Application of electromagnetic induction in life. What determines the inductive electric current? Modern theory of electromagnetic induction

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INTRODUCTION

It is no coincidence that the first and most important step in the discovery of this new side of electromagnetic interactions was made by the founder of the ideas about the electromagnetic field - one of the greatest scientists in the world - Michael Faraday (1791-1867). Faraday was absolutely sure of the unity of electric and magnetic phenomena. Shortly after Oersted's discovery, he wrote in his diary (1821): "Turn magnetism into electricity." Since then, Faraday, without ceasing, thought about this problem. They say that he constantly carried a magnet in his vest pocket, which was supposed to remind him of the task at hand. Ten years later, in 1831, as a result of hard work and faith in success, the problem was solved. He made a discovery that underlies the design of all generators of power plants in the world, converting mechanical energy into electric current energy. Other sources: galvanic cells, thermo- and photocells provide a negligible share of the generated energy.

Electric current, Faraday reasoned, is capable of magnetizing iron objects. To do this, just put an iron bar inside the coil. Could the magnet, in turn, cause the appearance of an electric current or change its magnitude? For a long time nothing could be found.

HISTORY OF THE DISCOVERY OF THE PHENOMENON OF ELECTROMAGNETIC INDUCTION

Sayings of Signors Nobili and Antinori from the magazine "Antologia"

« Mr. Faraday has recently discovered a new class of electrodynamic phenomena. He submitted a memoir about this to the Royal Society of London, but this memoir has not yet been published. We know about himonly a note communicated by Mr. Aclerk of the Academy of Sciences in ParisDecember 26, 1831, on the basis of a letter he received from Mr. Faraday himself.

This communication prompted Chevalier Antinori and myself to immediately repeat the basic experiment and study it from various points of view. We flatter ourselves with the hope that the results we have arrived at are of some significance, and therefore we hasten to publish them without having anypreviousmaterials, except for the note that served as the starting point in our research.»

"Mr. Faraday's memoir," as the note says, "is divided into four parts.

In the first, entitled "The Excitation of Galvanic Electricity," we find the following main fact: A galvanic current passing through a metal wire produces another current in the approaching wire; the second current is opposite in direction to the first and lasts only one instant. If the excitatory current is removed, a current arises in the wire under its influence, opposite to that which arose in it in the first case, i.e. in the same direction as the exciting current.

The second part of the memoir tells about the electric currents caused by the magnet. By approaching the coil magnets, Mr. Faraday produced electric currents; when the coils were removed, currents of the opposite direction arose. These currents have a strong effect on the galvanometer, passing, albeit weakly, through brine and other solutions. From this it follows that this scientist, using a magnet, excited the electric currents discovered by Mr. Ampère.

The third part of the memoir refers to the basic electrical state, which Mr. Faraday calls the electromonic state.

The fourth part speaks of an experiment as curious as it is unusual, belonging to Mr. Arago; as is known, this experiment consists in the fact that the magnetic needle rotates under the influence of a rotating metal disk. He found that when a metal disk rotates under the influence of a magnet, electric currents can appear in an amount sufficient to make a new electrical machine out of the disk.

MODERN THEORY OF ELECTROMAGNETIC INDUCTION

Electric currents create a magnetic field around them. Can a magnetic field cause an electric field? Faraday experimentally found that when the magnetic flux penetrating a closed circuit changes, an electric current arises in it. This phenomenon has been called electromagnetic induction. The current that occurs during the phenomenon of electromagnetic induction is called inductive. Strictly speaking, when the circuit moves in a magnetic field, not a certain current is generated, but a certain EMF. A more detailed study of electromagnetic induction showed that the induction EMF that occurs in any closed circuit is equal to the rate of change of the magnetic flux through the surface bounded by this circuit, taken with the opposite sign.

The electromotive force in the circuit is the result of the action of external forces, i.e. forces of non-electric origin. When a conductor moves in a magnetic field, the role of external forces is played by the Lorentz force, under the action of which the charges are separated, as a result of which a potential difference appears at the ends of the conductor. EMF of induction in a conductor characterizes the work of moving a unit positive charge along the conductor.

The phenomenon of electromagnetic induction underlies the operation of electric generators. If the wire frame is uniformly rotated in a uniform magnetic field, then an induced current arises, periodically changing its direction. Even a single frame rotating in a uniform magnetic field is an alternating current generator.

EXPERIMENTAL STUDY OF THE PHENOMENA OF ELECTROMAGNETIC INDUCTION

Consider the classical experiments of Faraday, with the help of which the phenomenon of electromagnetic induction was discovered:

When a permanent magnet moves, its lines of force cross the turns of the coil, and an induction current arises, so the galvanometer needle deviates. The readings of the device depend on the speed of movement of the magnet and on the number of turns of the coil.

In this experiment, we pass a current through the first coil, which creates a magnetic flux, and when the second coil moves inside the first, the magnetic lines intersect, so an induction current occurs.

When conducting experiment No. 2, it was recorded that at the moment the switch was turned on, the arrow of the device deviated and showed the value of the EMF, then the arrow returned to its original position. When the switch was turned off, the arrow again deviated, but in the other direction and showed the value of the EMF, then returned to its original position. At the moment the switch is turned on, the current increases, but some kind of force arises that prevents the increase in current. This force induces itself, so it was called the self-induction emf. At the time of shutdown, the same thing happens, only the direction of the EMF has changed, so the arrow of the device deviated in the opposite direction.

This experience shows that the EMF of electromagnetic induction occurs when the magnitude and direction of the current change. This proves that the EMF of induction, which creates itself, is the rate of change of current.

Within one month, Faraday experimentally discovered all the essential features of the phenomenon of electromagnetic induction. It only remained to give the law a strict quantitative form and fully reveal the physical nature of the phenomenon. Faraday himself already grasped the common thing that determines the appearance of an induction current in experiments that look different outwardly.

In a closed conducting circuit, a current arises when the number of magnetic induction lines penetrating the surface bounded by this circuit changes. This phenomenon is called electromagnetic induction.

And the faster the number of lines of magnetic induction changes, the greater the resulting current. In this case, the reason for the change in the number of lines of magnetic induction is completely indifferent.

This may be a change in the number of lines of magnetic induction penetrating a fixed conductor due to a change in the current strength in an adjacent coil, and a change in the number of lines due to the movement of the circuit in an inhomogeneous magnetic field, the density of lines of which varies in space.

LENTZ RULE

The induction current that has arisen in the conductor immediately begins to interact with the current or magnet that generated it. If a magnet (or a coil with current) is brought closer to a closed conductor, then the emerging induction current with its magnetic field necessarily repels the magnet (coil). Work must be done to bring the magnet and coil closer together. When the magnet is removed, attraction occurs. This rule is strictly followed. Imagine if things were different: you pushed the magnet towards the coil, and it would rush into it by itself. This would violate the law of conservation of energy. After all, the mechanical energy of the magnet would increase and at the same time a current would arise, which in itself requires the expenditure of energy, because the current can also do work. The electric current induced in the generator armature, interacting with the magnetic field of the stator, slows down the rotation of the armature. Only therefore, to rotate the armature, it is necessary to do work, the greater, the greater the current strength. Due to this work, an inductive current arises. It is interesting to note that if the magnetic field of our planet were very large and highly inhomogeneous, then fast movements of conducting bodies on its surface and in the atmosphere would be impossible due to the intense interaction of the current induced in the body with this field. The bodies would move as in a dense viscous medium and at the same time would be strongly heated. Neither airplanes nor rockets could fly. A person could not quickly move either his arms or legs, since the human body is a good conductor.

If the coil in which the current is induced is stationary relative to the adjacent coil with alternating current, as, for example, in a transformer, then in this case the direction of the induction current is dictated by the law of conservation of energy. This current is always directed in such a way that the magnetic field it creates tends to reduce changes in the current in the primary.

The repulsion or attraction of a magnet by a coil depends on the direction of the induction current in it. Therefore, the law of conservation of energy allows us to formulate a rule that determines the direction of the induction current. What is the difference between the two experiments: the approach of the magnet to the coil and its removal? In the first case, the magnetic flux (or the number of magnetic induction lines penetrating the turns of the coil) increases (Fig. a), and in the second case it decreases (Fig. b). Moreover, in the first case, the lines of induction B "of the magnetic field created by the induction current that has arisen in the coil come out of the upper end of the coil, since the coil repels the magnet, and in the second case, on the contrary, they enter this end. These lines of magnetic induction in the figure are shown with a stroke .

Now we have come to the main point: with an increase in the magnetic flux through the turns of the coil, the induction current has such a direction that the magnetic field it creates prevents the growth of the magnetic flux through the turns of the coil. After all, the induction vector of this field is directed against the field induction vector, the change of which generates an electric current. If the magnetic flux through the coil weakens, then the inductive current creates a magnetic field with induction, which increases the magnetic flux through the turns of the coil.

This is the essence of the general rule for determining the direction of the inductive current, which is applicable in all cases. This rule was established by the Russian physicist E.X. Lenz (1804-1865).

According to Lenz's rule, the inductive current that occurs in a closed circuit has such a direction that the magnetic flux created by it through the surface bounded by the circuit tends to prevent the change in the flux that generates this current. Or, the induction current has such a direction that it prevents the cause causing it.

In the case of superconductors, the compensation for changes in the external magnetic flux will be complete. The flux of magnetic induction through a surface bounded by a superconducting circuit does not change at all with time under any conditions.

LAW OF ELECTROMAGNETIC INDUCTION

electromagnetic induction faraday lenz

Faraday's experiments showed that the strength of the induced current I i in a conducting circuit is proportional to the rate of change in the number of magnetic induction lines penetrating the surface bounded by this circuit. More precisely, this statement can be formulated using the concept of magnetic flux.

The magnetic flux is clearly interpreted as the number of lines of magnetic induction penetrating a surface with an area S. Therefore, the rate of change of this number is nothing but the rate of change of the magnetic flux. If in a short time t magnetic flux changes to D F, then the rate of change of the magnetic flux is equal to.

Therefore, a statement that follows directly from experience can be formulated as follows:

the strength of the induction current is proportional to the rate of change of the magnetic flux through the surface bounded by the contour:

Recall that an electric current arises in the circuit when external forces act on free charges. The work of these forces when moving a single positive charge along a closed circuit is called the electromotive force. Consequently, when the magnetic flux changes through the surface bounded by the contour, external forces appear in it, the action of which is characterized by an EMF, called the EMF of induction. Let's denote it with the letter E i .

The law of electromagnetic induction is formulated specifically for EMF, and not for current strength. With this formulation, the law expresses the essence of the phenomenon, which does not depend on the properties of the conductors in which the induction current occurs.

According to the law of electromagnetic induction (EMI), the EMF of induction in a closed loop is equal in absolute value to the rate of change of the magnetic flux through the surface bounded by the loop:

How to take into account the direction of the induction current (or the sign of the induction EMF) in the law of electromagnetic induction in accordance with the Lenz rule?

The figure shows a closed loop. We will consider positive the direction of bypassing the contour counterclockwise. The normal to the contour forms a right screw with the bypass direction. The sign of the EMF, i.e., specific work, depends on the direction of external forces with respect to the direction of bypassing the circuit.

If these directions coincide, then E i > 0 and, accordingly, I i > 0. Otherwise, the EMF and current strength are negative.

Let the magnetic induction of the external magnetic field be directed along the normal to the contour and increase with time. Then F> 0 and > 0. According to Lenz's rule, the induction current creates a magnetic flux F" < 0. Линии индукции B"The magnetic field of the induction current is shown in the figure with a dash. Therefore, the induction current I i is directed clockwise (against the positive bypass direction) and the induction emf is negative. Therefore, in the law of electromagnetic induction, there must be a minus sign:

In the International System of Units, the law of electromagnetic induction is used to establish the unit of magnetic flux. This unit is called the weber (Wb).

Since the EMF of induction E i is expressed in volts, and time is in seconds, then from the Weber EMP law can be determined as follows:

the magnetic flux through the surface bounded by a closed loop is 1 Wb, if, with a uniform decrease in this flux to zero in 1 s, an induction emf equal to 1 V appears in the circuit: 1 Wb \u003d 1 V 1 s.

PRACTICAL APPLICATION OF THE PHENOMENA OF ELECTROMAGNETIC INDUCTION

Broadcasting

An alternating magnetic field, excited by a changing current, creates an electric field in the surrounding space, which in turn excites a magnetic field, and so on. Mutually generating each other, these fields form a single variable electromagnetic field - an electromagnetic wave. Having arisen in the place where there is a wire with current, the electromagnetic field propagates in space at the speed of light -300,000 km/s.

Magnetotherapy

In the frequency spectrum different places are occupied by radio waves, light, x-rays and other electromagnetic radiation. They are usually characterized by continuously interconnected electric and magnetic fields.

Synchrophasotrons

At present, a magnetic field is understood as a special form of matter consisting of charged particles. In modern physics, beams of charged particles are used to penetrate deep into atoms in order to study them. The force with which a magnetic field acts on a moving charged particle is called the Lorentz force.

Flow meters - meters

The method is based on the application of Faraday's law for a conductor in a magnetic field: in the flow of an electrically conductive liquid moving in a magnetic field, an EMF is induced proportional to the flow velocity, which is converted by the electronic part into an electrical analog / digital signal.

DC generator

In the generator mode, the armature of the machine rotates under the influence of an external moment. Between the poles of the stator there is a constant magnetic flux penetrating the armature. The armature winding conductors move in a magnetic field and, therefore, an EMF is induced in them, the direction of which can be determined by the "right hand" rule. In this case, a positive potential arises on one brush relative to the second. If a load is connected to the terminals of the generator, then current will flow in it.

The EMR phenomenon is widely used in transformers. Let's consider this device in more detail.

TRANSFORMERS

Transformer (from lat. transformo - transform) - a static electromagnetic device having two or more inductively coupled windings and designed to convert one or more AC systems into one or more other AC systems by electromagnetic induction.

The inventor of the transformer is the Russian scientist P.N. Yablochkov (1847 - 1894). In 1876, Yablochkov used an induction coil with two windings as a transformer to power the electric candles he invented. The Yablochkov transformer had an open core. Closed-core transformers, similar to those used today, appeared much later, in 1884. With the invention of the transformer, a technical interest arose in alternating current, which had not been applied until that time.

Transformers are widely used in the transmission of electrical energy over long distances, its distribution between receivers, as well as in various rectifying, amplifying, signaling and other devices.

The transformation of energy in the transformer is carried out by an alternating magnetic field. The transformer is a core of thin steel plates insulated from one another, on which two, and sometimes more windings (coils) of insulated wire are placed. The winding to which the source of AC electrical energy is connected is called the primary winding, the remaining windings are called secondary.

If three times more turns are wound in the secondary winding of the transformer than in the primary, then the magnetic field created in the core by the primary winding, crossing the turns of the secondary winding, will create three times more voltage in it.

Using a transformer with a reverse turns ratio, you can just as easily and simply get a reduced voltage.

Atideal transformer equation

An ideal transformer is a transformer that has no energy losses for heating the windings and winding leakage fluxes. In an ideal transformer, all lines of force pass through all turns of both windings, and since the changing magnetic field generates the same EMF in each turn, the total EMF induced in the winding is proportional to the total number of its turns. Such a transformer transforms all incoming energy from the primary circuit into a magnetic field and, then, into the energy of the secondary circuit. In this case, the incoming energy is equal to the converted energy:

Where P1 is the instantaneous value of the power supplied to the transformer from the primary circuit,

P2 is the instantaneous value of the power converted by the transformer, which enters the secondary circuit.

Combining this equation with the ratio of voltages at the ends of the windings, we get the equation for an ideal transformer:

Thus, we obtain that with an increase in the voltage at the ends of the secondary winding U2, the current of the secondary circuit I2 decreases.

To convert the resistance of one circuit to the resistance of another, you need to multiply the value by the square of the ratio. For example, the resistance Z2 is connected to the ends of the secondary winding, its reduced value to the primary circuit will be

This rule is also valid for the secondary circuit:

Designation on the diagrams

In the diagrams, the transformer is indicated as follows:

The central thick line corresponds to the core, 1 is the primary winding (usually on the left), 2.3 is the secondary windings. The number of semicircles in some rough approximation symbolizes the number of turns of the winding (more turns - more semicircles, but without strict proportionality).

TRANSFORMER APPLICATIONS

Transformers are widely used in industry and everyday life for various purposes:

1. For the transmission and distribution of electrical energy.

Typically, at power plants, alternating current generators generate electrical energy at a voltage of 6-24 kV, and it is profitable to transmit electricity over long distances at much higher voltages (110, 220, 330, 400, 500, and 750 kV). Therefore, at each power plant, transformers are installed that increase the voltage.

The distribution of electrical energy between industrial enterprises, settlements, in cities and rural areas, as well as within industrial enterprises, is carried out via overhead and cable lines, at a voltage of 220, 110, 35, 20, 10 and 6 kV. Therefore, transformers must be installed in all distribution nodes that reduce the voltage to 220, 380 and 660 V

2. To provide the desired circuit for switching on valves in converter devices and to match the voltage at the output and input of the converter. Transformers used for these purposes are called transformers.

3. For various technological purposes: welding (welding transformers), power supply of electrothermal installations (electric furnace transformers), etc.

4. For powering various circuits of radio equipment, electronic equipment, communication and automation devices, household appliances, for separating electrical circuits of various elements of these devices, for matching voltage, etc.

5. To include electrical measuring instruments and some devices (relays, etc.) in high voltage electrical circuits or in circuits through which large currents pass, in order to expand the measurement limits and ensure electrical safety. Transformers used for these purposes are called measuring.

CONCLUSION

The phenomenon of electromagnetic induction and its special cases are widely used in electrical engineering. Used to convert mechanical energy into electrical energy synchronous generators. Transformers are used to step up or step down AC voltage. The use of transformers makes it possible to economically transfer electricity from power plants to consumption nodes.

BIBLIOGRAPHY:

1. Physics course, textbook for universities. T.I. Trofimova, 2007.

2. Fundamentals of the theory of circuits, G.I. Atabekov, Lan, St. Petersburg, - M., - Krasnodar, 2006.

3. Electrical machines, L.M. Piotrovsky, L., Energy, 1972.

4. Power transformers. Reference book / Ed. S.D. Lizunova, A.K. Lokhanin. M.: Energoizdat 2004.

5. Design of transformers. A.V. Sapozhnikov. M.: Gosenergoizdat. 1959.

6. Calculation of transformers. Textbook for universities. P.M. Tikhomirov. Moscow: Energy, 1976.

7. Physics - textbook for technical schools, author V.F. Dmitriev, edition Moscow "Higher School" 2004.

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Practical application of electromagnetic induction

The phenomenon of electromagnetic induction is used primarily to convert mechanical energy into electric current energy. For this purpose, apply alternators(induction generators).

sin

BUT

With

Rice. 4.6

For the industrial production of electricity at power plants are used synchronous generators(turbo generators, if the station is thermal or nuclear, and hydro generators, if the station is hydraulic). The stationary part of a synchronous generator is called stator, and rotating - rotor(Fig. 4.6). The generator rotor has a DC winding (excitation winding) and is a powerful electromagnet. DC current applied to
the excitation winding through the brush-contact apparatus, magnetizes the rotor, and in this case an electromagnet with north and south poles is formed.

On the stator of the generator there are three windings of alternating current, which are offset one relative to the other by 120 0 and are interconnected according to a certain switching circuit.

When an excited rotor rotates with the help of a steam or hydraulic turbine, its poles pass under the stator windings, and an electromotive force that changes according to a harmonic law is induced in them. Further, the generator, according to a certain scheme of the electrical network, is connected to the nodes of electricity consumption.

If you transfer electricity from generators of stations to consumers via power lines directly (at the generator voltage, which is relatively small), then large losses of energy and voltage will occur in the network (pay attention to the ratios , ). Therefore, for economical transportation of electricity, it is necessary to reduce the current strength. However, since the transmitted power remains unchanged, the voltage must
increase by the same factor as the current decreases.

At the consumer of electricity, in turn, the voltage must be reduced to the required level. Electrical devices in which the voltage is increased or decreased by a given number of times are called transformers. The work of the transformer is also based on the law of electromagnetic induction.

sin

Then

In powerful transformers, the coil resistances are very small,
therefore, the voltages at the terminals of the primary and secondary windings are approximately equal to the EMF:

where k- transformation ratio. At k<1 () the transformer is raising, at k>1 () the transformer is lowering.

When connected to the secondary winding of a load transformer, current will flow in it. With an increase in electricity consumption according to the law
energy conservation, the energy given off by the generators of the station should increase, that is

This means that by increasing the voltage with a transformer
in k times, it is possible to reduce the current strength in the circuit by the same amount (in this case, the Joule losses decrease by k 2 times).

Topic 17. Fundamentals of Maxwell's theory for the electromagnetic field. Electromagnetic waves

In the 60s. 19th century English scientist J. Maxwell (1831-1879) summarized the experimentally established laws of electric and magnetic fields and created a complete unified electromagnetic field theory. It allows you to decide the main task of electrodynamics: find the characteristics of the electromagnetic field of a given system of electric charges and currents.

Maxwell hypothesized that any alternating magnetic field excites a vortex electric field in the surrounding space, the circulation of which is the cause of the emf of electromagnetic induction in the circuit:

(5.1)

Equation (5.1) is called Maxwell's second equation. The meaning of this equation is that a changing magnetic field generates a vortex electric field, and the latter, in turn, causes a changing magnetic field in the surrounding dielectric or vacuum. Since the magnetic field is created by an electric current, then, according to Maxwell, the vortex electric field should be considered as a certain current,
which flows both in a dielectric and in a vacuum. Maxwell called this current bias current.

Displacement current, as follows from Maxwell's theory
and Eichenwald's experiments, creates the same magnetic field as the conduction current.

In his theory, Maxwell introduced the concept full current equal to the sum
conduction and displacement currents. Therefore, the total current density

According to Maxwell, the total current in the circuit is always closed, that is, only the conduction current breaks at the ends of the conductors, and in the dielectric (vacuum) between the ends of the conductor there is a displacement current that closes the conduction current.

Introducing the concept of total current, Maxwell generalized the vector circulation theorem (or ):

(5.6)

Equation (5.6) is called Maxwell's first equation in integral form. It is a generalized law of the total current and expresses the main position of the electromagnetic theory: displacement currents create the same magnetic fields as conduction currents.

The unified macroscopic theory of the electromagnetic field created by Maxwell made it possible, from a unified point of view, not only to explain electrical and magnetic phenomena, but to predict new ones, the existence of which was subsequently confirmed in practice (for example, the discovery of electromagnetic waves).

Summarizing the provisions discussed above, we present the equations that form the basis of Maxwell's electromagnetic theory.

1. Theorem on the circulation of the magnetic field vector:

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

2. The electric field can be both potential () and vortex (), so the total field strength . Since the circulation of the vector is equal to zero, then the circulation of the vector of the total electric field strength

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

3. ,

where is the volume charge density inside the closed surface; is the specific conductivity of the substance.

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

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

Using known from vector analysis Stokes and Gauss theorems, one can imagine the complete system of Maxwell's equations in differential form(characterizing the field at each point in space):

(5.7)

Obviously, Maxwell's equations not symmetrical regarding electric and magnetic fields. This is due to the fact that nature
There are electric charges, but there are no magnetic charges.

Maxwell's equations are the most general equations for electrical
and magnetic fields in media at rest. They play the same role in the theory of electromagnetism as Newton's laws in mechanics.

electromagnetic wave called an alternating electromagnetic field propagating in space with a finite speed.

The existence of electromagnetic waves follows from Maxwell's equations, formulated in 1865 on the basis of a generalization of the empirical laws of electrical and magnetic phenomena. An electromagnetic wave is formed due to the interconnection of alternating electric and magnetic fields - a change in one field leads to a change in the other, that is, the faster the magnetic field induction changes in time, the greater the electric field strength, and vice versa. Thus, for the formation of intense electromagnetic waves, it is necessary to excite electromagnetic oscillations of a sufficiently high frequency. Phase speed electromagnetic waves is determined
electrical and magnetic properties of the medium:

In a vacuum ( ) the speed of propagation of electromagnetic waves coincides with the speed of light; in matter , That's why the speed of propagation of electromagnetic waves in matter is always less than in vacuum.

Electromagnetic waves are shear waves –
oscillations of the vectors and occur in mutually perpendicular planes, and the vectors , and form a right-handed system. It also follows from Maxwell's equations that in an electromagnetic wave the vectors and always oscillate in the same phases, and the instantaneous values E and H at any point are related by the relation

Plane electromagnetic wave equations in vector form:

(6.66)

Rice. 6.21

On fig. 6.21 shows a "snapshot" of a plane electromagnetic wave. It can be seen from it that the vectors and form a right-handed system with the direction of wave propagation. At a fixed point in space, the vectors of the electric and magnetic fields change with time according to a harmonic law.

To characterize the transfer of energy by any wave in physics, a vector quantity called energy flux density. It is numerically equal to the amount of energy transferred per unit time through a unit area perpendicular to the direction in which
the wave propagates. The direction of the vector coincides with the direction of energy transfer. The value of the energy flux density can be obtained by multiplying the energy density by the wave speed

The energy density of the electromagnetic field is the sum of the energy density of the electric field and the energy density of the magnetic field:

(6.67)

Multiplying the energy density of an electromagnetic wave by its phase velocity, we obtain the energy flux density

(6.68)

The vectors and are mutually perpendicular and form a right-handed system with the direction of wave propagation. Therefore the direction
vector coincides with the direction of energy transfer, and the modulus of this vector is determined by relation (6.68). Therefore, the energy flux density vector of an electromagnetic wave can be represented as a vector product

(6.69)

Vector call Umov-Poynting vector.

Vibrations and waves

Topic 18. Free harmonic vibrations

Movements that have some degree of repetition are called fluctuations.

If the values of physical quantities that change in the process of movement are repeated at regular intervals, then such a movement is called periodical (the movement of planets around the Sun, the movement of a piston in the cylinder of an internal combustion engine, etc.). An oscillatory system, regardless of its physical nature, is called oscillator. An example of an oscillator is an oscillating weight suspended on a spring or thread.

Full swingone complete cycle of oscillatory motion is called, after which it is repeated in the same order.

According to the method of excitation, vibrations are divided into:

· free(intrinsic) occurring in the system presented to itself near the equilibrium position after some initial impact;

· forced occurring under periodic external action;

· parametric, occurring when changing any parameter of the oscillatory system;

· self-oscillations occurring in systems that independently regulate the flow of external influences.

Any oscillatory movement is characterized amplitude A - the maximum deviation of the oscillating point from the equilibrium position.

Oscillations of a point occurring with a constant amplitude are called undamped, and fluctuations with gradually decreasing amplitude – fading.

The time it takes for a complete oscillation to take place is called period(T).

Frequency periodic oscillations is the number of complete oscillations per unit of time. Oscillation frequency unit - hertz(Hz). Hertz is the frequency of oscillations, the period of which is equal to 1 s: 1 Hz = 1 s -1 .

cyclicor circular frequency periodic oscillations is the number of complete oscillations that occur in a time 2p with: . \u003d rad / s.

The law of electromagnetic induction underlies modern electrical engineering, as well as radio engineering, which, in turn, forms the core of modern industry, which has completely transformed our entire civilization. The practical application of electromagnetic induction began only half a century after its discovery. At that time, technological progress was still relatively slow. The reason why electrical engineering plays such an important role in all of our modern lives is because electricity is the most convenient form of energy and it is precisely because of the law of electromagnetic induction. The latter makes it easy to obtain electricity from mechanical energy (generators), to flexibly distribute and transport energy (transformers) and convert it back into mechanical energy (electric motor) and other types of energy, and all this happens with a very high efficiency. Some 50 years ago, the distribution of energy between machine tools in factories was carried out through a complex system of shafts and belt drives - the forest of transmissions was a characteristic detail of the industrial "interior" of that time. Modern machine tools are equipped with compact electric motors fed through a hidden electrical wiring system.

Modern industry uses a single power supply system covering the entire country, and sometimes several neighboring countries.

The power supply system starts with a power generator. The operation of the generator is based on the direct use of the law of electromagnetic induction. Schematically, the simplest generator is a stationary electromagnet (stator), in the field of which a coil (rotor) rotates. The alternating current excited in the rotor winding is removed with the help of special movable contacts - brushes. Since it is difficult to pass large power through moving contacts, an inverted generator circuit is often used: a rotating electromagnet excites current in the stationary stator windings. Thus, the generator converts the mechanical energy of the rotation of the rotor into electricity. The latter is driven by either thermal energy (steam or gas turbine) or mechanical energy (hydro turbine).

At the other end of the power supply system are various actuators that use electricity, the most important of which is the electric motor (electric motor). The most common, due to its simplicity, is the so-called asynchronous motor, invented independently in 1885-1887. Httalian physicist Ferraris and the famous Croatian engineer Tesla (USA). The stator of such an engine is a complex electromagnet that creates a rotating field. The rotation of the field is achieved using a system of windings in which the currents are phase shifted. In the simplest case, it suffices to take a superposition of two fields in perpendicular directions, shifted in phase by 90° (Fig. VI.10).

Such a field can be written as a complex expression:

which represents a two-dimensional vector of constant length, rotating counterclockwise with a frequency o. Although formula (53.1) is similar to the complex representation of alternating current in § 52, its physical meaning is different. In the case of alternating current, only the real part of the complex expression had real value, but here the complex value represents a two-dimensional vector, and its phase is not only the phase of oscillations of the components of the alternating field, but also characterizes the direction of the field vector (see Fig. VI.10).

In technology, a somewhat more complex scheme of field rotation is usually used with the help of the so-called three-phase current, i.e. three currents, the phases of which are shifted by 120 ° relative to each other. These currents create a magnetic field in three directions, rotated one relative to the other by an angle of 120 ° (Fig. VI.11). Note that such a three-phase current is automatically obtained in generators with a similar arrangement of windings. The three-phase current, which was widely used in technology, was invented

Rice. VI.10. Scheme for obtaining a rotating magnetic field.

Rice. VI.11. Scheme of an asynchronous motor. For simplicity, the rotor is shown as a single turn.

in 1888 by the outstanding Russian electrical engineer Dolivo-Dobrovolsky, who built in Germany on this basis the world's first technical power line.

The rotor winding of an induction motor consists in the simplest case of short-circuited turns. An alternating magnetic field induces a current in the coils, which leads to the rotation of the rotor in the same direction as the magnetic field. In accordance with Lenz's rule, the rotor tends to "catch up" with the rotating magnetic field. For a loaded motor, the rotor speed is always less than the field, since otherwise the induction EMF and the current in the rotor would turn to zero. Hence the name - asynchronous motor.

Task 1. Find the speed of rotation of the rotor of an induction motor depending on the load.

The equation for the current in one turn of the rotor has the form

where - the angular velocity of the field sliding relative to the rotor, characterizes the orientation of the coil relative to the field, the location of the coil in the rotor (Fig. VI.12, a). Passing to complex quantities (see § 52), we obtain the solution (53.2)

The torque acting on a coil in the same magnetic field is

Rice. VI.12. On the problem of an asynchronous motor. a - a turn of the rotor winding in a "sliding" field; b - load characteristic of the engine.

Typically, the rotor winding contains a large number of evenly spaced turns, so that summation over 9 can be replaced by integration, as a result we get for the total torque on the motor shaft

where is the number of turns of the rotor. The dependency graph is shown in Fig. VI.12, b. The maximum torque corresponds to the slip frequency Note that the ohmic resistance of the rotor only affects the slip frequency, but not the maximum torque of the motor. The negative slip frequency (the rotor “overtakes” the field) corresponds to the generator mode. To maintain this mode, it is necessary to expend external energy, which is converted into electrical energy in the stator windings.

For a given torque, the slip frequency is ambiguous, but only the mode is stable

The main element of the systems for converting and transporting electricity is a transformer that changes the AC voltage. For long-distance transmission of electricity, it is advantageous to use the maximum possible voltage, limited only by insulation breakdown. At present, transmission lines operate with a voltage of about For a given transmitted power, the current in the line is inversely proportional to the voltage, and the losses in the line fall as the square of the voltage. On the other hand, much lower voltages are needed to power consumers of electricity, mainly for reasons of simplicity of design (insulation), as well as safety. Hence the need for voltage transformation.

Usually a transformer consists of two windings on a common iron core (Fig. VI. 13). An iron core is needed in a transformer to reduce stray flux and therefore better flux linkage between the windings. Since iron is also a conductor, it passes a variable

Rice. V1.13. Schematic of an AC transformer.

Rice. VI.14. Scheme of the Rogowski belt. The dashed line conditionally shows the integration path.

magnetic field only to a shallow depth (see § 87). Therefore, the cores of transformers have to be made laminated, that is, in the form of a set of thin plates electrically isolated from one another. For a power frequency of 50 Hz, the usual plate thickness is 0.5 mm. For transformers at high frequencies (in radio engineering), you have to use very thin plates (mm) or ferrite cores.

Task 2. What voltage should the transformer core plates be insulated to?

If the number of plates in the core and the voltage per turn of the transformer winding, then the voltage between adjacent plates

In the simplest case of the absence of a scattered flow, the EMF ratio in both windings is proportional to the number of their turns, since the induction EMF per turn is determined by the same flux in the core. If, in addition, the losses in the transformer are small, and the load resistance is large, then it is obvious that the ratio of the voltages on the primary and secondary windings is also proportional. This is the principle of operation of the transformer, which thus makes it easy to change the voltage many times over.

Task 3. Find the voltage transformation ratio for an arbitrary load.

Neglecting losses in the transformer and leakage (ideal transformer), we write the equation for currents in the windings in the form (in SI units)

where is the complex load resistance (see § 52) and the expression (51.2) is used for the induction EMF of a complex circuit. With the help of relation (51.6); you can find the voltage transformation ratio without solving equations (53.6), but simply by dividing them one by the other:

The transformation ratio turns out to be equal, therefore, simply to the ratio of the number of turns at any load. The sign depends on the choice of the beginning and end of the windings.

To find the current transformation ratio, you need to solve the system (53.7), as a result of which we get

In the general case, the coefficient turns out to be some complex value, i.e., a phase shift appears between the currents in the windings. Of interest is the special case of a small load. Then, i.e., the ratio of currents becomes the inverse of the ratio of voltages.

This transformer mode can be used to measure high currents (current transformer). It turns out that the same simple transformation of currents is also preserved for an arbitrary dependence of the current on time with a special design of the current transformer. In this case, it is called the Rogowski coil (Fig. VI.14) and is a flexible closed solenoid of arbitrary shape with uniform winding. The operation of the belt is based on the law of conservation of the circulation of the magnetic field (see § 33): where the integration is performed along the contour inside the belt (see Fig. VI.14), is the total measured current covered by the belt. Assuming that the transverse dimensions of the belt are small enough, we can write the induction emf induced on the belt as follows:

where is the cross section of the belt, a is the winding density, both values are assumed to be constant along the belt; inside the belt, if the density of the winding of the belt and its cross section 50 are constant along the length (53.9).

A simple conversion of electrical voltage is possible only for alternating current. This determines its decisive role in modern industry. In cases where direct current is required, significant difficulties arise. For example, in ultra-long-range power transmission lines, the use of direct current provides significant advantages: heat losses are reduced, since there is no skin effect (see § 87) and there are no resonant

(wave) transients when turning on - off the transmission line, the length of which is of the order of the wavelength of alternating current (6000 km for an industrial frequency of 50 Hz). The difficulty lies in rectifying high voltage alternating current at one end of the transmission line and inverting it at the other.

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