quanta of the electromagnetic field - photons - is large and a small number of photons participate in the processes.

Topic: Electromagnetic induction

Lesson: Electromagneticfield.TheoryMaxwell

Consider the above diagram and the case when a DC source is connected (Fig. 1).

Rice. 1. Scheme

The main elements of the circuit include a light bulb, an ordinary conductor, a capacitor - when the circuit is closed, a voltage arises on the capacitor plates equal to the voltage at the source terminals.

A capacitor consists of two parallel metal plates with a dielectric in between. When a potential difference is applied to the capacitor plates, they are charged, and an electrostatic field arises inside the dielectric. In this case, there can be no current inside the dielectric at low voltages.

When replacing direct current with alternating current, the properties of the dielectrics in the capacitor do not change, and there are still practically no free charges in the dielectric, but we observe that the light bulb is on. The question arises: what is happening? Maxwell called the current arising in this case the displacement current.

We know that when a current-carrying circuit is placed in an alternating magnetic field, an EMF of induction arises in it. This is due to the fact that a vortex electric field arises.

But what if a similar picture occurs when the electric field changes?

Maxwell's hypothesis: the time-varying electric field causes the appearance of a vortex magnetic field.

According to this hypothesis, the magnetic field after the circuit is closed is formed not only due to the current flow in the conductor, but also due to the presence of an alternating electric field between the capacitor plates. This alternating electric field generates a magnetic field in the same area between the capacitor plates. Moreover, this magnetic field is exactly the same, as if a current flowed between the plates of the capacitor, equal to the current in the rest of the circuit. The theory is based on four Maxwell equations, from which it follows that the change in electric and magnetic fields in space and time occur in a consistent manner. Thus, the electric and magnetic fields form a single whole. Electromagnetic waves propagate in space in the form of transverse waves with a finite speed.

The indicated relationship between an alternating magnetic and an alternating electric field suggests that they cannot exist separately from each other. The question arises: does this statement apply to static fields (electrostatic, created by constant charges, and magnetostatic, created by direct currents)? This relationship also exists for static fields. But it is important to understand that these fields can exist in relation to a certain frame of reference.

A charge at rest creates an electrostatic field in space (Fig. 2) relative to a certain reference frame. Relative to other reference systems, it can move and, therefore, in these systems the same charge will create a magnetic field.

Electromagnetic field- this is a special form of the existence of matter, which is created by charged bodies and manifests itself by the action on charged bodies. During this action, their energy state can change, therefore, the electromagnetic field has energy.

1. The study of the phenomena of electromagnetic induction leads to the conclusion that an alternating magnetic field generates a vortex electric field around itself.

2. Analyzing the passage of alternating current through circuits containing dielectrics, Maxwell came to the conclusion that an alternating electric field can generate a magnetic field due to the displacement current.

3. Electric and magnetic fields are components of a single electromagnetic field that propagates in space in the form of transverse waves with a finite speed.

Bukhovtsev B.B., Myakishev G.Ya., Charugin V.M. Physics Grade 11: Textbook. for general education institutions. - 17th ed., Converted. and additional - M.: Education, 2008.
Gendenstein L.E., Dick Yu.I., Physics 11. - M .: Mnemosyne.
Tikhomirova S.A., Yarovsky B.M., Physics 11. - M.: Mnemosyne.

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What electric field is formed when the magnetic field changes?
What current causes the glow of a light bulb in an alternating current circuit with a capacitor?
Which of Maxwell's equations indicates the dependence of magnetic induction on conduction current and displacement?

The basic laws of electrical and magnetic phenomena are a generalization of experimental facts. At the same time, they described separately electrical and magnetic phenomena. In the 60s of the last century, Maxwell, based on Faraday's ideas about electric and magnetic fields, generalized these laws and developed a complete theory of a unified electromagnetic field.

Maxwell's theory is a macroscopic theory. It considers electric and magnetic fields created by macroscopic charges and currents without taking into account the internal mechanisms associated with vibrations of atoms or electrons. Therefore, the distances from the sources of the fields to the considered points of space are assumed to be much larger compared to the sizes of the molecules. In addition, the frequency of oscillations of electric and magnetic fields in this theory is taken to be much lower than the frequency of intramolecular oscillations. In Maxwell's works, Faraday's idea of a close connection between electrical and magnetic phenomena was finally formalized in the form of two main provisions and was expressed in strict form in the form of Maxwell's equations (1873).

The main achievements of Maxwell's theory are substantiation of the idea that:

- an alternating electric field excites a vortex magnetic field;
- an alternating magnetic field excites a vortex electric field.

Bias current

Analyzing various electromagnetic processes, Maxwell came to the conclusion that any change in the electric field should cause the appearance of a magnetic field. This statement is one of the main provisions of Maxwell's theory and expresses the most important property of the electromagnetic field.

Consider the following experiment: we place a dielectric between the plates of a flat capacitor charged with a surface charge density.

Connect the capacitor plates with an external conductor. Since there is a potential difference between the capacitor plates, a current will flow through the conductor:. At the boundaries of the plates, the streamlines are perpendicular to their surfaces and the current density is equal to:

(2) if, then.

Taking into account formula (1), we obtain the formula for the conduction current density

As the capacitor discharges, the electric field in it weakens. Therefore, the derivative of induction will have a negative sign, and the vector will be directed oppositely. Those. the direction of the vector will coincide with the direction of the current density vector. Therefore, formula (3) can be written in vector form:

The left side of equation (4) characterizes the electric conduction current, and the right side characterizes the rate of change of the electric field in the dielectric. The equality of these two vectors at the metal-dielectric interface shows that the lines of the vector, as it were, continue the current lines through the dielectric and close the current. Therefore, the derivative of electric induction with respect to time is called by Maxwell the displacement current density

So, in the considered experiment, the conduction current passes in the dielectric into a displacement current (i.e., into a changing electric field).

If we use the formula for the relationship between induction, intensity and polarization P of a substance, then the following formula can be obtained for the displacement current density:

The first term on the right side of formula (6) determines the variable field of free charges (an alternating electric field in vacuum). The second term is the rate of change in the polarization of the dielectric with time, associated with the displacement of its charges when the field strength changes. The movement of charges in an electric field within molecular dimensions is ordered and is called the polarization component of the displacement current. This explains the origin of the term displacement current - the current due to the displacement of charges in a dielectric placed in an alternating electric field.

When the polarization is reversed, the molecules "turn" behind the changing field and collide with neighboring molecules. As a result of such collisions, the dielectric heats up. That. the displacement current can be registered by its thermal action. Also, like any current, the displacement current creates a magnetic field. Direct observation of the magnetic field generated by the displacement current was carried out by the Russian scientist Eichenwald.

In his experiment, a dielectric disk was placed between the plates of two flat capacitors, and rotated around an axis. The capacitor plates were connected to a voltage source so that the halves of the dielectric were polarized in opposite directions. With each revolution of the disk, the direction of polarization of each of the parts is reversed. As a result of such repolarization of the dielectric during its rotation, a polarization current arises in it, directed parallel to the axis of rotation. The magnetic field of this current was detected from the deflection of a magnetic needle placed near the axis of the disk.